Funct. Mater. 2025; 32 (4): 612-644.

doi:https://doi.org/10.15407/fm32.04.612

Magnetorotational and convective instabilities in a thin layer of electrically conductive <-30>

M.I. Kopp1, V.V. Yanovsky1,2

1 Institute for Single Crystals, NAS of Ukraine, Nauky Ave. 60, Kharkiv 61072, Ukraine
2 V. N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv 61022, Ukraine}

Abstract: 

This review summarizes recent advances in the theoretical analysis of the stability of magnetized flows in a non-uniformly rotating layer of electrically conductive nanofluid, incorporating the effects of Brownian diffusion and thermophoresis. In the absence of temperature gradients, different forms of magnetorotational instability (MRI): standard (SMRI), azimuthal (AMRI), and helical (HMRI) are investigated for nanofluid layers subjected to axial, azimuthal, and helical magnetic fields. The corresponding growth rates and instability regions are analyzed in relation to the rotation profile (quantified by the Rossby number Ro) and the radial wave number k. When temperature gradients and nanoparticle concentration effects are present, stationary convective modes in both axial and helical magnetic fields are examined under conditions of non-uniform rotation. Analytical expressions for the critical Rayleigh number Rast are derived, and neutral stability curves are constructed as functions of the angular velocity profile, the azimuthal magnetic field inhomogeneity (magnetic Rossby number Rb, and the wave number k. The study identifies and discusses key mechanisms responsible for the stabilization or destabilization of stationary convection in axial and spiral magnetic field configurations, highlighting the role of nanoparticle-driven effects in modifying classical magnetoconvective behavior.

Keywords: 
magnetoconvection, non-uniform rotation, magnetorotational instability (MRI), helical magnetic fields, nanofluid, critical Rayleigh number

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References: 

1. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME J. Heat Transfer 66 (1995) 99-105.

2. J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240-250. https://doi.org/10.1115/1.2150834

3. B. Ghasemi, S.M. Aminossadati, A. Raisi. Magnetic field effect on natural convection in a nanofluid-filledsquare enclosure, Int. J. Therm. Sci., 2011; 50, 1748-1756.

4. M. A. A. Hamada, I. Pop, A.I.Md. Ismail, Magnetic field effects on free convection flow of a nanofluid past avertical semi-infinite flat plate, Nonlinear Anal. R. World Appl. 12 (2011) 1338-1346.

5. D. Yadav, R. Bhargava, G.S. Agrawal, Thermal instability in a nanofluid layer with a vertical magnetic field,J. Eng, Math. 80 (2013)147-164.

6. S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability, Oxford Uni. Press, London, 1961.

7. G.Z. Gershuni, E.M. Zhukhovitskii, Convective Stability of Incompressible Fluids, Keter Publishing House,Jerusalem, 1976.

8. A.V. Getling, Rayleigh-Benard Convection: Structures and Dynamics, World Sci., Singapore, 1998.

9. D. Yadav, R. Bhargava, G.S. Agrawal, G.S. Hwang, J. Lee and M.C. Kim, Magneto-convection in a rotatinglayer of nanofluid, Asia-Pac. J. Chem. Eng. 9 (2014) 663-677. https://doi.org/10.1002/apj.1796

10. M.I. Kopp, A.V.Tur, V.V. Yanovsky, Instabilities in the non-uniformly rotating medium with temperaturestratification in the external uniform magnetic field, East Eur. J. Phys. 1 (2019) 4-33.https://doi.org/10.26565/2312-4334-2019-1-01

11. M.I. Kopp, A.V. Tur, V.V. Yanovsky, Chaotic magnetoconvection in a non-uniformly rotatingelectroconductive fluids, VANT 116 (2018) 230-234. arXiv:1805.11894v1 [astro-ph.EP](2018).https://doi.org/10.48550/arXiv.1805.11894

