Water flow down a porous channel (Navier-Stokes/Fluid dynamics)

[tome101]

Homework Statement

Water flows down a channel whose floor is porous, so that water seeps out of the bottom of the channel at a speed v, where v is constant and much less than the flow speed, U, along the surface of the channel. The seepage rate is slow so that H may be regarded as constant.

The x−component of the Navier-Stokes equation for such a system is

$\nu\frac{d^{2}u}{dy^{2}}+v\frac{du}{dy}+G=0$

where G is a constant.

Verify (i.e. no need to derive) that the general solution of this equation is

$u=A-\frac{G}{v}y+Be^{-vy/\nu}$

where A and B are integration constants.
State clearly what the boundary conditions are that determine A, B and G, and verify that the required solution is:

$u*=\frac{1-Re^{-R}y*-e^{-Ry*}}{1-(R+1)e^{-R}}$

where u = Uu*, y = Hy* and R = vH/$\nu$.

Homework Equations

u(0)=0

u(H)=U

The Attempt at a Solution

Ok, I managed the first part and have verified that the 2nd equation is indeed a solution of the first. However, I am having trouble removing the integration constants and G and rearranging to the required equation.

I know that at the top of the channel, u=U so u(H)=U. I'm also assuming that since v is 'much less' than the flow speed we can assum that at the bottom of the path the horizontal flow speed is equivalent to 0 (although I am less sure about this).

So that gives boundary conditions of:
u(H)=0
u(0)=0

Using u(0)=0 I can see that A+B=0

I can get an equation for U by using u(H).

However, I can't seem to find any useful way of rearranging these equations to a) remove A B and G or b) look anything like the final equation... I am also not 100% certain my boundary conditions are corect. Attached is a diagam of the problem. Any help would be greatly appreciated.

Thanks

 

[tome101]

290 views and no-one has any thoughts on this? I'm still completely stumped! I've also just noticed i should say u(H)=U under the attempted solution

 

[vela]

I'm not very familiar with fluid dynamics at all, but here are some thoughts:

Since you have three constants to eliminate, you'll need three equations. Your boundary conditions only give you two. You need to get one more constraint from somewhere.

The condition u(0)=0 is correct — you can see this by setting y^*=0 in the provided solution — but your reason for it is wrong. It doesn't have to do with the seepage rate. It's the no-slip condition. The water doesn't slide along the channel floor.