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Cornell University Library 
 TC 160.D491 
 
 A B C of hydrodynamics, 
 
 3 1924 004 931 709 
 
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ABC OF HYDRODYNAMICS 
 
ABC 
 
 OF 
 
 HYDRODYNAMICS 
 
 BY 
 
 LiEUT.-CoL. R. DE VILLAMIL 
 
 R. ENG. (ret.) 
 
 " Next to being right in this world, the best of all things is to be clearly and 
 definitely wrong, because you will come out somewhere. If you go buzzing 
 about between right and wrong, vibrating and fluctuating, you come out no- 
 where ; but if you are absolutely and thoroughly and persistently wrong, you 
 must, some of these days, have the extreme good fortune of knocking your 
 head against a fact, and that sets you all straight again." Huxlev, 
 
 FORTY-EIGHT ILLUSTRATIONS 
 
 Xon&on 
 
 E. & F. N. SPON, Ltd., 57 HAYMARKET 
 
 flew ISovft 
 
 SPON & CHAMBERLAIN, 123 LIBERTY STREET 
 
 1912 
 
 S 
 
TO THE 
 
 /TO e m r s 
 
 OF 
 
 LORD KELVIN 
 
 WITH 
 
 PROFOUND RESPECT 
 
 AND IN 
 
 ADMIRATION OF HIS CONTRIBUTIONS 
 
 TO 
 
 HYDRODYNAMICS 
 
CONTENTS 
 
 CHAP. 
 
 1. RESISTANCE OF LIQUIDS — CONFUSED STATE OF 
 
 THE SUBJECT 
 
 2. A LIQUID OF INFINITE DIMENSIONS 
 
 PAGE 
 
 PREFACE ... . ix 
 
 I 
 
 9 
 
 3. MOVEMENT OF A LIQUID AND THE LAW OF FLOW 1 8 
 
 4. RESISTANCE, CONSIDERED GENERALLY ... 26 
 
 5. VISCOSITY AND FLUID FRICTION . . -33 
 
 6 A VISCOUS FLUID FLOWING " BY OBLIGATION " — 
 
 DISCONTINUITY 43 
 
 7. VISCOUS LIQUID NOT FLOWING "BY OBLIGA- 
 
 TION," BUT FOLLOWING STREAM-LINES GIVEN 
 
 IN HYDRODYNAMICAL TEXT-BOOKS . . -52 
 
 8. RESISTANCE DUE TO VISCOSITY VARIES, NOT AS 
 
 THE " WETTED SURFACE," BUT AS THE " SHEAR- 
 ING SURFACE" AND AS THE VELOCITY ONLY . 62 
 
 9. RESISTANCE DUE TO VISCOSITY {continued) — 
 
 STOKES' LAW 73 
 
 10. HOW LIQUIDS CHANGE FROM STEADY TO "SINU- 
 
 OUS" MOTION — VORTICES — SENSITIVE FLAME — 
 SINGING FLAME 84 
 
 11. LIQUIDS MOVING IN TUBES — POISEUILLE'S LAW 
 
 — RING VORTICES 93 
 
 12. VORTICES IN A PERFECT LIQUID — HELMHOLTZ 
 
 RINGS . . 107 
 
viii Contents 
 
 CHAP. I'AGE 
 
 13, MOTION OF WATER IN REAR OF A BODY EXPOSED 
 TO A STREAM — RESISTANCE AT VERY HIGH 
 VELOCITIES 118 
 
 APPENDIX — CONNECTION BETWEEN THE LAW OF 
 THE RESISTANCE OF LIQUIDS AND OF AIR, 
 "DYNAMICAL similarity" I29 
 
 INDEX 133 
 
PREFACE 
 
 This little book has no pretensions to being a treatise on 
 hydrodynamics. It is, as its title implies, only intended 
 as an introduction to the study of that subject. There is 
 not very much that is new in it; some of the quotations 
 are so old that they have been forgotten, and so will appear 
 to the reader of the ordinary text-books as if they were 
 new. What is, I fancy, original is the way in which the 
 matter is arranged and the subject presented. Instead of 
 treating of the movement of a " perfect liquid " and then 
 informing the reader that ordinary liquids behave quite 
 differently, I have endeavoured to show that perfect and 
 imperfect (?) liquids follow exactly the same lav/s, and under 
 similar conditions move in a similar manner. 
 
 The difficulties are generally supposed to be very great. 
 Sir John Herschel told us that "if there be one part of 
 dynamic science more abstruse and unapproachable than 
 another, it is the doctrine of propagation of motion in 
 fluids, and especially in elastic fluids like the air, even 
 where the amount and application of the original acting 
 forces are known and calculable." These difficulties are, I 
 think, very largely artificial ones. It is well to remember 
 what Dubuat said more than a hundred years ago : " On 
 risque souvent de se tromper, quand on applique aux fluides 
 les lois du mouvement qui conviennent aux solides." I 
 might even go further and say that, when thinking of liquids, 
 what (to the untrained mind) appears " obvious " or " appeals 
 to common sense " is very frequently wrong — more frequently, 
 perhaps, than not. When the student has trained himself to 
 " think in pressures " many difficulties will disappear. 
 
X A B C of Hydrodynamics 
 
 Another difficulty is caused by tlie assumption of " con- 
 tinuity " in the perfect liquid, a property which certainly 
 does not pertain to mundane liquids. Avanzini saw and 
 pointed out this as long ago as 1 804, when he said {Istituto 
 Nazionale Italiano, Tomo i., Parte i): "The physico-mathe- 
 maticians who, with their investigations on the resistance 
 of fluids, arrived at the ordinary formula, all supposed, either 
 implicitly/ or tacitly — 
 
 " I. That the pressure of the fluid derived from its own 
 weight has no influence on the resistance. 
 
 " 2. ... If the first hypothesis is not true ordinarily, 
 it is so, nevertheless, in the case where there is no vacuum 
 formed"'^ [italics added]. 
 
 Avanzini's meaning is obviously that gravity has no 
 effect in producing resistance unless there is a vacuum. 
 Put slightly differently, if there is no free surface — some- 
 where — there can be no inertia resistance. 
 
 Fifty years elapsed, and then Lord Kelvin said, practically, 
 the same thing : " Thus it is that gravity, which could not 
 affect the motion of a liquid entirely filling a rigid closed 
 vessel, will exercise a most important influence on the motion 
 of a liquid contained in an open vessel, and exposing a free 
 surface to the atmospheric pressurt." 
 
 Some parts of the book are necessarily contentious. I 
 have, however, when I disagree with the ordinary modern 
 teaching, only quoted from authors of position and authority 
 to explain my objections. The quotations are, perhaps, 
 rather full, but I do not think it is fair to take a sentence out 
 of a paragraph and remove it from the context. 
 
 In all my arguments and reasoning I have closely followed 
 
 1 " I fisico-matematici, che colle loro, investigazioni intorno alia resis- 
 tenza de fluicli giunsero a trovare la formula ordinarla, supposero tutti o 
 espressamente o tacitamente. 
 
 " I. Che la pressione del fluido nascente dalla sua propria gravitk non 
 abbia influsso alcuno sopra la resistenza. 
 
 " 2. ... Si la prima ipotesi non h vera generalmente lo 6 pero nel case 
 che non succeda vacuo." 
 
 I have translated "vacuo" by vacuum; this I define as a space void 
 of liquid. 
 
Preface xi 
 
 the teaching of Lord Kelvin, and I have not, I trust, stated 
 anything that he would have disapproved of. 
 
 The views on viscosity are based on those of Newton. If, 
 as he taught, the resistance due to viscosity is a resistance to 
 shearing of the liquid, then, in the last analysis, it must be 
 caused by the rotating liquid molecules rubbing against one 
 another and so generating heat. If my hypothesis of the 
 way the molecules move be considered theoretical and 
 "fantastic," 1 must plead (\) my belief that "those who 
 refuse to go beyond fact rarely get as far as fact " (Huxley), 
 and (2) that it is not one bit more fantastic than the kinetic 
 theory of gases. Carlyle has said that the tree of knowledge 
 requires periodical shaking ; but for the shaking to be any 
 good it must be severe and thorough. I think the hydro- 
 dynamical tree will not be hurt by a shaking. By following 
 the train of argument adopted here I trust that a clearer 
 view on the subject will be attained ; and to the student I 
 would beg to repeat Professor Osborne's words {Huxley and 
 Education") : " Do not climb that mountain of learning in the 
 hope that when you reach the summit you will be able to 
 think for yourself: think for yourself while you are climbing." 
 
 The purist may object to the style, in consequence of my 
 very liberal employment of emphasis : I trust, however, that 
 the reader will find that what he loses on the swings he gains 
 on the round-abouts. 
 
 I wish to acknowledge my great indebtedness to Mr Lewis 
 R. Shorter, B.Sc, not only for reading the proofs, but for his 
 continual help and sympathy, valuable criticism and great 
 assistance in mathematical questions. He is responsible for 
 such " betterment " as gives him almost a tenant right in these 
 pages. My best thanks are also due to Mr Charles Spon for 
 the preparation of the index, which adds value to the little 
 book, as well as for many useful suggestions. 
 
 R. DE VILLAMIL. 
 
 December 191 1. 
 
ABC OF HYDRODYNAMICS 
 
 CHAPTER I 
 
 RESISTANCE OF LIQUIDS — CONFUSED STATE OF 
 THE SUBJECT 
 
 Lord RayleigH in one of his very earliest papers con- 
 tributed to the Philosophical Magazine (1876), said: "There 
 is no part of hydrodynamics more puzzling to the student 
 than that which treats of the resistance of fluids." It is 
 sad to have to admit that, after more than a quarter of a 
 century, this is as true to-day as when the remark was 
 made. My present object is to try and point out the 
 pitfalls to be avoided in this study, as well as to offer a 
 simple and non - mathematical explanation of how the 
 resistance of a liquid is actually caused, many of the 
 difficulties being more apparent than real. 
 
 Lord Rayleigh continues : " According to one school of 
 writers, a body exposed to a stream of perfect fluid would 
 experience no resultant force at all, any augmentation of 
 pressure on its face, due to the stream, being compensated 
 by equal and opposite pressure on its rear ; and indeed it 
 is a rigorous consequence of the usual hypothesis of perfect 
 fluidity and of the continuity of the motion, that the resultant 
 of the fluid pressures reduces to a couple tending to turn 
 the broader face of the body towards the stream." 
 
 We see in this remark, "according to one school of writers," 
 that Lord Rayleigh does not commit himself to this view, 
 though he admits that there is something to be said for it, 
 subject to the hypothesis of perfect fluidity and of "con- 
 tinuity of the motion." 
 
 I A 
 
2 A B C of Hydrodynamics 
 
 The statement that a body moving in a non-viscous or 
 perfect liquid would meet with no resistance, ;/ unqualified, 
 is in direct contradiction to the teaching of Newton. If, 
 however, it be qualified by the addition that the liquid 
 must be of infinite dimensions in every direction, then it is, 
 I believe, mathematically true. This is only saying, in a 
 different way, that all motion must be strictly continuous. 
 
 The whole of this paper is so very interesting that it 
 will be quoted from freely : " It was Helmholtz who first 
 pointed out that there is nothing in the nature of a perfect 
 fluid to forbid a finite slipping between contiguous layers, 
 and that the possibility of such occurrence is not taken 
 into account in the common mathematical theory, which 
 makes the fluid flow according to the same laws as determine 
 the motion of electricity in uniform conductors." 
 
 There is, of course, nothing in the nature of a perfect 
 fluid to " forbid slipping," but in the mathematical theory 
 all motion is assumed to be strictly continuous: since the 
 slipping would imply discontinuity, it is barred by the 
 premises. As the mathematicians' fluid is always assumed 
 AS of infinite dimensions, it is obvious that no crack or "rift" 
 could be made in it, as it is already occupying the maximum 
 of space possible. 
 
 Again : " Moreover, the electric law of flow (as it may 
 be called for brevity) would make the velocity infinite at 
 every sharp edge encountered by the fluid ; and this would 
 require a negative pressure of infinite magnitude. 
 
 " It is no answer to this objection that a mathematical 
 sharp edge is an impossibility, inasmuch as the electric law 
 of flow would require negative pressure in cases where the 
 edge is not perfectly sharp." 
 
 This difficulty is more apparent than real. It is quite 
 true that a mathematical sharp edge is an impossibility, but 
 so is an infinite velocity, or a negative pressure of infinite 
 magnitude. It will be quite sufficient to say that the sharper 
 the edge the greater will be velocity of flow of the liquid 
 past it, and, consequently, the greater will be the magnitude 
 of the negative pressure there; but to this should also be 
 
Resistance of Liquids — Confused State of the Subject 3 
 
 added, "and the greater the plus pressure somewhere else" 
 for a great velocity necessarily postulates a great pressure 
 somewhere. 
 
 The view I have taken up is that held by Lord Kelvin, 
 who wrote in 1894 {^Nature): 
 
 " § I. The doctrine that ' discontinuity,' that is to say, 
 finite difference of velocity on two sides of a surface in a 
 fluid, would be produced if an inviscid incompressible fluid 
 were caused to flow past a sharp edge of a rigid solid with 
 no vacant space between the fluid and the solid, was, I believe, 
 first given by Stokes in 1847. 
 
 " It is inconsistent with the now well-known dynamical 
 theorem that an incompressible fluid initially at rest, and 
 set in motion by pressure applied to its boundary, acquires 
 the unique distribution of motion throughout its mass, of 
 which the kinetic energy is less than that of any other motion 
 of the fluid with the same motion of its boundary. 
 
 " § 2. The reason assigned for the formation of a surface 
 of finite slip between fluid and fluid was the infinitely great 
 velocity of the fluid at the edge, and the corresponding 
 negative — infinite pressure — implied by the unique solution, 
 unless the fluid is allowed to separate itself from contact with 
 the solid. This an inviscid incompressible fluid certainly 
 would do, unless the pressure of the fluid was infinitely 
 great everywhere except at the edge." 
 
 It will be well here to point out that one must be very 
 careful in speaking of negative pressure, since, in the last 
 analysis, it will be found that the negative sign is only 
 apparent : speaking generally, there can be no such thing 
 as negative pressure. To explain what I mean, suppose a 
 small lamina to be moving broadside-on in a liquid — say 
 water — at such a depth of immersion that the hydrostatic 
 pressure would be 5 lbs. per square inch. At a moderately 
 high velocity the pressure on the leading face of the lamina, 
 near the edge, might very easily be minus 5 lbs. per square 
 inch. This is not, however, in reality a negative pressure, 
 since the pressure at the surface of the water is not zero, 
 but roughly 15 lbs. per square inch: 5 lbs. minus pressure 
 
 A 2 
 
4 A B C of Hydrodynamics 
 
 is thus in reality 15-5 = 10 lbs. pins pressure. This will 
 be considered and become more evident later, when the 
 subject is more developed. That there can be no negative 
 pressure must not be taken too literally ; ordinary mundane 
 liquids are certainly capable of resisting tension, and under 
 some exceptional circumstances actually do so. Whether 
 a perfect liquid, which offers no resistance to shearing, could 
 be under tension appears questionable. 
 
 Lord Rayleigh ha\ing said : "... It is well known that 
 in practice an obstacle does experience a force tending to 
 carry it down stream, and of magnitude too great to be the 
 direct effect of friction ; while in many of the treatises 
 calculations of resistance arc given leading to results depend- 
 ing on the inertia of the fluid without any reference to 
 friction," at last comes to the conclusion that " the application 
 of these ideas to the problem of the resistance of a stream to 
 a plane lamina immersed transversely amounts to a justi- 
 fication of the old theory, as at least approximately correct." 
 
 This leaves the subject in an extremely nebulous state ; 
 but the paper is full of exceedingly interesting and useful 
 details, some of which will be referred to later. 
 
 Another distinguished scientific author, Dr Fleming, in his 
 charming little book on Waves and Ripples, after speaking of 
 liquid resistance and a " perfect " fluid, says : " It is obvious, 
 from what has been said, that if a perfect fluid did exist, 
 it would be impossible by mechanical means to make eddies 
 in it ; but if they were created, they would continue for ever, 
 and have something of the permanence of material substances." 
 
 It is certainly not "obvious " to the ordinary reader: it is, 
 on the contrary, absolutely questionable; for in 1887 Lord 
 Kelvin published his classical paper " On the Formation of 
 Coreless Vortices by the Motion of a Solid through an Inviscid 
 incompressible Fluid," in which he showed how the vortices 
 would be formed. In 1904 also he stated: "After many 
 years of failure to prove that the motion in the ordinary 
 .Helmholtz circular ring is stable, I came to the conclusion 
 that it is essentially unstable, and that its fate must be to 
 become dissipated as now described." 
 
Resistance of Liquids — Confused State of the Subject 5 
 
 When experts like these disagree the subject is not made 
 clearer for the student. 
 
 A few years ago Dr Hele-Shaw, in his lecture on " The 
 Motion of a Perfect Liquid," said : " Probably one of the 
 most perplexing things in engineering science is the absence 
 of all apparent connection between the higher treatises on 
 hydrodynamics and the vast array of works on practical 
 hydraulics. The natural connection between the treatises 
 of mathematicians and experimental researches of engineers 
 would appear to be obvious, but very little, if any such 
 connection, exists in reality, and while at every step electrical 
 applications owe much to the theories which are common to 
 electricity and hydromechanics, we look in vain for such 
 applications in connection with the actual flow of water." 
 
 That this chasm exists is unfortunately only too true ; but 
 is it all the fault of the engineers? It does not appear to 
 be so. 
 
 Dr Hele-Shaw most rightly says : " All scientific advance 
 in discovering the laws of Nature has been made by first 
 simplifying the problem and reducing it to certain ideal 
 conditions " ; but to this he adds, " and this is what mathe- 
 maticians have done in studying the motion of a liquid." 
 
 Now, have the mathematicians " simplified " the problem ? 
 Unfortunately, from the hydraulic engineer's point of view, 
 that is what they have not done : they have actually com- 
 plicated matters ; so much so that many engineers look upon 
 this " perfect " liquid as a perfectly useless toy invented by, 
 and for the delectation of, the mathematicians. 
 
 It is evident that this is not quite a fair way of looking at 
 a "perfect" liquid, the only assumed properties in which it 
 differs from an ordinary liquid being that it is inviscid, or 
 frictionless, and incompressible. It is certainly impossible, 
 but its assumption is no more absurd than that of a frictionless 
 pulley, or a frictionless crane. 
 
 When engineers commence the study of mechanics they 
 are introduced to the frictionless pulley and are taught that 
 its "advantage" is, say, 4 to i, or that a cev\.a.m frictionless 
 crane has an "advantage" of 40 to i. It is only later that 
 
6 A B C of Hydnuhuauiics 
 
 they learn that friction modifies these results more or less. 
 Wh\- is not the same method adopted when treating of 
 liquids ? Dr Hele-Shaw says : " The reason for this appears 
 to be the immense difference between the flow of an actual 
 liquid and that of a perfect one oivitis to the /•ivpcrty of 
 viscosity." It is difficult to accept this as the correct explan- 
 ation. If, however, Dr Hele-Shaw's reason were the correct 
 one, it would surely be the very strongest justification for 
 the neglect which he complains of It would clearly be a 
 waste of time for an engineer to study the movements of 
 a perfect liquid, only to be told at the end that an ordinary 
 liquid behaved in a totally iliWarnt tnantier. No ! a most 
 emphatic negative reply must be given to this ; for the only 
 difference due to viscosity between a perfect liquid and water 
 (which is practically incompressible') is that the particles of 
 the former can ha\-e no tangfiitinl stress applied to them, 
 whilst those of the latter can. You cannot, therefore, rotate 
 the particles of a perfect liquid, so that all motion of a perfect 
 liquid must of necessity be " irrotational." 
 
 It is true that the mathematicians, by their assumption of 
 an inviscid (or, as they call it, " perfect ") liquid, liave, so far, 
 " simplified " the problem, but they have at the same time 
 introduced a special complication by the further assumption 
 that this liquid is of infinite dimcmions ; and it is in this 
 assumption that the cause lies of the discrepancy in the 
 behaviour of the perfect fluid as compared with the real fluid. 
 
 The motion of an infinite liquid is (at times) quite different 
 from that of water, which is not infinite in extent, and which 
 ordinarily has a free surface : so much so that the theory 
 " appears to be \ erj' tolerably eomplete and afTords the 
 means of calculating the results to be expected in almost 
 every case of fluid motion ; but while in man\- cases the 
 theoretical results agree with those actually obtained, in other 
 cases they are altogether difTerent" (Osborne Re\Miolds). 
 With a sound theory, under similar eonditions, the How of a 
 viscid liquid shouUl not be ."(Vi' difforont from that of a 
 perfect one : not much more than, sa_\-, the motion of an 
 ordinary pulley differs from a theoretical!)- frictionless one. 
 
Resistance of Liquids — Confused State of the Subject 7 
 
 What strikes one as extraordinary is that Dr Hele-Shaw 
 should have been the man who first, by suitable means, 
 caused a viscid fluid to flow like a perfect one : also, mirabile 
 dictu, far from friction being a hindrance, he found water was 
 not viscid enough for his purpose, and so he had to employ 
 glycerine. If a viscid liquid be compared with the perfect 
 liquid, it must be under exactly similar conditions, especially as 
 regards the presence or absence of a free surface. 
 
 It will be well here to define the terms "rotary" and 
 " rotational," since they may be, and frequently are, employed 
 in two different senses. They will be invariably used here as 
 implying rotation of the particles about an axis. When the 
 particles move in a circle, or other closed curve, the motion 
 will be called " cyclic " : cyclic motion may, of course, be 
 " rotational " or " irrotational," according to whether the 
 particles are turning on their axes or not. Many writers 
 unfortunately use the terms very loosely ; sometimes as de- 
 scribed here and sometimes as " rotary or vortex." This leads 
 to endless confusion ; since an " irrotational vortex " would 
 then be a contradiction of terms. 
 
 To show that I have not exaggerated the confused state of 
 the subject, I will close this chapter with a quotation from 
 Mr Lanchester's Aerodynamics, which is a very modern 
 treatise and, probably, contains most of the latest views : I 
 have italicised the parts to which I particularly wish to draw 
 attention : 
 
 " I 98. The deficiencies of hydrodynamic theory have 
 already been pointed out in several instances and partially 
 discussed. The forms of flow that result from the assumption 
 of continuity and the equations of motion bear in general but 
 scant resemblance to those that obtain in practice, and it is not 
 altogether easy to account for the cause of failure. 
 
 "If an actual fluid behaved anything like the ideal fluid of 
 theory, the necessity for the ichthyoid form would not exist (?); 
 any shape, however abrupt, short of producing cavitation, 
 would give rise to stream-line motion [?] and be destitute of 
 resistance. The actual phenomena of fluid resistance, discussed 
 in the two previous chapters, is characterised by features which 
 
S .ABC of ffydrodviiamiis 
 
 at present are not dipjolt- of co»iph-tc eluciiiatioH by analytic 
 methods." 
 
 In the next chapter we will consider what is implied h\- the 
 assiunption that a fluid is of infiMiU t\rUnf, as this is the key 
 which will unlock many difficulties and enable us to reconcile 
 many statements which appear, at first sight. :»bsolutel\- 
 contradictory. 
 
 Summary 
 
 The subject of liquid resistance appears to be in a very 
 confused state. This seems to be due to the assumption of 
 the mathematicians' " perfect " liquid, which is supposed to be 
 not only frictionless but also of infinite dimensions, and which 
 consequently /ms no /nr sntf<hi\ and the comparison of this 
 liquid, without a free surface, with an ordinary liquid, like 
 water, which /ms it free siir/.ur — conditions which are not 
 sifntViir. 
 
 REFERENCES 
 
 Lord Rayleigh's collected papers, vol. i. 
 
 Lord Kelvin's collected papers, vol. iv. 
 
 Dr Fleming, Wares and Hi/'/'Us. Tublished by the S.P.C.K., 
 
 IQ02. 
 
 Dr Hele-Shaw, "The Motion of a Perfect Liquid" (R. Inst. 
 1899). 
 
CHAPTER II 
 
 A LIQUID OF INFINITE DIMENSIONS 
 
 If a liquid (supposed incompressible)^ be of infinite extent 
 in every direction, it is evident that it is ^occupying the 
 maximum, space possible, and that no particle could move 
 without some other particle moving away to make room for 
 it. Any separation of the particles would be, obviously, 
 impossible. All motion in it would be, in fact, perfectly con- 
 tinuous, and all " di.scontinuity " would be not only non- 
 existent but impossible. All motion of the liquid would be 
 what is called " by obligation." If a body moving in this 
 liquid were suddenly stopped, all motion in the liquid would 
 cease instantly. Under these circumstances (as the mathe- 
 maticians have shown), if the liquid were inviscid, all resistance 
 -would be impossible, since the sum of all the anterior pressures 
 would exactly balance that of all the posterior pressures :' in 
 other words, " it would be found that an object of any shape 
 wholly submerged in the fluid could be moved about in any 
 way without experiencing the least resistance " (Dr Fleming, 
 Waves and Ripples'). This will not be disputed by anyone. 
 
 It is not even absolutely necessary to assume that the 
 liquid is quite infinite in extent for this to be true. Since a 
 stress requires time for its transmission to a distance, and 
 since it also takes a definite time for the formation of a 
 
 ' When I speak of a liquid as being an incompressible fluid, it must be 
 understood that I mean that it is nearly incompressible, compressibility 
 being a negligible factor. If one assumed (as is commonly done) a 
 liquid to be absolutely incompressible, it is diflficult to imagine how energy 
 could be communicated from one particle to another of it. 
 
 9 
 
lo A B C of HydrodynaDiics 
 
 vortex, it follows that if the free surface be so far away that 
 the stress cannot travel to this free surface in the time required 
 for the formation of the vortex, then the actual result would 
 be the same as if the liquid were infinite. For example, if we 
 imagine that the velocity with which a stress can travel in a 
 liquid to be 186,000 miles in a second — it is hardly probable 
 that it could possibly exceed this — and if we suppose that it 
 takes even half a second to form a vortex — which is probably 
 not less than ten times the actual period — then, if the body 
 moving in the liquid were 93,000 miles from the free surface, 
 there would be no inertia resistance. 
 
 If the non-viscous liquid, however, were not of infinite 
 extent — if it had a free surface within a reasonable distance — 
 if, in fact, it were like the Atlantic Ocean — then resistance 
 would be quite possible, and tlie behaviour of the liquid would 
 not be unlike that of water, for the flow would not be, neus- 
 sarily, one " by obligation.' 
 
 This is the view that was held by Lord Kelvin, who stated 
 {Naturi\ 1S94): "A stiff circular disc of lo-inch diameter 
 and xV of ^n inch thick in its middle, shaped truly to the 
 figure of an oblate ellipsoid of revolution, would cause a 
 vacuum ^ to be formed all round its edge, if moved at even so 
 small a \rclocity as / foot per second [italics added] under 
 water of anj- depth less than 63 feet, if water were invisCid ; 
 and at greater depths the motion would, on the same supposi- 
 tion, be wholly continuous, witii no vacuum, and would be 
 exactly in accordance with the unique minimum of energy 
 solution " ; and later : 
 
 " In the case of our circular disc, it would be circular vortex 
 rings that, if the water were inviscid, would be shed off from 
 its edge when the depth is less than 63 feet. If the depth is 
 very little less than 63 feet, these rings would be exceedingly 
 fine, and would follow one another at exceedingly short 
 intervals of time. Thus quite close to the edge there would 
 be something somewhat like to Stokes' ' rift," but with a rapid 
 succession of vacuum rollers, as it were, and no slipping be- 
 tween the portions of the fluid on its two sides." 
 
 ' Single word to denote space vacated by water. 
 
A Liquid of Infinite Dimensions 1 1 
 
 Since it is difficult to picture to oneself a fluid of infinite 
 extent — even if it be possible at all — it will perhaps be better 
 to put the case in a way which is easier to grasp. Suppose 
 an inviscid, incompressible liquid to completely fill a closed 
 tank whose sides zx& perfectly rigid. If the tank be assumed 
 to be large, the case will, in essence, be similar to that of a 
 liquid of infinite extent, since the liquid will have no free 
 surface, and it will be occupying the maximum space possible. 
 Any body moving in this liquid would meet with no resistance 
 of any kind whatever : all anterior pressures would be exactly 
 balanced by the posterior pressures. There would be no, 
 what we may call, " free " flow, since all flow would be " by 
 obligation " ; and if the body were stopped, all motion of the 
 liquid would cease instantly. 
 
 If the body moving in this liquid were a thin lamina, as 
 its velocity increased, so would the pressure on the walls of the 
 tank. Even at very moderate velocities this pressure might 
 become enormous ; whilst if the lamina had a mathematical 
 sharp edge, the pressure on the tank walls would necessarily be 
 infinite. This question has been very fully and ably treated 
 by Professor Lamb in his Hydrodynamics. 
 
 Next, let us imagine the roof of the tank to be removed. 
 The liquid now has a free surface, and any motion in it would 
 not, necessarily, be one " by obligation," since the pressure on 
 the tank walls could not be sensibly increased, but would 
 resemble that of an ordinary liquid, like water. Coreless 
 vortices would be formed, as has been pointed out by Lord 
 Kelvin in his papers referred to. 
 
 It is not even necessary that the whole of the roof of the 
 tank should be removed. If a small hole were bored in it 
 and a stand-pipe were attached, on moving the lamina, core- 
 less vortices would still be formed, whilst the level of the 
 liquid in the stand-pipe would rise. 
 
 Some writers appear to assume that what is true in the 
 closed tank will be equally true in the open tank. Mr 
 Lanchester (Aerodynamics), referring to this, says : 
 
 " When a body propelled through an incompressible fluid, con- 
 tained within a fixed enclosure, experiences resistance to its 
 
12 ABC 0/ Nvi/fvi/y/KVf/its 
 
 motion, t/iefonr lUt/Av/ dr the body on tlie jhtid docs not impart 
 momentum to tfu fluid, but is transmitted instant/v to t/ie 
 confines of the fluid, Iiozccvcr remote, ,i>id is wfwUy borne by its 
 bon/td<irr ztud/s." This is a ven- clear statement of the case, 
 and Mr Lanchester refers to it in his work as the " principle 
 of no momentum." Four pages later he continues : " When a 
 body of fish-shaped or ichthyoid form travels in the direction 
 of its axis through a frictionless fluid [no mention made of 
 • contained within a fixed enclosure "] there is no disturbance 
 in its wake. Now we have seen that in any (-(Jjy [italics added] 
 the fluid as a whole receives no momentum, so that it is perhaps 
 scarcely legitimate to argue that there is no resistance beciuse 
 there is no communication of momentum, although this is a 
 common statement. It is clear, however, that if there is no 
 residuary disturbance, there is no necessar\- expenditure of 
 energy, and this equall\- implies that the resistance is nil." 
 
