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

The fluid flow is continuous and the fluid velocity is continuous

The fluid flow is two dimensional

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 havewithand
To find the flow in a direction at angleanticlockwise from the real axis (since anticlockwise is taken to be in the positive sense) relative to the real axis we can rotateby the anglethen take the real part, so thatwhereis the complex conjugate ofand similarly
We are interested in the real part ofand this is the same if we take the complex conjugate ofThe conjugate ofis in some ways more useful than
Theorem
A steady two dimensional fluid flow with continuous velocity functionon a regionis 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 functionanalytic on a regioncan be considered as the conjugate velocity function of a continuous velocity functionon
Example: If (is analytic onthenis the model flow velocity function.