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K. V. Gubareva, E. Yu. Prosviryakov

NUMERICAL MODELING AND PARAMETRIC ANALYSIS OF COUETTE FLOW IN FLUIDS WITH COUPLE STRESSES

DOI: 10.17804/2410-9908.2026.3.035-048

A systematic parametric analysis of the generalized Couette flow within a micropolar fluid model with couple stresses is presented. The study is based on exact analytical solutions implemented in the Matlab R2023b environment, and it covers a wide range of dimensionless parameters, namely the couple viscosity coefficient s ∈ [0.03; 5.0], the Reynolds number Re ∈ [30; 250], and the Taylor number Ta ∈ [10; 250]. Point-wise and integral deviation metrics are used to compare the classical and micropolar flows. A dimensionless characteristic defined as the ratio of the average velocity to the maximum velocity gradient is introduced. The results allow us to assess quantitatively the
applicability limits for the Newtonian model and to reveal flow regimes where microstructural effects become determinant. The obtained correlations are of practical importance for modeling complex fluids in microfluidic and biomedical systems.

Keywords: Couette flow, micropolar fluid, couple stresses, parametric analysis, similarity numbers, nonlinear velocity profiles, velocity gradients, shear stress, mathematical modeling

References:

  1. Coles, D. Transition in circular Couette flow. Journal of Fluid Mechanics, 1965, 21 (3), 385–425. DOI: 10.1017/S0022112065000241.
  2. Pop, I., Groșan, T., Revnic, C., and Roșca, A.V. Unsteady flow and heat transfer of nanofluids, hybrid nanofluids, micropolar fluids and porous media: a review. Thermal Science and Engineering Progress, 2023, 46, 102248. DOI: 10.1016/j.tsep.2023.102248.
  3. Kocić, M., Stamenković, Ž., Petrović, J., and Bogdanović-Jovanović, J. MHD micropolar fluid flow in porous media. Advances in Mechanical Engineering, 2023, 15 (6), 16878132231178436. DOI: 10.1177/16878132231178436.
  4. Eringen, A.C. Theory of micropolar fluids. Journal of Mathematics and Mechanics, 1966, 16 (1), 1–18. DOI: 10.1512/iumj.1967.16.16001.
  5. Eringen, A.C. Microcontinuum Field Theories: I. Foundations and Solids, Springer, New York, NY, 1999, 325 p. DOI: 10.1007/978-1-4612-0555-5.
  6. Łukaszewicz, G. Micropolar Fluids. Theory and Applications, Series Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, MA, 1999, 253 p. DOI: 10.1007/978-1-4612-0641-5.
  7. Stokes, V.K. Theories of Fluids with Microstructure: An Introduction, Springer, Berlin, Heidelberg, 1984, 212 p.
  8. Ariman, T., Turk, M.A., and Sylvester, N.D. Microcontinuum fluid mechanics – a review. International Journal of Engineering Science, 1973, 11 (8), 905–930. DOI: 10.1016/0020-7225(73)90038-4.
  9. Jalili, B., Azar, A.A., Jalili, P., and Ganji, D.D. Analytical approach for micropolar fluid flow in a channel with porous walls. Alexandria Engineering Journal, 2023, 79, 196–226. DOI: 10.1016/j.aej.2023.08.015.
  10. Siddiqui, A.A. and Turkyilmazoglu, M. Film flow of nano-micropolar fluid with dissipation effect. Computer Modeling in Engineering & Sciences, 2024, 140 (3), 2487–2512. DOI: 10.32604/cmes.2024.050525.
  11. Ahmed, M.M., Eldesoky, I.M., Nasr, A.G., Abumandour, R.M., and Abdelsalam, S.I. The profound effect of heat transfer on magnetic peristaltic flow of a couple stress fluid in an inclined annular tube. Modern Physics Letters B, 2024, 38 (25), 2450233. DOI: 10.1142/s0217984924502336.
  12. Zhang, M., Jiang, X., and Arefi, M. Dynamic formulation of a sandwich microshell considering modified couple stress and thickness-stretching. The European Physical Journal Plus, 2023, 138 (3), 227. DOI: 10.1140/epjp/s13360-023-03753-4.
  13. Dhange, M., Devi, C.U., Jamshed, W., Eid, M.R., Ramesh, K., Shamshuddin, M.D., and Batool, K. Studying the effect of various types of chemical reactions on hydrodynamic properties of dispersion and peristaltic flow of couple-stress fluid: comprehensive examination. Journal of Molecular Liquids, 2024, 409, 125542. DOI: 10.1016/j.molliq.2024.125542.
  14. Maurya, P.K., Deo, S., and Maurya, D.K. Couple stress fluid flow enclosing a solid sphere in a porous medium: effect of magnetic field. Physics of Fluids, 2023, 35 (7), 072006. DOI: 10.1063/5.0155532.
