the development of poiseuille flow of a pseudoplastic fluid

106
M.A.M. Al Khatib
The constitutive Equation (3.1) becomes, in dimensionless form,
After linearization we find
τij =2 1+k 2 · γ 2 (n−1) /2
eij. (3.10)
∂u1
τ1 XX =2θ◦
∂X
∂v1
τ1YY =2θ◦
∂Y
τ1 XY = µ◦
∂u1
∂Y
∂v1
+
∂X
where θ◦ =(1+k 2 u ′ 2
◦ ) (n−1)/2 and µ◦ = θ◦
1+nk 2 u ′ 2
◦
1+k 2 u ′ 2
◦
.
(3.11)
From (3.9) and (3.11) with u1 =ΨY and v1 = −ΨX, where Ψ is the stream function, we have
∂2 ∂X∂Y (4θ◦ΨXY
2 ∂ ∂2
)+ −
∂Y 2 ∂X2
[µ◦(ΨYY − ΨXX)] = 0 (3.12)
and the boundary conditions on the walls Y = ±1 are
Now let
Ψ(X, ±1) = 0, ΨY (X, ±1) = 0. (3.13)
Ψ=Φ(Y ) exp(−αX) (3.14)
as in Wilson [1]. Substituting (3.14) in (3.12) we get
2 d
4α
dY
with boundary conditions
θ◦
2
2 dΦ d d Φ
+ − α2 µ◦
dY dY 2 dY 2 − α2
Φ = 0 (3.15)
Φ(±1) = Φ ′ (±1) = 0 (3.16)
and this determines the eigenvalues α. These are complex in general and we are interested in the decaying modes
for which Re(α) > 0. Note that when k =0,orn =1weget
θ◦ = µ◦ =1,
and we recover the Newtonian case
Φ iv +2α 2 Φ ′′
+ α 4 Φ = 0 (3.17)
studied by Wilson [1]. The eigenvalues satisfy sin 2α = ±2α, with four families of roots, one in each quadrant
of the complex plane. The two signs in sin 2α = ±2α correspond to odd and even solutions to Equation (3.15),
The Arabian Journal for Science and Engineering, Volume 31, Number 1A. January 2006