# Fluid Dynamics of Mixtures of Incompressible Miscible Liquids

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## Abstract

The velocity field of binary mixture of incompressible miscible liquids is non-solenoidal when the densities of the two liquids are different. If the mixture density is linear in the volume fraction, as is the case of liquids which satisfy the law of additive volumes, then the velocity can be decomposed into a solenoidal and expansion part. Here we propose a theory for liquids which do not satisfy the law of additive volumes. In this theory the mixture density is again given by a linear form but the densities of the liquids are scaled by the factor expressing the change of the volume of the mixture upon mixing. The dynamical theory of simple mixtures of incompressible liquids can be formed as the correct form of the Navier–Stokes equations in which the compressibility of the mixture is recognized. A rigorous form of the diffusion equation, different than the usual one, is also derived from first principles. The diffusion equation is based a non-linear form of Fick’s law, expressed in terms of gradients of the chemical potential. It is argued that the diffusion of species (of heat and in general) is impossible; signals must move with a finite speed though they may rapidly decay to diffusion. The underlying equation for the evolution of species and heat in the linear case is a damped wave equation rather than the conventional diffusion equation. The Navier–Stokes theory can be identified as a mass transport theory. The solenoidal part of the velocity satisfies an equation which can be shown to govern the transport of volume; it differs from the mass transport velocity by an irrotational expansion velocity associated with the dilitation of the mixture. The equations governing the transport of mass and volume differ from one another by well-defined mathematical transformations; the choice of one or the other is a matter of convenience. However, a genuine difference is associated with boundary conditions. The conventional assumption that the mass transport velocity vanishes is supported by calculations from molecular dynamics but these calculations employ entirely different assumptions and, hence, lack authority. The idea that gradients of composition ought to induce stresses and not just diffusion has been considered and is modeled by a second-order theory introduced by Korteweg 1901. There is not strong evidence that these stresses are important except in regions of strong gradients where a relaxation theory rather than a second-order theory ought to apply. A relaxation theory for stresses due to gradients of composition which relaxes into the second-order theory when the gradients are small is proposed and applied to explain observations of a transient interfacial tension which may be traced to a difference between the relaxation times for diffusion and stresses.

## Keywords

Interfacial Tension Mixture Density Additive Volume Simple Mixture Miscible Displacement## Preview

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