Unknown in PDE

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unknown in PDE!!

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 [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:

fresh_42

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As far as I see it, there is an equation ##u'=c\cdot u_B## which you can insert into your solution. This gives you an ordinary differential equation ##F(y,u,u')=0##.
 

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