On Theories for Reacting Immiscible Mixtures

On some small scale each constituent of an immiscible mixture occupies a separate region of space. Given sufficient time and computing power, we could solve the continuum field equations and boundary conditions for this het erogenous system. This usually represents an enormously difficult task that is well beyond today's computational ca- pabilities. Mixture theories approximate this complex heterogeneous formulation with a set of field equations for an equivalent homoge- neous mat erial. In this work, we compare the theory for immiscible mixtures by Drumheller and Bedford with the theory of Passman, Nunziato, and Walsh. We describe the conditions under which these theories reduce to an equivalent formulation, and we also investigate the differences in their microinertial descriptions. Two variables play special roles in both theories. They are t he true material density and the volume fraction. Here we use a kinematical approach based on two new variables-t he true deformation gradient and the distention gradient. We show how the true deformation gra- dient is connected to the true material density and, in the absence of chemical reactions, the volume fraction is the inverse of the deter- minant of the distention gradient. However, when chemical reactions occur, the distention gradient and the volume fraction are not directly connected. We ako present a mixture model for a granuIar expIosive. This model is based upon the work of Baer and Nunziato, but our theory differs from their work in that we Present a three-dimension-al rnodd, `.. ` - - we cast the constitutive postulates in terms of the distention gradient rather than the volume fraction, and we incorporate elastic-plastic effects into the constitutive description of the solid granules