»iAc/i-r;ti'/3H5' 
 
 ■kKLF 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1342 
 
 SPIRAL MOTIONS OF VISCOUS FLUIDS 
 
 By Georg Hamel 
 
 Translation of "Spiralformige Bewegungen zaher Fliissigkeiten." 
 Jahresber. d. deutschen Math. Ver. 25, 1917 
 
 Washington 
 January 1953 
 
 IJNIVERSITVC 
 
 HDA 
 
 ^ViaE.FL 32611-701 
 
 USA 
 
^(cO S7^( /y 
 
 / « 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 13ij-2 
 
 SPIRAL MOTIONS OF VISCOUS FLUIDS* 
 By Georg Hamel 
 
 INTRODUCTION 
 
 The equations for the plane motion of viscous fluids of constant 
 volume are, after elimination of the pressure and introduction of the 
 stream function ii by which the velocity components 
 
 V = ^ V =-^ 
 X Sy y 5x 
 
 are expressed, reduced to the one equation 
 
 S A^l^ + ^ At Si _ d M M = a A A*!' (l) 
 
 Bt Sx Sy Sy dx 
 
 therein a indicates the ratio between viscosity coefficient and spe- 
 cific mass n, and A signifies the Laplace operator. 
 
 This equation is satisfied by all potential motions 
 
 All' = 
 
 however, this fact is of little significance since viscous fluids adhere 
 to solid walls and, from well-known considerations of function theory, 
 there cannot exist a potential motion which would do so. Otherwise, 
 properly speaking, only Poiseuille's laminar motion is known as exact 
 solution of equation (l) and that solution does not even show the sig- 
 nificance of the quadratic terms because they identically disappear 
 there . 
 
 "Spiralformige Bewegungen zaher Flussigkeiten." Jahresber. d. 
 deutschen Math. Ver. 25, 1917, pp. 3l4-60. 
 
NACA TM 13i|2 
 
 Under these circvimstances it seems perhaps useful to know a few 
 more exact solutions of equation (l) for which the quadratic terms do 
 not disappear; such solutions will be indicated below according to two 
 methods . 
 
 In both cases, one deals with motions in spiral-shaped streamlines 
 (which are observed frequently) . 
 
 Third, we shall, in addition, investigate the neighborhood solu- 
 tions to pure radial flow. 
 
 FIRST PART 
 
 We raise the question: 
 
 Are there solutions of equation (l) which are not potential motions 
 for which, however, the stream paths are the same as for a potential 
 motion whereas the velocity distribution is to be different? 
 
 We shall be able to indicate such solutions, in fact all of them: 
 the streamlines are logarithmic spirals (including concentric circles 
 and pure radial flow); for the velocity distribution, one arrives at an 
 ordinary differential equation which for pure radial flow leads to 
 elliptic functions. In the discussion, the influence of the quadratic 
 terms becomes manifest in a considerable difference between inflow and 
 outflow (see paragraphs 7? 8, and 9) • 
 
 We require, therefore, solutions ^f of equation (l) for which 
 
 ilr = f (cp) 
 and A9 = 0, but not Ai = 0. The latter condition excludes 
 
 f • '(9) =0 
 
 We limit ourselves to steady motions —■ = 0. 
 
 ot 
 
 1. The calculation becomes clearer if first the aiixiliary problem 
 has been solved: 
 
NACA TM I3J+2 
 
 Transformation of equation (l) into isometric coordinates, that is, 
 such curvilinear coordinates '? ,X that 
 
 9 + iX = ■w(x + iy) = v{z) 
 
 Let us thus assume 
 
 i = t(cp,x) 
 
 Sx By dy ^x 
 
 If one denotes 2_i. + 2_1 by A'^, there resiilts first, with the abbre- 
 
 viation 
 
 dv 
 dz 
 
 = Q 
 
 Ai^ = Q A' 11/ 
 
 With the double integral extended over an arbitrary region, one has 
 
 f|^M.|A|iWdy = 
 \ ox oy ay ox/ 
 
 d A d\|/ 
 
 f 
 
 MQ A') M S(Q A 
 
 ^ 
 
 ^X 
 
 :i^1 
 ^J 
 
 d<P dX 
 
 ' [ ^(Q A') ^ _ S(Q A') ^l/^ ^ 
 |_ Sep SX ~ SX acpj\dx Sy 
 
 
 Since, however. 
 
 ^9 hX 
 hx by 
 
 By Sx 
 
 dw 
 dz 
 
 = Q 
 
NACA TM 13^4-2 
 
 there follows 
 
 Sx Sy ^y ^x 
 
 / d A' M _ a A' ai\ + A^./ S In Q ai _ a In Q. '^ 
 \ Sep Sx dx Sep/ \ Sep Sx SX Sep, 
 
 However , 
 
 ^ ^^ ^ = 2 ^ R In ^ 
 ^9 Sep dz 
 
 2R J- In ^ 
 dw dz 
 
 = 2R 
 
 d w 
 
 dz2 
 
 'dwf 
 
 .dzy 
 
 and 
 
 
 
 
 d^w 
 
 ^^^ - 2 ,^ R In ^^ - 
 dX dX dz 
 
 -2 ,^ J In <lw - 
 
 acp dz 
 
 -2J d In ^^ - 
 dw dz 
 
 ^ 2 
 -2J <1^ 
 /dw\2 
 
 \dz/ 
 
 
 
 
 are valid. If one puts the analytic function of z 
 
 ,2 
 
 d w 
 
 dz 
 
 'dwV 
 vdz/ 
 
 = a + bi 
 
 one obtains 
 
 d A M _ ^_A M = 
 Sx Sy Sy Sx 
 
 dw 
 dz 
 
 a A' a^ d A' 
 
 Sep Sx SX 
 
 ^UA'faM. + bii'^l 
 Sep/ \ Sx Sep/j 
 
NACA TM 13^2 
 
 Finally, there results 
 
 A A^ = Q A'(Q A'\|') = q2a' A' ^ + Q A' Q A' i^ + ZQ^^-^^ |^ + ^^^ ^] 
 
 V ^ Sep bx hxj 
 
 Q^ 
 
 A- A'1/ + A't ^^ + zfi ^'t a - ^^'^ b 
 Q, V Sep SX 
 
 In Q = In 
 
 dw 
 dz 
 
 is a harmonic fimction, thus 
 
 A' In Q = 
 
 hence, 
 
 Q \ Sep y V sx y 
 
 Thus one obtains as the result of the conversion of equation (l) to 
 isometric coordinates ep, X for steady motion 
 
 S A'^ ac£ _ S A'i M + A'irfa M + b M\ = a 
 
 S9 ax sx Sep y ax s*?/ 
 
 A' A't + 
 
 A'\|/(a2 + b^ 
 
 ).3( 
 
 S A'^ a _ S A'^ ^ 
 
 S9 
 
 Sx 
 
 (II) 
 
 therein, a + bi is the analytic fiinction 
 
 ,2 
 
 d V 
 
 dz 
 
 (w = cp + iX, z = X + iy) 
 
 and A' denotes the operator 
 
 ^\ b^ 
 
 Sep^ SX^ 
 
6 NACA TM 13^2 
 
 2. We retiirn to the question on page 2: if must be a mere function 
 of cp 
 
 ^ = f(T) 
 
 If derivatives, with respect to cp, are denoted by primes, equation (ll) 
 becomes 
 
 f ' 'f 'b = a f^ + f " (a^ + b^) + 2f ' ' 'a (ill) 
 
 f may depend only on cp but must not depend on X. 
 
 This is certainly possible if a and b do not depend on X, thus, 
 since a + bi is an analytic function of cp + xi, do not depend on T 
 either, if a + bi is, therefore 
 
 ,2 
 d w 
 
 a + bi = 2 -^ — = C 
 
 2 
 2 
 
 that is, constant. We shall see later (paragraph 3) that this is the 
 only possibility. 
 
 From a and b being constant, there follows 
 
 w=- 2 
 
 -^ in (z - zo) . wo 
 
 thus, after introduction of the polar coordinates 
 
 z - z^ = re 
 
 (a In r + bi3) + 9^ 
 
 2 ^2 ' 
 
 a + b 
 
NACA TM 13ij-2 
 
 Thus, the streamlines 9 = const are identical with the logarithmic 
 spirals 
 
 a In r + bi3 = const 
 
 a = signifies pure radial flow, b = flow in concentric circles. 
 The velocity distribution, however, is given by equation (ill): the 
 radial component is 
 
 Bi 
 
 _ f . a9 - 
 
 2b f, 1 
 
 r Si3 
 
 r b-3 
 
 a2 + b2 r 
 
 the circular component 
 
 M = _f ^ = 2a f, 1 
 Sr Sr a2 + b2 ^ 
 
 consequently — '^ f ' — the magnitude of the velocity. There- 
 
 v/~2 , 2 
 
 b 
 fore, f ' must disappear on solid walls. 
 
 Without restriction of the generality, one may presuppose left- 
 hand spirals so that r increases with t3, thus a and b have dif- 
 ferent signs; since, furthermore, -(9 + iX) is an analytical function 
 just as 9 + iX, and equation (ill) is actually invariant with respect 
 
 to a simultaneous sign change of 9,a,b, one may presupjpose 
 
 a ^ 
 b < 
 
 Therefore, positive velocity components "7 amd -^ signify for 
 
 r a-d or 
 
 , f ' > outflow, 
 
 in contrast for 
 
 f ' < inflow. 
 
 Translator's note: The original says "time change," obviously a 
 misprint . 
 
8 MCA TM 1342 
 
 Since T may be replaced by c<P, one may in addition impose a con- 
 dition on the constants a and b. 
 
 3- We now want to conduct the proof that on the basis of our require- 
 ments a and b must be constant, that therefore the flows in loga- 
 rithmic spirals are the only ones the flow patterns of which correspond 
 to a potential motion without themselves being a potential motion. 
 
 If a and b were not constant, the analytical function a + bi 
 would produce a conformal transformation of the 9 + iX-planej by virtue 
 of equation (ill) which with the abbreviations 
 
 A =-± B = ^ f ' C = i — 
 
 f' 2a f" 
 
 (f ' ' = is excluded) may also be written 
 
 a^ + b^ - 2A(9)a - 2B(cp)b + C(9) = (ill') 
 
 the circles (equation (ill')) would correspond to the straight lines 
 
 9 = const in this transformation. 
 
 These circles would therefore have to form an isometric curve 
 family. 
 
 However, if the family of curves 
 
 g(a,b,cp) = (III') 
 
 is to be an isometric one so that A9 = 0, the function g must satisfy 
 the equation 
 
 ^a Sb J 
 
 and this equation must either be identically satisfied, or be a conse- 
 quence of equation (ill'). 
 
NACA TM 1342 
 
 One has 
 
 -^a 
 
 =2(a-A), g^=2(b-B), Ag = U, g^^ = -2A' , g^^=-2B' 
 
 g = -2A'a - 2B'b + C, S^ = -2A"a - 2B"b + C" 
 
 g ^ + g^^ = k{a - A)2 + k{l) - B)2 = 4(a2 + b2 - C) 
 
 Thus, equation (iV) is quadratic in a and b; an easy calculation shows 
 the result that the quadratic terms are automatically eliminated. There- 
 fore, the coefficients of the two terms must be zero whence follow three 
 conditions 
 
 Q ^ All. _ C - 2AA' - 2BB' ^- Bl^ _ C - 2AA' - 2BB' 
 ^' C - A^ - B^ ^' C - A^ - B^ 
 
 _ C " C ' - 2AA ' - 2BB ' 
 
 ^' C - a2 - B^ 
 
 Hence, there follows further that A', B', C, must be proportional and 
 furthermore that 
 
 B' 
 
 ? 2 
 
 C - A^ - B 
 
 must be constant . 
 
 The final condition yields 
 
 C = a-B + p A = 7,B - 5 
 
 or with 
 
 fl = a li = - 
 
 2a 2a 
 
 f^^ = af'f ' ' + pf ' ' and f ' ' ' = yf 'f ' ' + 6f " 
 
10 NACA TM I3U2 
 
 vhich, integrated, yields 
 
 f'''=iai"^ + pf'+e and f ' = ^ yf'^ + bf + n 
 
 2 2 
 
 Comparison of the two values for f ' ' results in 
 
 ^ of ' ^ + pf ' + e 
 7f ' + & 
 
 which must be identical with the preceding value of f ' ' . The compari- 
 son requires 
 
 7 = a = 3 = 5^ 6=6t] 
 
 thus C = A^ = 5^ constant and f ' ' = 8f ' + r\. 
 The second condition 
 
 1 = 11 = const 
 
 2 2 2 
 A"^ + B - C B^ 
 
 f ' ' 
 however woiild yield - — = const and this together with f ' ' = 5f ' + r\ 
 
 43 1 2 
 would result in the contradiction 
 
 const 
 
 therewith, the proof has been produced. 
 
