MichielM
May23-10, 04:18 AM
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?!
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?!