Studying the Effect of Weak Convection on the Couette–poiseuille Flow of a Power-Law Fluid in a Planar Channel by the Perturbation Method

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

doi:10.17804/2410-9908.2026.2.056-072

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

STUDYING THE EFFECT OF WEAK CONVECTION ON THE COUETTE–POISEUILLE FLOW OF A POWER-LAW FLUID IN A PLANAR CHANNEL BY THE PERTURBATION METHOD

DOI: 10.17804/2410-9908.2026.2.056-072

The paper studies the influence of weak convective effects on the steady flow of a non-Newtonian fluid in a planar channel with a moving upper wall and a longitudinal pressure gradient. The fluid is described by the Ostwald–de-Waele power-law model. To analyze the slow development of the profile along the channel, an asymptotic perturbation method is used, based on a small parameter equal to the ratio of the channel height to the characteristic longitudinal scale. A first-order analytical solution is obtained, including a correction to the primary velocity profile and the shear stress distribution. The flow features for pseudoplastic, Newtonian, and dilatant fluids are analyzed. It is shown that convection disrupts the linearity of the stress field, causes longitudinal evolution of the velocity profile, and modifies the conditions for the onset of reverse flow. The results are important for accurate modeling of processes in devices with combined action of the pressure gradient and wall motion.

Keywords: power-law fluid, Couette–Poiseuille flow, convection, perturbation method, analytical solution, shear stress, non-Newtonian fluid

References:

