Effects of volume fraction condition on thermodynamic restrictions in mixture theory

Volume fraction condition is a true constraint that must be taken into consideration in deducing the thermodynamic restrictions of mixture theory applying the axiom of dissipation. For a process to be admissible, the constraints imposed by the volume fraction condition include not only the equation obtained by taking its material derivative with respect to the motion of a given phase, but also those by taking its spatial gradient. The thermodynamic restrictions are deduced under the complete constraints, the results obtained are consistent for the mixtures with or without a compressible phase, and in which the free energy of each phase depends on the densities of all phases.