\JI\cAl/^-Mf 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1349 
 
 ON A CLASS OF EXACT SOLUTIONS OF THE EQUATIONS 
 
 OF MOTION OF A VISCOUS FLUID 
 
 By V. I. Yatseyev 
 
 Translation 
 
 "Ob odnom klasse tochnykh reshenii uravnenii dvizheniya vyazkoi 
 zhidkosti." Zhurnal Eksperimental 'noi i Teoretisheskoi Fiziki, 
 
 vol. 20, no. 11, 1950. 
 
 Washington 
 February 1953 
 
%ft nil Y^ >//^bv^ 
 
 NACA TM 1349 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM NO. 1349 
 
 ON A CLASS OF EXACT SOLUTIONS OF THE EQUATIONS OF MOTION 
 
 OF A VISCOUS FLUID* 
 By V. I. Yatseyev 
 
 The general solution is obtained herein of the equations of motion 
 of a viscous fluid in which the velocity field is inversely proportional 
 to the distance from a certain point. Some particular cases of such 
 motion are investigated. 
 
 1. The motion of a viscous fluid with velocity field and pressiire 
 in spherical coordinates can he given by the following expressions: 
 
 Vv. = 
 
 F(9) _ f(e) p _ g{9) , . 
 ^0 = -^- ^ = p = —2- (1) 
 
 'r ~ r 
 
 r" 
 
 A particular solution of the equations of Navier-Stokes for this case 
 was obtained by Landau (reference 1). In the present .paper a general 
 solution is given of the equations of Navier-Stokes for the motion of 
 the class under consideration. 
 
 Substituting expressions (l) in the equations of Navier-Stokes and in 
 the equation of continuity yields the following system: 
 
 f2 + f2 _ fp. + 2g 4V 
 
 F" + F' cot - 2f ' - 2F - 2f cot 9 = 
 
 (2) 
 
 ff + 
 
 f" + f ' cot 9 + 2F' - f (1 + cot^ 0) 
 
 = (3) 
 
 F + f ' + f cot 9=0 (4) 
 
 Determining F from equation (4) and substituting in equations (2) and 
 (3) give 
 
 * Ob odnom klasse tochnykh reshenii uravnenii dvizheniya vyazkoi 
 zhidkosti." Zhurnal Eksperimental 'noi i Teoretisheskoi Fiziki, vol. 20, 
 no. 11, 1950, pp. I031-103ij-. 
 
f '2 + ff" + 3ff' cot e + 2i 
 
 V f" 
 
 NACA TM 1349 
 
 + 2f" cot e - f (2 + cot^ 9) + 
 
 f cot 9 (1 + cot2 9) 
 
 = 
 
 f f ' + g ' + U 
 
 f" + f ' cot - f (1 + cot^ 0) 
 
 = 
 
 (5) 
 (6) 
 
 Differentiating expression (6) 
 
 f'2 + ff" + g" + V 
 
 f" + f" cot 9 - 2f' (l + cot^ 9) + 
 
 2f cot 9 (1 + cot^ 9) 
 
 = 
 
 (7) 
 
 Eliminating the nonlinear terms f'^, ff", and ff from equation (5) 
 with the aid of equations (6) and (7) yields a linear equation in the 
 function g + 2uf ' : 
 
 (g + 2Uf')" + 3 cot (g + 2Vf')' - 2 (g + 2Vf') = (8) 
 the general solution of which is in the form 
 
 g + 2Uf ' = 2V 
 
 2 "b cos - a 
 
 2 
 sin 
 
 (9) 
 
 where 2u2a and ZV'^'b are constants of integration. 
 Integrating equation (6) 
 
 f2 + 2g + 2U (f + f cot 0) = - 2P^c 
 where 2V^c is the constant of integration. 
 
 (10) 
 
 The function g(0) is eliminated from equations (9) and (lO) to 
 give an equation of the Riccati type for the function fr^ 
 
 f • = ^ f2 + f cot + 2V 
 
 ( 
 
 b cos - a c 
 sin^ 9 2 
 
 (11) 
 
 lAfter sending the manuscript to press the author obtained from 
 L. D. Landau a communication on the work of N. Slezkin (reference 2) in 
 which he arrived at the same equation by a different method. 
 
NACA TM IS-iS 
 
 The substitution 
 
 f = - 2VX'(0)/X(0) 
 reduces equation (ll) to the linear equation: 
 
 (12) 
 
 X" - X' cot e + 
 
 "b cos - a 
 sin^ 9 
 
 ^^) 
 
 X = 
 
 which hy the substitution 
 
 z = cos' 
 
 (9/2) 
 
 is transformed into an equation of the Fuchsian type: 
 
 d'^X a + b-2(b + c) z + 2cz' 
 
 dz^ 
 
 4z'^ (z - 1) 
 
 X = 
 
 (13) 
 
 (14) 
 
 (15) 
 
 The usual computations (reference 3), which are omitted herein, give the 
 general solution of equation (15) as: 
 
 X(a)=^coa|]'^(Mn|j^«"^-Nc,F(a,p,r, 
 
 2 9 , 
 cos^ - J + 
 
 C2F 1 a + 1 - r^P + 1 - r^2 - r> cos^ - 
 
 (16) 
 
 where the parameters of the hypergeometric function a, P^-y (which can 
 also have complex values) are connected with the constants of integra- 
 tion a,b,c by the formulas: 
 
