Numerical simulation of Couette flow in a micropolar fluid using the projection method

Objectives. The study develops an efficient numerical algorithm for simulating a two-dimensional flow of a micropolar fluid in a plane channel with a moving upper wall (Couette flow) in order to investigate the influence of micropolarity parameters on the flow structure.

Methods. The equations governing the dynamics of a micropolar fluid are solved using the projection method with explicit time integration. Spatial discretization is performed by the finite difference method on a uniform 51 × 51 grid. The convective terms are approximated using a first-order upwind scheme to ensure stability at moderate Reynolds numbers. The microrotation and momentum conservation equations are solved separately, followed by a pressure correction to satisfy the incompressibility condition.

Results. Steady-state velocity and microrotation fields were obtained for a Couette flow at Reynolds number Re = 100 with micropolarity parameters N = 0.3 and m = 0.015. The numerical method demonstrated stable convergence within 5000 iterations to achieve residuals of 3 ∙ 10[−6] for velocity components and 1.5 ∙ 10[−6] for microrotation. Visualized results include the vector velocity field and streamlines, as well as distributions of microrotation, vorticity, and energy dissipation. A nonuniform microrotation field was formed having maximum values (~0.45 rad/s) localized in the corner regions of the channel.

Conclusions. The developed algorithm effectively simulates a micropolar fluid flow in a channel. The numerical method exhibits stable convergence to yield physically meaningful results. Micropolarity is confirmed to significantly alter the flow structure in comparison with the Newtonian case, leading to the development of a transverse velocity component and a nonlinear microrotation distribution. The obtained distributions of field characteristics can serve as a basis for further research into the rheological properties of micropolar media and for the verification of experimental data.

1. Eringen A.C. Simple Microfluids. Int. J. Eng. Sci. 1964;2(2):205–217. https://doi.org/10.1016/0020-7225(64)90005-9

2. Lukaszewicz G. Micropolar Fluids: Theory and Applications. Basel: Birkhäuser; 1999, 320 p. https://doi.org/10.1007/9781-4612-0641-5

3. Ariman T., Turk M.A., Sylvester N.D. Microcontinuum Fluid Mechanics A Review. Int. J. Eng. Sci. 1973;11(8):905–930. https://doi.org/10.1016/0020-7225(73)90038-4

4. Cowin S.C. The Theory of Polar Fluids. Adv. Appl. Mech. 1974;14:279–347. https://doi.org/10.1016/S0065-2156(08)70034-6

5. Squires T.M., Quake S.R. Microfluidics: Fluid Physics at the Nanoliter Scale. Rev. Mod. Phys. 2005;77(3):977–1026. https://doi.org/10.1103/REVMODPHYS.77.977

6. Carvalho V., Pinho D., Lima R.A., Teixeira J.C., Teixeira S. Blood Flow Modeling in Coronary Arteries: A Review. Fluids. 2021;6(2):53. https://doi.org/10.3390/fluids6020053

7. Mathur P., Soni S. Transient nonlinear modeling of lemon bore journal bearings with micropolar fluid and eccentricity effects. Multiscale and Multidiscip. Model. Exp. and Des. 2025;8:467. https://doi.org/10.1007/s41939-025-01065-x

8. Tassaddiq A. Impact of Cattaneo-Christov heat flux model on MHD hybrid nano-micropolar fluid flow and heat transfer with viscous and joule dissipation effects. Sci. Rep. 2021;11(1):67. https://doi.org/10.1038/s41598-020-77419-x

9. Kartashov E.M. Development of model representations of thermal reaction viscoelastic bodies on the temperature field. Russian Technological Journal. 2024;12(6):80–90. https://doi.org/10.32362/2500-316X-2024-12-6-80-9010.

10. Eringen A.C. Theory of Micropolar Fluids. J. Math. Mech. 1966;16(1):1–18. https://doi.org/10.1512/IUMJ.1967.16.16001

11. Ahmadi G. Self-Similar Solution of Incompressible Micropolar Boundary Layer Flow Over a Semi-Infinite Plate. Int. J. Eng. Sci. 1976;14(7):639–646. https://doi.org/10.1016/0020-7225(76)90006-9

12. Burmasheva N.V., Prosviryakov E.Y. 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 (DReaM). 2024;3:41–63. https://doi.org/10.17804/2410-9908.2024.3.041-063

13. Baranovskii E.S., Ershkov S.V., Prosviryakov E.Y., Yudin A.V. Exact Solutions to the Oberbeck–Boussinesq Equations for Describing Three-Dimensional Flows of Micropolar Liquids. Symmetry. 2024;16(12):1669. https://doi.org/10.3390/sym16121669

14. Batchelor G.K. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press; 1967, 615 p.

15. Alwawi F., Sulaiman I.M., Swalmeh M.Z., Yaseen N. Energy transport boosters of magneto micropolar fluid flowing past a cylinder: A case of laminar combined convection. Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science. 2022;236(22):10902–10913.

16. Ali N., Nazeer M., Javed T., Abbas F. A numerical study of micropolar flow inside a lid-driven triangular enclosure. Meccanica. 2018;53(13):3279–3299. https://doi.org/10.1007/s11012-018-0884-5

17. Burmasheva N.V., Prosviryakov E.Y. The Inhomogeneous Couette Flow of a Micropolar Fluid. Russ. J. Nonlin. Dyn. 2025;21(3):345–358. https://doi.org/10.20537/nd250601

18. Patankar S.V. Numerical Heat Transfer and Fluid Flow. New York: Hemisphere Publishing Corporation; 1980, 197 p.

19. Maeda M., Soga K. More on convergence of Chorin’s projection method for incompressible Navier–Stokes equations. J. Math. Fluid Mech. 2022;24(2):41. https://doi.org/10.1007/s00021-021-00652-5

20. Chorin A.J. Numerical solution of the Navier-Stokes equations. Mathematics of Computation. 1968;22(104):745–762. https://doi.org/10.1090/S0025-5718-1968-0242392-2

21. Venkatadri K., Maheswari S., Lakshmi C.V., Prasad V.R. Numerical simulation of lid-driven cavity flow of micropolar fluid. IOP Conf. Ser.: Mater. Sci. Eng. 2018;402:012168. https://doi.org/10.1088/1757-899X/402/1/012168

22. Batool S., Rasool G., Alshammari N., Khan I., Kaneez H., Hamadneh N. Numerical analysis of heat and mass transfer in micropolar nanofluids flow through lid driven cavity: Finite volume approach. Case Studies in Thermal Engineering. 2022;37:102233. https://doi.org/10.1016/j.csite.2022.102233

23. Baranovskii E.S., Prosviryakov E.Y., Ershkov S.V. Mathematical analysis of steady non-isothermal flows of a micropolar fluid. Nonlin. Anal.: Real World Appl. 2025;84:104294. https://doi.org/10.1016/j.nonrwa.2024.104294

24. Kartashov E.M. Method for splitting integral transformation in problems of complex heat transfer. Russian Technological Journal. 2025;13(6):104–115. https://doi.org/10.32362/2500-316X-2025-13-6-104-11525.

25. Gubareva K.V., Prosviryakov E.Yu. MATLAB Implementation of a Projection Method for 2D Micropolar Couette Flow. Mendeley Data. V1. 2026. https://doi.org/10.17632/mpjtfj7krc.1