Methods of mechanics of a continuous medium for the description of multiphase mixtures: PMM vol. 34, n≗6, 1970, pp. 1097–1112

Abstract

The principal assumptions in the construction of a general multivelocity model of a continuous multiples medium are examined and the fundamental equations (for mass, momentum and energy) of mechanics are obtained for each phase in the heterogeneous mixture. On the basis of these equations a closed system is proposed which describes the motion of a dispersed mixture of two compressible phases in the presence of phase changes. Energy transitions in phase transformations are analyzed. The fundamental relationships of the surface of the discontinuity are derived. Proceeding from the assumption of additivity of the internal energy of the mixture according to masses of components and from the assumption of local equilibrium within the limits of a phase, thermodynamic questions of a heterogenous mixture are analyzed. In particular, an explicit expression is obtained for the dissipation function. The basic ideas of utilizing interpenetrating multivelocity continua in the mechanics of mixtures were worked out in [1] and [2]. Questions of derivation of corresponding equations by means of averaging methods were examined in [3], where the earlier literature is also mentioned. It should be noted that some equations of multivelocity continua were also contained in [4,5]. It is necessary [6] to distinguish homogenous or multicomponent ( ∗) mixtures (solutions, alloys, mixture of gases) from heterogenous or multiphase mixtures (emulsions, suspensions, gaseous suspensions, soil saturated with water, mixtures of powders, etc.). In the homogeneous mixture each constituent (component) can be considered as occupying the entire volume of the mixture V 1 = V 2 = · = V m = V while each constituent (phase) in the heterogeneous mixture occupies only part of the volume of the mixture, so that V 1 + V 2 + · + V m = V A neglect of this situation and also the introduction of a temperature concept of the mixture in the case where the temperatures of the constituents are not equal, lead the authors of [7–9] to formal theories which are not supported by physical reality, at least in the case of heterogeneous media [10]. The results obtained in this work are generalization of [11] and [12], where an analogous mixtures was examined, however, for the case where one of the phases is incompressible (in [11] in the absence of phase transitions). As an example for the case where it is necessary to take into account the compressibility of both phases, we point out the flow of dispersed vapor-liquid application of results presented, is the investigation of propagation of strong shock waves in condensed heterogeneous mixtures. We note the work [13] where a model of two compressible phases with application to water saturated soil but in the absence of phase transitions was examined. We also note the work [14] where a single-velocity formulation was used and where equations of motion were obtained which are applicable to a two-phase solid body taking into consideration phase transitions.