Advertisement

Nonlinear Dynamics

, Volume 94, Issue 4, pp 2763–2784 | Cite as

Study of a periodically forced magnetohydrodynamic system using Floquet analysis and nonlinear Galerkin modelling

  • Arnab Basak
Original Paper
  • 178 Downloads

Abstract

We present a detailed study of Rayleigh–Bénard magnetoconvection with periodic gravity modulation and uniform vertical magnetic field. Linear stability analysis is carried out using Floquet theory to construct the stability boundaries in order to estimate the magnitude of forcing amplitude \(\epsilon \) required for having convection in the system for a fixed Rayleigh number Ra, wave number k and modulating frequency \(\Omega \). The effects of varying Prandtl number Pr and Chandrasekhar number Q on the threshold of convection are also investigated. A higher Pr value reduces the value of the threshold, whereas a higher Q value increases it. Bicritical states are also observed at which the minimum forcing amplitude needed for convection to begin occurs at two different k values in harmonic and sub-harmonic regions, respectively. We also construct a nonlinear Galerkin model and compare the results with those obtained from linear stability analysis. Two-dimensional (2D) oscillatory convection is observed at the onset, while quasiperiodic and chaotic behaviours are found at higher Ra values. 2D as well as nonlinear convective flow patterns are observed for primary and higher-order instabilities, respectively. Bifurcation diagrams with respect to different parameters such as \(\epsilon \), Ra and Q are provided for thorough understanding of the forced nonlinear system.

Keywords

Driven magnetoconvection Modelling Bifurcations 

Notes

Acknowledgements

I acknowledge the financial support from the Center of Excellence in Space Sciences India (CESSI) funded by the Ministry of Human Resource Development, Government of India. I am grateful to my Ph.D. supervisor Prof. Krishna Kumar for learning different numerical techniques from him and to my father Tushar Kanti Basak for fruitful discussions. I am obliged to the anonymous referees whose valuable suggestions have helped improve the standard of the paper.

Disclosure Of Potential Conflicts Of Interest

Funding

The study was funded by the Ministry of Human Resource Development, Government of India via the Center of Excellence in Space Sciences India (CESSI).

Conflict of interest

The author declares that there is no conflict of interest.