 k. We now turn to the determination of the velocity according to 
 differential equation (ill) which may be integrated once and assumes, 
 after introduction of the quantity proportional to the velocity at unit 
 distance 
 
 u = f'(9) 
 
NACA TM 13i4-2 11 
 
 the form 
 
 u' ' + 2au' + ufa^ + b^) _ -L u^ + C = 
 ^ ' 2a 
 
 This equation is identical with a damped oscillation which takes place 
 under the influence of the potential 
 
 _^ u3 + l(a2 + b2)u2 + Cu 
 6a 2^ ' 
 
 We start with the limiting cases: 
 
 1. The streamlines are concentric circles: b = 0. 
 
 Then 
 
 u =--^ + e"^''^(A + m) 
 a2 
 
 and, because of 
 
 cp = _ 2 In r 
 
 u = const + v^ik + B2_ In r) 
 
 whereby, the velocity distribution 
 
 V = 2 u 
 a r 
 
 is given. The exact solution of Conette's case is also contained therein: 
 the three constants here are determined from the two limiting values of 
 the velocity and from the fact that in case of a full turn around the 
 circular anniilus, the pressure must revert to its initial value. An 
 easy calculation yields B-]_ = and thus 
 
 %k 
 
 r2 - r^ 
 
 (More details on the determination of the pressure are seen in para- 
 graph 10 . ) 
 
12 NACA TM 13i4-2 
 
 2 . The flow is purely radial: a = 0. 
 The different iaJ. eqiiation reads 
 
 u " = b^u - ^ u^ + C = 
 2ct 
 
 and leads to elliptic f\inctions 
 
 u' =./- — , /-u3 + 3abu^ + const u + const 
 
 where the three e's are only subject to the one condition 
 
 e^^ + e^ + e = Bab 
 
 but otherwise are still at disposal. 
 
 Since, according to the remark on page 8 one relation between a,b 
 is still unused, it will be expedient to put 
 
 b = -2 
 so that one obtains, according to page 7 
 
 cp = t3 
 Then the conditional equation for the e reads 
 
 e-, + Co + ep = -6ff 
 
 and one has 
 
 "' =\/l\/(^i-"){=2-'')h-'') 
 
NACA TM I3U2 
 
 13 
 
 thus 
 
 u = -2o + 
 
 -t(^ - ^0)' S2^S3 
 \J6a 
 
 where -d^fg^jg-, are the three integration constants. For the pressure 
 (see paragraph 10) there results the eq^uation 
 
 A/E + I. v^\ =J^f'f'- +20.^,, ^_a_ Wl f ,2 + 2fff A 
 
 acp\u 2 y ^2 ^2 acp ^2^2 j 
 
 its uniqueness is a priori ensured, thus does not determine here any of 
 the constants. 
 
 Discussion of the Radial Flow 
 5. The condition 
 
 e^ + eg + e^ 
 
 -6a 
 
 (1) 
 
 requires at least one e to have a negatively real part, for instance 
 
 then 
 
 R/'e^N ^ -2a 
 
 ^(=3) 
 
 with the eqiiality sign being valid only when all three e's have the 
 same real part . 
 
 Furthermore, since this part is real, there must apply 
 
 (a) for three real e's either 
 
 -00 < u ^ e^ -2a 
 
 or 
 
 eg ^ u ^ ei 
 
Ik NACA TM 1314-2 
 
 (b) for one real e 
 
 -<» s u 
 
 < 
 
 where, however, this e may be positive. 
 
 Fvirthermore , two possible types of flow must be distinguished: 
 
 1. Either there are no solid walls, thus a source or sink in an 
 unlimited fluid. Then u must be a periodic function of 9, with a 
 period which is an integral paj-t of 2jt. u = -00 is excluded, u = 
 need not occur. Therefore, this case can occur only for three real e's, 
 and 
 
 62 ^ u ^ e-L 
 
 must be valid. 
 
 2. Or there are two solid walls, for instance for -3=0 and for 
 t3 = i3-| (which may also be equal 2n); then at these walls u must be 
 
 u = 0. 
 
 (a) In case of three real e's there must be, additionally 
 
 e^ < e^ > 
 
 and either 
 
 (a) e^ g u ^ 
 
 or 
 
 (P) ^ u ^ e^ 
 
 (b) In the case of one real e, this e must be positive and 
 
 ^ u ^ e 
 
 One remembers, furthermore, that according to page 7, paragraph 2, 
 u > signifies outflow, u < signifies inflow; so that one has 
 inflow in the case of 2(a) (a), and outflow in the case of 2(a)(3), and 
 2(b) above. For the case 1, both cases may occur. 
 
NACA TM 131^2 15 
 
 First Case: Free Flow 
 6. One must assume i3 = for u = e^ and has therefore 
 
 -^/ -^ 
 
 '2 ^(ei -u)(u-e2)(u-e3) 
 
 Hence, there must 
 
 du - I3_ IL 
 
 3cr n 
 
 '2 ^(e^ - u)(u - e2)(u - e^) 
 
 vith n being an integral. 
 
 By the known substitution 
 
 . (e - e\ —2 2 _ ^1 - eg 
 
 '2 ri ^2J eg - e^ 
 
 u = e„ + /'e, - eA sin'^^l^ ?< ' 
 
 3a n 
 
 equation (2) becomes 
 
 2 r2 di ._ r2~ Jt 
 
 \/^2 - ^3 ° \[l~i^^Ki^ 
 
 If one now introduces the mean velocity^ 
 
 u = i/'e^ + e_"\ 
 m 2\ 1 2; 
 
 p 
 
 and the velocity fluctuation 
 
 S = e^ - e^ 
 ''At the distance r = 1. 
 
