Abstract
The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton’s law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually defined is shown to be inappropriate as a bifurcation parameter since the temperature across the layer depends on the amplitude of convection and hence changes as convection evolves at fixed external parameter values. Moreover, the final state of the system is also different since it depends on the details of the applied boundary conditions. A modified Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one.
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References
C.M. Surko, P. Kolodner, A. Passner and R.W. WaldenFinite-amplitude traveling-waveconvection in binary fluid mixturesPhysica D, 23 (1986), 220–229.
T.S. Sullivan and G. AhlersHopf bifurcation to convection near the codimension-two point in a 3 He- 4 He mixturePhys. Rev. Lett., 61 (1988), 78–81.
V. Steinberg, J. Fineberg, E. Moses and I. RehbergPattern selection and transition to turbulence in propagating wavesPhysica D, 37 (1989), 359–383.
M.R.E. ProctorPlanform selection by finite-amplitude thermal convection between poorly conducting slabsJ. Fluid Mech., 113 (1981), 469–485.
A. Recktenwald and M. LückeThermoconvection in magnetized ferrofluids: the influence of boundaries with finite heat conductivityJ. Magn. Magn. Mater., 188 (1998), 326–332.
I. Mercader, J. Prat and E. KnoblochRobust heteroclinic cycles in two-dimensional Rayleigh-Bénard convection without Boussinesq symmetry.Preprint (2001).
T.L. Clune, Ph.D Thesis, University of California at Berkeley (1993).
J. Prat, J.M. Massaguer and I. MercaderLarge-scale flows and resonances in 2-D thermal convectionPhys. Fluids, 7 (1995), 121–134.
J. Prat, I. Mercader and E. KnoblochResonant mode interaction in Rayleigh-Bénard convectionPhys. Rev. E, 58 (1998), 3145–3156.
I. Mercader, J. Prat and E. KnoblochThe 1:2 mode interaction in Rayleigh-Bénard convection with weakly broken midplane symmetryInt. J. Bif. Chaos, 11 (2001), 27–41.
J. Prat, I. Mercader and E. KnoblochThe 1:2 mode interaction in Rayleigh-Bénard convection with and without Boussinesq symmetryInt. J. Bif. Chaos, in press.
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Prat, J., Mercader, I., Knobloch, E. (2003). Rayleigh-Bénard Convection with Experimental Boundary Conditions. In: Buescu, J., Castro, S.B.S.D., da Silva Dias, A.P., Labouriau, I.S. (eds) Bifurcation, Symmetry and Patterns. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7982-8_17
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DOI: https://doi.org/10.1007/978-3-0348-7982-8_17
Publisher Name: Birkhäuser, Basel
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