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The Uniform Motion of. a Sphere through a 
 Viscous Liquid 
 
 t A THESIS 
 
 Pfescnted to the Facalty of the Graduate School of Cornell University in Conformity 
 with the Requirements for the Degree of Doctor of Philosophy 
 
 BY 
 
 ROBERT WILBUR BURGESS 
 
 ■> Eeprinted from 
 Ambbican Jouenal of Mathematics 
 Vol. XXXVIII, Nq. 1 
 , January, 1916 
 
The Uniform Motion of a Sphere through a 
 Viscous Liquid 
 
 A THESIS 
 
 Presented to the Faculty of the Graduate School of Cornell University in Conformity 
 with the Requirements for the Degree of Doctor of Philosophy 
 
 BY 
 
 ROBERT WILBUR BURGESS 
 
 Eeprinted from 
 
 Amebican Jouenal of Mathematics 
 
 Vol. XXXVIII, No. 1 
 
 January, 1916 
 
?•/ 
 
 The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 By E. W. Buegess. 
 
 1. The problem of finding at any point in a viscous liquid the velocity 
 due to the uniform rectilinear motion of a sphere was first attacked by Stokes 
 in 1851 and an approximate solution obtained. From this was derived the 
 well-known Stokes' law, that a force SnfiWa is required to maintain a uniform 
 velocity TF of a sphere of radius a through a liquid whose coefficient of vis- 
 cosity is {I. This solution was proposed merely as a limiting form to which 
 the distribution of velocities may be supposed to tend as W approaches the 
 limit zero, but has nevertheless proved itself well in accord with experimental 
 facts, at least as a close first approximation. The serious mathematical incom- 
 pleteness of the solution has been recognized, however, and several attempts 
 have been made to improve on it. Whitehead, in 1888, attempted to modify 
 this solution by taking into account terms depending on squares of velocities, 
 but found it impossible by his method to satisfy the boundary conditions at 
 infinity. In 1893, Rayleigh showed that Stokes' solution would be accurate if 
 certain additional forces were introduced, the magnitude of the required forces 
 furnishing some indication of the error of the solution. In 1911, Oseen found 
 a solution which is satisfactory at infinity; it also gives a good approximation 
 near the sphere if the velocity or the radius of the sphere is small. Shortly 
 afterwards Lamb, using a different method, presented Oseen's results in a 
 simpler form, but still subject to the same limitations. In 1913, F. Noether 
 attempted to take into consideration the terms neglected by Stokes, but merely 
 derived Whitehead's correction over again. A few months later Oseen took up 
 the same problem and tried to apply the principle of his previous solution, but 
 fell into the same error that he had carefully pointed out in Whitehead's work. 
 In the following paper Oseen's results are obtained by a simpler method than 
 that of either Oseen or Lamb. The method is generalized, so as to give a solu- 
 tion that satisfies the boundary conditions at the sphere to any desired degree 
 of approximation. A correction is determined which takes account of all the 
 terms quadratic in velocities, and differs from the corrections of Whitehead, 
 Noether and Oseen in that it satisfies the boundary conditions at infinity. 
 11 
 
82 BuEGBSs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 I wish to express my thanks and appreciation for advice and assistance 
 given me by Professor E. R. Sharpe in the preparation of this paper. 
 
 2. Using cylindrical coordinates {r, s),\e\, u,w denote the comp'onents of 
 
 the velocity, p the pressure, d the density, ^ ( = ^j the kinematic coefficient of 
 
 viscosity, and V the potential of the impressed forces. The differential equa- 
 tions for the motion of a viscous liquid, in the case when the Z-axis is an axis 
 of symmetry, are* 
 
 du , .. 9m , _. du _dQ , _ /_2 , u 
 
 where 
 
 dt dr dz dr \ r 
 
 dw , dw , dw dQ , _, 
 ^+^-37 + ^^ = ^+^^^' 
 
 
 (1) 
 
 Q^-V-p/d. 
 
