Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity

We consider a system which describes the behavior of a binary mixture of immiscible incompressible fluids with shear dependent viscosity by means of the diffuse interface approach. This system consists of Navier–Stokes type equations, characterized by a nonlinear stress-strain law, which are nonlinearly coupled with a convective Cahn–Hilliard equation for the order parameter. We analyze the corresponding dynamical system and, by means of the short trajectory method, we prove the existence of global and exponential attractors. We also discuss the dependence of an upper bound of the fractal dimension on the physical parameters of the system.