  We consider a diffuse interface model for phase separation of an isothermal
incompressible binary fluid in a Brinkman porous medium. The coupled system
consists of a convective Cahn-Hilliard equation for the phase field $\phi$,
i.e., the difference of the (relative) concentrations of the two phases,
coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for
the fluid velocity $\mathbf{u}$. This equation incorporates a diffuse interface
surface force proportional to $\phi \nabla \mu$, where $\mu$ is the so-called
chemical potential. We analyze the well-posedness of the resulting
Cahn-Hilliard-Brinkman (CHB) system for $(\phi,\mathbf{u})$. Then we establish
the existence of a global attractor and the convergence of a given (weak)
solution to a single equilibrium via {\L}ojasiewicz-Simon inequality.
Furthermore, we study the behavior of the solutions as the viscosity goes to
zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw
(CHHS) system. We first prove the existence of a weak solution to the CHHS
system as limit of CHB solutions. Then, in dimension two, we estimate the
difference of the solutions to CHB and CHHS systems in terms of the viscosity
constant appearing in CHB.
