Steady Boundary Layer Equations for Non-Newtonian Fluids

9.3.2. Steady Boundary Layer Equations for Non-Newtonian Fluids

2 = ∂y k ∂x∂y ∂x ∂y ∂y 2 ∂y 3 .

This equation describes a boundary layer on a flat plane in the flow of a power-law non-Newtonian fluid; w is the stream function, x and y are the longitudinal and normal coordinates, and n and k are rheological parameters ( n > 0, k > 0).

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

w 1 = C 1 w(C 2− n 2 1 n−1 C 2 x+C 3 , C 2 y+C 4 )+ C 5 , w 2 = w(x, y + ϕ(x)),

where C 1 , ...,C 5 are arbitrary constants and ϕ(x) is an arbitrary function, are also solutions of the equation.

1 2 n−1

w(x, y) =

2 C 1 ( n − 1)y + C 2 + C 3 y+C 4 − kC 1 x if C n ≠ 1/2,

n−1

1 n(2n − 1)

w(x, y) = − 2 ln( C 1 y+C 2 )+ C 3 y+C 4 +2 kC 1 x

if n = 1/2.

3 ◦ . Multiplicative separable solutions:

1 w(x, y) = λ(2 − n)x + C 1 2− n

F (y) if n ≠ 2,

w(x, y) = C 1 e λx

F = F (y) is determined by the autonomous ordinary differential equation

λ(F ′ ) 2 − λF F ′′ = k(F ′′ ) y n−1 yy yy F yyy ′′′ ,

whose order can be reduced by two. The equation for

F has a particular solution in the form of a power-law function, F=A

n 2 ( y + C) β n , where β n = n−1

n−2 .

4 ◦ . Self-similar solution ( n ≠ 2 and λ is any):

2 λn−λ+1

w(x, y) = x 2− n ψ(z),

z=x λ y,

where the function ψ = ψ(z) is determined by the autonomous ordinary differential equation

zz = k(ψ ′′ zz ) n−1 ψ ′′′ zzz , (2) whose order can be reduced by two. ❅✂❆

Reference : Z. P. Shulman and B. M. Berkovskii (1966). Special case 1. The generalized Blasius problem on a translational flow with an incident velocity U i past a flat plate is

characterized by the boundary conditions ∂ x w=∂ y w=0 at y = 0,

∂ y w→U i as y → ∞,

∂ y w=U i at x = 0. A solution to this problem (in the domain x ≥ 0, y ≥ 0) is sought in the form (1) with λ = − 1 n+1 . The boundary conditions

for ψ(z) are the following:

ψ=ψ z ′ =0 at z = 0,

ψ ′ z →U i as z → ∞.

In Zaitsev and Polyanin (1989, 1994), exact solutions to problem (2)–(3) are specified for λ=− 1 n+1 with n= 1 5 , 1 4 , 1 2 , 3 5 ,

7 , 2. Special case 2. The generalized Schlichting problem on the symmetric flow of a plane laminar power-law fluid jet out

of a thin slit is characterized by the boundary conditions

∂ y w → 0 as y → ∞, which are supplemented with the integral condition of conservation of momentum

∂ x w=∂ yy w=0 at y = 0,

2 0 ∂ y w) dy = A

A = const).

A solution to this problem (in the domain x ≥ 0, y ≥ 0) is sought in the form (1) with λ = − 2 3 n . A solution to equation (2) for ψ(z) with appropriate boundary conditions and integral condition (see the conditions in Special case 3, Subsection 9.3.1, where

F should be replaced by ψ) can be found in the books by Shulman and Berkovskii (1966) and Polyanin, Kutepov, et al. (2002).

5 ◦ . Self-similar solution for n = 2 (λ is any):

w(x, y) = x λ U (z),

z = yx −1 /3 ,

where the function U = U (z) is determined by the autonomous ordinary differential equation ( λ− 1 )( U 3 ′ z ) 2 + λU U ′′ zz = kU ′′ zz U ′′′ zzz ,

whose order can be reduced by two.

w(x, y) = e λ(2n−1)x Φ( τ ),

τ=e λ(2−n)x y,

where the function Φ = Φ( τ ) is determined by the autonomous ordinary differential equation

τ ′ ) − λ(2n − 1)ΦΦ ′′ ττ = k(Φ ′′ ττ ) n−1 Φ τττ ′′′ , whose order can be reduced by two. ❇✂❈

λ(n + 1)(Φ 2

Reference : Z. P. Shulman and B. M. Berkovskii (1966).

