Nonstationary Hydrodynamic Equations (Navier–Stokes equations)

10.3.3. Nonstationary Hydrodynamic Equations (Navier–Stokes equations)

( ∆w) = ν∆∆w,

Preliminary remarks. The two-dimensional nonstationary equations of a viscous incompressible fluid,

are reduced to the equation in question through the introduction of a stream function w such that u 1 = ∂w ∂y and u 2 =− ∂w ∂x followed by the elimination of the pressure ✹✂✺ p (with cross differentiation) from the first two equations. Reference : L. G. Loitsyanskiy (1996).

For stationary solutions, see equation 10.3.2.1.

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

w 1 =− w(y, x, t),

w 5 = w(x + ϕ(t), y + ψ(t), t) + ψ ′ t ( t)x − ϕ ′ t ( t)y + χ(t), where C 1 , ...,C 4 , α, and β are arbitrary constants and ϕ(t), ψ(t), and χ(t) are arbitrary functions, ✹✂✺ are also solutions of the equation.

References : V. V. Pukhnachov (1960), B. J. Cantwell (1978), S. P. Lloyd (1981), L. V. Ovsiannikov (1982).

2 ◦ . Any solution of the Poisson equation ∆ w = C is also a solution of the original equation (these are “inviscid” solutions). For details about the Poisson equation, see, for example, the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

Example of an inviscid solution involving five arbitrary functions:

2 w = ϕ(t)x 2 + ψ(t)xy + [C − ϕ(t)]y + a(t)x + b(t)y + c(t).

3 ◦ . Solution dependent on a single space variable:

w = W (x, t),

where the function W satisfies the linear nonhomogeneous heat equation

∂W 2 ∂ W

2 = f 1 ( t)x + f 0 ( ∂t t), ∂x

and f 1 ( t) and f 0 ( t) are arbitrary functions. Solutions of the form w = V (y, t) are determined by a similar equation.

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 fourth-order one- dimensional equations

∂t∂y 2

3 2 3 ∂ 4 G ∂G ∂ F ∂ G ∂ G

∂t∂y

∂y

∂y 2

∂y 3 ∂y 4 ∂y 3 ∂y 4

F (y + ψ(t), t) + ψ t ′ ( t),

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

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

F = F (y, t), (6) and the function h = h(t) satisfies the linear ordinary differential equation

G=

U dy − hF + h ′ t y,

where U = U (y, t),

(7) brings (5) to the linear homogeneous parabolic second-order equation

Thus, whenever a particular solution of equation (2) or (4) is known, determining the function G is reduced to solving the linear equations (7)–(8) followed by computing integrals by formula (6). Exact solutions of equation (2) are listed in Table 13 (two more complicated solutions are specified at the end of Item 4 ◦ ). The ordinary differential equations in the last two rows, which determine a traveling-wave solution and a self-similar solution, are autonomous and, therefore, its order can be reduced. Note that solutions of the form (1) with

F (y, t) = Cy/t were treated in Pukhnachov (1960); these solutions correspond to ϕ(t) = C/t in the first row. The general solution of the linear nonhomogeneous equation (7) is expressed as

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 (in this case, W 0 = const). Table 14 lists fundamental solutions of the homogeneous equation (7) corresponding to the exact solutions of (2) specified in Table 13. Equation (8) with any

F = F (y, t) has the trivial solution. The expressions in Tables 13–14 together with formulas (6) and (9) with U = 0 describe some exact solutions of the form (1). Nontrivial solutions of equation (8) generate a wider class of exact solutions.

Table 15 presents transformations that simplify equation (8) for some of the solutions to (2) or (4) listed in Table 13. It is apparent that solutions to (8) are expressed via solutions to the linear constant-coefficient heat equation in the first two cases. Equation (8) admits the application of the method of separation of variables in three other cases.

The third equation in Table 15 has the following particular solutions ( B 1 and B 2 are arbitrary constants):

Z(η) = B 1 + B 2 Φ( η) dη, Φ( η) = exp

νλ Z

2 dη

Z(η, t) = B 1 νλ t+B 1 Φ( η)

dη.

Solutions of equations (2) and (4); ϕ(t) and ψ(t) are arbitrary functions, and

A and λ are arbitrary constants Function

Determining coefficients No.