 (2) 
 
16 
 
 NACA TM 131^2 
 
 there 1)60011168 because of equation (l) 
 
 e„ - e, = 6a + 3u - - S > 
 23 m 2 
 
 (3) 
 
 thus 
 
 %^ = 
 
 6a + 3uj^ - ^ & 
 
 and 
 
 2 Pz 
 
 d^lf 
 
 yi + ;>f2sin2>tf 
 
 ^ ^6a ^ 2u^ - I 6 
 
 (2') 
 
 From this, one may draw several interesting conclusions, 
 One has 
 
 2 d>lf 
 
 k d^ 
 
 \fVx 
 
 h d\tf 
 
 ^sin^t 
 
 wi + x^sin^^^ y: 
 
 1 + Pi^cos^t 
 
 dlf 
 
 \r^ 
 
 (1 - cos 2i) 
 
 \/ 
 
 1 + I X^(l + cos 2i|/) 
 
 > 2 
 
 4 d^lf 
 
 \pF 
 
NACA TM 13i+2 17 
 
 thus 
 
 2 Ts d>l' _ 1 
 
 ° ^1 + X^Bi-nh Jl + I ex' 
 
 where 
 
 < e < 1 
 
 Thus the relation (2') between u^,6,n,cr reads, due to the significance 
 
 of x^, 
 
 y^6a + 3u^-|(l - 05 
 
 = 2 J^ 
 n 6a 
 
 or 
 
 6a + 3% - - Ti^S = 6a S^ (2") 
 
 with T] being a proper fraction. 
 Since, furthermore 
 
 ^2 Xdl^ > ^2_^di .sinh-l;,| 
 
 ° Wl + X^sin^i ° \/l + ?t' 
 
 2^2 
 
 thus becomes arbitrarily large with increasing H, one has lim € = 0, 
 
 X=oo 
 
 thus. 
 
 lim T] = 1 
 
18 MCA TM I3I+2 
 
 From equation (2'') there follows 
 
 u > -2a (1 - si^ 
 
 which, with u = 1, gives as the minimum value 
 
 ^>-2 " 
 
 The mean inflow velocity is therefore considerably limited upward, the 
 more so, the easier movable the fluid. 
 
 However, this is the only restriction: If u and the integer n 
 are selected so that 
 
 2 
 
 6a + 3\ijjj > 6>o ^^ 
 
 there exists, certainly, a pertaining 6. 
 
 For if — 6 increases from zero to the value 6a + 3^) — t] 6 
 
 lies between zero and 6a + 3^^ (because for the second value % 
 
 2 12 
 
 becomes infinite and, hence, r\ = l) so that certainly sometime p- t] 6 
 
 j^2 
 becomes equal to 6a + })\x^ - 6a — which is presupposed to be positive. 
 
 One sees, furthermore, that for a prescribed period number n the 
 fluctuation 5 and for a prescribed fluctuation 6 the period number n 
 must increase to infinity with the mean velocity Uj^. 
 
 Second Case : Outflow Between Solid Walls 
 7- The cases 2(a)(3), sind 2(b) may be summarized thus 
 
 -'^i: — ^ — - 
 
 \/(e - u) (u^ + 2au + p) 
 
NACA TM 13^+2 19 
 
 e > is the maximum velocity (at the distance r = l); because of 
 
 2a = -e - e = 6a + e 
 
 and p = e e > 0, otherwise, however, arbitrary 
 
 u^ + 2au + p 
 may for prescribed e assume all values from u + 2au to 00, so that 
 
 -#, 
 
 ^^ _ o /^- / du 
 
 \/(e - u)(u^ + 2au + p) 
 appears not at all restricted downward, but upward restricted by 
 
 l,max V 2 Jo 
 
 \ /(e - u)u(u + e + 6ct) 
 
 since 
 
 ^e 
 
 ^ \/(e - u)u 
 
 one has 
 
 ^^ = 2n / 32 (1,) 
 
 l,inax y2e(l + c) + 12a 
 
 where e signifies a positive proper fraction. 
 
 For the outflow, the width of the wall opening appears therefore 
 restricted, according to the preceding equation, by the maximum value e 
 of the velocity. For small velocity and large viscosity, the maximum 
 lies near n, otherwise, however, lower; with increasing e it drops 
 below all limits. 
 
20 NACA TM 131+2 
 
 If, therefore, an angle opening smaller than n is prescribed, it 
 permits an outflow only up to a certain maximum value. If a greater 
 outflow quantity is prescribed, the jet will, therefore, actually prob- 
 ably separate from the walls. 
 
 Also, there is, of course, for any prescribed angle ,3^ a flow 
 possible where partly inflow partly outflow occurs . 
 
 Third Case: Inflow Between Solid Walls 
 8. There remains the case 2(a) (a) 
 
 e ^ u ^ 
 2 ~ 
 
 all three roots e real, e , e negative, e positive, 
 
 J ^ 1 
 
 Here 
 
 »i = \Y 
 
 a 
 
 3^ / du 
 
 e 
 
 2 \/(--^2)(--.^3)(^-") 
 
 
 du 
 
 e 
 
 2 
 
 ^(u - e2)(-u2 - 2au - p) 
 
 where 
 
 2a = - [e, + e '\ = 6a + e 
 
 3 = e e < 
 1 3 
 
 otherwise, however, arbitrary. Thus, the angle i3, may be made arbi- 
 trarily small for prescribed e . On the other hand, however, it may 
 also be made arbitrarily large: one takes, for prescribed e , the 
 
NACA TM 13i^2 21 
 
 negative value e,, sufficiently close to e„, as far as this is not made 
 impossible by e < 0. The sole relation between the e 
 
 e^ + e„ + e = -6a 
 
 however, results with e, > in 
 
 -e^ - 63 > 6a 
 
 If -e„ > 30'; e^ may actually be assumed arbitrarily close to e„. 
 
 If the maximum inflow velocity is larger than 2^) any angle ^, 
 is possible between the solid walls. 
 
 If, however, -e„ < ^a, say eScr, where e is a positive proper 
 fraction, only 
 
 -e = e^ + (6 - 3e)a = -e^ + e-j^ + 6(l - e)a 
 
 is possible and 
 
 pO 
 ^. = v/6^ / ^ 
 
 1 
 
 "^2 
 
 ^(u - e^)!^^ (6 - 3e)a + e^](e^ -u) 
 
 attains its highest value for e = 
 
 ^0 
 
 1 .max , _ 
 
 ' ^-36a 
 
 \/(u + 3ea)[u + (6 - 30a](-u) 
 
 Ti\/6a It rt 
 
 6 
 
 J^ - 36(2 + n)](J y/i - I ^(1 + n) 
 
22 NACA TM 13^2 
 
 where t\ and 6 are positive proper fractions. Thus, the maximum of 
 ■d is larger than n . 
 
 When the maximum inflow velocity is smaller than 3^ ^ the angle 
 openings of the solid walls also may attain any magnitude up to « . 
 
 Flow in Spirals 
 
 9. Because of the damping 2au' (see paragraph h, page ll), a 
 periodic solution, aside from u = const, is not possible. 
 
 A free motion in logarithmic spirals is always a potential motion. 
 In contrast, there exist other flows on logarithmic spirals between 
 solid walls. 
 