References

  1. 1.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London (1961)zbMATHGoogle Scholar
  2. 2.
    Proctor, M.R.E., Weiss, N.O.: Magnetoconvection. Rep. Prog. Phys. 45, 1317–1379 (1982)zbMATHGoogle Scholar
  3. 3.
    Nakagawa, Y.: Experiments on the inhibition of thermal convection by a magnetic field. Proc. R. Soc. Lond. A 240, 108–113 (1957)Google Scholar
  4. 4.
    Nakagawa, Y.: Experiments on the instability of a layer of mercury heated from below and subject to the simultaneous action of a magnetic field and rotation. II. Proc. R. Soc. Lond. A 249, 138–145 (1959)Google Scholar
  5. 5.
    Busse, F.H., Clever, R.M.: Stability of convection rolls in the presence of a vertical magnetic field. Phys. Fluids 25, 931–935 (1982)zbMATHGoogle Scholar
  6. 6.
    Clever, R.M., Busse, F.H.: Nonlinear oscillatory convection in the presence of a vertical magnetic field. J. Fluid Mech. 201, 507–523 (1989)zbMATHGoogle Scholar
  7. 7.
    Cioni, S., Chaumat, S., Sommeria, J.: Effect of a vertical magnetic field on turbulent Rayleigh–Bénard convection. Phys. Rev. E 62, R4520–R4523 (2000)Google Scholar
  8. 8.
    Aurnou, J.M., Olson, P.L.: Experiments on Rayleigh–Bénard convection, magnetoconvection and rotating magnetoconvection in liquid gallium. J. Fluid Mech. 430, 283–307 (2001)zbMATHGoogle Scholar
  9. 9.
    Dawes, J.H.P.: Localized convection cells in the presence of a vertical magnetic field. J. Fluid Mech. 570, 385–406 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Podvigina, O.: Stability of rolls in rotating magnetoconvection in a layer with no-slip electrically insulating horizontal boundaries. Phys. Rev. E 81, 056322 (2010)MathSciNetGoogle Scholar
  11. 11.
    Basak, A., Raveendran, R., Kumar, K.: Rayleigh–Bénard convection with uniform vertical magnetic field. Phys. Rev. E 90, 033002 (2014)Google Scholar
  12. 12.
    Fauve, S., Laroche, C., Libchaber, A.: Effect of a horizontal magnetic field on convective instabilities in mercury. J. Phys. Lett. 42, L455–L457 (1981)Google Scholar
  13. 13.
    Fauve, S., Laroche, C., Libchaber, A., Perrin, B.: Chaotic phases and magnetic order in a convective fluid. Phys. Rev. Lett. 52, 1774–1777 (1984)Google Scholar
  14. 14.
    Meneguzzi, M., Sulem, C., Sulem, P.L., Thual, O.: Three-dimensional numerical simulation of convection in low-Prandtl-number fluids. J. Fluid Mech. 182, 169–191 (1987)zbMATHGoogle Scholar
  15. 15.
    Busse, F.H., Clever, R.M.: Traveling-wave convection in the presence of a horizontal magnetic field. Phys. Rev. A 40, 1954–1961 (1989)Google Scholar
  16. 16.
    Burr, U., Müller, U.: Rayleigh–Bénard convection in liquid metal layers under the influence of a horizontal magnetic field. J. Fluid Mech. 453, 345–369 (2002)zbMATHGoogle Scholar
  17. 17.
    Yanagisawa, T., Yamagishi, Y., Hamano, Y., Tasaka, Y., Takeda, Y.: Spontaneous flow reversals in Rayleigh–Bénard convection of a liquid metal. Phys. Rev. E 83, 036307 (2011)Google Scholar
  18. 18.
    Hurlburt, N.E., Matthews, P.C., Proctor, M.R.E.: Nonlinear compressible convection in oblique magnetic fields. Astrophys. J. 457, 933–938 (1996)Google Scholar
  19. 19.
    Julien, K., Knobloch, E., Tobias, S.M.: Nonlinear magnetoconvection in the presence of strong oblique fields. J. Fluid Mech. 410, 285–322 (2000)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Busse, F.H.: Generation of planetary magnetism by convection. Phys. Earth Planet. Inter. 12, 350–358 (1976)Google Scholar
  21. 21.
    Kuang, W., Bloxham, J.: An Earth-like numerical dynamo model. Nature 389, 371–374 (1997)Google Scholar
  22. 22.
    Glatzmaier, G.A., Roberts, P.H.: A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle. Phys. Earth Planet. Inter. 91, 63–75 (1995)Google Scholar
  23. 23.
