Filtration in porous media of fluids and gases containing associated with them (dissolved, particulate) solid substances is accompanied by the diffusion of these substances and mass transfer between the liquid (gas) and solid stages. The most common types of mass transfer are sorption and desorption, ion exchange, dissolution and crystallization, mudding, sulfation and suffusion, waxing. We consider the system of equations modeling the process of non-equilibrium sorption. We formulate a difference approximation of the differential problem by an implicit scheme. The solution to the difference problem is constructed by the sweep method. Basing on the numerical results, we can conclude the following: as the relaxation time decreases, the solution to the non-equilibrium problem tends to the solution of the equilibrium problem as the time increases.

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All Russian mathematical portal I. A. Kaliev, G. S. Sabitova, The third boundary value problem for the system of equations of non-equilibrium sorption, Sib.Abstract. In this paper, we investigate the system of equations modeling the process of non-equilibrium sorption. In particular, such systems are used in mathematical modeling of the production process of the useful component by the method of borehole underground leaching. The theorem of existence and uniqueness of the solution of the third boundary value problem in the multidimensional case in Hölder classes of functions is proved. The obtained maximum principle plays an important role in the proof of the theorem. The uniqueness of the solution is a consequence of this principle. The existence of a solution to the problem is shown by Schauder's fixed point theorem of a completely continuous operator. The description of the corresponding operator is given. Estimates are obtained to ensure the continuity of the constructed operator, and it is shown that the operator maps the original set of functions into itself at a small period of time. Then the estimates are given, allowing to continue the solution to any finite value of time.Keywords: process of non-equilibrium sorption, third boundary value problem, global single-valued solvability.Kaliev, I.A., Sabitova, G.S., The third boundary value problem for the system of equations of non-equilibrium sorption.c ⃝ 2018 Kaliev I.A., Sabitova G.S.

Filtration of liquids and gases containing associated (dissolved, suspended) solids in porous media is accompanied by diffusion of these substances and mass transfer between the liquid (gas) and solid phases. In this work, we study the system of equations modeling the process of a non-equilibrium sorption. We prove an existence and uniqueness theorem for a multi-dimensional Neumann initial-boundary value problem in the Hölder classes of functions. We obtain a maximum principle, which plays an important role in the proof of the theorem. The uniqueness of the solution follows this principle. The existence of a solution to the problem is shown by Schauder fixed point theorem for a completely continuous operator; we describe a corresponding operator. We obtain estimates ensuring the complete continuity of the constructed operator and the mapping of some closed set of functions into itself over a small time interval. Then we obtain the estimates allowing us to continue the solution up to arbitrary finite time.

The separation of a viscous fluid flow by means of a device operating on the centrifugal principle was modeled. Parameters of the device with a specified separation size were obtained in a series of numerical experiments.