Navier Stokes two infinite parallel plates

[gfd43tg]

Homework Statement

Compare the COMSOL results to the analytical solution for laminar flow between flat plates. Assume no effect of gravity on the flow (g = 0). The comparison will involve obtaining the velocity at a point in the flow field and the ΔP/L term. For example, you can compare the velocity at the center of the flow (d/2), where d (not L) is the distance between the two plates. Keep in mind the entrance boundary condition for the CFD simulation is a uniform velocity profile, which is NOT the same as the analytical solution.

Homework Equations

The Attempt at a Solution

I am working on the analytical portion of this problem. I have done the COMSOL simulation. I just want to see if I did my velocity profile correctly, and also how to do a pressure profile. I don't know if anyone here is familiar with COMSOL, but I don't know if it gives the equation of the velocity profile, it seems to just give a graph.

 

[maajdl]

It would help if you could ad some legends to you graphs.
What are the x and y-axis on these graphs, what are the different curves, ...
I could guess what it is, but writing it down would be better.

It should be clear that when the initial profile gets smoothed by viscosity, the linear pressure drop should be higher. Once this transient is over, you should be able to recover the traditional formula for the linear pressure drop.
How do you control the initial velocity profile in comsol? (Vy(y=0))

You assumed vy dvy/dy = 0 .
This is only true is the "stationary regime", not in the transient regime.
Your stationary solution is very likely correct.
You might try to estimate what happens in the transient phase by not assuming dvy/dy = 0.
just for a quick and dirty approximation, you may just replace vy dvy/dy by vyo dvy/dy,
where vyo is a constant. (or a similar approximation)

 

[gfd43tg]

We assumed that it is at steady state.

 

[maajdl]

You can have a steady state with vy dvy/dy =/= 0 .
Note in the question:

"Keep in mind the entrance boundary condition for the CFD simulation is a uniform velocity profile, which is NOT the same as the analytical solution."

You could have fun to also estimate analytically the stronger pressure drop at the entrance of the pipe.

 

[Chestermiller]

We assumed that it is at steady state.

Your steady state solution was very nicely done. One correction: you lost a factor of x in your final equation. It should be -xd, not -d. Also, the dp/dy should be an ordinary derivative, not a partial, since p varies only with y.

Chet

 

[gfd43tg]

Thank you, I found the missing x. Also, thanks for clearing up the pressure gradient as far as when its partial and when ordinary. One grad student writes as a partial and the other as a full, so I wasn't sure which one was correct, and thought it was too minor of a detail to ask in class.