Multiphase mixtures and singular surfaces

This paper provides an alternative formulation of the balance equations for multiphase mixtures and the associated jump relations across singular surfaces. It is motivated by inconsistencies arising in some recent work on singular surfaces in mixtures when one or more of the constituents is absent on one side of the singular surface: for example, a phase change front where a phase is annihilated or created, or a porous matrix boundary where the contained fluid continues into the exterior region. Setting the volume fraction and other physical variables of the constituent to zero when that constituent vanishes or is absent has led to inconsistencies in commonly used constituent jump conditions. In particular, the constituent jump relations for a binary mixture consisting of a single constituent on each side of the interface do not, with this procedure, lead to the usual jump conditions for a sharp interface in a single material. The inconsistency is due to the lack of surface production terms in the usual constituent jump conditions. It is shown that decomposing all constituent diffusive fluxes into two parts - one representing the flux purely within the constituent and the other representing the flux into that constituent from the other constituents, the generalization of a previous view of heat flux - leads to a consistent theory both for continuous regions and for the jump relations across a singular surface. The required surface production terms now arise directly from interaction terms in the continuous regions, which we view as an interesting alternative to a formal postulate of their presence. It is demonstrated that the surface production terms allow consistent treatment of a porous matrix boundary and phase creation and annihilation fronts, and also of an interface separating single phases. The flux decomposition also leads to a new interpretation of the stress appearing in the standard momentum balance, and in turn to a new definition of intrinsic constituent fluxes.