A phase-field model of solidification with convection

We develop a phase-field model for the solidification of a pure material th at includes convection in the liquid phase. The model permits the interface to have an anisotropic surface energy, and allows a quasi-incompressible t hermodynamic description in which the densities in the solid and liquid pha ses may each be uniform. The solid phase is modeled as an extremely viscous liquid, and the formalism of irreversible thermodynamics is employed to de rive the governing equations. We investigate the behavior of our model in t wo important simple situations corresponding to the solidification of a pla nar interface at constant velocity: density change flow and a shear flow. I n the former case we obtain a non-equilibrium form of the Clausius-Clapeyro n equation and investigate its behavior by both a direct numerical integrat ion of the governing equations, and an asymptotic analysis corresponding to a small density difference between the two phases. In the case of a parall el sheer flow we are able to obtain an exact solution which allows us to in vestigate its behavior in the sharp interface limit, and for large values o f the viscosity ratio. (C) 2000 Elsevier Science B.V. All rights reserved.