Stationary Hydrodynamic Equations (Navier–Stokes Equations)

B.7.1. Stationary Hydrodynamic Equations (Navier–Stokes Equations)

The two-dimensional equations of steady-state motion of a viscous incompressible fluid (stationary Navier–Stokes equations) Ó

are reduced to the equation under consideration. To this end, one introduces the stream function × ß by the formulas 1 = à✗á

and à✗â × 2 =−

and eliminates the pressure , using cross differentiation, from the first two equations.

. Exact solutions in additive form:

( ➼ , ➪ )= ä 1 exp( å ➼ )− æ å ➼ + ä 2 exp( å ➪ )+ æ å ➪ + ä 3 ,

( ➼ , ➪ )= ä 1 exp( å ➼ )+ æ å ➼ + ä 2 exp(− å ➪ )+ æ å ➪ + ä 3 , where ä 1 , ç✗ç✗ç , ä 5 , and å are arbitrary constants.

2 ➺ . Exact solutions:

, ➪ )= ➶PÏ sinh( î ➼ )+ Ð cosh( î ➼ ) ➴✗è + ( î 2 + å å 2 ) ➼ + ä ,

, ➪ )= ➶PÏ sin( î ➼ )+ Ð cos( î ➼ ➴✗è − é ) + å ( å 2 − î 2 ) ➼ + ä ,

+ , where Ï , Ð , ä , í , ì , î , and å are arbitrary constants.

3 ➺ . Exact solution:

where the functions ô = ô ( ➪ ) and õ = õ ( ➪ ) are determined by the system of fourth-order ordinary differential equations

(2) Integrating yields the system of third-order equations

It is not difficult to verify that equation (1) has the particular solutions

(7) where ➹ , ö , and å are arbitrary constants.

In general, equation (4) can be reduced by the change of variable ÷ = õ ➬ ✒ to the second-order nonhomogeneous linear equation

where ÷ = õ ➬ ✒ . (8) The corresponding homogeneous equation (with Ð = 0) has two linearly independent particular

. (9) (The first solution is apparent from comparing equations (1) and (8) with Ð = 0.) The general

solutions of equations (8) and (2) are given by

3 =− æ . (10) The general solution of equation (2) corresponding to the particular solution (6) is represented

1 ( ➪ + ➹ ) 3 + ä ü 2 + ä ü 3 ( ➪ + ➹ ) + ä ü 4 ( ➪ + ➹ ) −2 , where ä ü 1 , ä ü 2 , ä ü 3 , and ä ü 4 are arbitrary constants (these are expressed in terms of ä 1 , ä 2 , ä 3 , and ä 4 ).

The general solutions of (2) corresponding to the particular solutions (5) and (7) are determined from (9) and (10).

4 ➺ . Exact solution of a more general form:

where the functions ô = ô ( ý ) and õ = õ ( ý ) are determined by the system of fourth-order ordinary differential equations

Integrating yields the system of third-order equations

where Ï and Ð are arbitrary constants. The order of the autonomous equation (13) can be reduced by one.

It is not difficult to verify that equation (11) has the particular solutions

where ➹ , ö , and å are arbitrary constants. In general, equation (14) can be reduced by the change of variable ÷ = õ ➬ þ to a second-order nonhomogeneous linear equation. ÿ✁

Reference : A. D. Polyanin (2001d).

This equation describes plane flow of a viscous incompressible fluid under the action of a transverse Ó force ( ➚ is the stream function). The case ô ( ➪ )= ➹ sin( å ➪ ) corresponds to A. N. Kolmogorov’s

model which is used to describe subcritical and transcritical (laminar-turbulent) modes of flow.

1 ➺ . Exact solution in additive form for arbitrary ✄ ( ➪ ):

where ä 1 , ä 2 , ä 3 , ä 4 , and å are arbitrary constants.

Example. In the case ✆ ( )= ✝✟✞ cos( ✞ ), which corresponds to ✠ ( )= ✝ sin( ✞ ), it follows from the previous formula with ✡ 1 = ✡ 2 = ✡

=0 and ☛ Ú =− Ý ☞ that Ú

( , )=− 2 2 2 2 ☛ sin( ✞ )+ Ý ✞ cos( ✞

where ☛ and ✡ are arbitrary constants. This solution was indicated by Belotserkovskii and Oparin (2000); it describes the flow with a periodic structure.

. Exact solution in additive form for ✄ ( ✎ )= Ï è é + Ð è − é :

where ä 1 and ä 2 are arbitrary constants.

3 ➺ . Generalized separable solution for arbitrary ✄ ( ✎ ):

where the functions ✑ = ✑ ( ✎ ) and ✒ = ✒ ( ✎ ) are determined by the system of fourth-order ordinary differential equations

(2) Integrating yields the system of third-order equations

(4) where ✖ and ✗ are arbitrary constants. The order of the autonomous equation (3) can be reduced

by one. It is not difficult to verify that equation (1) has the particular solutions

(7) where ✘ , ✙ , and ✜ are arbitrary constants.

In general, equation (4) can be reduced by the change of variable ✢ = ✒ ✓ ✒ to the second-order nonhomogeneous linear equation

+ ✑ ✢ ✒ ✓ − ✑ ✓ ✒ ✢ + ✣ = 0, where ✢ = ✒ ✓ ✒ , ✣ = ù ✄ ( ✎ ) ✎ + ✗ . (8)

The corresponding homogeneous equation (with ✣ = 0) has two linearly independent particular solutions: ✤

for

. (The first solution is apparent from comparing equations (1) and (8) with ✣ = 0.) The general

solutions of equations (8) and (2) are given by

The equations of steady-state flow of a viscous incompressible fluid (stationary Navier–Stokes equations) written in polar ✭

coordinates ( ✲ = ✳ cos ✴ , ✵ = ✳ sin ✴ ) reduce to this equation. The radial and tangential components of the fluid velocity are

expressed in terms of the stream function ✶ by the formulas ✷ ✸ = 1 ✸ à✺✹ à✟✻ and ✷ ✻ =− à✺✹ à ✸ .

1 ➺ . Exact solution in additive form:

( ✽ , ✾ )= ✕ ✧ 1 ✾ + ✧ 2 ✽ ✿ 1 +2 + ✧ 3 ✽ 2 + ✧ 4 ln ✽ + ✧ 5 ,

where ✧ 1 , ❀✺❀✺❀ , ✧ 5 are arbitrary constants.

2 ➺ . Exact solution:

The functions ❁ = ❁ ( ✽ ) and ❂ = ❂ ( ✽ ) are determined by the system of ordinary differential equations

− ❁ ❃ ✓ L( ❁ )+ ❁ [L( ❁ )] ✓ ❃ = ✕ ✽ L 2 ( ❁ ),

(2) where L( ❁ )= ✽ −1 ( ✽ ❁ ❃ ✓ ) ✓ ❃ .

− ❂ ❃ ✓ L( ❁ )+ ❁ [L( ❂ )] ✓ ❃ = ✕ ✽ L 2 ( ❂ ),

Exact solution of system (1)–(2):

1 , ❀✺❀✺❀ , ● 6 are arbitrary constants.