
Kuzmina Ludmila Ivanovna 
National Research University Higher School of Economics
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, 101000, Russian Federation.

Osipov Yuri Viktorovich 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
Subject: a groundwater filtration affects the strength and stability of underground and hydrotechnical constructions. Research objectives: the study of onedimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck in the pores. It is assumed that size distributions of the solid particles and the pores overlap. In this case, the main mechanism of particle retention is a sizeexclusion: the particles pass freely through the large pores and get stuck at the inlet of the tiny pores that are smaller than the particle diameter. The concentrations of suspended and retained particles satisfy two quasilinear differential equations of the first order. To solve the filtration problem, methods of nonlinear asymptotic analysis are used. Results: in a mathematical model of filtration of suspensions, which takes into account the dependence of the porosity and permeability of the porous medium on concentration of retained particles, the boundary between two phases is moving with variable velocity. The asymptotic solution to the problem is constructed for a small filtration coefficient. The theorem of existence of the asymptotics is proved. Analytical expressions for the principal asymptotic terms are presented for the case of linear coefficients and initial conditions. The asymptotics of the boundary of two phases is given in explicit form. Conclusions: the filtration problem under study can be solved analytically.
DOI: 10.22227/19970935.2017.11.12781283

Kuzmina Ludmila Ivanovna 
Higher School of Economics
Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Higher School of Economics, 20 Myasnitskaya str., Moscow, 101000, Russian Federation;
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.

Osipov Yuri Viktorovich 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe Shosse, Moscow, 129337, Russian Federation;
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.
The mechanicalgeometric model of the suspension filtering in the porous media is considered. Suspended solid particles of the same size move with suspension flow through the porous media  a solid body with pores  channels of constant cross section. It is assumed that the particles pass freely through the pores of large diameter and are stuck at the inlet of pores that are smaller than the particle size. It is considered that one particle can clog only one small pore and vice versa. The particles stuck in the pores remain motionless and form a deposit. The concentrations of suspended and retained particles satisfy a quasilinear hyperbolic system of partial differential equations of the first order, obtained as a result of macroaveraging of microstochastic diffusion equations. Initially the porous media contains no particles and both concentrations are equal to zero; the suspension supplied to the porous media inlet has a constant concentration of suspended particles. The flow of particles moves in the porous media with a constant speed, before the wave front the concentrations of suspended and retained particles are zero. Assuming that the filtration coefficient is small we construct an asymptotic solution of the filtration problem over the concentration front. The terms of the asymptotic expansions satisfy linear partial differential equations of the first order and are determined successively in an explicit form. It is shown that in the simplest case the asymptotics found matches the known asymptotic expansion of the solution near the concentration front.
DOI: 10.22227/19970935.2015.1.5462
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Kuzmina Ludmila Ivanovna 
Higher School of Economics
Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Higher School of Economics, 20 Myasnitskaya str., Moscow, 101000, Russian Federation;
This email address is being protected from spambots. You need JavaScript enabled to view it
.

Osipov Yuri Viktorovich 
Moscow State University of Civil Engineering (National Research University) (MGSU)
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe Shosse, Moscow, 129337, Russian Federation;
This email address is being protected from spambots. You need JavaScript enabled to view it
.
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 quasilinear 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.
DOI: 10.22227/19970935.2016.2.4961
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