Melting of a nonequilibrium vortex crystal in a fluid film with polymers : elastic versus fluid turbulence

Abstract

We perform a direct numerical simulation (DNS) of the forced, incompressible two-dimensional Navier-Stokes equation coupled with the FENE-P equations for the polymer-conformation tensor. The forcing is such that, without polymers and at low Reynolds numbers \mbox{Re}, the film attains a steady state that is a square lattice of vortices and anti-vortices. We find that, as we increase the Weissenberg number \mbox{Wi}, a sequence of nonequilibrium phase transitions transforms this lattice, first to spatially distorted, but temporally steady, crystals and then to a sequence of crystals that oscillate in time, periodically, at low \mbox{Wi}, and quasiperiodically, for slightly larger \mbox{Wi}. Finally, the system becomes disordered and displays spatiotemporal chaos and elastic turbulence. We then obtain the nonequilibrium phase diagram for this system, in the \mbox{Wi} - \Omega plane, where \Omega \propto {\mbox{Re}}, and show that (a) the boundary between the crystalline and turbulent phases has a complicated, fractal-type character and (b) the Okubo-Weiss parameter $\Lambda$ provides us with a natural measure for characterizing the phases and transitions in this diagram.Comment: 16 pages, 17 figure