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

  1. May 24, 2012 #1
    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
     

    Attached Files:

    Last edited: May 24, 2012
  2. jcsd
  3. May 27, 2012 #2
    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
     
    Last edited: May 27, 2012
  4. May 27, 2012 #3

    vela

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    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.
     
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