Any differentiable complex function F(z) is the complex potential for a 2D incompressible potential flow. Further is the complex conjugate velocity.
Reason: differentiability requires that is the same whichever direction we take . In particular if we take , a change in x only, we should get the same as when we take , a change in y only. That means, if :
Since , from which it follows that u and v satisfy both the continuity equation and irrotationality:Differentiable complex functions are easy to find:
Not differentiable:
In other words, taking real or imaginary parts, absolute values, complex conjugates, ..., all make the expression nondifferentiable.
Exercise:
Give as many 2d incompressible potential flows as you can.
You should now be able to do 18.2, 3