Open AccessAsymptotic solution of the filtration equation
Author(s) -
Liudmila Kuzmina,
Yu. V. Osipov
Publication year - 2016
Publication title -
vestnik mgsu
Language(s) - English
Resource type - Journals
eISSN - 2304-6600
pISSN - 1997-0935
DOI - 10.22227/1997-0935.2016.2.49-61
Subject(s) - filtration (mathematics) , suspension (topology) , mechanics , method of matched asymptotic expansions , partial differential equation , particle (ecology) , boundary value problem , ordinary differential equation , constant (computer programming) , porous medium , flow (mathematics) , viscosity , differential equation , mathematical analysis , physics , materials science , thermodynamics , mathematics , porosity , composite material , statistics , oceanography , homotopy , computer science , pure mathematics , programming language , geology
The problem of filtering a suspension of tiny solid particles in a porous medium is considered. 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 ahead of the front; behind the front of the fluid flow solid particles interact with the porous medium. The geometric model of filtration without effects caused by viscosity and electrostatic forces is considered. Solid particles in the 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 considered that one particle can clog only one small pore and vice versa. The precipitated particles form a fixed deposit increasing over time. The filtration problem is formed by the system of two quasi-linear differential equations in partial derivatives with respect to the concentrations 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 discontinuous. 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 condition in integral form. An asymptotic solution of the filtration equation is constructed near the concentration front. The terms of the asymptotic expansions satisfy linear ordinary differential equations of the first order and are determined successively in an explicit form. For verification of the asymptotics the comparison with the known exact solutions is performed.