Solving a streamline function PDE

**Ehsan** · May 5th 2011, 09:12 PM

Hi,

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

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

with the boundary conditions of:

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

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

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

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

where $a$ , $b$ and $h$ are constants. $f$ and $F$ are functions of (x,z). Does anybody know if this PDE can be solved? Thanks.

**Danny** · May 6th 2011, 06:09 AM

Do you know the forms of $f$ and $F$ ?

**Ehsan** · May 7th 2011, 12:21 AM

No, the problem is that the functions $f$ and $F$ are actually functions of another variable say $Sw$ which is a function of $x$ and $z$

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

The ultimate goal is to find $S_w$ but I need to solve this equation in this general form if possible, then I use the results in another equation to solve for $S_w$ . Please let me know if it is not possible to solve it like this so I try to find another method to solve $S_w$ . Thanks.

**Danny** · May 7th 2011, 06:03 AM

I don't think that I can solve it but if you post the original problem there might be another approach.

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