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Unknown in PDE

  1. Sep 3, 2011 #1
    unknown in PDE!!


    I'm solving a problem which determines the flow between a porous material and an impermeable material, using the slip-flow boundary conditions as proposed by Beavers and Joseph in '67. I can solve the whole problem as stated below, which gives the velocity [itex]u[/itex] of the fluid in the x-direction over the gap height (y-direction). However, in this equation I still have one unknown namely [itex]u_{B}[/itex] which is the slip velocity. How can I write this [itex]u_{B}[/itex] in function of the other variables so that this unknown disappear in my equation of [itex]u[/itex] ? A hint can maybe be enough!

    Poiseuille motion:

    [itex]\frac{d^2u}{dy^2} = \frac{1}{\mu}\frac{dP}{dx}[/itex]

    boundary conditions:

    1. [itex]u = 0[/itex] at [itex]y = h[/itex]

    2. [itex]\frac{du}{dy} = \frac{\alpha}{\sqrt{k}}u_{B}[/itex] at [itex]y = 0[/itex]

    Solution of this PDE is:

    [itex]u = \frac{1}{2\mu}\frac{dP}{dx}(y^2-h^2) + \frac{\alpha}{\sqrt{k}} u_{B} (y-h)[/itex]
    Last edited: Sep 3, 2011
  2. jcsd
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