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: \(\partial _1V_1+\partial _2V_2=0\) . 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 \(-(\partial _1^2+\partial _2^2) \psi\) . A simple solution is discussed for the two different limits of viscosity.

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