Nonstationary Hydrodynamic Equations

B.7.2. Nonstationary Hydrodynamic Equations

The two-dimensional equations of steady-state flow of a viscous incompressible fluid (nonstationary ■ Navier–Stokes equations) can be reduced to this equation by introducing a stream function ✼ .

1 ➺ . Exact solution:

(1) where the functions ▼ ( , ▲ ) and ◆ = ◆ ✪ ( , ▲ ) are determined from the system of fourth-order one- ✪ ✪

dimensional equations ✪

Equation (2) is independent of (3). Integrating (2) and (3) with respect to ❖ ❖ ❖ ❖ yields

(5) where ❁ 1 ( ▲ ) and ❁ 2 ( ▲ ) are arbitrary functions. Equation (5) is linear in ✪ ✪ ✪ ✪ ✪ ◆ . The change of variable

where P = P ( , ▲ ), ▼ = ▼ ( , ▲ ), (6)

with ✪

= ◗ ( ▲ ) satisfying the linear ordinary differential equation

(7) brings (5) to the second-order homogenous linear equation ❖ ❖ ❖ ❖

So, if a particular solution of equation (2) or (4) is known, then determining the function ✪ ✪ ✪ ◆

reduces to solving the linear equations (7)–(8) followed by integrating in accordance with (6). Table B7 lists exact solutions of equation (2). The ordinary differential equations in the last two rows, which determine a traveling wave solution and a self-similar solution, are autonomous and hence admit reduction of order.

The general solution of the nonhomogeneous equation (7) can be found with the aid of the fundamental system of solutions for the corresponding homogeneous equation (with ❁ 2 ≡ 0). The necessary formulas and fundamental solutions of the homogeneous equation (7) that correspond to all exact solutions of equation (2) listed in Table B7 can be found in the handbooks by Kamke (1977) and Polyanin and Zaitsev (1995).

Equation (8) for any function ▼ = ▼ ( , ▲ ) has the trivial solution, P = 0. The expressions in Table B7 and relation (6) with P = 0 define some exact solutions of the form (1). By analyzing ✪

nontrivial solutions of equation (8), one can obtain a wider class of exact solutions. Table B8 lists transformations that simplify equation (8) for some of the solutions of equation (2) [or (4)] given in Table B7 . One can see that in the first two cases, solutions to equation (8) are expressed in terms of solutions to the classical constant coefficient heat equation. In the remaining three cases, the equation reduces to a separable equation.

2 ➺ . Exact solution of a more general form:

where the functions ✪ ( , ) and ◆ = ◆ ( ❯ , ▲ ) are determined from the system of fourth-order one- dimensional equations ❖ ❖ ❖

Integrating equations (9) and (10) with respect to ❖ ❖ ❖ ❯ yields

TABLE B7 Exact solution of equations (2) and (4); ❲ ( ▲ ), ❳ ( ▲ ) are arbitrary functions and ❨ , ❩ , ❬ are arbitrary constants

Function ▼ = ▼ ( , ▲ ) No

Determining coefficients (or general form of solution) ✪

Function ❁ 1 ( ▲ )

in equation (4)

(or determining equation)

1 N/A + ❨ exp[− ❬ + ❬ ❳ ( ▲

is an arbitrary constant 1+

where ❁ 1 ( ▲ ) is an arbitrary function and

) is any].

Equation (12) is linear in ◆ . The change of variable P = ❥ ❦ ❜ takes it to the second-order linear equation ❥ ❖ ❖ ❖ ❖

So, if a particular solution of equation (9) or (11) is known, then determining the function ◆

reduces to solving the linear equation (13). Scaling the independent variables by the formulas

=( ❱ 2 + 1) ❧ and ▲ =( ❱ 2 + 1) ♠ , one can reduce equation (9) to equation (2) in which ♥ and ▲ must

be replaced by ❧ and ♠ , respectively. Exact solutions of equation (2) are listed in Table B7 .

TABLE B8 Transformations of equation (8) for the corresponding exact solutions of equation (4) [the number in the first column corresponds to the number of the exact solution ▼ = ▼ ( ♥ , ▲ ) in Table B7 ]

No Transformations of equation (8) Resulting equation

3 ➺ . Exact solution [special case of (1)]:

( ), and ( ) are arbitrary functions and 1 , 2 , and are arbitrary parameters.

4 ❿ . Exact solution:

( ), and ( ) are arbitrary functions and 1 , 2 , , and are arbitrary parameters.

5 ❿ . Exact solution:

( ⑧ ) ⑨ + ❾ ( ⑧ ) ♥ + ⑩ ( ⑧ ), where ( ⑧ ❻ ), ❾ ( ⑧ ), and ⑩ ( ⑧ ) are arbitrary functions, ❺ and ➋ are arbitrary parameters, and the functions

, ➃✁➄ ⑧ )= ➀ − ( ⑧ ) sin( ➋ ⑨ )+ ➈ ( ⑧ ) cos( ➋ ⑨ ) ➉ +

( ⑧ ) and ➈ ( ⑧ ) are determined by the nonautonomous system of linear ordinary differential equations

The general solution of system (14) is given by

( ⑧ ) = exp ❸➊✕ ( ❺ 2 − ➋ 2 ) ⑧ − ❺

1 sin ➏ ➋

+ ❶ 2 cos ➏ ➋

( ⑧ ) = exp ❸➊✕ ( ❺ 2 − ➋ 2 ) ⑧ − ❺

1 cos

− 2 sin

( ⑧ )= ❶ 1 sin( ➓ ➋ ⑧ )+ ❶ 2 cos( ➓ ➋ ⑧ ),

( ⑧ )= ❶ 1 cos( ➓ ➋ ⑧ )− ❶ 2 sin( ➓ ➋ ⑧ ).