  15. Siddheshwar, P.G. and Sri Krishna, C.V. Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium. International Journal of Non-Linear Mechanics, 2003, 38 (10), 1561–1579. DOI: 10.1016/S0020-7462(02)00120-8.
  16. Glora, R.S.R. Unsteady mixed convection in micropolar boundary layer flow on a vertical plate. Fluid Dynamics Research, 1995, 15, 237–250. DOI: 10.1016/0169-5983(95)94957-U.
  17. Nadeem, S., Haq, R.U., and Khan, Z.H. Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles. Journal of the Taiwan Institute of Chemical Engineers, 2014, 45 (1), 121–126. DOI: 10.1016/j.jtice.2013.04.006.
  18. Takhar, H.S., Bhargava, R., Agrawal, R.S., and Balaji, A.V.S. Finite element solution of micropolar fluid flow and heat transfer between two porous discs. International Journal of Engineering Science, 2000, 38 (17), 1907–1922. DOI: 10.1016/S0020-7225(00)00019-7.
  19. Srinivasacharya, D., Mishra, M., and Rao, A.R. Peristaltic pumping of a micropolar fluid in a tube. Acta Mechanica, 2003, 161 (3), 165–178. DOI: 10.1007/s00707-002-0993-y.
  20. Prakash, J. and Tripathi, D. Melting heat transfer in electroosmotic augmented stagnation point flow of a micropolar fluid over stretching porous sheet. Numerical Heat Transfer, Part A: Applications, 2026, 87 (1). DOI: 10.1080/10407782.2025.2521423.
  21. Murashkin, E.V. and Radayev, Yu.N. On strong and weak discontinuities of the coupled thermomechanical field in micropolar thermoelastic type-II continua. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2014, 18 (4), 85–97. (In Russian). DOI: 10.14498/vsgtu1331.
  22. Bykova, S. and Ivanova, E. A micropolar continuum and equations of electrodynamics of moving media. Continuum Mechanics and Thermodynamics, 2025, 37 (4), 67. DOI: 10.1007/s00161-025-01398-5.
  23. Pattnaik, P.K., Jena, S., Dei, A., and Sahu, G. Impact of chemical reaction on micropolar fluid past a stretching sheet. JP Journal of Heat and Mass Transfer, 2019, 18 (1), 207–223. DOI: 10.17654/HM018010207.
  24. Burmasheva, N.V. and Prosviryakov, E.Yu. The inhomogeneous Couette flow of a micropolar fluid. Russian Journal of Nonlinear Dynamics, 2025, 21 (3), 345–358. DOI:10.20537/nd250601.
  25. Burmasheva, N.V. and Prosviryakov E.Yu. Exact solutions to the Navier-Stokes equations for unidirectional flows of micropolar fluids in a mass force field. Diagnostics, Resource and Mechanics of materials and structures, 2024, 3, 41–63. DOI: 10.17804/2410-9908.2024.3.041-063. Available at: http://dream-journal.org/issues/2024-3/2024-3_438.html
  26. Burmasheva, N.V. and Prosviryakov, E.Yu. Polynomial exact solutions for describing unidirectional flows of micropolar incompressible media. Procedia Structural Integrity, 2024, 65, 39–43. DOI: 10.1016/j.prostr.2024.11.007.
  27. Baranovskii, E.S., Ershkov, S.V., Prosviryakov, E.Yu., and Yudin, A.V. Exact solutions to the Oberbeck–Boussinesq equations for describing three-dimensional flows of micropolar liquids. Symmetry, 2024, 16 (12), 1669. DOI: 10.3390/sym16121669.
  28. Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solution for nonuniform unidirectional Nusselt flow taking into account couple stresses. Procedia Structural Integrity, 2024, 65, 44–47. DOI: 10.1016/j.prostr.2024.11.008.
  29. Baranovskii, E.S., Prosviryakov, E.Yu., and Ershkov, S.V. Mathematical analysis of steady non-isothermal flows of a micropolar fluid. Nonlinear Analysis: Real World Applications, 2025, 84, 104294. DOI: 10.1016/j.nonrwa.2024.104294.
  30. Burmasheva, N., Ershkov, S., Prosviryakov, E., and Leshchenko, D. Inhomogeneous gradient Poiseuille flows of a vertically swirled fluid. Journal of Applied and Computational Mechanics, 2024, 10 (1), 1–12. DOI: 10.22055/jacm.2023.43959.4150.
  31. Baranovskii, E.S., Burmasheva, N.V., and Prosviryakov, E.Yu. Exact solutions to the Navier-Stokes equations with couple stresses. Symmetry, 2021, 13 (8), 1355. DOI: 10.3390/sym13081355.
  32. Burmasheva, N. and Prosviryakov, E. Exact solutions to Navier–Stokes equations describing a gradient nonuniform unidirectional vertical vortex fluid flow. Dynamics, 2022, 2 (2), 175–186. DOI: 10.3390/dynamics2020009.