 In order to have, for r = const, the variable cp agree with the 
 angle d, one may fiirthermore prescribe for the constants a,b, in such 
 a manner that one obtains 
 
 2b ^ -L 
 
 2 ^,2 
 
 a + b 
 
 thus 
 
 b = -1 ± yi - a^ 
 
 a must be a proper fraction, otherwise it remains arbitrary. 
 Equation III, once integrated, (see page 10) then reads 
 
 u ' ' + 2au ' + p^u + ^ u^ + C = 
 
 where p = -2b = 2+2\Jl-a < k, but > a^ 
 
NACA TM 13^2 23 
 
 The velocity at unit distance is 
 
 2 
 -^ u = / ^ u 
 
 \/a^ * b^ Vl.^/ 
 
 1 -a2 
 
 u is, therefore, the velocity at the distance 
 
 r = / = 2 
 
 P 
 
 n/^ 
 
 1 + \/l - a^ 
 
 If one first omits the damping, one has exactly the aajne case one 
 had before except that 
 
 3^ / 2 
 
 instead of ' 
 
 is in front of the square root (see page 12). The relation for the e 
 
 remains the former one. Since p < k, the ajigle opening is increased 
 
 by this influence tS, . 
 
 The damping, however, takes effect in the same sense. Nevertheless, 
 the main result remains correct . 
 
 For outflow the admissible angle opening t3, is restricted by the 
 
 maximum flow velocity in such a manner that it tends toward zero when 
 this velocity increases beyond all limits . 
 
 If one puts 
 
 .-acp 
 the above differential equation becomes 
 
 ,2 
 
 u = ve 
 
 v" + (p^ . a^^v + ^ v^e 
 
 ka 
 
 2.-aCP ^ ^^aq^ ^ ^ 
 
2k NACA TM I3U2 
 
 If cp = is assumed to be the location of the maximum v,. for v, 
 multiplication by 2v' and integration yields 
 
 ,2 /„2 2\f 2 2\ p^ r^ 2 -acp^ ^^ P^ a9, 
 
 v' +(B -av -v^+i— / ve dv + 2C / e dv = 
 
 (,^ - a^)(v^ - ./) . f, 
 
 ^0 ^^ 
 
 From the corresponding equation for u 
 
 u'^ + k& r u' du + p^(u^ - Uq^) + ^(u^ - Uq^) + 2C(u - Uq) =0 
 
 one can see that for eqtial u^ the u-curve will be the steeper, i3-, 
 
 therefore the smaller, the larger C. For value close to u^ this is 
 
 immediately clear from the differential equation, for in case of u' = 0, 
 u' ' will be the smaller, the larger C, thus |u'| the larger^ from 
 the preceding equation one may see, however, that for larger C 
 
 u"^ + iva / u' du = u"^ - 4a / u' du 
 *^Uq vJ u 
 
 has the higher value. Hence, follows directly for the ascending branch 
 (u' > 0) that aJ-ways |u'| has the higher value when C is the larger 
 value. For if u' would once reach for the initially flatter curve 
 
 pU 
 
 (C smaller) the value of the steeper curve, / u' du would have to 
 
 w 11 _ 
 
 ^0 
 
 .Uq 
 
 have for the former the smaller, thus / u' du the higher value 
 
 ^u 
 which for u' > immediately leads to contradiction since, up to then, 
 that is between u and u^, u' had been the smaller value. If, how- 
 ever, u' < 0, one has in case of a variation of the C by /!!£! 
 
 A —^ + 4a 
 
 Uq - u 
 
 P 
 
 i / aIu'I du = 2 AC 
 
 Uq-uJ^ 
 
NACA TM 13^2 25 
 
 or, since 
 
 Z^'^ = A|u'| (2|u'| + A|u' I) 
 
 2 AC 
 
 Uq - u Uq - u.^ 
 
 If there vere at one point a|u'| = 0, then at this point the first 
 term woiild^ for fixed AC, decrease with decreasing u, that is, go over 
 from positive to negative values; the second term also woiild decrease 
 
 since the part of the integral / ^P ( '^^ supervening with 
 
 ^u 
 decreasing u woiild be negative. This is impossible, however, since 
 the sum of both terms is supposed to be constant 2 AC . 
 
 Therewith, it has been generally proved that the angle opening t3, 
 decreases with increasing C (for fixed u^,); since the maximum possible 
 tS-, is desired, the minimum admissible value for C may be assumed. 
 
 This value is determined from v > 
 
 ■> n ofo / o oV / o c>\ n2 P^Q 2 -sSP 
 
 3c/0A.S-(,^-a^)(v/-/)-|i 
 
 V e dv 
 
 V 
 
 whence, one may see that the minimiim admissible value of C is zero or 
 negative^. 
 
 The above inequality must be valid for all v ' s between v_^ and 
 
 zero and for the positive and negative 9 attained. One may write them 
 
 ^n ^ t o'^ „'^\ I,. . „>„~'^'*^u ^ /C, 2 , „ „ , „2\_ ^2 0/ 
 
 S.(p3.,2)(.^,.ya'P0.|.(,^2,,^,,,2)3 
 
 60 
 
 According to page 22 y p,^ - a,^ > 0. 
 
26 
 
 NACA TM 131+2 
 
 where cp^,^^ are certain mean values. One must further note that, for 
 
 V = v^, 1^0 I ^^^ 1*^0 I must be zero whereas they have maximum values 
 
 for V = 0. The severest restriction is due to the absolutely smaller 
 value of the right side, thus the minimum admissible C is given by 
 
 2C 
 
 .2 .2V_.-0 p-^, 2/K'^2'^^0') 
 
 = -(p^-a2)v, 
 
 -m 
 
 a Vr,e 
 
 6^^0 
 
 Vn e 
 
 9(^',<P„' are positive and the maximum of 9^ and 9_ which occur for 
 V = 0. 
 
 Consequently, one has for the maximum possible i3. 
 
 v'2 = (p2 _ ^2) 
 
 (vo^ - ^) - "0-^' X 
 
 V 
 
 2a 
 
 "°vV-V.lv„V^(''^'^''0') Pa-'dv 
 
 V 
 
 = (32- 
 
 (^0^ - ^') - ^0^0 - ^)< 
 
 ^(^o'-^o) 
 
 2a 
 
 l(v„3 . .3)e-^^2 . 1 .^3(.„ . .)e 
 
 -'P2'-{''o'-''o) 
 
 Since cp ' > cp and 9^' > 9^ 
 
 ^'^ >K - ^y 
 
 (,^ - a^) . |(v . v„)e 
 
 -a9. 
 
NACA TM 13^2 27 
 
 and since cpp ' ^ 'S-\ 
 
 v'2> 
 
 2 , . -aT3^ 
 
 (^0 - ^y {»' - ^') * B^ ^ ■'o> ' 
 
 whence follows 
 
 ■8^ < 2n 
 
 y(p^ - a 
 
 ^2 - -a-a. 
 