 Eliminating Q from these equations, we obtain 
 
 9 /du du , dw\ d /dw , dw , dw 
 
 a^U +" ai; + *" -ajh si »■ +""37 + " ^ 
 
 i§ih-^)-rM <^' 
 
 This equation is simplified by making use of the equation of continuity t 
 
 If we put 
 
 du u dw _ 
 dr r dz 
 
 1 d'^ 1 94- ,o^ 
 
 U= 5-, W = — 5—, (d) 
 
 r dz r or 
 
 this relation is satisfied ; hence the velocities may be found by differentiation 
 from this function 1^, known as Stokes' current function, and the problem is 
 reduced to the determination of i//. The curves of intersection of planes through 
 the ^-axis with the surface 4'= a constant, will be the lines of flow or stream 
 lines ; that is, the lines indicating the direction of the resultant velocity at any 
 point. This system of curves provides a convenient method of representing 
 graphically the motion given by any function ■4^. 
 
 * Bassett, "Hydrodynamics," Vol. II, p. 244. f Bassett, "Hydrodynamics," Vol. I, pp. 8, 12. 
 
BuEGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 83 
 Substituting in (2) the values of u and w given in (3), we have* 
 
 where 
 
 dr^ r dr dz' 
 
 The preceding equation applies to any case in which the motion of the 
 liquid is symmetrical with respect to the s-azis. We desire to discuss in this 
 paper the motion due to a sphere which has been moving for an infinite time 
 with uniform velocity W along the is-axis, and whose center is at (0, Wt) at 
 time t. Since the conditions at a point depend only on its position relative to 
 the sphere, 4' is evidently a function of r and z — Wt only. If z'=z—Wt, we 
 have 
 
 d'^' _ 3i|' 32!' _ d^^ d"\/ _ d-^ dz' _ d'4/ 
 
 dz dz' dz dz' ' dt dz' dt ~ dz' ' 
 
 Equation (4) then becomes 
 
 («^ + (^_TF)A_^_^Z>)z?^=0, (5) 
 
 a form on which we shall base our further discussion. 
 
 3. The problem we shall consider is the determination of the function ■»// 
 of r and z' which satisfies (5) and is such that the velocity of the liquid at the 
 surface of the sphere is the same as the velocity of the sphere, and that the 
 velocity of the liquid at an infinite distance from the sphere vanishes. This 
 problem was first attacked by Stokesf in 1851 and solved, approximately, for 
 the case in which the liquid is very viscous ; that is, v large, and the veloci- 
 ties sufficiently small so that the terms involving the products of velocities and 
 accelerations are comparatively unimportant. On these assumptions the 
 approximate form of (5) is 
 
 vD-D^^O, (6) 
 
 for which we may easily obtain Stokes' solution 
 
 \a p / 
 where a is the radius of the sphere, p=Vr^+z^ and 
 
 tan0=-. (7) 
 
 z 
 
 * Cf. Bassett, " Hydrodynamics," Vol. II, p. 259. 
 
 f Cambridge Transactions, Vol. IX (1851) ; Sci. Papers, Vol. Ill, p. 55. 
 
84 BuEGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 From this is derived Stokes' well-known formula F=6nfiWa for the force 
 required to maintain the motion. This formula has proved so accurate for 
 small velocities that the approximations by which it was obtained have not 
 always been fully recognized. Stokes himself, however, proposed the solution 
 (7) only as a limiting form to which the velocities may be supposed to tend as 
 W approaches zero. 
 
 In 1888, Whitehead * attempted to modify this solution to take account of 
 the neglected quadratic terms. The attempt was successful in finding correction 
 terms satisfactory as far as the differential equation (5) and the boundary con- 
 ditions at the sphere were concerned, but these terms failed, as Whitehead 
 himself noticed, to satisfy the condition that the velocity at an infinite distance 
 vanishes. The correction terms for the velocities along and perpendicular to 
 the radius vector were found to be 
 
 E,= -jg— (--l)(^ + -+2JP.(cos0), 
 
 (8) 
 
 It is obvious that when p is infinite these terms do not vanish as required. 
 
