The thermo-mechanical description of two-phase mixtures, as it is presented in Ishii’s text-book for example, is quite general but not particularly suited for mixtures in which one phase is dispersed into the second one : in fact, the introduction of two stress tensors and two heat fluxes (related to the velocity gradients and temperature gradients of the respective phases) is in conflict with the hydrodynamic description of suspensions in which there is a single stress tensor and a single heat flux, related to the gradients of the volume-weighted average velocity and of the volume-weighted average temperature respectively. The problem of a two-phase description suitable for suspensions of particles was initiated among others by Buyevich, Nigmatulin and Prosperetti who focused mainly on the two momentum equations but did not consider in so much detail the two energy equations. In this forum on multiphase flows we will suggest the way these energy (and entropy) equations are to be written. We will start from a “mean-field” type of approach, in which the energy exchange between a particle and the surrounding fluid is related in part to the mean energy exchanges and to the mean energy dissipation that occur in the medium surrounding the particle. In fact we extend to energy the reasoning applied to momentum when one says that a large part of the force exerted by the fluid on a particle is due to the mean stress that exists around that particle, leading to the concept of a generalized Archimede’s force. That mean-field approach results in evolution equations (for momentum, energy and entropy) that are common to all types of particles but which can be simplified when the particles are supposed to be rigid or to have a constant volume. Phase transitions between the particles and the fluid will be taken into account, and the role of collisions between particles will be considered briefly. It must be clear at the outset that we are not so much interested in the constitutive relations as in the structure of (and the links between) the various evolution equations. But the proposed description has the merit of requiring a minimum number of constitutive relations for the system of equations to be closed.

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