 ^ "^) ^ I- V„(X . e)e-"l 
 
 (Compare formula (4), page 19)- 
 
 Hence follows that with increasing v , thus also with increasing 
 u^, t3, must drop below all limits: a certain width of the spiral 
 permits only a limited outflow velocity. 
 
 SECOND PART 
 
 10. Since the only spiral motion, possible without walls, of the 
 type used so far, lead to be a potential motion, exact steady and non- 
 steady two-dimensional motions in free spirals will be investigated 
 according to another method. 
 
 In polar coordinates the differential equation (l) reads 
 
 d A^l^ ^ 1/ S A^ M _ ^ Af ^ = A Ai 
 dt rl hr ^9 d9 br 
 
 where 
 
 ^ Sr ^r r^ ^9^ 
 
28 
 
 NACA TM I3I+2 
 
 Obviously this equation permits solutions which are linear in 
 
 i|/ = u + cp« 
 
 in order to make the velocity which has the con^jonents 
 
 V = « 
 r r 
 
 and 
 
 V =_^ _ cp ^ 
 
 unique and thus enable a free motion^ K must be constant. By this 
 statement, the differential equation becomes 
 
 a_M + xb_M = a A Au 
 
 dt ^ ar 
 
 with 
 
 Au = I A ^ 
 
 Here it is necessary also to investigate the pressure lest perhaps 
 in a motion about the singular point r = a multivaluedness of the 
 pressure becomes evident . 
 
 Now one may write the equations of motion without elimination of 
 the press\ire in the following form 
 
 dfP + i v2 
 
 M 2 
 
 = At d^lf + 
 
 Sfa A - ^U 
 St, 
 
 dx 
 
 afo A - ^u 
 
 ^t, 
 
 Sx 
 
 dy 
 
 or because of the invariance of the last term 
 
 = At d^^ + 
 
 r S9 
 
 dr 
 
 dr 
 
 r d9 
 
NACA TM 13^2 29 
 
 Hence, follows 
 
 ^[l + lA = Ai ^ - r |-(a A - l-V = ^ Au - or ^ + r ^^ 
 
 By virtue of the differential equation for u the right side is con- 
 stant; thus it must be zero to make the pressure in case of a revolution 
 about r = revert to its former value, so that one obtains 
 
 r ^ ^ - L ^ u - 2^ Au = 
 Br <^ Sr at ^ 
 
 which by introduction of r ^ = v assumes the form 
 
 or 
 
 h^v - fl + ^^ ^ = ^ (V) 
 
 ^^2 V <^/r Sr St 
 
 Steady Motions 
 11. The solution independent of t is 
 
 a 
 u = c r + c„ In r + c^ 
 
 when ^ ^ -2, otherwise, when — = -2 
 a ^ a 
 
 u = c, (in r) + c„ In r + c. 
 
 '2 -^ ^3 
 
30 NACA TM 13i4-2 
 
 If one disregards the trivial case of potential motion, a spiral motion 
 the velocity of which disappears at infinity exists when 
 
 ^ + 1 < 
 
 that is, K < - a, thus a sufficiently strong inflow takes place. 
 The spirals then have the form 
 
 cp =-^ u = C^r + C_ In r + C_ 
 K 1. tL J 
 
 If ^2^ + 2 < 0, they approach at infinity the logarithmic spirals; near 
 a 
 
 the sink, in contrast, they converge considerably less pronouncedly 
 
 toward the sink point, and the vortex velocity is considerably higher 
 
 than in case of potential flow in logarithmic spirals . 
 
 Unsteady Motions 
 12. If one uses the formulation 
 
 V = e'^^X (r) 
 n 
 
 one obtains from equation (V) 
 
 ^ r \ a/ n an ,^ 
 
 X. 
 
 thus, with the abbreviation X, = 1 + -^^ 
 
 2a 
 
 ^n " ^'^+x'- '-- ^ 
 
NACA TM 13^4-2 
 
 31 
 
 where the J are the Bessel fiinctions 
 
 Al'o 
 
 J I / _ i± r 1 = const r" 
 
 1 + ^^ 
 
 4 1(1 + X) 
 
 n!/r\^ 
 
 (-) ^ - 
 
 ^^J 21(1 + \)(2 + \) 
 
 If \ does not happen to be an integer, r J and r J may be 
 
 A. "K 
 
 regarded as independent solutions. 
 
 13- Similarly to the case of the heat conduction equation there 
 exist also of equation (V) integrals which show for r = and t = 
 an indeterminate point. 
 
 Since the differential equation (V) remains unchanged if v is 
 multiplied by an arbitrary factor, r by a similar factor, t by its 
 square, there must exist solutions of the form 
 
 = '^""(i^) ■ '■'^'"'^) 
 
 After substitution, one obtains for w the differential equation 
 
 w' ' + V ' /cL + 1 - ^ \ + ^j /g^ - 2X,a + M = 
 
 (VI) 
 
 When does this equation permit a solution of the form 
 
 w = eV-^ 
 
 A simple calculation yields 
 
 la = -1 
 
32 NACA TM 1342 
 
 and then either 
 
 a = 3 = X-1 
 or 
 
 a = 2X (3 = -X, - 1 
 
 One thus has two simple integrals of the required type 
 
 1 r 
 V = t'' -^e 
 
 and 
 
 1 r2 
 
 2H-I-X3 ^- t 
 
 V = r 
 
 for X = 0, both are transformed into the known integral of the heat 
 conduction eqiiation. 
 
 Let us continue the discussion of the differential eqixation (Vl) . 
 
 The singular point z = is a determinate point. The determining 
 equation reads 
 
 p2 + p(a - X) + "^ - ^^'^ = 
 
 and has the roots 
 
 1 2 2 2 
 
NACA TM 13^2 
 
 33 
 
 so that generally there exist developments of the form 
 
 X-^. 
 
 \ = ' ^(l-c^ 
 
 z + c z + . 
 2 
 
 and 
 
 thus 
 
 a 
 
 w_ = z ^[l + c 'z + c 'z^ + . . .^ 
 
 a 
 
 v-^ = r^H ^ 
 
 1 + Ct 7^— + c^ 
 
 '1 li(Tt 2 p 
 
 and 
 
 V = t 
 2 
 
 3*1 
 
 2 1^ 
 
 1 + c, ' -i^ + c„' ^ 
 
 with the power series continuously converging since z = is the only 
 singular point of the differential equation. 
 
