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).