## A Complex Velocity Potential

Two dimensional fluid flow is naturally modelled by complex functions, which operate on complex numbers (each complex number has real and imaginary components, so is two dimensional) and return complex numbers, which are the components of the velocity in the real and imaginary directions.

If we assume that

1. The fluid flow is continuous and the fluid velocity is continuous

2. The fluid flow is two dimensional

3. The fluid flow is steady

then we can represent the velocity of the fluid at each point by a continuous complex function whose domain is the region occupied by the fluid.

In fact we must have with and To find the flow in a direction at angle anticlockwise from the real axis ( since anticlockwise is taken to be in the positive sense) relative to the real axis we can rotate by the angle then take the real part, so that where is the complex conjugate of and similarly We are interested in the real part of and this is the same if we take the complex conjugate of The conjugate of is in some ways more useful than Theorem

A steady two dimensional fluid flow with continuous velocity function on a region is a model flow ( circulation and flux free, so that the fluid has constant density) if and only the conjugate velocity function bar v is analytic on Any function analytic on a region can be considered as the conjugate velocity function of a continuous velocity function on Example: If ( is analytic on then is the model flow velocity function. 