Nonstationary Equations

9.4.2. Nonstationary Equations

Preliminary remarks. The two-dimensional nonstationary equations of an ideal incompressible fluid (Euler equations)

∂y and u 2 =− ∂x followed by the elimination of the pressure p (with cross differentiation) from the first two equations.

are reduced to the equation in question by the introduction of a stream function w such that u 1 = ∂w

∂w

For stationary equation, see Subsection 9.4.1.

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

w 1 =− w(y, x, t), w 2 = C 1 w(C 2 x+C 3 , C 2 y+C 4 , C 1 C 2 2 t+C 5 )+ C 6 , w 3 = w(x cos α + y sin α, −x sin α + y cos α, t),

w 4 = w(x cos βt + y sin βt, −x sin βt + y cos βt, t) − 1 2 β(x 2 + y 2 ),

w 5 = w(x + ϕ(t), y + ψ(t), t) + ψ ′ t ( t)x − ϕ t ′ ( t)y + χ(t), where C 1 , ...,C 6 , α, and β are arbitrary constants and ϕ(t), ψ(t), and χ(t) are arbitrary functions,

are also solutions of the equation.

2 ◦ . Any solution of the Poisson equation ∆ w = C is also a solution of the original equation. Solutions of the Laplace equation ∆ w = 0 describe irrotational (potential) flows of an ideal incompressible fluid.

w(x, y, t) = Q(z) + ψ ′ t ( t)x − ϕ ′ t ( t)y, z = C 1 [ x + ϕ(t)] + C 2 [ y + ψ(t)]; w(x, y, t) = Q(z) + ψ ′ t ( t)x − ϕ ′ t ( t)y, z = [x + ϕ(t)] 2 +[ y + ψ(t)] 2 ;

where Q(z), ϕ(t), and ϕ(t) are arbitrary functions; C 1 and C 2 are arbitrary constants. Likewise, the formulas of Item 1 ◦ can be used to construct nonstationary solution based on other, stationary solutions (see Subsection 9.4.1).

4 ◦ . Generalized separable solution linear in x:

(1) where the functions

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

F (y, t) and G = G(y, t) are determined by the system of one-dimensional third-order equations

(3) Equation (2) is solved independently of (3). If

2 + ∂y ∂y 2 − ∂t∂y F ∂y 3 = 0.

F = F (y, t) is a solution of equation (2), then the functions

F (y + ψ(t), t) + ψ t ′ ( t),

2 = C 1 1 F (C y+C 1 C 2 t+C 3 , C 1 t+C 4 )+ C 2 ,

where ψ(t) is an arbitrary function and C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.

Integrating (2) and (3) with respect to y yields the system of second-order equations

(5) where f 1 ( t) and f 2 ( t) are arbitrary functions. Equation (5) is linear in G. Then the substitution

F = F (y, t), (6) and the function h = h(t) is determined by the second-order linear ordinary differential equation

G=

U dy − hF + h ′ t y, where U = U (y, t),

(7) brings (5) to the first-order linear homogeneous partial differential equation

G is reduced to solving the linear equations (7) and (8) followed by integrating by formula (6).

Thus, whenever a particular solution of equation (2) or (4) is known, finding

Solutions of equation (2) are listed in Table 11. The ordinary differential equations in the last two

V (H), to first-order separable equations. Table 12 presents the general solutions of equation (5) that correspond to exact solutions of equation (2) in Table 11.

rows can be reduced, with the substitution H ′ z =

Solutions of equations (2) and (4)

Determining functions No.

