A Melnikov function for the break-up of closed streamlines in steady Navier–Stokes flows

Abstract

A Melnikov approach is proposed to study the existence/nonexistence of closed streamlines and asymptotic orbits in solutions of the steady Navier–Stokes equations. In certain limiting cases the method gives important insight into the physical processes that cause the break-up of closed and heteroclinic streamlines. The approach sheds light on the important issue of transport in three-dimensional Navier–Stokes flows. Specifically we show that the criterion for the existence of closed streamlines is that the line integral of the viscous force, induced from the perturbation of the flow along the unperturbed closed streamline, must vanish. We apply the approach to wavy Taylor vortex flow and show that the space averaged Melnikov function is related to the effective axial diffusivity.