Analytical Solutions of the Transport Equation for a Drift Model in an Unsteady Flow with Discontinuous Parameters

Abstract

An effective model for describing the relative motion of phases is the drift model, which uses simplified momentum equations that do not take into account inertial forces. For this model, in this paper, we study solutions for which various physical flow patterns are realized. The propagation velocity of the volume concentration of phases is analyzed, which has the most obvious physical meaning at zero phase volume velocity. Solutions with piecewise linear distributions are investigated. The evolution of the state in which at the initial moment of time the volume concentration of phase 1 on the left and right is constant and equal to \(0.5+\Delta\) and \(0.5-\Delta\), respectively, and in the transition zone with a width L linearly varies from the values on the left to the values on the right is studied. Two qualitatively different development scenarios are found. A problem is considered with a continuous distribution of phase volume concentrations at the initial moment of time at which a shock wave is formed (the graph is reversed): the propagation velocity of perturbations from the rear particles turns out to be greater than the velocity of propagation of perturbations from the front particles. A transition from a continuous distribution of volume concentrations of phases to a discontinuous distribution is constructed. The transition of the volume concentration profile of the first phase in the vicinity of the shock wave to a continuous distribution is analyzed taking into account diffusion terms proportional to the second derivative with respect to the coordinate. For this case, the volume concentration profile was studied. The main classes of solutions are found.