 The usual assumption is here frankly stated. The argu- 
 ment is something as follows : — 
 
 1. A fish in 9. fixed enclosure, filled with perfect liquid, when 
 moving would generate no momentum. 
 
 2. Therefore, in any case, "the fluid as a whole receives no 
 momentum." The cases are not similar and the one does 
 not follow from the other. Of course momentum can be used 
 in two senses. If Mr Lanchester had said that the fish could 
 generate no linear niomcntnm, then no fault could, perhaps, 
 have been found with his statement. It is contended here 
 that //(' bod\- can generate it in a liquid — perfect or im- 
 perfect — for a liquid does not obey Newton's second law of 
 motion. This will be referred to later. There is, however, 
 another kind of momentum — c\clic momentum — which a 
 bod\- can generate in a liquid. It is' the generation of this 
 momentum which causes resistance in liquids, and which 
 would cause resistance in a " perfect " liquid, if such a thing 
 existed. 
 
 As if the subject were not sufficiently complicated already, 
 Mr Lanchester also sa\s : " There are certain cases in which 
 the principle of the continuous communication of momentum 
 [linear is implied] can be a/'flied. A most striking example is 
 
A Liquid of Infinite Dimensions 13 
 
 to be found in the theory of marine propulsion founded by 
 Rankine and Froude." . . . "In general it would appear 
 that the Newtonian ^ method is applicable in cases where the 
 volume of the fluid handled is great, but where the impressed 
 velocity is small in comparison with the velocity of motion, 
 and where there are well-defined conditions on which to compute 
 the amount of fluid dealt with. . . ." 
 
 If this were not stated in all seriousness one would feel 
 inclined to treat it as a joke. Surely the volume of water 
 handled by the Mauretania ought to be sufficient to comply 
 with the pushing requirements. There are also well-defined 
 conditions for computing the amount of fluid dealt with. In 
 this case, however (according to Mr Lanchester), the small 
 propeller can apparently do what the leviathan can not. 
 
 1 It is not exactly evident why Mr Lanchester refers to this as the 
 " Newtonian method." I cannot see that it is in the least in accord with 
 the teaching of Newton, any more than it is with the writings of Lord 
 Kelvin. In the Principia (Book II. Sec. vii. Prop, xxxvi. Problem viii. 
 Cor. 7 — Motte's translation) Newton refers to the flow of water past a 
 small circular disc, and he shows that the pressure 
 may be expressed, graphically, by a conoid of revolu- 
 tion, as in the sketch. The pressure at D is repre- 
 sented by C D, the hydrostatic pressure ; and the 
 curve of pressure by A C and C B. Since this conoid 
 is convex, he says the total pressure, on the disc, will 
 exceed one-third of the hydrostatic presstcre. 
 
 In Cor. 8 he points out that, as the angles at A 
 and B are less than right angles, the total pressure 
 will be less than that represented, graphically, by a 
 sphoeroid : in other words, that the total pressure will 
 be less than two-thirds of the hydrostatic pressure. 
 
 In Cor. 9, Newton further says that when the disc 
 is very small the total pressure will be ''very nearly equal" to one-half 
 of the hydrostatic pressure. 
 
 These statements are made subject to the condition that the vel ocity 
 of flow of the water past the disc may be represented hy v= ^2gh, 
 where ^= depth of immersion of the disc. 
 
 How all this accords with what Mr Lanchester calls the " Newtonian 
 method" is not at all clear to me. If a plate drives water it would 
 appear to be necessary that the pressure on the " driving surface " should 
 be in excess of the hydrostatic pressure— a.xiA not the reverse. Besides, 
 the " volume of fluid handled" here is not " great," but, on the contrary, 
 very small. 
 
14 A B C of Hvdrodynainhs 
 
 There can be onl\- cue laic for <»// bodies. Either a body 
 can push water or it can tiot ; and by "' push " Is meaiit 
 to generate linear momentum by pressure. I know of 
 no evidence that a propeller does drive the water back- 
 wards. Water ^vt-j back, of course, and cjclic momentum is 
 generated ; but all the longitudinal acceleration of the water 
 is produced in front of t)ic propeller — i.e. towards the ship — 
 and not behind it. 
 
 Mr Lanchester's fish, if swimming near the surface of the 
 perfect liquid, would undoubtedl>- cause waves. 
 
 I cannot do better than close this chapter by an extract 
 from Lamb's Hydrodynamics (,1895): 
 
 " Ji. We have, in the preceding pages, had several instances 
 of tl\e flow of a liquid round a sharp projecting edge, and it 
 appeared in each case that the velocity there was infinite. 
 This is indeed a necessary consequence of the assumed ir- 
 rotational character of the motion, whether the fluid be in- 
 compressible or not, as may be seen by considering the 
 configuration of the equipotential surfaces (^ which meet the 
 boundary at right angles) in tlie immediate neighbourhood. 
 
 "The occurrence of infinite values of the velocity may be 
 avoided by supposing the edge to be slightly rounded, but 
 even then the velocity near the edge will much exceed that 
 which obtains at a distance great in comparison with the 
 radius of curvature. 
 
 " In order that the motion of a fluid may conform to such 
 conditions, it is necessary that the pressure at a distance 
 should great!}- exceed that at the edge. This excess of 
 pressure is demanded by the inertia of the fluid, which cannot 
 be guided round a sharp curve, in opposition to centrifugal 
 force, except b\- a distribution of pressure increasing with a 
 very rapid gradient outwards. 
 
 " Hence, unless the pressure at a distance be very great, 
 the maintenance of the motion in question would require a 
 negative pressure at the corner, such as fluids under ordinar)- 
 conditions are unable to sustain. 
 
 " To put the matter in as definite a form as possible, let us 
 imagine the following case. Let us suppose that a straight 
 
A Liquid of Infinite Dimensions 
 
 15 
 
 tube, whose length is large compared with the diameter, is 
 fixed in the middle of a large closed vessel filled with friction- 
 less liquid, and that this tube contains, at a distance from the 
 ends, a sliding plug, or piston P, which can be moved in any 
 required manner by extraneous forces applied to it. The 
 thickness of the walls of the tube is supposed to be small in 
 comparison with the diameter, and the edges, at the two 
 ends, to be rounded off so that there are no sharp angles. 
 Let us further suppose that at some point of the walls of the 
 
 Fig. 2. 
 
 vessel there is a lateral tube, with a piston Q, by means of 
 which the pressure in the interior can be adjusted at will. 
 
 " Everything bping at rest to begin with, let a slowly 
 increasing velocity be communicated to the plug P, so that 
 (for simplicity) the motion at any instant may be regarded as 
 approximately steady. At first, provided a sufficient force 
 be applied to Q, a continuous motion of the kind indicated 
 in the diagram on p. 83 will be produced in the fluid, there 
 being, in fact, only one type of motion consistent with the 
 conditions of the question. As the acceleration of the piston 
 P proceeds, the pressure on Q may become enormous, even 
 with very moderate velocities of P ; and if Q be allowed to 
 yield, an annular cavity will be formed at each end of the 
 tube." 
 
1 6 A B C of Hydrodynamics 
 
 Besides this, " Gravity, which could not aflFect the motion of 
 a liquid entirel>- filling a rigid closed vessel, will exercise a 
 most important influence on the motion of a liquid contained 
 in an open vessel, and exposing a free surface to the 
 atmospheric pressure " (Lord Kelvin, " Notes on H\'dro- 
 djn amies," Cambridge and Dublin Mathcnuiticat Journal, 
 
 1849). 
 
 This question of the perfect liquid being assumed as of 
 itifinitc extent has been referred to at considerable length ; 
 but it is a verj- important one, and one which is, frequentl}-, 
 not properly grasped. Even that scientific ijiant, Lord Kelvin, 
 was nodding when he wrote his paper on coreless vortices, for 
 he there assumed the fluid to be " of infinite extent in all 
 directions." He admitted in a letter, before his death, that 
 his meaning was that " the infinite liquid extends through all 
 space except certain given portions which are the hollows of 
 the supposed coreless vortices " : in other words, that the 
 liquid was nearly infinite only. I am afraid this does not get 
 over the difficulty, though it certainly shifts jt away another 
 step. To form the coreless vortices it would be necessary to 
 shift a (virtually) infinite column of liquid a finite distance in 
 a finite time, which is unimaginable. 
 
 SUMM.-VRY 
 
 " All the forces which are observed in Nature to act upon 
 the mass of a liquid at rest, whatever may be the agencies to 
 which it is subjected, are such that if the liquid be enclosed 
 in a fixed envelope the\- cannot disturb its equilibrium, but 
 are in all cases balanced by the resistance which the fluid 
 pressure experiences from the bounding solid, . . . and 
 therefore, in all cases in which the form of the bounding 
 surface is susceptible of alteration by the pressure of the 
 fluid, the forces through the mass will, by the effect they 
 may thus produce on the form of the bounding surface, 
 exercise an indirect influence on the motion which takes 
 place within it. Thus it is that gravit\-, which could not 
 affect the motion of a liquid entirely filling a rigid closed 
 
A Liquid of Infinite Dimensions 17 
 
 vessel, will exercise a most important influence on the motion 
 of a liquid contained in an open vessel, and exposing a 
 free surface to the atmospheric pressure" (Lord Kelvin, 
 Cambridge and Dublin Mathematical Journal, 1849). 
 
 REFERENCES 
 
 H. Lamb, Hydrodynamics. 
 Lanchester, Aerodynamics. 
 Newton, Principia, vol. ii. 
 
CHAPTER III 
 
 MOVEMENT OF A LIQUID AND THE LAW OF FLOW 
 
 Having now a clear idea of the difference between a " free " 
 flow and a flow " by obligation," it will be well to consider 
 what a liquid is, and what are the laws regulating its move- 
 ment. 
 
 Newton defined a fluid as " any body whose parts yield to 
 any force impressed on it, and, by yielding, are easily moved 
 among themselves." He seems to have laid stress on the 
 difference between a solid and a liquid: that the former 
 "resists" a force, whilst a liquid "yields" to it, a liquid 
 being an incompressible fluid. 
 
 The Encyclopcedia Britanrtica defines — " a fluid, as the name 
 implies, is a substance which flows, or is capable of flowing." 
 This may appear a little tautological ; but it conveys a clear 
 idea, though it might be necessary to precise what " flowing " 
 means. 
 
 A careful examination of the subject will satisfy anyone 
 
 that a fluid cannot be " pushed " or " driven " : a body moving 
 
 in a liquid can onl}- dimde it. This was pointed out by 
 
 Dr Wallis, who was a contemporary of Newton and who is 
 
 referred to in the Principia, with Sir Christopher Wren and 
 
 Huygens, as "the greatest geometers of our times." The 
 
 Chevalier Dubuat, in referring to this point, says : " Quand un 
 
 corps r^siste a Taction d'un courant, le fluide se d^vie k une 
 
 certaine distance en avant de lui, et qu'il se forme une espkt 
 
 de protte fluide, dans laquelle les filets, sans perdre toute leur 
 
 Vitesse, en ont cependant moins que le reste du fluide, dans le 
 
 sens du mouvement g^ndral." In short, a fluid, as its name 
 
 iS 
 
Movement of a Liquid and the Law of Flow 19 
 
 implies,y?ozyj: it can ont)/ How, and it must How, from any region 
 of higher pressure to any region of lower pressure. You can 
 induce a fluid, by suitable means, to flow in any given direc- 
 tion, but you cannot mechanically drive it in any direction. 
 
 It is well that I should explain exactly what I mean. It 
 may be objected that water is forced, or ■' driven out," of a 
 squirt, and that, like a solid, it obeys Newton's second law 
 of motion.. This is not so, pressure being equal in all 
 directions. " Every particle of the stagnant water is equally 
 pressed on all sides, and, yielding to the pressure, tends all 
 ways with an equal force, whether it descends through the 
 hole in the bottom of the vessel, or gushes out in a horizontal 
 direction through a hole in the side " {Principid). If a small 
 hole be bored in the side of the squirt, the water will flow but 
 just as readily at right angles as along the axis of the piston ; 
 if the packing be loose, it will even flow as readily backwards 
 as forwards. It cannot be contended that there is any force 
 which drives the water at right angles or backwards. 
 
 It has been objected to me (by a Professor of Mathematics) 
 that the force driving the water in the squirt is not necessarily 
 in the direction of the motion of the piston ; that the real force 
 is the resultant of many forces, and that no attempt has been 
 made to calculate the direction of this resultant, etc. etc. 
 This is quite true ; but if no attempt has been made to 
 calculate the direction of this resultant, it is for the very 
 best of reasons, viz., that there is not — and cannot be — any 
 resultant. Whether the stress be applied from the N., S., E., W. 
 Zenith or Nadir, the result is exactly the same ; each particle 
 of water is equally stressed in every direction. If the force 
 applied to the piston be equal to 10 lbs. per square inch, every 
 particle of water will have a pressure equal to 10 lbs. per 
 square inch over the whole of its surface. If we could imagine 
 that the force was applied from the north, and that the 
 resultant force was calculated to pass through some particular 
 hole in the syringe; if we next applied the force from the 
 south — or any other point of the compass — the resultant 
 could hardly be the same ; yet the water would flow out in 
 exactly the same manner. There being no resultant, there is 
 
 B 2 
 
20 A B C of Hydrodynamics 
 
 no force "driving" the water out All that you do in the 
 squirt — all that you cau do — is to produce a region of higher 
 pressure in the instrument, and the liquid then flows from this 
 region to any region of lower pressure. Similarly, if you puU 
 up the plug of a bath, or turn on a water-tap, you produce a 
 r^on of lower pressure, and the water flows towards this. 
 
 I may appear to have rather laboured this point, but the 
 statement that a liquid does not obey Nezvion's second laiv of 
 motion is generally received with a smile of surprise, if not of 
 contempt The assertion having been made, it was as well 
 to be precise. It is not denied that each individual particle 
 obeys this law : what is maintained is that in a body of water 
 the particles in their " corporate capacity " do not obey this 
 law. The two things are quite different ; and parallel cases, 
 in everyday life, amongst men might be quoted. 
 
 In the first place, Newton never said that a liquid obeyed 
 this law, and his teaching in the Prindpia appears to be dis- 
 tinctly against it To take a few examples : — Prop. xlL 
 theorem xxxiii. : " A pressure is not prop^ated through a 
 fluid in rectilinear directions, unless where the particles of 
 the fluid be in a right line." Prop, xlii theorem xxxiii.: 
 " All motion propagated through a fluid diverges from a 
 rectilinear progress into the unmoved spaces." 
 
 Other propositions might be quoted, but I should like 
 specially to point out that when he is speaking about a 
 circular disc moving in water, at the speed corresponding to tJie 
 depth {i^ = 2.gK), he says the pressure on its leading surface 
 would be only about half the hydrostatic pressure (Book II. 
 Sec. vii. Prop, xxxvi. Cor. 9). 
 
 The statement about " flow," made previously, admits of no 
 exceptions ; for when there are, apparently, cases where it 
 does not apply, it wUl be found that there is something which 
 is preventing the flow. For example, a gas may be under 
 pressure in a bag and so not able to flow to the r^on of 
 lower pressure outside ; if the pressure be increased, the bag 
 may burst, and then the flow will tcike place. This example 
 may appear childish, but there are many cases where the 
 surface of a liquid acts like the material of a gas bag and so 
 
Movement of a Liquid and the Law of Flow 2 1 
 
 prevents the flow of the liquid, as long as the pressure is not 
 too high ; if the pressure increases too much the film breaks 
 and the liquid flows. 
 
 If we admit that a fluid flows, that it can only flow, and 
 that it must flow from a region of higher to one of lower 
 pressure, what is the law regulating this flow ? 
 
 The law connecting the pressure in a liquid and its velocity, 
 when the flow is steady, can be expressed by the very simple 
 formula 
 
 P + \p'^^ = constant, 
 
 where / = pressure in the liquid, jo = density, and 2^ = velocity ; 
 or, expressed slightly differently, 
 
 where Po = pressure at zero velocity. "By steady motion I 
 mean motion which at any and every time is precisely similar 
 to what it is at one time " (Lord Kelvin, Vortex Statics). 
 
 Quoting from Lord Rayleigh's paper previously referred to : 
 " The relation between the velocity and pressure in a steady 
 stream of incompressible fluid may be obtained immediately 
 by considering the transference of energy along an imaginary 
 lube bounded by stream-lines. 
 
 "In consequence of the steadiness of the motion, there 
 must be the same amount of energy transferred in a given 
 time across any one section of the tube as across any other. 
 
 " Now, \i p and v be the pressure and velocity respectively 
 at any point and p be the density of the fluid, the energy 
 corresponding to the passage of the unit of volume is 
 p-\-\pv^, of which the first term represents the potential zx^d. 
 the second the kinetic energy ; and this / + |joz^^ must retain 
 the same value at all points of the same stream-line." 
 
 This may be considered the formula in hydrodynamics 
 which especially interests engineers, and it will be frequently 
 referred to. Not only is the energy (potential + kinetic) 
 ■constant all along the same stream-line, but it is the same 
 for all the stream-lines in the same horizontal plane. 
 
 " It is further true, though not required for our present 
 purpose or to be proved so simply, thsit p + ^pv^ retains a con- 
 
22 A B C of Hydrodynamics 
 
 stant value, not only in the satne stream-line, but also when we 
 pass from one stream-line to another [? in the same horizontal 
 plane], provided that the fluid flows throughout the region con- 
 sidered in accordance with the electric law " (Lord Rayleigh). 
 
 A natural corollary of this is that, in the same horizontal 
 plane all the parts of a liquid in steady motion must be under 
 lower pressure than those parts which are at rest, or are 
 moving at a lower velocity. Also, the greater the velocity the 
 lower will be pressure. 
 
 To put the matter familiarly, if we consider the potential 
 energy of the liquid as its bank balance and its kinetic energy 
 as its cash, the total capital of the stream-line (liquid tube) 
 will be bank balance plus cash, and this is constant. If the 
 liquid increases its velocity, or cash, it can only do so by 
 drawing on its bank balance ; and its cheques will only be 
 honoured up to this balance, overdrafts not being permissible 
 in the hydrodynamical bank. Similarly, if it reduces its 
 velocity, or cash, it does so by paying into the bank and so 
 increasing the balance there. As a perfect fluid has no 
 viscosity, it may be said to have no " expenses " : its capital 
 will never diniinish. 
 
 Water obeys the same law exactly, with one exception, and 
 that is \}a.aX.p-\-\pv^ is not constant, but is always diminishing 
 slightly. Its viscosity causes the particles to rotate and so 
 rub against one another. Energy is converted into the heat 
 form, which is dissipated and so lost — to the water. In other 
 words, water has " expenses " and so is living on its capital, 
 which is therefore constantly decreasing. 
 
 In conclusion, kinetic energy cannot be directly transferred 
 to a liquid. It must be paid into the bank as potential energy, 
 which can be drawn on by the liquid in the usual manner. 
 
 A few examples will now be given illustrating the action 
 of fluids in some very simple cases. 
 
 I. In the common scent spray, or the "atomiser," used in 
 horticultural work, a powerful blast of air is caused to flow 
 along the tube A B and just over the top of the tube B C, 
 which is dipping into a liquid. Since the pressure in the air 
 issuing from B is considerably less than the atmospheric 
 
Movement of a Liquid and the Law of Flow 23 
 
 pressure, the liquid will rise from C to B, when it is " atomised," 
 as shown in the sketch. 
 
 c 
 
 Fig. 3. 
 
 2. If water, under a high head, be flowing rapidly through 
 a lead pipe, and a small hole be bored in this pipe, the water 
 will not flow out by this hole, but, on the contrary, air will 
 flow in. As in Venturi's experiment, if a small pipe be fixed 
 to this hole, and the end of the pipe be dipped in liquid, this 
 liquid will be sucked into the lead pipe. 
 
 3. Another very pretty and well-known lecture table 
 experiment, which used to be a favourite one of Lord Rayleigh, 
 is to arrange two vessels of water, as in the sketch, the plates 
 
24 A B C of Hydrodynamics 
 
 C C being simply placed together, but not fastened. When 
 the water is flowing freely, the vessels can be separated at 
 C C, when the water will spring across the gap just as if the 
 tube were continuous. 
 
 4. The ordinary " injector " works, of course, on the same 
 principle. 
 
 5. A very pretty experiment due to Mr W. Child may 
 not be so well known. Two half " ping-pong " balls were 
 mounted on a lath as anemometer cups, turning on the 
 pivot, as shown in the sketch. If we now strike or poke 
 one of the cups with a stick (in the direction of the plane of 
 
 
 Fig. S. 
 
 the paper) along the arrow, the cup will, of course, turn 
 towards A. If, however, we cause a blast of air to strike it 
 in the direction of the arrow, the cup will move towards B. 
 In the one case the pressure is increased, whilst in the other 
 it is decreased. In the same way, if the lath be suspended 
 by a string instead of being fixed on a pivot, and then the 
 lath be raised and lowered by means of the string, the whole 
 arrangement will rotate in the direction of the circular arrows. 
 A very simple and pretty experiment showing the differ- 
 ence between the action of very fine dry sand, which does 
 not flow, and that of water, was exhibited at the Royal 
 Institution by C. E. S. Phillips in 1910. Take a small tube 
 of about the diameter in the sketch and open at both ends. 
 Cover the bottom with a cigarette paper, fastened on by an 
 elastic band. The tube is then partially filled with fine dry 
 sand and a piston is inserted on the top of the sand. As 
 
Movement of a Liquid and the Law of Flow 25 
 
 the sand does not "flow," by carefully applying weight to 
 the piston (as shown by arrows), the weight of a man can 
 be suspended from the piston. It is hardly necessary to say 
 
 ^?:^^=^=i^^^^?'^i^^'^^^'^^'^^^ 
 
 \\\\\\\\^\\\\\\\\ \^^^^^^^ 
 
 Elastic band-' 
 
 ', ( Cigarette papen 
 '/ ' fastened by an 
 ^ Elastic band 
 
 Fig. 6. 
 
 what would happen if water had been put into the tube 
 instead of fine dry sand. 
 
 Very many more examples might be given, but the reader 
 can supply these for himself : some special cases will also be 
 referred to later. 
 
 Summary 
 
 A liquid flows — can only flow — and must flow from a 
 region of higher pressure to a region of lower pressure. 
 
 A liquid does not, under ordinary conditions, obey Newton's 
 second law of motion. When it is stressed it does not 
 move in a straight line, in any direction. 
 
 As the velocity of a liquid increases its pressure decreases. 
 
 A stream of liquid can only increase its velocity by con- 
 verting some of \ts potential energy into kinetic energy. 
 
 Kinetic energy cannot be directly conveyed to a liquid. 
 An energy transferred must be potential. 
 
 REFERENCES 
 
 Newton's Principia, translated by A. Motte, 1729, vol. ii. 
 DUBUAT, Principes d^Hydraulique. 
 
CHAPTER IV 
 
 RESISTANCE, CONSIDERED GENERALLY 
 
 Having explained how a liquid flows and the law regulating 
 this flow, it will be well now to get a few clear ideas about 
 liquid resistance. I propose to examine the subject from 
 three points of view: (i) Newton's view; (2) that of the 
 mathematicians ; and (3) that commonly taught in the present 
 day, and which I will call, for brevity, Dr Froude's view, 
 since he appears to have been one of the first who taught it. 
 
 1. Newton, in his Principia, says : " In mediums void of all 
 tenacity, the resistances made to bodies are in the duplicate 
 ratio of the velocities " ; also, at page 54 : " And that part of 
 the resistance which arises from the density of the fluid is, as 
 I said, in the duplicate ratio of the velocity." He is even 
 more precise, for he says, in another place, that it " cannot be 
 less." 
 
 2. The mathematicians who treat of the stream-line theory 
 have shown that in a " perfect " liquid of infinite dimensions a 
 body moving would meet with no resistatice at all. 
 
 These two statements appear, at first sight, as absolutely 
 contradicting one another. The mathematicians must, of 
 course, be right : granted the data, the result cannot well be 
 in error ; but " it is important to remember that the mathe- 
 matical method as applied to physics must always be trust- 
 worthy, or untrustworthy, according to the trustworthiness 
 of the data which are employed ; that the most complete 
 presentation of symbols and processes will only serve to 
 enlarge the consequences of error hidden in the original 
 
 premises, if such there be" (Langley, Aerodynamics). 
 
 26 
 
Resistance, Considered Generally 27 
 
 It will be quite evident, from what has been said previously, 
 that Newton and the mathematicians are speaking of different 
 things. The latter only treat of flow " by obligation," whilst 
 Newton refers to an ordinary liquid which is capable of "free" 
 flow and consequently of "discontinuity." Both may be 
 correct — and, indeed, both are. To reconcile Newton's view 
 with that of the mathematicians it would appear to be 
 necessary to somewhat amplify his formula and to say that 
 
 R varies as pV^xF(Yr), where D = depth, and F(t^) is 
 
 some indetermined inverse function of D which would become 
 zero at some depth, depending on the shape of the body 
 moving in the liquid, and which would remain zero for all 
 greater depths. 
 
 For example, in an ocean of perfect liquid a body moving 
 near the surface would experience a resistance varying as the 
 square of the velocity. At greater depths this resistance would 
 decrease ; whilst at infinite depth (even with mathematical 
 sharp edges) it would meet with no resistance at all. 
 
 This is in accordance with Lord Kelvin's views (quoted 
 previously) when he says an oblate spheroid of 10 inches 
 diameter and -^ inch thickness would, if immersed in an 
 inviscid water, meet with resistance at a less depth than 63 
 feet, but not beyond that depth of immersion. 
 
 3. Dr Froude's view is that, if the ocean were a " perfect " 
 liquid, a fish or torpedo would meet with no resistance at all, 
 and, once moving, would go on for ever. 
 
 Dr Fleming, who is one of the eminent men who appear to 
 teach this, says, in his Waves and Ripples: ". . . If we could 
 obtain a perfect fluid in practice, it would be found that an 
 object 0/ any shape wholly submerged in the fluid [italics 
 added] could be moved about in any way without experienc- 
 ing the least resistance." I have italicised " of any shape " ; 
 and the employment of the words " wholly submerged " shows 
 that Dr Fleming admits the possibility of its not being wholly 
 submerged : evidently he postulates a "free surface." 
 
 Naval architects apparently hold views which are most 
 curious and incomprehensible. They would appear to be 
 
28 A B C of Hydrodynamics 
 
 rather like Jekyll and Hyde. When they speak of the 
 forward motion of a ship, a fish-like body, or a torpedo, then 
 they say the body divides the water and does not push it 
 forwards. When they reach the stern of the ship, however — 
 having presumably allowed the necessary quarter of an hour 
 to elapse — and speak of the propeller, then they say the 
 water is driven backwards by it. Apparently they would 
 seem to believe that a spherical body cannot push water, 
 whilst a circular flat plate can. If, however, the sphere be 
 supposed to be gradually flattened and to pass through the 
 orange and muffin shape to that of a flat plate, which would 
 be the exact shape when the pushing would begin ? Where 
 does the law change, and why? It has always appeared 
 extraordinary to me that Professor Rankine, who was one of 
 the leading exponents of the stream-line theory, should also 
 have been the author of the theory of the screw propeller, and 
 to have stated that all propellers act by driving water back- 
 wards, and so causing the forward reaction to propel the ship. 
 
 It has been said that to understand an idea it is necessary 
 to study its pedigree. What is the pedigree of this idea that 
 a perfect fluid could offer no resistance ? It would appear to 
 be somewhat as follows. The stream-line theory was first 
 propounded by certain mathematicians, who, for convenience, 
 treated their " perfect " fluid as of infinite extent : they found 
 that a body moving in this perfect fluid would not encounter 
 any resistance. This point no one disputes. 
 
 In 1875 Dr Froude, when president of Section G at the 
 British Association, selected for the subject of his masterly 
 address the motion and resistance of liquids. He may be said 
 to have, on this occasion, popularised the " stream-line " theory. 
 
 Dr Froude made no pretensions to being a mathematician, 
 for he said the mathematics "are beyond my ken and my 
 purpose," but he was obviously following the lead of the 
 mathematicians. 
 
 We notice that, in the opening part of his address, he was 
 most careful to speak of an " unlimited ocean of fluid " — an 
 " infinite ocean of perfect fluid," and " a submerged body in 
 the midst of an ocean of perfect fluid, unlimited in every 
 
Resistance, Considered Generally 29 
 
 direction." All this was most proper and correct. As the 
 address proceeds we see less and less insistence on the 
 " infinity " of the ocean, until, towards the end, we find Dr 
 Froude frankly stating : " I have shown that a submerged 
 body, such as a fish or torpedo, travelling in a perfect fluid 
 would experience no resistance at all!' This is, by no means, 
 exactly what he had said previously, which was that in an 
 infinite ocean of perfect fluid, etc. The two things are not 
 the same at all, although at first sight they may appear to be 
 so. The mathematicians had proved one, but the other 
 appears to be a pure assumption, in support of which not a 
 particle of evidence has ever been produced. 
 
 This statement of Dr Froude 's appears to have been 
 accepted without question, and is very commonly taught to 
 this day. 
 
 The case may be put briefly thus : — 
 
 1. The flow causing "no resistance'' must, necessarily, be 
 an " electric flow.'' 
 
 2. An electric flow necessitates an infinite velocity of the 
 fluid at every sharp edge. 
 
 3. This infinite velocity of the fluid necessitates an infinite 
 potential ; or, in other words, 
 
 V-\-\pv^— infinite. 
 
 4. As the immersion assumed is finite, 
 
 P+^/oz'^= constant a.nd finite. 
 
 .: The flow cannot be electric, and the assumption of " no 
 resistance " is an unsound one. 
 
 When Dr Froude, in 1875, had satisfied himself that a fish 
 or torpedo — why should naval architects lay such stress on 
 the fish or torpedo ? — moving in a perfect liquid would meet 
 with no resistance, he, very naturally, supposed that all the 
 resistance encountered by a body moving in a viscous liquid, 
 like water, was caused by the liquid friction. This has led to 
 other complications and errors ; for, as he had found by 
 experiment that the resistance of a body moving through 
 water varied — more or less — as the square of the velocity, the 
 
30 A B C of Hydrodynamics 
 
 inference was that the resistance due to friction varied in this 
 same ratio. As will be shown later, this is not correct, since 
 it only varies as the velocity, as was pointed out by Newton 
 more than 200 years ago. 
 