Middleman, S. Fundamentals of Polymer Processing, McGraw-Hill, 1977, 525 p.
Wilson, S.D.R. The drag-out problem in film coating theory. Journal of Engineering Mathematics, 1982, 16, 209–221. DOI: 10.1007/BF00042717.
Aristov, S.N. and Skulskiy, O.I. Viscoelastic effects of blood flow in nondeformable. Russian Journal of Biomechanics, 1999, 3 (4), 24–33.
Bird, R.B., Armstrong, R.C., and Hassager, O. Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, 2nd ed., Wiley, 1987, 672 p.
Tanner, R.I. Engineering Rheology, 2nd ed., OUP Oxford, 2000, 586 p.
Schowalter, W.R. Mechanics of Non-Newtonian Fluids, Pergamon Press, 1978, 300 p.
Skulsky, O.I. and Aristov, S.N. Mekhanika anomalno vyazkikh zhidkostey [The Anomalous-Viscous Fluid Dynamics]. RKhD Publ., Moscow, Izhevsk, 2004, 154 p. (In Russian).
Chhabra, R.P. and Richardson, J.F. Non‐Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications, 1st ed., Butterworth‐Heinemann, Oxford, UK, 1999, 436 p.
Vinogradov, G.V. and Malkin, A.Y. Rheology of Polymers: Viscoelasticity and Flow of Polymers, Springer, 1980, 468 p.
Nádai, A. Plasticity: A Mechanics of the Plastic State of Matter, 4th ed., McGraw-Hill, 1931, 349 p.
Truesdell, C. and Noll, W. The Non-Linear Field Theories of Mechanics, 3rd ed., Springer, 2004, 602 p.
Denier, J.P. and Dabrowski, P.P. On the boundary-layer equations for power-law fluids. In: Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, 2004, 460, 3143–3158. DOI: 10.1098/rspa.2004.1349.
Málek, J., Rajagopal, K.R., and Růžička, M. Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Mathematical Models and Methods in Applied Sciences, 1995, 05 (06), 789–812. DOI: 10.1142/S0218202595000449.
Frigaard, I.A. and Ryan, D.P. Flow of a visco-plastic fluid in a channel of slowly varying width. Journal of Non-Newtonian Fluid Mechanics, 2004, 123 (1), 67–83. DOI: 10.1016/j.jnnfm.2004.06.011.
Chang, H.-C. and Demekhin, E.A. Studies in Interface Science, vol. 14: Complex Wave Dynamics on Thin Films, Elsevier, 2002.
Kalliadasis, S., Ruyer-Quil, C., Scheid, B., and Velarde, M.G. Falling Liquid Films, series Applied Mathematical Sciences, Springer, London, 2012. DOI: 10.1007/978-1-84882-367-9.
Yih, C.-S. Dynamics of Nonhomogeneous Fluids, Macmillan, 1965, 306 p.
Hartnett, J.P. and Kostic, M. Heat transfer to a viscoelastic fluid in laminar flow through a rectangular channel. International Journal of Heat and Mass Transfer, 1985, 28 (6), 1147–1155. DOI: 10.1016/0017-9310(85)90122-X.
Whiteman, J.R., ed. The Mathematics of Finite Elements and Applications: Mafelap 1984, Academic Press, 1985, 650 p.
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.
Poole, R.J. The Deborah and Weissenberg numbers. In: The British Society of Rheology, Rheology Bulletin, 2012, 53 (2), 32–39.
Rajagopal, K.R. On boundary conditions for fluids of the differential type. In: Sequeira, A., ed., Navier-Stokes Equations and Related Nonlinear Problems, Springer, Boston, MA, 1995. DOI: 10.1007/978-1-4899-1415-6_22.
Rao, I.J. and Rajagopal, K.R. The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mechanica, 1999, 135, 113–126. DOI: 10.1007/BF01305747.
Gubareva, K.V. and Prosviryakov, E.Yu. Exact analytical solution to the problem of stationary convection in the Boussinesq approximation with account for viscous dissipation. Diagnostics, Resource and Mechanics of materials and structures, 2025, 6, 23–38. DOI: 10.17804/2410-9908.2025.6.023-038. Available at: http://dream-journal.org/issues/2025-6/2025-6_528.html
Gubareva, K.V., Prosviryakov, E.Yu., and Eremin, A.V. Inhomogeneous Couette-Poiseuille flow of a viscous incompressible fluid in an infinite horizontal layer with permeable boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2025, 5, 6–28. DOI: 10.17804/2410-9908.2025.5.006-028. Available at: http://dream-journal.org/issues/2025-5/2025-5_523.html
Aristov, S.N. and Skulskii, O.I. Exact solution of the problem on a six‐constant Jeffreys model of fluid in a plane channel. Journal of Applied Mechanics and Technical Physics, 2002, 43, 817–822. DOI: 10.1023/A:1020752101539.
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.
Mekheimer, Kh.S. and El Kot, M.A. The micropolar fluid model for blood flow through a tapered artery with a stenosis. Acta Mechanica Sinica, 2008, 24, 637–644. DOI: 10.1007/s10409-008-0185-7.
Baranovskii, E.S. Analytical solutions to the unsteady Poiseuille flow of a second grade fluid with slip boundary conditions. Polymers, 2024, 16 (2), 179. DOI: 10.3390/polym16020179.
Massoudi, M. and Phuoc, T.X. Fully developed flow of a modified second grade fluid with temperature dependent viscosity. Acta Mechanica, 2001, 150, 23–37. DOI: 10.1007/BF01178542.
Ershkov, S.V., Baranovskii, E.S., Prosviryakov, E.Yu., and Yudin, A.V. Non-Newtonian rivulet-flows on unsteady heated plane surface. International Journal of Non-Linear Mechanics, 2025, 170, 104984. DOI: 10.1016/j.ijnonlinmec.2024.104984.
Prosviryakov, E.Yu. A new class of exact solutions to the Navier–Stokes equations with the Boussinesq approximation for describing convective flows of multilayer fluids. Fluid Dynamics, 2020, 55 (6), 798–809.
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.
Burmasheva, N.V., Privalova, V.V., and Prosviryakov, E.Yu. Layered Marangoni convection with the Navier slip condition. Sādhanā, 2021, 46, 55. DOI: 10.1007/s12046-021-01585-5.
Burmasheva, N.V. and Prosviryakov, E.Yu. A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2017, 21 (4), 736–751. (In Russian). DOI: 10.14498/vsgtu1568.
Gubareva, K.V., Prosviryakov, E.Yu., and Eremin, A.V. An exact solution with inhomogeneous boundary conditions for a steady non-uniform Couette flow between permeable plates. Diagnostics, Resource and Mechanics of materials and structures, 2025, 5, 66–86. DOI: 10.17804/2410-9908.2025.5.066-086. Available at: http://dream-journal.org/issues/2025-5/2025-5_522.html
Sofonea, M. and Matei, A. Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer, 2009, 230 p.
Aristov, S.N. and Keller, I.E. Beltrami stress fields in an elastic body. Doklady Physics, 2016, 61 (7), 343–346. DOI: 10.1134/S1028335816070065.
Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Oberbeck–Boussinesq equations for shear flows of a viscous binary fluid with allowance made for the Soret effect. The Bulletin of Irkutsk State University. Series Mathematics, 2021, 37, 17–30. DOI: 10.26516/1997-7670.2021.37.17.
Prosviryakov, E.Yu., Mikhailov, S.A., Ledyankina, O.A., and Goruleva, L.S. Exact solutions to the Navier-Stokes equations with the Boussinesq approximation for describing binary fluid flows. Russian Aeronautics, 2023, 66, 500–509. DOI: 10.3103/S106879982303011X.

