# Solving a streamline function PDE

• May 5th 2011, 09:12 PM
Ehsan
Solving a streamline function PDE
Hi,

I am trying to solve an streamline function. I have found this PDE for the steamline function, $\displaystyle \psi$.

$\displaystyle \frac{\partial}{\partial z}\left(f\frac{\partial \psi }{\partial z} \right)+ \frac{\partial}{\partial x}\left(a.f\frac{\partial \psi }{\partial x} \right)-\frac{\partial F}{\partial z}=0$

with the boundary conditions of:

$\displaystyle \frac{\partial \psi }{\partial x}=0$ at $\displaystyle z=0$

$\displaystyle \frac{\partial \psi }{\partial x}=0$ at $\displaystyle z=h$

$\displaystyle \frac{\partial \psi }{\partial z}=b$ at $\displaystyle x=0$

$\displaystyle \frac{\partial \psi }{\partial z}=b$ at $\displaystyle x=\infty$

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle h$ are constants. $\displaystyle f$ and $\displaystyle F$ are functions of (x,z). Does anybody know if this PDE can be solved? Thanks.
• May 6th 2011, 06:09 AM
Jester
Do you know the forms of $\displaystyle f$ and $\displaystyle F$?
• May 7th 2011, 12:21 AM
Ehsan
No, the problem is that the functions $\displaystyle f$ and $\displaystyle F$ are actually functions of another variable say $\displaystyle Sw$ which is a function of $\displaystyle x$ and $\displaystyle z$

$\displaystyle F=F(S_w)=F(S_w(x,z))=F(x,z)$
$\displaystyle f=f(S_w)=f(S_w(x,z))=f(x,z)$

The ultimate goal is to find $\displaystyle S_w$ but I need to solve this equation in this general form if possible, then I use the results in another equation to solve for $\displaystyle S_w$. Please let me know if it is not possible to solve it like this so I try to find another method to solve $\displaystyle S_w$. Thanks.
• May 7th 2011, 06:03 AM
Jester
I don't think that I can solve it but if you post the original problem there might be another approach.