PSI - Issue 65

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2024) 000–000 Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2024) 000–000

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Procedia Structural Integrity 65 (2024) 177–184

The 17th International Conference on MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS 2023) Exact solutions of the Oberbeck–Boussinesq equations for describing the creeping flows of multicomponent fluids E. Yu. Prosviryakov a,b,c, *, O. A. Ledyankina c , L. S. Goruleva a,b a Institute of Engineering Science, Ural Branch of the RAS, 34 Komsomolskaya St., Ekaterinburg, 620049, Russia b Ural Federal University, 19 Mira St., 620002, Ekaterinburg, Russia c Tupolev Kazan National Research Technical University, 10 Karla Marksa St., Kazan, 420111, Russia The 17th International Conference on MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS 2023) Exact solutions of the Oberbeck–Boussinesq equations for describing the creeping flows of multicomponent fluids E. Yu. Prosviryakov a,b,c, *, O. A. Ledyankina c , L. S. Goruleva a,b a Institute of Engineering Science, Ural Branch of the RAS, 34 Komsomolskaya St., Ekaterinburg, 620049, Russia b Ural Federal University, 19 Mira St., 620002, Ekaterinburg, Russia c Tupolev Kazan National Research Technical University, 10 Karla Marksa St., Kazan, 420111, Russia A new class of exact solutions of magnetohydrodynamic equations for description of creeping flows of multilayer incompressible fluids with the Rayleigh dissipative function is found. The velocity field is described by linear forms with respect to two spatial (horizontal, longitudinal) coordinates. The coefficients of the linear forms depend on the third (vertical, transverse) coordinate. The pressure field and the temperature field are quadratic forms with similar dependence on coordinates and coefficients as in the Lin-Sidorov-Aristov class. © 2024 The Authors, Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of MRDMS 2023 organizers Keywords: exact solutions, convection, Oberbeck–Boussinesq equations, dissipative function, creeping flow, Stokes flow; © 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of MRDMS 2023 organizers Abstract Abstract A new class of exact solutions of magnetohydrodynamic equations for description of creeping flows of multilayer incompressible fluids with the Rayleigh dissipative function is found. The velocity field is described by linear forms with respect to two spatial (horizontal, longitudinal) coordinates. The coefficients of the linear forms depend on the third (vertical, transverse) coordinate. The pressure field and the temperature field are quadratic forms with similar dependence on coordinates and coefficients as in the Lin-Sidorov-Aristov class. © 2024 The Authors, Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of MRDMS 2023 organizers Keywords: exact solutions, convection, Oberbeck–Boussinesq equations, dissipative function, creeping flow, Stokes flow;

1. Introduction 1. Introduction

Finding exact solutions of hydrodynamic equations is a topical problem [Drazin et al.(2006)), Aristov et al.(2009), Ershkov et al.(2021), Prosviryakov et al.(2024), Ershkov et al.(2023)]. When integrating the Navier Stokes equations to describe flows in various force fields, symmetries and approximations are used to zero the convective (nonlinear) derivative [Drazin et al.(2006)), Aristov et al.(2009), Ershkov et al.(2021), Prosviryakov et Finding exact solutions of hydrodynamic equations is a topical problem [Drazin et al.(2006)), Aristov et al.(2009), Ershkov et al.(2021), Prosviryakov et al.(2024), Ershkov et al.(2023)]. When integrating the Navier Stokes equations to describe flows in various force fields, symmetries and approximations are used to zero the convective (nonlinear) derivative [Drazin et al.(2006)), Aristov et al.(2009), Ershkov et al.(2021), Prosviryakov et

* Corresponding author. Tel.: +7-982-654-5223. E-mail address: evgen_pros@mail.ru * Corresponding author. Tel.: +7-982-654-5223. E-mail address: evgen_pros@mail.ru

2452-3216 © 2024 The Authors, Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of MRDMS 2023 organizers 2452-3216 © 2024 The Authors, Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of MRDMS 2023 organizers

2452-3216 © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of MRDMS 2023 organizers 10.1016/j.prostr.2024.11.028