К. В. Губарева, Е. Ю. Просвиряков, А. В. Еремин

ИССЛЕДОВАНИЕ ВЛИЯНИЯ СЛАБОЙ КОНВЕКЦИИ НА ПОЛЗУЩЕЕ ТЕЧЕНИЕ КУЭТТА – ПУАЗЕЙЛЯ СТЕПЕННОЙ ЖИДКОСТИ В ПЛОСКОМ КАНАЛЕ МЕТОДОМ ВОЗМУЩЕНИЙ

Исследуется влияние слабых конвективных эффектов на стационарное течение неньютоновской жидкости в плоском канале с движущейся верхней стенкой и продольным градиентом давления. Жидкость описывается степенной моделью Оствальда – Вейля. Для анализа медленного развития профиля вдоль канала используется асимптотический метод возмущений по малому параметру, равному отношению высоты канала к характерному продольному масштабу. Получено аналитическое решение в первом порядке, включающее поправку к основному профилю скорости и распределению касательного напряжения. Проанализированы особенности течения для псевдопластичных, ньютоновских и дилатантных жидкостей. Показано, что конвекция нарушает линейность поля напряжений, вызывает продольную эволюцию профиля скорости и модифицирует условия возникновения обратного течения. Результаты важны для корректного моделирования процессов в аппаратах с совмещенным действием градиента давления и движения границ.

Ключевые слова: степенная жидкость, течение Куэтта – Пуазейля, конвекция, метод возмущений, аналитическое решение, касательное напряжение, неньютоновская жидкость

Библиография:

