On a class of exact solutions of the equations of motion of a viscous fluid

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Material Information

Title:
On a class of exact solutions of the equations of motion of a viscous fluid
Series Title:
TM
Physical Description:
7 p. : ; 27 cm.
Language:
English
Creator:
Yatseyev, V. I
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Equations of motion   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The general solution is obtained 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.
Bibliography:
Includes bibliographic references (p. 7).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by V.I. Yatseyev.
General Note:
"Report No. NACA TM 1349."
General Note:
"Report date February 1953."
General Note:
"Translation of "Ob odnom klasse tochnykh reshenii uravnenii dvizheniya vyazkoi zhidkosti." Zhurnal Eksperimental 'noi i Teoretisheskoi Fiziki, vol. 20, no. 11, 1950."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003770594
oclc - 85871986
sobekcm - AA00006180_00001
System ID:
AA00006180:00001

Full Text









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 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 (1) in the equations of Navier-Stokes 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. 1031-1034.







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 0-a + = 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 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 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, M-L,
sec. 19, 1944, p. 77.

2. Slezkin, N. A.: Uch. zap. MGU, no. 2, 1934.

3. Smirnov, V. I.: Course in Higher Mathematics. GTTI, M-L, vol III,
sec. 162, 1933.


NACA-Langley 2-11-53 1000









































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