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Solving PDE for slip flow in porous media

Solving PDE for slip flow in porous media

Solving PDE for slip flow in porous media

(OP)
Hi,

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 'u' of the fluid in the x-direction over the gap height (y-direction). However, in this equation I still have one unknown namely 'ub' which is the slip velocity. How can I write this 'ub' in function of the other variables so that this unknown disappear in my equation of  'u' ? A hint can maybe be enough!

Poiseuille motion:

d^2u/dy^2 = 1/mu dP/dx

boundary conditions:

1. u = 0 at y = h

2. du/dy = alpha/k^(1/2) ub at y  = 0

Solution of this PDE is:

u = 1/2mu dP/dx (y^2-h^2) + alpha/k^(1/2) ub (y-h)  

RE: Solving PDE for slip flow in porous media

Seems like continuity ought to apply somewhere.  And if you knew something about shear stress it might help, too.  You doing this for fun or profit?  

 

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