Function

F = F (y, t)

Function f 1 ( t)

(or general form of solutions)

in equation (4)

(of determining equation)

1 F = ϕ(t)y + ψ(t)

f 1 ( t) = ϕ ′ t + ϕ 2 ϕ(t) and ψ(t) are arbitrary

2 F = A exp[−λy − λψ(t)] + ψ t ′ ( t)

f 1 ( t) = 0

ψ(t) is arbitrary; A and λ are any

3 F = A sinh[λy + λψ(t)] + ψ ′ t ( t)

f 1 ( t) = A 2 λ 2 ψ(t) is arbitrary; A and λ are any

4 F = A cosh[λy + λψ(t)] + ψ ′ t ( t)

f 1 ( t) = −A 2 λ 2 ψ(t) is arbitrary; A and λ are any

5 F = A sin[λy + λψ(t)] + ψ ′ ( t)

f 1 ( t) = A t 2 λ 2 ψ(t) is arbitrary; A and λ are any

6 F = A cos[λy + λψ(t)] + ψ ′ t ( t)

f 1 ( t) = A 2 λ 2 ψ(t) is arbitrary; A and λ are any

7 F=t −1 H(z) + ψ t ′ ( t), z = y + ψ(t)

H(z) − 3 1

Solutions of equation (5); Θ( ξ) is an arbitrary function everywhere; the number in the first column corresponds to the number of an exact solution in Table 11

No. General solution of equation (5) Notation

1 G= Φ 2 ( t) Θ( ξ) + y Φ( t) f 2 ( t)Φ(t) dt, ξ = yΦ(t) + ψ(t)Φ(t) dt

Φ( t) = exp ϕ(t) dt

2 Formula (6), where U=e − λz Θ( ξ), ξ = t + 1 e Aλ λz

z = y + ψ(t)

3 Formula (6), where

U = sinh(λz)Θ(ξ), ξ = t + 1

Aλ ln

λz

2 z = y + ψ(t)

4 Formula (6), where U = cosh(λz)Θ(ξ), ξ = t + 2 Aλ arctan e λz

z = y + ψ(t)

2 z = y + ψ(t) Formula (6), where

5 Formula (6), where U = sin(λz)Θ(ξ), ξ = t + 1 Aλ ln

λz

6 1 U = cos(λz)Θ(ξ), ξ = t +

Aλ ln

λz

2 + π 4 z = y + ψ(t)

7 Formula (6), where U = Θ(ξ)H(z), ξ = t exp

dz H(z)

z = y + ψ(t)

8 Formula (6), where U = Θ(ξ)H(z) exp −

2 H(z) , ξ = t exp

dz

dz H(z)

z= √ y t

The general solution of the linear nonhomogeneous equation (7) can be obtained by the formula

h(t) = C 1 h 1 ( t) + C 2 h 2 ( t) +

h 2 ( t) h 1 ( t)f 2 ( t) dt − h 1 ( t) h 2 ( t)f 2 ( t) dt , (9)

where h 1 = h 1 ( t) and h 2 = h 2 ( t) are fundamental solutions of the corresponding homogeneous equation (with f 2 ≡ 0), and W 0 = h 1 ( h 2 ) ′ t − h 2 ( h 1 ) ′ t is the Wronskian determinant ( W 0 = const).

for solution 2;

for solutions 3, 5, 6;

h 1 = cos( Aλt), h 2 = sin( Aλt), W 0 = Aλ

for solution 4;

W 0 =2 µ = (1 + 4A) 2 for solutions 7, 8 in formula (9).

w(x, y, t) = F (ζ, t)x + G(ζ, t),

ζ = y + kx,

where the functions

F (ζ, t) and G = G(ζ, t) are determined from the system of one-dimensional third-order equations

2 − F 3 = 2 F ∂t∂ζ (11) ∂ζ ∂ζ k ∂ζ 2 − ∂ζ . +1 ∂t∂ζ Integrating (10) and (11) with respect to ζ yields

− F = Q(ζ, t),

where f 1 ( t) is an arbitrary function, and the function Q(ζ, t) is given by

2 k ∂F

Q(ζ, t) = − 2 + 2 F 2 dζ + f 2 ( t),

f 2 ( t) is any.

G. Consequently, the substitution U = ∂G ∂ζ brings it to the first-order linear equation

Equation (13) is linear in

U + Q(ζ, t).

