- #1
hermano
- 38
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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 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)
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)
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