F = F (y, t)

Function f 1 ( t)

(or general form of solution)

in equation (4)

(or determining equation)

1 F = ϕ(t)y + ψ(t)

y+ψ(t) + ψ t ′ ( t)

f 1 ( t) = 0

N/A

3 F = A exp − λy − λψ(t) + ψ t ′ ( t) + νλ

f 1 ( t) = 0

N/A

2 4 2 F = Ae βt sin[ λy + λψ(t)] + ψ

F = Ae − βt cos[ λy + λψ(t)] + ψ ′ ( t)

6 F = Ae βt sinh[ λy + λψ(t)] + ψ ′

2 2 t) 2 f 1 ( t) = Be βt β = νλ , B=A λ >0

2 2 7 2 F = Ae cosh[ λy + λψ(t)] + ψ

ψ ′ ( 8 t) F = ψ(t)e λy − Ae

βt−λy

9 F = F (ξ), ξ = y + λt

4 − A−2U ξ ′ +( U ξ ′ ) − UU ξξ ′′ = νU ξξξ ′′′

For other exact solutions of this equation, see the book by Polyanin (2002), where a more general solution of the form ✻✂✼ ∂ t w = f (x)∂ xx w + g(x)∂ x w was considered.

References : R. Berker (1963), A. D. Polyanin (2001 c, 2002), A. D. Polyanin and V. F. Zaitsev (2002). Special case 1. Solution exponentially dependent on time:

w(x, y, t) = f (y)x + e Z − λt g(y) dy,

where the functions f = f (y) and g = g(y) are determined by the system of ordinary differential equations

( f y ′ ) 2 − ff yy ′′ = νf yyy ′′′ + C 1 , − λg + gf y ′ − fg ′ y = νg ′′ yy + C 2 ,

and ✻✂✼ C 1 and C 2 are arbitrary constants. Reference : N. Rott (1956). Special case 2. Periodic solution:

w(x, y, t) = f (y)x + sin(λt) Z g(y) dy + cos(λt) h(y) dy, where the functions

f = f (y), g = g(y), and h = h(y) are determined by the solution of ordinary differential equations

( f y ′ ) 2 − ff yy ′′ = νf yyy ′′′ + C 1 , − λh + f y ′ g−fg ′ y = νg yy ′′ + C 2 , λg + f y ′ h−fh ′ y = νh yy ′′ + C 3 .

Below are another two exact solutions of equation (2):

2 A cosh +

B sinh ,

F (y, t) = − t y+γ exp − ν

3 dt

A cos

B sin ,

Fundamental system of solutions determining the general solution (9) of the nonhomogeneous equation (7); the number in the first column corresponds to the respective number of an exact solution in Table 13

No. Fundamental system of solutions Wronskian W 0 Notation and remarks

Φ( t) = exp ϕ(t) dt

I ( z), K ( z) are modified Bessel

I ( z), K 0 ( z) are modified Bessel

I 0 1 0 = 0 , 2 = 0 0 ( z), K ( z) are modified Bessel

J ( z), Y ( z) are Bessel functions;

2 √ A βt/2

8 h 1 = I 0 β e , h 2 = K 0 A β βt/2 e W 0 =− β 2 functions; 2 β = 2νλ

I 0 ( z), K 0 ( z) are modified Bessel

h 1 = cosh( kt), h 2 = sinh( kt)

h 1 = cos( kt), h 2 = sin( kt)

h 1 =| t| 2 cos( µ ln |t|), h 2 =| t| 2 1

sin( 1 µ ln |t|) W 0 = µ 1 if 1 A<−

4 ; µ= 2 |1 + 4 A| 2

where

A and B are arbitrary constants, and γ = γ(t) is an arbitrary function. The first formula of the two displayed after (3) allows us to generalize the above expressions to obtain solutions involving two arbitrary functions.

5 ◦ . Solution (generalizes the solution of Item 4 ◦ ):

w(x, y, t) = F (ξ, t)x + G(ξ, t),

ξ = y + kx,

where k is an arbitrary constant and the functions F (ξ, t) and G = G(ξ, t) are determined from the system of one-dimensional fourth-order equations

F − . (11) ∂t∂ξ

2 − F 3 = ν(k + 1)

+4 νk

∂ξ 2 ∂t∂ξ Integrating (10) and (11) with respect to ξ yields

+ f 1 ( t), (12)

∂t∂ξ

Transformations of equation (8) for the corresponding exact solutions of equation (4); the number in the first column corresponds to the respective