 If the resistance of a body moving in water were due 
 to viscosity only, it would be natural to expect that the 
 resistance would be the same whether the "fish or torpedo'' 
 were towed head first or tail first, which is by no means the 
 case — a sailor knows that a spar towed with the butt foremost 
 offers less resistance than if towed with the point first. To 
 get over this new difficulty the term " eddy-making resistance " 
 was introduced ; and at present the accepted view appears to 
 be that the resistance of a " fish or torpedo " is caused by 
 fluid friction plus eddy-making resistance. As these appear 
 to be the only two possible sources of resistance, no objection 
 can be raised to this view, the only difference of opinion 
 being on the point of how much is due to fluid friction and 
 how much to "eddy-making resistance." Apparently, the 
 ordinary method is to take the resistance due to the viscosity 
 at a certain amount (found by experiment) per square foot of 
 the wetted surface, and varying as the 1-83 power of the 
 velocity, the remainder being put down to "eddy-making 
 resistance." This is fairly simple, if not exactly scientific ; 
 but there are cases where even this does not agree very well 
 with facts, so a further complication has been added, called 
 the " augmented wetted surface." " This augmented surface 
 was intended to represent the plane area, having the same 
 resistance as the actual wetted surface, moving at the same 
 speed. It might be termed ' the equivalent plane surface ' " 
 {Atherton and Mellanby's Resistance and Power of Steam- 
 ships, 1903). This leaves the subject in a state which can 
 only mildly be expressed as " perplexing." In practice, if a 
 naval architect wants to know the resistance of a ship, he 
 makes an exact model to scale and finds the resistance of 
 this model by experiment. Knowing this, a very simple 
 calculation, by Dr Froude's well-known laws of proportion, 
 will give the resistance of the full-sized ship with an extra- 
 ordinary degree of accuracy. 
 
Resistance, Considered Generally 31 
 
 If the accepted view of the resistance of a ship be some- 
 what obscure, the resistance of a flat plate, moving broadside 
 on to the water, is vastly more so. The whole theory applied 
 to the " fish or torpedo " crumbles away like a child's house of 
 cards. 
 
 Mr R. E. Froude, at Greenock in 1894, said : " In these 
 questions I believe that we are treading on very difficult 
 ground, but this much I imagine may be safely said in reply : 
 in the cases here supposed [a propeller blade and the keel of 
 a ship] the flatwise reaction is not due to the friction of the 
 water, nor in any sense proportional to it. 
 
 " The resistance arises directly out of the inertia of the fluid, 
 in virtue of the establishment of a system of stream-lines of a 
 totally different character from those of no resistance, which is 
 theoretically proper to a frictionless fluid, and the presence of 
 a certain frictional quality in the fluid — perhaps in a very 
 moderate degree — makes the resisting system of stream-lines 
 possible" [italics added]. 
 
 This " explanation " leaves one very little wiser than before. 
 Certainly, if the author had a clear idea of what he meant, he 
 appears to have been most careful to keep it to himself. 
 From this macidoine of words, however, we can gather 
 that — 
 
 (i) The resistance of a flat plate is not due to friction, but 
 is directly due to the inertia of the liquid. 
 
 (2) This inertia resistance would not be possible but for the 
 liquid having some — possibly very small — frictional quality. 
 
 Of these we may accept (i) as correct, but it will be shown 
 later that (2) is not — Lord Kelvin's view, quoted previously, 
 being in direct contradiction qf (2). It would appear to be 
 true that, if a perfect liquid existed, a screw propeller would 
 work as well in it — if not better — than it does in water. 
 
 Mr R. E. Froude appears, at times, to almost regret the 
 old theory of fluid resistance, for he says: "There are 
 undoubtedly cases of fluid resistance in which the old theory, 
 or something like it, is still applicable," thus almost echoing 
 Lord Rayleigh's view, quoted previously. 
 
 The old theory may have been bad; may have been — 
 
32 A B C of Hydrodynamics 
 
 almost undoubtedly was — wrong ; but it had, at least, one 
 merit: it was intelligible, which is perhaps more than can 
 be said of the present accepted view. 
 
 We therefore come back to Newton's view — which is the 
 same as Lord Kelvin's — that " in mediums void of all tenacity 
 the resistances made to bodies are in the duplicate ratio of 
 the velocities." 
 
 In the next chapter the subject of viscosity and fluid 
 friction will be gone into, and we will see how this friction 
 modifies the results. Subsequently the question of resistance 
 will be gone into in more detail and comparison made between 
 theory and experiment. 
 
 Summary 
 
 The resistance of a body moving in a " perfect " or inviscid 
 liquid would vary as the density of the liquid and as the 
 square of the velocity of the body, and would continually 
 diminish as the depth of immersion ; would vanish at a 
 certain finite depth, and at all greater depths the resistance 
 would remain zero. If, however, the body be supposed to 
 have mathematical sharp edges, then the immersion must be 
 to an infinite depth for the resistance to become zero. 
 
 REFERENCES 
 
 Report of Brit. Association, 1875. 
 
 R. E. Froude, Ship Resistance (Greenock Philosophical Society, 
 1894). 
 
 Atherton and Mellanby, Resistance and Power of Steamships, 
 1903. 
 
 S. P. Langley, Experiments in Aerodynamics. 
 
 Thomson and Tait, Natural Philosophy. 
 
 Helmholtz, Phil. Mag., xliii. 
 
CHAPTER V 
 
 VISCOSITY AND FLUID FRICTION 
 
 Besides the resistance which is due to the density of the 
 liquid, there is another which is due to the stickiness or 
 " treacliness " of the liquid. Newton refers to it as the 
 " want of lubricity," and he says : " The resistance arising 
 from the want of lubricity in the parts of the fluid is, cateris 
 paribus, proportional to the velocity with which the parts 
 of the fluid are separated from each other " [italics added]. 
 It is clear that Newton considered this resistance as a stress 
 in the liquid to resist shearing. Text-books on hydraulics 
 do not, as a rule, devote very much space to liquid friction, 
 but it may be advisable to recapitulate their statements 
 thereon ; and probably the very best description is that 
 found in Perry's Applied Mechanics. 
 
 He says : " Water flowing in a certain pipe at the velocities 
 I, 2, 3, etc., inches per second, the friction is proportional to 
 the numbers i, 2,. 3, etc. ; whereas at the velocities i, 2, 3, etc., 
 yards per second the friction is proportional to the numbers 
 I, 4, 9, etc. At small velocities three times the speed means 
 three times the friction ; whereas at great velocities, such as 
 those of ships, three times the speed means nine, or more, 
 times the friction." 
 
 In this paragraph it is necessary to understand what is 
 the exact meaning of the word "friction." If, as is probable, 
 Perry meant it as equivalent to " resistance," then it is indis- 
 putable. If it be employed as resistance due to viscosity, then 
 it cannot be accepted as a correct statement of the facts. 
 In the second case, the total resistance is that due to viscosity 
 
 33 C 
 
34 A B C of Hydrodynamics 
 
 plus " eddy-making resistance." Because the total resistance 
 has increased as i, 4, 9, etc (which is a fact), it does not 
 follow, as a consequence, that therefore the resistance due 
 to viscosity has increased in this ratio. If the viscosity were 
 the cause of the eddy-making resistance, then the statement 
 might be true; the exact reverse, is, however, the case: 
 viscosity tends to prevent eddy-making. 
 
 To continue : " At constant temperatures below a certain 
 critical speed, I found that the friction was proportional to 
 the velocity, so that jx could be found. At the critical 
 speed I found there was a sudden change in the law [italics 
 added], and above that speed the friction [? total resistance] 
 is proportional to a higher power of the speed than i. We 
 know that above the critical speed the plane motion, which 
 I described above, would become unstable and eddies would 
 be formed. . . ." 
 
 " At speeds below the critical, I measured fi at many 
 different temperatures, and noted the rapid decrease of it 
 as the temperature increased." 
 
 If we assume, as before, that the word "friction" has 
 been employed in two rather different senses, we may put 
 Perry's statement somewhat as follows: The resistance due 
 to viscosity varies as the velocity, but that at a " critical speed " 
 the motion of the liquid becomes unstable, eddies are formed, 
 and a new resistance — which is an inertia resistance'^ — may 
 be said to be " switched in " suddenly. It is well to remember, 
 also, that the resistance due to viscosity decreases rapidly as 
 the temperature increases. 
 
 Professor Perry's summary is very much to the point, and 
 is as follows : — 
 
 " I. The force of friction very much depends on the velocity 
 and is indefinitely small when the speed is very low. 
 
 " 2. The force of friction does not depend on the pressure. 
 
 " 3. The force of friction is proportional to the area of the 
 wetted surface. 
 
 ^ Inertia resistance is that resistance which is due to the continuous 
 communication of momentum to the liquid surrounding the body. Mr 
 Lanchester speaks of it as "dynamic resistance." 
 
Viscosity and Fluid Friction 35 
 
 " 4. The force of friction at moderate speeds does not much 
 depend on the nature of the wetted surface." 
 
 In all these oases it is plain that by " friction " is meant 
 the resistance due to viscosity. It is obvious that the friction 
 of mercury in a glass tube is very small ; but it does not 
 follow that the viscosity of the mercury is also small. Rules 
 Nos. 3 and 4 do not apply here, for there is no wetting of 
 the glass at all ; No. 4 will be referred to again later, for the 
 statement requires qualification. 
 
 From what Professor Perry has said we would infer that 
 the resistance is composed of two terms, one of which varies 
 as the velocity, whilst the other varies as the square of the 
 velocity. 
 
 In 1884 Professor Osborne Reynolds said at the Royal 
 Institution: "In rivers, and all pipes of sensible size, ex- 
 perience has shown that the resistance increased as the 
 square of the velocity, whereas in very small pipes, such as 
 represent the smaller veins of animals, Poiseuille has proved 
 the resistance increased as the velocity. 
 
 " Now, since the resistance would be as the square of the 
 velocity with sinuous motion, and as the velocity, if direct, 
 it seemed that this discrepancy could be accounted for if the 
 motion could be shown to become unstable for a sufficiently 
 large velocity." 
 
 Osborne Reynolds also found experimentally that the 
 eddies were formed suddenly at a critical velocity when 
 the law of resistance changed. 
 
 Stokes said : " In Coulomb's experiments it appeared that 
 the resistance was composed of two terms, one involving the 
 first power and the other the square of the velocity." 
 
 In 1808 Dr Young wrote (Phil. Trans!): " I began by ex- 
 amining the velocities of water discharged through pipes of a 
 given diameter, with different degrees of pressure ; and I found 
 that the friction could not be represented by any single power of 
 the velocity, although it frequently approached to the propor- 
 tion of that power, of which the exponent is i-8, but that it 
 appeared to consist of two parts, the one varying simply 
 as the velocity, the other as the square. The proportion of 
 
 C2 
 
36 A B C of Hydrodynamics 
 
 these parts to each other must, however, be considered as 
 different in pipes of different diameters, the first being less 
 perceptible in very large pipes, or in rivers, but becoming 
 greater than the second in very small tubes ; while the second 
 also becomes greater, for each given portion of the internal 
 surface of the pipe, as the diameter is diminished." 
 
 The latter part of this last paragraph is, perhaps, a little 
 obscure, but there can be no doubt of what Dr Young meant, 
 for he gives the formula as 
 
 f= a—v'' + 2C-V, 
 a d 
 
 which may be expressed 
 
 f=-Xav^-\-2cv\ 
 a 
 
 where /= length of pipe, ^= diameter, whilst a and c are 
 constants. 
 
 Lord Kelvin's views are similar to these, for in " Thomson 
 and Tait " we find under the head of friction : 
 
 " 340. According to the approximate knowledge which 
 we have from experiment, these forces are independent of 
 the velocities when due to the friction of solids, but are simply 
 proportional to the velocities when due to fluid viscosity, directly 
 [italics added], or to electric or magnetic disturbances, with 
 corrections depending on varying temperature and on the 
 varying conditions of the system. In consequence of the 
 last-mentioned cause, the resistance of a real liquid (which 
 is always more or less viscous) against a body moving 
 rapidly enough through it to leave a great deal of irregular 
 motion, in the shape of ' eddies,' in its wake, seems, when 
 the motion of the solid has been kept long enough uniform, 
 to be nearly in proportion to the square of the velocity; 
 although, as Stokes has shown, at the lowest speeds the 
 resistance is probably in simple proportion to the velocity, 
 and for all speeds, after long enough time of one speed, may, 
 it is probable, be approximately expressed as the sum of two 
 terms, one simply as the velocity, and the other as the square 
 of the velocity. If a solid is started from rest in an incom- 
 
Viscosity and Fluid Friction 37 
 
 pressible fluid, the initial law of resistance is no doubt simply 
 proportiottal to velocity (however great, if suddenly given), until, 
 by the gradual growth of eddies, the resistance is increased 
 gradually till it comes to fulfil Stokes' law." 
 
 All this evidence — and more might have been added — 
 confirms the view that the resistance due to viscosity varies 
 as the velocity, and that the total resistance experienced by 
 a body moving in a liquid may be expressed by the simple 
 formula 
 
 R = AV + BV2, 
 
 where A and B are constants. 
 
 In the face of all this authority it is certainly " perplexing " 
 to read in the Bluebook on Aeronautics over the signature of 
 F. W. Lanchester: 
 
 " Notes on the Resistance of Planes in Normal and Tangential 
 Presentation, and on the Resistance of Ichthyoid Bodies. 
 
 " The law of the pressure reaction as a function of velocity 
 is approximately the same whatever the form of the immersed 
 body, namely, RooV^" 
 
 This would appear fairly clear, even though it may not 
 agree too closely with facts ; but later on we see : 
 
 ".The V-squared law is also subject to error on account 
 of the viscosity of the fluid. In this case the error becomes 
 greater the less the velocity. For very low velocities, as shown 
 by Stokes and more recently by Allen, it breaks down altogether. 
 For low velocities the index becomes less than 2, and there 
 appear to be two distinct stages or ranges of velocity during 
 which the index approximates to i'5 (Allen) and to i (Stokes) 
 respectively." 
 
 This should be perplexing enough to satisfy anybody. 
 We have a law which is 
 
 1. Approximately, RooV^. 
 
 2. Which is "subject to error," on account of the viscosity. 
 
 3. The error " becomes greater the less the velocity," and 
 
 4. At very low velocities it " breaks down altogether." 
 The law appears unfortunate ; for not only does it not 
 
 hold good for low velocities, but it appears to be equally 
 
38 A B C of Hydrodynamics 
 
 unhappy for high velocities ; for " as soon as any considerable 
 variation in the velocity is contemplated it is necessary (as is 
 well known to naval architects) to make a correction in the 
 coefficients." Certainly, if one allows oneself sufficient 
 latitude in the change of the coefficients, one can straighten 
 out almost any law. In the last resort one can always blame 
 viscosity ! 
 ' What would be thought of a law of gravity, say, which 
 
 was approximately Goot^ but which was "subject to error" 
 
 for some cause or other, which broke down utterly at small 
 distances, and which had to be corrected by different co- 
 efficients for planetary or stellar distances ? 
 
 It has been pointed out that on reaching a " critical speed " 
 a liquid becomes unstable and suddenly breaks into eddies, or, 
 as Osborne Reynolds calls it, " sinuous motion." Why is this ? 
 
 Dr Hele-Shaw says these eddies are produced — in some 
 way which he does not attempt to explain — by viscosity. 
 That eddies can be produced by viscosity is undoubted ; but 
 that they are ordinarily so produced does not appear correct. 
 
 Professor Osborne Reynolds in 1884 (Royal Institution) 
 compared the flow of a stream of water to the movement of 
 a body of troops, and the laws of hydrodynamics to a drill 
 book. He then continued : " Suppose this science [of War] 
 proceeds on the assumption that the discipline of the troops 
 is perfect, and hence takes no account of such moral effects as 
 may be produced by the presence of an enemy. 
 
 " Such a theory would stand in the same relation to the 
 movement of troops as that of hydrodynamics does in the 
 movements of water. For although only the disciplined 
 motion is recognised in military tactics, troops have another 
 manner of motion when anything disturbs their order. All 
 this is precisely how it is with water; it will move in a 
 perfectly direct disciplined manner under some circumstances, 
 while under others it becomes a mass of eddies and cross 
 streams which may well be likened to the motion of a 
 whirling, struggling mob where each individual particle is 
 obstructing the others. 
 
Viscosity and Fluid Friction 39 
 
 " Nor does the analogy end here : the circumstances which 
 determine whether the motion of troops shall be a march or 
 a scramble are closely analogous to those which determine 
 whether the motion of water shall be direct or ' sinuous.' 
 
 " In both cases there is a certain influence necessary for 
 order : with troops it is discipline ; with water it is viscosity 
 or treacliness. 
 
 "The better the discipline of the troops, or more treacly 
 the fluid, the less likely is steady motion to be disturbed 
 under' any circumstances. On the other hand, speed and 
 size are in both cases influences conducive to unsteadiness. 
 The larger the army, and the more rapid the evolutions, the 
 greater the chance of disorder. So with fluids, the larger the 
 channel, and the greater the velocity, the more the chance of 
 eddies. 
 
 " With troops some evolutions are much more difficult to 
 effect with steadiness than others, and some evolutions which 
 would be perfectly safe on parade would be sheer madness 
 in the presence of an enemy. So it is with water." 
 
 This very excellent analogy only lacks one thing, and that 
 is the "enemy" of the water. It cannot be accepted that 
 eddies are produced by " chance " — there must be some definite 
 cause, and that appears to be undoubtedly the differences of 
 pressure in different parts of the liquid. 
 
 Osborne Reynolds uses the expression "steady motion" 
 as such movement of the stream-line which has not broken, 
 or is not breaking into eddies^ and " sinuous motion " when 
 the stream-line is breaking, or has broken into eddies. He 
 shows that viscosity is not only not the cause of eddies, but 
 that it is the great restraining influence in preventing them. 
 He also adds : " The effect of these influences is subject to 
 one perfectly definite law, which is that a particular evolution 
 becomes unstable for a definite value of viscosity divided by 
 the product of the velocity and space. This law explains 
 a vast number of phenomena which have hitherto appeared 
 paradoxical. One general conclusion is, that with sufficiently 
 slow motion all manners of motion are stable." 
 
 To prevent any misunderstanding as to what he means 
 
40 A B C of Hydrodynamics 
 
 by a stream, he says : " Solid walls are not necessary to form 
 a stream ; the jet from a fire-hose, the falls of Niagara, are 
 streams bounded by a free surface. A river is a stream half 
 bounded by a solid surface." He further gives the "circum- 
 stances conducive to direct or steady motion " as — 
 
 " I. Viscosity, or fluid friction which continually destroys 
 disturbances (treacle is steadier than water). 
 
 " 2. A free surface. 
 
 " 3. Converging solid boundaries. 
 
 " 4. Curvature with the velocity greatest on the outside." 
 
 " Those conducive to sinuous or unsteady motion are — 
 
 " 5. Particular variation of velocity across the stream, as 
 when a stream flows through still water. 
 
 "6. Solid boundary walls. 
 
 " 7. Diverging solid boundaries. 
 
 " 8. Curvature, with the velocity greatest on the inside." 
 
 From all the foregoing we see that the less the viscosity the 
 more difficult it is to prevent a liquid from breaking into eddies. 
 
 Professor Perry (^Applied Mechanics') also says : " A very 
 frictionless fluid is very unstable." 
 
 Lord Kelvin's views are, as we should expect, clear and 
 definite {Phil. Mag., 1887): "It seems probable, almost 
 certain indeed, that analysis similar to that of §§ 38 and 39 
 will demonstrate that the steady motion is stable for any 
 viscosity, however small ; and that the practical unsteadiness 
 pointed out by Stokes forty-four years ago, and so admirably 
 investigated experimentally five or six years ago by Osborne 
 Reynolds, is to be explained by limits of stability becoming 
 narrower and narrower the smaller is the viscosity " [italics 
 added]. 
 
 The natural deduction from all this would appear to be 
 that if the viscosity were eliminated, or became ^ero — as in 
 the imaginary perfect liquid — that the fluid would be perfectly 
 unstable, and that it would be impossible to move it in any 
 manner " conducive to sinuous or unsteady motion," without 
 it breaking into eddies ; and yet some authors would appear 
 to wish the student to believe that though " a very frictionless 
 liquid is very unstable^' an absolutely frictionless liquid is 
 
Viscosity and Fluid Friction 41 
 
 perfectly stable. Great faith is required to believe this : more 
 especially as not a scrap of evidence is produced in support 
 of it. 
 
 Helmholtz is very clear on this point {Phil. Mag., xliii.) : 
 " Such great differences between what actually takes place 
 and the deductions from the theoretical analysis hitherto 
 accepted must cause physicists to regard the hydrodynamical 
 equations as a practically very imperfect approximation to 
 the truth. The cause of this discrepancy might be supposed to 
 lie in the internal friction of the liquid, although the divers 
 strange and saltatory irregularities which everyone has 
 encountered who has experimented upon the motions of 
 fluids can in no wise be accounted for by the continuous and 
 uniform action of the friction!' 
 
 " The investigation of the case vfh.&re. periodical motions result 
 from a continuous air current, as, for instance, in organ pipes, 
 convinced me that such an action could only arise by a dis- 
 continuous m.otion of the air, or at least by an approximately 
 discontinuous one. Hence I was led to the discovery of a con- 
 dition ..." involving what he calls " surfaces of separation." 
 
 " The moment the pressure passes zero and commences to 
 become negative, a discontinuous change in the density takes 
 place : the fluid is broken asunder." ^ 
 
 These remarks should be pondered over deeply, for when 
 the reader has thoroughly grasped Helmholtz's meaning, he 
 will be well on the road towards understanding the leading 
 features of liquid resistance. 
 
 Summary 
 
 Fluid friction is caused by a stress set up in a liquid to 
 resist shearing. 
 
 It increases as the velocity only. 
 It is not affected by pressure.^ 
 
 1 Avanzini appears to have known this more than a hundred years ago. 
 
 2 This is only approximately correct. "It has been shown by Warburg 
 and Sachs, and also by Rontgen, that water at ordinary temperatures 
 becomes more mobile when subjected to pressure : in other words, its 
 viscosity is lowered by pressure" (T. E. Thorpe, Roy. Inst., 1898). 
 
42 A B C of Hydrodynamics 
 
 At a " critical speed " the liquid breaks suddenly into eddies. 
 The resistance of a body moving in a viscous liquid can 
 therefore be expressed by 
 
 R = AV + BV2, 
 
 where AV is the resistance due to viscosity, and BV^ that 
 due to eddy-making. 
 
 Viscosity is not in general the cause of eddies ; it actually 
 tends to prevent their formation and to stop them when 
 formed. 
 
 REFERENCES 
 
 F. W. Lanchester, Aerodynamics, 1907. 
 
 Sir G. Stokes, collected papers. 
 
 Report of Advisory Committee for Aeronautics, 1909-10. 
 
 Helmholtz, Phil. Mag., xliii. 
 
 Professor Perry, Applied Mechanics. 
 
 Professor Osborne Reynolds (R. Inst., 1877 and 1884). 
 
 Dr Young, Phil Trans., 1808. 
 
CHAPTER VI 
 
 A VISCOUS FLUID FLOWING " BY OBLIGATION "— 
 DISCONTINUITY 
 
 In Chapter I. it was stated that the flow of a viscous liquid 
 should not be very different from that of a " perfect " one 
 under similar conditions. If this is true, it follows naturally 
 that if a viscous liquid be caused to flow " by obligation," it 
 should behave almost exactly as the mathematicians say their 
 liquid would — the motion being subject to some small varia- 
 tion caused by the viscosity. 
 
 Dr Hele-Shaw has, by a very beautiful piece of apparatus, 
 actually caused a viscous liquid to flow " by obligation," and 
 has thus thoroughly confirmed experimentally what the 
 mathematicians had predicted theoretically. His description 
 of the experiment is as follows : — 
 
 " If we take two sheets of glass and bring them nearly 
 close together, leaving only a space the thickness of a thin 
 card or piece of paper, and then by suitable means cause the 
 liquid to flow under pressure [italics added] between them, the 
 very property of viscosity, which, as before noted, is the cause 
 of the eddying motion in large bodies of water, in the present 
 case greatly limits the freedom of motion of the fluid between 
 the two sheets of glass, and thus prevents not only the eddy- 
 ing motion, but also counteracts the effect of inertia." 
 
 In examining this statement, and before proceeding further, 
 it may be pointed out that (i) this is a case of the closed tank 
 referred to in Chapter II. ; the liquid is under pressure, and 
 the flow (within limits) will be an "obligation" flow and not 
 a "free" flow. (2) As shown in the last chapter, it cannot be 
 
 43 
 
44 
 
 A B C of Hydrodynamics 
 
 accepted as a fact that viscosity is the cause (or chief cause, 
 for it may in special circumstances be the cause) of eddying 
 motion in " large bodies " of water : it acts in exactly the 
 opposite manner, and prevents, or tends to prevent, eddies. 
 Dr Hele-Shaw, of course, actually employs it for this very 
 purpose. As previously remarked, since water is not sufficiently 
 viscous, he makes use of glycerine. It is hardly reasonable to 
 expect anyone to believe (without proof) that viscosity causes 
 eddies in large bodies of liquid, whilst it at the same time 
 
 prevents their formation in small 
 bodies of liquid. (3) The vis- 
 cosity counteracts the effect of 
 inertia by preventing eddies, 
 and so (practically) eliminating 
 the mass of the liquid. 
 
 " If we now, by suitable means, 
 allow distinguishing bands of 
 coloured liquid to take part in 
 the general flow, we are able to 
 imitate- exactly the conditions 
 in the diagrams " [of flow of the 
 mathematicians' fluid]. 
 
 The imitation is most perfect, 
 whether the flow be past an 
 obstacle, through a narrow gap, 
 '■ source and sink " flow, etc., 
 etc. To give one example only, 
 the flow past the obstacle is as shown in the sketch (fig. 7), 
 and not in the least in the usual manner of a viscous liquid, 
 like water. The velocity is, of course, kept very low ; the 
 pressure, also, is sufficient to cause the liquid to "turn the 
 corners" properly. 
 
 If, however, the flow be allowed a little freedom, a very 
 different result is produced. This Dr Hele-Shaw effects by 
 setting the plates of glass a little further apart, when, as he 
 says, " the water refuses to flow round to the back, and spreads 
 on either side," as in the sketch (fig. 8). 
 
 This flow, which is no longer " by obligation," is what 
 
 Fig. 7. 
 
A Viscous Fluid Flowing "by Obligation" 45 
 
 Mr R. E. Froude calls a "potential cleavage" flow ; it might 
 also be called the Helmholtz-Kirchhoff " flow, as they were the 
 first to study it. It is also referred to by Lord Rayleigh in 
 the paper previously quoted from. 
 
 In this flow it will be observed that there has been a 
 permanent cleavage of the liquid reaching A B, and that the 
 liquid then flows right and left past the stagnant pool ABCD. 
 Since the liquid is flowing along A C and B D at a higher 
 velocity than it had before reaching A B, and since the liquid 
 in the pond is at rest, if what has been stated previously as 
 regards the law of flow be 
 correct, the pressure in the 
 pond must be greater than 
 in the streams along A C 
 and B D. This, of course, 
 follows from the law of 
 flow referred to in Chapter 
 III., and which was ex- 
 pressed in " algebraical 
 shorthand " as 
 
 or 
 
 The flow in the case under 
 consideration is no longer 
 an '' obligation " flow, and 
 the mass of the liquid has not been " practically eliminated " 
 — hence p, the density, is important. The pressure in 
 the pond is here expressed as P^, which is the pressure 
 in the imaginary stream-line when v = o. In the streams 
 along A C and B D the pressure is Pq — i Vp^^, the value 
 of which can be arrived at when we know P^ and v. 
 The two pressures on the opposite sides of the " surfaces 
 of discontinuity " are therefore as Po : Pq — ^ V/ow^ i.e. the 
 pressure on the pond side is greater than that in the 
 moving liquid. 
 
 If what was said in Chapter III. is true, that a liquid must 
 flow from any region of higher pressure to any region of lower 
 
 Fig. 8. 
 
46 A B C of Hydrodynamics 
 
 pressure, why is there no ^ov^ from the pond into the flowing 
 stream ? What is preventing this flow ? 
 
 The reason is that the liquid has been "torn asunder," and 
 that the tear extends along A C and B D. Along A C and 
 B D there are '' surfaces of discontinuity," or free surfaces. As 
 the free surface is capable of sustaining tension, although it 
 is " stressed " the film does not break, and so allow the liquid 
 to flow. The case is similar to that of a soap-bubble, where 
 the film is retaining the air inside, which is under greater 
 pressure than the atmospheric. As long as the "excess of 
 pressure " on the pond side of A C and B D is not too great, 
 the flow continues ; but if this excess be increased, which 
 A B can easily be done by increasing the 
 
 jl^^^^^^^j^ velocity and so reducing the pressure 
 
 in the flowing stream, the breaking 
 point is soon reached. 
 
 If the velocity of flow be slightly 
 
 increased the result is rather curious : 
 
 the lines AC and BD gradually 
 
 approach one another, as in the 
 
 sketch. The pressure is still not 
 
 sufficient to break the film : a little 
 
 FiQ g more velocity, the film breaks and 
 
 the liquid flows from the pond into 
 
 the stream, as shown in the sketch (fig. 9). This is exactly 
 
 what we should have expected for the reason previously 
 
 given. 
 
 It may be asked, why should the stream-lines A C and B D 
 gradually approach one another? I am not aware that any 
 explanation has ever been given ; so I will venture the 
 following, which will serve as a working hypothesis until a 
 better one is offered. The stream-lines along A C and B D 
 act like the well-known Sprengel pump — in the manner 
 pointed out by Venturi many years ago. We thus have — 
 
 1. The liquid in the pond evaporates through the film into 
 the void space. 
 
 2. The streams AC and BD absorb this vapour and so 
 increase the evaporation. 
 
A Viscous Fluid Flowing " by Obligation " 47 
 
 3. This action continuing, the pond gets smaller, and, 
 erg-o, A C and B D approach one another, and eventually 
 meet. 
 
 All Dr Hele-Shaw's beautiful experiments show that a 
 viscous liquid, when made to flow "by obligation," will move 
 almosi ex3iCt\y as the mathematicians' fluid would under similar 
 conditions. 
 
 I have dwelt very long over this experiment, and the reason 
 is that I think we are here assisting at the birth of a vortex. 
 This is, I believe, the first stage in the formation of vortices : 
 the diagram given by Lord Kelvin in his paper on " Coreless 
 Vortices " I consider to be the second stage. As the resist- 
 ance of bodies moving in, liquids is chiefly caused by the 
 formation of vortices, it is well to think carefully about this 
 experiment. 
 
 This " potential cleavage " flow was imagined, as previously 
 stated, by Helmholtz and Kirchhofif as a possible flow of a 
 perfect liquid past a circular lamina. 
 
 There are certain grave objections to this form of flow. 
 Some of these have already been referred to ; but there is 
 another pointed out by Lord Kelvin. In his paper on " Core- 
 less Vortices," he says, about the fluid, when it passes the 
 sphere : " It might be thought that the result of this collision 
 is a 'vortex-sheet' which, in virtue of its instability, gets 
 drawn out and mixed up indefinitely, and is carried away by 
 the fluid further and further from the globe. A definite 
 amount of kinetic energy would thus be practically annulled 
 in a manner which I hope to explain in an early communica- 
 tion to the Royal Society of Edinburgh. 
 
 " But it is impossible, either in our ideal inviscid incom- 
 pressible fluid, or in a real fluid, such as water or air, to form 
 a vortex-sheet, that is to say, an interface of finite slip, by 
 any natural action." 
 