6 ❿ . Exact solutions:

( ➄ ⑨ , ♥ , ⑧ )= ( ⑧ ) exp( ➔ 1 ⑨ + ❺ 1 ♥ )+ ➈ ( ⑧ ) exp( ➔ 2 ⑨ + ❺ 2 ♥ )+

( ⑧ ) ⑨ + ❾ ( ⑧ ) ♥ + ⑩ ( ⑧ ), where ( ⑧ ), ( ⑧ ), and ⑩ ❻ ❾ ( ⑧ ) are arbitrary functions and ➔ 1 , ❺ 1 , ➔ 2 , and ❺ 2 are arbitrary parameters that

satisfy one of the two relations

1 + ❺ 2 1 = ➔ 2 2 + ❺ 2 2 (first family of solutions),

1 ❺ 2 = ➔ 2 ❺ 1 (second family of solutions),

and the functions ➄ ( ⑧ ) and ➈ ( ⑧ ) are determined by the ordinary differential equations

These equations are easy to integrate:

7 ❿ . Exact solution:

( ⑧ ) ⑨ + ⑩ ( ⑧ ), where

( ➃→➄ , ♥ , ⑧ )= ❶ 1 sin( ❺ ⑨ )+ ❶ 2 cos( ❺ ⑨ ) ➉ ( ⑧ ) sin( ➋ ♥ )+ ➈ ( ⑧ ) cos( ➋ ♥ ) ➉ +

are arbitrary parameters, and the functions ( ⑧ ) and ➈ ( ⑧ ) are determined by the nonautonomous system of linear ordinary differential equations

( ➄ ) and ( ) are arbitrary functions, 1 , 2 , , and

The general solution of system (15) is given by

3 cos ➏ ➋ ❣ ❻ ❤ ⑧ ➐ + ❶ 4 sin ➏ ➋ ❣ ❻ ❤ ⑧ ➐ ❼ , where ❶ 3 and ❶ 4 are arbitrary constants.

8 ❿ . Exact solution:

, ➄ ♥ , ⑧ )= ❶ 1 sinh( ❺ ⑨ )+ ❶ 2 cosh( ❺ ⑨ ) ➉ ( ⑧ ) sin( ➋ ♥ )+ ➈ ( ⑧ ) cos( ➋ ♥ ) ➉ +

( ⑧ ) ⑨ + ⑩ ( ⑧ ), where ❻ ( ⑧ ) and ⑩ ( ⑧ ) are arbitrary functions, ❶ 1 , ❶ 2 , ❺ , and ➋ are arbitrary parameters, and the

functions ➄ ( ⑧ ) and ➈ ( ⑧ ) are determined by the nonautonomous system of linear ordinary differential equations

The general solution of system (16) is given by

, where ❶ 3 and ❶ 4 are arbitrary constants.

( ➃ ⑧ ) = exp ✕ ( ❺ 2 − ➋ 2 ) ⑧ ➉ ❸ − ❶

3 cos ➏ ➋

+ 4 sin

9 ❿ . Exact solution:

= ➔ ⑨ + ❺ ♥ , where ( ⑧ ) and ( ⑧ ) are arbitrary functions, ➔ ❻ ❾ and ❺ are arbitrary parameters, and the function ♣ ( q , ⑧ )

( ⑨ , ♥ , ⑧ )= ♣ ( q , ⑧ )+ ❻ ( ⑧ ) ⑨ + ( ⑧ ) ♥ ❾ ,

is determined by the fourth-order linear equation

2 + ➔ ( ⑧ )− ❺ ( ⑧ ) ➉ ➣ 3 = ✕ ( ➔ + ❾ ❺ ❻ q ) ➣ q 4 .

The transformation

2 ➃ , ↕ = q − ➙ ➔ ( ➛ )− ❺ ( ➛ ) ➉ ➜ ➛

brings it to the customary heat equation ↔

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

The two-dimensional equations of steady-state flow of a viscous incompressible fluid written in ➦ polar coordinates are reduced to this equation ( ➩ is the stream function).

Exact solution:

The functions ➯ = ➯ ( ➫ , ➛ ) and ➲ = ➲ ( ➫ , ➛ ) satisfy the system of equations

L( ➯ ➍ )− ➫ −1 ➯ ➳ L( ➯ )+ ➫ −1 ➯ [L( ➯ )] ➳ = ➝ L 2 ( ➯ ),

(2) where the subscripts ➫ and ➛ denote partial derivatives; L( ➯ )= ➫ −1 ( ➫ ➯ ➳ ) ➳ and L 2 ( ➯ ) = LL( ↔ ➯ ).

L( ➲ ➍ )− ➫ −1 ➲ ➳ L( ➯ )+ ➫ −1 ➯ [L( ➲ )] ➳ = ➝ L 2 ( ➲ ),

For the particular solution ➯ = ➵ ( ➛ ) ln ➫ + ➸ ( ➛ ) of equation (1), with ➵ and ➸ arbitrary, equation (2) can be reduced by the change of variable = L( ➲ ) to a second-order linear equation.