×

zbMATH — the first resource for mathematics

Dynamical transitions of a low-dimensional model for Rayleigh-Bénard convection under a vertical magnetic field. (English) Zbl 1415.34073
Summary: In this article, we study the dynamic transitions of a low-dimensional dynamical system for the Rayleigh-Bénard convection subject to a vertically applied magnetic field. Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. We find that, as the Rayleigh number increases, the system undergoes two successive transitions: the first one is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. More precisely, for large magnetic field, the second transition is of continuous type and gives to a stable limit cycle; on the other hand, for low magnetic field or small height-to-width aspect ratio, a jump transition occurs where an unstable periodic orbit eventually collides with the stable steady state, leading to the loss of stability at the critical Rayleigh number. Finally, numerical results are presented to corroborate the analytic predictions.

MSC:
34C23 Bifurcation theory for ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
35Q35 PDEs in connection with fluid mechanics
76E06 Convection in hydrodynamic stability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kadanoff, L. P., Turbulent heat flow: structures and scaling, Phys Today, 54, 8, 34-39, (2001)
[2] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, (2013), Courier Corporation · Zbl 0142.44103
[3] Drazin, P. G.; Reid, W. H., Hydrodynamic stability, (2004), Cambridge University Press · Zbl 1055.76001
[4] Kirchgässner, K., Bifurcation in nonlinear hydrodynamic stability, SIAM Rev, 17, 4, 652-683, (1975) · Zbl 0328.76035
[5] Rabinowitz, P., Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch Ration Mech Anal, 29, 1, 32-57, (1968) · Zbl 0164.28704
[6] Iudovich, V., Free convection and bifurcation, J Appl Math Mech, 31, 1, 103-114, (1967)
[7] Ma, T.; Wang, S., Dynamic bifurcation and stability in the Rayleigh-Bénard convection, Commun Math Sci, 2, 2, 159-183, (2004) · Zbl 1133.76315
[8] Ma, T.; Wang, S., Attractor bifurcation theory and its applications to Rayleigh-Bénard convection, Commun Pure Appl Anal, 2, 4, 591-599, (2003) · Zbl 1210.37056
[9] Ma, T.; Wang, S., Rayleigh Bénard convection: dynamics and structure in the physical space, Commun Math Sci, 5, 3, 553-574, (2007) · Zbl 1133.35426
[10] Sengul T., Wang S.. Pattern formation in Rayleigh Benard convection. arXiv preprint 2011; arXiv:11095655. · Zbl 1291.35235
[11] Ma, T.; Wang, S., Geometric theory of incompressible flows with applications to fluid dynamics, (2005), American Mathematical Soc. · Zbl 1099.76002
[12] Ma, T.; Wang, S., Bifurcation theory and applications, 53, (2005), World Scientific
[13] Proctor, M.; Weiss, N., Magnetoconvection, Rep Prog Phys, 45, 11, 1317, (1982) · Zbl 1331.85001
[14] Andreev, O.; Thess, A.; Haberstroh, C., Visualization of magnetoconvection, Phys Fluids, 15, 12, 3886-3889, (2003) · Zbl 1186.76027
[15] Basak, A.; Raveendran, R.; Kumar, K., Rayleigh-Bénard convection with uniform vertical magnetic field, Phys Rev E, 90, 3, 033002, (2014)
[16] Dawes, J., Localized convection cells in the presence of a vertical magnetic field, J Fluid Mech, 570, 385-406, (2007) · Zbl 1105.76026
[17] Wang, Q., Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders, Discrete Contin Dyn Syst-B, 19, 2, 543-563, (2014) · Zbl 1292.76078
[18] Pal, P.; Kumar, K., Role of uniform horizontal magnetic field on convective flow, Eur Phys J B, 85, 6, 201, (2012)
[19] Yadav, D.; Bhargava, R.; Agrawal, G., Thermal instability in a nanofluid layer with a vertical magnetic field, J Eng Math, 80, 1, 147-164, (2013) · Zbl 1367.76059
[20] Yadav, D.; Kim, C.; Lee, J.; Cho, H. H., Influence of magnetic field on the onset of nanofluid convection induced by purely internal heating, Comput Fluids, 121, 26-36, (2015) · Zbl 1390.76900
[21] Yadav, D.; Lee, J., The onset of MHD nanofluid convection with Hall current effect, Eur Phys J Plus, 130, 8, 162, (2015)
[22] Yadav, D.; Mohamed, R.; Cho, H. H.; Lee, J., Effect of Hall current on the onset of MHD convection in a porous medium layer saturated by a nanofluid, J Appl Fluid Mech, 9, 5, (2016)
[23] Yadav, D.; Wang, J.; Bhargava, R.; Lee, J.; Cho, H. H., Numerical investigation of the effect of magnetic field on the onset of nanofluid convection, Appl Therm Eng, 103, 1441-1449, (2016)
[24] Yanagisawa, T.; Yamagishi, Y.; Hamano, Y.; Tasaka, Y.; Yano, K.; Takahashi, J., Detailed investigation of thermal convection in a liquid metal under a horizontal magnetic field: suppression of oscillatory flow observed by velocity profiles, Phys Rev E, 82, 5, 056306, (2010)
[25] Lorenz, E. N., Deterministic nonperiodic flow, J Atmos Sci, 20, 2, 130-141, (1963) · Zbl 1417.37129
[26] Gotoda, H.; Takeuchi, R.; Okuno, Y.; Miyano, T., Low-dimensional dynamical system for Rayleigh-benard convection subjected to magnetic field, J Appl Phys, 113, 12, 124902, (2013)
[27] Ma, T.; Wang, S., Phase transition dynamics, (2016), Springer
[28] Dijkstra, H.; Sengul, T.; Shen, J.; Wang, S., Dynamic transitions of quasi-geostrophic channel flow, SIAM J Appl Math, 75, 5, 2361-2378, (2015) · Zbl 1329.35232
[29] Hsia, C.-H.; Lin, C.-S.; Ma, T.; Wang, S., Tropical atmospheric circulations with humidity effects, Proc R Soc A, 471, 2173, 20140353, (2015) · Zbl 1371.86020
[30] Kaper, H. G.; Wang, S.; Yari, M., Dynamical transitions of Turing patterns, Nonlinearity, 22, 3, 601, (2009) · Zbl 1157.92003
[31] Liu, H.; Sengul, T.; Wang, S., Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility, J Math Phys, 53, 2, 023518, (2012) · Zbl 1274.82039
[32] Sengul, T.; Shen, J.; Wang, S., Pattern formations of 2d Rayleigh-Bénard convection with no-slip boundary conditions for the velocity at the critical length scales, Math Methods Appl Sci, 38, 17, 3792-3806, (2015) · Zbl 1333.76036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.