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The two-dimensional Navier-Stokes equations are rewritten as a system ofcoupled nonlinear ordinary differential equations. These equations describe theevolution of the moments of an expansion of the vorticity with respect toHermite functions and of the centers of vorticity concentrations. We prove theconvergence of this expansion and show that in the zero viscosity and zero coresize limit we formally recover the Helmholtz-Kirchhoff model for the evolutionof point-vortices. The present expansion systematically incorporates theeffects of both viscosity and finite vortex core size. We also show that alow-order truncation of our expansion leads to the representation of the flowas a system of interacting Gaussian (i.e. Oseen) vortices which previousexperimental work has shown to be an accurate approximation to many importantphysical flows [9].

Generalized Helmholtz-Kirchhoff model for two dimensional distributed vortex motion