Equation (10) coincides, up to renaming, with equation (2), whose exact solutions are listed in Table 11. In these cases, solutions of the corresponding equation (14) can be found by quadrature.

6 ◦ . Solution [special case of a solution of the form (1)]:

w(x, y, t) = exp − λy − λ ϕ(t) dt C 1 x+C 2 − C 1 ψ(t) dt + ϕ(t)x + ψ(t)y, where ϕ(t) and ψ(t) are arbitrary functions and C 1 , C 2 , and λ are arbitrary constants.

7 ◦ . Generalized separable solution:

w(x, y, t) = e − λy A(t)e βx +

B(t)e − βx + ϕ(t)x + ψ(t)y,

A(t) = C 1 exp − β

ψ(t) dt − λ ϕ(t) dt ,

B(t) = C 2 exp β ψ(t) dt − λ

ϕ(t) dt , where ϕ(t) and ψ(t) are arbitrary functions and C 1 , C 2 , λ, and β are arbitrary constants.

w(x, y, t) = e − λy A(t) sin(βx) + B(t) cos(βx) + ϕ(t)x + ψ(t)y,

A(t) = exp − λ ϕ dt C 1 sin β

ψ dt + C 2 cos β ψ dt ,

Z B(t) = exp − λ ϕ dt C 1 cos β ψ dt − C 2 sin β ψ dt ,

where ϕ = ϕ(t) and ψ = ψ(t) are arbitrary functions and C 1 , C 2 , λ, and β are arbitrary constants.

9 ◦ . Generalized separable solutions: w(x, y, t) = A(t) exp(k 1 x+λ 1 y) + B(t) exp(k 2 x+λ 2 y) + ϕ(t)x + ψ(t)y,

A(t) = C 1 exp λ 1 ϕ(t) dt − k 1 ψ(t) dt ,

B(t) = C 2 exp λ 2 ϕ(t) dt − k 2 ψ(t) dt ,

where ϕ(t) and ψ(t) are arbitrary functions; C 1 and C 2 are arbitrary constants; and k 1 , λ 1 , k 2 , and λ 2 are arbitrary parameters related by one of the two constraints

1 + λ 1 = 2 k 2 + 2 λ 2 (first family of solutions), k 1 λ 2 = k 2 λ 1 (second family of solutions).

10 ◦ . Generalized separable solution: w(x, y, t) = C 1 sin( λx) + C 2 cos( λx) A(t) sin(βy) + B(t) cos(βy) + ϕ(t)x,

A(t) = C 3 cos β

ϕ dt + C 4 , where ϕ = ϕ(t) is an arbitrary function and C 1 , ...,C 4 , λ, and β are arbitrary constants.

ϕ dt + C 4 , B(t) = C 3 sin β

11 ◦ . Generalized separable solution: w(x, y, t) = C 1 sinh( λx) + C 2 cosh( λx) A(t) sin(βy) + B(t) cos(βy) + ϕ(t)x,

A(t) = C 3 cos β

ϕ dt + C 4 , where ϕ = ϕ(t) is an arbitrary function and C 1 , ...,C 4 , λ, and β are arbitrary constants.

ϕ dt + C 4 , B(t) = C 3 sin β

12 ◦ . Solution: w(x, y, t) = f (z) + g(t)z + ϕ(t)x + ψ(t)y,

z = kx + λy +

λϕ(t) − kψ(t) dt,

where f (z), g(t), ϕ(t), and ψ(t) are arbitrary functions and k and λ are arbitrary constants.

