  In this paper we analyze a fully discrete scheme for a general Cahn-Hilliard
equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes
two immiscible, incompressible and electrically conducting fluids with
different mobilities, fluid viscosities and magnetic diffusivities. A typical
fully discrete scheme, which is comprised of conforming finite element method
and the Euler semi-implicit discretization based on a convex splitting of the
energy of the equation is considered in detail. We prove that our scheme is
unconditionally energy stability and obtain some optimal error estimates for
the concentration field, the chemical potential, the velocity field, the
magnetic field and the pressure. The results of numerical tests are presented
to validate the rates of convergence.
