The Hamilton principle for fluid binary mixtures with two temperatures

– For binary mixtures of fluids without chemical reactions, but with components having different temperatures, the Hamilton principle of least action is able to produce the equation of motion for each component and a balance equation of the total heat exchange between components. In this nonconservative case, a Gibbs dynamical identity connecting the equations of momenta, masses, energy and heat exchange allows to deduce the balance equation of energy of the mixture. Due to the unknown exchange of heat between components, the number of obtained equations is less than the number of field variables. The second law of thermodynamics constrains the possible expression of a supplementary constitutive equation closing the system of equations. The exchange of energy between components produces an increasing rate of entropy and creates a dynamical pressure term associated with the difference of temperature between components. This new dynamical pressure term fits with the results obtained by classical thermodynamical arguments in [1] and confirms that the Hamilton principle can afford to obtain the equations of motions for multitemperature mixtures of fluids. Keyword: Fluid mixtures, multi-temperatures, dynamical pressure, variational principle. MSC: 02.30.Xx; 58E30 1. – Introduction. The theory of mixtures considers generally two different kinds of continua: homogeneous mixtures (each component occupies the whole mixture volume) and heterogeneous ones (each component occupies only a part of the mixture volume). At least four approaches to the construction of two-fluids models are known. The first one for studying the heterogeneous two-flows is an averaging method (Ishii [2]; Nigmatulin [3]). The averaged equations of motion are obtained by applying an appropriate averaging operator to the balance laws of mass, energy, etc..., valid inside each phase [4, 5]. A second approach known as Landau method [6, 7] was used for the construction of a quantum liquid model and was purposed for the homogeneous mixtures of fluids [8]. For the total mixture, the method requires the balances of mass, momentum, energy, complemented with the Galilean invariance principle and the second law of thermodynamics [9]. A third approach is presented University of Aix-Marseille & M2P2, C.N.R.S. U.M.R. 6181, Avenue Escadrille NormandieNiemen, Box 322, 13397 Marseille Cedex 20 France. E-mail: [email protected] 2 Department of Mathematics and Research Center of Applied Mathematics, C.I.R.A.M., University of Bologna, Via Saragozza 8, 40123-I Bologna Italy. E-mail: [email protected]; url: http://www.ciram.unibo.it/ruggeri