Abstract
The pure conduction state of a horizontal layer of fluid heated from below becomes unstable with respect to a convecting state when the temperature difference exceeds a critical value. We examine the question of how real, physical systems evolve from conduction to convection. Most experimental cells contain geometric or thermal inhomogeneities which render the bifurcation to convection imperfect. In that case the pure conduction state never exists and the convecting state evolves continuously and smoothly as the temperature difference is raised. When a sufficiently perfect experimental cell is constructed to eliminate this route to convection, then dynamic imperfections will usually prevail. When the temperature difference across the cell is raised, the vertical gradients in the sidewalls evolve at a rate which differs from that in the fluid. The resultingtransient horizontal thermal gradients initiate the convective flow. This phenomenon can be eliminated by providing sidewalls which have the same thermal diffusivity as that of the fluid. When that is done, the convective flow is started by random noise which exists in any experimental system. Analysis of experiments shows that the noise source is considerably stronger than thermal noise, but its origin is unclear at this time.
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References
S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability (Oxford University Press, 1961).
R. E. Kelly and D. Pal,J. Fluid Mech. 86:433 (1978); P. G. Daniels,Proc. R. Soc. A 358:173 (1977); P. Hall and I. C. Walton,Proc. R. Soc. A 358:199 (1977); E. L. Reiss, inApplication of Bifurcation Theory (Academic Press, 1977), p. 37; S. N. Brown and K. Stewartson,Proc. R. Soc. Lond. A 360:455 (1978); J. Tavantzis, E. L. Reiss, and B. Matkowsky,SIAM J. Appl. Math. 34:322 (1978).
E. L. Koschmieder and S. G. Pallas,Int. J. Heat Mass Transfer 17:991 (1974).
V. Steinberg, G. Ahlers, and D. S. Cannell,Phys. Scripta 32:534 (1985);T9:97 (1985).
G. Ahlers, D. S. Cannell, and C. Meyer, unpublished; G. Ahlers, D. S. Cannell, and V. Steinberg,Phys. Rev. Lett. 54:1373 (1985).
G. Ahlers, inFluctuations, Instabilities, and Phase Transitions, T. Riste, ed. (Plenum, New York, 1975);J. Fluid Mech. 98:137 (1980).
A. Aitta, G. Ahlers, and D. D. Cannell,Phys. Rev. Lett. 54:673 (1985).
R. P. Behringer and G. Ahlers,J. Fluid Mech. 125:219 (1982).
G. Ahlers, M. C. Cross, P. C. Hohenberg, and S. Safran,J. Fluid Mech. 110:297 (1981).
G. Ahlers, P. C. Hohenberg, and M. Lücke,Phys. Rev. Lett. 53:48 (1984);Phys. Rev. A 32:3493, 4519 (1985).
M. C. Cross, P. C. Hohenberg, and M. C. Lücke,J. Fluid Mech. 136:269 (1983).
M. S. Heutmaker, P. N. Fraenkel, and J. P. Gollub,Phys. Rev. Lett. 54:1369 (1985); and references therein.
M. C. Cross,Phys. Rev. A 25:1065 (1982); and references therein.
C. W. Meyer, G. Ahlers, and D. S. Cannell,Phys. Rev. Lett. 59:1577 (1987).
V. M. Zaitsev and M. I. Shliomis,Zh. Eksp. Teor. Fiz. 59:1583 (1970) [Sov. Phys. JETP 32:866 (1971)]; R. Graham,Phys. Rev. A 10:1762 (1974).
J. B. Swift and P. C. Hohenberg,Phys. Rev. A 15:319 (1977).
H. van Beijeren and E. G. D. Cohen,Phys. Rev. Lett. 60:1208 (1988).
J. B. Swift and P. C. Hohenberg,Phys. Rev. Lett. 60:75 (1988).
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Ahlers, G., Meyer, C.W. & Cannell, D.S. Deterministic and stochastic effects near the convective onset. J Stat Phys 54, 1121–1131 (1989). https://doi.org/10.1007/BF01044706
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DOI: https://doi.org/10.1007/BF01044706