    Glatzmaier, G.A.: Numerical simulations of stellar convective dynamos. I. The model and method. J. Comput. Phys. 55, 461–484 (1984)Google Scholar
  24. 24.
    Cattaneo, F.: On the origin of magnetic fields in the quiet photosphere. Astrophys. J. 515, L39–L42 (1999)Google Scholar
  25. 25.
    Schekochihin, A.A., Cowley, S.C., Taylor, S.F., Maron, J.L., McWilliams, J.C.: Simulations of the small-scale turbulent dynamo. Astrophys. J. 612, 276–307 (2004)Google Scholar
  26. 26.
    Braithwaite, J.: A differential rotation driven dynamo in a stably stratified star. Astron. Astrophys. 449, 451–460 (2006)zbMATHGoogle Scholar
  27. 27.
    Kim, D.H., Adornato, P.M., Brown, R.A.: Effect of vertical magnetic field on convection and segregation in vertical Bridgman crystal growth. J. Cryst. Growth 89, 339–356 (1988)Google Scholar
  28. 28.
    Ji, H., Terry, S., Yamada, M., Kulsrud, R., Kuritsyn, A., Ren, Y.: Electromagnetic fluctuations during fast reconnection in a laboratory plasma. Phys. Rev. Lett. 92, 115001 (2004)Google Scholar
  29. 29.
    Reimerdes, H., Chu, M.S., Garofalo, A.M., Jackson, G.L., La Haye, R.J., Navratil, G.A., Okabayashi, M., Scoville, J.T., Strait, E.J.: Measurement of the resistive-wall-mode stability in a rotating plasma using active MHD spectroscopy. Phys. Rev. Lett. 93, 135002 (2004)Google Scholar
  30. 30.
    Hadad, K., Rahimian, A., Nematollahi, M.R.: Numerical study of single and two-phase models of water/Al2O3 nanofluid turbulent forced convection flow in VVER-1000 nuclear reactor. Ann. Nucl. Energy 60, 287–294 (2013)Google Scholar
  31. 31.
    Pal, P., Kumar, K.: Role of uniform horizontal magnetic field on convective flow. Eur. Phys. J. B 85, 201 (2012)Google Scholar
  32. 32.
    Pal, P., Kumar, K., Maity, P., Dana, S.K.: Pattern dynamics near inverse homoclinic bifurcation in fluids. Phys. Rev. E 87, 023001 (2013)Google Scholar
  33. 33.
    Basak, A., Kumar, K.: A model for Rayleigh–Bénard magnetoconvection. Eur. Phys. J. B 88, 244 (2015)Google Scholar
  34. 34.
    Basak, A., Kumar, K.: Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid. Chaos 26, 123123 (2016)MathSciNetGoogle Scholar
  35. 35.
    Stribling, T., Matthaeus, W.H.: Relaxation processes in a low-order three-dimensional magnetohydrodynamics model. Phys. Fluids B 3, 1848–1864 (1991)Google Scholar
  36. 36.
    Ma, X., Karniadakis, G.E.: A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181–190 (2002)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Linz, S.J., Lücke, M.: Convection in binary mixtures: a Galerkin model with impermeable boundary conditions. Phys. Rev. A 35, 3997–4000 (1987)Google Scholar
  38. 38.
    Sobh, N., Huang, J., Yin, L., Haber, R.B., Tortorelli, D.A., Hyland Jr., R.W.: A discontinuous Galerkin model for precipitate nucleation and growth in aluminium alloy quench processes. Int. J. Numer. Methods Eng. 47, 749–767 (2000)zbMATHGoogle Scholar
  39. 39.
    Gloerfelt, X.: Compressible proper orthogonal decomposition/Galerkin reduced-order model of self-sustained oscillations in a cavity. Phys. Fluids 20, 115105 (2008)zbMATHGoogle Scholar
  40. 40.
    Rogers, J.M., McCulloch, A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41, 743–757 (1994)Google Scholar
  41. 41.
    Mynard, J.P., Nithiarasu, P.: A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method. Commun. Numer. Methods Eng. 24, 367–417 (2008)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Dao, T.-S., Vyasarayani, C.P., McPhee, J.: Simplification and order reduction of lithium-ion battery model based on porous-electrode theory. J. Power Sources 198, 329–337 (2012)Google Scholar
  43. 43.
    Nair, R.D., Thomas, S.J., Loft, R.D.: A discontinuous Galerkin global shallow water model. Mon. Weather Rev. 133, 876–888 (2004)Google Scholar
  44. 44.
