The Hamilton Principle for Fluid Binary Mixtures with Two Temperatures

For binary mixtures of fluids without chemical reactions, but when the components have different entropies and temperatures, Hamilton's principle of least action is able to produce the equation of motion for each component and a global equation of energy. Nevertheless the number of equations is one less than the number of field variables. This fact is due to the unknown exchange of heat between components of different temperatures. To obtain a supplementary constitutive equation such that the number of equations is the same than the number of field variables, we use a dynamics Gibbs identity connecting equations of momenta, masses, energy and entropies; the second law of thermodynamics constraints the expression of the supplementary constitutive equation. For such a non-equilibrium process, the exchange of energy between components produces an increasing rate of entropy for the total mixture and in linear approximation creates a dynamical pressure associated with the difference of temperature between components. This new non-equilibrium dynamical pressure term fits with the results obtained by generalized classical arguments