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

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.