Steady-state non-uniform Poiseuille shear flows with Navier boundary condition

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Abstract

This study presents an exact solution to the Navier–Stokes equations for a steady non‑uniform Poiseuille shear flow in an infinite horizontal fluid layer. For this class of flows, the governing system reduces to a nonlinear overdetermined set of partial differential equations. A nontrivial exact solution is constructed within the Lin–Sidorov–Aristov class, wherein the velocity field is given by linear forms of two horizontal coordinates with coefficients depending on the vertical coordinate. The boundary‑value problem employs the Navier slip condition at the lower wall and a non‑uniform velocity profile at the upper boundary. The resulting polynomial solution is analyzed, revealing that counter‑flows can emerge due to the presence of stagnation points. It is shown that the Navier condition can lead to a maximum stratification of the velocity field into four distinct zones (three stagnation points). In the limiting case of perfect slip, the analysis demonstrates the possibility of two stagnation points.

About the authors

Natalya V. Burmasheva

Ural Federal University named after the first President of Russia B. N. Yeltsin; Institute of Engineering Science, Ural Branch of RAS

Author for correspondence.
Email: nat_burm@mail.ru
ORCID iD: 0000-0003-4711-1894
Scopus Author ID: 57193346922
ResearcherId: E-3908-2016
https://www.mathnet.ru/eng/person52636

Cand. Tech. Sci.; Associate Professor; Dept. of Information Technology and Automation; Senior Researcher; Sect. of Nonlinear Vortex Hydrodynamics

Russian Federation, 620002, Ekaterinburg, Mira st., 19; 620049, Ekaterinburg, Komsomolskaya st., 34

Evgeniy Yu. Prosviryakov

Ural Federal University named after the first President of Russia B. N. Yeltsin; Institute of Engineering Science, Ural Branch of RAS

Email: evgen_pros@mail.ru
ORCID iD: 0000-0002-2349-7801
SPIN-code: 3880-5690
Scopus Author ID: 57189461740
ResearcherId: E-6254-2016
https://www.mathnet.ru/eng/person41426

Dr. Phys. & Math. Sci.; Professor; Dept. of Information Technology and Automation; Head of Sector; Sect. of Nonlinear Vortex Hydrodynamics

Russian Federation, 620002, Ekaterinburg, Mira st., 19; 620049, Ekaterinburg, Komsomolskaya st., 34

Mikhail Yu. Alies

Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences

Email: aliesmy@mail.ru
ORCID iD: 0000-0001-8853-5365
https://www.mathnet.ru/eng/person37555

