# Thread: Velocity Functions & streamlines (general & degenerate)

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

Note: I have included the mark scheme for each element of the question to give an indication of the amount of work involved.

Thanks

2. Ok, let's find the complex potential just to get you going

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

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

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

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

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

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

Now set $\displaystyle \psi=$constant for the equation of streamlines, etc.