    Tanaka, S., Bunya, S., Westerink, J.J., Dawson, C., Luettich Jr., R.A.: Scalability of an unstructured grid continuous Galerkin based hurricane storm surge model. J. Sci. Comput. 46, 329–358 (2011)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Noack, B.R., Niven, R.K.: Maximum-entropy closure for a Galerkin model of an incompressible periodic wake. J. Fluid Mech. 700, 187–213 (2012)zbMATHGoogle Scholar
  46. 46.
    Chen, Z., Dai, S.: Adaptive Galerkin methods with error control for a dynamical Ginzburg–Landau model in superconductivity. SIAM J. Numer. Anal. 38, 1961–1985 (2001)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Gresho, P.M., Sani, R.L.: The effects of gravity modulation on the stability of a heated fluid layer. J. Fluid Mech. 40, 783–806 (1970)zbMATHGoogle Scholar
  48. 48.
    Wadih, M., Roux, B.: Natural convection in a long vertical cylinder under gravity modulation. J. Fluid Mech. 193, 391–415 (1988)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Murray, B.T., Coriell, S.R., McFadden, G.B.: The effect of gravity modulation on solutal convection during directional solidification. J. Cryst. Growth 110, 713–723 (1991)Google Scholar
  50. 50.
    Wheeler, A.A., McFadden, G.B., Murray, B.T., Coriell, S.R.: Convective stability in the Rayleigh–Bénard and directional solidification problems: high-frequency gravity modulation. Phys. Fluids 3, 2847–2858 (1991)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Clever, R., Schubert, G., Busse, F.H.: Two-dimensional oscillatory convection in a gravitationally modulated fluid layer. J. Fluid Mech. 253, 663–680 (1993)zbMATHGoogle Scholar
  52. 52.
    Kumar, K., Tuckerman, L.S.: Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68 (1994)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Kumar, K.: Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 1113–1126 (1996)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Volmar, U.E., Müller, H.W.: Quasiperiodic patterns in Rayleigh–Bénard convection under gravity modulation. Phys. Rev. E 56, 5423–5430 (1997)Google Scholar
  55. 55.
    Christov, C.I., Homsy, G.M.: Nonlinear dynamics of two-dimensional convection in a vertically stratified slot with and without gravity modulation. J. Fluid Mech. 430, 335–360 (2001)zbMATHGoogle Scholar
  56. 56.
    Li, B.Q.: Stability of modulated-gravity-induced thermal convection in magnetic fields. Phys. Rev. E 63, 041508 (2001)Google Scholar
  57. 57.
    Venezian, G.: Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35, 243–254 (1969)zbMATHGoogle Scholar
  58. 58.
    Rosenblat, S., Tanaka, G.A.: Modulation of thermal convection instability. Phys. Fluids 14, 1319–1322 (1971)zbMATHGoogle Scholar
  59. 59.
    Ahlers, G., Hohenberg, P.C., Lücke, M.: Externally modulated Rayleigh–Bénard convection: experiment and theory. Phys. Rev. Lett. 53, 48–51 (1984)Google Scholar
  60. 60.
    Roppo, M.N., Davis, S.H., Rosenblat, S.: Bénard convection with time-periodic heating. Phys. Fluids 27, 796–803 (1984)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Bhadauria, B.S., Bhatia, P.K.: Time-periodic heating of Rayleigh–Bénard convection. Phys. Scr. 66, 59–65 (2002)zbMATHGoogle Scholar
  62. 62.
    Bhadauria, B.S.: Combined effect of temperature modulation and magnetic field on the onset of convection in an electrically conducting-fluid-saturated porous medium. J. Heat Transf. 130, 052601 (2008)Google Scholar
  63. 63.
    Singh, J., Bajaj, R.: Temperature modulation in ferrofluid convection. Phys. Fluids 21, 064105 (2009)zbMATHGoogle Scholar
  64. 64.
    Paul, S., Kumar, K.: Effect of magnetic field on parametrically driven surface waves. Proc. R. Soc. Lond. A 463, 711–722 (2006)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Yasir, M., Ahmad, S., Ahmed, F., Aqeel, M., Akbar, M.Z.: Improved numerical solutions for chaotic-cancer-model. AIP Adv. 7, 015110 (2017)Google Scholar