Middleman S. Fundamentals of Polymer Processing. – McGraw-Hill, 1977. – 525 p.
Wilson S. D. R. The drag-out problem in film coating theory // Journal of Engineering Mathematics. – 1982. – Vol. 16. – P. 209–221. – DOI: 10.1007/BF00042717.
Aristov S. N., Skulskiy O. I. Viscoelastic effects of blood flow in nondeformable // Russian Journal of Biomechanics. – 1999. – Vol. 3 (4). – P. 24–33.
Bird R. B., Armstrong R. C., Hassager O. Dynamics of Polymeric Liquids. Vol. 1. Fluid Mechanics. – 2nd ed. – Wiley, 1987. – 672 p.
Tanner R. I. Engineering Rheology. – 2nd ed. – OUP Oxford University Press, 2000. – 586 p.
Schowalter W. R. Mechanics of Non-Newtonian Fluids. – Pergamon Press, 1978. – 300 p.
Скульский О. И., Аристов С. Н. Механика аномально вязких – М. ; Ижевск : РХД, 2004. – 154 с.
Chhabra R. P., Richardson J. F. Non‐Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications. – 1st ed. – Oxford, UK : Butterworth‐Heinemann, 1999. – 436 p.
Vinogradov G. V., Malkin A. Y. Rheology of Polymers. Viscoelasticity and Flow of Polymers. – Springer, 1980. – 468 p.
Nádai A. Plasticity: A Mechanics of the Plastic State of Matter. – 4^th ed. – McGraw-Hill, 1931. – 349 p.
Truesdell C., Noll W. The Non-Linear Field Theories of Mechanics. – 3rd ed. – Springer, 2004. – 602 p.
Denier J. P., Dabrowski P. P. On the boundary-layer equations for power-law fluids // Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. – 2004. – Vol. 460. – P. 3143–3158. – DOI: 10.1098/rspa.2004.1349.
Málek, J., Rajagopal, K.R., Růžička, M. Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Mathematical Models and Methods in Applied Sciences, 1995, 05 (06), 789–812. DOI: 10.1142/S0218202595000449.
Frigaard I. A., Ryan D. P. Flow of a visco-plastic fluid in a channel of slowly varying width // Journal of Non-Newtonian Fluid Mechanics. – 2004. – Vol. 123 (1). – P. 67–83. – DOI: 10.1016/j.jnnfm.2004.06.011.
Chang, H.-C., Demekhin E. A. Studies in Interface Science. Vol. 14. Complex Wave Dynamics on Thin Films. – Elsevier, 2002.
Falling Liquid Films. Series Applied Mathematical Sciences / S. Kalliadasis, C. Ruyer-Quil, B. Scheid, M. G. Velarde. – London : Springer, 2012. – DOI: 10.1007/978-1-84882-367-9.
Yih, C.-S. Dynamics of Nonhomogeneous Fluids. – Macmillan, 1965. – 306 p.
Hartnett J. P., Kostic M. Heat transfer to a viscoelastic fluid in laminar flow through a rectangular channel // International Journal of Heat and Mass Transfer. – 1985. – Vol. 28 (6). – P. 1147–1155. – DOI: 10.1016/0017-9310(85)90122-X.
The Mathematics of Finite Elements and Applications: Mafelap 1984 / ed. by J. R. Whiteman. – Academic Press, 1985. – 650 p.
Towards understanding the algorithms for solving the Navier-Stokes equations / S. V. Ershkov, E. Yu. Prosviryakov, N. V. Burmasheva, V. Christianto // Fluid Dynamics Research. – 2021. – Vol. 53 (4). – P. 044501. – DOI: 10.1088/1873-7005/ac10f0.
Poole R. J. The Deborah and Weissenberg numbers // Rheology Bulletin. – 2012. – 53 (2). – P. 32–39.
Rajagopal K. R. On boundary conditions for fluids of the differential type // Navier-Stokes equations and related nonlinear problems / ed. by A. Sequeira. – Boston, MA : Springer, 1995. – P. 273–278. – DOI: 10.1007/978-1-4899-1415-6_22.
Rao I. J., Rajagopal K. R. The effect of the slip boundary condition on the flow of fluids in a channel // Acta Mechanica. – 1999. – Vol. 135. – P. 113–126. – DOI: 10.1007/BF01305747.
Gubareva K. V., Prosviryakov E. Yu. Exact analytical solution to the problem of stationary convection in the Boussinesq approximation with account for viscous dissipation // Diagnostics, Resource and Mechanics of materials and structures. – 2025. – Iss. 6. – P. 23–38. – DOI: 10.17804/2410-9908.2025.6.023-038. – URL: http://dream-journal.org/issues/2025-6/2025-6_528.html
Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. Inhomogeneous Couette-Poiseuille flow of a viscous incompressible fluid in an infinite horizontal layer with permeable boundaries // Diagnostics, Resource and Mechanics of materials and structures. – 2025. – Iss. 5. – P. 6–28. – DOI: 10.17804/2410-9908.2025.5.006-028. – URL: http://dream-journal.org/issues/2025-5/2025-5_523.html
