Solving the Unknown in PDE: Finding u_B!

[hermano]

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

Poiseuille motion:

$\frac{d^2u}{dy^2} = \frac{1}{\mu}\frac{dP}{dx}$

boundary conditions:

1. $u = 0$ at $y = h$

2. $\frac{du}{dy} = \frac{\alpha}{\sqrt{k}}u_{B}$ at $y = 0$

Solution of this PDE is:

$u = \frac{1}{2\mu}\frac{dP}{dx}(y^2-h^2) + \frac{\alpha}{\sqrt{k}} u_{B} (y-h)$

 

[fresh_42]

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 $F(y,u,u')=0$.