6.2: Complex Potential- Basic examples

Basic examples

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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 $ψ$, which is the imaginary component of (1).

Rewriting (1) 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 $k≥0$.

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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 (2) 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 $c=a+ib$.

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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 $\PageIndex{1}$

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

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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 $U≥0$, and a source $\frac{Q}{2\pi}\log z$, with $Q≥0$. 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 (3). Drag the sliders to change parameters.

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Exercise $\PageIndex{2}$

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.