Fluid Dynamics of Two Miscible Liquids with Diffusion and Gradient Stresses

Abstract

This chapter is based on papers by Joseph [1990b], Galdi, Joseph, Preziosi and Rionero [1991], Joseph anu [1991] and Hu and Joseph [1992]. The density of incompressible fluids can vary with concentration and temperature, but not with pressure. The velocity field u of such incompressible fluids is not in general solenoidal: div u ≠ 0. We require that the mass per unit total volume of one of the liquids in a material volume is conserved in the absence of diffusion. This yields the diffusion equation for the mass fraction ψ. Alternatively, if we obtain an equation for the volume fraction φ, then the left hand side of the diffusion equation differs from the usual substantial derivative of φ by the addition of φ div u.