Vortex flow and maximum principles

Maximum principles of partial differential equations can give information on general properties of vortices and vorticity fields and the corresponding pressure and temperature fields independently of the solutions proper. They also reveal fundamental differences between two- and three-dimensional motions and peculiar flow properties in limiting cases: In steady two-dimensional viscous flows a maximum principle for the vorticity-transport equation exists and its consequences are discussed. Steady concentrated vortices with an extremum of vorticity occur only in the three-dimensional nonlinear case, and in linear slow motion vortices exhibit peculiar properties. In unsteady flows local extrema of vorticity are possible, even in two-dimensional flows. In all cases studied, the Navier–Stokes equation in the form of the vorticity-transport equation is used as the basic equation for incompressible homogeneous fluid flows.