K. V. Gubareva, E. Yu. Prosviryakov, A. V. Eremin

AN EXACT SOLUTION WITH INHOMOGENEOUS BOUNDARY CONDITIONS FOR A STEADY NON-UNIFORM COUETTE FLOW BETWEEN PERMEABLE PLATES

The paper offers an analytical solution for a non-uniform isobaric disturbed unidirectional stationary flow of a viscous incompressible fluid in an infinitely long horizontal layer. The boundaries of the infinite fluid layer are represented by permeable plates. A generalized Couette flow is discussed, where the longitudinal (horizontal) velocity is the linear form of the other longitudinal (horizontal) coordinate. The coefficients of the linear form depend on the transverse (vertical) coordinate. The effect of the normal component of the velocity vector (uniform permeability) via the inertial terms of the Navier–Stokes equation is taken into account. The velocity, vorticity, and shear stress fields are computed from the exact solution to the Navier–Stokes equations. This exact solution belongs to the Lin–Sidorov–Aristov family. The hydrodynamic fields are parametrically visualized in the Matlab software. It is shown that permeability results in the nonlinear deformation of the velocity profile and that the inhomogeneity of the boundary conditions causes the spatial variability of the hydrodynamic fields. The solution can be used to verify numerical methods and to model flows in channels with complex boundary conditions.

References:

