The paper discusses the laws behind the filtering procedures of process liquids through porous materials. At metalwork finishing operations, the use of cutting fluids is of particular importance. During operation, liquids are continuously and intensively contaminated with solid metal parts. To restore the original properties, process fluids are cleaned of mechanical admixtures. The most widely used methods for purifying process liquids are those by filtration. The use of filtration for the purification of process fluids is most effective, since filtering through a layer of porous materials results in complete extraction and removal of solids from liquids. However, the structural features of the pores in the porous environment trigger a number of specific phenomena that arise when liquids move in the porous channels of a porous medium.

The research purpose is to discuss and establish the laws behind the filtering procedures of process fluids through porous materials. When filtering process liquids through a layer of porous materials, the porous medium of the filter membrane expands with a change in porosity. The change in porosity occurs due to a decrease in the pore volume of the porous environment, since the solid parts together with the liquid penetrate into the porous channels of the porous environment and hover in them. The conducted studies permitted the authors to identify and study the laws of the filtering process and establish the law of change in porosity of the porous environment. Based on the established law, a differential equation is derived. It allowed, for given initial and border-line conditions, stating the problem of filtering the liquid through a layer of solid particles of a variable porous medium of the filtering membrane. The solution of the non-stationary problem with initial and border-line conditions by the finite difference method allowed determining the hydrodynamic parameters of fluid filtration through a layer of particles of the porous environment of the filter membrane and to obtain a solution of the non-stationary boundary problem of fluid filtration in a deformed porous medium. The use of research results promotes complete clarification of the process fluid and thorough removal of the solid parts of valuable secondary raw materials of metal processing, in particular for powder metallurgy, facilitates the launching of waste-free production, and increases the level of environmental cleanliness in the operating area of cutting fluids.

[1] V. F. Molchanov, “Teoreticheskie issledovaniya veroyatnosti zaklinivaniya tverdyh chastic pri shlifovanii” [“Theoretical studies of the probability of jamming of solid particles during grinding”]*, Vestnik Zhitomirskogo gosudarstvennogo tehnologicheskogo universiteta. Seriya: Tehnicheskie nauki* *[Bulletin of Zhytomyr State Technological University. Series: Technical Sciences]*, no. 2 (80), pp. 114–119, 2017. [in Russian].

[2] L. V. Hudobin, and E. P. Gulnov, “Vliyanie zagryazneniya SOZh othodami shlifovaniya na prizhogoobrazovanie” [“Effect of lubricoolant contamination by grinding waste on burn formation”], *Vestnik mashinostroeniya* *[Bulletin of Mechanical Engineering]*, no. 1, pp. 67–68, 1978. [in Russian].

[3] Yu. V. Polyanskov, and E. A. Karev, “K metodike issledovaniya effektivnoj ochistki SOZh ot mehanicheskih primesej pri abrazivnoj obrabotke” [“On the methodology for the study of effective cleaning of lubricoolant of mechanical admixture during abrasive treatment”], *Trudy Ulyanovskogo politehnicheskogo instituta [Proceedings of the Ulyanovsk Polytechnic Institute]*, vol. 1, pp. 46–54, 1976. [in Russian].

[4] V. F. Molchanov, “Issledovanie filtracii zhidkostej cherez poristye materially” [“The study of fluid filtration through porous materials”], in *Proc. of* *Scientific Conference “Matematicheskie problemy tehnicheskoj mehaniki” [“Mathematical problems of technical mechanics”], *Dneprodzerzhinsk, Ukraine, 2003, pp. 71–72. [in Russian].

[5] V. F. Molchanov, “Postanovka i reshenie nestacionarnoj zadachi filtracii zhidkostej v poristoj srede” [“Formulation and solution of the non-stationary problem of fluid filtration in a porous medium”], *Vestnik Harkovskogo nacionalnogo avtomobilno-dorozhnogo universiteta i Severo-Vostochnogo nauchnogo centra Transportnoj akademii Ukrainy* *[Bulletin of the Kharkov National Automobile and Highway University and the Northeast Scientific Center of the Transport Academy of Ukraine]*, vol. 33, pp. 112–115, 2006. [in Russian].

[6] V. F. Molchanov, “Postanovka nestacionarnoj zadachi filtracii zhidkostej v poristoj srede” [“The formulation of the non-stationary problem of fluid filtration in a porous medium”], *Matematicheskoe modelirovanie *[*Mathematical modeling]*, no. 1, pp. 28–30, 2014. [in Russian].

[7] L. S. Pontryagin, *Obyknovennye differencialnye uravneniya* *[Ordinary Differential Equations]*. Moscow, Russia: Nauka Publ., 1977. [in Russian].

[8] A. Shejdegger, *Fizika techeniya zhidkostej cherez poristye sredy* *[Physics of the flow of fluids through porous media]*. Moscow, Russia: Gostoptekhizdat Publ., 1960. [in Russian].

[9] V. F. Molchanov, “Issledovanie propusknoj sposobnosti filtrovalnyh setok” [“Investigation of filter grids throughput”], *Sbornik nauchnyh trudov Dneprodzerzhinskogo gosudarstvennogo tehnicheskogo universiteta (tehnicheskie nauki)* *[Collection of scientific papers of the Dneprodzerzhinsk State Technical University (technical sciences)]*, vol. 3 (20), pp. 61–65, 2012. [in Russian].

[10] V. A. Zhuzhikov, *Filtrovanie. Teoriya i praktika razdeleniya suspenzij [Filtration. Theory and practice of separation of suspensions]*. Moscow, Russia: Himiya Publ., 1968. [in Russian].

[11] S. K. Godunov, and V. S. Ryabenkij, *Raznostnye shemy. Vvedenie v teoriyu* *[Difference schemes. Introduction to Theory]*. Moscow, Russia: Nauka Publ., 1977. [in Russian].

[12] V. F. Molchanov, “Postanovka i reshenie nestacionarnoj granichnoj zadachi filtracii zhidkostej v poristoj srede” [“Statement and solution of the non-stationary boundary value problem of fluid filtration in a porous medium”], *Sbornik nauchnyh trudov Dneprovskogo gosudarstvennogo tehnicheskogo universiteta (tehnicheskie nauki)* *[**Collection of scientific papers **of the Dniprovsky State Technical University (technical sciences)]*, vol. 1 (32), pp. 53–58, 2018. [in Russian].