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Physica D 151 (2001) 305331 A phase-field model with convection: sharp-interface asymptotics

Summary: Physica D 151 (2001) 305­331
A phase-field model with convection: sharp-interface asymptotics
D.M. Andersona,, G.B. McFaddenb, A.A. Wheelerc
a Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
b Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8910, USA
c Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, UK
Received 2 October 2000; received in revised form 19 January 2001; accepted 29 January 2001
Communicated by C.K.R.T. Jones
We have previously developed a phase-field model of solidification that includes convection in the melt [Physica D 135
(2000) 175]. This model represents the two phases as viscous liquids, where the putative solid phase has a viscosity much
larger than the liquid phase. The object of this paper is to examine in detail a simplified version of the governing equations
for this phase-field model in the sharp-interface limit to derive the interfacial conditions of the associated free-boundary
problem. The importance of this analysis is that it reveals the underlying physical mechanisms built into the phase-field
model in the context of a free-boundary problem and, in turn, provides a further validation of the model. In equilibrium,
we recover the standard interfacial conditions including the Young­Laplace and Clausius­Clapeyron equations that relate
the temperature to the pressures in the two bulk phases, the interface curvature and material parameters. In nonequilibrium,
we identify boundary conditions associated with classical hydrodynamics, such as the normal mass flux condition, the
no-slip condition and stress balances. We also identify the heat flux balance condition which is modified to account for the
flow, interface curvature and density difference between the bulk phases. The interface temperature satisfies a nonequilibrium


Source: Anderson, Daniel M. - Department of Mathematical Sciences, George Mason University


Collections: Mathematics