12. M.I. Kopp, A.V. Tur, V.V. Yanovsky, Magnetic convection in a nonuniformly rotating electrically conductivemedium in an external spiral magnetic, Fluid Dyn. Res. 53 (2021) 015509. https://doi.org/10.1088/1873-7005/abd8dc

13. P.G. Drazin and W.H. Reid, Hydrodynamic Stability, 2nd edition, Cambridge University Press, 2004.https://doi.org/10.1017/CBO9780511616938

14. D. Coles, Transition in circular Couette flow, J. Fluid Mech. 21 (1965) 385-425.https://doi.org/10.1017/S0022112065000241

15. C.D. Andereck, S.S. Liu, H.L. Swinney, Flow regimes in a circular Couette system with independently rotatingcylinders, J. Fluid Mech. 164 (1986)155-183. https://doi.org/10.1017/S0022112086002513

16. S.A. Balbus and J.F. Hawley, A Powerful Local Shear Instability in Weakly Magnetized Discs. I. LinearAnalysis, Astrophys. J. 376 (1991) 214-222. http://dx.doi.org/10.1086/170270

17. D.R. Sisan, N. Mujica, W.A. Tillotson, Y. Huang, W. Dorland, A.B. Hassam, T.M. Antonsen, and D.P. Lathrop,Experimental Obser-vation and Characterization of the Magnetorotational Instability, Phys. Rev. Lett. 93(2004) 114502.

18. Z. Wang, J. Si, and W. Liu, Equilibrium and magnetic properties of a rotating plasma annulus, Phys. Plasmas.15 (2008) 102109.

19. F. Stefani, G. Gerbeth, T. Gundrum, J. Szklarski, G. R¨udiger, and R. Hollerbach, Results of a modifiedPROMISE experiment”, Astron. Nachr. 329 (2008) 652-658. https://doi.org/10.1002/asna.200811023

20. J.M. Zayan, A.K. Rasheed, A. John, S. Muniandy, L.B. Fen, and A.F. Ismail, Synthesisand Characterization of Novel Ternary Hybrid Nanoparticles as Thermal Additives in H2O”, 2021.https://doi.org/10.26434/chemrxiv.13710130.v1

21. M.I. Kopp, V.V. Yanovsky. Magnetic convection in a nonuniformly rotating layer of conductive nanofluidin an external helical magnetic field. December 13, 2024; Zurich, Switzerland: VII International Scientific andPractical Conference ”GRUNDLAGEN DER MODERNEN WISSENSCHAFTLICHEN FORSCHUNG”. Bookof Abstracts, p. 271-274. https://doi.org/10.36074/logos-13.12.2024.059

22. G. Moffat, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, Cambridge,1978.

23. G. R¨udiger and M. Schultz, Tayler instability of toroidal magnetic fields in MHD Taylor-Couette flows, Astron.Nachr. 331 (2010) 121-129.

24. M. Gopal, Control Systems: Principles and Design, McGraw Hill, 2002.

25. B. Eckhardt, D. Yao, Local stability analysis along Lagrangian paths, Chaos, Solitons & Fractals 5(11) (1995)2073-2088.

26. G. R¨udiger, M. Schultz, M. Gellert and F. Stefani, Azimuthal magnetorotational instability with superrotation,J. Plasma Phys. 84 (2018) 735840101. https://doi.org/10.1017/S0022377818000065

27. D.H. Michael, The stability of an incompressible electrically conducting fluid rotating about an axis whencurrent flows parallel to the axis, Mathematika 1 (1954) 45-50.

28. S. Chandrasekhar, On the stability of the simplest solution of the equations of hydromagnetics, Proc. NatlAcad. Sci. USA 42 (1956) 273-276.

29. D.A. Nield, A.V. Kuznetsov, The onset of convection in a horizontal nanofluid layer of finite depth, Eur. J.Mech. B Fluid. 29 (2010) 217-223. https://doi.org/10.1016/j.euromechflu.2010.02.003

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