Variational Theory of Mixtures in Continuum Mechanics

TL;DR

This paper develops a variational framework for deriving the equations of motion in continuum mixtures, incorporating thermodynamics and spatial gradients, and extends classical results to complex interfaces with surface tension effects.

Contribution

It introduces a universal thermodynamic form of the equations of motion for mixtures using Hamilton's principle, including spatial gradients and interface effects, which was not previously formulated.

Findings

Derived equations of motion for inviscid and dissipative mixtures

Reproduced classical results for compressible mixtures

Provided a thermodynamic interpretation of interface surface tension

Abstract

In continuum mechanics, the equations of motion for mixtures are derived through the use of Hamilton's extended principle which regards the mixture as a collection of distinct continua. The internal energy is assumed to be a function of densities, entropies and successive spatial gradients of each constituent. We first write the equations of motion for each constituent of an inviscid miscible mixture of fluids without chemical reactions or diffusion. Our work leads to the equations of motion in an universal thermodynamic form in which interaction terms subject to constitutive laws, difficult to interpret physically, do not occur. For an internal energy function of densities, entropies and spatial gradients, an equation describing the barycentric motion of the constituents is obtained. The result is extended for dissipative mixtures and an equation of energy is obtained. A form of Clausius-Duhem's inequality which represents the second law of thermodynamics is deduced. In the particular case of compressible mixtures, the equations reproduce the classical results. Far from critical conditions, the interfaces between different phases in a mixture of fluids are layers with strong gradients of density and entropy. The surface tension of such interfaces is interpreted.