Dr. Phys. & Math. Sci., Professor; Director

Russian Federation, 426067, Izhevsk, T. Baramzina str., 34

References

  1. Hagen G. Über die Bewegung des Wassers in engen cylindrischen Röhren, Ann. Phys., 1836, vol. 122, no. 3, pp. 423–442. DOI: https://doi.org/10.1002/andp.18391220304.
  2. Poiseuille J.-L.-M. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres, C. R. Hebd. Séances Acad. Sci., 1840, vol. 11, pp. 961–967.
  3. Poiseuille J.-L.-M. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres, C. R. Hebd. Séances Acad. Sci., 1840, vol. 11, pp. 1041–1048.
  4. Poiseuille J.-L.-M. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres, C. R. Hebd. Séances Acad. Sci., 1841, vol. 12, pp. 112–115.
  5. Drazin P. G., Riley N. The Navier–Stokes Equations. A Classification of Flows and Exact Solutions, London Mathematical Society Lecture Note Series, vol. 334. Cambridge, Cambridge Univ. Press, 2006, x+196 pp. DOI: https://doi.org/10.1017/CBO9780511526459.
  6. Ershkov S. V., Prosviryakov E. Yu., Burmasheva N. V., Christianto V. Towards understanding the algorithms for solving the Navier–Stokes equations, Fluid Dyn. Res., 2021, vol. 53, no. 4, 044501. EDN: ICXFMY. DOI: https://doi.org/10.1088/1873-7005/ac10f0.
  7. Ershkov S. V., Prosviryakov E. Yu., Burmasheva N. V., Christianto V. Solving the hydrodynamical system of equations of inhomogeneous fluid flows with thermal diffusion: a review, Symmetry, 2023, vol. 15, no. 10, 1825. EDN: NLVEBF. DOI: https://doi.org/10.3390/sym15101825.
  8. Pedley T. J. The Fluid Mechanics of Large Blood Vessels, Cambridge Monographs on Mechanics. Cambridge, Cambridge Univ. Press, 2008, xv+446 pp. DOI: https://doi.org/10.1017/CBO9780511896996.
  9. Knyazev D. V., Kolpakov I. Yu. The exact solutions of the problem of a viscous fluid flow in a cylindrical domain with varying radius, Nelin. Dinam., 2015, vol. 11, no. 1, pp. 89–97 (In Russian). EDN: TKLIJH.
  10. Aristov S. N., Knyazev D. V. New exact solution of the problem of rotationally symmetric Couette–Poiseuille flow, J. Appl. Mech. Tech. Phys., 2007, vol. 48, no. 5, pp. 680–685. EDN: LKHUXN. DOI: https://doi.org/10.1007/s10808-007-0087-7.
  11. Borzenko E. I., Shrager G. R. Non-isothermal steady flow of power-law fluid in a planar/axismetric channel, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2018, no. 52, pp. 41–52 (In Russian). EDN: XNHSEP. DOI: https://doi.org/10.17223/19988621/52/5.
  12. Mamazova D. A., Ryltseva K. E., Shrager G. R. The structure and kinematics of a non-Newtonian fluid flow in a pipe with a sudden expansion, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2021, no. 74, pp. 113–126 (In Russian). EDN: DFIEZD. DOI: https://doi.org/10.17223/19988621/74/12.
  13. Fomin A. A., Fomina L. N. Numerical simulation of the viscous incompressible fluid flow and heat transfer in a plane channel with backward-facing step, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, vol. 25, no. 2, pp. 280–294 (In Russian). EDN: TYFOGV.
  14. Eringen A. C. Simple microfluids, Int. J. Eng. Sci., 1964, vol. 2, pp. 205–217. DOI: https://doi.org/10.1016/0020-7225(64)90005-9.
  15. Eringen A. C. Theory of micropolar fluids, J. Math. Mech., 1966, vol. 16, no. 1, pp. 1–18. https://www.jstor.org/stable/24901466.
  16. Stokes V. K. Couple stresses in fluids, Phys. Fluids, 1966, vol. 9, no. 9, pp. 1709–1715. DOI: https://doi.org/10.1063/1.1761925.
  17. Stokes V. K. Effects of couple stresses in fluids on hydromagnetic channel flows, Phys. Fluids, 1968, vol. 11, no. 5, pp. 1131–1133. DOI: https://doi.org/10.1063/1.1692056.
  18. Baranovskii E. S., Prosviryakov E. Yu., Ershkov S. V. Mathematical analysis of steady non-isothermal flows of a micropolar fluid, Nonlinear Anal. Real World Appl., 2025, vol. 84, 104294. EDN: RLNDEQ. DOI: https://doi.org/10.1016/j.nonrwa.2024.104294.
  19. Baranovskii E. S., Burmasheva N. V., Prosviryakov E. Y. Exact solutions to the Navier–Stokes equations with couple stresses, Symmetry, 2021, vol. 13, no. 8, 1355. EDN: KMTTYI. DOI: https://doi.org/10.3390/sym13081355.
  20. Stankevich Yu. A., Fisenko S. P. Reorganization of the Poiseuille profile in nonisothermal flows in the reactor, J. Eng. Phys. Thermophys., 2011, vol. 84, no. 6, pp. 1318–1321. EDN: XKMOVT. DOI: https://doi.org/10.1007/s10891-011-0600-y.
  21. Chefranov S. G., Chefranov A. G. Solution to the paradox of the linear stability of the Hagen-Poiseuille flow and the viscous dissipative mechanism of the emergence of turbulence in a boundary layer, J. Exp. Theor. Phys., 2014, vol. 119, no. 2, pp. 331–340. EDN: UFGCNN. DOI: https://doi.org/10.1134/S1063776114070127.
  22. Savenkov I. V. Axisymmetric instability of the Poiseuille–Couette flow between concentric cylinders at high Reynolds numbers, Comput. Math. Math. Phys., 2015, vol. 55, no. 2, pp. 291–297. EDN: UFLNKP. DOI: https://doi.org/10.1134/S0965542515020177.
  23. Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to Navier–Stokes equations describing a gradient nonuniform unidirectional vertical vortex fluid flow, Dynamics, 2022, vol. 2, no. 2, pp. 175–186. EDN: CVZIZI. DOI: https://doi.org/10.3390/dynamics2020009.
  24. Burmasheva N. V., Dyachkova A. V., Prosviryakov E. Yu. Inhomogeneous Poiseuille flow, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2022, no. 77, pp. 68–85 (In Russian). EDN: LGYZKM. DOI: https://doi.org/10.17223/19988621/77/6.
  25. Lin C. C. Note on a class of exact solutions in magneto-hydrodynamics, Arch. Ration. Mech. Anal., 1957, vol. 1, no. 1, pp. 391–395. DOI: https://doi.org/10.1007/BF00298016.
  26. Sidorov A. F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory, J. Appl. Mech. Tech. Phys., 1989, vol. 30, no. 2, pp. 197–203. EDN: LZHPWE. DOI: https://doi.org/10.1007/bf00852164.
  27. Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations describing stratified fluid flows, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2021, vol. 25, no. 3, pp. 491–507. EDN: JKXFDQ. DOI: https://doi.org/10.14498/vsgtu1860.
  28. Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations for describing the convective flows of multilayer fluids, Rus. J. Nonlin. Dyn., 2022, vol. 18, no. 3, pp. 397–410. EDN: UGFCUL. DOI: https://doi.org/10.20537/nd220305.
  29. Burmasheva N. V., Prosviryakov E. Yu. Inhomogeneous Nusselt–Couette–Poiseuille flow, Theor. Found. Chem. Eng., 2022, vol. 56, no. 5, pp. 662–668. EDN: TBSKKW. DOI: https://doi.org/10.1134/s0040579522050207.
  30. Burmasheva N. V., Larina E. A., Prosviryakov E. Yu. A Couette-type flow with a perfect slip condition on a solid surface, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2021, no. 74, pp. 79–94 (In Russian). EDN: PAKEV. DOI: https://doi.org/10.17223/19988621/74/9.
  31. Burmasheva N. V., Prosviryakov E. Yu. Studying the stratification of hydrodynamic fields for laminar flows of vertically swirling fluid, Diagn. Resour. Mech. Mater. Struct., 2020, no. 4, pp. 62–78 (In Russian). EDN: WDCCSD. DOI: https://doi.org/10.17804/2410-9908.2020.4.062-078.
  32. Mase G. E. Theory and Problems of Continuum Mechanics. New York, McGraw-Hill, 1970, 221 pp.

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2. Figure 1. Profile of the velocity $\overline{U}$ in the perfect slip case

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3. Figure 2. Profile of the velocity $\overline{U}$ in the slip case

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