the development of poiseuille flow of a pseudoplastic fluid

3. EQUATIONS OF MOTION
M.A.M. Al Khatib
We consider a region far downstream from the entrance of a straight channel of width 2h, with the origin
in the centre of the channel. The velocity components of the flow in the downstream, x, and transverse, y,
directions are u and v respectively.
The smoothed-out power law model which is going to be used here, the Carreau Model, is given in
3-dimensional form by Equation (2.2) with · γ= 2eijeij (see Bird [4]). With η∞ = 0 for simplicity, we get
where e ∗ ij
τ ∗ ij =2η◦
is the strain-rate tensor.
1+(λ · γ) 2(n−1) /2
e ∗ ij
By neglecting the inertial effects, the equations of motion in the steady state in the x and y directions are
given respectively by
0=− ∂P∗
∂x + ∂τ∗ xx
∂x + ∂τ∗ xy
∂y ,
0=− ∂P∗
∂y + ∂τ∗ xy
∂x + ∂τ∗ yy
∂y .
First we determine the basic parallel flow which is approached far downstream. Here the pressure gradient is
uniform and we put − ∂P
∂x = Gr > 0 for convenience. Then from (3.2) we get
0=Gr + ∂τ∗ xy
∂y
January 2006 The Arabian Journal for Science and Engineering, Volume 31, Number 1A. 103
(3.1)
(3.2)
. (3.3)
Integrating (3.3) with respect to y using the symmetry condition τ ∗ xy = 0 when y =0weget
τ ∗ xy = −Gr y . (3.4)
Using Equations (3.1) and (3.4) we get
−Gr y = η◦
1+(λ (u ∗ ◦) ′ ) 2 (n−1) /2
where (u ∗ ◦) ′ = ∂u ∗ ◦/∂y and the subscript ◦ refers to the parallel flow solution.
(u ∗ ◦) ′ . (3.5)
Now we choose dimensionless variables; we put Y = y/h, X = x/h, P = P ⋆ /Gr h, τij = τ ∗ ij /Grh, and
u◦ = η◦u∗ ◦/Grh2 . Then the dimensionless form of the solution is given by the following equations
t◦ = −Y, (3.6)
−Y =
1+k 2 ( u ′
◦) 2 (n−1)/2
u ′
◦ , (3.7)
where t◦ and u ′
◦ are the stress and velocity derivative in the case of parallel flow and k =(λGrh)/η◦. Note
that when k is large, (3.7) represents a shear thinning fluid and when k is small, (3.7) represents a Newtonian
fluid. Equation (3.7) is solved numerically with the boundary condition u◦ = 0 when Y = −1, giving u ′
◦ and u◦.
Examples of velocity profiles are given in Figures 2, 3, and 4.