F = F (y, t) in Table 13 No.

number of an exact solution

Transformations of equation (8) Resulting equation

U= Φ( t) u(z, τ ), τ = Φ ( t) dt + C 1 ,

∂u

z = yΦ(t) + ψ(t)Φ(t) dt + C 2 , Φ( t) = exp ϕ(t) dt

∂τ ∂z

∂ 2 2 U=ζ u(ζ, t), ζ = y + ψ(t) = ν u

Z(η, t), η = −λy − λψ(t) 2 ∂Z = νλ ∂ Z 2 +( νλ − Aλe η ) ∂Z

∂t

9 U = u(ξ, t), ξ = y + λt

u(ξ, τ ), ξ = yt −1 /2 , τ = ln t

∂u = ν ∂ 2 u

∂ξ 2 + H(ξ) ∂u ∂ξ + 1− H ξ ′ ( ξ) u

3 + Q(ξ, t), (13)

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

2 ∂ 2 F 2 k ∂F 2 k Z ∂ F

f 2 ( t) is any. Equation (13) is linear in

Q(ξ, t) = 4νk

G. The substitution U = ∂G ∂ξ brings (13) to the second-order linear equation

U + Q(ξ, t).

Thus, whenever a particular solution of equation (10) or (12) is known, determining the func- tion

G is reduced to solving the second-order linear equation (14). Equation (10) is reduced, by scaling the independent variables so that

ζ and t = (k 2 + 1) τ , to equation (2) in which y and ✽✂✾ t should be replaced by ζ and τ ; exact solutions of equation (2) are listed in Table 13.

ξ = (k 2 + 1)

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

6 ◦ . Solutions:

3 w(x, y, t) = Az 2 + Bz + Cz + ψ

t ′ ( t)x, z = y + kx + ψ(t);

t ′ ( t)x, where

w(x, y, t) = Ae 2 λz

+ 2 Bz + Cz + νλ(k + 1) x+ψ

A, B, C, k, and λ are arbitrary constants and ψ(t) is an arbitrary function.

7 ◦ . Generalized separable solution [special case of a solution of the form (1)]:

w(x, y, t) = e − λy

f (t)x + g(t) + ϕ(t)x + ψ(t)y,

f (t) = C 1 E(t), E(t) = exp νλ t−λ

ϕ(t) dt ,

Z g(t) = C 2 E(t) − C 1 E(t) ψ(t) dt,

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

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

− λy

βx

2 A(t) = C 2

1 exp ν(λ + β ) t−β

ψ(t) dt − λ

ϕ(t) dt ,

2 B(t) = C 2

2 exp ν(λ + β ) t+β

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

ψ(t) dt − λ

9 ◦ . Generalized separable solution: w(x, y, t) = e − λy A(t) sin(βx) + B(t) cos(βx) + ϕ(t)x + ψ(t)y,

where ϕ(t) and ψ(t) are arbitrary functions, λ and β are arbitrary constants, and the functions A(t) and B(t) satisfy the linear nonautonomous system of ordinary differential equations

t = ν(λ − β )− λϕ(t)

A + βψ(t)B,

B ′ t = ν(λ − β )− λϕ(t)

B − βψ(t)A.

The general solution of system (15) is expressed as

2 A(t) = exp 2 ν(λ − β ) t−λ ϕ dt C 1 sin β ψ dt + C 2 cos β ψ dt ,

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

where ϕ = ϕ(t) and ψ = ψ(t); C 1

and 2 C 2 are arbitrary constants. In particular, for 2 ϕ= ( λ − β ) λ

and ψ = a, we obtain the periodic solution

A(t) = C 1 sin( aβt) + C 2 cos( aβt), B(t) = C 1 cos( aβt) − C 2 sin( aβt).

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

10 ◦ . Generalized separable solution: 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, where ϕ(t) and ψ(t) are arbitrary functions, k 1 , λ 1 , k 2 , and λ 2 are arbitrary constants, constrained

by one of the two relations

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

and the functions A(t) and B(t) satisfy the linear ordinary differential equations

t = ν(k 1 + λ 1 )+ λ 1 ϕ(t) − k 1 ψ(t) A,

= 2 ν(k

2 + λ 2 )+ λ 2 ϕ(t) − k 2 ψ(t) B.