 In 1893, Eayleight showed that Stokes' solution would be accurate if cer- 
 tain additional forces were introduced; the ratio of the forces thus required 
 to the viscous forces actually considered gives a means of estimating the accu- 
 racy of the approximation. This ratio can easily be shown to be — £ , which 
 
 V 
 
 increases indefinitely with p ; Stokes' solution is therefore not valid at infinity. 
 4. In 1911, Oseent introduced a new approximate solution, valid at 
 
 infinity. Consider the term {w — W) ^ in equation (5). Near the surface of 
 
 the sphere it is true that w — PF is very small, and the neglect of this term is 
 
 justified, but at any considerable distance it is not; in fact, w — W is very 
 
 nearly — W, not zero. Oseen shows that the ratio of the neglected terms to the 
 
 . Ws 
 terms giving the viscous forces is , which becomes infinite with z, no matter 
 
 V 
 
 how small we take W. On account of these considerations at a large distance 
 from the sphere, we take as an approximate form of (5) 
 
 [-W^-vD^D^=0. 
 
 (9) 
 
 * Quarterly Journal of Mathematics, Vol. XXIII (1888) , pp. 143-152. 
 
 ^Philosophical Magazine, Ser. 4, Vol. XXXVI (1893), p. 354. 
 
 I Arkiv for Matematik, Astronomi, och Fysik, Bd. 6 (1911), No. 29. 
 
Burgess: The Uniform Motion of a Sphere through a Viscous Liquid. 85 
 
 Oseen's method is not founded on a consideration of equation (5) or (9), nor 
 does he use the current function; but the above remarks embody his argument. 
 He obtains a solution which, expressed as a current function, satisfies (9) and 
 the boundary conditions at infinity, but satisfies the boundary conditions at the 
 
 surface of the sphere only to the first power of the quantity — — , which is 
 
 V 
 
 assumed to be small. 
 
 5. The essential points of this work of Oseen were restated by Lamb* 
 shortly afterwards, with the use of a less intricate method of analysis. Lamb 
 points out especially the significance of the result, showing that the equation 
 (9), or rather the similar equations used by Oseen, represents more accurately 
 than does Stokes' equation (6) the nature of the motion at a large distance 
 from the sphere. Lamb shows, moreover, that Oseen's solution more nearly 
 represents the wake which follows a moving object, in»that the velocity preced- 
 ing the sphere is markedly smaller than that following. 
 
 Two years later F. Noether,t referring in his work to that of Oseen, deter- 
 mined as a correction to Stokes' current function (7) 
 
 ^^^~^sm'6cose(^^^\2p' + ap+a'). (10) 
 
 Corresponding to this, the corrections for the velocities are 
 
 16 
 
 p / \ p p' 
 
 which are exactly (8), as found by Whitehead. 
 
 In the same year Oseen? again published an article on the subject criti- 
 cizing certain portions of Noether's work, and giving as the value of Stokes^ 
 current function 
 
 + l^cos e (2p^ + ap + a^) ]- y p^ sin^ Q. (11) 
 
 But this function yields values of B and © which fail to vanish at infinity, 
 
 * Philosophical Magazine, Ser. 6, Vol. XXI (1911), pp. 112-121. 
 
 f Zeitschrift fiir Mathematik und Physik, Bd. 62, 
 
 J Arkiv for Matematik, Astronomi, och Fysik, Bd. 9, No. 10. 
 
86 BuBGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 exactly as Whitehead's resolution did, and therefore, as Oseen pointed out in 
 his earlier paper, is not admissible. 
 
 6. We shall now consider in detail equation (9), 
 
 (—W~-vd)d^=0. (9) 
 
 As these operations are commutative, a solution will be given by 
 
 ^=^' + V, (12) 
 
 where 
 
 D^j^' = (13) 
 
 and 
 
 (vD+W^y"=0. (14) 
 
 We shall obtain the sqlution of these separately. Equation (13) has been 
 treated very fully by Sampson,* but for convenience of reference we shall here 
 give in detail the solution so far as is necessary for our purposes. Transform- 
 ing to polar coordinates 
 
 our operation D becomes 
 
 8^ sin e 8 / 1 d 
 dp^ + p' aeVsin 6 de 
 
 We desire, then, the solution of the equation 
 
 8N^ sin_0_a/_J_^\ 
 
 dp' + p' aaUne dd)~^- ^^'^> 
 
 A particular solution is 
 
 ^«=^n~3^-9^. (13 A) 
 
 where P„ is the zonal harmonic of degree n, and therefore satisfies the differ- 
 ential equation 
 