 If one assumes p = 0, that is, if one desires solution of equa- 
 tion (V) of the form 
 
 ■-{£> 
 
 an integration by definite integrals is possible. 
 
 The differential equation (Vl) reads after introduction of the 
 
 roots p ,p 
 l' 2 
 
 dz 
 
 (VI •) 
 
3^^ NACA TM 131^2 
 
 The connection with Gauss' equation for the hypergeometric fiinction can 
 be easily recognized. If one makes the Euler transformation 
 
 w = / e 
 
 '^1 - j)\(s) ds 
 
 with the integral extended over a suitable closed path, one finds for y 
 a differential equation which may be satisfied for 
 
 and for 
 
 -1+Pp 
 n = -p^ by y - s 
 
 -1+p, 
 n = -Pg by y = 6 
 
 Therefore 
 
 " - M - 1) 
 
 and 
 
 M^ - 1) ^ '^' 
 
 ■Pi -i+P, 
 
 s "^ds 
 
 P2 -1+p, 
 
 ds 
 
 are integrals of eqiiation (VI'). The integrals are extended best over 
 a path which leads from R(s) = +» around the points 6=0 and s = z 
 back to R(s) = +c». 
 
 Since 
 
 -p -l+p„ 
 r^(z - s) -^s ^ds 
 
NACA TM 13i<-2 35 
 
 is analyliically regular in the neighborhood of z = 0, there is 
 
 and 
 
 -P2 -1+p 
 
 ^2 = 0^/ e-^fl - ^^ s Ms 
 
 One can show that 
 
 V = > r^-ICw 
 
 a= - 00 
 
 "Fi"i(i^) * ^2-3(1^ 
 
 are the general solutions of equation (V) and likewise are represented 
 
 by definite integrals in closed form. I shall perhaps refer back to 
 
 this and to the connection with the representation and the development 
 in terms of Bessel functions elsewhere. 
 
 THIRD PART 
 Neighborhood Solutions to Radial Flow 
 
 Ik. We shall first look for steady neighborhood solutions to the 
 radial flow (pages 12 and I3) by putting 
 
 11/ = f(9) + p(cp,r) 
 
 where p is assumed to be a small quantity, the square of which is 
 neglected. 
 
 We then obtain for f the former equation 
 
36 MCA TM 13^+2 
 
 with f ' = u and integrating once 
 
 u''+Uu+-u^+C=0 
 
 For p we obtain 
 
 S Ap ^ u S Ap _ Sul Sp _ uJJ. ^ = a A Ap 
 
 St r Sr 4 S9 3 br 
 r r~^ 
 
 where 
 
 n2 
 
 A = li-r A + X^ 
 r dr dr ^2 ^cpS 
 
 Since the differential equation in the steady case remains unchanged 
 when r and p each are multiplied by an arbitrary factor, there must 
 exist solutions of the form 
 
 p = r\f(cp) 
 One obtains for w the differential equation 
 
 w^ + w' ' (ZX^ . kX + k + ^\i-^u\+^ u'w' + 
 \ a a J a 
 
 w/x^ - U3 + 2X2 + ^X^ - x3 ^ ^ ^ ^, ^ ^ Q (VII) 
 
 We are particiilarly interested in free flows and thus in periodic solu- 
 tions in 9. 
 
 As concerns the uniqueness of the pressiire (see paragraph 10, 
 page 28), one obtains by a simple calculation, the condition 
 
 ^2rt 
 
 \^ / wfu - (X - 2)0ld'P = 
 
NACA TM 13^4-2 37 
 
 On the other hajid, there follows from the above differential eqiiation 
 itself, by integration over the interval from to 2jt with assiomp- 
 tion of periodicity 
 
 «2n 
 
 
 A,2(X. - 2) / w[u - (X - 2)a]dT = 
 
 so that in general the uniqueness of the pressvire follows from the peri- 
 odicity except for the case when X = Z. 
 
 For 'free flow, u itself is a periodic function of 9; the period 
 is an integral w part of 2« . However, it is not necessary that w 
 have the same period as u; but this period must likewise be an integral 
 part of Zn . 
 
 Since 
 
 1+U - i u2 
 
 ■as well as 
 
 U.2 = _2Cu - l.u2 - ^ u3 . D = ^(e^ - u) (u - e^) (u - e3) 
 
 may be rationally expressed by u, it will be useful to introduce u 
 instead of cp as independent variable in equation (VIl) . Because of 
 
 w' = ^ u' 
 
 du 
 
 2 
 w' • = d^ u'2 + ^ u" 
 du2 ^^ 
 
 3 2 
 
 w' ' • = ^_J£ u'3 + 3 d^-w; u'u' ' +^u'" 
 
 ^3 ^2 du 
 du-" du 
 
 ^IV ^ d^ ^,1+ + 5 d\; ^_^,2^, , + 3 d^i ^, .2 + 1^ dfw ^,^, , , + dw ^IV 
 
 A ^ A 3 ^2 ,2 du 
 
 du du du du 
 
38 NACA TM 13^2 
 
 and because of 
 
 u^'^ = Lk - ^ u\n' ' - I u'2 u'u' ' ' = {-k - '^ u]u'2 
 
 all coefficients of the new equation are integral and rational in u; 
 indicating the degree, one writes them 
 
 cLy: + R d^w + R dw + j^ dw + R (VII') 
 
 with 
 
 du du du 
 
 9a 
 
 From the form (VIl) one can see that w possesses singularities 
 
 only where they occur for u, thus certainly not in the real part of 9 
 
 (which is of interest); equation (VII') shows that, as a function of u, 
 
 w becomes singular only at the branch points e ,e ,e . 
 ( 1 "^ J 
 
 Since R^ = 6u'^u'' is divisible by (e-|^ - u)(u - e2)(u - e^), 
 the points e, ,e„,e-, are determinate points, and since the degree of 
 
 the coefficients decreases steadily by 1 with the order of the deriva- 
 tives, u = oo also is a determinate point; the differential equa- 
 tion (VII') belongs to the Fuchs class. 
 
 A well-known calculation yields as the four roots of the determining 
 equation for the points e the values 
 
 p=0 p=l p = — p, = ^ 
 
 1 2 3 2 k Z 
 
 Although, therefore, two root differences here are integral, no loga- 
 rithmics appear in the developments: For from the form (VIl) there fol- 
 lows that at the points 'P for which u becomes = e, where, therefore, 
 u and u' are regular functions of 9, w also must be such a func- 
 tion, whereas ln(u - e) does not possess this regiilar character. 
 
