On the derivation of dynamical equations for systems with an interface

Transport equations for the Gibbs superficial densities at the interface between two visco-elastic media are derived using response theory and the surface multipole expansion developed in first two papers of this series. Boundary conditions linking the surface behavior to that in the bulk phases are obtained in an expansion in the interfacial width over a characteristic macrosopic lenght scale. To lowest order, the standard phenomenological boundary conditions are obtained (i.e., no velocity, temperature, heat current or stress jumps), independent of the precise nature of the phases. The next order corrections yield correlation function expressions for slipping lengths, thermal creep coefficients, etc. The nature of the first order boundary conditions linking the dynamics of a fluid to an elastic solid are considered as a specific limit of the theory.