These equations can be readily integrated to obtain

A(t) = C 1 2 exp 2 ν(k 1 + λ 1 ) t+λ 1 ϕ(t) dt − k 1 ψ(t) dt ,

B(t) = C 2 exp ν(k 2 + λ 2 ) t+λ 2 ϕ(t) dt − k 2 ψ(t) dt .

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

w(x, y, t) = C 1 sin( λx) + C 2 cos( λx) A(t) sin(βy) + B(t) cos(βy) + ϕ(t)x, where ϕ(t) is an arbitrary function, C 1 , C 2 , λ, and β are arbitrary constants, and the functions A(t)

and B(t) satisfy the linear nonautonomous system of ordinary differential equations

t =− ν(λ + β )

A − βϕ(t)B,

B t ′ =− ν(λ + β )

B + βϕ(t)A.

The general solution of system (16) is expressed as

2 A(t) = exp 2 − ν(λ + β ) t C 3 sin β

ϕ dt + C 4 cos β

ϕ dt , ϕ = ϕ(t),

Z B(t) = exp 2 − ν(λ + 2 β ) t − C 3 cos β ϕ dt + C 4 sin β ϕ dt ,

❁✂❂ where C 3 and C 4 are arbitrary constants.

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

12 ◦ . 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,

where ϕ(t) is an arbitrary function, C 1 , C 2 , λ, and β are arbitrary constants, and the functions A(t) and B(t) satisfy the linear nonautonomous system of ordinary differential equations

2 A 2 ′ t = ν(λ − β )

A − βϕ(t)B,

B t = ν(λ − β )

B + βϕ(t)A.

The general solution of system (17) is expressed as

2 A(t) = exp 2 ν(λ − β ) t C

3 sin β ϕ dt + C 4 cos β ϕ dt , ϕ = ϕ(t),

ϕ dt , ❁✂❂ where C 3 and C 4 are arbitrary constants.

2 B(t) = exp 2 ν(λ − β ) t − C 3 cos β ϕ dt + C 4 sin β

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

13 ◦ . “Two-dimensional” solution:

z = kx + λy, where ϕ(t) and ψ(t) are arbitrary functions, k and λ are arbitrary constants, and the function u(z, t)

w(x, y, t) = u(z, t) + ϕ(t)x + ψ(t)y,

is determined by the fourth-order linear equation

2 + kψ(t) − λϕ(t)

3 = ν(k + λ ∂t∂z ) ∂z ∂z 4 .

The transformation

U (ξ, t) =

ξ=z−

kψ(t) − λϕ(t) dt

∂z

brings it to the linear heat equation

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

w(x, y, t) = W (ρ 1 , ρ 2 )+ c 1 x+c 2 y,

ρ 1 = a 1 x+a 2 y+a 3 t,

ρ 2 = b 1 x+b 2 y+b 3 t.

15 ◦ +. “Two-dimensional” solution ( a, b, and c are arbitrary constants):

x+a

y+b

w(x, y, t) = Z(X, Y ), X= √

, Y= √

t+c where the function Z = Z(X, Y ) is determined by the differential equation

Reference : V. V. Pukhnachov (1960).

16 ◦ +. “Two-dimensional” solution:

w(x, y, t) = Ψ(ξ, η), −1 ξ=t /2 x cos(k ln t) − y sin(k ln t) , η=t /2 x sin(k ln t) + y cos(k ln t) , where k is an arbitrary constant and the function Ψ(ξ, η) is determined by the differential equation

Reference : B. J. Cantwell (1978).

17 ◦ +. “Two-dimensional” solution:

2 ϕ 2 t ( x − y +2 ϕxy) w(x, y, t) = y − ϕx +

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

Reference : D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (1999).

F = F (ζ, t) is a solution to (18), the function

Equation (18) is solved independently of equation (19). If

F (y + σ(t), t) − σ t ′ ( t),

where σ(t) is an arbitrary function, is also a solution of the equation. Integrating (18) and (19) 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 second-order parabolic linear equation (with known

F ).

Note that equation (18) admits particular solutions of the forms

F (ζ, t) = a(t)ζ + b(t),

F (ζ, t) = a(t)e λζ

+ 2 t + t − νλ(1 + ϕ ),

( t)

λa(t) λ(1 + ϕ 2 )

where a(t) and b(t) are arbitrary functions and λ is an arbitrary constant. ◮ For other exact solutions, see equation 10.3.3.3.

