Approximation of the Long-Term Dynamics of the Dynamical System Generated by a 3D NS-$α$ System with Phase Transition

Abstract

In this article we study an approximate model for a binary fluid flow in a three-dimensional bounded domain. The governing equations consist of the Allen–Cahn equation for the order (phase) parameter $\phi$ coupled with the Navier–Stokes-$α$ (NS-$α$) system for the velocity $u$. We discretize these equations in time using the implicit Euler scheme and we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.