## Complex Velocity Functions

Suppose thatis a model flow velocity function with domainThe conjugate velocity functionis analytic onmay not be simple but ifis a simply connected subregion ofhas has a primitivesatisfying for

A functionwhich is a primitive of a complex velocity functionis called a complex potential function for the flow. Taking the complex conjugate of (1) above givesfor

The circulation ofalong a curvedrawn in the fluid iswhere the integral is evaluated alongand the flux ofacrossiswhere the integral is evaluated along

Hencewhere the integral is evaluated along

Provided that %GAMMA lies within the simply connected subregion S on which the complex potential %OMEGA is defined, we may apply the Fundamental Theorem of Algebra to deduce

whereandare the start and end points of the curve

Henceand

Hence the circulation or flux along any contour betweenanddepends only on the points %alpha and %beta and is independent of the contour.

If the flow at each point is tangential to a curvethen the flow acrossat each point ofis zero. Curves with this property are called streamlines and represent snapshots of the motion of the fluid. If a curve is a streamline thenfor all pointsandonso that