Stationary Equations

9.4.1. Stationary Equations

Preliminary remarks. The stationary two-dimensional equations of motion of an ideal fluid (Euler equations) ∂u 1 ∂u 1 u 1 1 + u 2 =− ∂p ,

are reduced to this equation by the introduction of a stream function, w, such that u 1 = ∂w and u 2 =− ∂y ∂w ∂x followed by the elimination of the pressure p (with the cross differentiation) from the first two equations; the third equation is then satisfied automatically.

1 ◦ . Suppose w(x, y) is a solution of the equation in question. Then the functions

w 1 = C 1 w(C 2 x+C 3 , C 2 y+C 4 )+ C 5 , w 2 = w(x cos α + y sin α, −x sin α + y cos α),

where C 1 , ...,C 5 and α are arbitrary constants, are also solutions of the equation.

2 ◦ . Solutions of general form:

w(x, y) = ϕ 1 ( ξ), ξ = a 1 x+b 1 y; p

w(x, y) = ϕ 2 ( r), r= ( x−a 2 ) 2 +( y−b 2 ) 2 ; where ϕ 1 ( ξ) and ϕ 2 ( r) are arbitrary functions; a 1 , b 1 , a 2 , and b 2 are arbitrary constants.

3 ◦ . Any solutions of the linear equations ∆ w=0

(Laplace equation),

∆ w=C

(Poisson equation),

∆ w = λw

(Helmholtz equation),

∆ w = λw + C (nonhomogeneous Helmholtz equation), where

C and λ are arbitrary constants, are also solutions of the original equation. For details about the Laplace, Poisson, and Helmholtz equations, see the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

The solutions of the Laplace equation ∆ w = 0 correspond to irrotational (potential) solutions of the Euler equation. Such solutions are discussed in detail in textbooks on hydrodynamics (e.g., see Sedov, 1980, and Loitsyanskiy, 1996), where the methods of the theory of functions of a complex variable are extensively used.

4 ◦ . The Jacobian of the functions w and v = ∆w appears on the left-hand side of the equation in question. The fact that the Jacobian of two functions is zero means that the two functions are functionally dependent. Hence, v must be a function of w, so that

(1) where f (w) is an arbitrary function. Any solution of the second-order equation (1) for arbitrary

∆ w = f (w),

f (w) is a solution of the original equation. The results of Item 3 ◦ correspond to special cases of the linear function f (w) = λw + C. For solutions of equation (1) with some nonlinear f = f (w), see 5.1.1.1, 5.2.1.1, 5.3.1.1, 5.3.2.1, 5.3.3.1,

5.4.1.1, and Subsection S.5.3 (Example 12).

2 w(x, y) = A 2

1 x + A 2 x+B 1 y + B 2 y + C,

w(x, y) = A 1 exp( λx) + A 2 exp(− λx) + B 1 exp( λy) + B 2 exp(− λy) + C, w(x, y) = A 1 sin( λx) + A 2 cos( λx) + B 1 sin( λy) + B 2 cos( λy) + C,

C, and λ are arbitrary constants. These solutions are special cases of solutions presented in Item 3 ◦ .

where A 1 , A 2 , B 1 , B 2 ,

6 ◦ . Generalized separable solutions: w(x, y) = (Ax + B)e − λy + C,

w(x, y) = A 1 sin( βx) + A 2 cos( βx) B 1 sin( λy) + B 2 cos( λy) + C, w(x, y) = A 1 sin( βx) + A 2 cos( βx) B 1 sinh( λy) + B 2 cosh( λy) + C, w(x, y) = A 1 sinh( βx) + A 2 cosh( βx) B 1 sin( λy) + B 2 cos( λy) + C, w(x, y) = A 1 sinh( βx) + A 2 cosh( βx) B 1 sinh( λy) + B 2 cosh( λy) + C,

w(x, y) = Ae αx+βy + Be γx+λy + C,

where

A, B, C, D, k, β, and λ are arbitrary constants. These solutions are special cases of solutions presented in Item 3 ◦ .

7 ◦ . Solution:

w(x, y) = F (z)x + G(z),

z = y + kx,

where k is an arbitrary constant and the functions F = F (z) and G = G(z) are determined by the autonomous system of third-order ordinary differential equations:

On integrating the system once, we arrive at the following second-order equations:

where A 1 and A 2 are arbitrary constants. The autonomous equation (4) can be reduced, with the change of variable Z(F ) = (F z ′ ) 2 , to a first-order linear equation. The general solution of equation (2), or (4), is given by

F (z) = B 2

1 z+B 2 ,

F (z) = B 1 exp( λz) + B 2 exp(− λz), A 1 = −4 λ 2 B 1 B 2 ;

A 1 = 2 λ 2 ( B 1 + B 2 ), where B 1 , B 2 , and λ are arbitrary constants.

F (z) = B 2

1 sin( λz) + B 2 cos( λz),

The general solution of equation (3), or (5), is expressed as

F = F (z),

ψ= 2 FF zz ′′ dz + A 2 ,

k +1

where C 1 and C 2 are arbitrary constants.

w(x, y) = x a U (ζ),

ζ = y/x;

w(x, y) = e ax

V (ρ),

ρ = bx + cy;

w(x, y) = W (ζ) + a ln |x|, ζ = y/x;

where

a, b, and c are arbitrary constants. ◮

For other exact solutions, see equation 9.4.1.2 . References for equation

9.4.1.1: A. A. Buchnev (1971), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Ro- dionov (1999), A. D. Polyanin and V. F. Zaitsev (2002).

