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Math Help - Velocity Functions & streamlines (general & degenerate)

  1. #1
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    Velocity Functions & streamlines (general & degenerate)

    Velocity Functions & streamlines (general & degenerate)-q2a.jpg
    Note: I have included the mark scheme for each element of the question to give an indication of the amount of work involved.

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  2. #2
    Super Member Rebesques's Avatar
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    Ok, let's find the complex potential just to get you going

    If q(x,y)=q_x(x,y)+iq_y(x,y)=3x+(6-3y){\rm i}, we have

    {\rm div}q=\frac{\partial q_x}{\partial x}(x,y)+\frac{\partial q_y}{\partial y}(x,y)=3-3=0, so q is a model flow.

    Now to find the complex potential \Phi, we find its real part as the function \phi to satisfy q_x=\frac{\partial \phi}{\partial x}(x,y)=3x, \ q_y=\frac{\partial \phi}{\partial y}(x,y)=6-3y. The first equation gives \phi=\frac{3x^2}{2}+g(y), and substituting into the second one we get \frac{\partial (\frac{3x^2}{2}+g(y))}{\partial y}(x,y)=6-3y or g'(y)=6-3y, so g(y)=6y-\frac{3y^2}{2} and \phi(x,y)=\frac{3x^2}{2}-\frac{3y^2}{2}+6y.

    Now we find the streamline function \psi as the imaginary part of \Phi. Since \psi and \phi are conjugate harmonic, we get that \frac{\partial \phi}{\partial x}=\frac{\partial \psi}{\partial y}, \ \frac{\partial \phi}{\partial y}=-\frac{\partial \psi}{\partial x}.

    The first one gives \frac{\partial \psi}{\partial y}=3x or \psi=3xy+h(x), and by substituting this into the second one 3y+h'(x)=-6+3y, from which h(x)=-6x and finally \psi(x,y)=3xy-6x.

    We obtain the complex potential as \Phi(x,y)=\phi(x,y)+{\rm i}\psi(x,y)=\phi(x,y)=\frac{3x^2}{2}-\frac{3y^2}{2}+6y+{\rm i}x(3y-6).

    Now set \psi=constant for the equation of streamlines, etc.
    Last edited by Rebesques; August 27th 2007 at 10:07 AM. Reason: old age :(
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