( ∆w) + 2a∆w = ν∆∆w. ∂t

Preliminary remarks. The system

describing the motion of a viscous incompressible fluid induced by two parallel disks moving towards each other is reduced to the given equation. Here,

a is the relative velocity of the disks, u 1 and u 2 are the horizontal velocity components, and u 3 = −2 az is the vertical velocity component. The introduction of a stream function w such that u 1 = ax + ∂w ∂y and

u 2 = ay − ∂w ∂x followed by the elimination of the pressure p (with the help of cross differentiation) leads to the equation in question. For ❅✂❆

a = 0, see equation 10.3.3.1. Reference : A. Craik (1989).

For stationary solutions, see equation 10.3.2.3.

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 = w(x cos β + y sin β, −x sin β + y cos β, t), w 3 = w x + ϕ(t), y + ψ(t), t + C + ψ t ′ ( t) − aψ(t) x+ aϕ(t) − ϕ ′ t ( t) y + χ(t),

where ϕ(t), ψ(t), and χ(t) are arbitrary functions and C and β are arbitrary constants, are also solutions of the equation.

2 ◦ . Any solution of the Poisson equation ∆ w = C is also a solution of the original equation (these are “inviscid” solutions). For details about the Poisson equation, see, for example, the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

3 ◦ . Solution dependent on a single coordinate x:

w(x, t) =

( x − ξ)U (ξ, t) dξ + f 1 ( t)x + f 0 ( t),

where f 1 ( t) and f 0 ( t) are arbitrary functions and the function U (x, t) satisfies the linear nonhomo- geneous parabolic equation

which can be reduced to a linear constant-coefficient heat equation; see Polyanin (2002, page 93). Solutions of the form w = w(y, t) can be obtained likewise.

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 fourth-order equations

∂y Equation (2) is solved independently of equation (3). If

F = F (y, t) is a solution to (2), then the function

F y + ψ(t), t + ψ t ′ ( t) − aψ(t),

where ψ(t) is an arbitrary function, is also a solution of the equation.

3 + f 2 ( t), (5) ∂t∂y

where f 1 ( t) and f 2 ( t) are arbitrary functions.

Equation (2) has a particular solution

(6) where f 1 = f 1 ( t) and f 0 = f 0 ( t) are arbitrary functions. On substituting (6) into (5), we arrive at a

F (y, t) = f 1 ( t)y + f 0 ( t),

linear equation whose order can be reduced by two:

2 ∂Q 2 ∂ Q ∂ G

. The equation for Q can be reduced to a linear constant-coefficient heat equation; see Polyanin (2002,

page 135). Note that equation (2) has the following particular solutions:

F (y, t) = ay + 2 C 1 exp(− λy) + C 2 exp( λy) exp ( νλ −4 a)t ,

F (y, t) = ay + C 1 cos( λy) + C 2 sin( λy) exp −( νλ +4 a)t , where C 1 , C 2 , and λ are arbitrary constants.

Solutions of the form w(x, y, t) = f (x, t)y + g(x, t) can be obtained likewise. Remark. The results of Items 1 ◦ –4 ◦ exclusive of formula (7) remain true if a = a(t) is an

arbitrary function in the original equation (in this case, one should set

C = 0 in Item 1 ◦ ). ◮ For other exact solutions, see equation 10.3.3.4.

Preliminary remarks. Equation 10.3.3.1 is reduced to the given equation by passing to polar coordinates with origin at a

point ( x 0 , y 0 ), where x 0 and y 0 are any numbers, according to

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

x−x (inverse transformation).

The radial and angular fluid velocity components are expressed in terms of the stream function = w as follows: u 1 r ∂w and u θ =− ∂w

r ∂θ

∂r .

1 ◦ . Solutions with axial symmetry

w = W (r, t)

are described by the linear nonhomogeneous heat equation

∂W

∂W

= ϕ(t) ln r + ψ(t),

where ϕ(t) and ψ(t) are arbitrary functions. For particular solutions of equation (1) that occur in fluid dynamics, see Pukhnachov (1960) and Loitsyanskiy (1996).

2 ◦ . Generalized separable solution linear in θ:

w(r, θ, t) = f (r, t)θ + g(r, t),

L( 2 f t )− r f r L( f)+r f [L(f )] r = νL ( f ),

(4) Here, the subscripts −1 r and t denote partial derivatives with respect to r and t, L(f ) = r ( rf ) , and

L( 2 g t )− r g r L( f)+r f [L(g)] r = νL ( g).

L ( f ) = LL(f ). Equation (3) has a particular solution of the form

f = ϕ(t) ln r + ψ(t),

where ϕ(t) and ψ(t) are arbitrary functions. In this case, equation (4) is reduced by the change of variable U = L(g) to a second-order linear equation.

Remark. Equation (3) has also a particular solution f=−

2( t + C)

References : R. Berker (1963), D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (1999).

3 ◦ . Let us consider the case f = ψ(t) in Item 2 ◦ in more detail. This case corresponds to w= ψ(t)θ + g(r, t); the existence of such an exact solution was established by Pukhnachov (1960). For g, we have the equation

∂U

ψ(t) ∂U

where U=

r ∂r ∂r Below are some exact solutions of equation (5):

1 Z ψ(t)

U=r 2 +4 νt − 2 ψ(t) dt + a,

2 ν − ψ(t) p(t) dt + b, where a and b are arbitrary constants. The second and the third solutions are special cases of

4 U=r 2 + p(t)r + q(t), p(t) = 16νt − 4 ψ(t) dt + a, q(t) = 2

solutions having the form

with n arbitrary constants. The function g(r, t) can be expressed in terms of U (r, t) by

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

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

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

4 ◦ . “Two-dimensional” solution:

2 2 w(r, θ, t) = Ar 2 t + νH(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

Reference : D. K. Ludlow, P. A. Clarkson, and A. P. Bassom (1999).

where Q = ∆w =

r ∂r ∂r

r 2 ∂θ 2

Equation 10.3.3.2 is reduced to the given equation by passing to polar coordinates r, θ: x = r cos θ, y = r sin θ.

1 ◦ . Solutions with axial symmetry,

w = W (r, t),

are described by the linear parabolic equation ∂Q

2 ◦ . Generalized separable solution linear in θ:

w(r, θ, t) = f (r, t)θ + g(r, t),

where the functions f = f (r, t) and g = g(r, t) are determined by the differential equations

L( 2 f t )+ ar[L(f )] r +2 aL(f ) − r f r L( f)+r f [L(f )] r = νL ( f ), (2)

L( 2 g t )+ ar[L(g)] r +2 aL(g) − r g r L( f)+r f [L(g)] r = νL ( g). (3) Here, the subscripts −1 r and t denote partial derivatives with respect to r and t, L(f ) = r ( rf ) , and

L ( f ) = LL(f ). Equation (2) has particular solutions of the form

f = ϕ(t) ln r + ψ(t),

where ϕ(t) and ψ(t) are arbitrary functions. In this case, equation (3) is reduced by the change of variable U = L(g) to a second-order linear equation.

where E 2 w=r +

2 , E ∂r w = E(Ew). r ∂r ∂z

Preliminary remarks. The nonstationary Navier–Stokes equations written in cylindrical coordinates for the axisymmetric case are reduced to the equation in question by the introduction of a stream function w such that u r = 1 r ∂w ∂z and u z =− 1 r ∂w ∂r ,

where r= px 2 + y 2 , and u r and u z are the radial and axial fluid velocity components.

Reference : J. Happel and H. Brenner (1965).

1 ◦ . Any function w = w(r, z, t) that solves the second-order linear stationary equation Ew = 0 is also a solution of the original equation.

2 ◦ . Solution with axial symmetry:

w = U (r, t) + ϕ(t)r 2 + ψ(t),

where ϕ(t) and ψ(t) are arbitrary functions and the function U = U (r, t) is determined by the linear parabolic equation

∂U

1 ∂U

− νr

∂t

∂r r ∂r ∂r r ∂r

Here, f = f (r, t) and g = g(r, t) satisfy the system

L( 2 f t )+ r f [L(f )] r − r f r L( f ) − 2r f L(f ) = νL ( f ), (1)

f L(g) = νL ( g), (2) where L(

f [L(g)] r − r g r L( f ) − 2r

f)=f −1 rr − r f r ; the subscripts denote the corresponding partial derivatives. Particular solution of equation (1):

f (r, t) = C 2

1 ( t)r + C 2 ( t),

where C 1 ( t) and C 2 ( t) are arbitrary functions. In this case, the change of variable U = L(g) brings (2) to a second-order linear equation.

Reference : A. D. Polyanin and V. F. Zaitsev (2002).