  In this paper the motion of two-phase, incompressible, viscous fluids with
surface tension is investigated. Three cases are considered: (1) the case of
heat-conducting fluids, (2) the case of isothermal fluids, and (3) the case of
Stokes flows. In all three situations, the equilibrium states in the absence of
outer forces are characterized and their stability properties are analyzed. It
is shown that the equilibrium states correspond to the critical points of a
natural physical or geometric functional (entropy, available energy, surface
area) constrained by the pertinent conserved quantities (total energy, phase
volumes). Moreover, it is shown that solutions which do not develop
singularities exist globally and converge to an equilibrium state.
