 
Summary: Physica D 135 (2000) 175194
A phasefield model of solidification with convection
D.M. Anderson a
, G.B. McFadden b,
, A.A. Wheeler c
a Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
b Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD 208998910, USA
c Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, UK
Received 29 October 1998; accepted 15 March 1999
Communicated by H. MüllerKrumbhaar
Abstract
We develop a phasefield model for the solidification of a pure material that includes convection in the liquid phase.
The model permits the interface to have an anisotropic surface energy, and allows a quasiincompressible thermodynamic
description in which the densities in the solid and liquid phases may each be uniform. The solid phase is modeled as an
extremely viscous liquid, and the formalism of irreversible thermodynamics is employed to derive the governing equations.
We investigate the behavior of our model in two important simple situations corresponding to the solidification of a planar
interface at constant velocity: density change flow and a shear flow. In the former case we obtain a nonequilibrium form of the
ClausiusClapeyron equation and investigate its behavior by both a direct numerical integration of the governing equations,
and an asymptotic analysis corresponding to a small density difference between the two phases. In the case of a parallel shear
flow we are able to obtain an exact solution which allows us to investigate its behavior in the sharp interface limit, and for