 Another objection is that as the pressure on the face 
 of the lamina (the side in presentation) must be less than 
 the hydrostatic pressure, there is only one small spot 
 in the centre where the pressure attains the H.P. ; at all 
 other parts it is less than this ; and as the pressure on the 
 
48 A B C of Hydrodynamics 
 
 rear of the lamina must be the hydrostatic pressure, called in 
 the formula of flow P,,, it follows that the lamina would 
 propel itself and we should have a fine example of perpetual 
 motion. 
 
 A third objection is that the action could not be reversed. 
 This argument will appeal to engineers ; for one can hardly 
 imagine a lamina moving through a liquid and dragging an 
 indefinite, (not to say infinite) body of water behind it like a 
 tail. Just imagine what force it would require to start this 
 tail moving, without entering into the mechanical difficulties 
 of keeping the particles of the tail together. 
 
 Possibly the severest condemnation of this flow was that 
 made by Lord Kelvin in Nature in 1894: "The assumption 
 to which I object as being inconsistent with hydrodynamics, 
 and very far from any approximation to the truth for an 
 inviscid incompressible fluid in any circumstances, and utterly 
 at variance with observation of discs or blades (as oar 
 blades) caused to move through water, is that, starting 
 from the edge, as represented by the two continuous curves 
 in the diagram, and extending indefinitely rearwards, there 
 is a ' surface of discontinuity,' on the outside of which 
 the water flows relatively to the disc, with velocity V, 
 and on the inside of which there is a rearless mass of 
 ' dead water.' 
 
 " The supposed constancy of the velocity outside of the 
 supposed surface of discontinuity entails for the inside a 
 constant pressure, and therefore quiescence relatively to the 
 disc, and rearlessness of the ' dead water.' How could such 
 a state of motion be produced? and what is it in respect 
 of its rear? are questions which I may suggest to the 
 teachers of the doctrine, but which, happily, not going in 
 for an examination in hydrokinetics, I need not try to 
 answer." 
 
 When all has been said against it, one must admit that Dr 
 Hele-Shaw has actually shown, experimentally, that such a 
 flow is possible in a liquid of practically two dimensions and at 
 very low velocities. It is, at least, imaginable in a non-viscous 
 liquid of three dimensions, with the lainina at rest and the 
 
A Viscous Fluid Flowing ''by Obligation" 49 
 
 liquid flowing at an exceedingly low velocity, provided 
 that— 
 
 1. The liquid has a free surface, or 
 
 2. The boundaries of the liquid are not fixed, but extensible, 
 or 
 
 3. The liquid is compressible — any one of these conditions 
 is sufficient — and 
 
 4. Provided that the surface of the inviscid liquid is capable 
 of sustaining surface tension. 
 
 I do not know the original paper by Helmholtz and 
 Kirchhoff, but I rather fancy that the liquid is assumed as 
 incompressible and of infinite dimensions, i.e. that it has not 
 a free surface and that its boundary walls prefixed. 
 
 If this Helmholtz-Kirchhoff theory of flow has been 
 discussed at such length, it is only because, in consequence 
 of the eminence of the authors, it is so commonly referred to 
 by writers ; it has thus acquired a notoriety which I can only 
 think is far beyond its merits. 
 
 It is generally admitted that it is not a " no-resistance " flow : 
 it therefore only tends to support an argument that flat plates 
 obey different laws to round or " fish-like " bodies, which we 
 cannot admit. This theory may appeal to mathematicians, 
 but it is hardly likely to do so to engineers. 
 
 Discontinuity 
 
 In the course of this chapter the word " discontinuity " has 
 been used, and it has been employed in the sense of the 
 particles of the liquid having been " torn asunder.'' It has 
 been objected to me, and, doubtless, may be again, that this 
 " discontinuity " is not physical discontinuity, but kinetic : in 
 other words, that there is no separation between the layers of 
 liquid, but that they are moving at different speeds — in the 
 case here referred to one is actually at rest. This appears to 
 be rather a case for the objectors to show that such kinetic 
 discontinuity is possible without the production of a free 
 surface. It is about as easy to imagine that in a bar of iron 
 one layer of particles were red-hot and the next layer at the 
 
 D 
 
50 A B C of Hydrodynamics 
 
 temperature of liquid hydrogen. Professor Lamb always 
 speaks of a " surface of discontinuity, or, what is the same 
 thing, a free surface." Sir George Stokes spoke of discontin- 
 uity or " rift." Helmholtz refers to " surfaces of separation." 
 There is no dou'bt about what Lord Kelvin meant. In 
 using the term ''discontinuity," I always mean to imply 
 that the fluid has been torn asunder so as to produce 
 actual physical discontinuity. This physical " cracking " of a 
 fluid is very common in everyday life, and explains many 
 things. The spark of a Riihmkorff coil " cracks " the air, 
 just as it will crack a sheet of glass. Lightning similarly 
 cracks the air, and the resulting noise is thunder. 
 
 When a gun or a rifle is fired the hot gases issuing very 
 rapidly "crack" the air: even when you crack a whip you do 
 the same thing. The loudness of the report depends, very 
 largely, on the suddenness with which the crack is made. 
 To prevent any possibility of being misunderstood, it may be 
 necessary to point out that it is not the physical crack of the 
 air which directly causes the sound : this is produced by the 
 rush of air which takes place afterwards. 
 
 There are many cases where such " cracking " of the air 
 would hardly be suspected — e.g. organ pipes, whistles, etc. ; 
 but in these cases it may be that physical separation is 
 never complete, but that there is only high rarefaction. 
 
 " It is probable that much of the foam seen near the sides 
 and in the wake of a steamer going at a high speed through 
 glassy-calm water is due to ' vacuum ' behind edges and 
 roughnesses causing dissolved air to be extracted from the 
 water" (Lord Kelvin, Nature, 1894). 
 
 Summary 
 
 A viscous liquid, when made to flow " by obligation," does 
 so in a similar manner to that laid down by the mathe- 
 maticians for their " perfect " liquid. 
 
 The form of flow imagined by Helmholtz and Kirchhoff, 
 and called by Mr R. E. Froude the " potential cleavage " 
 flow, is not a possible one, except under such very narrow 
 
A Viscous Fluid Flowing ''by Obligation" 51 
 
 conditions as are not usually found in Nature, because it is 
 essentially unstable. The liquid flowing past a lamina does, 
 undoubtedly, commence flowing in this manner ; but the flow 
 immediately breaks down and eddies are formed, This would 
 be equally true in a " perfect " liquid, or a viscous one, like 
 water. 
 
 " Discontinuity " is a physical separation of the liquid, which 
 then has a free surface which is capable of sustaining tension. 
 
 REFERENCES 
 
 Dr Hele-Shaw, "On the Motion of a Perfect Liquid" (R. Inst., 
 1899). 
 
 Lord Kelvin, "Coreless Vortices," etc., Phil. Mag., 1887. 
 ,, ,, Nature, 1894. 
 
 D 2 
 
CHAPTER VII 
 
 VISCOUS LIQUID NOT FLOWING "BY OBLIGATION," BUT 
 FOLLOWING STREAM -LINES GIVEN IN HYDRO- 
 DYNAMICAL TEXT-BOOKS 
 
 In the last chapter it was pointed out that a viscous liquid, 
 
 if made to flow "by obligation," would move in stream- 
 lines similar to those of the 
 mathematicians'perfect liquid. 
 There are cases where it does 
 so, even when it has a free 
 surface, as there are cases 
 where it does not. 
 
 When water is flowing to 
 meet a body immersed in it, 
 the flow before arriving at 
 the body is a perfectly " con- 
 tinuous flow," and so should, 
 of course, follow the orthodox 
 hydrodynamical stream-lines. 
 It is notorious that it does so ; 
 and these stream-lines have 
 been photographed by M. 
 Marey and others, who have 
 employed various devices for 
 
 making the stream-lines visible. It should also follow the 
 
 same laws as they do. 
 
 It is, I believe, generally accepted that if a thin lamina — or 
 
 even any other body — moves in a perfect liquid, the pressure 
 
 52 
 
Viscous Liquid not Flowing "by Obligation" 53 
 
 on the anterior face will be less than it would be were the 
 lamina at rest. 
 
 In reference to a circular lamina exposed to an infinite 
 perfect fluid moving past it (fig. 10), Professor Lamb says: 
 " The velocity at a distance r from the centre of the disc 
 (rad=t) will be 
 
 1 ?^ u 
 
 when U is the general velocity of the stream. 
 " Consequently the pressure 
 
 {absolute units) when p^ is what the pressure would be if the 
 fluid were at rest." The pressure curve would be something 
 like that in the sketch (fig. 10). 
 
 It will be seen at once that p=p^ when r=o, or at the 
 centre of the lamina : on all other parts of the disc the 
 pressure is less than p^, which is 
 the hydrostatic pressure. It is 
 also well to remember that the 
 pressure at the centre of a 
 lamina — or other body — is al- 
 ways the hydrostatic pressure, 
 for the liquid at the " dividing 
 
 point" is at rest, referred to 
 , , , Fig. II. 
 
 the body. 
 
 Is this the same for water ? From what has been previously 
 said, since all the water in front of the lamina is flowing, it 
 must be under lower pressure than if it were at rest : it can 
 be shown, experimentally, that such is the case. 
 
 Some years ago a company had on sale a nozzle for fire 
 purposes call,ed a " ball-nozzle." This nozzle was cone-shaped 
 (as in the sketch, fig. 11), and inside this cone was a hollow 
 ebonite ball, which was quite loose. When the water was 
 turned on to the hose from the main, the ball remained firmly 
 fixed in the mouth of the cone, and remained fixed as long as 
 the water was flowing. This is a practical proof that the 
 
54 A B C of Hydrodynamics 
 
 pressure of the water against the ball was less than the 
 atmospheric (!), though the " head " on the main may have 
 been sufficient to produce a pressure of 30 or 40 lbs. per 
 square inch. 
 
 What is true for water is equally true for air. If a small 
 cone be made of tin (about the size of the sketch, fig. 1 2), and 
 if a tube of about the size of a small quill be fitted into the 
 apex of the cone, on blowing hard through this, a ball of light 
 wood can be kept suspended, against gravity, in the cone. 
 If, in fact, the ball be placed in the palm of the hand and the 
 cone (with a stream of air passing through it) be made to 
 approach the ball, the latter can be caused to jump off the 
 
 Fig. 12. Fig. 13. 
 
 hand into the cone. This experiment can be made by anyone 
 with a minimum of trouble. 
 
 To prevent any possible misunderstanding, when it is stated 
 that the ball is " fixed " in the mouth of the cone, it is not 
 pretended that the ball ever touches the cone: it does not. 
 The pressure of the atmosphere forces it in ; but it will be 
 evident that, if the ball gets too near the cone, the supply of 
 water will be reduced : the flow of the water between the 
 cone and the ball decreasing, the pressure behind will increase, 
 and the ball will move outwards again. There is probably a 
 very small oscillation of the ball in the mouth of the cone. 
 
 It is interesting to note that the same result is not arrived 
 at if the ball is pointed (towards the cone), i.e. if it is a cone 
 with a hemispherical base. In such a case the form is 
 highly unstable and there is what Mr Bairstow (Bluebook, 
 Aeronautics, 1910-11) calls a "negative righting moment," 
 which forces the point against the side of the cone and so 
 spoils the effect. This is an argument against the supposed 
 advantage of a body having a fine '' entry." 
 
Viscous Liquid not Flowing ''by Obligation" 55 
 
 It may be argued that these are "round" bodies and that 
 what is true for them may not be true for "flat" bodies- 
 Quite so. Almost exactly the same experiment may be 
 made with a flat body. If a small glass tube be fixed to a 
 brass plate (fig. 14), on blowing through the tube and bring- 
 ing the glass plate near to a sheet of paper, the latter can 
 be lifted up and kept suspended as long as the blowing is 
 continued. 
 
 How is the pressure distributed ? 
 
 More than a century ago the Chevalier Dubuat (Colonel 
 of the French Royal Engineers) published his Principes 
 
 ■Glass tube 
 
 v/j/iiji/iiiiiii/i/in 
 
 
 Sheet of paper 'Bhass plate 
 
 Fig. 14, 
 
 Fig. is. 
 
 d'Hydraulique, where he says : " Quand I'eau coule le long 
 d'une surface en vertu d'une charge sup6rieure, la hauteur 
 due a la pression, qu'eprouve cette surface est 6gale a celle 
 qui avait lieu, si le fluide etait en repos, mains la hauteur due 
 d la vttesse moyenne reelle du fluide dans le sens parallele a la 
 surface." From this he deduced : " D'apres, ce principe, la 
 pression occasionn6e par le choc contre une surface doit 
 diminuer du centre a la circonference." 
 
 He examined this experimentally by exposing a sub- 
 merged plate (fig. 15) — in reality a very thin box — to a 
 moving stream, and measuring the actual pressures at diff"erent 
 parts of the surface by means of open holes and the well- 
 known Pitot's tubes. This is, of course, far better than 
 taking the whole resistance of the plate, which is really only 
 
56 A B C of Hydrodynamics 
 
 the difference between the pressures on the front and on the 
 back, and which teaches nothing about the distribution of 
 the pressures on the anterior surface. 
 
 He was prepared, as we have seen, to find a greater 
 pressure at the centre of the plate than at the edges; but 
 when he made the experiment he was surprised to find that 
 a negative pressure was recorded at the edges. He says : 
 " On remarque d'abord dans ces experiences, un efTet bien 
 singulier, et que nous ne pouvions pas prevoir: c'est que la 
 pression, qui diminue du centre de la surface vers les bords, 
 devient nulle a une certaine distance, et ensuite negative 
 au bord mSme." 
 
 His figures are — 
 
 (The measurements are old French ones, but are useful for 
 comparison one with another.) 
 
 No. I. 
 
 Centre hole of the box 
 open, the other four 
 closed : the height 
 corresponding to 
 the pressure at this 
 
 
 
 
 hole was 
 
 -f32-8 lignes. 
 
 
 No. 2. 
 
 (Half-way centre to 
 
 
 
 
 edge) . 
 
 -f-27"8 lignes. 
 
 
 No. 3. 
 
 (Three-quarters centre 
 
 
 
 
 to edge) . . = 
 
 + 20-8 
 
 
 No. 4. 
 
 (Close to edge) . . = 
 
 - s-s „ (?)] 
 
 -? reversed, 
 
 No. 5. 
 
 (Close to corner) . = 
 
 - 8-6 „ (?)j 
 
 
 With alt the holes open =4-17 lignes. 
 
 Of these figures No. 4 and No. 5 appear open to question ; 
 No. 5 should be less — measured negatively — than No. 4. 
 This progression is not confirmed by his other experiments, 
 and it would appear that -there must be a clerical error 
 somewhere. Comparison with his next experiment would 
 lead one to think that the figures are probably reversed — 4 for 
 5 and 5 for 4. The true average pressure over the whole 
 plate may, or may not, be = -h 17 lignes. Too much import- 
 ance should not be attached to this exact figure: it is 
 
Viscous Liquid not Flowing ''by Obligation" 57 
 
 perhaps curious that it should be so near Newton's estimate 
 of what it should be when a circular plate moved through 
 water at "the speed corresponding to the depth" {v= J2gh), 
 viz., about half the hydrostatic pressure. Dubuat's experi- 
 ment certainly may be taken as showing that the total 
 pressure on the plate was considerably less than the 
 hydrostatic. 
 
 The next point Dubuat examined was, if the pressure on 
 the anterior surface was affected by the shape of the posterior 
 part of the body. There appears x\o prima facie reason why 
 it should be, except that the velocity of the stream at the 
 edges would probably be slightly reduced when the body is 
 lengthened. Dubuat fixed a cube at the back of his thin box 
 and found no essential difference. Referring to the previous 
 •diagram, the pressures recorded were — 
 
 No. I. -I-32-8 lignes. No. 3. = + 17-8 lignes. 
 
 No. 2. +30-2 „ No. 4. = —1 2- 1 „ 
 
 All the holes open = + 14"3 lignes. 
 
 Again, with a parallelepiped of proportions 3x1x1: 
 
 All the holes open = + i6'o lignes. 
 
 The conclusion he arrived at was : " En tenant compte 
 des erreurs inevitables dans ces sortes d'experiences, on 
 peut affirmer que la longueur des corps n'influe point sur 
 la pression ant^rieure." 
 
 It may be argued that in all these cases the bodies are 
 at rest and the liquid is moving. Would the same thing 
 occur if the plate were moving and the water were at 
 rest? It is difficult to imagine why it should not be 
 so, and Professor Osborne Reynolds has shown a very 
 beautiful little experiment which will prove that the 
 phenomena depend only on the relative motion of body 
 and fluid. 
 
 To a small wooden float attach a little circular disc by 
 means of fine wires (as in the sketch, fig. 16). If this be put 
 in water and the float be given a sharp push — releasing it 
 instantly — the combination will be found to travel across 
 the tank as if it were meeting with no resistance. This may 
 
58 A B C of Hydrodynamics 
 
 be said to prove nothing ; but if the float be stopped and 
 released at once, the combination will start again, and go 
 on merrily. Why does it start itself? Obviously because 
 the pressure on the rear face is in excess of that on the 
 front face. As the pressure on the rear face cannot, con- 
 ceivably, be greater than the hydrostatic pressure, it is 
 clear that the pressure on the front face must be less than 
 this, more especially as the float undoubtedly causes some 
 resistance. 
 
 There is evidently something curious happening at the 
 rear side of the disc, but what that is will be considered 
 later. 
 
 Notwithstanding all the evidence to the contrary, it is 
 
 ^my/my//////////y^/yy/////////z^^ ^ 
 
 I Disc of Tin 
 Fig. 16. 
 
 commonly taught that the pressure is augmented on the 
 front face when a body is moving in a liquid. For 
 example : " The resistance experienced by bodies of 
 imperfect \sic\ form is due to the work done on the fluid, 
 which is not subsequently given back, as in the case of 
 the stream-line body. This resistance can be traced to 
 two causes, namely, excess pressure on the surface in pre- 
 sentation and diminished pressure in the dead-water region " 
 (Lanchester, Aerodynamics, 1907). 
 
 How can the small lamina and float experiment be ex- 
 plained by this? I suppose the lamina is an "imperfect" 
 shape, since it is very far removed from the ichthyoid. 
 The pressure is not greater in front than behind, but the 
 reverse. 
 
 Also : " Now, as the body advances, the head being 
 
Viscous Liquid not Flowing " by Obligation " 59 
 
 subject to pressure in excess of that due to the hydrostatic 
 head, etc." (same author and book). Examples might be 
 multiplied, but these should suffice, as the work referred to 
 is of late date and by a recognised author. 
 
 Since the motion of water on the anterior side of a body- 
 is always " steady motion," and since " it seems probable, 
 almost certain indeed, . . . that the steady motion is stable 
 for any viscosity, however small " (Kelvin), it may be taken 
 that it will behave in an exactly similar manner to a perfect 
 fluid, on the anterior side of a body. 
 
 The deductions that we may fairly draw from the foregoing 
 are — 
 
 1. That the shape of the head of a body is not of very 
 great importance in increasing or reducing resistance — -pro- 
 vided always that there are no sharp angles and that the head 
 is symmetrical 
 
 Colonel Beaufoy, long ago, remarked the same thing, for he 
 said : " A convincing proof of the advantage curved lines 
 have over rectilineal ones in dividing the fluid; and this 
 circumstance strongly corroborates the assertion made by 
 many professional men that a full bow has the preference 
 to a lean one in point of sailing ; but it is to be understood 
 that the bow should be made nearly circular to gain this 
 superiority." 
 
 2. That whenever an immersed body and the water are 
 in relative motion, what may be called a "liquid prow" is 
 formed : to interfere with this formation cannot be right. 
 A sharp bow, or point, on the front of a body must 
 frequently act as a "sharp edge," and so, if not cause, at 
 least increase the " negative righting moment " previously 
 referred to. 
 
 It has been found, practically, that a torpedo with a blunt 
 nose travels faster than one with a sharp point, and it also 
 steers better. 
 
 What may we expect would be the best head for a torpedo, 
 say? Obviously the one which generates the minimum 
 amount of kinetic energy. This should be calculable ; and 
 the shape of the head would appear to be some form of 
 
6o A B C of Hydrodynamics 
 
 spheroid.! Mr Bairstow (Bluebook, Aeronautics, 1910-11) 
 says that " a comparatively short head of about one and a 
 half diameter was satisfactory" 
 
 M. Albert Bazin has lately suggested a method (which 
 may or may not be original) of finding the shape of least re- 
 sistance practically. He 
 says : " Prenez un pain 
 de savon paralldlipi- 
 pedique: fixez-le au bout 
 F'G- I?- d'une longue ficelle A B 
 
 comme ci-contre (fig. 17): puis remorquez-le (autant que 
 possible hors du sillage) dans I'eau, derriere un bateau rapide ; 
 au bout de quelques minutes retirez-le ; il aura pris naturelle- 
 ment une forme de fuseau analogue a celles que vous avez 
 dessinees et il le conservera tant qu'il restera du savon autour 
 de la ficelle" (Ernoult, L' Aviation de Demain, 1910). 
 
 Summary 
 
 Water flowing to meet a body, even when the liquid has a 
 free surface, behaves like the mathematicians' perfect fluid — 
 subject to some slight variations caused by viscosity. The 
 difference might, in this case, be popularly described as the 
 
 ' Mr Lewis R. Shorter has very kindly calculated for me the equatorial 
 velocity of a stream flowing past an ellipsoid of revolution, the direction 
 of flow at infinite distance being parallel to the axis of revolution and its 
 magnitude V, from the very formidable-looking formula of Lord Kelvin. 
 
 EY- ('^'- 
 
 b'-)V 
 
 ( 
 
 
 
 a 
 
 
 X^d'-b-' 
 
 
 
 Ratio of Axes. 
 
 
 Equatorial Velocity 
 
 I. Prolate spheroids 
 
 00 : I 
 
 
 roc V. 
 
 
 3: I 
 
 
 I-I2 V. 
 
 
 2 : I 
 
 
 1-21 V. 
 
 2. Sphere 
 
 I : I 
 
 
 1-50 V. 
 
 3. Oblate spheroids 
 
 I : 2 
 
 
 2-12 V. 
 
 
 I =3 
 
 
 274 V. 
 
 
 I : 100 
 
 
 64-47 V. 
 
 
 I : 00 
 
 
 00. 
 
Viscous Liquid not Flowing "by Obligation" 6i 
 
 difference between the reflection from a flat mirror and from' 
 one which is vety slightly curved cylindrically. 
 
 The pressure on the anterior side of a body is less than the 
 hydrostatic pressure, whether the body moves past the liquid,, 
 or the reverse. 
 
 The pressure on the centre of a circular plate broadside on 
 to a flowing stream is the hydrostatic pressure : this pressure 
 decreases from the centre to the edge of the plate. 
 
 Any sharp edge in the head of an immersed body appears 
 to produce increased resistance and to cause, or increase, a 
 " negative righting moment " : i.e. tends to make the motion 
 unstable. 
 
 REFERENCES 
 As before. 
 
CHAPTER VIII 
 
 RESISTANCE DUE TO VISCOSITY VARIES, NOT AS THE 
 "WETTED SURFACE," BUT AS THE "SHEARING SUR- 
 FACE" AND AS THE VELOCITY ONLY 
 
 It will be well now to consider how far this theory agrees 
 with experiment. 
 
 In 1872 Dr Froude published the results of some very 
 fine experiments he had conducted on the resistance of long 
 flat boards towed edgewise. These are still the classical 
 experiments which are always referred to when mention is 
 made of liquid friction. He gives the curves of resistance 
 at varying speeds. 
 
 Selecting curve B on Plate IV. (I select this as a specially 
 good curve, since all the observed spots are accurately 
 on the curve), the following resistances can be measured 
 off at the speeds of 100, 200, 300, 400, 500, 600, 700, 
 and 800. 
 
 (Board, 28 ft. long, varnished surface.) 
 
 At 100 ft. p.m. resistance = -9 lbs. 
 
 200 , 
 
 » )J 
 
 = 3'4 ., 
 
 300 , 
 
 ) )) 
 
 = 7-1 ■. 
 
 400 , 
 
 1 )> 
 
 = "•7 „ 
 
 500 , 
 
 J >J 
 
 = 17-5 >. 
 
 600 , 
 
 ) >) 
 
 = 24-6 „ 
 
 700 , 
 
 > 5) 
 
 = 327 .> 
 
 800 , 
 
 ) 9) 
 
 = 42-4 » 
 
 If we examine these in the ordinary way — assuming that 
 
 62 
 
Resistance due to Viscosity Varies 
 
 63 
 
 they vary as some power of the velocity — it will be seen 
 that they vary — 
 
 Between 100 and 200 as V^ (nearly). 
 
 „ 200 
 
 , 300 „ 
 
 Yl-82_ 
 
 >. 300 
 
 , 400 „ 
 
 Yl-73_ 
 
 400 „ soo „ 
 
 yiw 
 
 500 , 
 
 , 600 „ 
 
 Yl-80_ 
 
 600 
 
 , 700 „ 
 
 Yl-84_ 
 
 700 
 
 , 800 „ 
 
 yi-gi 
 
 No special law is apparent, though it is customary to say 
 the variation is as V^'^ (some American books put it more 
 neatly as VV), the differences being put down to " error." 
 Not only is this unscientific, but it completely masks the 
 dimension law. 
 
 If we examine them by the formula AV + BV^ = R, putting 
 V as 200, 400, 600, and 800, it will be found that A and B 
 are constants, as they should be, and that they have approxi- 
 mate values — 
 
 A = -oo 5 , B = -00006. 
 
 Applying these constants to the formula for the velocities 
 300, 500, and 700, we get 
 
 At 300 R= 7*x calculated as against 6'9 from curve. 
 „ 500 R= 17-5 „ „ 170 
 
 „ 700 R = 327 „ „ 32-9 
 
 An exceedingly close agreement for such very small curves, 
 which were probably drawn by means of a bent lath. 
 
 To show that this is not an accidental coincidence, let us 
 take the next curve, C C, also Plate IV. (16 ft. long only, 
 but also with varnished surface) — 
 
 At 200 ft. R= 1-95 lbs. 
 
 300 „ R= 4'iS >. 
 
 400 „ R= 70 „ 
 
 500 „ R=io'6 „ 
 
 600 „ R= iS'o 
 
 700 „ R = 20-o 
 
 800 „ R = 2S-4 „ 
 
64 A B C of Hydrodynamics 
 
 Putting the different values of V and R in the equation 
 R = AV + BV^ as before, it will be found that 
 
 A = -00325, B = -000037, 
 
 will satisfy the equation within limits of error. 
 
 To compare the constants in these two curves it must be 
 remembered that B B is the curve of resistance of a board 
 28 ft. long, whilst C C is that of a board of 16 ft. only. 
 Reduced to the same length, it will be found that A 
 is practically the same for both experiments, whilst B 
 is not. 
 
 In the curve A A, Plate IV. (board 50 ft. long, varnished 
 surface), we find 
 
 At 100 ft. R= 1-4 lbs. 
 
 „ 200 „ R= 5-3 „ 
 
 „ 300 „ R=ir2 „ 
 
 „ 400 „ R=i9-o „ 
 
 „ 500 „ R = 28-3 „ 
 
 „ 600 „ R = 39-8 „ 
 
 Where A =0089 and B = -000095 will satisfy the curve 
 very nicely. Here, again, A has increased in the ratio of 
 SO : 28 from the value in the curve B B, or as the length : 
 B is less than it should be if this ratio were maintained. 
 
 Curve DD, Plate IV. (5 ft. long, varnished surface), 
 examined in the same manner — 
 
 A = -0009, B = 
 
 0000155. 
 
 From curve. 
 
 Calculated. 
 
 At 400 ft. R= 2-7 lbs. 
 
 2-87 lbs. 
 
 „ SOO „ R= 4-3 „ 
 
 4-32 „ 
 
 „ 600 „ R= 6-2 „ 
 
 6-12 „ 
 
 „ 700 „ R= 8-4 „ 
 
 8-22 „ 
 
 „ 800 „ R=io-7 „ 
 
 10-67 » 
 
 Here, again, A is reduced as 5 : 28 when compared with the 
 B B curve. 
 
 Plate v., curve A A (2 ft. 6 ins. long, varnished surface). 
 As before — 
 
Resistance due to Viscosity Varies 65 
 
 
 From curve. 
 
 Calculated. 
 
 At 300 ft. 
 
 .R= -9 
 
 •924 
 
 .. 400 „ 
 
 R=i-S6 
 
 1-376 
 
 „ 500 „ 
 
 R = 2-4 
 
 2-375 
 
 „ 600 „ 
 
 R = 3-4 
 
 3-366 
 
 „ 700 „ 
 
 R = 4-S6 
 
 4-529 
 
 „ 800 „ 
 
 R = S-88 
 
 5-864 
 
 When A = -00045 and B = -0000086. 
 
 Here, again, A is reduced proportionally to the reduction 
 in the length of the board. 
 
 Plate v., curve B B (i ft. 6 ins. long, varnished surface) — 
 
 A = -00027 and B = -000005 1 6. 
 
 Plate v., curve C C (i ft. long, varnished surface) — 
 
 A = -0001 8 and B = -0000035. 
 
 An examination of all these results will show that in all 
 cases A varies as the length of the plane. This is commonly 
 stated as varying as the wetted surface. This is not correct, 
 for it should be as the shearing surface. It is true that in 
 this case the wetted and shearing surfaces are the same ; 
 but there are cases where they are very different, as will 
 appear later. We may, therefore, replace A in the equations 
 by «L, where « is a small constant for all the curves, and 
 L = length ; and we may rewrite the equation 
 
 R = «LV + BV2, 
 
 where, however, the value of B is different for different curves. 
 
 In this equation ia: = -oooi8 or 18 X io~^ per foot of length. 
 Since the girth of the boards was 3 ft. 2 ins., the value of 
 a per square foot would be 5-7 x lO"^ 
 
 To understand the subject properly it will be necessary to 
 enter into a little history. 
 
 Coulomb first made experiments with discs caused to 
 oscillate in oil. 
 
 " In Coulomb's experiments it appeared that the resistance 
 was composed of two terms, one involving the first power, 
 and the other the square of the velocity" (Stokes, Math. 
 Works, vol. iii.). 
 
66 
 
 A B C of Hydrodynamics 
 
 Coulomb believed that at very low velocities the part of the 
 resistance depending upon the square of the velocity was so 
 small that it might be neglected. Stokes appears to have held 
 the same view, for he says that " in the case of minute globules 
 falling with their terminal velocity, the part of the resistance 
 depending upon the square of the velocity, as determined by 
 
 Fig. i8. 
 
 the common theory, is quite insignificant compared with the 
 part which depends on the internal friction of the air." 
 