 M(n + l)P„sin0+^(sin0^)=O. (15) 
 
 For substituting in (13) the given value of 4^^, we have for the left-hand side 
 n(n + l)c„ 8P, casing 8 f 1 8/. 8P„\1 
 
 * Philosophical Transactions, Vol. CLXXXII (1891), p. 449. 
 
BuBGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 87 
 that is, using (15), 
 
 a 
 
 c„ sin e r 
 
 n{n-{-l) 
 
 dP. 
 
 n{ 
 
 n+l)P,.\\ 
 
 .(16) 
 
 p— \_ ■ dB dd 
 
 which is identically zero. It is well known that 
 
 T, A ( , n ^(l^ — 1) 9^ 9 n . I^i^ — 1) (** — 2) (W — 3) „ .- 
 
 P„=^„[cos" e- 2(^ COS-- 6 + ^.,(2;-l)(2L3) '''^- 
 
 Hence, by (14), 
 , c„sin^0r „ ,. {n—l){n—2) .. , 
 
 ,( .y jn-l) {n-2) . . . . {n-2r) ^2r+ig , 1 (13B) 
 
 + ^ ^^ 2'lr^[2»i-l) . . . . (2n-2r+l) ''''^ P+ . . . . j U^is; 
 
 for all positive integral values of n. For %=0, we have 4* a function of 6 only ; 
 
 then 
 
 1 3^0 , fl 
 
 ^i^W='<" ^0 = ^0 «««^- 
 
 Placing w successively 1, 2, 3, ... . in (13B), we have 
 
 sin^0 
 
 '4'3 = C3 
 
 sin^ cos 6 
 
 -^4 = ^4 
 
 sin"e(5cos^e— 1) 
 sin^0cos0(7cos2 0— 3) 
 
 (17) 
 
 7. We desire also the solution of equation (14), which is, if 2k-- 
 
 W 
 
 / 3^ _1^ 3^ ^J_\ 
 
 (14A) 
 
 Letting 4'"=e~*'''^(r, s'), we have at once 
 
 W~rd~r'^W~''r ' 
 or, in polar coordinates, 
 
 d'^ , sine 3/ 1 3^\ , „ 
 
 (18) 
 
88 BuEGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 Assume ^ = R@, where R and are respectively functions of p and 6 only. 
 ^d'R ,„ sine 3/ 1 dQ\ ,,„„ „ 
 
 P^^'-R ^2 2_ sine 3/ 1 36 
 R'd^~''^^' 
 
 
 We have, then, to determine R and 0, the two equations : 
 
 9p' P 
 
 K. + l)0+sine|(^^§)=O. 
 
 (19) 
 (20) 
 
 In this form, for n any positive integer, B is a solution in finite form of 
 Riccati's equation; that is, 
 
 and is sin 6 
 
 dB 
 
 /I 3 \"+^ 
 ^»=P'"^\-f) Ue^'+Be-""), 
 
 We have, therefore, 
 
 >3P„ 
 
 ^"= 2 e-^'-^^^sine^E^. 
 
 (21) 
 
 (22) 
 
 We have, then, as a general function satisfying the differential equation (9) 
 
 ^=:e-*''<i+™=«rC+I> cos e+b^ sin^ e(k+ ~) 
 
 + b2sin^eGose(¥+ ^ + ^) + . . . . 1 
 
 ,^ „^Msin^O , ^sin^osQ Psin^e .. ,. ^^, ,„„. 
 
 + L cos 6-1 1 ^ 1 3 — (5 008^6—1) + (23) 
 
 8. To adapt this solution to the problem of the sphere moving with uni- 
 form velocity W, we must have, if 
 
 1 3i// . 
 R = "5 — -. — 7f -~7r is the radial velocity 
 
 p^ sm 6 dd •' 
 
 and = — 
 
 d^ 
 
 — -. — fl-g— is the velocity perpendicular to this. 
 
