Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields
In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.
S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability (Oxford Uni. Press, London, 1961), p. 652.
G.Z. Gershuni, and E.M. Zhukhovitckii, Convective Stability of Incompressible Fluids (Nauka, Moscow, 1972), pp. 392 (in Russian)
A.V. Getling, Rayleigh-Benard Convection: Structures and Dynamics (URSS, Moscow, 1999), p. 235. (in Russian)
M. Lappa, Rotating thermal flows in natural and industrial processes. (A John Wiley & Sons, Ltd., Publication, 2012), pp. 544.
S. Chandrasekhar, Proc. R. Soc. Lond. A217, 306-327 (1953), https://doi.org/10.1098/rspa.1953.0065.
S. Chandrasekhar, and D.D. Elbert, Proc. R. Soc. Lond. A231, 198-210 (1955), https://doi.org/10.1098/rspa.1955.0166.
I.A. Eltayeb, Proc. R. Soc. Lond. A326, 229-254 (1972), https://doi.org/10.1098/rspa.1972.0007.
I.A. Eltayeb, J. Fluid Mech. 71(1), 161–179 (1975), https://doi.org/10.1017/S0022112075002480.
R. Avila and A. Cabello, Mathematical Problems in Engineering, 2013, 1-15 (2013), https://doi.org/10.1155/2013/236901.
E. Kurt, F.H. Busse and W. Pesch, Theoret. Comput. Fluid Dynamics, 18, 251-263 (2004), https://doi.org/10.1007/s00162-004-0132-6.
M.I. Kopp, A.V. Tour, and V.V. Yanovsky, JETP 127, 1173-1196 (2018), https://doi.org/10.1134/S106377611812018X.
M.I. Kopp, A.V. Tur, and V.V. Yanovsky, Problems of Atomic Science and Technology, 4(116), 230-234 (2018), https://arxiv.org/abs/1805.11894.
M. Kopp, A. Tur, and V. Yanovsky, East Eur. J. Phys. 1, 4-33 (2019), https://doi.org/10.26565/2312-4334-2020-1-01.
M.I. Kopp, A.V. Tur, and V.V. Yanovsky, https://arxiv.org/abs/1905.05472.
P. Vadasz, and S. Olek, Int. J. Heat Mass Transfer 41, 1417-1435 (1999), https://doi.org/10.1016/S0017-9310(97)00265-2.
V.K. Gupta, B.S. Bhadauria, I. Hasim, J. Jawdat, and A.K. Singh, Alexandria Engineering Journal, 54, 981-992 (2015), https://doi.org/10.1016/j.aej.2015.09.002.
V.K. Gupta, R. Prasad, and A.K. Singh, International Journal of Energy and Technology, 5(28), 1-9 (2013).
V.K. Gupta, and A.K. Singh, A Study of Chaos in an Anisotropic Porous Cavity, International Journal of Energy and Technology, 5 (27), 1-27 (2013).
R. Prasad, and A.K. Singh, International Journal of Applied Mathematics and Informatics, 7(3), 87-96 (2013).
J.M. Jawdat, and I. Hashim, International Journal on Advanced Science, Engineering and Technology, 2(5), 346-349 (2012), https://doi.org/10.18517/ijaseit.2.5.220.
R. Prasad, and A. K. Singh, Journal of Applied Fluid Mechanics, 9(6), 2887-2897 (2016). https://doi.org/10.29252/jafm.09.06.24811.
G. Moffat, Возбуждение магнитного поля в проводящей среде [Magnetic Field Generation in Electrically Conducting Fluids], (Mir, Moscow, 1980), pp. 343. (in Russian)
T. Rikitake, Proc. Cambridge Philos. Soc. 54, 89 (1958).
A.E. Cook, and P.H. Roberts, Proc. Cambridge Philos. Soc. 68, 547-569 (1970).
Y. Gholipour, A. Ramezani, and M. Mola, Bulletin of Electrical Engineering and Informatics, 3(4), 273-276 (2014).
Xuedi Wang, Tianyu Yang, Wei Xu, International Journal of Nonlinear Science, 14(2), 211-215 (2012).