К. В. Губарева, Е. Ю. Просвиряков

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ И ПАРАМЕТРИЧЕСКИЙ АНАЛИЗ ТЕЧЕНИЯ КУЭТТА В ЖИДКОСТЯХ С МОМЕНТНЫМИ НАПРЯЖЕНИЯМИ

В работе проведен систематический параметрический анализ обобщенного течения Куэтта в рамках микрополярной модели жидкости с моментными напряжениями. Исследование основано на точных аналитических решениях, реализованных в среде Matlab R2023b,
и охватывает широкий диапазон безразмерных параметров: коэффициента моментной вязкости s ϵ [0,03; 5,0], числа Рейнольдса Re ϵ [30; 250] и числа Тейлора Ta ϵ [10; 250]. Сравнение классического и микрополярного течений выполнено с использованием поточечных и интегральной метрик отклонения. Введена безразмерная характеристика, определяемая как отношение средней скорости к максимальному градиенту. Результаты позволяют количественно оценить границы применимости ньютоновской модели и выявить режимы, в которых микроструктурные эффекты становятся определяющими. Полученные зависимости имеют прикладное значение для моделирования сложных жидкостей в микрофлюидных и биомедицинских системах.

Ключевые слова: течение Куэтта, микрополярная жидкость, моментные напряжения, параметрический анализ, числа подобия, нелинейные профили скорости, градиенты скорости, касательные напряжения, математическое моделирование

Библиография:

