Unsteady Boundary Layer Equations for Non-Newtonian Fluids

9.3.4. Unsteady Boundary Layer Equations for Non-Newtonian Fluids

2 = ∂t∂y k ∂y ∂x∂y ∂x ∂y ∂y 2 ∂y 3 .

This equation describes an unsteady boundary layer on a flat plate in a power-law fluid flow; w is the steam function, and x and y are coordinates along and normal to the plate.

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

1 w(C n−2 1 C n−1 x+C 2 n−2 1 C 2 n−1 C 3 t, C 2 y+C 2 C 5 t, C n−1 1 C 2 n t) + C 5 x−C 3 y,

w 2 = w(x + C 6 , y+C 7 , t+C 8 )+ C 9 , Z

w 3 = w x, y + ϕ(x, t), t +

ϕ(x, t) dx + ψ(t),

∂t

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

2 ◦ . Generalized separable solution linear in x:

ψ(t) dt, where ψ(t) is an arbitrary function, and the function U (z, t) is determined by the second-order

w(x, y, t) = ψ(t)x +

U (z, t) dz, z=y+

differential equation

For details about this equation, see 1.6.18.2 with f (x) = const and 1.6.18.3 with f (U ) = kU n−1 (for n = 2, see Special case in equation 8.1.1.2).

xy

w(x, y, t) =

+ ψ(t)x +

U (y, t) dy,

t+C 1

where ψ(t) is an arbitrary function, C 1 is an arbitrary constant, and the function U (y, t) is determined by the second-order differential equation

t+C With the transformation

U= 1 u(ζ, τ ), τ=

3 ( t+C 1 ) 3 + C 2 , ζ = (t + C 1 ) y+ ψ(t)(t + C 1 ) dt + C t+C 3 1

one arrives at the simpler equation

For details about this equation, see 1.6.18.2 with f (x) = const and 1.6.18.3 with f (U ) = kU n−1 .

4 ◦ . “Two-dimensional” solution:

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

w(x, y, t) =

v(η, t) dη + ϕ(t)y + ψ(t)x,

is determined by the second-order differential equation

∂v n−1 ∂v ∂ 2

+ kϕ(t) − λψ(t)

With the transformation

v = R(ζ, t) − ϕ(t),

kϕ(t) − λψ(t) dt

one arrives at the simpler equation

Reference for equation

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

∂t∂y ∂y ∂x∂y

∂x ∂y 2 ∂y

∂y 2

This equation describes an unsteady boundary layer on a flat plate in a non-Newtonian fluid flow; w is the stream function, and x and y are coordinates along and normal to the plate.

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

w 1 = w(x, y + ϕ(x, t), t) +

ϕ(x, t) dx + ψ(t),

∂t

2 = C 1 w(C 1 x+C 1 C 2 t+C 3 , C 1 y+C 1 C 4 t+C 5 , C 1 t+C 6 )+ C 4 x−C 2 y+C 7 , where ϕ(x, t) and ψ(t) are arbitrary functions and the C n are arbitrary constants, are also solutions

of the equation.

ψ(t) dt, where ψ(t) is an arbitrary function and the function U (z, t) is determined by the second-order

w(x, y, t) = ψ(t)x +

U (z, t) dz, z=y+

differential equation

It admits, for any f = f (v), exact solutions of the following forms: U (z, t) = H(ζ),

ζ = kz + λt =⇒ equation λH = kf (kH ζ ′ )+ C;

U (z, t) = az + H(ζ), ζ = kz + λt =⇒ equation λH = kf (kH ζ ′ +

a) + C; √

=⇒ equation 1 2 1 H− 2 ζH ζ ′ =[ f (H ζ ′ )] ′ ζ , where

U (z, t) = t H(ζ),

ζ = z/ t

a, k, C, and λ are arbitrary constants. Solutions of the first two equations with H = H(ζ) can

be obtained in parametric form; see Kamke (1977) and Polyanin and Zaitsev (2003).

