# Velocity profile between parallel plates with velocity dependent force

1. May 23, 2010

### MichielM

1. The problem statement, all variables and given/known data
I have a channel of height H (y-direction) with three different layers. A bottom layer of height L where only viscous forces play a role and the lower boundary is a stationary plate, a middle layer of height L (so from L to 2L) where a force F depending linearly on the fluid velocity (F=A+B u) works in the positive x-direction, and a top layer of height H-2L (so from 2L to H) where again only viscous forces act and the upper boundary is stationary. I can assume to have laminar flow without pressure gradient

Question: calculate the flow profile ux(y) and the flow rate Q

2. Relevant equations
The equation of motion for laminar flow only dependent on the y-direction:
$$0=\mu \frac{\partial^2 ux_i}{\partial y^2}+\frac{\partial P}{\partial x}+F_i$$

where subscript i denotes the layer (bottom, middle or top) and thus Fb and Ft are 0 and the pressure gradient term is always zero (and can be left out)

with boundary conditions:
$$ux_b(0)=0$$
$$ux_t(H)=0$$
$$ux_b(L)=ux_m(L)$$
$$ux_t(2L)=ux_m(2L)$$
$$\frac{\partial ux_b}{\partial y}(L)=\frac{\partial ux_m}{\partial y}(L)$$
$$\frac{\partial ux_t}{\partial y}(2L)=\frac{\partial ux_m}{\partial y}(2L)$$

3. The attempt at a solution
Unfortunately, I have not gotten to this part yet, because I think my equation of motion is incorrect since the units of my force term are Newton while the other terms are in Newton per cubic metre.

I have tried using $$\frac{F}{W_{channel} H_{channel} L_{channel}}$$ but then my solution contains the width of the channel in such a way that I have to know the numerical value of W to solve the flow rate, and I don't have that value.

Can anyone give me a hint on the way to approach this problem?!

Last edited: May 23, 2010
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