Domain comparison theorems for flows with vorticity

Steady plane flows of an ideal incompressible fluid are considered. Each flow possesses a vorticity function which is constant along the streamlines (lines on which the streamfunction is constant) of the flow. The qualitative dependence of the flow speed on variations of the bounding streamline is discussed using maximum principles for elliptic differential equations. The main theorems, involving the comparison of different flow domains, are the inclusion theorem and the under-over theorem. The simpler inclusion theorem involves the comparison of two flows in which the domain of one flow is included in the domain of the other. Both theorems are generalizations of the Lavrentiev-Gilbarg-Serrin comparison theorems for irrotational flows.