3 ◦ . Generalized separable solution linear in x:

Z xy

w(x, y, t) =

+ ψ(t)x +

U (y, t) dy,

t+C

where ψ(t) is an arbitrary function, C is an arbitrary constant, and the function U (y, t) is determined by the second-order differential equation

t+C With the transformation

U= u(ζ, τ ), τ= 1 3 3 ( t+C 1 ) + C 2 , ζ = (t + C 1 ) y+ ψ(t)(t + C 1 ) dt + C 3 t+C 1

one arrives at the simpler equation

For details about this equation, see Item 2 ◦ .

4 ◦ . “Two-dimensional” solution:

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

w(x, y, t) =

v(η, t) dη + ϕ(t)y + ψ(t)x,

is determined by the second-order differential equation

∂v

2 ∂v

+ kϕ(t) − λψ(t)

With the transformation

v = R(ζ, t) − ϕ(t),

kϕ(t) − λψ(t) dt

one arrives at the simpler equation

References for equation 9.3.4.2: A. D. Polyanin (2001 b, 2002), A. D. Polyanin and V. F. Zaitsev (2002).

f + g(x, t).

∂t∂y ∂y ∂x∂y

∂x ∂y 2 ∂y

∂y 2

This is an unsteady boundary layer equation for a non-Newtonian fluid with pressure gradient.

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

w 1 = w(x, y + ϕ(x, t), t) +

ϕ(x, t) dx + ψ(t),

∂t

where ϕ(x, t) and ψ(t) are arbitrary functions, is also a solution of the equation.

2 ◦ . There are degenerate solutions; see Item 3 ◦ in 9.3.3.2, where f (x, t) should be substituted by g(x, t).

3 ◦ . For g(x, t) = g(t), the transformation Z t

w = u(ξ, y, t) − ϕ ′ t ( t)y,

ξ = x + ϕ(t),

where ϕ(t) = −

( t − τ )g(τ ) dτ ,

leads to a simpler equation of the form 9.3.4.2:

∂y ∂ξ∂y

Note that g = g(t) and ϕ = ϕ(t) are related by the simple equation ϕ ′′ tt =− g.

4 ◦ . “Two-dimensional” solution (linear in x) for g(x, t) = g(t):

Z w(x, y, t) = a(t)x + U (y, t) dy,

where the function U = U (y, t) is determined by the second-order differential equation

− a(t)

f + g(t).

With the transformation

U = u(ξ, t) + g(t) dt,

ξ=y+ a(t) dt

one arrives at the simpler equation

For details about this equation, see 9.3.4.2, Item 2 ◦ .

5 ◦ . “Two-dimensional” solution (linear in x) for g(x, t) = s(t)x + h(t):

w(x, y, t) = a(t)y + ψ(t) x+

Q(y, t) dy,

where ψ(t) is an arbitrary function and a = a(t) is determined by the Riccati equation

a ′ t + 2 a = s( t),

and the function Q = Q(y, t) satisfies the second-order equation

∂Q

∂Q

f + a(t)y + ψ(t)

− a(t)Q + h(t).

With the transformation

Z Q= 2 Z(ξ, τ ) + h(t)Φ(t) dt , τ= Φ ( t) dt + A, ξ = yΦ(t) + ψ(t)Φ(t) dt + B, Φ( t)

where Φ( t) = exp

a(t) dt , one arrives at the simpler equation

For details about this equation, see 9.3.4.2, Item 2 ◦ .

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

w(x, y, t) =

v(η, t) dη + ϕ(t)y + ψ(t)x,

is determined by the second-order differential equation ∂v

∂v

f λ 2 − ϕ ′ ( t) + ∂t

+ kϕ(t) − λψ(t)

g(t). λ

With the transformation

v = R(ζ, t) − ϕ(t) +

g(t) dt,

kϕ(t) − λψ(t) dt

one arrives at the simpler equation

References for equation 9.3.4.3: A. D. Polyanin (2001 b, 2002), A. D. Polyanin and V. F. Zaitsev (2002).