  In this work, we study an optimal boundary control problem for a Cahn -
Hilliard -Navier-Stokes (CHNS) system in a two dimensional bounded domain. The
CHNS system consists of a Navier-Stokes equation governing the fluid velocity
field coupled with a convective Cahn - Hilliard equation for the relative
concentration of the fluids. An optimal control problem is formulated as the
minimization of a cost functional subject to the controlled CHNS system where
the control acts on the boundary of the Navier-Stokes equations. We first prove
that there exists an optimal boundary control. Then we establish that the
control-to-state operator is Frechet differentiable and derive first-order
necessary optimality conditions in terms of a variational inequality involving
the adjoint system.
