Two approaches to modeling the dynamics of multiphase media are possible.One of them is focused on studying a continuous medium, whereas a dispersed phase is considered as an additional phenomenon. The mathematical modeling of the medium is based on Navier-Stokes equations in which the effect of dispersed inclusions is considered through the concentrations of phases, the interaction force between them, and so on. Significant progress has been achieved in this area [1]. However, the use of the complete system of conservation equations written in the Eulerian frame of reference for each of the phases poses difficulties in solving the problem. In addition, the determination of physicochemical coefficients calls for additional information.The second approach is focused on the dispersed medium, the particle motion being calculated based on a particle-dynamics equation written in the Lagrangian coordinate system [2]. In this case, the continuum effect is derived from the particle drag coefficient in a heterogeneous medium, which is determined in the experiment. In continuum equations, the existence of particles is taken into account by introducing the effective viscosity.Actually, the particle trajectory cannot coincide with the trajectory of the mean main-flow velocity since the local stress-tensor components for a two-dimensional vortex flow, which have an effect on the particle trajectory, are not homogeneous. In addition, such an approach cannot present the full pattern of the particle trajectory in a vortex flow. This is an important drawback of the given method.A rather attractive and promising method is a combination of these two approaches when a vector equation of motion of a dispersed particle in the Lagrangian frame of reference is solved jointly with a continuum equation of motion in the Eulerian frame of reference.Let us write the vector equation for a dispersed particle in the general form (1) where k = 0.75 C ρ 1 / ρ 2 d , and e = ( e 1 , e 2 , e 3 ) = cons α · i + sin α · j + 0 · k is the unit vector of the direction of the particle relative velocity V rel . The problem is set as twodimensional; therefore, e 3 = 0. The position of the vector V rel is specified by the angle of rotation α from the unit vector i of the x 1 coordinate to the vector e (Fig. 1). The counter-clockwise direction is taken as the positive angle direction (trigonometric convention). Thus, the change in α from π to 2 π corresponds to the sedimentation of a dispersed particle and that from 0 to π , to its rising to the surface (Fig. 1). The body-force acceleration vector F is determined by the sum of the external force vector and the Archimedean buoyancy force vector.The velocity of dispersed particles can be presented as the sum of the continuous phase velocity and the relative velocity: V 2 = V 1 + V rel . The components of the dV 2 /dt kw 2 e F, + -= Abstract -A vector equation of motion in the Lagrangian frame of reference is rearranged into a set of two scalar equations to determine the relative-velocity magnitude of a particle...
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