  We consider a model for the evolution of a mixture of two incompressible and
partially immiscible Newtonian fluids in two dimensional bounded domain. More
precisely, we address the well-known model H consisting of the Navier-Stokes
equation with non-autonomous external forcing term for the (average) fluid
velocity, coupled with a convective Cahn-Hilliard equation with polynomial
double-well potential describing the evolution of the relative density of atoms
of one of the fluids. We study the long term behavior of solutions and prove
that the system possesses a pullback exponential attractor. In particular the
regularity estimates we obtain depend on the initial data only through fixed
powers of their norms and these powers are uniform with respect to the growth
of the polynomial potential considered in the Cahn-Hilliard equation.