 All this is ancient history, for Stokes wrote in 1850. We 
 know now, from Professor Perry's experiments, that at very 
 low velocities (below a critical speed) the part of the resistance 
 depending upon the square of the velocity does not exist. 
 R = AV accurately, but above this critical speed the resist- 
 ance suddenly increases. Referring to this suddenness, 
 Osborne Reynolds says : "It was a matter of surprise to me 
 to see the sudden force with which the eddies sprang into 
 
Resistance dtte to Viscosity Varies 67 
 
 existence, showing a highly unstable condition to have existed 
 at the time the steady motion broke down." 
 
 It is clear, therefore, that as the resistance varies as the 
 velocity only, to begin with, and as the BV^ factor is only, 
 what I may call, " switched in " at the critical velocity, the 
 inertia resistance does not commence at the origin but at a 
 point on the R = AV line which corresponds to the critical 
 velocity. The curve of velocity-resistance, which is repre- 
 sented by R = AV-1-BV^, is, of course, a parabola, which 
 passes through the origin and which is tangential to the line 
 R = AV at the origin. This may be represented graphically, 
 as in the diagram (fig. 18), where MOBD is the parabola, 
 tangential at O to the line O V. C represents the point on 
 the line R = AV which corresponds to the critical velocity. 
 The resistance will therefore be as follows : — Alto C (critical 
 velocity), R = AV. At C the resistance increases, suddenly, to 
 B, and it then follows the parabolic curve along B D. From 
 A to C the liquid is stressed and is in the highly unstable 
 condition referred to by Osborne Reynolds. 
 
 The diagram is not to scale and is only intended to illus- 
 trate my meaning. 
 
 The law of the variation of the parameter B in the BV^ 
 term is decidedly puzzling. 
 
 For I ft. o ins. B = 3'5 X lO"" per foot of board. 
 
 I ,- 
 
 6 
 
 
 B = 3-44X lo' 
 
 2 „ 
 
 6 
 
 
 B = 3-44 X 10 
 
 5 >. 
 
 
 
 
 B = 3"i X 10 
 
 16 „ 
 
 
 
 
 B = 2-3i X 10 
 
 28 „ 
 
 
 
 
 B = 2i4X 10 
 
 SO „ 
 
 
 
 
 B = i-9 X 10 
 
 In other words, up to 2 ft. 6 ins. B increases as the length 
 ■of the board almost exactly. After the length 2 ft. 6 ins. is 
 reached B no longer increases proportionately to the length, 
 but at a slower rate; and the rate decreases as the length 
 is increased. 
 
 The only explanation I can offer is that the planes vibrate 
 like the " Bull roarer," and that, as they vibrate more slowly 
 
 E 2 
 
68 A B C of Hydrodynamics 
 
 with increased length, the inertia resistance, caused by the 
 formation of eddies or vortices, also increases more slowly. 
 
 Until a better explanation is ofTered, this will serve as a 
 good working hypothesis. That the plate does vibrate may 
 be taken as a fact. It was observed by Beaufoy, who says : 
 " Almost all the figures, when drawn through the water with 
 a certain velocity, acquired a violent tremulous motion, 
 which, in a small degree, may be illustrated by observing a 
 large pot on the fire when it simme'rs." Mr Mallock (Aero- 
 nautics, Bluebook, 1909-10) also says: "A flat surface when 
 towed by the leading edge is exceedingly unstable." 
 
 That the boards must vibrate can easily be shown to be 
 a necessary consequence of theory. If we imagine A B in 
 
 c 
 ,,— — — ~ ^ 
 
 . ''' "^"^^ 
 
 A^t: ■ ^B 
 
 c 
 
 Fig. 19. 
 
 the diagram (fig. 19) to represent the board, which is held fast 
 at B and being towed in the direction of the arrow. We next 
 imagine that, for some reason or another the stream-lines flow 
 faster on one side than the other — say past C. There will 
 be less pressure on the C side of the board than on the other, 
 and the board will bend towards ACB. This will further 
 increase the velocity of flow past C, and so further reduce the 
 pressure : the board will bend more towards C. Eventually 
 the water on the other side will be " cracked "at B ; discon- 
 tinuity will be produced and the board will fly over to A C B 
 and so on. A vibratory movement of the board is thus set 
 up, its period being a function of the dimensions of the board. 
 
 From the foregoing it may be seen that the resistance due to 
 viscosity (which in the last analysis is a stress set up to resist 
 shearing of the liquid) of a plate drawn edgewise through 
 water varies as tfie velocity and as the shearing surface. In this 
 case the wetted surface and shearing surfaces are the same. 
 
 The law of this resistance, as commonly taught, is that it 
 
Resistance due to Viscosity Varies 69 
 
 varies as V^'^^; it may, of course, vary as V^'^^ at certain 
 speeds, but certainly not for all speeds, or beyond very narrow 
 limits. I maintain that this V^'^s law (??) has a further 
 defect, that it completely masks the simple " law of dimension " 
 which is here pointed out : hence it is commonly stated that 
 the resistance due to viscosity does not vary as the length, 
 whereas I here show it does. 
 
 There is another law of liquid resistance which has latterly 
 been introduced by Mr Lanchester in his Aeronautics. He 
 says the curve of resistance commences as a straight line, 
 (i) thus following " Stokes' law " ; after a critical speed, which 
 he does not state, it changes to RooV^'*, (2) which he calls 
 " Allen's law " ; it later changes again to RooV^, (3) which 
 he calls " Newton's law." 
 
 (i) I am unable to find that Stokes ever said that the 
 resistance varied as the velocity. He said that at very low 
 speeds the resistance depending upon the square of the velocity 
 was so small that it might be neglected; but that is not the 
 same thing at all. It is curious to find Mr Lanchester speak- 
 ing approvingly of Stokes' law, for he certainly gives it a very 
 severe condemnation when he says : " Stokes' law is based on 
 a system of motion of the fluid that has been mathematically 
 investigated and the lines of flow plotted from an equation. 
 It may be remarked that this system of motion can never exist 
 in its entirety, for it involves an infinite quantity of momentum 
 and an infinite quantity of energy : in other words, the steady 
 state involves a force applied for an infinite time through 
 an infinite distance ; it also constitutes a violation of the 
 principle of no momentum." So much for Buckingham ! 
 
 (2) Allen commences by saying that "if th.e resistance can 
 be represented by a single term it must be proportional to 
 {aY.y." He then proceeds to show that it varies approxi- 
 mately as V^°. His experiments ^ are not convincing, for they 
 are subject to the following errors, which he admits : — 
 
 I. The reduction of pressure, as the bubbles rise, causes 
 the bubbles to enlarge. 
 
 > On the time taken by small bubbles of air in rising through a 
 cylindrical vessel of water. 
 
•JO A B C of Hydrodynamics 
 
 2. The gradual solution of the gas in the liquid causes the 
 bubbles to diminish in size. 
 
 3. The bubbles are skimmed off the surface of the liquid 
 and measured. The range of velocities is exceedingly small : 
 from -12 inches per second (leaving very little room for 
 Stokes' law) up to a maximum of 2-2 inches per second. The 
 law (?) appears to have no very great practical importance. 
 
 (3) Newton's law does not appear to have been that the 
 resistance varied as the square of the velocity, except in fluids 
 which have no "lubricity." The actual curve of resistance 
 does not follow this law except in special cases. The general 
 approach to it is about V^'^. Newton undoubtedly says, in 
 places, what might be supposed to mean that RooV^ ; but 
 nothing can be clearer than his scholium : " The resistance of 
 sphcErical bodies in fluids arises partly from the tenacity, 
 partly from the attrition, and partly from the density of the 
 medium. And that part of the resistance which arises from 
 the density of the fluid is, as I said, in a duplicate ratio of 
 the velocity, the other part [italics added], which arises from 
 the tenacity of the fluid, is uniform, or as the moment of the 
 time. " 
 
 Why, then, should it be taught that the resistance varies as 
 V^*'.? It is very difficult to say. There appears to be a 
 fascination about the idea that the resistance varies as some 
 single power of the velocity. Undoubtedly, within narrow 
 limits, it is always possible to find some logarithmic curve 
 which is nearly the same as R = AV + BV^; but the whole 
 process reminds me of the way carpenters strike an ellipse (?) 
 with a compass ; by piecing together small pieces of circles 
 with different radii, they describe a curve which — to the un- 
 trained eye — looks very like an ellipse ! 
 
 Mr Lanchester's ideas appear clear enough : " In all cases 
 oS. purely viscous resistance the law of viscosity requires that 
 the resistance shall vary directly as the velocity ; and the whole 
 of the energy expended disappears at once into the thermo- 
 dynamic system. In cases where the resistance is dynamic — 
 that is to say, where it is due to the continuous setting of new 
 masses of fluid in motion — the whole of the energy expended 
 
Resistance due to Viscosity Varies Ji 
 
 remains in the fluid in the kinetic form (being only sub- 
 sequently frittered away), and the resistance varies as the 
 square of the velocity. When the resistance is due to both 
 causes combined, as in the case of skin friction, the portions of 
 the total resistance varying directly, and as the square, are 
 respectively proportional to the energy expended in the two 
 directions" {^Aerodynamics, italics added). 
 
 Nothing could apparently be clearer ; but in case anyone 
 should think the end of the last paragraph the least bit vague, 
 he says later : 
 
 "Let Rj be the resistance varying as V^, and R^ be the 
 resistance varying as V^, then R = R^+R2.'' 
 
 Notwithstanding this exceedingly clear presentation of the 
 case, Mr Lanchester, by some curious train of reasoning, comes 
 to the conclusion that R = V", the argument commencing: 
 " Now for any particular velocity ^^ toXsX resistance — that is, 
 the sum of the viscous and dynamic resistances — may be 
 expressed as varying as the i^^ power of the velocity." This, 
 of course, is a truism ; for N^ = anything you like. If we take, 
 say, V=ioo; then, if n = 2, V"= io,ooo;if n=i, V"=ioo; 
 and ii n = -s, then V"= lo. 
 
 What appears fairly certain is that if R = Ri + R2, then R 
 varies as no single power of the velocity, and Allen's law is not 
 true. 
 
 In the next chapter Stokes' law will be specially referred 
 to, as it is very pertinent to the law of resistance which has 
 been discussed here. 
 
 Summary 
 
 The resistance of a long flat board drawn edgewise through 
 water may be expressed by R = A V + B V^, where A and B 
 are constants for each flat plane, being what are called 
 " parameters." 
 
 For A we may substitute «L, where a is a constant for all 
 the planes and L is the length of the plane, the resistance 
 due to viscosity (AV) varying as the velocity and as the 
 shearing area. 
 
72 A B C of Hydrodynamics 
 
 The coefficient B of the inertia resistance (or dynamic 
 resistance), term BV^ varies in a curious manner which is not 
 easy to explain. The variation appears to depend on the 
 length of the board, and — (?) probably — on the. period oi the 
 vibration of the board transversely to the line of motion, this 
 vibration being a necessary consequence of the theory of 
 hydrodynamics. 
 
 REFERENCES 
 
 Report of British Association, 1872. 
 
 Colonel Beaufoy's Nautical and Hydraulic Experiments. 
 
CHAPTER IX 
 
 RESISTANCE DUE TO VISCOSITY {continued) — 
 
 stokes' law 
 
 In the last chapter it was shown that when thin plates are 
 drawn through water that the resistance due to viscosity 
 varied as the velocity and as the shearing surface. All the 
 curves of resistance of boards of the same kind, in Dr Froude's 
 paper, were examined and gave the same results. These 
 planes were all varnished ; but others, covered with Hay's 
 composition, give almost exactly the same results. There are, 
 however, some curves, in the same paper, of the resistance of 
 planes covered with tin-foil where A is very much less than in 
 the curves examined. For instance, the plane corresponding 
 to curve C" C", Plate IV., is of the same length as that referred 
 to in curve C C ; and the same holds for curves B' B' and 
 C C, "Plate v., as compared with the curves B B and C C in 
 the same plate. In all these cases it will be found that A 
 is, roughly, from 45 to 50 per cent, less than it is in the 
 " varnished plane " curves. The reason for this appears to be 
 that the tin-foil is very imperfectly " wetted " by the water. 
 Put in other words, the adhesion of the water to the tin is less 
 than it is to a varnished surface. When this adhesion is 
 imperfect, the liquid slides on or rolls over the solid, and the 
 resistance is no longer the same. It is probably rolling and 
 sliding a little as it rolls. I should liken it to the action of a 
 bicycle tyre on a sticky road. The tyre is always in contact 
 with the road, but whilst it rolls along it sometimes slips as 
 well when going down a slope. 
 
 We see, therefore, that Professor Perry's rule 4 : " The force 
 
 73 
 
74 A B C of Hydrodynamics 
 
 of friction at moderate speeds does not much depend on 
 the nature of the wetted surfaces," is not strictly accurate. It 
 requires to be qualified hy— provided the surfaces are equally or 
 perfectly wetted. 
 
 His rule 3 also requires a trifling modification : " The force 
 of friction is proportional to the shearing [not wetted] surface " ; 
 though in the cases he refers to the two are the same. 
 
 It may be well to try and get some clear mental image of 
 how this treacliness or viscosity acts in causing a resistance 
 to a plate. Amongst naval architects, who speak of " skin 
 friction," it is commonly supposed to be caused by the water 
 "rubbing" against the side of the ship, and tables have 
 
 Direction of motion of plate 
 Fig. 20. 
 
 been made of the friction of water against different 
 substances. Even Dr Fleming, when referring to the 
 resistances of a boat ( Waves and Ripples^ says : " These are, 
 first, the skin of friction between the ship's surface and the 
 water " — he was probably speaking in a very " popular 
 manner." 
 
 Professor Lamb {Hydrodynamics) goes a little to the other 
 extreme, for he says : " It is probable that in all ordinary 
 cases there is no motion, relative to the solid, of the fluid 
 immediately in contact with it. The contrary supposition 
 would imply infinitely greater resistance to the sliding of one 
 portion of the liquid past another than in the sliding of the 
 fluid over the solid." The truth would appear to be probably 
 somewhere between these two views. 
 
 Let us imagine the sketch (fig. 20) to represent (very 
 diagrammatically) some particles of water and a moving 
 plate. We may assume the layer next to the plate to be 
 
Resistance due to Viscosity 
 
 75 
 
 absolutely at rest, relatively to it, and consequently moving at 
 the same velocity. The next layer will move slightly slower 
 than the plate, rolling along the particles at rest. The next 
 layer will roll on this layer, and so on. It will be evident that 
 these particles in rolling will rub against one another and so 
 generate heat which will be dissipated. 
 
 We may put the matter in a different way. Let us imagine, 
 in this case (fig. 21), the water to be moving and the wall to 
 be at rest. Let A B C D represent diagram matically a longi- 
 
 rvater at rest ^ 
 
 Fig. 21. 
 
 tudinal section of a body of water moving in the direction of 
 the arrow ; C D representing the surface of the molecules of 
 water which are " at rest.'' 
 
 After a small period of time this body of water will have 
 changed its position to B C D E, when all the particles of 
 water in contact with those at rest will have rolled a finite 
 distance along C D, and all the particles will have rotated 
 through the same angle, as shown by the small arrows.^ 
 
 Let us now imagine the water, which is now at A B C D,^ to 
 be moved in the same manner. The " first layer " particles 
 will roll still more along C D in the direction of arrow, and 
 
 ' The diagram shows the particles as deformed : such is not intended, 
 for the angle A C B is supposed to be exceedingly small. 
 ^ Not necessarily the same particles. 
 
76 A B C of Hydrodynamics 
 
 all the particles will rotate still more through the same angle, 
 as shown by the small horizontal arrows. 
 
 Repeat this process indefinitely, and the reader can imagine 
 all the particles moving along in the direction A B, and all 
 rotating at one uniform velocity. 
 
 If the particles near A B move too fast the " first layer " 
 particles will slip,, as well as roll over, the particles at rest. 
 This is what has actually been found to be the case in very 
 large rivers. 
 
 If we next suppose the adhesion of the liquid to be im- 
 perfect, we can imagine the layer marked " at rest," as also 
 rolling slightly along the plate — not necessarily at the same 
 rate as the other layers. If the adhesion is very small we can 
 further imagine the particles next to the plate as, not only 
 rolling, but actually slipping along the plate. 
 
 A step further: if the adhesion be quite negligible — as 
 between clean mercury and glass — then the liquid will slide 
 along the plate and there will not necessarily be any rolling 
 at all. 
 
 This appears a fairly satisfactory working hypothesis of 
 the molecular action of a liquid against a plate. This 
 explanation is consistent with the well-known fact that a 
 slight roughness of the surface will not increase the resistance. 
 Poiseuille placed a small quantity of very finely powdered 
 shellac in a glass tube, and took the time required to dis- 
 charge a certain amount of water through the tube with a fixed 
 pressure. He then heated the tube so as to melt the shellac and 
 make it quite smooth : the time of discharge was unaltered. 
 
 If the liquid do not wet the body, as, for example, in the 
 case of mercury flowing through a glass tube, then a layer of 
 air would be interposed between the mercury and the glass, 
 and the friction would be an " air-air " one, depending upon 
 the viscosity of the air. This Girard found so small, in fine 
 tubes, that he could not measure it, and he gives the resistance 
 varying strictly as B V^ only. 
 
 If the foregoing be correct, there are certain practical 
 deductions we can draw. 
 
 If frictional resistance depends on the body immersed 
 
Resistance due to Viscosity yy 
 
 in the liquid being wetted, then, if we could prevent a ship 
 from being wetted — or even reduce the perfection of the 
 wetting — it ought to meet with less resistance. This is, of 
 course, done, to some effect, by copper sheathing (though this 
 was adopted for other reasons) ; but if we could coat the ship 
 with a layer of air, the resistance, being " air-air," should be 
 very much less than " water-water." This was actually tried 
 some years ago, a layer of compressed air being interposed 
 between the bottom of the vessel and the water. Sir 
 Frederick Bramwell, who described the experiment, said the 
 effect was peculiar and "soda-watery," but the resistance 
 was materially reduced. Scientifically the experiment was a 
 success, though it was a failure ^.. „. 
 
 ■(, ^nin Pipe 
 
 commercially ; for the power 
 
 expended in compressing the 
 
 air was so great that there 
 
 was no advantage measured '4^^,^w^^sv . s^^^^■^ 
 
 in £ s. d. ^^^§ 
 
 Very much the same result 
 is achieved in the racing boats pj^^ 22. 
 
 called " hydroplanes." If the 
 
 diagram (fig. 22) represent a small piece of the bottom of the 
 boat, travelling to the right, as represented by the arrow, 
 the water moving quickly past the " step " of the hydroplane 
 will cause air to flow through the air-pipe into the step. 
 The boat may be said to be partly " floating on air." These 
 boats have achieved a great measure of success. Sir John 
 Thornycroft's last pattern (the maple leaf) is said to have 
 attained a speed of 48'8 knots. 
 
 There is another way in which reduction of resistance 
 might be tried : we might imitate Nature. A fish is covered 
 with oil glands which keep its body well greased. It would 
 not be very difficult to arrange small oil tubes round a ship, 
 with very fine holes in them (fig. 23). This might be worth 
 trying (on yachts or torpedo boats), as the consumption of oil 
 need only be comparatively trifling. The old whaling captains 
 used to declare that whalers were always faster than sister 
 ships ; and they attributed this to the sides being always oily. 
 
78 A B C of Hydrodynamics 
 
 It must not be imagined that if the wetted surface of 
 a body be increased that therefore the resistance will be 
 increased. Very often the reverse is true, and Beaufoy 
 .^Oilhote ^^y^'- "Among the conclu- 
 sions suggested by the tables 
 [in his book], one of the most 
 Side of Ship curious is, that the increasing 
 
 ^''^- ^3- the length of a solid of almost 
 
 any form by the addition of a cylinder in the middle ex- 
 ceedingly diminishes the resistance with which it moves, 
 provided the weight in water continues to be the same — a 
 fact, I apprehend, that cannot be easily explained." One 
 of the examples he gives is — 
 
 Resistance of a sphere = 62"85 lbs. 
 
 Sphere cut in half and cylinder inserted {l=d), where 
 /= length of cylinder and if = diameter of sphere — 
 
 Resistance = 49' 7 1 lbs. 
 
 It is well known, also, that many ships have been lengthened 
 by the insertion of a middle piece, with the result that they 
 attained a greater speed. 
 
 A very curious example of the difference between the 
 action of a body in a liquid, when the body was wetted and 
 when it was not, was 
 first shown by Professor 
 Worthington at the 
 Royal Institution in 
 1894 {Splash of a Drop, 
 S.P.C.K.), and is as 
 follows : — Take a small 
 very smooth marble of '^^ 
 
 about \ inch diameter : dry it thoroughly and slightly warm 
 it. If this marble be dropped from a height of 2 or 3 feet into 
 a bucket of water, the water will spread " over the sphere so 
 rapidly that it is sheathed with the liquid even before it has 
 passed below the general level of the surface " (fig. 24, A). " The 
 splash is insignificantly small and of very short duration." 
 
Stokes" Law 79 
 
 If the marble be wet, however — the same marble picked 
 out of the bucket — it will give a very considerable splash 
 (fig. 24, B). 
 
 I think I have observed the same sort of effect on armour 
 plates struck by steel and bronze shot. In the latter case the 
 " splash " of the plate was very much less than when it was 
 struck by the steel projectile. The bronze shot was probably 
 not so much " wetted " (if one may use the word) by the steel 
 of the armour plate as in the case of the steel projectile. In 
 other words, the adhesion of steel to bronze was less than steel 
 to steel. 
 
 Stokes' Law 
 
 It is very curious how many people refer to Stokes' law and 
 how few really know what it is. It appears to be generally 
 believed that Stokes said that at very low velocities the 
 resistance to bodies varied as the velocity only. I am not 
 aware that Stokes ever said anything of the sort : I have not 
 seen it after careful study of his papers. What he did say 
 was {Math. Papers, vol. iii.) : " The resistance of a sphere 
 moving uniformly in a fluid may be obtained as a limiting 
 case of the resistance of a ball pendulum, provided the 
 circumstances be such that the square of the velocity may be 
 neglected [italics added]. The resistance thus determined 
 proves to be proportional, for a given fluid and a given 
 velocity, not to the square, but to the radius of the sphere ; and 
 therefore the accelerating force of the resistance increases 
 much more rapidly, as the radius of the sphere decreases, 
 than if the resistance varied as the surface, as would follow 
 from, the common theory. Accordingly the resistance to a 
 minute globule of water falling through the air with its 
 terminal velocity depends almost wholly on the internal friction 
 of air. Since the index of friction of air is known from 
 pendulum experiments, we may easily calculate the terminal 
 velocity of the globule of given size, neglecting the part of the 
 resistance which depends upon the square of the velocity. . . . 
 Since in the case of minute globules falling with their 
 terminal velocity the part of the resistance depending upon 
 
8o A B C of Hydrodynamics 
 
 the square of the velocity, as determined by the common theory 
 [? R = AV + BV^], is quite insignificant compared with the part 
 which depends on the internal friction of the air, it follows that 
 were the pressure equal in all directions in air in the state of 
 motion, the quantity of water which would remain suspended 
 in the state of cloud would be enormously diminished." 
 
 The paper is exceedingly mathematical and difficult to 
 follow, but the " law " is quite clear from the foregoing quota- 
 tion. Put in other words, when the velocity is very low the 
 resistance of a sphere varies {practically) as the velocity and as 
 the radius of the sphere. The reader will see at once that 
 this law is a special case of the general law here enunciated 
 that the resistance varies as the velocity and as the shearing 
 surface. In the case of a sphere the shearing surface is 
 obviously a band round the equator, which, of course, varies 
 as the radius. If the resistance due to viscosity varied as the 
 wetted surface, then it would vary as r^. 
 
 Stokes' law may have all the vices attributed to it by 
 Mr Lanchester, and quoted in the last chapter ; but it at least 
 has the merit of agreeing with experiment, which some laws 
 occasionally do not. It appears capable of a rational ex- 
 planation ; and it further indicates why Newton's experiments 
 with spheres tended to show that the resistance varied as the 
 square of the velocity. It is clear that since the "shearing 
 surface " of an oblate spheroid moving along its axes of revolu- 
 tion is exceedingly small — smaller than that of any other 
 body (a sphere may be considered as an oblate spheroid of 
 revolution where the axes are as i : i) — then in the formula 
 R = AV-fBV^ AV, the resistance due to the viscosity, will 
 be very small ; and the resistance of the sphere will ap- 
 proximate to BV^. If anyone wants to prove that the 
 resistance of bodies varies as the square of the velocity, he 
 should select a sphere or a thin lamina to experiment with. 
 
 The whole subject of viscosity wants overhauling very 
 badly. If we accept Newton's hypothesis that "the resist- 
 ance arising from the want of lubricity in the parts of a fluid 
 is, cceteris paribus, proportional to the velocity with which the 
 parts of the fluid are separated from each other,'' then it will 
 
Stokes' Law 8i 
 
 be evident that the method of measuring the coefficient of 
 viscosity by allowing liquids to flow through capillary tubes 
 and observing the time of flow is not a sound one. It is clear 
 that the flow of water through a glass tube will not be the 
 same as that through a tin tube of the same dimensions. Some 
 liquids adhere to glass much more strongly than water does, 
 and so would be credited with a higher coefficient of viscosity 
 than they actually possess. For example, Girard found that 
 a strong syrup of sugar and water flowed more freely through 
 a small glass tube than alcohol. If the liquid slips along the 
 glass, then we are not measuring the cohesion of the liquid, but 
 something very different. 
 
 It will probably be objected that Professor Lamb, having 
 discussed the " steady flow of a liquid through a straight pipe," 
 says in reference to his formulae : 
 
 "This last result is of great importance as furnishing a 
 conclusive proof xhsX there is in these experiments [of Poiseuille} 
 no appreciable slipping of the fluid in contact with the wall" 
 [italics added]. Also : " The assumption of no slipping 
 being thus justified, the comparison of the formula with 
 experiment gives a very direct m.eans of determining the value 
 of /A for various fluids." 
 
 Professor Lamb is a very eminent mathematician and the 
 distinguished author of the great classic on Hydrodynamics ; 
 it requires therefore considerable audacity to question any- 
 thing he says on his special subject. Still, " there is no more 
 common error than to assume that, because prolonged and 
 accurate mathematical calculations have been made, the 
 application of the result to some fact of nature is absolutely 
 certain. The conclusion of no argument can be more certain 
 than the assumption from which it starts" (Whitehead, 
 Introduction to Mathematics). 
 
 If we grant Professor Lamb's premises, there is no getting 
 away from his conclusions. His view appears to be as follows : 
 — If we imagine the diagram (fig. 25) to represent the longi- 
 tudinal section of a piece of the tube, and A, B, C, D, etc., to 
 represent sections of the very thin co-axial tubes of water. 
 The tube A, not slipping, will have a velocity = 0, referred to 
 
 F 
 
82 
 
 A B C of Hydrodynamics 
 
 &c. 
 
 &c. 
 
 2S- 
 
 the wall. Expressing the velocity of B as i ; then, if this be 
 supposed very small, the velocities of C, D, .etc., may be 
 represented^by 2, 3, etc. 
 
 If we next imagine A to slip, so as to have a velocity = i, 
 then the velocities of B, C, D, etc., will be represented as 
 2, 3, 4, etc. This flow would be similar 
 to that which would occur in a larger 
 tube where there was another liquid 
 cylinder, outside A, which did not slip. 
 
 Now, what are the assumptions im- 
 plied in this argument? Clearly, that 
 there is attraction between the different 
 layers of liquid, and that there is attrac- 
 tion between the layer A and the wall.^ 
 But if there is attraction between A and 
 the wall, there must also be attraction 
 between B and the wall (though in a 
 less degree), and similarly between C, D, 
 etc., and the wall. This does not appear 
 to have been taken into account. If we bring this factor in, 
 we can easily imagine that A is slipping (to some appreciable 
 extent), whilst B, C, D, etc., are being slightly retarded by 
 the wall, so that the effect of A's slipping might be completely 
 masked. The result might be exactly the same as that supposed 
 by Professor Lamb, though the reasons for this might be 
 vastly different. 
 
 Referring to the second point, it appears only reasonable 
 to suppose that different liquids exert more or less attraction 
 on the wall. If we suppose some liquid to h^ greatly attracted 
 by the wall, we might imagine that not only the velocity of 
 A is zero, but also that of B, and possibly of C. In this case 
 we would have a flow similar to that which would occur in 
 a smaller tube, where the outside layer did not slip — as if, 
 in fact, the layer A (or the layers A and B) had been 
 "peeled off." 
 
 ' It may be that even attraction of the layer A to the wall is not postu- 
 lated. The external friction — between the liquid and the wall — may be 
 possibly what is intended. 
 
Stokes' Law 83 
 
 Applying this to the measure of viscosity: suppose we 
 compare two liquids, where, in the first place, B= i, and, in 
 the second case, B = o. It would be said that the second 
 liquid was more viscous than the first, because it flowed through 
 the tube more slowly. Of course it flowed more slowly — it 
 was flowing through a smaller tube ! 
 
 The further discussion of this point will be deferred until 
 Poiseuille's law has been explained in Chapter XI. 
 
 Summary 
 
 The resistance of a body in a liquid depends, to a certain 
 extent, on how far the body is " wetted " by the liquid. 
 
 Stokes' law appears to be a special case of the general law 
 that, at low velocities — where the resistance is entirely due 
 to viscosity, and consequently varies as the velocity only — the 
 resistance varies as the shearing surface. In the case of a 
 sphere the shearing surface is a band round the " equator " of 
 the sphere. The resistance varies as the shearing surface, not 
 as the " wetted surface," although it frequently happens that 
 the shearing and wetted surfaces are the same. 
 
 The statement that a liquid flowing through a tube does 
 not "slip" along the walls of the tube appears questionable; 
 the subject requires further investigation.^ 
 
 REFERENCES 
 
 Whitehead, An Introduction to Mathematics. 
 WoRTHiNGTON, The Splash of a Drop. (S.P.C.K.) 
 
 „ A Study of Splashes. (Cambridge Press.) 
 
 Stokes, Math. Papers, vol. iii. 
 
 * " According to Helmholtz and Pietrowski, there is slipping between 
 the fluid and the solid, but more between water and silver than between 
 -water and glass." — J. G. Butcher, On Viscous Fluids in Motion. 
 
 F 2 
 
CHAPTER X 
 
 HOW LIQUIDS CHANGE FROM STEADY TO "SINUOUS" 
 MOTION — VORTICES — SENSITIVE FLAME — SINGING FLAME 
 
 It is well to define what we mean by " sinuous motion '' and 
 vortices. 
 
 Osborne Reynolds employs the term sinuous motion for 
 all forms of motion where the liquid has broken, or is 
 breaking, into eddies: it will be used here strictly in that 
 sense. 
 
 Lord Kelvin says : " I now define a vortex as a portion of 
 fluid having any motion that it could not acquire by fluid 
 pressure transmitted through itself from its boundary" [italics 
 added]. It will be evident that Lord Kelvin's definition will 
 include all forms of sinuous motion : it is necessary to reflect 
 well over it, for it contains the key to a great difficulty. 
 
