UDK 532.133

DOI: 10.15507/0236-2910.028.201802.164-174

** Transverse Oscillatory Motion in Viscous Fluid in Contact with Porous Medium **

**Elvira N. Egereva **Associate Professor, Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia) Ph.D. (Physics and Mathematics), ResearcherID: F-9071-2018, ORCID: https://orcid.org/0000-0002-5474-924X, This email address is being protected from spambots. You need JavaScript enabled to view it.

** Artem Yu. Egerev **Student, National Research University Higher School of Economics, (20 Myasnitskaya St., Moscow 101000, Russia), ResearcherID: F-8420-2018, ORCID: https://orcid.org/0000-0001-6928-3764, This email address is being protected from spambots. You need JavaScript enabled to view it.

**Alexander O. Zubov **Master’s Student, National Research University of Civil Engineering (26 Yaroslavskoye Shosse, Moscow 129337, Russia), ResearcherID: F-7655-2018, ORCID: http://orcid.org/0000-0002-4909-228X, This email address is being protected from spambots. You need JavaScript enabled to view it.

**Introduction.** We consider the solution of two problems on transverse oscillations in a viscous incompressible homogeneous fluid in contact with a porous medium (matrix) saturated with the same liquid. The surface of the section of the porous medium and the liquid in contact with it is the plane in all the cases considered. ** Materials and Methods.** To describe the motion of a liquid in a porous medium, the nonstationary Brinkman equation is used. In the boundary conditions, the possible slip of a liquid in a porous medium along a solid impermeable surface, which limits the porous medium, is taken into account. ** Results.** Exact analytical solutions of two problems on internal transverse waves in a viscous fluid located on a layer of a porous medium are obtained. The first problem shows that damped transverse waves exist in a viscous fluid. The velocity of the wave is perpendicular to its direction. The amplitude of the velocity decreases monotonically as it moves deeper into the porous medium. Damped transverse waves can exist both in a free liquid and in a porous medium. The amplitude of these waves attenuates with distance from the oscillating plane into the interior of the liquid. In those cases where the transverse waves exist, their length in regions 1 and 2 is equal to 2πσ_{2}√ Г and 2πσ_{2}, respectively. The strong attenuation of the wave occurs at a distance of the order of its length. Therefore, the motion is concentrated in a layer of thickness on the order of the wavelength. That the wave could penetrate from the free liquid into the porous medium of the thickness of the layers h_{1} and h_{2} should be comparable with the wave lengths. In the second problem, it is obtained that in the case of ε^{2} << 1 damped transverse waves exist only in a free liquid, and in the case of ε^{2} >> 1 damped transverse waves exist in both a liquid and a porous medium.

** Conclusions.** For the case of low frequencies, damped transverse waves can exist only in a free liquid, and in the case of high oscillation frequencies, both in a liquid and in a porous medium. The lengths of these waves in regions 1 and 2 are the same as in the first problem. Vibrational motion of a porous sphere with a solid impermeable core in a viscous fluid could be usefully explored in further research.

**Keywords:** porous medium, viscous fluid, transverse oscillatory, inner transverse waves, Brinkman equation, exact analytical solutions

**For citation: ** Egereva E. N., Egerev A. Yu., Zubov A. O. Transverse Oscillatory Motion in Viscous Fluid in Contact with Porous Medium. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(2):164–174. DOI: 10.15507/0236-2910.028.201802.164-174

**Authors’ contribution: ** E. N. Egereva – collection and analysis of theoretical material, analysis of the results obtained, writing the draft; A. Yu. Egerev – numerical data processing of and revision of the text; A. O. Zubov – word processing, analysis of literature data.

All authors have read and approved the final version of the paper.

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