  1. Coles D. Transition in circular Couette flow // Journal of Fluid Mechanics. – 1965. – Vol. 21 (3). – P. 385–425. – DOI: 10.1017/S0022112065000241.
  2. Unsteady flow and heat transfer of nanofluids, hybrid nanofluids, micropolar fluids and porous media: a review / I. Pop, T. Groșan, C. Revnic, A. V. Roșca // Thermal Science and Engineering Progress. – 2023. – Vol. 46. – P. 102248. – DOI: 10.1016/j.tsep.2023.102248.
  3. MHD micropolar fluid flow in porous media / M. Kocić, Z. Stamenkovic, J. D. Petrović, J. B. Bogdanovic-Jovanovic // Advances in Mechanical Engineering. – 2023. – Vol. 15 (6). – P. 16878132231178436. – DOI: 10.1177/16878132231178436.
  4. Eringen A. C. Theory of micropolar fluids // Journal of Mathematics and Mechanics. – 1966. – Vol. 16 (1). – P. 1–18. – DOI: 10.1512/iumj.1967.16.16001.
  5. Eringen A. C. Microcontinuum Field Theories: I. Foundations and Solids. – New York, NY : Springer, 1999. – 325 p. – DOI: 10.1007/978-1-4612-0555-5.
  6. Łukaszewicz G. Micropolar Fluids. Theory and Applications, Series Modeling and Simulation in Science, Engineering and Technology. – Boston, MA : Birkhäuser, 1999. – 253 p. – DOI: 10.1007/978-1-4612-0641-5.
  7. Stokes V. K. Theories of Fluids with Microstructure: An Introduction. – Berlin, Heidelberg : Springer, 1984. – 212 p. – DOI: 10.1007/978-3-642-82351-0.
  8. Ariman T., Turk M. A., Sylvester N. D. Microcontinuum fluid mechanics – a review // International Journal of Engineering Science. – 1973. – Vol. 11 (8). – P. 905–930. – DOI: 10.1016/0020-7225(73)90038-4.
  9. Analytical approach for micropolar fluid flow in a channel with porous walls / B. Jalili, A. A. Azar, P. Jalili, D. D. Ganji // Alexandria Engineering Journal. – 2023. – Vol. 79. – P. 196–226. – DOI: 10.1016/j.aej.2023.08.015.
  10. Siddiqui A. A., Turkyilmazoglu M. Film flow of nano-micropolar fluid with dissipation effect // Computer Modeling in Engineering & Sciences. – 2024. – Vol. 140 (3). – P. 2487–2512. – DOI: 10.32604/cmes.2024.050525.
  11. The profound effect of heat transfer on magnetic peristaltic flow of a couple stress fluid in an inclined annular tube / M. M. Ahmed, I. M. Eldesoky, Ahmed G. Nasr, Ramzy M. Abumandour, Sara I. Abdelsalam // Modern Physics Letters B. – 2024. – Vol. 38 (25). – P. 2450233. – DOI:  10.1142/s0217984924502336.
  12. Zhang M., Jiang X., Arefi M. Dynamic formulation of a sandwich microshell considering modified couple stress and thickness-stretching // The European Physical Journal Plus. – 2023. – Vol. 138 (3). – P. 227. – DOI: 10.1140/epjp/s13360-023-03753-4.
  13. Studying the effect of various types of chemical reactions on hydrodynamic properties of dispersion and peristaltic flow of couple-stress fluid: comprehensive examination / M. Dhange, C. U. Devi, W. Jamshed, M. R. Eid, K. Ramesh, M. D. Shamshuddin, F. Aslam, K. Batool // Journal of Molecular Liquids. – 2024. – Vol. 409. – P. 125542. – DOI: 10.1016/j.molliq.2024.125542.
  14. Maurya P. K., Deo S., Maurya D. K. Couple stress fluid flow enclosing a solid sphere in a porous medium: effect of magnetic field // Physics of Fluids. – 2023. – Vol. 35 (7). – 072006. – DOI: 10.1063/5.0155532.
  15. Siddheshwar P. G., Sri Krishna C. V. Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium // International Journal of Non-Linear Mechanics. – 2003. – Vol. 38 (10). – P. 1561–1579. – DOI: 10.1016/S0020-7462(02)00120-8.
  16. Gorla R. S. R. Unsteady mixed convection in micropolar boundary layer flow on a vertical plate // Fluid Dynamics Research. – 1995. – Vol. 15. – P. 237–250. – DOI: 10.1016/0169-5983(95)94957-U.
  17. Nadeem S., Haq R. U., Khan Z. H. Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles // Journal of the Taiwan Institute of Chemical Engineers. – 2014. – Vol. 45 (1). – P. 121–126. – DOI: 10.1016/j.jtice.2013.04.006.
  18. Finite element solution of micropolar fluid flow and heat transfer between two porous discs / H. S. Takhar, R. Bhargava, R. S. Agrawal, A. V. S. Balaji // International journal of engineering science. – 2000. – Vol. 38 (17). – P. 1907–1922. – DOI: 10.1016/S0020-7225(00)00019-7.
