Fluid dynamics of two miscible liquids with diffusion and gradient stresses

Abstract

The density of incompressible fluids can vary with concentration φ and temperature, but not with pressure. The velocity field u of such incompressible fluids is not in general solenoidal, div u ≠0. A conservation form for the left hand side of the diffusion equation which differs from the usual substantial derivative of φ by the addition of φ div u, is implied by requiring that the mass per unit total volume of one liquid in a material volume is conserved in the absence of diffusion. The possibility that stresses are induced by gradients of concentration and density in slow diffusion of incompressible miscible liquids, as in the theory of Korteweg [1901] is considered. Such stresses could be important in regions of high gradients giving rise to effects which can mimic surface tension. The small but interesting history of thought about interfacial tension between miscible liquids is collected here. The presence of a sharp interface in the case of slow diffusion in rising bubbles and falling drops has been documented in many experiments and in the experiments reported here. The shape of such interfaces can scarcely be distinguished from the shapes of bubbles and drops of immiscible liquids with surface tension. The usual description of interface problems for miscible liquids with classical interface conditions but with zero interfacial tension misses out on slow diffusion on the one hand and gradient stresses on the other. The usual description of diffusion with div u=0 is also inexact, though it is a good approximation in some cases.