 (24) 
 
 (1) jR and always finite, 
 
 (2) R = and 0=0 at infinity, 
 
 and (3) i2 = l^ cos 0, 0=— TF sin on the surface of the sphere, 
 
 i. e., when p = a. 
 
 (25) 
 
Burgess: The Uniform Motion of a Sphere through a Viscous Liquid. 89 
 
 The terms we have chosen are all such that R and © derived from them vanish 
 at an infinite distance, and, if we take C+D cos Q—ho{l — cos 0), are everywhere 
 
 finite, smce -^.- and -^ are both divisible by sin 6. The last condition [25 (3) ] 
 
 remains to be satisfied by a proper combination of the functions and determi- 
 nation of the constants. 
 
 Oseen's solution is equivalent to taking the terms in (23) whose coefficients 
 are &o > L, and M ; that is to say, 
 
 if.=b,e-*>'q+''°°'') (1— cos 6) +L cos 6+ . (26) 
 
 P 
 
 Using the conditions [25(3)] at the surface of the sphere to determine the 
 
 constants, we have 
 
 W cos 0= 2" "I 3 1 2 \l + ka{l—Gos6) 
 
 0/0/ 0/ 
 
 -W sin 0= ^^ + M e-*«a+oos«) sin e. 
 
 (27) 
 
 Equating the constant terms in each of these two equations and the coefficients 
 of cos 6 on each side of the first, we have 
 
 It is to be noticed that we have neglected all squares and higher powers of the 
 quantity Jca, which must therefore be assumed to be small. Hence we derive 
 
 ^ ^ 3av ,^ Wa' 
 L = bo= 2"' ^= 4~- 
 
 Substituting these values in (26), we have 
 
 4,= _^ [e-*pa+-«(i_cos0)+cos0]-^5^4^^- ^^^^ 
 
 If we transform this value of •J- into cylindrical coordinates, find u and w from 
 equations (3 ) , and resolve u in the directions of the x and y axes, we get expres- 
 sions for the velocities which, except for notation, are exactly those derived 
 by Oseen. As explained in section 4, this solution differs from Stokes' solu- 
 tion (7) fundamentally in that it satisfies the differential equation (9), which 
 for large values of p is a close approximation to the complete equation (5). 
 "We can see its significance by throwing it into several different forms. In the 
 first place, since a constant makes no difference in a function used solely in the 
 12 
 
90 BuEGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 form of its derivatives, we may write it 
 
 '^'~ 4 L — A;a(l + cose) p J' ^ ' 
 
 Expanding the exponential term, this is 
 
 ^=i^(^_«)sin^0-^-^p^sin^0(l + cos0) + (28B) 
 
 The first term of this expansion is Stokes' solution ; the following terms con- 
 tain powers of ^ and are therefore less important than the first when p is 
 
 small {i. e., near the sphere) and W is small {i. e., for slow motion). As 
 Oseen says, then, in the neighborhood of the sphere the solution approaches 
 that of Stokes as a limit when W approaches zero. It is important to notice, 
 however, that the character of the solution is essentially different from that of 
 Stokes. If we plot the stream-lines for Stokes' solution, we notice that we 
 have symmetry with regard to the plane s' = ; for Oseen's, however, the stream- 
 lines are more crowded following the sphere. If, for instance, we set the first 
 two terms of (28) equal to a constant; that is, e-^'^'+^'^^Cl— cos 6) -|-cos d—c, 
 the curves thus resulting, more conveniently expressed in the form 
 
 , 1 — cos d 
 
 P A;(l-|-cos0) ' 
 are asymptotic to the lines 0=cos~'c before the sphere and run off in a para- 
 bolic form in the wake of the sphere. The simplest way of seeing the differ- 
 ence between the conditions before and after the sphere, is to notice that the 
 velocities will each have an exponential factor — A;p(l-|-cos 6) ; when is in the 
 first quadrant, this term will be smaller than when 6 is in the second quadrant, 
 for the same values of p. Since this corresponds with our observations of fluid 
 motion, the solution derived by Oseen appears worthy of further consideration 
 and improvement. 
 
