  In this paper we report on 2D numerical simulations concerning linear and
nonlinear evolution of surface-tension-driven instability in two-fluid systems
heated from below using classical and phase-field models. In the phase-field
formalism, one introduces an order parameter called phase-field function to
characterize thermodynamically the phases. All the system parameters are
assumed to vary continuously from one fluid bulk to another (as linear
functions of the phase-field). The Navier-Stokes equation (with some extra
terms) and the heat equation are written only once for the whole system. The
evolution of the phase-field is described by the Cahn-Hilliard equation. In the
sharp-interface limit the results found by the phase-field formalism recover
the results given by the classical formulation.