13 ◦ +. There is a “two-dimensional” solution of the form

14 ◦ +. “Two-dimensional” solution: w=t (2− k)/k Ψ( ξ, η), ξ = t −1 /k x cos(λ ln t) − y sin(λ ln t) , η=t −1 /k x sin(λ ln t) + y cos(λ ln t) , where k and λ are arbitrary constants and the function Ψ(ξ, η) is determined by the differential

equation ∂Ψ 1 ∂ e ∂Ψ 1 ∂ e ∂ e 2 ∂ 2

−e ∆Ψ +

ϕ ′ t ( x 2 − y 2 +2 ϕxy) y − ϕx

w(x, y, t) =

F (ζ, t) − 2G(ζ, t),

where ϕ = ϕ(t) is an arbitrary function and the functions F = F (ζ, t) and G = G(ζ, t) are determined by the differential equations

(1 + ϕ 2 ) 2 ∂ζ 2 Equation (15) is solved independently of equation (16). If

F = F (ζ, t) is a solution to (15), then the function

F (y + σ(t), t) − σ ′ t ( t),

where σ(t) is an arbitrary function, is also a solution of the equation. Integrating (15) and (16) with respect to ζ yields

(1 + ϕ 2 ) 2 ∂ζ where ψ 1 ( t) and ψ 2 ( t) are arbitrary functions. The change of variable u = ∂G ∂ζ brings the last equation

to a first-order linear equation (for known

F ).

Note that equation (15) admits particular solutions of the following forms:

F (ζ, t) = a(t)ζ + b(t),

F (ζ, t) = a(t)e λζ

( t)

λa(t)

where ❥✂❦ a(t) and b(t) are arbitrary functions and λ is an arbitrary constant.

References for equation 9.4.2.1: A. A. Buchnev (1971), B. J. Cantwell (1978), P. J. Olver (1986), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov (1999), D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (1999), A. D. Polyanin and V. F. Zaitsev (2002).

Preliminary remarks. Equation 9.4.4.1 is reduced to the equation in question by passing to the polar coordinate system

with center at a point ( x 0 , y 0 ), where x 0 and y 0 are any, by the formulas

x = r cos θ + x 0 ,

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

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

x−x (inverse transformation). 0

The radial and angular components of the fluid velocity are expressed via the stream function w as follows: u

1 ◦ . Generalized separable solution linear in θ:

(1) where the functions f = f (r, t) and g = g(r, t) satisfy the system of equations

w(r, θ, t) = f (r, t)θ + g(r, t),

L( f t −1 )− r −1 f r L( f)+r f [L(f )] r = 0,

t )− r g r L( f)+r

f [L(g)] r = 0.

Here, the subscripts r and t denote the corresponding partial derivatives, L(f ) = r −1 ( rf r ) r .

(4) where ϕ = ϕ(t) and ψ = ψ(t) are arbitrary functions, equation (3) can be reduced, with the change

f = ϕ(t) ln r + ψ(t)

of variable U = L(g), to the first-order linear equation U t −1 + r fU r = 0. Two families of particular solutions to this equation are given by

Z U = Θ(ζ), ζ = r 2 −2 ψ(t) dt

(first family of solutions, ϕ = 0),

Z r dr Z

U = Θ(ζ), ζ =

ϕ(t) dt

(second family of solutions, ψ = 0),

ln r

where Θ( ζ) is an arbitrary function. The second term in solution (1) is expressed via U = U (r, t), provided the first term has the form (4), as follows:

g(r, t) = C 1 ( t) ln r + C 2 ( t) +

r where C 1 ( t) and C 2 ( t) are arbitrary functions.

Remark. Equation (2) has also a solution f=−

2( t + C)

3 ◦ . “Two-dimensional” solution:

2 2 w(r, θ, t) = Ar 2 t + H(ξ, η), ξ = r cos(θ + At ), η = r sin(θ + At ), where

A is an arbitrary constant and the function H(ξ, η) is determined by the differential equation

For other exact solutions, see equation 9.4.2.1 . References for equation

9.4.2.2: A. A. Buchnev (1971), P. J. Olver (1986), V. K. Andreev, O. V. Kaptsov, V. V. Pukhna- chov, and A. A. Rodionov (1999), D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (1999), A. D. Polyanin and V. F. Zaitsev (2002).