NACA TM I3I+2 39 
 
 Therefore, the solutions of equation (VII') have at every point e the 
 form 
 
 u = P-|^(u - e) + v/u - eP^Cu - e) 
 
 other singularities do not exist in a finite domain. 
 
 For u = a> there results the determining equation 
 
 (2^2 + ^ - i^jL^ + ^ ^ + lx\ =0 
 
 which has the roots 
 
 ^1 = ^ ^2=-| 
 
 which are independent of \, and the roots 
 
 ^3 
 
 = -5, 1^25 - 2kX 
 
 k ~ k 
 
 which are dependent on X. 
 
 Continuation 
 
 15. Solutions with the real period 2jt (this period must be present 
 at least in case of free flow) will exist only for certain A,. In 
 analogy with Hermite's method for Laine's differential equation, one can 
 proceed as follows: 
 
 If w ,w ,w_,w. are a fundamental system of equation (VIl), the 
 w(cp + 2n) are expressed homogeneously linearly by the w 
 
 w (cp + 2rt) = ^ a w (cp) (v = 1,2,3A) 
 
 11=1 '^ ^ 
 
ko 
 
 NACA TM 1314-2 
 
 There certainly exist periodic functions of the second kind, that 
 is, there exist solutions w for which 
 
 v(cp + 2rt) = aw(9) 
 
 This a is a root of the eq^uation of the foiirth degree 
 
 D(a; X) H 
 
 ^ 
 
 '21 
 
 ^31 
 
 %1 
 
 - a a 
 
 12 
 
 22 
 
 32 
 
 \Z 
 
 ^13 
 
 a a 
 
 23 
 
 33 
 
 ^^^3 
 
 'ik 
 
 '2k 
 
 a a 
 
 3h 
 
 ai. 1. - a 
 
 \k 
 
 = 
 
 If a periodic solution is to exist, a = 1 must be a root, ajid one 
 obtains for \ the equation 
 
 D(l,\) = 
 
 The characteristic exponents which were calculated suggest the 
 attempts of putting 
 
 w = u + const, w = . /u - e etnd w = ^jTe ^~li77ir~^^~e7T 
 
 Elementary calculation yields the following particular solutions: 
 
 1. The trivial possibility w = u for X, = 
 
 2 . w = u for X. = 2 , that is 
 
 p = ar'^u 
 
 \|' = f(cp) + or^f'Ccp) 
 
NACA TM 13it.2 kl 
 
 where a must be small and therefore with the same approximation 
 
 4 = f (cp + ar^^ 
 so that the streamlines are approximately the spirals 
 
 cp = cp - or^ 
 
 
 f ' remains the same elliptic fiinction discussed before in the case of 
 radial flow- It is true that this flow now cannot exist as free flow, 
 since this is precisely the exceptional case X, = 2 (see page 37) J and 
 the condition for the imiqueness of the pressure caxi certainly not be 
 satisfied for w = u. 
 
 3. w = u + 3a, when X = 1 and C = 3<^ whence for e, - e_ < a\/3 
 no contradiction results. 
 
 k. V = ,/u-eT for X = -1 and e = 0. This solution has a 
 period twice that of u; likewise, w = ^/el - u for X, = -1 and e, = 0. 
 
 5. w = \/(eZ - uwu - e \ or w = >/m - e wu - e \ when X = 1 
 and e_ = or e, = 0. This solution too has a period twice that of u. 
 
 The large X, may be easily calculated approximately from equa- 
 tion (VIl) . For such large X. there is in first approximation 
 
 w-"-^ + ZX^v' ' + xK = 
 
 that is, w = e- ^^ (we restrict ourselves to the periodic solutions), 
 so that — is the period. The large set -apart X, -values are therefore 
 approximately integral. 
 
 Finally, one case may be calculated quite elementarily: the case 
 when u is constant, the basic flow therefore an all around uniformly 
 distributed flow. 
 
kZ NACA TM 1314-2 
 
 This case is also of significance for the more general one since 
 
 1+ 
 according to a well-known theorem by Cauchy and Boltzmann the period 
 
 of w in first approximation is obtained if the constant mean value is 
 
 inserted for the periodic u, under the presupposition that the larger 
 
 fluctuation e - e be s\iff iciently small. 
 
 For constant u there follows from equation (VII), page 36 
 w' ' (ZX^ . kX + k + ^ ~ ^ u) + v(\^ - U3 + k\^ + X.2 ^ - ^ \i\ = 
 
 w^ + w' ' IZX'^ - kX 
 
 thus with the formulation 
 
 w = e^i^ 
 
 U^ - ^2 hx'^ - 2X + k + ^-^:-A u\ + X^ - kX^ + kx"^ + X^ ^^^-t^ u = 
 
 This equation has four roots 
 
 [X = ±X \? = {X - 2)"^ - "Ljz-k u 
 
 a 
 
 so that all integral positive sjid negative X are possible (potential 
 motions) as well as all X which are calciilated from 
 
 ^fe^\A^^ 
 
 with integral n. 
 
 For the case u = const, that is: 
 
 16. For the radial flow which is uniform all around, the unsteady 
 neighborhood solutions also can be given. 
 
 Boltzmann, Ges. Abh., Bd. 1, S. kj,. 
 
MCA TM 131^2 1^3 
 
 The differential equation now reads (see page 36) 
 
 d Ap u ^ Ap . . 
 
 One may integrate it either by means of the formulation 
 
 U+ni9 , V 
 Ap = e "w(r) 
 
 (n integral) and thus arrives at the differential equation 
 
 2 1 - "" / 2 \ 
 ^-Jt + 2. dw _ f n_ ^ M^ ^ 
 
 dr2 ^ ^ lr2 '' 
 
 which may be solved by Bessel functions, or by means of the formulation 
 (compare page 32) 
 
 Ap = e^^'^r^U^\ = e^^'^T^iz) 
 
 whereby one obtains for w(z) the differential equation 
 
 z^w' ■ + zw'fm + 1 - i^ + z] + w F^ : '^^ - Su ) = 
 
 r'fm 
 
 2a \ k ho 
 
 For z = this equation has a determinate point, the determining equa- 
 tion has the real roots 
 
 2 k a 2\l k ^Z 
 
kk NACA IM 13^2 
 
 By introduction of the roots p^ ajid p the differential equation 
 assumes the form 
 
 z^w' '+zw'(l-p -p +z)+ppw=0 
 ' 12 / 12 
 
 This is, however, exactly the differential equation (VI') of page 33 so 
 that everything said about it there is also valid here. 
 
 Translated by Mary L . Mahler 
 National Advisory Committee 
 for Aeronautics 
 
 NACA-Langley - 1-16-53 - 1000 
 
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