6.2: Complex Potential- Basic examples
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Basic examples
Uniform flow
The complex potential
F(z)=Ue−iαz
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
V=(u(x,y),v(x,y)).
But first we need to obtain the stream funcion ψ, which is the imaginary component of (1).
Rewriting (1) we obtain
F(z)=Ue−iαz=U(cosα−isinα)(x+iy)=U(xcosα+ysinα)+iU(ycosα−xsinα).
Thus
ψ=U(ycosα−xsinα).
Finally, since u=∂ψ∂y and v=−∂ψ∂x, we have that
u=Ucosα,v=Usinα.
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.
Stagnation point flow
The complex potential
F(z)=kz22
corresponds to the stagnation point flow with strength k≥0.
Source & Sink
A source of strength Q>0 at the origin is represented by the complex potential
F(z)=Q2πlogz.
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.
It is easy to generalise (2) for an arbitrary point (a,b) in the complex plane. The required complex potential is
F(z)=Q2πlog(z−c).
where c=a+ib.
Vortex
A vortex of strength C at the origin is represented by the complex potential
F(z)=−iC2πlogz.
This is again a multi-valued function. For C>0, rotation is anticlockwise, and for C<0 rotation is clockwise.
A vortex at an arbitrary point c∈C is represented by the complex potential
F(z)=−iC2πlog(z−c).
where c=a+ib.
Exercise 6.2.1
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≥0, and a source Q2πlogz, with Q≥0. Thus we can produce the complex potential
F(z)=Uz+Q2πlogz.
The following applet shows the flow produced by (3). Drag the sliders to change parameters.
Exercise 6.2.2
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.