 Expanding the boundary conditions as given in (27), we have 
 
 ^ . L. 2MCOS0 fcof-, , /^ ., k^a\-. 
 
 WgosB^ ^ + 3 — 4--4U-A;a(l+cos0)-h-nr (l + eos0)^ 
 
 x\l+ka{l—Gos6)\, (■ (27A) 
 
 —W sm 6= 3 — + \l—ka{l+Gos 0) + —^ (1-l-cos 6)' 
 
 d d {, \ Ji 
 
 From these equations it is clear that on the surface of the sphere the velocity 
 components found in (28) differ from the velocity components of the sphere 
 
 l2 
 
BuEGESs : The Uniform Motion of a Sphere through a Viscous Liquid. 91 
 
 by terms whose ratio to the terms W cos 6 and — W sin $, is ka. We proceed 
 to improve this approximation by using two more terms from (20), and by this 
 means making the above-mentioned ratio equal to A;V . 
 9. Let 
 
 „-fcp(l+cos9)_]_ _ /_ 1\ ,„„^_,, . M . iVcOSei 
 
 il/=sin''0 &o 
 
 iTfCOSO \ p/ p 
 
 p' 
 
 "} 
 
 (29) 
 
 :sin2 
 
 ■Mp+fci(A;+-)+--4[^-M(l+A;p)+4]a+cose) 
 + [_^+W'(,+ l)](l+cos(^)^+...' 
 
 + 
 
 L \n+l In \ p/J 
 
 (29A) 
 
 = sin^0[5o+^i(l + cos0) +B,{l + Gos6y+ .... +B„(l+cos0)''+ . . . . J.(29B) 
 The boundary conditions (21) and (22) at the surface of the sphere give 
 
 p^ sm d ov 
 1 d^ 
 
 = 
 
 = — W sin 6, when p—a. 
 
 p sin 6 3p 
 As the first of these is to be true, for all values of 6, we may integrate, giving 
 
 4,= - Wa^ sin^ 6, when p=a. We have, therefore, 
 
 ^Wa" sin^ e=sin^ 0[Bo+-Bi(H-cds 0) + ....] , 
 TFa sin^ 0=sin^ ^[^ + ^ (1+cos 0) + ... .J. 
 Dividing by sin^ 6 and equating like powers of (1 + cos 0), we obtain 
 
 \wa^- 
 
 op Jp=a 
 
 (30) 
 
 Using the values of Bo, B^, etc., given above in equation (29A), we find that 
 we can satisfy the four equations for Bo, Bi,-^ and ^-^ , neglecting squares 
 and higher powers of ka, with the following values for the constants : 
 
 3 
 
 4 
 
 ^o=-t1^«\^ + 
 
 M-. 
 
 Wa^ 
 
 ■ka 
 
 b, = -^Wa'(l-^ka^, N = -^Wa'-k.a. 
 
 (31) 
 
92 BuBGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 To satisfy the boundary conditions (30) exactly, we should have B^ and 
 
 -~ =0 when p = a; it is obvious that, while the given values of the constants 
 
 do not effect this, the terms which should but do not vanish all contain at least 
 the square of ka as a factor, and our error in approximately satisfying the 
 boundary conditions at the sphere is proportional to the square instead of the 
 first power of the small quantity ka. We see, therefore, that while solution 
 (28) holds as valid in the limit when ka approaches zero, the solution just 
 found is a better approximation when ka is sufficiently small so that its square, 
 but not its first power, may be neglected. If a closer approximation is desired, 
 two more terms may be added, namely : 
 
 ,^,-.a.e„.>3i,.e ,,,e/,.+ Sk ^ ^X Psin-e(5cos-e-l) ^ ^^^^ 
 
 \ p pV p 
 
 and we shall then be able to satisfy exactly the boundary conditions ^2 = 0, 
 
 dB 
 
 -^ =0 when p = o, making our error proportional to k^a^W. Similarly, we can 
 
 add any number of pairs of functions and make our error proportional to as 
 high a power of ka as we desire. On account also of the \n_ in the denominator 
 of the n-th term of (29), this error is more and more negligible as we go on. 
 We have shown, then, how to obtain as accurate a solution of equation (9) as 
 may be desired. 
 
