Complex Analysis

Complex Potential

Basic examples


Uniform flow

The complex potential \begin{eqnarray}\label{uniform} F(z)=Ue^{-i\alpha}z \end{eqnarray} corresponds to uniform flow at speed $U$ in a direction making an angle with the $x$-axis.

Here we are interested in finding the velocity field $$\mathbf V = \left(u(x,y), v(x,y)\right).$$ But first we need to obtain the stream funcion $\psi$, which is the imaginary component of ($\ref{uniform}$).

Rewriting ($\ref{uniform}$) we obtain \begin{eqnarray*} F(z)&=& Ue^{-i\alpha}z \\ &=& U\left( \cos \alpha - i \sin \alpha \right)\left( x+iy\right) \\ &=& U\left( x\cos \alpha + y \sin \alpha \right) + i U\left( y\cos \alpha - x \sin \alpha \right). \end{eqnarray*} Thus \begin{eqnarray*} \psi = U\left( y\cos \alpha - x \sin \alpha \right). \end{eqnarray*}

Finally, since $u=\frac{\partial \psi}{\partial y}$ and $v=-\frac{\partial \psi}{\partial x}$, we have that \begin{eqnarray*} u=U \cos \alpha ,\quad v=U\sin \alpha. \end{eqnarray*}

The applet below shows a simulation of the uniform flow. Drag the sliders to change parameters. Click on Trace button to show streamlines. Click on Field button to show vector field.



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Stagnation point flow

The complex potential \begin{eqnarray*} F(z)=\frac{k z^2}{2} \end{eqnarray*} corresponds to the stagnation point flow with strength $k\geq 0$.



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Source & Sink

A source of strength $Q>0$ at the origin is represented by the complex potential \begin{eqnarray}\label{source-sink} F(z)=\frac{Q}{2\pi}\log z . \end{eqnarray} Note that this is a multi-valued function, with a branch point at the origin. If $Q<0$, then the complex potential corresponds to a sink.



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It is easy to generalise (\ref{source-sink}) for an arbitrary point $(a, b)$ in the complex plane. The required complex potential is \begin{eqnarray*} F(z)=\frac{Q}{2\pi}\log(z-c). \end{eqnarray*} where $c= a+ib$.


Vortex

A vortex of strength $C$ at the origin is represented by the complex potential \begin{eqnarray*} F(z)=\frac{-iC}{2\pi}\log z . \end{eqnarray*} This is again a multi-valued function. For $C>0$, rotation is anticlockwise, and for $C < 0$ rotation is clockwise.



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A vortex at an arbitrary point $c\in \mathbb C$ is represented by the complex potential \begin{eqnarray*} F(z)=\frac{-iC}{2\pi}\log(z-c). \end{eqnarray*} where $c= a+ib$.

Exercise: Find the velocity fields of the Stagnation point, Source & Sink and Vortex flows.


Combining complex potentials

The basic flows presented above can be combined by simply superimposing the corresponding complex potentials.

For example, consider a uniform flow $Uz$, with speed $U\ge 0$, and a source $\frac{Q}{2\pi}\log z $, with $Q\ge 0$. Thus we can produce the complex potential \begin{eqnarray}\label{comb} F(z)=Uz+\frac{Q}{2\pi}\log z . \end{eqnarray} The following applet shows the flow produced by (\ref{comb}). Drag the sliders to change parameters.



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Exercise: Find the velocity field of the flow produced by a source of strength $Q$ in a uniform flow at speed $U$ in the $x$-direction.


NEXT: Uniform flow around a circle