## Laplace's Equation and Fluid Flow

Ifis a continuous complex velocity function on a regionand suppose thatand have partial derivatives with respect toandwhich are continuous onThen is a model flow velocity function onif and only ifand

Proof

By the Cauchy – Riemann equations, if the partial derivatives of the real functionsandexist and are continuous on a regionthenis analytic onif and only ifandon

The velocity functionhas conjugate velocity functionandis a model flow velocity function onif and only ifis analytic onPuttingand we have thatis a model flow velocity function onif and only if

(1) and(2) on R.

Differentiating (1) with respect togivesand differentiating (2) with respect togives

Subtracting these equations and using thatfor analytic functions gives(3)

A similar equation can be derived for

Equation (30 is Laplace's equation.