Diffuse-interface methods in fluid mechanics

We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been ap- plied successfully to situations in which the physical phenomena of interest have as- sociated 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 occur- ring 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 free- boundary problem, computational approaches to describe interfacial phenomena, and models of fully-miscible fluids.