We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been applied successfully to situations in which the physical phenomena of interest have associated with them a length scale commensurate with the thickness of the interfacial region, (e.g. near-critical interfacial phenomena or small scale flows such as those occurring near contact lines), and fluid flows involving large interface deformations and/or topological changes (e.g. breakup and coalescence events associated with fluid jets, droplets, and large-deformation waves). We discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, we address issues including sharp-interface analyses that relate these models to the classical freeboundary problem, computational approaches to describe interfacial phenomena, and models of fully-miscible fluids.

We develop a phase-field model for

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Karma and Rappel [1] recently developed a new sharp interface asymptotic analysis of the phase-field equations that is especially appropriate for modeling dendritic growth at low undercoolings. Their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis, and has the added advantage that interfacial kinetic effects can also be eliminated. However, their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases; when applied to a standard phase-field model with unequal conductivities, anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp-interface solidification models, as discussed further by Almgren [2]. In this paper we apply their asymptotic methodology to a generalized phase-field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities. The 1 additional degrees of freedom associated with the generalized phase-field equations can be chosen to eliminate the anomalous terms that arise for unequal conductivities.