Sources and Sinks
Letbe a model flow velocity function (so thatfor every simple closed contourinwhose inside also lies in) with domainand letbe a punctured open disc inwith centreThen
is a source of strengthiffor every simple closed contoursurrounding
is a sink of strengthiffor every simple closed contoursurrounding
In each case the contour is traversed in the positive counterclockwise sense.
Example: Find the strength of the source at the pointfor the complex velocity function
The source strength iswhere C is the unit circle soandThe integral becomes
by Cauchy's Residue Theorem.
Example: Show that there is no source or sink at the pointfor the complex velocity function
The source strength iswhere C is the circle centre the origin, radius 1 over 2 soand
The functionis analytic insidesoby Cauchy's Theorem. The integral becomes
hence there is no source of sink at the origin.