 10. In this differential equation (9) which we have just discussed, we have 
 neglected certain terms which were included in our fundamental equation (5). 
 We shall now include these and proceed to obtain a solution of the complete 
 equation (5) accurate as far as the terms involving k^a^ or A;^ as a factor. 
 When transformed to polar coordinates, equation (5) becomes 
 
 [ 
 
 ■ /a;// 8 d^ d\ 2 / d^ sin 6 8^ \ 
 
 p^ sin d\de dp dp del + p" sin^ dV^^ dp p 36 ) 
 
 — (9' sine a / 1 8\] 
 
 "Up' ^ p' aeVsineae/j 
 ^j „a sin e a \ ya^^ sin a / d^ \-\ ^ ,_, 
 
 Let us assume that the desired approximate solution of this equation is of the 
 form 
 
 4'=sm^B[Ao+AiGosd], (33) 
 
 where Ao and Ai are functions of p, and A-^ contains ka as a factor. This 
 assumption appears appropriate in view of the fact that, to the desired degree 
 
BuEGESS: The Uniform Motion of a Sphere through a Viscous Liquid. 93 
 
 of approximation, each of the solutions (7), (28) and (29) already found is 
 of this type. Substituting this value of '^ in the differential equation (5A), 
 omitting all terms which contain fc^o^ as a factor, we have, after some simplifi- 
 cation, 
 
 ^-f--)[m-T)--iw-T)\-^^ 
 
 
 4 d^Ao ^ dA, 8Ao 
 
 I _3 
 
 dp' p' dp^ ^ p« dp p* 
 
 In order to satisfy this relation, we must in the first place have 
 
 d'A, 4 d'Ao , 8 dA, 8 A n n^^ 
 
 -d^-y-d^ + j^-Y 
 
 That is to say, any value assumed for Aq must be such that Aq sin^ d satisfies 
 Z)^'i|'=0, since (35) results from this part of (5A), which was used by Stokes. 
 The complete solution of this equation can easily be shown to be 
 
 Ao=— +aip + a2p*+a3p*- 
 
 In order to satisfy (34), we must also have 
 
 U2A, \/d_ 2_\(lA_2Ao\_d^_12.d^ '2^dA,, /o.v 
 
 V \ p' "^Aap pV\ dp' p' )- dp' p' ap'' ^ p ap • ^ ' 
 
 Hence we may easily determine the value of ^i corresponding to any value 
 
 of A. 
 
 11. The first obvious assumption is to take A^ from Stokes' solution (7), 
 
 Then 
 
 d'^Ao 2Ao 6bo 
 
 and (34) becomes 
 
 (37) 
 
 Assume a solution, 
 
 A,=d,p' + d,p+ d,+ ^ + ^, (39) 
 
 r P 
 
 in which the third and fifth terms are parts of the complementary function, 
 
 the fifth term being essentially •4'2 of equation (17). We could now easily 
 
94 Bukgbsb: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 obtain the value of Ai ; our work is simplified, however, by first determining 
 bo . As our solution must satisfy the boundary conditions at the surface of 
 the sphere, we have, as in (25), 
 
 -A-M] =Weos6, -^^] =W.me. (40) 
 
 a' sm y du Jp=o a smd op Jp=a 
 
 Since i// = sin^ ^(^o+A cos 6), 
 
 Wa^ cos e=2Ao cos 6+ Ad— 1+3 cos' 6} , 
 
 -„ dAo , dAi . 
 
