The Flow Mapping Theorem

The Flow Mapping Theorem makes the flow of a fluid around an object easier to find. Ifis an obstacle andis a conformal mapping fromontowhere then it is possible to use the solutionof the Obstacle problem forto solve the obstacle problem for

Theorem

Letbe an obstacle and letbe a one to one conformal mapping fromontowheresuch that

for

The velocity functionis the unique solution to the obstacle problem forwith complex potential function

Example: Letbe the intervalwith circulationFind the complex velocity function and the conjugate velocity function.

By the Flow Mapping Theorem the the required complex potential function iswithand

We have

The velocity function is

Hence