Nonstationary One-Dimensional Matematical Model of the Dynamics of Incompressible Two-Phase Medium

Abstract

One of the developing branches of the modern fluid and gas mechanics is the mechanics of multiphase and multicomponent media. Experimental study of many processes of dynamics of inhomogeneous media is difficult; therefore, mathematical modeling is of great importance. Moreover, many models are essentially nonlinear; for this reason, numerical methods are used to integrate such models. The aim of this work is to obtain an exact solution for one of the special cases, assuming a number of simplifications (one-dimensionality of the flow, incompressibility of the carrier and dispersed components, and the linear nature of inter-component momentum exchange). At the same time, to obtain an exact solution, a model has been used, which implements the continual approach of the mechanics of multiphase media, i.e., taking into account the inter-component exchange of momentum and heat transfer. The paper presents a mathematical model of a one-dimensional unsteady flow of an incompressible two-component medium. The equations of dynamics are derived from the equations of the dynamics of the flow of a two-component inhomogeneous medium, taking into account the inter-component exchange of momentum and heat. In the model under consideration, the inter-component force interaction takes into account the Stokes force. The system of partial differential equations is reduced to a nonlinear system of ordinary differential equations due to the condition of incompressibility of the flow. The system of nonlinear differential equations is reduced to the sequential solution of three linear systems of ordinary differential equations with respect to six unknown functions. The analytical solution of the mathematical model of the gas–suspension flow is implemented in the form of a computer program. The exact solution for the continual model of aerosol dynamics can be used to test numerical models of aerosol dynamics.