Preliminary remarks. Equation 9.4.1.1 is reduced to this equation by passing to polar coordinates r, θ with origin at a

point ( x 0 , y 0 ), where x 0 and y 0 are any, such that

x = r cos θ + x 0 ,

y = r sin θ + y 0 (direct transformation),

r= 0 ( x−x 0 ) 2 + ( y−y 0 ) 2 , tan θ= y−y

x−x 0 (inverse transformation). The radial and angular components of the fluid velocity are expressed in terms of the stream function w as follows: u r = 1 r ∂w ∂θ

and u θ =− ∂w ∂r .

1 ◦ . Multiplicative separable solution:

w(r, θ) = r λ U (θ),

where the function U = U (θ) is determined by the second-order autonomous ordinary differential equation

λ−2 U θθ ′′ + 2 λ U = CU λ

( λ and C are any).

Its general solution can be written out in implicit form. In particular, if

C = 0, we have

U=A 1 sin( λθ) + A 2 cos( λθ) if λ ≠ 0, U=A 1 θ+A 2 if λ = 0.

To λ = 0 there corresponds a solution dependent on the angle θ only.

2 ◦ . Multiplicative separable solution:

w(r, θ) = f (r)g(θ),

where the functions f = f (r) and g = g(θ) are determined by the linear ordinary differential equations

L( f ) = (β − λr −2 ) f,

g ′′ θθ = λg,

where β and λ are arbitrary constants; L(f ) = r −1 ( rf r ′ ) ′ r .

3 ◦ . Solution:

(1) where the function U = U (ξ) is determined by the ordinary differential equation

w = bθ + U (ξ),

ξ = θ + a ln r,

abU ξξξ ′′′ =2 bU ξξ ′′ +2 U ξ ′ U ξξ ′′ .

The onefold integration yields

abU ξξ ′′ =( U 2 ξ ′ ) +2 bU ξ ′ + C 1 ,

4 ◦ . Generalized separable solution linear in θ:

w(r, θ) = f (r)θ + g(r).

Here, the functions f = f (r) and g = g(r) are determined by the system of ordinary differential equations

− f r ′ L( f ) + f [L(f )] ′ r = 0,

− g r ′ L( f ) + f [L(g)] ′ r = 0,

where L( f)=r −1 ( rf r ′ ) ′ r . System (3) admits first integrals, which allow us to obtain the following second-order linear ordinary differential equations for f and g:

A and B are arbitrary constants. For A = 0, the solutions of equations (4) are given by

f (r) = C 1 ln r+C 2 ,

1 g(r) = 2

4 Br + C 3 ln r+C 4 .

For

A ≠ 0, the solutions of equations (4) are expressed in terms of Bessel functions.

For other exact solutions, see equation 9.4.1.1 . References for equation

9.4.1.2: A. A. Buchnev (1971), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Ro- dionov (1999), A. D. Polyanin and V. F. Zaitsev (2002).

Preliminary remarks. The stationary Euler equations written in cylindrical coordinates for the axisymmetric case are reduced to the equation in question by the introduction of a stream function

r 1 = w such that u 1 ∂w r ∂z and u z =− ∂w r ∂r , where r= px 2 + y 2 , and u r and u z are the radial and axial fluid velocity components.

1 ◦ . Any function w = w(r, z) that solves the second-order linear equation Ew = 0 will also be a solution of the given equation.

2 ◦ . Solutions:

w = ϕ(r),

2 w = (C 2

1 z + C 2 z+C 3 ) r + C 4 z+C 5 ,

where ϕ(r) is an arbitrary function and C 1 , ...,C 5 are arbitrary constants.

3 ◦ . Generalized separable solution linear in z:

w(r, z) = ϕ(r)z + ψ(r). Here, ϕ = ϕ(r) and ψ = ψ(r) are determined by the system of ordinary differential equations ϕ[L(ϕ)] ′ r − ϕ ′ L( ϕ) − 2r −1 r ϕ L(ϕ) = 0,

ϕ[L(ψ)] r ψ r ϕ) − 2r ϕ L(ψ) = 0,

′ − ′ L(

where L( ϕ) = ϕ ′′ rr − r −1 ϕ ′ r .

ordinary differential equations for ϕ and ψ:

where C 1 and C 2 are arbitrary constants. The substitution ξ=r 2 brings (2) to the linear constant- coefficient equations

ϕ ′′ ξξ = C 1 ϕ, ψ ′′ ξξ = C 1 ψ+C 2 .

Integrating yields

  A 1 cosh( kξ) + B 1 sinh( kξ) if C 1 = k 2 > 0,

ϕ= A 1 cos( kξ) + B 1 sin( kξ)

if C 1 =− k 2 < 0,

 A 1 ξ+B 1 if C 1 = 0, 

A 2 cosh( kξ) + B 2 sinh( kξ) − C 2 /C 1 if C 1 = k  2 > 0, ψ= A 2 cos( kξ) + B 2 sin( kξ) − C 2 /C 1 if C 1 =− k 2 < 0,

2 C 2 ξ 2 + A 2 ξ+B 2 if C 1 = 0,

where ❤✂✐ A 1 , B 1 , A 2 , and B 2 are arbitrary constants.

References for equation 9.4.1.3: A. A. Buchnev (1971), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Ro- dionov (1999), A. D. Polyanin and V. F. Zaitsev (2002).