Numerical simulation of two-dimensional flow of a micropolar fluid in an enclosed cavity with the full dissipation function
DOI:
https://doi.org/10.7242/1999-6691/2026.19.2.11Keywords:
micropolar fluid, Eringen equations, energy dissipation, numerical simulation, lid-driven cavity flow, projection methodAbstract
The paper studies numerically a steady two-dimensional flow of incompressible micropolar fluid in a square cavity with a moving top wall. The steady solution is obtained by the time-marching method - time integration of the full system of unsteady Eringen equations. A distinctive feature of this work is a detailed calculation and analysis of the total dissipation function, which includes three mechanisms: classical viscous dissipation, dissipation due to microrotation gradients, and dissipation resulting from the interaction between macroscopic vorticity and the microrotation field. A numerical algorithm is developed based on the Chorin–Temam projection method; its correctness is verified by comparison with the classical Newtonian limit. A parametric study is carried out over wide ranges of the governing dimensionless parameters - the Reynolds number, the micropolarity parameter, and the microscale parameter. It is shown that the integral dissipation increases with the Reynolds number and the micropolarity parameter, although there are local non-monotonicities due to a nonlinear interaction of dissipative mechanisms. The influence of the microscale parameter is found to be considerably weaker than that of the other two parameters. It is shown that as the Reynolds number increases, the thickness of the boundary layer at the moving wall decreases, and the microrotation velocity maximum diminishes and is localized in the near-wall layers. Analysis of the velocity and microrotation distributions confirms the formation of thin boundary layers near solid walls, where dissipation is maximal. Micropolar effects lead to a marked increase in dissipation compared with a classical Newtonian fluid, which demonstrates the importance of considering the internal rotational degrees of freedom in the energy balance. The results may find application in microfluidics and modeling of complex fluid flows.
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Squires T.M., Quake S.R. Microfluidics: Fluid physics at the nanoliter scale. Reviews of Modern Physics. 2005. Vol. 77, no. 3. P. 977–1026. DOI: 10.1103/REVMODPHYS.77.977
Cosserat E., Cosserat F. Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils, 1909. 226 p.
Eringen A.C. Simple microfluids. International Journal of Engineering Science. 1964. Vol. 2, no. 2. P. 205–217. DOI: 10.1016/0020-7225(64)90005-9
Eringen A.C. Theory of Micropolar Fluids. Indiana University Mathematics Journal. 1966. Vol. 16. P. 1–18. DOI: 10.1512/iumj.1967.16.16001
Ariman T., Turk M.A., Sylvester N.D. Microcontinuum fluid mechanics—A review. International Journal of Engineering Science. 1973. Vol. 11, no. 8. P. 905–930. DOI: 10.1016/0020-7225(73)90038-4
Lukaszewicz G. Micropolar Fluids: Theory and Applications. Birkhäuser Boston, 1999. 320 p. . DOI: 10.1007/978-1-4612-0641-5
Cowin S.C. The Theory of Polar Fluids. Vol. 14. 1974. P. 279–347. DOI: 10.1016/S0065-2156(08)70034-6
Rees D.A.S., Bassom A.P. The Blasius boundary-layer flow of a micropolar fluid. International Journal of Engineering Science. 1996. Vol. 34, no. 1. P. 113–124. DOI: 10.1016/0020-7225(95)00058-5
Ghia U., Ghia K.N., Shin C.T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics. 1982. Vol. 48, no. 3. P. 387–411. DOI: 10.1016/0021-9991(82)90058-4
Shankar P.N., Deshpande M.D. Fluid Mechanics in the Driven Cavity. Annual Review of Fluid Mechanics. 2000. Vol. 32, no. 1. P. 93–136. DOI: 10.1146/annurev.fluid.32.1.93
Ahmadi G. Self-similar solution of imcompressible micropolar boundary layer flow over a semi-infinite plate. International Journal of Engineering Science. 1976. Vol. 14, no. 7. P. 639–646. DOI: 10.1016/0020-7225(76)90006-9
Jena S.K., Mathur M.N. Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a non-isothermal vertical flat plate. International Journal of Engineering Science. 1981. Vol. 19, no. 11. P. 1431–1439. DOI: 10.1016/0020-7225(81)90040-9
Burmasheva N.V., Prosviryakov E.Y. 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
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. 2024. Vol. 3. P. 41–63. DOI: 10.17804/2410-9908.2024.3.041-063
Batchelor G.K. An Introduction to Fluid Dynamics. Cambridge University Press, 1967. 615 p.
