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Differentiability

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


Next: Some Manipulations Up: Introduction Previous: Complex variables