  66. 66.
    Aqeel, M., Ahmad, S.: Analytical and numerical study of Hopf bifurcation scenario for a three-dimensional chaotic system. Nonlinear Dyn. 84, 755–765 (2016)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Aqeel, M., Azam, A., Ahmad, S.: Control of chaos: Lie algebraic exact linearization approach for the Lü system. Eur. Phys. J. Plus 132, 426 (2017)Google Scholar
  68. 68.
    Dias, F.S., Mello, L.F.: Hopf bifurcations and small amplitude limit cycles in Rucklidge systems. Electron. J. Differ. Equ. 2013, 1–9 (2013)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Dias, F.S., Mello, L.F., Zhang, J.-G.: Nonlinear analysis in a Lorenz-like system. Nonlinear Anal. Real World Appl. 11, 3491–3500 (2010)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Messias, M., de Carvalho Braga, D., Mello, L.F.: Degenerate Hopf bifurcations in Chua’s system. Int. J. Bifurc. Chaos 19, 497–515 (2009)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Marques, F., Lopez, J.M.: Taylor–Couette flow with axial oscillations of the inner cylinder: Floquet analysis of the basic flow. J. Fluid Mech. 348, 153–175 (1997)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Brambilla, A., Gruosso, G., Gajani, G.S.: Determination of Floquet exponents for small-signal analysis of nonlinear periodic circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 28, 447–451 (2009)Google Scholar
  73. 73.
    Watanabe, G., Mäkelä, H.: Floquet analysis of the modulated two-mode Bose–Hubbard model. Phys. Rev. A 85, 053624 (2012)Google Scholar
  74. 74.
    Shin, J.-Y., Lee, H.-W.: Floquet analysis of quantum resonance in a driven nonlinear system. Phys. Rev. E 50, 902–909 (1994)Google Scholar
  75. 75.
    Lundh, E.: Directed transport and Floquet analysis for a periodically kicked wave packet at a quantum resonance. Phys. Rev. E 74, 016212 (2006)Google Scholar
  76. 76.
    Staliunas, K., Longhi, S., de Valcárcel, G.J.: Faraday patterns in Bose–Einstein condensates. Phys. Rev. Lett. 89, 210406 (2002)Google Scholar
  77. 77.
    Carias, H., Beratan, D.N., Skourtis, S.S.: Floquet analysis for vibronically modulated electron tunneling. J. Phys. Chem. B 115, 5510–5518 (2011)Google Scholar
  78. 78.
    Luter, R., Reichl, L.E.: Floquet analysis of atom-optics tunneling experiments. Phys. Rev. A 66, 053615 (2002)Google Scholar
  79. 79.
    Tanner, J.J., Maricq, M.M.: Floquet analysis of the far-infrared dissociation of a Morse oscillator. Phys. Rev. A 40, 4054–4064 (1989)Google Scholar
  80. 80.
    Guérin, S.: Complete dissociation by chirped laser pulses designed by adiabatic Floquet analysis. Phys. Rev. A 56, 1458–1462 (1997)Google Scholar
  81. 81.
    Cavagnero, M.J.: Floquet analysis of inelastic collisions of ions with Rydberg atoms. Phys. Rev. A 52, 2865–2875 (1995)Google Scholar
  82. 82.
    Safaenili, A., Chimenti, D.E., Auld, B.A., Datta, S.K.: Floquet analysis of guided waves propagating in periodically layered composites. Compos. Eng. 5, 1471–1476 (1995)Google Scholar
  83. 83.
    Skjoldan, P.F., Hansen, M.H.: Implicit Floquet analysis of wind turbines using tangent matrices of a non-linear aeroelastic code. Wind Energy 15, 275–287 (2012)Google Scholar
  84. 84.
    Lee, B., Liu, J.Z., Sun, B., Shen, C.Y., Dai, G.C.: Thermally conductive and electrically insulating EVA composite encapsulants for solar photovoltaic (PV) cell. Express Polym. Lett. 2, 357–363 (2008)Google Scholar
  85. 85.
    Thual, O.: Zero-Prandtl-number convection. J. Fluid Mech. 240, 229–258 (1992)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Center of Excellence in Space Sciences IndiaIndian Institute of Science Education and Research KolkataMohanpurIndia

Personalised recommendations