Landau L.D., Lifshitz E.M. Fluid Mechanics. 2nd. Oxford: Pergamon Press, 1987. 539 p.
Stokes V.K. Theories of Fluids with Microstructure: An Introduction. Springer Berlin Heidelberg, 1984. 205 p. . DOI: 10.1007/978-3-642-82351-0
Peddieson J., McNitt R.P. Boundary Layer Theory for a Micropolar Fluid. Recent Advances in Engineering Science. 1970. Vol. 5. P. 405–426.
Chikulaev D.G., Shvarts K.G. Effect of rotation on the monotonic instability mode of advective flow in a horizontal incompressible fluid layer with rigid boundaries in the case of normal spiral perturbation. Computational Continuum Mechanics. 2016. Vol. 9, no. 1. P. 27–37. DOI: 10.7242/1999-6691/2016.9.1.3
Chikulaev D.G., Shvarts K.G. Numerical study of the fast rotation effect on the stability of advective flow in a horizontal fluid layer with rigid boundaries at small Prandtl numbers. Computational Continuum Mechanics. 2014. Vol. 7, no. 2. P. 135–141. DOI: 10.7242/1999-6691/2014.7.2.14
Alekseev M.S., Ponomarev R.R., Shelistov V.S., Popov V.A., Morshneva I.V., Demekhin E.A. Modeling of nanochannels in synthetic membranes. Computational Continuum Mechanics. 2024. Vol. 17, no. 1. P. 5–14. DOI: 10.7242/1999-6691/2024.17.1.1
Gad-el-Hak M. The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture. Journal of Fluids Engineering. 1999. Vol. 121. P. 5–33. DOI: 10.1115/1.2822013
Chang H.- C., Yossifon G., Demekhin E.A. Nanoscale Electrokinetics and Microvortices: How Microhydrodynamics Affects Nanofluidic Ion Flux. Annual Review of Fluid Mechanics. 2012. Vol. 44, no. 1. P. 401–426. DOI: 10.1146/annurev-fluid-120710-101046
Stone H.A., Stroock A.D., Ajdari A. Engineering Flows in Small Devices: Microfluidics Toward a Lab-on-a-Chip. Annual Review of Fluid Mechanics. 2004. Vol. 36, no. 1. P. 381–411. DOI: 10.1146/annurev.fluid.36.050802.122124
Khanukaeva D.Y., Filippov A.N. Isothermal Flows of Micropolar Liquids: Formulation of Problems and Analytical Solutions. Colloid Journal. 2018. Vol. 80, no. 1. P. 14–36. DOI: 10.1134/S1061933X18010040
Chorin A.J. Numerical solution of the Navier-Stokes equations. Mathematics of Computation. 1968. Vol. 22, no. 104. P. 745–762.
Témam R. Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I). Archive for Rational Mechanics and Analysis. 1969. Vol. 32. P. 135–153. DOI: 10.1007/BF00247678
Kochinov A.Y., Shvarts K.G. Finite-amplitude perturbations of the advective flows in a horizontal layer of incompressible liquid with a free upper boundary at low rotation. Computational Continuum Mechanics. 2015. Vol. 8, no. 2. P. 174–187. DOI: 10.7242/1999-6691/2015.8.2.15
Knutova N.S., Shvarts K.G. Numerical simulation of finite-amplitude spatial perturbations of advective thermocapillary flow in a weakly rotating liquid layer under microgravity conditions. Computational Continuum Mechanics. 2018. Vol. 11, no. 3. P. 263–274. DOI: 10.7242/1999-6691/2018.11.3.20
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