 Wa= -75 1- ^s-^cos I). 
 
 op Op 
 
 We must therefore have 
 
 
 (41) 
 
 From these conditions on Aq, we find 6o= j^*^^- Substituting this value for 
 bo, (38) becomes 
 
 f*^-l¥^+24,3^=^[-V+3«p.-<.,. (42) 
 
 Hence we easily obtain 
 
 A,=-^^[2p'-3ap' + d,pV-a'p + d,]. (43) 
 
 Using (41), we have 
 
 ^3 = ^5=1- (44) 
 
 The solution given by this assumption of the value of Aq is therefore 
 
 ^=^Wa sin'^eKsp-— )-^(l--y(2p^ + ap + a')cos0J, (45) 
 
 which is exactly the correction found by Noether, as stated in (10) ; the veloci- 
 ties derived from this are given in (8), and were first found by Whitehead. 
 •The objection which Whitehead himself raised against this approximation still 
 stands, however ; that is, the velocities do not vanish at infinity as required by 
 the conditions of the problem. 
 
 12. In order to avoid this difficulty at infinity, let us take Aq from the 
 solution of an equation which is, as Stokes' is not, a good approximation at a 
 
BuEGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 95 
 
 large distance from tlie sphere. The simplest one of this type is (28). We 
 shall consider this as our basis and determine a correction, accurate as far as 
 terms involving h^a^ or hy as a factor, which shall take account of the quad- 
 ratic terms. Accordingly we take 
 
 A, = }>,(^~l-^-^-haWf. (46) 
 
 Determining &o at once from equations (41), we have 
 
 —^ =26o— -A;oTFa^ 
 
 Wa=^-\kaWa; 
 a 4 
 
 that is 
 
 &o= 
 
 {l + lua). 
 
 Substituting the value of A^ from (46) in (36), and neglecting, as we are doing 
 throughout, all terms involving the square of ka, we again obtain (42). The 
 approximate solution '4/=sin^ 0(^o+^i cos 0) derived by this method is then 
 
 ^=lTrsin^0[(l+|^a)(3pa-|)-^p^ 
 
 — I A;a cos (!—-)'( 2p''-fap + a^) 1, (47) 
 
 which is in substance exactly equation (11), as derived by Oseen. In this form, 
 however, the solution is still subject to the objection that it gives velocities 
 which do not vanish at infinity. 
 
 Solution (47) is therefore unsatisfactory; but if we notice that the cor- 
 rection taking into account the quadratic terms is based on solution (28), and 
 
 must therefore be added to it to give the required result, this difificulty disap- 
 
 3 
 pearsv The term — - Wka^^ sin B cos Q of (47) is already included in the expo- 
 nential term of (28); combining with (28) the terms of (47) not already 
 included, we obtain 
 
 ^ 2 \ 4 / l-fcosfl 
 
 4-(H-4A;aj-— + -jg-sm^0cos0(3ap-a^+--— ^j. (48) 
 
96 BuEGESs: The Uniform Motion of a Sphere through a Viscous Liquid. 
 
 Hence, as usual, 
 
 l+^kaj [e-'"'^'+'""'^l + Jcp{l—Gos 6) f — 1] 
 
 „_ 1 d-^ _ 3va 
 
 Wa' L , 3- \cos0 , 3Wka. , , „ ^.J^ ^ a^ a'\ ,,„, 
 
 2 \ ' 4 / p« ' 16p^ ' ' '\ '^ P P' 
 
 6 = ^|^=-^(l+^A;a)sin0e-''C^--« 
 
 psmd dp 4 \ 4 / 
 
 Using these velocities, we find with Oseen that Stokes' law should be 
 
 3Wda 
 
 F = 6nfiWa(l + 
 
 8iJ. 
 
 As is obvious, these velocities, (49) and (50), both vanish when p is infinite; 
 when p — a, they are, except for terms involving the squares or higher powers 
 of ka, equal to W cos 6 and — W sin 6, respectively; and since, omitting squares 
 and higher powers of ka and kp, (48) is the same as (47), to that approxima- 
 tion they satisfy the complete differential equation (5). The solution, then, is 
 superior to any previously given. The difficulty first raised by Whitehead, 
 that a certain term gives velocities which do not vanish at infinity, is met by 
 showing that this term is included in an exponential expression. 
 
 CoBNELL University, May, 1915. 
 
arY2 
 
 ,. 3 1924 032 226 197 
 
 olin.anx