Momentum flux in multiphase mixtures

Abstract

Bodies which are mixtures, in the sense that reasonably small elements of such bodies contain particles of two or more materials or phases, are of major interest in many physical applications. For the purpose of constructing mathematical models, mixtures may be divided into two classes: solutions, or homogeneous mixtures, in which the phases are so intimately intermixed that all phases may be thought of as occupying the same points in space at the same time, and multiphase mixtures, in which each phase occupies a portion of space that is different from each other phase. Physical materials appropriate to multiphase mixture theories include suspensions of solid particles in a fluid, and fluids in the interstices of deformable porous media. The theory of solutions is well‐established. For multiphase mixtures, the governing physical principles still are being debated, though generally it is agreed that at least part of the theoretical structure should be similar to that for solutions. Here we discuss multiphase mixture theory.There is a very substantial literature in which models specific to particular multiphase mixtures are constructed. Such modeling efforts are not economical in that they may give the impression that a new model is needed for each different type of physical system. A more systematic approach is to organize the work in such a fashion that general principles, that is, ones common to large classes of multiphase materials, are separated from constitutive assumptions, that is, models of specific materials in that class. Then different mixtures corresponds to different constitutive assumptions.The structures of theories for solutions and for multiphase mixtures are sufficiently complicated so that a certain amount of arbitrariness in placing terms in the individual equations of the theories is possible. That is, placing a certain term in one equation or another in the theoretical development ultimately may lead to the same field equations. This has been pointed out by Adkins, Atkin and Craine, and Craine and Atkin, among others. That does not mean that such choices are meaningless. For example, differences in placement of terms may affect the nature of the boundary conditions required. The theory of solutions is well‐established. For multiphase mixtures, however, the theories are relatively new, and the general principles are not completely agreed upon. Therefore, often, aspects of the exact structure of a specific case which is successful in describing a range of physical phenomena well may be worthy of incorporation into general principles. Then the form of particular intermediate steps may have significance with respect to the formulation of these principles.