7 ◦ . Solution for n ≠ 1/2:

w(x, y) = C 1 ln | x| + C 2 + g(ξ),

ξ=x 1−2 n y,

where the function g = g(ξ) is determined by the autonomous ordinary differential equation

whose order can be reduced by two.

8 ◦ . Solution for n = 1/2:

w(x, y) = C 1 x+C 2 + h(ζ),

ζ=e λx y,

where the function h = h(ζ) is determined by the autonomous ordinary differential equation

λ(h −1 ′

ζ ) 2 − C 1 h ζζ /2 ′′ = k(h ′′ ζζ ) h ′′′ ζζζ ,

whose order can be reduced by two.

9 ◦ . Conservation law:

x nw y + D y − nw x w y − kw n yy = 0,

where D x = ∂ ∂x and D y = ∂ ∂y .

2 3 + ∂y f (x). ∂x∂y ∂x ∂y ∂y ∂y This is a steady boundary layer equation for a power-law fluid with pressure gradient.

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

w 1 = w(x, y + ϕ(x)) + C,

where ❇✂❈ ϕ(x) is an arbitrary function and C is an arbitrary constant, is also a solution of the equation.

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

2 ◦ . Degenerate solutions (linear and quadratic in y) for any f (x):

w(x, y) = ❉ y 2 f (x) dx + C 1 + ϕ(x),

w(x, y) = C 1 y 2 + ϕ(x)y +

ϕ 2 ( x) − 2

f (x) dx + C 2 ,

where ϕ(x) is an arbitrary function and C 1 and C 2 are arbitrary constants. These solutions are independent of k and correspond to inviscid fluid flows.

2 nm+2n−m+1

2 m−n−nm

w(x, y) = x

2( n+1)

ψ(z),

z=x 2( n+1) y,

where the function ψ = ψ(z) is determined by the autonomous ordinary differential equation

zz = k(ψ n−1 zz ′′ ) ψ ′′′ zzz + a. Note that solving the generalized Falkner–Skan problem on a symmetric power-law fluid flow

❊✂❋ past a wedge is reduced to solving the equation just obtained.

Reference : Z. P. Shulman and B. M. Berkovskii (1966).

4 ◦ . Generalized self-similar solution for f (x) = ae βx :

2 n−1

2− n

w(x, y) = exp β

where the function Φ = Φ( τ ) is determined by the autonomous ordinary differential equation

Reference : Z. P. Shulman and B. M. Berkovskii (1966).

5 ◦ . Additive separable solution for f (x) = a:

w(x, y) = C 1 x + h(y), where the function h = h(y) is determined by the autonomous ordinary differential equation

Its general solution can be written out in parametric form: Z t u n−1 du

Z u v n dv y = −k

Z t u n−1 ϕ(u) du

, h=k 2 ,

where ϕ(u) = .

6 ◦ . Multiplicative separable solution for f (x) = ax 2− n , n ≠ 2:

w(x, y) = x 2− n

F (y),

where the function

F = F (y) is determined by the autonomous ordinary differential equation

7 ◦ . Self-similar solution for f (x) = ax m , n = 2:

w(x, y) = x 1 2 m+ 5 6 U (z),

z = yx −1 /3 ,

where the function U = U (z) is determined by the autonomous ordinary differential equation

1 ( m + 1)(U

z 2 ′ ) 2 + 1 6 (3 m + 5)U U zz ′′ = kU zz ′′ U zzz ′′′ + a.

8 ◦ . Multiplicative separable solution for f (x) = ae λx , n = 2:

w(x, y) = e 2 1 λx G(y),

where the function

G = G(y) is determined by the autonomous ordinary differential equation

1 λ(G ′ ) 2

− 1 2 λGG ′′ yy = k(G ′′ yy ) n−1 G ′′′ yyy + a.