  19. Srinivasacharya D., Mishra M., Rao A. R. Peristaltic pumping of a micropolar fluid in a tube // Acta Mechanica. – 2003. – Vol. 161 (3). – P. 165–178. – DOI: 10.1007/s00707-002-0993-y.
  20. Prakash J., Tripathi D. Melting heat transfer in electroosmotic augmented stagnation point flow of a micropolar fluid over stretching porous sheet // Numerical Heat Transfer, Part A: Applications. – 2026. – Vol. 87 (1). – DOI: 10.1080/10407782.2025.2521423.
  21. Мурашкин E. В., Радаев Ю. Н. О сильных и слабых разрывах связанноготермомеханического поля в термоупругих микрополярных континуумах второго типа // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. – 2014. – Vol. 18 (4). – P. 85–97. – DOI: 10.14498/vsgtu1331.
  22. Bykova S., Ivanova E. A micropolar continuum and equations of electrodynamics of moving media // Continuum Mechanics and Thermodynamics. – 2025. – Vol. 37 (4). – P. 67. – DOI: 10.1007/s00161-025-01398-5.
  23. Impact of chemical reaction on micropolar fluid past a stretching sheet / P. K. Pattnaik, S. Jena, A. Dei, G. Sahu // JP Journal of Heat and Mass Transfer. – 2019. – Vol. 18 (1). – P. 207–223. – DOI: 10.17654/HM018010207.
  24. Burmasheva N. V., Prosviryakov E. Yu. The inhomogeneous Couette flow of a micropolar fluid // Russian Journal of Nonlinear Dynamics. – 2025. – Vol. 21 (3). – P. 345–358. – DOI: 10.20537/nd250601.
  25. Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations for unidirectional flows of micropolar fluids in a mass force field // Diagnostics, Resource and Mechanics of materials and structures. – 2024. – Iss. 3. – P. 41–63. – DOI: 10.17804/2410-9908.2024.3.041-063. – URL: http://dream-journal.org/issues/2024-3/2024-3_438.html
  26. Burmasheva N. V., Prosviryakov E. Yu. Polynomial exact solutions for describing unidirectional flows of micropolar incompressible media // Procedia Structural Integrity. – 2024. – Vol. 65. – P. 39–43. – DOI: 10.1016/j.prostr.2024.11.007. 
  27. Exact solutions to the Oberbeck–Boussinesq equations for describing three-dimensional flows of micropolar liquids / E. S. Baranovskii, S. V. Ershkov, E. Yu. Prosviryakov, A. V. Yudin // Symmetry. – Vol. 16 (12). – P. 1669. – DOI: 10.3390/sym16121669.
  28. Burmasheva N. V., Prosviryakov E. Yu. Exact solution for nonuniform unidirectional Nusselt flow taking into account couple stresses // Procedia Structural Integrity. – 2024. – Vol. 65. – P. 44–47. – DOI: 10.1016/j.prostr.2024.11.008.
  29. Baranovskii E. S., Prosviryakov E. Yu., Ershkov S. V. Mathematical analysis of steady non-isothermal flows of a micropolar fluid // Nonlinear Analysis: Real World Applications. – 2025. – Vol. 84. – P. 104294. – DOI: 10.1016/j.nonrwa.2024.104294.
  30. Inhomogeneous gradient Poiseuille flows of a vertically swirled fluid / N. Burmasheva, S. Ershkov, E. Prosviryakov, D. Leshchenko // Journal of Applied and Computational Mechanics. – 2024. – Vol. 10 (1). – P. 1–12. – DOI: 10.22055/jacm.2023.43959.4150.
  31. Baranovskii E. S., Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations with couple stresses // Symmetry. – 2021. – Vol. 13 (8). – P. 1355. – DOI: 10.3390/sym13081355.
  32. Burmasheva N., Prosviryakov E. Exact solutions to Navier–Stokes equations describing a gradient nonuniform unidirectional vertical vortex fluid flow // Dynamics. – 2022. – Vol. 2 (2). – P. 175–186. – DOI: 10.3390/dynamics2020009.

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Библиографическая ссылка на статью

Gubareva K. V., Prosviryakov E. Yu. Numerical Modeling and Parametric Analysis of Couette Flow in Fluids with Couple Stresses // Diagnostics, Resource and Mechanics of materials and structures. - 2026. - Iss. 3. - P. 35-48. -
DOI: 10.17804/2410-9908.2026.3.035-048. -
URL: http://dream-journal.org/issues/2026-3/2026-3_537.html
(accessed: 26.09.2026).

 

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