Mathematical simulation of the dynamics of a dispersed phase in free gravitational convection of a viscous incompressible liquid in a square cavity

Abstract

The problem of motion of monodisperse spherical particles in a heterogeneous medium in nonisothermal free convection of a carrying incompressible liquid in a square cavity with inhomogeneous distribution of temperature on the walls is considered. The problem is solved by the finite-difference method via joint solution of equations for the carrying phase in Euler variables and the equation for a disperse particle in Lagrange variables. Introduction. In power engineering and in a number of industries (chemical, food, pharmaceutical, metallur- gical), the phenomena and problems associated with simulation of inhomogeneous multiphase media play a decisive role. They are currently central in solving a number of environmental problems (1). In investigation of multiphase inhomogeneous media, most important is the problem of interaction of the rela- tive motion of a dispersed phase with a carrying solid medium. This is achieved by mathematical simulation by nu- merical solutions, on electronic computers, of stationary and nonstationary problems of hydro- and gas dynamics under the conditions of forced and free-convective motion of heterogeneous media. Among the various methods of mathematical simulation of the dynamics of heterogeneous media two ap- proaches should be noted. 1. The calculation of the motion of a particle is based on the equation of the dynamics of a material point written in the Lagrangian coordinate system (1, 2). Here, the influence of the carrying medium is taken into account in terms of its average velocity on the assumption that the trajectory of particles coincides with the direction of the average velocity of flow. However, in a real case the trajectory of the particle cannot coincide with the trajectory of the main flow, since the local components of the tensor of stresses of a two-dimensional swirled flow are inhomogene- ous. Moreover, such an approach cannot reproduce the full picture of the trajectory of the particle in a swirled flow. Therein lies the substantial drawback of the approach. 2. The main object of investigation is a continuous carrying medium, whereas a dispersed phase is taken into account in terms of its concentration and the force of interaction between the continuous medium and dispersed phase. The mathematical simulation of the medium is based on the Navier-Stokes type equations. Here, substantial achieve- ments have been attained in this direction (3). In our opinion, most fruitful is the method detailed out in (4) and based on the combination of these two ap-