A variational theory of immiscible mixtures

A continuum theory is presented for mixtures whose constituents remain physically separate, such as a mixture of immiscible liquids or a fluid containing a distribution of particles, droplets, or bubbles. The theory reported does model local volume changes of the constituents, although it does not model more complex changes in the local structure. The change in the local volume of a constituent is measured in the theory by the volume fraction which, in the case of a compressible constituent, is independent of the partial density of the constituent. A theory of mixtures with a microstructure of a particularly simple type is, therefore, obtained. The equations of motion for the constituents are obtained by an application of Hamilton's extended variational principle.