Let
be a model flow velocity function (so that
for every simple closed contour
in
whose inside also lies in) with domain
and let
be a punctured open disc inwith centre
Then
is a source of strength
if
for every simple closed contoursurrounding
is a sink of strengthif
for 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 point
for the complex velocity function
The source strength is
where C is the unit circle so
and
The 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 is
where C is the circle centre the origin, radius 1 over 2 so
and
The function
is analytic inside
so
by Cauchy's Theorem. The integral becomes
hence there is no source of sink at the origin.