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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 pressure in spherical coordinates can be given by the following expressions: S F(O) r r f(9) V9 = r p g(9) P = 0 P r 2 PP r2 r A particular solution of the equations of NavierStokes for this case was obtained by Landau (reference 1). In the present .paper a general solution is given of the equations of NavierStokes for the motion of the class under consideration. Substituting expressions (1) in the equations of NavierStokes and in the equation of continuity yields the following system: F2+ f2 fF' + 2g+V F" + F' cot 2f' 2F 2f cot 9 =0 (2) ff' + g' f" + f' cot 9 + 2F' f (1 + cot2 )] = 0 (3) F + f' + f cot e = 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. 10311034. NACA TM 1349 f'2 + ff" + 3ff' cot 8 + 2g [i"' + 2f" cot e f' (2 + cot2 ) + f cot 9 (1 + cot2 e) = 0 (5) ff' + g' + I f" + f' cot 9 f (1 + cot2 )] = 0 (6) Differentiating expression (6) f'2 + ff" + g" +V f1" + f" cot 0 2f' (1 + cot2 e) + 2f cot e (1 + cot2 )] = 0 (7) Eliminating the nonlinear terms f'2, ff", and ff' from equation (5) with the aid of equations (6) and (7) yields a linear equation in the function g + 2Vf': (g + 2Vf')" + 3 cot e (g + 2Vf')' 2 (g + 2Vf') = 0 (8) the general solution of which is in the form g + 2Vf' = 2V2 b cos 2 a (9) sin e where 2v2a and 2V2b are constants of integration. Integrating equation (6) f2 + 2g + 2V (f' + f cot e) = 2P2c (10) where 212c is the constant of integration. The function g(e) is eliminated from equations (9) and (10) to give an equation of the Riccati type for the function f:l 1L ffrrt o plb cos 0 a c f = f2 + f cot 0 + 2 b cos 0 a + ) (11ii) 2V ( sin2 e 2 iAfter 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 1349 The substitution f = 2,x'(9)/x(e) (12) reduces equation (ll) to the linear equation: X" ' cot + (b cos 0a + = 0 (13) sin20 2; which by the substitution z = cos2 (0/2) (14) is transformed into an equation of the Fuchsian type: d2X a + b 2 (b + c) z + 2cz2 X = o (15) dz2 4z2 (z 1) The usual computations (reference 3), which are omitted herein, give the general solution of equation (15) as: () (cos 7 (sin ) c (la,P,y, cos2 + c2F (a + 1 r,P + 1 y,2 r, cos2 (16) where the parameters of the hypergeometric function a,4,y (which can also have complex values) are connected with the constants of integra tion a,b,c by the formulas: (a + 3)2 1 a = 2 (1 + a + ) y + 2 2 2 2 (C + 8) l (17) c = (a p)2 1 2 NACA TM 1349 Formulas (4), (9), (16), and (17) give the general solution, depend ing on the four constants a, b, c, and A = c2/cl, of the NavierStokes 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 momentum transfer: lik = Pbik (18) Carrying out the computations 2V2p nee = 2L 2v2 Ir0 = r r b cos 9 a) Ssin2 a a b cos 0 c\ Sin2 . (c cos a b \ sin2 6 (19) The streamlines are determined by the equation: dr/vr = rdO/ve (20) the integration of which gives const/r = f sin 0 (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 = I. In this case a =b = =0 (22) and therefore by equations (19) ripcp= lee = TI"r = 0 +pVi(kvi + PViVk PV +xk (23) NACA TM 1349 The particular solution of equation (15) X(9) = 2z 1 A (24) leads by formulas (4), (9), and (12) to the solution found by Landau: F() = A2 1 F(9) = 2V A 1) 1 (A cos 0) 2 1 2f() sin 9 cos 9 A g(e) = 4V2 1 A cos (25) (cos 0 A)2 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 8. (b) Equation (15) has only two regular points z = 0 and z =w. In this case it follows from equation (15) that a = b =c 0 and equation (11) becomes Euler's equation 2z2 (d2X/dz2) aX = 0 (26) (27) the general solutions of which are Correspondingly, f(e): X(B) = ex/2 cosh (nx + A) for a> 1/2 X(8) = ex/2 cos (nx + A) for a< 1/2 X(9) = ex/2 (1 + Ax) for a = 1/2 the following equations are obtained for I (21 the function 3) sin 0 f 1+ s n tan (nx + A); for a < 1/2 sin e A 1 + cos 0 1 + Ax for a= 1/2 (29) (30) (31) NACA TM 1349 where x = In (1 + cos 8), n = 1/ + 2a (For a = 0, n = 1/2 in equation (29) the solution of Landau is again obtained.) For the solution of equation (29) by formula (4) F(e) = 2 n tanh (nx + A) + 1/2 + 22 1 cos 0 n SCose cosh (nx + A) (32) while g(e) is determined by formula (9). The equation of the streamlines is in the form const/r = (1 cos e) n tanh (nx + A) + 1/2 (33) where the values of the constants n and A are determined from the conditions f( 2) 2 (n tanh A + 1/2) n2 (34) 2 2 cosh2 A The obtained solution corresponds to the problem of the stream flowing from the half line 9 = v 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) [1/2 n tan (nx + A)] 8 = arc cos (ex 1) (35) The function (35) for e. t 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 Lifshits, E.: Mechanics of Dense Media. GTTI, ML, sec. 19, 1944, p. 77. 2. Slezkin, N. A.: Uch. zap. MGU, no. 2, 1934. 3. Smirnov, V. I.: Course in Higher Mathematics. 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