About vortex equations of two dimensional flows

Abstract

A method to obtain a time-independent vortex solution of a nonlinear differential equation describing two-dimensional flow is investigated. In the usual way, starting from the Navier-Stokes equation the vortex equation is derived by taking a curl operation. After rearranging the equation of the vortex, we get a continuity equation or a divergence-free equation: partial derivative V-1(1) + partial derivative V-2(2) = 0. Additional irrotationality of V-1 and V-2 leads us to the Cauchy-Riemann condition satisfied by a newly introduced stream function Psi and velocity potential Phi. As a result, if we know V-1 and V-2 or a combination of two, the differential equation is mapped to a lower-order partial differential equation. This differential equation is the one satisfied by the stream function psi where the vorticity vector omega is given by -(partial derivative(2)(1)+partial derivative(2)(2))psi. A simple solution is discussed for the two different limits of viscosity.