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Thread: Solving a streamline function PDE

  1. #1
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    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.
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  2. #2
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    Do you know the forms of $\displaystyle f$ and $\displaystyle F$?
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  3. #3
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    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.
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  4. #4
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    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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