 Now, how does steady motion change into sinuous motion ? 
 
 Stokes says {Math. Works, vol. i.) : " Except in the case 
 of capillary tubes, or, in case the tube be somewhat wider, of 
 excessively slow motions, the main part of the resistance 
 depends upon the formation of eddies. This much appears 
 clear ; but the precise way in which the eddies act is less 
 evident." 
 
 Speaking of waves, he says {Math. Papers, vol. ii.) : " Waves 
 are usually produced either by some sudden disturbing cause, 
 which acts at a particular part of the fluid in a manner too 
 complicated for calculation, or by the wind meeting the surface 
 in a manner which cannot be investigated'' 
 
 These statements, made in 1849, ^""^ certainly not promising. 
 
 In 1899 (Royal Institution) Dr Hele-Shaw remarked: 
 
 84 
 
Change from Steady to ''Sinuous'' Motion 85 
 
 " Whatever the real nature of viscosity is, it results in pro- 
 ducing in water eddying motion, which would be perfectly 
 impossible if viscosity were absent" ; but it must be remembered 
 that Lord Kelvin has stated that vortices would be produced 
 in a non-viscous liquid in certain circumstances, and he has 
 also shown how he conceives they would be formed. 
 
 If we refer to Mr Lanchester's Aerodynamics, we see that 
 his description of the formation of eddies leaves much to be 
 desired. " If, in the first place, the fluid be taken as inviscid, 
 and if, for the purpose of this argument, we assume (}.^) that 
 the system of flow in the figure (fig. 26) is possible in an inviscid 
 fluid. . . . Let us next introduce viscosity as a factor. The 
 
 Fig. 26. 
 
 conditions are now altered, for the fluid in the dead-water 
 region is no longer motionless, but is in active circulation. . . ." 
 
 This has almost the appearance of a conjuring trick. The 
 perfect liquid behind the body is at rest ; introduce a tea- 
 spoonful of glycerine, when, presto ! the liquid commences 
 to boil. Why ? 
 
 It is not denied that eddies can be produced by viscosity, 
 under special circumstances, but it is contended that they are, 
 nearly always, produced in water in a totally different manner ; 
 and that there is no reason why they could not, in similar 
 circumstances, be produced in a perfect liquid, if such a thing 
 existed. Professor Lamb appears perfectly clear on the 
 subject of the possibility of producing vortices, for he says 
 {Hydrodynamics') : " It is obvious that cyclic irrotational motion 
 of a liquid cannot be reproduced by any arrangement of 
 simple sources. It is easily seen, however, that it may be 
 represented by a certain distribution of double sources." 
 
86 
 
 A B C of Hydrodynamics 
 
 Remembering Professor Perry's dictum that with liquids 
 it is pressure, pressure — always pressure, it is evident that 
 what is required to produce sinuous motion is a cross pressure 
 to act as a second " source." A few examples will now be 
 given of the change of direct into sinuous motion. 
 
 To take the case of water issuing from, or entering a 
 trumpet mouth. This problem is said to have bothered 
 Dr Froude for a long time ; for whilst the liquid could be 
 caused to flow into the trumpet at almost any reasonable 
 velocity, and still act according to the recognised laws of the 
 hydrodynamical "drill book," if the motion were reversed 
 
 ^ small tube discharging 
 coloured liquid 
 
 Fig. 27. 
 
 it was hardly possible to cause it to move without breaking 
 into some " nasty " eddies. 
 
 Professor Osborne Reynolds, by the use of a little coloured 
 liquid, showed that the motion of the water at very low speeds 
 was along A B and C D (fig. 27). This is what he would call 
 "steady" or direct motion, but it comes under his rule No. 5 — 
 " a stream flowing through still water " — and this, as he most 
 rightly says, is conducive to " unsteady motion." It is steady 
 motion (in the sense in which he employs the word), but it is 
 unstable motion, since a very slight increase of the velocity 
 will cause the stream to break into sinuous motion or eddies. 
 
 If what has been said previously be accepted, the pressure 
 in A B C D must be less than in the undisturbed water 
 surrounding it. Why does not the undisturbed water flow 
 into A B C D, «j- it should} It is evident that along A B and 
 C D there is a surface of discontinuity, or, what is the same 
 
Sensitive Flame 87 
 
 thing, a free surface. This surface is capable of sustaining a 
 certain amount of tension , and so is just able to prevent the 
 flow which would otherwise take place. If the velocity of the 
 stream be increased, the difference of pressures on the two 
 sides of the film will become so great that the film will not 
 be able to resist it : the film breaks and sinuous motion 
 commences at once — suddenly. Doubtless the friction will 
 assist in breaking this surface, but this result would occur 
 even if there were no friction. Viscosity can, therefore, in no 
 sense be said to be the cause of the eddies, though it may 
 somewhat accelerate their production. 
 
 There appears no valid reason for supposing that a perfect 
 liquid, if such a thing existed, would not behave in the same 
 manner in similar conditions, provided its surface were 
 capable of exerting tension. 
 
 Sensitive Flame 
 
 The well-known " sensitive flame," so largely employed in 
 acoustical experiments, will come under Osborne Reynolds' 
 rule No. S, and is a very good example of " unstable motion " 
 (in the sense employed previously) kept " steady " by surface 
 tension. 
 
 " If ordinary coal-gas stored in a gasometer is burnt at 
 a small jet under considerable pressure, we are able to produce 
 a tall flame about 18 to 24 inches in height. The jet used 
 is one with a steatite top and a small pinhole gas exit about 
 ■^ inch in diameter. [Professor Barrett said it was very 
 essential that this aperture should be V-shaped : he recom- 
 mends a |-inch glass tube drawn out to -^-^ inch.] The 
 pressure of gas must be equal to about 10 inches of water, 
 and it cannot be drawn straight off the house gas-pipes, but 
 must be supplied from a special gasometer or gas-bag under 
 a pressure sufficient to make a flame 10 inches or so in height. 
 If the pressure is too great, the flame roars ; if the pressure is 
 slightly reduced, the flame can be made to burn quietly and 
 form a tall reed-like flame (A, fig. 28). This flame, when 
 properly adjusted, is curiously sensitive to shrill chirping 
 
88 
 
 A B C of Hydrodynamics 
 
 sounds. You may shout or talk loudly to it, and it takes no 
 notice of the voice, but if you chirrup or whistle in a shrill 
 tone, or clink your keys or clink a few coins in your hand, the 
 flame at once shortens itself to about 6 or 7 inches in height, 
 and exhibits a peculiarly ragged edge, whilst at the same 
 time it roars (B, fig. 28). When in adjustment, the clink of a 
 couple of coins will affect this sensitive flame on the other 
 side of the room. The flame is also very sensitive to a shrill 
 whistle or bird-call. It will be clear to you, 
 from previous explanations, that the flame 
 responds, therefore, to very short air waves 
 forming high notes. The particular flame 
 I shall now use responds with great readi- 
 ness to air waves of i inch to ^ inch in length. 
 "It may be well to explain that the 
 sensitive portion of the flame is in the root, 
 just where it emerges from the burner ; it 
 is the action of the sound wave in throw- 
 ing this portion of the flame into vibrations 
 which is the cause of its curious action. 
 
 " If you think what the reaction must be, 
 you will easily see that the operation of 
 the sound wave is to throw the particles 
 of the gas, just as they escape from the hole 
 in the jet, into vibration in a direction trans- 
 verse, or at right angles, to the direction of their movement in 
 the flame. The gas molecules are, when unacted upon by 
 the sound wave, rushing out of the jet in an upward direction. 
 When the sound wave impinges on them they are, so to speak, 
 caught, and caused to rock to and fro in a direction across the 
 flame. The combination of these two motions results in a 
 spreading action of theflam.e, so that instead of being a thin 
 lance-like shape, it becomes more blunt, stumpy, and ragged at 
 the edges " [italics added] (Dr Fleming, Waves and Ripples). 
 
 This description has been quoted at full length because it 
 cannot well be improved upon ; but the explanation is highly 
 unconvincing. If the action is caused by the " rocking of the 
 particles " of the flame at a definite period, depending on the 
 
 1 
 
 Fig. 28. 
 
Sensitive Flame 89 
 
 shrillness of the sound, and the longitudinal motion of the 
 particles be steady, then the combination of these two motions 
 should be a regular wave and not a ragged figure. We might 
 put the following questions : — 
 
 1. Why is the same result not produced when a deeper 
 note is sounded ? 
 
 2. Why, if the flame is caused to rock, should it get 
 shorter! There does not appear to be any " machinery" for 
 shortening it. 
 
 3. Why should exactly the same result be produced when 
 the particles are not being rocked! — i.e. when the supply of gas 
 is increased. The rocking machinery does not exist in this 
 case. 
 
 4. Why should the length of the flame be decreased when 
 the supply of gas is increased! We might expect the very 
 opposite result. 
 
 5. Why should "rocking the particles" cause the flame to 
 roar and become ragged! 
 
 6. Why should the root of the flame be the most sensitive 
 part? We should expect that the rocking action would be 
 least apparent there, since the gas is moving fastest at this 
 point. 
 
 The explanation breaks down under cross-examination. 
 It appears to no more "explain" the action than if one were 
 to say that pulling the trigger of a rifle causes the bullet to 
 fly out of the muzzle. One follows from the other, but can 
 hardly be said to be the cause of the other. You cannot say 
 you cause a ball to fall to the ground because you let go of it. 
 You allow it to fall. 
 
 r venture to suggest that the correct explanation is that 
 the flame is at " full cock," and the bird-call " pulls the trigger," 
 or " presses the button," and, by so doing, allows the forces, 
 all ready, to " do the rest." 
 
 r. The pressure inside the flame is less than that outside. 
 
 2. The film round the surface of the ?^cLm& prevents the air 
 from rushing in. 
 
 3. The pressure of the gas is so regulated that the film is 
 just at the breaking point. 
 
90 
 
 A B C of Hydrodynamics 
 
 4. A deep note throws the surface of the flame into long 
 waves (fig. 29), thereby stretching the film, but only to such 
 an extent that it is jusi able to bear the strain. 
 
 5. Short waves (fig. 29), caused by shrill notes, stretch the 
 film much more : not being able to support this extra strain, 
 the film breaks, and the air rushes in. 
 
 6. When the air rushes in, since violent eddies are produced, 
 it is only natural to expect that the flame would be very 
 
 Straight edge of flame 
 
 Long wave 
 
 Very short wave 
 
 Fig. 29. 
 
 materially shortened ; that the edges would be ragged ; also 
 that the flame would roar. 
 
 7. If the supply of gas be increased, the pressure inside the 
 flame would be reduced, and this the film is unable to support. 
 
 8. The root of the flame is where the gas is moving fastest, 
 and consequently where the difference of pressures on the 
 
 two sides of the film would be 
 the greatest. It would conse- 
 quently be the most sensitive 
 part of the flame. 
 
 There is another peculiarity 
 about these flames, which has 
 been pointed out by Lord 
 Rayleigh. That is that the 
 flame is not symmetrically 
 affected ; or rather that it is 
 polar. If, for example, it is very sensitive from the side we 
 may call N, it is also very sensitive on the side S ; but much 
 less so on the sides E and W. This fact has been known 
 for a good many years, but Lord Rayleigh has never, that I 
 am aware of, offered any explanation of it. The cause would 
 appear to be that the particles of gas are issuing in a vortex 
 which is not circular, but slightly elliptical. It will be 
 evident that the flame (fig. 30) should be more sensitive in 
 the direction C D than in the direction A B. 
 
Singing Flame 
 
 91 
 
 Singing Flame 
 
 Let us take another very similar case. Why does the 
 well-known "singing flame" sing? The general arrange- 
 ment is shown in the sketch (fig. 31); and the explanation is 
 almost a repetition of what was said for the " sensitive flame," 
 with one exception : the sensitive flame roars, whilst the sing- 
 ing flame is more genteel and " sings." What one may call the 
 " trigger " of the arrangement is the air flowing 
 up past the flame. Partly by the friction 
 between this air and the flame, and partly by 
 the difference of pressure, the film over the flame 
 is broken and the flame gives a small roar. 
 When this takes place the pressure is equalised 
 across the tube and another film is formed, 
 which is in its turn broken. This action going 
 on very rapidly and periodically, a musical note 
 is produced, and the glass tube acting as a 
 resonator makes this audible. It is, in fact, a 
 kind of very rapid making and breaking of the 
 film, causing a periodic change of pressure in the 
 tube, which causes the waves of sound. That 
 the flame actually does go up and down — like 
 the sensitive flame, ov\y periodically — cannot, of 
 course, be seen by the unaided eye : with the help 
 of a rapidly revolving mirror, however, this is seen to be the case. 
 
 It may be argued that these two explanations are based on 
 the assumption (with a very fair share of reason, it is hoped) 
 that the pressure of the gas inside the flame was less than the 
 pressure outside. Whether this is true or not can easily be 
 decided by experiment. I regret that I have been unable to 
 get any information on this subject about the sensitive flame. 
 No one appears to know, and, apparently, no one has had the 
 curiosity to try the experiment, which requires delicate appa- 
 ratus. In the case of the "singing flame," however, thanks 
 to the courtesy of Mr Leonard Bairstow of the National 
 Physical Laboratory, I have been more successful. It 
 appears that the pressure inside the gas flame was from y^^ 
 
 Glass tube ^ 
 Fig. 31. 
 
92 A B C of Hydrodynamics 
 
 to 3-15^ of an inch of water measured negatively. At these 
 pressures the flame was only " humming." As the gas supply 
 was increased, the humming got louder and the pressure, 
 measured negatively, increased. All this thoroughly agrees 
 with what one would expect. 
 
 When the resonator was removed, the pressure went down 
 to jpyVs- of an inch. The pressure of the gas at the main was, 
 roughly (measured next day, only), plus y^^nnr of a" \'ac\\. 
 
 Since the apparatus for measuring the pressure damps the 
 vibrations, what was really measured was only the mean 
 pressure — say A B (fig. 32) — and not the maxima or minima 
 of the curve (see sketch). The minimum would consequently 
 
 Fig. 32. 
 
 be less than even this (greater, if measured negatively). If the 
 supply of gas had been increased, and the flame had been 
 actually " singing," instead of humming, it is reasonable to 
 suppose that the pressure, measured negatively, would have 
 been still greater. 
 
 If the pressure inside a singing flame is less than the atmo- 
 spheric, it is not, I hope, unreasonable to suppose that it would 
 be the same in the sensitive flame. 
 
 In these cases viscosity has not been called in to produce 
 the sinuous motion. 
 
 In the next chapter the motion of a liquid in a tube will be 
 considered. 
 
 Summary 
 
 For the formation of sinuous motion in a stream a cross 
 pressure is necessary — a pressure across the line of motion of 
 the stream. 
 
 The pressure inside the " sensitive " and " singing " flames 
 is less than the atmospheric pressure. 
 
 REFERENCES 
 As before. 
 
CHAPTER XI 
 
 LIQUIDS MOVING IN TUBES — POISEUILLE'S LAW — 
 RING VORTICES 
 
 In Professor Osborne Reynolds' list of " circumstances con- 
 ducive to sinuous or unsteady motion," he gives (No. 6) 
 " solid boundary walls." We will examine this next. If 
 we imagine the sketch (fig. 33) to represent a solid boundary 
 wall, which is supposed to be perfectly wetted, and a stream of 
 water flowing past it. Since the wall is perfectly wetted, 
 there will be a thin layer of water adhering to it. This 
 
 water has been shown as at rest ; 
 
 but this maybe taken as actually 
 so, or, if moving at all, as moving 
 so slowly that it may be re- 
 
 •' , . 1 Water at rest 
 
 earded as at rest relatively to 
 
 the boundary wall. At fairly ^^^^^^^ 
 
 . . , .,, , Solid Boundary Wall 
 
 high velocities there will be a 
 
 free surface between this layer 
 
 and the moving water. From what has been said previously, 
 
 it will be evident that the water " at rest " is under greater 
 
 pressure than the moving water and that a flow will take 
 
 place as shown. Eddies or sinuous motion are represented 
 
 as starting. 
 
 If we next imagine the boundary wall to be a small 
 piece of a tube, the action will, of course, be the same, and 
 we can, in imagination, see the water in the tube breaking 
 into small whirls or eddies. Since the formation of these 
 causes inertia resistance, the expression for R changes from 
 R = AV to R = AV+BV2. 
 
 93 
 
94 A B C of Hydrodynamics 
 
 As previously pointed out,! Dr Young appears to have 
 been the first to state that the resistance "could not be 
 represented by any single power of the velocity," but that 
 "it appeared to consist of two parts, the one varying 
 simply as the velocity, the other as the square." His 
 formula was — 
 
 f=a-,v'--\-2c-,v. 
 a a 
 
 Girard made very many experiments and found the same 
 thing. Y{\s formule generale was — 
 
 4/ 
 
 where " le premier terme au, de la force retardatrice, repre- 
 sente la portion de la resistance qui est due d. la cohesion 
 des conches fiuides entre elks [italics added] tandis que le 
 deuxieme terme bi^ represente la portion de la resistance 
 qui est produite par les asperit^s de la surface sur laquelle 
 le fluide se meut" (Institut de France, 18 16). 
 
 So many other writers in the early part of the nineteenth 
 century appear to have arrived at the same conclusion, that 
 Poncelet et Lesbros (Capitaines du G6nie) in 1832 wrote 
 {Savans Stranger, vol. iii.) : " Le mouvement uniforme de I'eau 
 dans les canaux et les tuyaux de conduite reguliers, d'une 
 grande longueur, a pu dtre soumis au calcul dans ces derniers 
 temps, de la maniere la plus keureuse et la plus satisfaisante, 
 par MM. Girard, de Prony, Navier et Etylwein ; les formules 
 ainsi obtenues ne semblent plus rien laisser a desirer du c6t6 
 de V exactitude des applications " [italics added]. 
 
 It will not appear necessary for me to labour this point 
 any more. 
 
 It may occur to anyone who reflects on the matter that 
 since the water in a lead pipe is in a violent state of turmoil, 
 caused by the eddies and rotational movement, it very 
 naturally issues from the tap somewhat as in sketch A (fig. 34). 
 The water is not acting at all like a liquid, but as a solid, 
 or series of solid particles. The particles strike any object 
 1 Chap. V. p. 35. 
 
Liquids Moving in Tubes 
 
 95 
 
 they meet with considerable violence and rebound from 
 it, causing what is commonly called "splashing." Possibly 
 some of the spreading of the particles may be due to 
 electrical conditions, but it is chiefly caused by the particles 
 of water striking against one another. 
 
 Now, if we could, by some means or other, " comb all the 
 curl" out of this water and cause each small filament to 
 issue at. the same velocity, the result should be quite different 
 and more in accord with the stream-line theory. This is 
 
 i" , , •' 
 
 IVMMM 
 
 
 '■•^''/ !fe'/' 
 
 ;-)' 
 
 //■,■/ 
 
 '/;/■; 
 
 
 Fig. 34. 
 
 done daily in many a scullery by means of the well-known 
 "anti-splash" tap. The essentials of this tap are that the 
 water is caused to pass through a very fine screen with a 
 piece of sponge above it. As is well known, the water 
 issuing from this tap has the appearance of a glass rod, 
 and when it falls on any object it does not splash (fig. 34, B). 
 
 POISEUILLE'S Law 
 
 So far we have only considered liquids flowing in ordinary 
 tubes, at ordinary speeds. When the tubes are very fine, 
 or the velocity of flow exceedingly small, then the resistance 
 
96 A B C of Hydrodynamics 
 
 varies as the velocity only. In these cases there is no " dis- 
 continuity" in the liquid, and the flow may be considered 
 as due to the motion of an infinite number of infinitely 
 thin co-axial cylindrical elements, the outermost of which 
 is at rest, in contact with the inner surface of the tube ; 
 the next moving with small velocity inside this one ; the 
 next with greater velocity inside the last, and so on to the 
 axis, where the velocity is a maximum.. 
 
 In the middle of the nineteenth century, Poiseuille, who 
 was a doctor, being desirous of finding out the velocity of 
 flow in the small veins of animals, made a series of most 
 remarkable experiments on the flow of water in fine glass 
 tubes, which he published in the Compies Rendus, vols. xi. 
 and xii. With tubes of about •14 millimetre he found 
 that with a constant pressure the discharge varied as the 
 fourth power of the diameter and inversely as the length of the 
 tube. This is now known as " Poiseuille's law." Poiseuille 
 also put the law in another way : " La vitesse dans les tubes 
 de tres-petit diametre est proportionnelle a la pression, en 
 raison inverse de leurs longueurs, et proportionnelle au 
 carre de leurs diametres " (Poiseuille, Comptes Rendus, xi.). 
 
 On the assumption of the flow made above, it is not 
 difficult to show that, theoretically, the mean velocity of 
 the liquid in the tube must vary as the square of the diameter 
 of the tube ; and consequently the discharge will be as 
 ih& fourth power of the diameter, which was what Poiseuille 
 found experimentally. He points out another curious thing, 
 and that is, that whilst the tube was 51 mm. in length, 
 the relation between time of discharge and pressures always 
 held good; but when it was reduced in length to 25-55 mm., 
 157s mm., 9-55 mm., and 675 mm., "cette relation n'a 
 plus lieu ; les temps, pour des pressions de plus en plus 
 considerables, sent plus grands que ceux que donnerait la 
 relation dont il est question " (Poiseuille). That is to say, 
 the resistance is greater than it ought to be according to 
 his law. The reason for this is, of course, that the velocity, 
 exceeding the "critical," discontinuity has commenced and, 
 ergo, sinuous motion. We have therefore the curious fact 
 
Poiseuilles Law 97 
 
 that the resistance decreases as the tube is shortened; but 
 at a certain length the resistance suddenly increases, as we 
 should expect. He also points out that the addition of 
 iodide of potassium reduces the resistance — and consequently 
 time of flow of a fixed quantity of liquid — very considerably. 
 This is true for all amounts even up to one-third salt and 
 two-thirds water (by weight). Many other salts act in the 
 same manner, viz., bromide of potassium, nitrates of potassium 
 and ammonium, chlorides of ammonium and potassium, 
 cyanide of potassium, and acetate of ammonium. Other 
 salts act in the opposite manner, viz., nitrate of sodium, 
 chloride of sodium, sulphates of potassium and ammonium, 
 and phosphates of potassium and ammonium. Some salts 
 act in one way at certain temperatures and in the opposite 
 manner at other temperatures. It is ordinarily taught that 
 these salts increase or decrease the viscosity. It is difficult 
 to believe that a very syrupy liquid is less viscous than 
 water. It appears more probable that the liquids have 
 their adhesion to the glass decreased or increased. 
 
 In Chapter IX. the question of a liquid slipping on a 
 boundary wall was discussed, and it was said that this would 
 be referred to again after the explanation of Poiseuille's law. 
 I quoted Professor Lamb as having stated that this law of 
 Poiseuille furnished "conclusive proof" that the liquid did 
 not slip "to any appreciable extent," and I gave reasons 
 showing that another explanation of the facts appeared to 
 me to be equally satisfactory. The subject will here be 
 considered in another manner. 
 
 Poiseuille's law is that the discharge through a small tube 
 
 d^ 
 at constant pressure = A-j . This, however, is not the whole 
 
 law, for there are certain corrections required for temperature, 
 and when these are added the formula becomes 
 
 D = A^(i+aT-|-/3T2), 
 
 where T = temperature in degrees Centigrade, and a and ^ 
 are constants. 
 
98 A B C of Hydrodynamics 
 
 If there were no slipping, the law connecting viscosity and 
 temperature ought to be expressed by 
 
 where ^ui is the coefficient of viscosity at temperature T (Cent.), 
 which does not appear to be the case. 
 
 " O. E. Meyer finds that for a considerable number of 
 liquids the simpler formula ^, = ^„(i +«0"^ answers well enough 
 for all practical purposes" [>? is the coefficient of viscosity] 
 (Castell-Evans' Physico-chemical Tables). 
 
 An examination of a list of the viscosities of over a hundred 
 different liquids at different temperatures (made by Rellstab 
 and Pribram, and Handl) shows that this law of Meyer's is 
 exactly true for all temperatures from io° C. to 50° C. in the 
 case of the following liquids : — acetone, ethyl ether, methyl 
 acetate, methyl alcohol, methyl butyrate, methyl valerate, 
 propyl acetate {n), propyl (e) acetate, propyl (n) iodide, 
 propyl (n) nitrite, propyl (?) propionate, and valeral. 
 
 If we consider the effect caused by the attraction of the 
 different layers of the liquid to the tube, these peculiarities 
 appear to me to be capable of a satisfactory explanation. 
 
 Heat appears to act on the liquid in two different ways : — 
 
 (i) It reduces the viscosity, or treacliness, of the liquid, 
 and so the flow is increased by some small fraction. 
 
 (2) It reduces the attraction of the liquid to the walls of 
 the tube, and so increases the effective bore of the tube, either 
 by allowing the liquid to flow more freely through the tube, 
 or by allowing the liquid to slip along the walls. 
 
 The law of (i) is not very exactly known, but there appear 
 to be very fair reasons for supposing that it is something 
 like yUT = /xo(i+aT)"\ when the discharge would increase as 
 some function of ^t first power of the temperature. 
 
 Since in (2) the temperature acts by practically enlarging 
 the effective bore of the tube, if we put this enlargement of the 
 diameter as (i +j8T), the discharge will be increased by some 
 function of the square of the temperature, something like 
 
 D = A^*(i+/3T)2, 
 
Poiseuille s Law 99 
 
 and if both actions be taken into account the formula should 
 be 
 
 D = ^0+aT)(I+;8T)^ 
 
 or 
 
 = ~(^ 1 + AT + A'T2 + A"T3). 
 
 The term A"T* is so small as to be negligible, so (follow- 
 ing Poiseuille) if we omit this we get 
 
 D = A^'(i-|-AT-|-A'T2). 
 
 It would appear, therefore, that the rational expression of 
 Poiseuille's formula should be 
 
 D = A' 
 
 ^(l+aT)(l+/3T)2, 
 
 where a and /8 are not necessarily interdependent. 
 
 As the matter is important I will put the case in another 
 way and apply a crucial test. It is quite commonly stated 
 that the viscosity of alcohol is very much greater than that 
 of water. This is not strictly correct. What is meant by the 
 statement is, that alcohol and water flows less freely through 
 a tube than water does. Dubuat was most cautious in his 
 statement of this fact : " L'esprit de vin coule sensiblement 
 moins vite que I'eau a cause de sa viscosite, ou de sa plus 
 grande adherence aux parois" [emphasis added]. The truth is 
 that the coefficients of viscosity of water and (pure) alcohol are 
 almost exactly the same. The " most probable value " of that of 
 water, at 0° C, being •0181456, whilst the coefficient for alcohol, 
 at the same temperature, is "018430, the difference is not very 
 great. If, however, you mix water and alcohol, you get a liquid 
 which flows much more slowly through a fine tube, and is there- 
 fore said to have much greater viscosity (?). If the mixture 
 contains 35'ii percent, of alcohol, the coefficient of viscosity 
 (?) is m,ore than three times as great, viz., "05703 at 10° C. ; and 
 probably greater than this at 0° C. By adding more alcohol 
 and making the proportion 49 per cent., the viscosity (?) falls 
 
 G2 
 
lOO A B C of Hydrodynamics 
 
 to •04133 at 10° C. At 70 per cent, it is still lower, being- 
 only -03279 at the same temperature. " Traube found in every 
 case [? at all temperatures] the maximum viscosity fell between 
 40 and 50 per cent, (inclusive)" (Castell- Evans' Physico- 
 chemical Tables). 
 
 It is difficult to believe that by mixing two liquids of small 
 viscosity you will produce a liquid which has such an enormous 
 increase of viscosity — cases where chemical combination 
 takes place, must, of course, be excluded. The correct 
 explanation would appear to be that the mixture has much 
 greater attraction to the glass and so "chokes" the tube. 
 
 It may be thought that this is a case of substituting one 
 improbability for another. That is not so. It is well known 
 that if you mix alcohol and water they contract : the molecules 
 are closer together and their attraction to the walls of the 
 tube might be expected to increase. If this be the true ex- 
 planation, then the maximum attraction to the glass should 
 correspond with this m.aximum. contraction ; and this is found 
 to be the case. The m.aximum viscosity (?) is found in a 
 liquid containing nearly 50 per cent, of alcohol ; whilst the 
 maximum contraction is found in a liquid containing 52 per 
 cent, of alcohol. The agreement is very close for a first 
 approximation and the relation is regular. If you add alcohol 
 to water the viscosity (?) gradually increases until the mixture 
 contains 52 per cent, of alcohol ; on continuing the addition of 
 alcohol the viscosity (?) decreases with the increased proportion 
 of the spirit. It appears to be a curious fact that there is more 
 contraction when you add water toalcohol than if you add alcohol 
 to water. For example, if you add 10 parts of water to 90 parts 
 of alcohol, the contraction exceeds 175 per cent. ; whilst if you 
 add 10 parts of alcohol to 90 parts of water the contraction 
 is only 714 per cent, or considerably less than half that 
 in the other case. 
 
 A short time ago a chemist friend very kindly took the flow 
 of absolute and 50 per cent, alcohol through a Boverton 
 Redwood viscometer. He could find no appreciable difference ! 
 and yet the latter is supposed to have between three and four 
 times inore viscosity. 
 
Ring Vortices loi 
 
 Take another case. Fill a V-tube (fig. 35) about two-thirds 
 full of water and boil it well, sealing up the tube whilst the 
 water is boiling freely. Putting one arm horizontally and care- 
 fully tapping the tube — to allow the water to settle well down 
 in it — you can gently raise this arm, as in the sketch, and the 
 water will not flow out of it. Now, how is that water kept 
 up, except by its " stiction" to the tube ? (I must apologise 
 for this word, but it expresses forcibly what I mean.) If the 
 water is kept up by viscosity, and if it cannot slip on the 
 -walls of the tube, why should the 
 water, if the tube is tapped, fall 
 with great velocity ? 
 
 It is not pretended that the fore- 
 going is an absolute proof that a 
 liquid may slip along the walls of 
 a tube, but I certainly think that it 
 points to the probability that it does 
 
 so in some cases — and that the „ 
 
 Fig. 35. 
 
 effect may be completely masked. 
 
 In any case, it shows that the subject of viscosity requires 
 more work being done upon it : a new method of measuring 
 viscosity is much wanted. The view I have advanced gives 
 a rational explanation of the empirical law of Poiseuille 
 regulating flow through small tubes and temperature. 
 
