(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

[itex]\nu\frac{d^{2}u}{dy^{2}}+v\frac{du}{dy}+G=0[/itex]

where G is a constant.

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

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

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:

[itex]u*=\frac{1-Re^{-R}y*-e^{-Ry*}}{1-(R+1)e^{-R}}[/itex]

where u = Uu*, y = Hy* and R = vH/[itex]\nu[/itex].

2. Relevant equations

u(0)=0

u(H)=U

3. The attempt at a solution

Ok, I managed the first part and have verified that the 2nd eqation 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

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Water flow down a porous channel (Navier-Stokes/Fluid dynamics)

**Physics Forums | Science Articles, Homework Help, Discussion**