Асимптотика уравнения фильтрации

Вестник Московского государственного строительного университета. 2016. № 2. С. 49–61.

Кузьмина Л.И., Осипов Ю. В.

The problem of filtering a suspension of tiny solid particles in a porous medium is con-sidered. The suspension with constant concentration of suspended particles at the filter inlet moves through the empty filter at a constant speed. There are no particles before the front; behind the front of the fluid flow solid particles interact with of the porous medium. The ge-ometric model of filtration without effects caused by viscosity and electrostatic forces is considered. Solid particles in suspension pass freely through large pores together with the fluid flow and are stuck in the pores that are smaller than the size of the particles. It is con-sidered that one particle can clog only one small pore and vice versa. The precipitated par-ticles form a fixed deposit increasing over time. The filtration problem is formed by the sys-tem of two quasi-linear differential equations in partial derivatives with respect to the con-centrations of suspended and retained particles. The boundary conditions are set at the filter inlet and at the initial moment. At the concentration front the solution of the problem is dis-continuous. By the method of potential the system of equations of the filtration problem is reduced to one equation with respect to the concentration of deposit with a boundary condi-tion in integral form. An asymptotic solution of the filtration equation is constructed near the concentration front. Terms of the asymptotic expansions satisfy linear ordinary differential equations of the first order and are determined successively in an explicit form. For verifi-cation of the asymptotics the comparison with known exact solutions is performed.

Keywords: asymptotic solution, асимптотическое решение, фильтрация, пористая среда, suspension, суспензия, porous media, filtration problem, filtration coefficient, коэффициент фильтрации