By F. Plunian, Ph. Marty, and A. Alemany, Proc. R. Soc. Lond. A. 454, 1835-1842 (1998).
I.A. Ilyin, D.S. Noshchenko, and A.S. Perezhogin, Vestnik KRAUNC. Fiz.-Mat. Nauki 2(7), 43-45 (2013).
V.I. Potapov, Rus. J. Nonlin. Dyn. 6, 255-265 (2010).
W.V.R. Malkus, and G. Veronis, J. Fluid Mech. 4(3), 225-260 (1958), https://doi.org/10.1017/S0022112058000410.
J.K. Bhattacharjee, J. Phy. A: Math. Gen. 22(24), L1135-L1189 (1989), https://doi.org/10.1088/0305-4470/22/24/001.
J.K. Bhattacharjee, Phy. Rev. A. 41, 5491-5494 (1990), https://doi.org/10.1103/PhysRevA.41.5491.
B.S. Bhadauria, and P. Kiran, Ain Shams Eng. J. 5(4), 1287-1297 (2015), https://doi.org/10.1016/j.asej.2014.05.005.
R. Ramya, E.J. Shelin, and G.K. Sangeetha, International Journal of Mathematics Trends and Technology, 54(6), 477-484 (2018), https://doi.org/10.14445/22315373/IJMTT-V54P558
P. Kiran, Ain Shams Eng. J. 7(2), 639-651 (2016), https://doi.org/10.1016/j.asej.2015.06.005
P.G. Siddheshwar, B.S. Bhadauria, and A. Srivastava, Transp. Porous Media, 91(2), 585- 604 (2012), https://doi.org/10.1007/s11242-011-9861-3
B.S. Bhadauria, P.G. Siddheshwar, J. Kumar, and O.P. Suthar, Trans. Porous Med. 73(3), 633-647 (2012), https://doi.org/10.1007/s11242-011-9925-4
P.G. Siddheshwar, B.S. Bhadauria, Pankaj Mishra, and A.K. Srivastava, Int. J. Non Linear Mech. 47, 418-425 (2012), https://doi.org/10.1016/j.ijnonlinmec.2011.06.006.
B.S. Bhadauria, and P. Kiran, Int. J. Eng. Math. 1, 2014 (2014), https://doi.org/10.1155/2014/296216.
B.S. Bhadauria, and P. Kiran, Transp. Porous Media. 100, 279-295 (2013), https://doi.org/10.1007/s11242-013-0216-0.
B.S. Bhadauria, and P. Kiran, Phys. Scr. 89(9), 095209 (2014), https://doi.org/10.1088/0031-8949/89/9/095209.
S. Aniss, M. Belhaq, and M. Souhar, J. Heat Transfer, 123(3), 428-433 (2001), https://doi.org/10.1115/1.1370501.
B.J. Geurts, and R. Kunnen, International Journal of Heat and Fluid Flow, 49, 62-68 (2014).
S.D. Alessio, and K. Ogden, WIT Transactions on Engineering Sciences, 74, 453-463 (2012).
G. Venezian, J. Fluid Mech. 35, 243-254 (1969).
P. Goldreich, and D. Lynden-Bell, Mon. Not. R. Astron. Soc. 130 (2), 125-158 (1965), https://doi.org/10.1093/mnras/130.2.125.
E. Knobloch, and K. Jullien, Physics of Fluids, 17(9), 094106 (2005), https://doi.org/10.1063/1.2047592.
R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th ed. (Pearson/Prentice Hall, N.J., 2004), p. 769.
O.N. Kirillov, F. Stefani, and Y. Fukumoto, J. Fluid Mech. 760, 591- 633 (2014), https://doi.org/10.1017/jfm.2014.614.
R.J. Donnelly. Proc. R. Soc. Lond. Ser. A281, 130139 (1964).
Jin-Qiang Zhong, Sebastian Sterl, and Hui-Min Li, J. Fluid Mech. 778, R4 (2015).
Copyright (c) 2020 Michael I. Kopp, Anatoly V. Tur, Volodymyr V. Yanovsky
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).