Aristov S. N., Skulskii O. I. Exact solution of the problem on a six‐constant Jeffreys model of fluid in a plane channel // Journal of Applied Mechanics and Technical Physics. – 2002. – Vol. 43. – P. 817–822. – DOI: 10.1023/A:1020752101539.
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.
Mekheimer Kh. S., El Kot M. A. The micropolar fluid model for blood flow through a tapered artery with a stenosis // Acta Mechanica Sinica. – 2008. – Vol. 24. – P. 637–644. – DOI: 10.1007/s10409-008-0185-7.
Baranovskii E. S. Analytical Solutions to the unsteady Poiseuille flow of a second grade fluid with slip boundary conditions // Polymers. – 2024. – Vol. 16 (2). – P. 179. – DOI: 10.3390/polym16020179.
Massoudi M., Phuoc T. X. Fully developed flow of a modified second grade fluid with temperature dependent viscosity // Acta Mechanica. – 2001. – Vol. 150. – P. 23–37. – DOI: 10.1007/BF01178542.
Non-Newtonian rivulet flows on unsteady heated plane surface / S. V. Ershkov, E. S. Baranovskii, E. Yu. Prosviryakov, A. V. Yudin // International Journal of Non-Linear Mechanics. – 2025. – Vol. 170. – P. 104984. – DOI: 10.1016/j.ijnonlinmec.2024.104984.
Prosviryakov E. Yu. A new class of exact solutions to the Navier-Stokes equations with the Boussinesq approximation for describing convective flows of multilayer fluids // Fluid Dynamics. – 2020. – Vol. 55 (6). – P. 798–809.
Exact solutions of the Oberbeck–Boussinesq equations for the description of shear thermal diffusion of Newtonian fluid flows / S. Ershkov, N. Burmasheva, D. D. Leshchenko, E. Yu. Prosviryakov // Symmetry. – 2023. – Vol. 15 (9). – P. 1730. – DOI: 10.3390/sym15091730.
Burmasheva N. V., Privalova V. V., Prosviryakov E. Yu. Layered Marangoni convection with the Navier slip condition // Sādhanā. – 2021. – Vol. 46. – 55. – DOI: 10.1007/s12046-021-01585-5.
Бурмашева Н. В., Просвиряков Е. Ю. Крупномасштабная слоистая стационарная конвекция вязкой несжимаемой жидкости под действием касательных напряжений на верхней границе. Исследование полей температуры и давления // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. – 2017. – Т. 21 (4). – С. 736–751. – DOI: 10.14498/vsgtu1568.
Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. An exact solution with inhomogeneous boundary conditions for a steady non-uniform Couette flow between permeable plates // Diagnostics, Resource and Mechanics of materials and structures. – 2025. – Iss. 5. – P. 66–86. – DOI: 10.17804/2410-9908.2025.5.066-086. – URL: http://dream-journal.org/issues/2025-5/2025-5_522.html
Sofonea M., Matei A. Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems. – Springer, 2009. – 230 p.
Aristov S. N., Keller I. E. Beltrami stress fields in an elastic body // Doklady Physics. – 2016. – Vol. 61 (7). – P. 343–346. – DOI: 10.1134/S1028335816070065.
Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Oberbeck-Boussinesq equations for shear flows of a viscous binary fluid with allowance made for the Soret effect // The Bulletin of Irkutsk State University. Series Mathematics. – 2021. – Vol. 37. – P. 17–30. – DOI: 10.26516/1997-7670.2021.37.17.
Exact solutions to the Navier-Stokes equations with the Boussinesq approximation for describing binary fluid flows / E. Yu. Prosviryakov, S. A. Mikhailov, O. A. Ledyankina, L. S. Goruleva // Russian Aeronautics. – 2023. – Vol. 66. – P. 500–509. – DOI: 10.3103/S106879982303011X.

Библиографическая ссылка на статью

Gubareva K. V., Prosviryakov E. Yu., Eremin A. V. Studying the Effect of Weak Convection on the Couette–poiseuille Flow of a Power-Law Fluid in a Planar Channel by the Perturbation Method // Diagnostics, Resource and Mechanics of materials and structures. - 2026. - Iss. 2. - P. 56-72. -
DOI: 10.17804/2410-9908.2026.2.056-072. -
URL: http://dream-journal.org/issues/2026-2/2026-2_554.html
(accessed: 26.09.2026).