 Ring Vortices 
 
 Let us now examine what should be the motion of a liquid 
 behind a thin lamina, which we will suppose to have rounded 
 edges. If A B (fig. 36) be a section of the lamina, the water 
 flowing past it will proceed at first in the directions A E and B F. 
 This may be taken as a case of Professor Osborne Reynolds 
 {No. 8): "Circumstances conducive to unsteady motion" — 
 viz., " Curvature with the velocity greatest on the inside." 
 At any reasonable velocity the liquid will be " cracked," and 
 a surface of discontinuity formed in the directions A E andB F. 
 It is not important to consider how far this surface extends, 
 nor is it necessary for my present purpose ; it is sufficient that 
 
I02 
 
 A B C of Hydrodynamics 
 
 it is formed. This free surface has, as usual, a film which is 
 capable of sustaining tension. At very low speeds we might 
 imagine the crack as extending for some distance — as in Dr 
 Hele-Shaw's experiment quoted previously ; at any ordinary 
 speeds, however, the film would break at once. Now, if we 
 suppose that when this takes place the water in the rear of 
 the lamina is at rest; the pressure at C being very much 
 greater than at A and B, a flow of liquid will commence 
 outwards, as shown by the arrows. The pressure at C 
 will then be reduced and a flow will commence from D to C. 
 
 Similarly, a flow towards D, as 
 shown by the arrows, and eventu- 
 ally a complete ring vortex will 
 be produced, without calling in 
 the aid of viscosity or liquid 
 friction. 
 
 The action has been described, 
 for simplicity, as taking place 
 step by step. The vortex would, 
 however, be formed by all these 
 flows taking place at the same 
 time, or, at least, in the most 
 rapid succession. It will be 
 evident that since this ring vortex has been produced without 
 the assistance of viscosity that it could equally well be formed 
 in water or in a " perfect" liquid. In the latter case it would 
 necessarily be an irrotational vortex, and consequently a 
 coreless vortex. In water it would not necessarily be an 
 irrotational vortex, but it would equally be a coreless one. 
 Further reasons for this last remark will be given later. 
 
 There are four objections which may be, very reasonably, 
 raised against this explanation. 
 
 1. No evidence has been produced that such a vortex 
 actually is formed behind a lamina: ordinarily one sees 
 nothing but a confused mass of eddies. 
 
 2. If a vortex were formed, since, practically, no work 
 would be required to keep it rotating, all resistance should 
 cease when the vortex was once formed. 
 
Ring Vortices 103 
 
 3. As a ring vortex is a complete entity (acting, as it does, 
 like an elastic solid), the pressure on the rear of the lamina 
 would be that transmitted, from the liquid in rear, through 
 the vortex. Such pressure would be the hydrostatic pressure ; 
 and, since it has been shown that the pressure on the front of 
 the lamina is less than this, the lamina should move forwards 
 — against the stream.. 
 
 4. That the formation of a vortex in a "perfect" liquid 
 would be equivalent to an act of creation ; that therefore no 
 vortex could be produced ; and, equally, if a vortex existed, 
 it would be impossible to destroy it. 
 
 These objections are all perfectly reasonable and will 
 require to be dealt with separately. 
 
 To consider No. i first. Can a vortex be formed behind 
 a lamina? An experiment of Osborne Reynolds was 
 described previously, where a small disc was suspended from 
 a float. When this was given a sudden push, and was released 
 the combination travelled along as if meeting with no resistance. 
 Further, if this combination were stopped, and released 
 instantly, it would restart itself. It was remarked that what 
 was taking place would be considered later. It will now 
 be evident that a vortex had been formed, and that this 
 vortex will explain what appeared mysterious. That this is 
 so can be easily made evident by placing a small quantity 
 of coloured liquid immediately in front of the lamina, before 
 moving it, when a very beautiful ring vortex will be made 
 evident by the coloured liquid which, owing to the motion of 
 the lamina, is now behind it. 
 
 Professor Osborne Reynolds describes this vortex as 
 " bearing something the same proportion to the visible ring 
 as a ball made by wrapping string (in and out) round a 
 circular curtain ring until the aperture was entirely filled up. 
 
 " The disc, when it was there, formed the front of this ball 
 or spheroid of water, but the rest of the surface of the ball 
 had nothing to separate it from the surrounding water but 
 its own integrity. Yet when the motion was very steady, 
 the surface of the ball was definite, and the entire moving 
 mass might be rendered visible by colour. The water within 
 
I04 A B C of Hydrodynamics 
 
 the ball was everywhere gyrating round the central ring, as 
 if the coils of string were each spinning round the curtain 
 ring as an axis, the water moving forwards through the 
 interior of the ring and backwards round the outside, the 
 velocity of gyration gradually diminishing as the distance 
 from the central ring is increased. 
 
 " The way in which the water moves to let the ball pass 
 can also be seen, either by streaking the water with colour 
 or suspending small balls in it. In moving to get out of the 
 way and let the ball of water pass, the surrounding water 
 partakes, as it were, of the gyratory motion of the water 
 within the ball, the particles moving in a horse-shoe fashion, 
 so that at the actual surface of the ball the motion of the 
 water outside is identical with that within, and there was no 
 rubbing at the surface and consequently no friction. 
 
 " It is impossible to have a ring in which \.h.e gyratory motion 
 is great and the velocity of progression slow. As the one 
 motion dies out so does the other, and any attempt to 
 accelerate the velocity of the ring by urging forward the disc 
 invariably destroys it" (Royal Institution, February 1877). 
 
 I have quoted this description very fully, as it gives a most 
 vivid and graphic picture of what takes place behind the 
 disc. The explanation which follows, however, hardly seems 
 to me quite satisfying. 
 
 " The striking ease with which the vortex ring, or the disc 
 with the vortex ring behind it, moves through the water 
 naturally raised the question as to why a solid should 
 experience resistance. Could it be that there was something 
 in the particular spheroidal shape of these balls of water 
 which allowed them to move freely ? To try this, a solid of 
 the same shape as the fluid ball was constructed and floated 
 after the same manner as the disc. But when this was set 
 in motion, it stopped directly — it would not move at all. 
 What was the cause of this resistance? Here were two 
 objects of the same shape and weight, the one of which 
 moved freely through the water, and the other experienced 
 
Ring Vortices 105 
 
 very great resistance. As already explained, there is no 
 friction at the surface of the water, whereas there must be 
 friction between the water and the solid. But it could 
 be easily shown that the resistance of the solid is much 
 greater than can be accounted for by its surface friction 
 or skin resistance. The only [?] other respect in which 
 these two surfaces differ is that the one is flexible while 
 the other is rigid, and this seems to be the cause of the differ- 
 ence in resistance. 
 
 " Colouring the water behind the solid shows that instead 
 of passing through the water without disturbing it, there is 
 a very great disturbance in its wake" (Royal Institution, 
 February 1877). 
 
 As stated previously, this explanation is hardly convincing. 
 I will endeavour to give what I think is the true explanation 
 later, after discussing " Helmholtz rings." 
 
 We have now data to enable us to reply to objections i, 2, 
 and 3. 
 
 1. Vortices are formed behind a circular lamina. 
 
 2. When the vortex is formed, the resistance actually does 
 cease, the disc moving with no appearance of resistance. This 
 is only half the truth, however, for " if the speed of the disc 
 were maintained imiform., the ring gradually dropped behind 
 and broke up" (Osborne Reynolds). When, therefore, the 
 disc is fixed and the stream moves uniformly past it, or if the 
 disc be moved uniformly through the water, the vortices are 
 being perpetually " sloughed off." Other vortices are then 
 formed, to be sloughed off in their turn, and so on. It is this 
 continual formation of vortices .which causes the inertia 
 resistance, and the breaking of these which causes the 
 appearance of " eddies," or the confused turmoil of the 
 water. 
 
 We may imagine the action of " sloughing off" the vortices 
 to be somewhat as follows : — The disc commences to move 
 a "little too fast" for the vortex, which consequently lags 
 behind slightly. A fresh crack is formed in the liquid at A and 
 
io6 
 
 A B C of Hydrodynamics 
 
 B (fig. 37), just in front of the vortex. A new vortex is formed 
 
 in the liquid, as shown in the sketch, and this " peels " the old 
 
 one off the disc, destroying it in 
 the process : just as you can peel 
 off soap-bubbles. 
 
 It must be remembered that 
 though this vortex may be said 
 (popularly) to be able to propel 
 itself, it will only do so at its 
 own pace, which is not, neces- 
 sarily, the pace you wish it to 
 move at. 
 
 These three objections may 
 
 therefore be considered as having been satisfactorily answered. 
 
 The question of the formation of vortices in a perfect liquid 
 
 will be dealt with in the next chapter. 
 
 Fig. 37. 
 
 Summary 
 
 Liquids flowing in tubes are subject to the same law of 
 resistance R=AV-|-BV^. If the tubes are very small 
 (capillary), or the velocity of flow is very low, then " inertia 
 resistance" (dynamic resistance) does not occur, and the 
 resistance may be expressed as R = AV only. 
 
 In the case of capillary tubes the discharge of liquid taking 
 place through them (with a fixed pressure) varies as ^^ fourth 
 power of the diameters of the tubes and inversely as their length. 
 
 Some salts added to water cause it to flow more freely through 
 the tubes : other salts act in exactly the opposite manner. 
 
 When a circular disc moves in water ring vortices are 
 formed behind it. These are only stable if the disc is moved 
 with a " staccato movement " and then released. They will 
 only persist if allowed to move at their own speed. 
 
 REFERENCES 
 
 PoiSEUiLLE, Comptes Rendus, xi. and xii. 
 ,, M(m, des Sav. Strangers, ix. 
 
CHAPTER XII 
 
 VORTICES IN A PERFECT LIQUID — HELMHOLTZ RINGS 
 
 Just as it is commonly taught that a body moving in a 
 perfect liquid would meet with no resistance, so, and prob- 
 ably for the same reasons, it is taught that it would be im- 
 possible to produce any vortices in a perfect liquid. Dr 
 Fleming is quite clear on this point, for he says ( Waves and 
 Ripples) : " If they [eddies] were created [in a perfect fluid], 
 they would continue for ever, and have something of the 
 permanence of material substances." 
 
 This was, for many years, the view held by Lord Kelvin, 
 for in 1869 he wrote: "The simple circular Helmholtz ring 
 is a case of stable steady motion, with energy maximum — 
 minimum for given vorticity and given impulse. A circular 
 vortex ring, with an inner irrotational annular core, surrounded 
 by a rotationally moving annular shell (or endless tube), with 
 irrotational circulation outside all, is a case of motion which 
 is steady, if the outer and inner contours of the irrotational 
 shell are properly shaped, but certainly unstable " if the shell 
 be too thin. It was essentially on this view that he framed 
 his famous theory of matter. In 1875 he still held the view- 
 that to make a vortex in an inviscid fluid would be equivalent 
 to an act of creation ; but in the PAi'l Mag., 1 880, he admits 
 that "hitherto I have not indeed succeeded in rigorously 
 demonstrating the stability of the Helmholtz ring in any case" 
 [italics added]. In 1887 he apparently changed his views, for 
 he published his classical paper " On the Formation of Coreless 
 Vortices by the Motion of a Solid through an Inviscid 
 
 Incompressible Fluid " (Phil. Mag., vol. xxiii.). Finally, 
 
 107 
 
io8 A B C of Hydrodynamics 
 
 in 1904 ( Waves in Water), he frankly says : " After many years 
 of failure to prove that the motion in the ordinary Helmholtz 
 circular ring is stable, I came to the conclusion that it is 
 essentially unstable, and that its fate must be to become 
 dissipated as now described." He had finally, with great 
 regret, abandoned his vortex theory of matter. 
 
 It is not possible to experiment with a perfect liquid ; so 
 that what it would or would not do is a good deal a matter 
 of opinion — or perhaps, more correctly, it depends on the 
 properties you postulate. In all that has been said here 
 about the vortex ring, it will be noticed that viscosity has 
 been ignored. The vortex was not considered as (i) formed 
 by the aid of viscosity ; (2) retarded by viscosity, there being 
 no friction between the water flowing round it and the vortex ; 
 and (3) it was not dissipated by viscosity. There appears, 
 therefore, no reason why similar vortices should not be formed, 
 in a similar manner, in a perfect liquid, if. such a thing existed. 
 Of course the perfect liquid must possess all the properties of 
 an ordinary liquid, with the single exception that its particles 
 cannot be rotated. Without this assumption the perfect liquid 
 could only be considered by engineers a " curiosity," the 
 most suitable place for which would be a Mathematical 
 Museum. 
 
 Helmholtz Rings 
 
 These rings are probably about the easiest form of vortex 
 to experiment with, and are commonly known as " smoke 
 rings." They can be seen occasionally coming out of the 
 funnel of a locomotive, out of the mouth of a smoker or the 
 muzzles of heavy guns — the old 40 pr. Armstrong sometimes 
 produced very fine ones of a lemon-chrome tint. As is well 
 known, they are most commonly formed by means of a box 
 with a round hole in front and an elastic back (fig. 38). On 
 filling the box with smoke and striking A B sharply, a smoke 
 ring issues from C D, as in the sketch. 
 
 Dr Fleming ( Waves and Ripples') describes the action very 
 graphically : " The motion of the air or smoke particles com- 
 posing the ring is like that of an indiarubber umbrella ring 
 
Fig. 38. 
 
 Helmholtz Rings 109 
 
 fitted tightly on a round ruler and pushed along." It must 
 be remembered, however, that, though this gives a most 
 graphic idea of the motion, it is in reality exactly the reverse. 
 The motion described would be as at A (fig. 39), which repre- 
 sents a central section 
 of an umbrella ring per- 
 pendicular to its plane, 
 whereas the real move- 
 ment is as in B (fig. 39). 
 If it were as in A, the 
 ring would not travel 
 forwards (as it does), 
 but would tend to go 
 backwards : it is im- 
 portant to remember that this ring vortex propels itself. 
 How this is effected will be considered later. 
 
 How is this smoke ring, or ring vortex, formed? Dr 
 Fleming says : " This rotary motion is set up by the friction 
 of the smoky air against the hole in the box, as the puff of 
 air emerges from it when the back of the box is thumped." 
 
 This description is 
 not very full, but the 
 impression conveyed 
 by it is that (i) a 
 couple is formed by 
 the friction of the 
 moving air, and (2} 
 that the smoke ring 
 is formed in the open- 
 ing of the box. It is not difificult to show that the smoke 
 ring is not formed in this opening, but outside the box 
 altogether. 
 
 Now, if the vortical motion were caused by the friction of 
 the air against the hole in the box, we should expect that 
 the greater the friction the better would be the vortex, and 
 vice versa. Such is not the case, for the very best rings can 
 be produced when the side of the box is very thin and the 
 hole has very sharp edges. In this case the friction would be 
 
 Fig. 39. 
 
no 
 
 A B C of Hydrodynamics 
 
 quite insignificant. Reusch has also shown (Pogg. Ann., ex.) 
 that if a tube, or nozzle, be fixed to the hole in the box, the 
 ring formation goes on very well until the length of the tube 
 is five times the diameter — then the rings are not formed at 
 
 Cohured liquid 
 
 Fig. 40. 
 
 ^ 'Va/er tank 
 Fig. 41. 
 
 ^^~^\:k: "•■■■■■■; 
 
 " " " ^" ~'.-:^^~i-^s'^^$^$§i^ 
 
 
 
 
 ^^^^^^^^ 
 
 all. In this case we have very great increase of friction, but 
 no smoke ring. 
 
 The correct explanation would appear to be that the ring 
 is formed outside the box and clear of the opening. The 
 sketch (fig. 40) and what was said about the formation of the 
 
 vortex behind the lamina will 
 explain clearly the action in 
 forming these " smoke rings." 
 The reference has been 
 made to these Helmholtz 
 rings in air, because the ex- 
 periment can be so easily 
 repeated. Professor Osborne Reynolds produced them in a 
 tank of water by means of the arrangement in the sketch 
 (fig. 41). They behave exactly as do the smoke rings, 
 though they do not travel so fast. 
 
 Probably the foregoing is the reason why heavy guns are 
 corroded at A and B (fig. 43) after much firing. 
 
 In the formation of smoke rings, by means of a smoke box, 
 it is essential that a sudden blow be given to the back of 
 the box, so as to cause discontinuity of the fluid. It is not 
 
 Fig. 42. 
 
Helmholtz Rings 1 1 1 
 
 specially necessary that the opening should be very round. 
 Reusch has shown that circular rings can be produced when 
 the opening is triangular, square, or even a not over-long 
 rectangle. With a rectangular opening of 2 : i round rings 
 are always produced ; but if the proportion of the sides be 
 4 or 5:1, then two rings are formed. When the proportion 
 reaches 6 or 7 : i, with care, three rings can be produced, the 
 central ring being frequently crippled. 
 
 When the smoke ring has been formed and is moving 
 away, the elastic back of the box swinging back causes a 
 " rarefaction " in the box, and air rushes in from the outside. 
 Another surface of discontinuity is formed inside the box, and 
 an " air ring " is formed which travels backwards in the box. 
 We have thus a " smoke ring " travelling in clean air away 
 fr;om the box ; whilst a clean " air ring " travels through the 
 smoky air in the opposite direction. This very curious fact, 
 which was first pointed out by Reusch, can easily be seen if 
 the box is transparent and the smoke not too dense. No 
 English authors, that I am aware of, appear to have observed 
 this. 
 
 When you look at an ordinary smoke ring, you really only 
 see part of it — what I may call its skeleton. The outer part, 
 being composed of clean air, is not visible. If you put smoke 
 outside the box (in front of the hole) instead of inside, then 
 quite a different view will be obtained. The ring will not be 
 seen, but the outer part of the vortex will ; and this will show 
 that it is spheroidal in shape, reminding one somewhat of 
 the dandelion seed. This very curious fact was apparently 
 first pointed out by Sir Robert Ball (Royal Dublin Society) 
 in May 1868. He caused "air rings" (smoke rings without 
 smoke) to issue from a "smoke box," then to pass through 
 smoke, outside the box. He described the result as follows : — 
 " The air ring had penetrated the smoke uninjured ; it had not 
 apparently left any of its particles behind, nor had it admitted 
 an atom of smoke into it ; but it had drawn with it sufficient 
 of the smoke to form a complete shell, which enclosed it, and 
 thus rendered the air visible [italics added]. . . . The appear- 
 ance is one of great beauty and suggested the name ' negative 
 
X 1 2 A B C of Hydrodynamics 
 
 smoke ring.'"^ It will now be apparent that there is no 
 essential difference between the small vortex formed behind 
 Osborne Reynolds' disc and the vortex formed by what 
 Reusch calls the " staccato movement " of a fluid through a 
 circular opening. Both are " spherical vortices." 
 
 Sir Robert Ball measured the velocity of these air rings at 
 different parts of their flight, and found that the velocity 
 increased until they had got about 4 or 5 feet from the 
 smoke box ; the speed then remained fairly constant for some 
 distance, after which it decreased. Osborne Reynolds found 
 practically the same thing when he made these rings in water, 
 for he says : " It is found that the force of the blow they will 
 strike is nearly independent of the distance of the object 
 struck from the orifice." 
 
 I am not aware of any explanation having been offered of 
 this curious motion, so I venture the following. 
 
 If we refer to the sketch of the vortex formed behind the 
 lamina and remember what was said about its formation, it 
 will be evident that cyclic energy has been generated behind 
 the disc. Since it is impossible to create, or destroy, energy ; 
 this energy can only have been generated at the expense of 
 the kinetic energy of the stream-lines; their \pv'' has been 
 reduced, and they will consequently be travelling more slowly 
 round the rear of the vortex than they did in front of it. They 
 will therefore exert more pressure there. Then, since the 
 vortex is a complete entity, with greater pressure behind it 
 than in front, it will travel forwards with a gradually acceler- 
 ating velocity. The vortex travels faster and faster for a 
 certain distance and then the velocity becomes steady. All 
 this while, it may be said, familiarly, that the vortex is sucking 
 kinetic energy out of the stream-line. Whilst it is doing this 
 to increase its velocity, it has also been absorbing fluid from the 
 stream-line so as to increase its size. These two actions are 
 opposed to one another. At first the velocity increases, but 
 as the size also increases, after a certain time the acceleration 
 
 1 The term "negative smoke ring" would appear to be more suitable 
 for Reusch's rings, which are really rings, and which travel in the opposite 
 direction to the ordinary ring. 
 
Helmholtz Rings 113 
 
 ceases, and the velocity remains constant; ultimately the 
 vortex gets "too fat" and the cyclic velocity diminishes. 
 Remembering Professor Osborne Reynolds' statement of the 
 connection between the gyratory motion and the velocity of 
 progression, it is evident that as the gyratory motion decreases 
 so will the velocity of progression, and the vortex will event- 
 ually be dissipated. We now see why the disc in Osborne 
 Reynolds' experiment, previously referred to, when stopped 
 and released instantly, starts itself again. 
 
 That these vortices increase in size is confirmed by 
 Osborne Reynolds, when referring to those made in water : 
 " Their velocity gradually diminishes ; but this would appear 
 to be accounted for by their growth in size, for they are thus 
 continually taking up fresh water into their constitution, with 
 which they have to share their velocity." 
 
 The question of the smoke ring having a maximum velocity 
 has been specially referred to, as Professor Osborne Reynolds 
 (who is generally right) says, in reference to ring vortices in 
 water : " These rings are much more definite than smoke 
 rings, and although they cannot move with higher velocities, 
 since that of the smoke ring is unlimited . . ." This cannot be 
 admitted, and it certainly does not appear to be confirmed by 
 experiment. It must be remembered that this statement was 
 made in 1877 — a very long time ago. If the explanation of 
 the formation of the vortex be correct, it will be evident that 
 the strength of the vortex is regulated by the pressure in the 
 still water. As this is fixed by the depth of immersion 
 + atmospheric pressure, it will be obvious that there is a 
 maximum gyratory motion, and, ergo, progressive motion — 
 which cannot possibly be exceeded. We might indeed imagine 
 an immersion of infinite depth \ but — and it is a very big but 
 — at this immersion a crack in the fluid could not be made, 
 and the vortex could never be formed. 
 
 These vortices are always coreless. It is difficult to produce 
 actual proof of this ; but here again an experiment of Osborne 
 Reynolds supports the statement : " Yet a still more striking 
 spectacle may be shown, if, instead of coloured water, a few 
 bubbles of air be injected into the box from which the puff is 
 
 H 
 
114 A B C of Hydrodynamics 
 
 sent ; a beautiful ring of air is seen to shoot along through 
 the water. . . ." (Royal Institution, 1877). This " ring of air " 
 would appear to be the core — vacant of water — of the vortex 
 
 ring- 
 There is another argument from a theoretical point of 
 view. I have been informed by a professor, who has' 
 made a speciality of vortices, that an ordinary vortex (in 
 the strict mathematical sense) can be cut across, whilst a 
 coreless vortex cannot. It is very well known that you 
 cannot cut a "smoke ring." In any case it appears very 
 fairly certain that no crack in fluid, no vortex; and no vortex, 
 no inertia resistance. 
 
 The whole subject of smoke rings is a most fascinating one. 
 To watch the first stages of the motion, when at very low 
 velocity the smoke emerges from the hole in the form of a 
 mushroom ; how this mushroom develops into a sort of 
 " ionic capital " ; to the perfect smoke ring, when the fluid is 
 cracked by a sudden blow on the box: all this is most 
 beautiful. The reader must, however, be referred to Reusch, 
 as well as Rogers {American Journal of Science and Arts, 
 vol. xxvi., 1858), for further details. 
 
 I cannot do better, in closing this chapter, than finish with 
 a quotation from Professor Osborne Reynolds : 
 
 " That the vortex takes a systematic part in almost every 
 form of fluid motion was now evident. Any irregular 
 solid moving through the water must from its angles send 
 off lines of vortices such as those behind the oblique 
 vane. As we move about we must be continually causing 
 vortex rings and vortex bands in the air. Most of these 
 will probably be irregular, and resemble more the curls in 
 a smoke cloud than systematic rings. But from our mouths 
 as we talk we must produce numberless rings" (Royal In- 
 stitution, 1877). 
 
 In this chapter the words " vortex " and " ring vortex " have 
 been employed. What was meant will, I trust, have been 
 fairly evident ; but to prevent any misunderstanding, it may 
 be well to be precise and to give a little more explanation, 
 especially as " coreless vortices " are not well understood and 
 
Helmholtz Rings 115 
 
 they differ from the ordinary mathematicians' vortex referred 
 to in their treatises. To begin with the latter:— Let us 
 imagine the sketch (fig. 43) to represent, diagrammatically, 
 the section of a vortex and of its liquid cylinders. 
 
 " The velocity of the fluid is everywhere inversely as the 
 length of its path of flow ; consequently, if we suppose the 
 cylinder be made smaller, the velocity at its surface will be 
 proportionally greater, so that in the limit, if we suppose 
 
 Fig. 43. 
 
 the cylinder to become evanescent, the velocity becomes 
 infinite." 
 
 " Such a motion is known as vortex motion, and the system 
 figured constitutes a vortex filament. It will be seen that if 
 r represents the radius of the path of flow and v the cor- 
 responding velocity, z'r= constant, and if the angular velocity 
 <a = vlr, we have wr^ = constant — that is to say, for any circuit 
 of flow the area x angular velocity is constant, which is the 
 relation for vortex motion established ^^^^ra//?' by the theorem 
 of Helmholtz and Kelvin "_ (Lanchester, Aerodynamics). 
 
 This excellent description of Mr Lanchester's gives a very 
 
 H2 
 
1 16 A B C of Hydrodynamics 
 
 clear idea of the mathematicians' vortex, which, however,, 
 differs in some ways from the vortices which I have described 
 as being formed in water or air. In these latter the velocity 
 V is strictly limited by the energy (potential + kinetic) of the 
 stream-line. Since z/r= constant, it also follows that r is 
 limited. If we imagine v, in the diagram (fig. 43), to represent 
 the section of the cylinder having the maximum velocity 
 attainable, all the shaded part, inside v, will become void 
 of water and be a " hollow core." This is not necessarily 
 a vacuum, since it may be filled with air, or water vapour — 
 the essential is that it is void of water: in the "smoke ring" 
 it is full of smoke. All the remainder of the vortex will, 
 of course, obey the law t/r= constant. This will, I hope, 
 give a clear idea of what I mean when I speak of a " core- 
 less vortex." 
 
 The mathematicians' vortex is imaginable \ but it would, I 
 think, be beyond the power of human imagination to conceive 
 how it could be formed. It would require the consumption 
 of infinite energy to form a single one. 
 
 In the next chapter we will see, by reference to Dubuat's 
 experiments, how far theory is confirmed by these experiments. 
 
 Summary 
 
 Vortex rings are formed by what Reusch calls a " staccato 
 movement '' of the fluid past a sharp edge. The word " sharp " 
 must be taken not in an absolute, but in a relative sense ; for 
 rings can be formed when the edge is not very sharp. The 
 fluid is cracked and the cross pressure assists in generating a 
 cyclic movement. The visible part of a smoke ring may be 
 described as the " skeleton " of the vortex, the complete 
 vortex being melon-shaped. Smoke rings cannot, ordinarily, 
 be formed when the air is driven through a tube whose 
 length exceeds five times its diameter. For every smoke 
 ring formed by a smoke-box, there is a "clear-air" ring 
 formed in the box, and which travels backwards through the 
 smoky air. 
 
 All these vortices, formed in water or air, are necessarily 
 
Helmholtz Rings 1 1 7 
 
 coreless ; but the hollow core is not necessarily a vacuum ; 
 it may be filled with air, water vapour, smoke, etc. 
 
 These " rings " cannot be cut ; they travel along at their 
 own speed, and are eventually dissipated. 
 
 REFERENCES 
 
 Sir Robert Ball (Royal Dublin Society, 1 868). 
 
 E. Reusch, Pogg. Ann., ex. 
 
 Professor W. B. Rogers, American Journal of Science and Arts, 
 
 1853. 
 
 Dr Thom.\s Young, Phil. Trans., 1800. 
 
CHAPTER XIII 
 
 MOTION OF WATER IN REAR OF A BODY EXPOSED TO A 
 STREAM — RESISTANCE AT VERY HIGH VELOCITIES 
 
 It was shown previously that if a plate was fixed and exposed 
 to a moving liquid — or if the body was moving and the liquid 
 was at rest — the pressure on the anterior face was less than 
 the hydrostatic. As the plate undoubtedly meets with resist- 
 ance, it is evident that the posterior pressure must be less than 
 that on the anterior face. Since, further, the pressure on the 
 anterior face diminishes as the velocity increases, it is also 
 evident that the posterior pressure must also diminish, and 
 more rapidly, than it does on the anterior face ; for experiment 
 shows that the resistance of the plate increases very largely as 
 the velocity increases. 
 
 To submit the matter to experiment, Dubuat made measure- 
 ments of the pressure on the posterior surface of a plate (fig. 
 44) by reversing his thin box in the stream, with the following 
 results : — 
 
 No 
 No 
 No 
 No 
 No. s 
 
 (Centre) 
 (Half-way) . 
 (Three-quarters) 
 (Close to edge) 
 (Bottom corner) 
 
 — 127 lignes 
 
 -I2-0 „ 
 
 - 14-9 ., 
 
 -1 57 .> 
 
 -15-3 „ 
 
 As was to be expected, the pressure is very much less than 
 
 it was on the anterior side. Attention may be directed, in 
 
 examination of these figures, to a curious thing : though the 
 
 pressure at the centre is less {measured negatively) than it is 
 
 at the edge — as one would suppose it should be — still, the 
 
 118 
 
Motion of Water in Rear of a Body 119 
 
 progression is not regular. This might be considered as due 
 to unavoidable experimental errors; but, against this, the 
 experiments appear to have been conducted with very great 
 care, and, also — this peculiarity recurs in other cases. What 
 appears evident is that the water at the rear of the plate was 
 in very rapid motion, and that the velocity opposite hole No. i 
 was greater than opposite hole No. 2. 
 
 Now what should we expect if the thin box were fixed 
 behind a cube A B C D (fig. 45) ? We know that the pressure 
 
 Fig. 44. 
 
 Fig. 45. 
 
 on the face of the cube is not, materially, different from that on 
 the thin plate. There appears no reason to suppose that there 
 should be any essential difference between the action at the 
 rear of the plate and at the rear of the cube. Of course the 
 water flowing along A C and B D must be somewhat retarded 
 by the friction ; so that the velocity of the stream-lines should 
 be rather less at C and D than it is at A and B. The vortex 
 formed at the back of C D ought, therefore, to be less strong 
 than the one formed behind the plate. 
 
 On placing the thin box behind the cube and facing towards 
 the rear, Dubuat found the following pressures : — 
 
 No. I. = 
 No. 2. = 
 
 — 7'2 lignes. 
 -7-1 .. 
 
 No. 3. = 
 No. 4. = 
 
 — 8"0 lignes. 
 -6-0 „ (?). 
 
I20 
 
 A B C of Hydrodynamics 
 
 This confirms what we should have expected from theoreti- 
 cal reasoning. The progression is again irregular, that at the 
 centre being in excess (measured negatively) of that opposite 
 hole No. 2, thus confirming the results obtained behind the 
 plate. The change of velocity (?) from 3 to 4 appears 
 quite inexplicable. Dubuat notices this irregularity, which 
 
 Fig. 46. 
 
 he calls " une marche tres irreguliere, et mSme contraire. 
 Mais cet effet peut avoir k.th produit par une faible devia- 
 tion au trou du bord." It seems a pity that he did not 
 repeat the experiment more carefully. 
 