 2 /-, o\ (a- + P)2 1 "^ 
 = r'^ - (1 + a + p) r + -i^ —^ 
 
 / n T \ (Ct + P) 1 
 
 b = (a+p-l)r- -^ 2~^ ^ 2 
 
 (17) 
 
 c = 
 
 ^ (g - p)2 - 1 
 
 2 
 
 y 
 
NACA TM 1349 
 
 Formulas (4)^ (9)^ (l6)> and (17) give the general solution, depend- 
 ing on the four constants a, b, c, and A = cg/cj^, of the Navier-Stokes 
 
 equations for the class of motion of a viscous fluid under consideration. 
 The constants of integration a., "b, and c are expressed in terms of the 
 corresponding tensor components of the density of the momentvun transfer: 
 
 Sv^ 5v^ 
 
 
 (18) 
 
 Carrying out the computations 
 
 n 
 
 CPCP- ^2 
 
 2v^p (\} cos - a 
 
 sin^ 
 
 _ SV^p / a - b cos c 
 
 ^^ ~ 2 1 . 2 a ~ 2 
 
 r \ sm 8 
 
 n 
 
 2V p /c cos - b 
 
 re = -2" I TT" — 
 
 r \ sm 
 
 > 
 
 (19) 
 
 The streamlines are determined by the equation: 
 
 dr/v^ = rd0/vg 
 
 (20) 
 
 the integration of which gives 
 
 const/r = f sin 
 
 (21) 
 
 2. Attention is now given to two particular examples for which the 
 equation of Fuchs degenerates. 
 
 (a) Equation (15) has only one regular singular point, z = oo. In 
 this case 
 
 a = b = c = 
 and therefore by equations (19) 
 
 (22) 
 
 n:cpq3= tlqq = iip0 
 
 (23) 
 
NACA TM 1349 
 
 The particular solution of equation (15) 
 
 \{e) = 2z - 1 - A (24) 
 
 leads by formulas (4), (9), and (12) to the solution found by Landau: 
 
 F(0) = 2V 
 
 (A - cos 9)2 
 
 f(e) = 
 
 2V sin 9 ■ 
 cos 9 - A 
 
 g(e) = 4V2 
 
 1 - A cos Q 
 (cos 9 - A)2 
 
 (25) 
 
 This solution is analogous to the problem of a stream flowing out of the 
 end of a thin pipe into a region filled with the same fluid. It is the 
 only regular solution for all values of the angle 9. 
 
 (b) Equation (l5) has only two regular points z = and z =oo. 
 In this case it follows from equation (15) that 
 
 a = b = c 5^ 
 and equation (ll) becomes Euler's equation 
 
 (26) 
 
 2z2 (d2x/dz2) - aX = 
 
 the general solutions of which are 
 
 X(0) = e^/2 cosh (nx + A) f or a > - l/2 
 X(0) = e^/2 cos (nx + A) for a < - l/2 } 
 X(e) = eV2 (1 + Ax) for a = - l/2 
 
 (27) 
 
 (28) 
 
 Correspondingly, the following equations are obtained for the function 
 f(9): 
 
 f = 2V 
 
 sm 
 
 1 + cos 
 
 n tanh (nx + A) + l/2> for a > - l/2 (29) 
 
 f = 2V ., ^^^ ^ ^ < Q - n tan (nx + A) > for a < - l/2 (30) 
 
 1 + cos 9)2 '' M ' 
 
 f = 2V 
 
 sm 9 
 1 + cos 9 "l 1 + Ax 
 
 + 1/2 
 
 f or a = - 1/2 (31) 
 
6 NACA TM 1349 
 
 where x = In (l + cos 9), n = -^ L/l + 2a . (For a = 0, n = l/2 in 
 equation (29) the solution of Landau is again obtained.) 
 
 For the solution of equation (29) by formula (4) 
 
 2 
 1 - cos 6 n 
 
 F(0) =- 2V ntanh(nx + A) -Hl/2W2vi-^^2|^ 
 
 J cosh (nx + A) 
 
 (32) 
 while g(9) is determined by formula (9). 
 
 The equation of the streamlines is in the form 
 
 r 1 
 
 const/r = (l - cos 9) <n tanh (nx + A) + l/2 ^ (33) 
 
 where the values of the constants n and A are determined from the 
 conditions 
 
 f f I j = 2U (n tanh A + l/2) ] 
 
 ? r (34) 
 
 The obtained solution corresponds to the problem of the stream flowing 
 from the half line 9 = n into a region filled with the same fluid. 
 
 For solution (30), the parametric equation for the streamlines is 
 in the form 
 
 const/r = (2 - e^) l/2 - n tan (nx + A) 
 
 9 = arc cos (e^ - l) (35) 
 
 The function (35) for -v it is a strongly oscillating one. It can 
 therefore be concluded that solution (30) has no physical sense. 
 
 The author acknowledges the suggestions and help received from 
 Professor Y. B. Rumer. 
 
 Translated by S. Reiss 
 
 National Advisory Committee for Aeronautics 
 
NACA TM 1349 
 
 REFERENCES 
 
 1. Landau, L., and Lifshlts, E. : Mechanics of Dense Media. GTTI, M-L, 
 
 sec. 19, 1944, p. 77. 
 
 2. Slezkin, N. A.: Uch. zap. MGU, no. 2, 1934. 
 
 3. Smimov, V. I.: Course in Higher Mathematics. GTTI, M-L, vol III, 
 
 sec, 162, 1933. 
 
 NACA-Langley - 2-II-53 - 1000 
 
Digitized by tlie Internet Archive 
 
 in 2011 with funding from 
 
 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
 http://www.archive.org/details/onclassofexactsoOOunit 
 
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