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tome101
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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
[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].
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
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