 Dubuat fixed the thin box behind a parallelepiped of 
 proportions 3x1x1, and found pressures at — 
 
 No. I (fig. 44). = — i'5 lignes. ' No. 2. = — 3"2 lignes. 
 
 These figures tend to show that the strength of the vortex 
 was considerably less than that behind the cube. 
 
Motion of Water in Rear of a Body 121 
 
 Judging from the pressures measured by Dubuat on the 
 rear face of the plate, and remembering that any change in / 
 involves an opposite change in V^, the vortex behind the 
 plate must be something like, though possibly differing from, 
 the sketch, fig. 46. It would be a form of what Professor 
 Lamb would appear to call a " spherical vortex," with the core 
 very close to the edge 
 of the lamina. Since 
 sketching out this vortex 
 from theoretical reason- 
 ing, I have become 
 aware that Avanzini in 
 1804 {Istituto Nazionale 
 Italiano, Tomo i.. Parte 
 i) found experimentally 
 that the water actually 
 moves in this manner. 
 By an arrangement of 
 silk threads attached to 
 fine wires he found the 
 direction of flow at very 
 many points and then 
 drew the sketch, in fig. 
 47 (traced from the 
 original), of the vortical 
 motion. The agreement 
 between theory and ex- 
 periment is most close. 
 
 It was pointed out 
 that in Dubuat's experiments the pressure at the back of the 
 plate was " irregular " ; that the pressure {measured negatively) 
 increased from the centre to the edge, but that it commenced 
 by decreasing, so that the negative pressure at the centre was 
 greater than it was at hole No. 2 (fig. 44). This was very 
 puzzling at first, as the only possible explanation was that the 
 water was in more rapid motion in the centre than half-way 
 towards the edge. If one considers, however, that the pressures 
 recorded are those when the vortex has just broken — when the 
 
 Fig. 47. 
 
122 
 
 A B C of Hydrodynamics 
 
 »- 
 
 vortex is formed it acts as an elastic solid — it will be evident 
 that as the water flowing through the centre of the ring 
 (fig. 48) has to spread out right and left {viewed in section, 
 but in reality in all directions), it should be flowing faster in 
 the centre than when it has " turned the corner," so to speak ; 
 and this would account for the differences of pressure. This 
 is not insisted on, but it appears a reasonable explanation ; 
 the records appear reliable, and this peculiarity recurs.^ 
 
 It is hardly necessary to point out that the pressure 
 
 measured behind the 
 plate is less {measured 
 negatively) than it ought 
 to be. Since vortices 
 are being rapidly formed 
 and broken, the pressure 
 is not a constant one, 
 but is varying rapidly. 
 This variation is damped, 
 and it is the " damped 
 pressure" which is re- 
 corded and not the real 
 negative pressure. 
 
 Since the strength of 
 the vortex is dependent on the velocity of the stream-lines 
 passing the edge of the plate, and this is dependent on the 
 velocity of the stream, it will be apparent that the resistance 
 caused by forming these vortices will vary as the density of the 
 liquid and as the square of the velocity of the stream. The 
 total resistance of the plate may, therefore, be expressed by 
 the formula already discussed, i.e. R = A V + BV^. 
 
 If we refer to Beaufoy's experiments on the resistance of a 
 square flat plate: area = 2'97i8 square feet; immersed 9 feet 
 below a float. The figures given are the resistance of the 
 combination less the resistance of the float and iron bar 
 supporting the plate. 
 
 ' Fig. 48 has been sketched from a model made by winding tarred 
 string round a hollow curtain ring ; the hollow in the ring representing 
 the empty " core." 
 
 f-FLocu 
 
 Fig. 48. 
 
Resistance at Very High Velocities 123 
 
 Resistance of a Square Flat Plate 
 
 Area = 2'97i8 sq. ft. Immersion g ft. 
 
 R = Total resistance less that of float and bar. 
 
 V. 
 
 Resistance 
 
 R 
 
 Resistance 
 
 Ft. per 
 Second. 
 
 (observed). 
 
 V^2- 
 
 (calculated). 
 A = -26, 6 = 3-515. 
 
 I 
 
 3-5925 
 
 3-5925 
 
 3-765 
 
 -> 
 
 14-312 
 
 3-578 
 
 14-56 
 
 3 
 
 32130 
 
 3-570 
 
 32-38 
 
 4 
 
 57-017 
 
 3-563 
 
 57-24 
 
 5 
 
 88-96 
 
 3-558 
 
 89-12 
 
 6 
 
 127-94 
 
 3-554 
 
 128-04 
 
 7 
 
 173-96 
 
 3-55 
 
 173-98 
 
 8 
 
 227-00 
 
 3-547 
 
 226-96 
 
 9 
 
 287-03 
 
 3-54 
 
 286-96 
 
 10 
 
 354-09 
 
 3-54 
 
 354-00 
 
 II 
 
 428-14 
 
 3-538 
 
 428-06 
 
 12 
 
 509-19 
 
 3-536 
 
 509-16 
 
 13-527 
 
 546-36 
 
 3-532 
 
 546-55 
 
 A=-26, 3 = 3-515. 
 (Beaufoy's experiments.) 
 
 An examination of the table, giving observed and calculated 
 resistance, will show the correctness of the formula. 
 
 By a similar line of reasoning it can be shown that the 
 resistance of all bodies moving in liquids may be expressed 
 by the formula R= AV + BV^; but, as we have seen, in the 
 special cases where 
 
 1. The liquid {supposed incompressible) is of infinite extent, 
 
 or 
 
 2. Completely fills a closed tank, the walls of which are 
 
 inextensible ; 
 (These two might be classed under one heading " When 
 the liquid has no free surface," but are separated to 
 make my meaning clearer.) 
 
 3. When the motion is excessively slow ; 
 
 4. When the liquid is flowing through very fi,ne tubes, or 
 
 between plates very close to one another, 
 the inertia resistance does not exist. The term BV2 = o, 
 and the resistance = A V only — the resistance due to viscosity ; 
 
1 24 A B C of Hydrodynamics 
 
 which varies as thi shearing surface (not, as usually stated, as 
 the wetted surface). 
 
 A in the formula is not necessarily positive : there are cases 
 where the resistance may be expressed as R = BV^— AV. 
 For example, in Beaufoy's experiments (p. 474) we find the 
 resistances of a globe of I3"5 inches diameter encountered at 
 different velocities. If these be examined by the formula 
 R = AV + BV2 it will be found that A = -245 and B = -328 will 
 satisfy the equation. 
 
 When the sphere was lengthened by the insertion of a 
 middle piece 12 inches long the resistance was so much 
 reduced that A became =—-094 and B=2g. But the dis- 
 cussion of this point lies outside the scope of this book. 
 
 As there is a " critical velocity " delou/ which the resistance 
 
 varies as the velocity, so also there is a superior " critical 
 
 velocity " above which the resistance again increases in this 
 
 ratio.^ This, of course, follows logically from what has been 
 
 said about the " law of flow." Referring to the formula 
 
 '?^=p-\-\pv'^, we know /-|-^/oZ''' = constant, and that/ cannot 
 
 be negative. It follows that v has a maximum value, beyond 
 
 which it cannot increase. The maximum value of v, in a 
 
 gravitating fluid, varies as the depth of immersion ; thus, if, at 
 
 a given depth, the total pressure, hydrostatic and atmospheric, 
 
 is Pq, then the maximum value of v for this depth is found 
 
 from the equation Jyo2;^ = P„ {j> having become zero) or 
 
 /2P 
 z'= / — - When the velocity of the stream-lines, moving 
 
 past the lamina, has reached this maximum, the strength of 
 the vortex — and, ergo, the minimum pressure on the back of 
 the lamina — will remain constant and the resistance will then 
 be expressed as = const. -|-AV, when the increase of resist- 
 ance is as the velocity. 
 
 This idea of the pressure remaining constant is not new, 
 for Dubuat says : 
 
 " Quand I'effet de la non-pression est parvenu, par un 
 certain degre de vitesse, a dgaler I'effort naturel de I'air, la 
 poupe doit Stre entierement vide, et le defaut de pression est 
 ■ C in fig. 18, p. 66. 
 
Resistance at Very High Velocities 125 
 
 parvenu a son plus haut degr6 ; alors quelque augmentation 
 que puisse recevoir la vitesse du mobile, la non-fression reste 
 constante" [italics added]. 
 
 Experiment will be found to confirm this view. Some 
 years ago, through the favour of Mr Yarrow, I was enabled 
 to examine the H.P. speed curves of the torpedo boat 
 destroyers Havock and Sokol. I found the " critical speed " 
 was between twenty-four and twenty-five knots; beyond 
 these speeds the resistance appeared to increase as the 
 velocity, the increase in H.P. being very nearly the same for 
 each knot above this speed. Theoretically the critical velocity 
 should have been more than this speed; but then theory 
 assumes the pressure in the vortices to be zero, which, practi- 
 cally, it never is. There is always a residual pressure. 
 
 This law holds also for air, where the '' critical velocity " 
 appears to be about 1 100 feet per second — less than theory 
 again. In the case of a bullet: "Beyond noo feet per 
 second we may take P being in pounds, d the diameter 
 of the shot in feet, v the velocity in feet per second, 
 F = »^(z;— 800), where/'=3 for spherical and 2 for elongated 
 shots with ogee-shaped heads " (Perry, Applied Mechanics), 
 the resistance increasing as the velocity. In parentheses I 
 may remark that if Stokes' law holds for high velocities — 
 and there appears no reason why it should not — the resistance 
 should vary as the first power of d and not as the second : 
 certainly for the spherical shot, and I think equally for the 
 elongated projectile. 
 
 Still more remarkable than this, I am inclined to believe 
 (though I have much hesitation in making the statement) 
 that at still higher velocities the resistance may even increase 
 at a slower rate. There is indeed no h priori reason why 
 the resistance due to viscosity should continue increasing 
 indefinitely. Some of Professor Langley's experiments seem 
 to support this view, for he says : 
 
 " Further than this [referring to the possibilities of flight], 
 these experiments (and the theory also when reviewed in 
 their light) show that if, in such aerial motion, there be 
 given a plane of fixed size and weight, inclined at such 
 
126 A B C of Hydrodynamics 
 
 an angle, and moved forward at such speed, that it shall 
 be sustained in horizontal flight, then the more rapid the 
 motion the less will be the power required to support and 
 advance it. This statement may, I am aware, present an 
 appearance so paradoxical that the reader may ask himself 
 if he has rightly understood it. To make the meaning 
 quite indubitable, let me repeat it in another form, and 
 say that the experiments show that a definite amount of 
 power so expended at any constant rate will attain more 
 economical results at high speeds than at low speeds, e.g. 
 one horse-power thus employed will transport a larger 
 weight at 20 miles an hour than at 10, a still larger at 
 40 than 20, and so on, with an increasing economy of power 
 with each higher speed, up to some remote limit not yet 
 attained in experiment, but probably represented by higher 
 speeds than have as yet been reached in any other mode 
 of transport " (Langley, Aerodynamics). It is not necessary, 
 however, to press this point. 
 
 From the foregoing it will be evident that since inertia 
 resistance is caused by the formation of vortices, and these 
 vortices are produced at the edges or angles of a body 
 moving in the liquid ; then, the more sharp edges a body 
 has the greater will be the resistance to its motion. In 
 other words, a plate having more periphery — in proportion 
 to its area — ought to experience greater resistance per unit 
 ■of area than a plate of more compact form. It is well 
 known that such is the case. 
 
 There are cases where resistance is an advantage. Mr 
 John Bourne, C.E. {Screw Propellers), says: "In order that 
 a vessel may be able to sail very close to the wind, the 
 surface of the sail should be quite flat, and the sails should 
 have holes in them, or be made like a Venetian blind, as 
 the sails of vessels are made in China." I have been 
 informed that in China some of the boats have rudders 
 which are perforated. Mr Alexander has made a small 
 model screw propeller with small holes in the blades. He 
 says it gives a greater thrust than one without holes, though 
 it obviously takes more power to drive it. Even 400 years 
 
Resistance at Very High Velocities 127 
 
 ago that great genius, Leonardo da Vinci, was convinced 
 of the advantages of having small holes {sportellini is the 
 word he used) in the wings of flying machines. No birds' 
 wings are air-tight ; bats, whose wings are, do not appear 
 to be able to fly as steadily and gracefully as birds. 
 
 In conclusion, it will be seen that, if examined under similar 
 conditions, the motion of an ordinary liquid does not diff^er 
 very much from that of an inviscid liquid, if such a thing 
 existed. The difficulties of the subject are chiefly caused 
 by the assumptions made by the mathematicians, and the 
 consequences leading from these assumptions. 
 
 1. The ideal liquid is supposed to be continuous. This 
 leads to the maze of the infinite negative pressure at every 
 sharp edge, which, of course, involves the assumption that 
 the liquid is capable of sustaining infinite tension. This 
 assumption appears unnecessary, and, I am afraid, against 
 common sense. To assure that the flow shall be continuous, 
 it is only necessary to have a sufficient vis a tergo — sufficient 
 "push from behind" — to keep it so. If the "push" be not 
 sufficient, the liquid (which I assume with a free surface), 
 having inertia, will " tear " away and cause " discontinuity." 
 
 2. The next assumption is " incompressibility." This leads 
 to even a worse tangle. In the first place, the deduction is 
 that a stress would be transmitted instantly to an infinite 
 distance; per contra, however, the stress could not be trans- 
 mitted at all. How can one molecule transfer energy to 
 another incompressible molecule} What, for example, would 
 be the result if an incompressible shot were to strike an 
 incompressible target ? 
 
 3. Inviscidity. — It is difficult to find out what is 
 generally meant by the term " viscosity." If it means treacli- 
 ness or stickiness, then it implies attraction between the 
 molecules of the liquid. " Inviscidity " would then imply that 
 there is no attraction between the molecules. Yet this same 
 inviscid liquid is supposed to be capable of being under 
 infinite negative pressure ! 
 
 I am aware that some physicists define viscosity as "a 
 kinetic effect, due to the interpenetration, in course of time, 
 
128 A B C of Hydrodynamics 
 
 of the various layers of molecules." This description, however, 
 though very soothing from its profundity, leaves no very clear 
 impression on my mind. 
 
 4. Infinite size of the liquid. — This has been dealt with 
 at some length, for it appears to be the chief cause of most 
 of the puzzles put before the student. 
 
 The subject of flat plates moving at an angle in liquids, 
 the screw propeller (which is really only a modification of 
 this), and some other peculiarities of liquid motion must be 
 deferred to some future occasion. 
 
 Summary 
 
 The resistance experienced by bodies moving in liquids 
 is chiefly caused by the formation of vortices behind them ; 
 these vortices are being continuously formed and as con- 
 tinuously " peeled off"" and destroyed by the new vortices 
 formed in front of them. In the case of a circular lamina 
 the vortices are of a spherical form, with the core very near 
 the edge of the lamina, the particles of liquid, moving round 
 the core, travelling in elliptical (?) orbits, like comets moving 
 round the sun. 
 
 Since " discontinuity '' is necessary for the formation of the 
 vortex, it follows that if there is sufficient vis a tergo, or " push 
 from behind," in the liquid to enable it to get round the 
 corners of the body nicely, there will be no inertia resistance. 
 
 The laws that apply to inviscid liquids apply equally to 
 viscid ones — special reservation made of the question of 
 " rotation " in the sense specially defined by me. 
 
 There are no special laws which apply to "flat bodies" 
 which do not equally apply to "round," or even "ichthyoid" 
 bodies. 
 
 REFERENCE 
 J. Bourne, Screw Propellers. 
 
APPENDIX 
 
 It has been suggested to me that in these days, when everyone 
 appears to be more interested in Aerodynamics than in Hydro- 
 dynamics, it might be useful to indicate the connection between 
 the law of the resistance of liquids — incompressible fluids — and 
 compressible fluids, like air. 
 
 It is commonly taught that (certainly up to speeds of loo miles 
 per hour) the laws of the resistance of liquids and gases is the same 
 — subject, of course, to the densities being roughly as 800 : i, 
 and the coefficients of viscosity being different. Dr Stanton 
 (A.C.A., 1909-10) says: "In the case of models placed in a 
 current of air, the resistance is found to be proportional to the 
 square of the speed within ordinary limits, so that there is not the 
 same necessity to make the tests on models at corresponding 
 speeds." Mr F. W. Lanchester {A.C.A., 1909-10) says: "... In 
 connection with flight, or, more broadly speaking, with aerial navi- 
 gation, this effect of the compressibility of the fluid can be ignored." 
 
 It is true that the same laws apply when compression does not 
 take place; but it is certainly not true that "the effect of com- 
 pressibility can be ignored." In other words, under similar con- 
 ditions it is true, but not otherwise. 
 
 To make my meaning perfectly clear, I will first refer to cases 
 where the conditions are similar, and then to cases where they 
 are not. 
 
 (i) '^ The surf ace friction of thin plates. — The most reliable work 
 
 in this direction appears to have been done by Zahm. . . . The 
 
 resistances bear a striking resemblance to those obtained by 
 
 Mr W. Froude (Table II.) for plates in water, in that the resistance 
 
 per square foot diminishes with the length, and for smooth surfaces 
 
 varies as V'*^ where V is the velocity" (Dr Stanton, A.C.A., 
 
 1909-10). 
 
 129 I 
 
130 A B C of Hydrodynamics 
 
 In this case compression does not occur, or, to be accurate, only 
 to an extent which is apparently negligible; the conditions are 
 exactly similar. It is unnecessary to labour this case, for if the 
 reader will refer to Chapter VIII., whatever was said for water will 
 apply equally to air. What I have likened to the "bull-roarer" 
 action is produced, and this accounts for the "resistance per square 
 foot diminishing with the length." The resistance varies as 
 AV+BV2. 
 
 (2) When compression occurs this law no longer holds good ; 
 another term has to be addedj when R = AV + BV^ + CV^. 
 
 In the bluebook oxi Aeronautics for 1909-10 (which for brevity 
 I refer to as A.C.A), Lord Rayleigh has a note on "Dynamical 
 Similarity," in which we find the formula 
 
 "P = P-'^ •/£,)•• A" 
 
 for the pressure on a plate in normal presentation, which he gives 
 as " the expression for the mean force per unit area normal to the 
 plate." 
 
 !< = kinematic viscosity, /= length of side of plate, & = velocity, 
 
 and " where/. ( —\ is an arbitrary function of the one variable v/w/. 
 
 It is for experiment to determine the form of this function, or in 
 the alternative to show that the facts cannot be represented at all 
 by an equation of form (A)" {A.C.A., 1909-10). 
 
 In the report {A.C.A., 1910-11) we read: "Messrs Bairstow and 
 Booth have shown that a formula can be found [emphasis mine] 
 falling under the general type indicated by Lord Rayleigh, which 
 accurately represents the results, both of Eiffel and Stanton, over 
 the whole range to which their experiments extended, when both 
 the dimensions of the plate used and the air velocity at which 
 the results were obtained are taken into account." 
 
 "The question is one which is at present mainly of theoretical 
 interest, and the importance of which lies in the light it may throw 
 on the comparison of water and air resistances^' 
 
 In the appendices to the report (A.C.A., 1910-11), when 
 referring to resistance of square plates in normal presentation, 
 Messrs Bairstow and Booth say : " After some trial and error, it 
 appeared that a formula of the type F = a(w/)2 + b(^rf was a 
 satisfactory empirical approximation." There is nothing specially 
 
Appendix 131 
 
 new about this type of formula, for it was employed by Colonel 
 Duchemin, as far back as 1842, for the resistance of projectiles in 
 the Memorial dArtilkrie.^ It was also used by Zahm {Phil. Mag., 
 1901) in his paper on the resistance of bullets. 
 
 On page 29 {A.C.A.), Dr Stanton gives another formula, 
 
 for the law of resistance in rough and smooth pipes. He adds : 
 " It is interesting to notice that the dimensional relation for these 
 artificially roughened pipes is precisely similar to that found by 
 Mr Bairstow and Mr Booth, in regard to the experiments -on the 
 normal resistance of flat plates of different sizes, i.e. the relation 
 necessitates the introductio7i of the third term in the above equation," 
 
 Since this equation of Messrs Bairstow and Booth refers to unit 
 of area, we may put the whole force on the plate as 
 
 If I may then take the liberty of changing the v in the last term to 
 
 p (for reasons which will be obvious — it will only necessitate 
 
 change in the value of B, which is a constant), and substituting 
 /i/p for V, we then get 
 
 F = /xA(z;/) + (,k{vlf + ^{vlf. 
 
 (Stokes' law.) (Newton's law.) (Term represent- 
 
 ing resistance due 
 to compression.) 
 
 or, very generally, as previously stated, 
 
 R = AV-|-BV2 + CV3. 
 
 Since the first term depends on the "shearing area," it will be 
 exceedingly small for a thin plate, and this formula will not differ 
 very much (in this case) from 
 
 R = BV2-t-CV3. 
 
 Why the resistance due to compression should vary as P" and as 
 ifi, I do not propose to enter into at present ; it has, in reality, 
 
 ^ Not in the British Museum, and I know of only one copy in England. 
 
 I 2 
 
132 A B C of Hydrodynamics 
 
 nothing to do with Hydrodynamics, which treats of liquids which 
 are practically incompressible. 
 
 Expressed in the same "dynamical similarity" notation, the 
 resistance of liquids, and what is commonly called the "friction" 
 of flat surfaces in gases, 
 
 or 
 
 R = M(^») + pW'vf 
 
 (Stokes' law.) (Newton's law.) 
 
INDEX 
 
 Air and water, resistances compared, 
 
 129. 
 Alcohol, coefficient of viscosity, 99. 
 
 — and water, attraction for glass, 100. 
 
 contraction on mixing, 100. 
 
 viscosity of, 100. 
 
 Allen's experiments examined, 69. 
 
 — law (Lanchester), 69. 
 Anti-splash tap, action of, 95. 
 Atomiser, action of, 22. 
 Augmented wetted surface defined, 30. 
 Avanzini's experiment, 121. 
 
 Bairstow and Booth's formula, 131. 
 
 — on negative righting moment, 54. 
 Ball, Sir Robert, smoke ring experi- 
 ments, III. 
 
 Ball-nozzle, action of, 53. 
 Bazin's experiment, 60. 
 Beaufoy's experiments on lengthened 
 sphere, 124. 
 
 on sphere, 124. 
 
 on square plates, 122. 
 
 — shape of bows of ship, 59. 
 Bows of ship, correct shape for, 59. 
 Bullet, resistance of, 125. 
 
 (Zahm), 131. 
 
 Child's experiment with half balls, 24. 
 Coulomb's experiments, 35, 65. 
 Critical speed of destroyers, 125. 
 of liquids, 42. 
 
 Definitions, augmented wetted surface, 
 
 — discontinuity, 49. 
 
 — dynamic resistance, 34. 
 
 — electric law of flow, 2. 
 
 — flow by obligation, 9 
 
 — fluid (Newton), 18. 
 
 — inertia resistance, 34. 
 
 — negative pressure, 3. 
 
 — perfect liquid, 5. 
 
 — rotary and rotational, 7. 
 
 — sinuous motion, 84. 
 
 — steady motion (Kelvin), 21. 
 
 — vortex (Kelvin), 84. 
 
 Disc, distribution of pressure on, 55. 
 Discontinuity (Stokes), 3. 
 
 Discontinuity defined, 49. 
 
 — surfaces of, 41, 45. 
 
 D\ibuat, on viscosity of alcohol, 99. 
 
 — experiments, 55. 
 
 on square plates, 118. 
 
 Duchemin's formula for resistance of 
 
 projectiles, 131. 
 Dynamic resistance, 106. 
 
 defined, 34. 
 
 Dynamical similarity (Raleigh), 1 30. 
 
 Eddies, formation of, 42, 84, 85. 
 
 — prevented by viscosity, 44. 
 Electric flow, Froude, 29. 
 
 — law of flow (Helniholtz), 2. 
 
 Fleming on liquid of infinite extent, 9. 
 
 — on "perfect liquid," 4. 
 ■ — on resistance, 27. 
 
 — on sensitive flame, 87, 88. 
 
 — on smoke rings, 108. 
 Flow " by obligation," 43. 
 defined, 9. 
 
 Fluid, definition {Ency. Brit.), 18. 
 (Newton), 18. 
 
 — friction and viscosity, 33. 
 Friction of water. Perry on, 34-35- 
 
 — Kelvin, 36. 
 
 Froude on electric flow, 29. 
 
 — on perfect fluid, 29. 
 
 — on resistance, 27. 
 
 — experiments of, 62. 
 
 Girard's formula, 94. 
 
 Heat, action of, on liquid, 98. 
 Hele-Shaw on "perfect liquid," 5. 
 
 — experiments of, 43. 
 
 Helmholtz on hydrodynamical theory, 
 
 41- 
 
 — rings, 108. 
 
 explained (Kelvin), 107. 
 
 unstable (Kelvin), 108. 
 
 " Helmholtz-Kirchhoff" flow, 45. 
 
 Inertia resistance caused by vortices, 
 105. 
 
 defined, 34. 
 
 in tubes, 93. 
 
 133 
 
134 
 
 A B C of Hydrodynamics 
 
 Injector, action of, 24. 
 Inviscid liquid and ordinary liquid com- 
 pared, 127. 
 
 Kelvin on " discontinuity," 3. 
 
 — on electric law of flow, 2. 
 
 — on friction, 36. 
 
 — on Helmholtz rings, 107. 
 
 — on influence of gravity, 16. 
 
 — on instability of Helmholtz rings, 108. 
 
 — on resistance, 27. 
 
 — on resistance and velocity, 36, 
 
 Lanchester, principle of no momentum, 
 12. 
 
 — on hydrodynamic theory, 7. 
 
 — on resistance and velocity, 37. 
 
 — on resistance of bodies in water, 58. 
 
 — on resistance of bodies in air, 129. 
 
 — on V-squared law, 37. 
 
 — on vortex motion, 115- 
 Langley, experiments on flight, 125. 
 Law of flow, 18. 
 
 Liquids do not obey Newton's second 
 law of motion, 20. 
 
 — moving in tubes, 93. 
 
 Meyer's formula, 98. 
 Movement of a liquid, 18. 
 
 Negative pressure defined, 3. 
 
 — righting moment, 54- 
 Newton on resistance, 26. 
 Newton's law (Lanchester), 69. 
 
 Osborne Reynolds, experiment with 
 lamina, 57. 
 
 resistance and velocity, 35. 
 
 sinuous motion, 38, 40. 
 
 steady motion, 38, 40. 
 
 Perfect fluid (Froude), 29. 
 • — liquid and ordinary liquid compared, 
 6, 127. 
 
 defined, 5. 
 
 Perry on friction of water, 34-35- 
 
 — on resistance of bullet, 125. 
 Phillips' experiment with sand, 24. 
 Poiseuille's experiments, 96. 
 
 — law, 95. 
 
 Poncelet et Lesbros, on resistance in 
 
 tubes, 94. 
 " Potential cleavage " flow, 45. 
 
 objections to, 47, 48. 
 
 Pressure, effect on viscosity, 41. 
 
 — relation to velocity, 22. 
 
 — on square lamina (Dubuat), 118. 
 Principle of no momentum (Lanchester), 
 
 12. 
 
 Projectile, resistance of (Duchemin), 
 
 Rankine on action of propellers, 28. 
 Rayleigh on dynamical similarity, 130. 
 
 perfect fluid, i. 
 
 resistance of fluid, I. 
 
 Relation of pressure and velocity, 22. 
 Resistance and velocity. Coulomb, 35. 
 
 Coulomb's experiments, 65. 
 
 Froude's experiments, 62. 
 
 Kelvin, 36. 
 
 Lanchester, 37. 
 
 Osborne Reynolds, 35. 
 
 Perry, 34-35. 
 
 Poiseuille, 35. 
 
 — — Stokes, 35. 
 Young, 35. 
 
 — due to viscosity, 62. 
 
 — Fleming's view, 27. 
 
 — Froude's view, 27. 
 
 — in liquid, caused by generation of 
 
 cyclic momentum, 12. 
 
 — Kelvin on, 27. 
 
 — Newton's view, 26. 
 
 — of bullets, 125. 
 Zahm, 131. 
 
 — of projectiles (Duchemin), 131. 
 
 — of water and air compared, 129. 
 
 — view of the mathematicians, 26. 
 Reusch, experiments on smoke rings, 
 
 no. 
 Ring vortices, loi. 
 Rotary and rotational defined, 7. 
 
 Scent spray, action of, 22. 
 Sensitive flame, 87. 
 
 explained, 89. 
 
 Shape of least resistance (Bazin), 60. 
 Singing flame, 91. 
 Sinuous motion defined, 84. 
 
 of Osborne Reynolds, 38, 40. 
 
 Skin friction, 74. 
 
 methods of reducing, 77. 
 
 Smoke rings, 108. 
 
 Fleming, 108. 
 
 formation explained, no. 
 
 Reusch's experiments, no. 
 
 Sir Robert Ball'sexperiments, III. 
 
 Sphere, resistance of (Beaufoy), 1 24. 
 Squirt, action of, explained, 19. 
 Stanton, resistance of bodies in air, 129. 
 Stanton's formula, 131. 
 Steady motion of Lord Kelvin. 21. 
 
 of Osborne Reynolds, 38, 40. 
 
 Stokes on discontinuity, 3. 
 
 — law, 73. 
 
 explanation of, 79. 
 
 j (Lanchester), 69. 
 
 — resistance and velocity, 35. 
 
Index 
 
 135 
 
 Surface friction of thin plates in air 
 (Zahm), 129. 
 
 — of discontinuity, 41, 45. 
 
 Torpedo-boat destroyers, critical speed, 
 125. 
 
 — head, shape of, 59. 
 Traube, on viscosity, 100. 
 Tubes, resistance in, 93. 
 
 V-squared law (Lanchester), 37. 
 Velocity, relation to pressure, 22, 
 Venturi's experiment, 23. 
 Viscosity, action of, 74. 
 
 — and fluid friction, 33. 
 
 — and resistance, 29, 30. 
 
 — difficulty of measurement, 8 1 . 
 
 — law of resistance due to, 62. 
 
 — of alcohol, 99. 
 
 41. 
 
 Viscosity of alcohol and water, 100. 
 
 — of water, 99. 
 
 — prevents eddies, 44. 
 Vortex defined, 84. 
 
 — method of formation, 102. 
 
 — motion (Lanchester), 115. 
 Vortices, ring, loi. 
 
 — cause of inertia resistance, 105. 
 
 Water and air, resistance compared, 
 129. 
 
 — coetticient of viscosity, 99. 
 
 Young, resistance and velocity, 35. 
 Young's formula, 94. 
 
 Zahm, experiments on surface friction, 
 129. 
 
 — on resistance of bullets, 131. 
 
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