We first consider a simple velocity field which satisfies Laplace's equation and has zero vorticity. The velocity field is
and
This velocity field satisfies Euler's equation if the pressure gradient is given by the equations:
and
where
is the gravity acceleration constant. From the two equations above, the pressure is given by the equation
where
is an integration constant defined by the boundary conditions. Then, this velocity field satisfies the hypothesis in Kelvin's theorem, so the circulation on any closed material contour should be constant, and in this special case the value of the circulation is zero. One way to check this affirmation is just to compute the circulation on a closed material contour, and the reader can use the vector field manipulation application. Another way is to take a look on the vorticity of this velocity field. The vorticity for this case is given by the equation
The circulation around any material contour is related to the vorticity flux across a surface limited by the integration contour according to Stoke's theorem. Therefore, for this particular velocity field the circulation on any closed material contour is zero. The reader can use the vector field manipulation application to check this by doing the following exercise: