  In this paper, we introduce a diffuse interface model for describing the
dynamics of mixtures involving multiple (two or more) phases. The coupled
hydrodynamical system is derived through an energetic variational approach. The
total energy of the system includes the kinetic energy and the mixing
(interfacial) energies. The least action principle (or the principle of virtual
work) is applied to derive the conservative part of the dynamics, with a focus
on the reversible part of the stress tensor arising from the mixing energies.
The dissipative part of the dynamics is then introduced through a dissipation
function in the energy law, in line with the Onsager principle of least energy
dissipation. The final system, formed by a set of coupled time-dependent
partial differential equations, reflects a balance among various conservative
and dissipative forces and governs the evolution of velocity and phase fields.
To demonstrate the applicability of the proposed model, a few two-dimensional
simulations have been carried out, including (1) the force balance at the
three-phase contact line in equilibrium, (2) a rising bubble penetrating a
fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The
effects of slip at solid surface have been examined in connection with contact
line motion and a pinch-off phenomenon.