9 ◦ . Solution f (x) = ax 1−2 n , n ≠ 1/2:

w(x, y) = C 1 ln | x| + C 2 + g(ξ),

ξ=x 1−2 n y,

where the function g = g(ξ) is determined by the autonomous ordinary differential equation

10 ◦ . Solution f (x) = ae λx , n = 1/2:

w(x, y) = C 1

1 x+C 2 + h(ζ),

ζ=e 2 λx y,

where the function h = h(ζ) is determined by the autonomous ordinary differential equation

11 ◦ . Conservation law:

x nw y − nF (x) + D y − nw x w y − kw n yy = 0,

where D x =

∂y ∂x∂y

This is an equation of a steady boundary layer on a flat plate in the flow of a non-Newtonian fluid of general form; w is the steam function, and x and y are the coordinates along and normal to the plate. Preliminary remarks. The system of non-Newtonian fluid boundary layer equations

where u 1 and u 2 are the longitudinal and normal fluid velocity components, can be reduced to the equation in question by

the introduction of a stream function w such that u 1 = ∂w ∂y and u 2 =− ∂w ∂x .

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

w 1 = C −2 1 w(C 3 1 x+C 2 , C 1 y+C 3 )+ C 4 , w 2 = w(x, y + ϕ(x)),

where C 1 , ...,C 4 are arbitrary constants and ϕ(x) is an arbitrary function, are also solutions of the equation. ●✂❍

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

2 ◦ . Solutions involving arbitrary functions:

w(x, y) = C 1 y 2 + 1 ϕ(x)y + ϕ 2 ( x) + C 2 ,

4 C 1 w(x, y) = g(z) + C 1 x+C 2 , z = y + ϕ(x),

where C 1 and C 2 are arbitrary constants, and ϕ(x) is an arbitrary function. The function g = g(z) in the second formula is determined by the autonomous ordinary differential equation

f (g ′′ zz )+ C 1 g z ′ = C 3 ,

3 ◦ . Self-similar solution:

w(x, y) = x 2 /3

ψ(ξ), −1 ξ = yx /3 ,

where the function ψ = ψ(ξ) is determined by the autonomous ordinary differential equation ( ψ ′ ξ ) 2 −2 ψψ ξξ ′′ = 3[ f (ψ ′′ ξξ )] ′ ξ .

4 ◦ . The von Mises transformation

∂w

ξ = x, η = w, U (ξ, η) =

where w = w(x, y),

∂y

leads to the second-order nonlinear equation

It admits, for example, a traveling-wave solution U = U (aξ + bη).

5 ◦ . Conservation law:

D x w 2 y + D y − w x w y − f (w yy ) = 0, where D x = ∂ ∂x and D y = ∂ ∂y .

f + g(x).

∂y ∂x∂y

This is a steady boundary layer equation for a non-Newtonian fluid of general form with pressure gradient.

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

w 1 = w(x, y + ϕ(x)) + C,

where ϕ(x) is an arbitrary function and C is an arbitrary constant, is also a solution of the equation.

2 ◦ . There are degenerate solutions; see Item 2 ◦ in 9.3.2.2, where f (x) should be replaced by g(x).

3 ◦ . Solution for g(x) = a:

w(x, y) = ζ(z) + C 1 x+C 2 , z = y + ϕ(x), where ϕ(x) is an arbitrary function and C 1 and C 2 are arbitrary constants. The function ζ = ζ(z) is

determined by the ordinary differential equation

f (ζ ′′ zz )+ C 1 ζ ′ z + aζ = C 3 .

4 ◦ . Self-similar solution for g(x) = a(x + b) −1 /3 :

w(x, y) = (x + b) 2 /3 ψ(ξ),

ξ = y(x + b) −1 /3 ,

where the function ψ = ψ(ξ) is determined by the autonomous ordinary differential equation ( ψ ′ ξ ) 2 −2 ψψ ξξ ′′ = 3[ f (ψ ′′ ξξ )] ′ ξ +3 a.

5 ◦ . Conservation law:

D w x 2 y − G(x) + D y − w x w y − f (w yy ) = 0, Z

where D x =