Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

A Phase-Field Model of Solidification With Convection



D M. Anderson, Geoffrey B. McFadden, A A. Wheeler


We develop a phase-field 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 quasi-incompressible 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 non-equilibrium form of the Clausius-Clapeyron 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 large values of the viscosity ratio.
Physica D-Nonlinear Phenomena


convection, diffuse interface, Navier-Stokes equations, phase-field model, solidification


Anderson, D. , McFadden, G. and Wheeler, A. (1999), A Phase-Field Model of Solidification With Convection, Physica D-Nonlinear Phenomena, [online], (Accessed July 25, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created February 28, 1999, Updated October 12, 2021