Nonlinear Pattern Formation near the Onset of Rayleigh-Bénard Convection

Greenside, Henry; Coughran, W.M.; Schryer, N. L.

doi:10.1103/physrevlett.49.726

Cited by 92 publications

(41 citation statements)

References 14 publications

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“…An alternative approach has used models of the characterizing field [49], such as the Swift-Hohenberg equation; however, this approach cannot describe individual bifurcations such as the Turing bifurcation. In Sec.…”

Section: Multiple Scale Analysismentioning

confidence: 99%

Pattern formation in the presence of symmetries

Gunaratne

Ouyang

Swinney³

1994

Phys. Rev. E

191

142

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We present a detailed theoretical study of pattern formation in planar continua with translational, rotational, and reflection symmetry. The theoretical predictions are tested in experiments on a quasitwo-dimensional reaction-diffusion system. Spatial patterns form in a chlorite-iodide-malonic acid reaction in a thin gel layer reactor that is sandwiched between two continuously refreshed reservoirs of reagents; thus, the system can be maintained inde6nitely in a well-defined nonequilibrium state. This physical system satis6es, to a very good approximation, the Euclidean symmetries assumed in the theory. The theoretical analysis, developed in the amplitude equation formalism, is a spatiotemporal extension of the normal form. The analysis is identical to the Newell-Whitehead-Segel theory [J. Fluid Mech. 38, 203 (1969); 38, 279 (1969)] at the lowest order in perturbation, but has the advantage that it exactly preserves the Euclidean symmetries of the physical system. Our equations can be derived by a suitable modification of the perturbation expansion, as shown for two variations of the Swift-Hohenberg equation [Phys. Rev. A 15, 319 (1977)]. Our analysis is complementary to the Cross-Newell approach [Physica D 10, 299 (1984)] to the study of pattern formation and is equivalent to it in the common domain of applicability. Our analysis yields a rotationally invariant generalization of the phase equation of Pomeau and Manneville [J. Phys. Lett. 40, L609 (1979)]. The theory predicts the existence of stable rhombic arrays with qualitative details that should be system independent. Our experiments in the reaction-diffusion system yield patterns in good accord with the predictions. Finally, we consider consequences of resonances between the basic modes of a hexagonal pattern and compare the results of the analysis with experiments.

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Section: Multiple Scale Analysismentioning

confidence: 99%

Pattern formation in the presence of symmetries

Gunaratne

Ouyang

Swinney³

1994

Phys. Rev. E

191

142

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“…Recent work has been done by Swift and Hohenberg (1977), Wesfreid et al (1978), Cross (1980Cross ( ,1982, Cross et aL (1980Cross et aL ( ,1983, Pomeau and Manneville (1981), Pomeau and Zaleski (1981), , Greenside et al (1982), and Greenside and Coughran (1983). Here, the idea originated by Newell and Whitehead (1969) and Segel (1969), is that v and 6 just above R c will have a spatial dependence which strongly resembles the critical mode.…”

Section: A Simple Examplementioning

confidence: 99%

Rayleigh-Bénard convection and turbulence in liquid helium

Behringer

1985

Rev. Mod. Phys.

139

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Recent advances in the understanding of Rayleigh-Benard convection and turbulence are reviewed in light of work using liquid helium. The discussion includes both experiments which have probed the steady flows preceding time dependence and experiments which have been directed toward understanding the ways in which turbulence evolves. Comparison is made where appropriate to the many important contributions which have been obtained using room-temperature fluids, and a discussion is given explaining the advantages of cryogenic techniques. Brief reviews are given for recent experimental investigations of convection in 3 He-4 He mixtures-in both the superfluid and the normal states-and investigations of convection in rotating layers of liquid helium. C. Superfluid 3 He-4 He mixtures D. Convection with rotation VI. Summary and Future Directions Acknowledgments References

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“…Further details of the bifurcation structure and Eckhaus instability mechanism were discussed by Tsiveriotis and Brown [8], whilst Hernandez-Garcia et al [9] considered the influence of noise on pattern selection. A comprehensive numerical study of the two-dimensional Swift-Hohenberg equation was reported by Greenside et al [10] and Greenside and Coughran [11], and Kuwamura [12] considered the stability of roll solutions of the two-dimensional equation using a spectral analysis.…”

Section: Introductionmentioning

confidence: 99%