1. Drazin, P.G. and Riley, N. The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, Cambridge University Press, Cambridge, 2006, 196 p.
2. Pukhnachev, V.V. Symmetries in the Navier–Stokes equations. Uspekhi Mekhaniki, 2006, 1, 6–76. (In Russian).
3. Aristov, S.N., Knyazev, D.V., and Polyanin, A.D. Exact solutions of the Navier–Stokes equations with the linear dependence of velocity components on two space variables. Theoretical Foundations of Chemical Engineering, 2009, 43 (5), 642–662. DOI: 10.1134/S0040579509050066.
4. Ershkov, S., Burmasheva, N., Leshchenko, D.D., and Prosviryakov, E.Yu. Exact solutions of the Oberbeck–Boussinesq equations for the description of shear thermal diffusion of Newtonian fluid flows. Symmetry, 2023, 15 (9), 1730. DOI: 10.3390/sym15091730.
5. Ershkov, S.V., Prosviryakov, E.Yu, Burmasheva, N.V, and Christianto, V. Towards understanding the algorithms for solving the Navier–Stokes equations. Fluid Dynamics Research, 2021, 53 (4), 044501. DOI: 10.1088/1873-7005/ac10f0.
6. Wang, C.Y. Exact solutions of the unsteady Navier–Stokes equations. Appl. Mech. Rev., 1989, 42 (11S), 269–282. DOI: 10.1115/1.3152400.
7. Wang, C.Y. Exact solutions of the steady-state Navier-Stokes equations. Annu. Rev. Fluid Mech., 1991, 23, 159–177. DOI: 10.1146/annurev.fl.23.010191.001111.
8. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions of the Navier–Stokes equations for describing an isobaric one-directional vertical vortex flow of a fluid. Diagnostics, Resource and Mechanics of materials and structures, 2021, 2, 30–51. DOI: 10.17804/2410-9908.2021.2.030-051. Available at: http://dream-journal.org/issues/2021-2/2021-2_316.html
9. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to Navier–Stokes equations describing a gradient nonuniform unidirectional vertical vortex fluid flow. Dynamics, 2022, 2, 175–186. DOI: 10.3390/dynamics2020009.
10. Goruleva, L.S. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2023, 1, 41–53. DOI: 10.17804/2410-9908.2023.1.041-053. Available at: http://dream-journal.org/issues/2021-2/2021-2_316.html
11. Berman, A.S. Laminar flow in channels with porous walls. J. Appl.Phys., 1953, 24 (9), 1232–1235. DOI: 10.1063/1.1721476.
12. Yuan, S.W. Further investigation of laminar flow in channels with porous walls. J. Appl. Phys., 1956, 27 (3), 267. DOI: 10.1063/1.1722355.
13. Yuan, S.W. and Finkelstein, A.B. Laminar pipe flow with injection and suction through a porous wall. Transactions of the American Society of Mechanical Engineers, 1956, 78, 719–724.
14. Sellars, J.R. Laminar flow in channels with porous walls at high suction Reynolds numbers. J. Appl. Phys., 1955, 26 (4), 489–490.
15. Berman, A.S. Concerning laminar flow in channels with porous walls. J. Appl. Phys., 1956, 27 (12), 1557. DOI: 10.1063/1.1722307.
16. Regirer, S.A. On the approximate theory of the flow of a viscous incompressible fluid in pipes with porous walls. Izv. Vyssh. Uchebn. Zaved. Mat., 1962, 5, 65–74. (In Russian).
17. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solution of Navier–Stokes equations describing spatially inhomogeneous flows of a rotating fluid. Trudy IMM UrO RAN, 2020, 26 (2), 79–87. DOI: 10.21538/0134-4889-2020-26-2-79-87. (In Russian).
18. Burmasheva, N.V. and Prosviryakov, E.Yu. A class of exact solutions for two-dimensional equations of geophysical hydrodynamics with two Coriolis parameters. Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika, 2020, 32, 33–48. DOI: 10.26516/1997-7670.2020.32.33. (In Russian).
19. Prosviryakov, E.Yu. New class of exact solutions of Navier–Stokes equations with exponential dependence of velocity on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, 53 (1), 107–114. DOI: 10.1134/S0040579518060088.
20. Aristov, S.N. and Prosviryakov, E.Yu. Large-scale flows of viscous incompressible vortical fluid. Russian Aeronautics, 2015, 58 (4), 413–418. DOI: 10.3103/S1068799815040091.
21. Aristov, S.N. and Prosviryakov, E.Yu. Inhomogeneous Couette flow. Nelineynaya Dinamika, 2014, 10 (2), 177–182. DOI: 10.20537/nd1402004. (In Russian).
22. Aristov, S.N. and Prosviryakov, E.Yu. Unsteady layered vortical fluid flows. Fluid Dynamics, 2016, 51 (2), 148–154. DOI: 10.1134/S0015462816020034.
23. Zubarev, N.M. and Prosviryakov, E.Yu. Exact solutions for layered three-dimensional nonstationary isobaric flows of a viscous incompressible fluid. Journal of Applied Mechanics and Technical Physics, 2019, 60 (6), 1031–1037. DOI: 10.1134/S0021894419060075.
24. Goruleva, L.S. and Prosviryakov, E.Yu. Nonuniform Couette–Poiseuille shear flow with a moving lower boundary of a horizontal layer. Technical Physics Letters, 2022, 48 (7), 258–262. DOI: 10.1134/s1063785022090024.
25. Prosviryakov, E.Yu. Layered gradient stationary flow vertically swirling viscous incompressible fluid. In: G.A. Timofeeva and A.V. Martynenko, eds., Proceedings of 3rd Russian Conference on Mathematical Modeling and Information Technologies (MMIT 2016), Yekaterinburg, Russia, November 16, 2016, 1825, 164–172. Available at: http://ceur-ws.org
26. Privalova, V.V., Prosviryakov, E.Yu., and Simonov, M.A. Nonlinear gradient flow of a vertical vortex fluid in a thin layer. Rus. J. Nonlin. Dyn., 2019, 15 (3), 271–283. DOI: 10.20537/nd190306.
27. Privalova, V.V. and Prosviryakov, E.Yu. Nonlinear isobaric flow of a viscous incompressible fluid in a thin layer with permeable boundaries. Vychislitelnaya Mekhanika Sploshnykh Sred, 2019, 12 (2), 230–242. (In Russian). DOI: 10.7242/1999-6691/2019.12.2.20.
28. Polyanin, A.D. and Zhurov, A.I. Metody razdeleniya peremennykh i tochnye resheniya nelineynykh uravneniy matematicheskoy fiziki [Methods of Separation of Variables and Exact Solutions of Nonlinear Equations of Mathematical Physics]. IPMekh RAN Publ., Moscow, 2020, 383 p. (In Russian).
29. Couette, M. Études sur le frottement des liquids. Ann. Chim. Phys., 1890, 21, 433–510.
30. Stokes, G.G. On the effect of the internal friction of fluids on the motion of pendulums. From the Transactions of the Cambridge Philosophical Society, Pitt Press, Cambridge, 1851, vol. IX, part II, 106 p.
31. Aristov, S.N. and Gitman, I.M. Viscous flow between two moving parallel disks: exact solutions and stability analysis. Journal of Fluid Mechanics, 2002, 464, 209–215. DOI: 10.1017/S0022112002001003.
32. Goruleva, L.S. and Prosviryakov, E.Yu. Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate. Diagnostics, Resource and Mechanics of materials and structures, 2022, 3, 47–60. DOI: 10.17804/2410-9908.2022.3.047-060. Available at: http://dream-journal.org/issues/2022-3/2022-3_367.html
33. Bogoyavlenskij, O. The new effect of oscillations of the total angular momentum vector of viscous fluid. Physics of Fluids, 2022, 34 (8), 083108. DOI: 10.1063/5.0101870.
34. Bogoyavlenskij, O. The new effect of oscillations of the total kinematic momentum vector of viscous fluid. Physics of Fluids, 2022, 34 (12), 123104. DOI: 10.1063/5.0127990.
35. Aristov, S.N. and Prosviryakov, E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theoretical Foundations of Chemical Engineering, 2016, 50 (3), 286–293. DOI: 10.1134/S0040579516030027.
36. Aristov, S.N and Shvarts, K.G. Vikhrevye techeniya advektivnoy prirody vo vrashchayushchemsya sloe zhidkosti [Vortical Flows of the Advective Nature in a Rotating Fluid Layer]. Permskiy Universitet Publ., Perm, 2006, 153 pp. (In Russian).

Article reference