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On exchanges between convective modes in a slightly tilted porous cavity

Published online by Cambridge University Press:  24 October 2008

M. D. Impey  and
D. S. Riley
M. D. Impey
Affiliation:
School of Mathematics, University of Bristol
D. S. Riley
Affiliation:
School of Mathematics, University of Bristol
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Abstract

We use Liapunov–Schmidt reduction and singularity-theory methods to investigate the bifurcation structure of steady free convection in a finite two-dimensional saturated porous cavity heated from below. In particular, we develop a qualitative model describing the modal exchanges that occur as the aspect ratio of a slightly tilted cavity varies. ℤ2-symmetry breaking bifurcations are involved in these exchanges and the mechanism is significantly different from those in previously studied physical systems. The work is akin to, but distinct from, that of Schaeffer's on the Taylor–Couette problem. We also derive and clearly state conditions for the application of this model to modal exchange in other systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 2 , September 1991 , pp. 395 - 416
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Riley, D. S. and Winters, K. H.. The onset of convection in a porous medium: a preliminary study. Harwell Report R.12586 (1987).Google Scholar
[2]Riley, D. S. and Winters, K. H.. Modal exchange mechanisms in Lapwood convection. J. Fluid Mech. 204 (1989), 325–358.CrossRefGoogle Scholar
[3]Riley, D. S. and Winters, K. H.. A numerical bifurcation study of natural convection in a tilted two-dimensional porous cavity. J. Fluid Mech. 215 (1990), 309–329.CrossRefGoogle Scholar
[4]Benjamin, T. B.. Bifurcation phenomena in steady flows in a viscous fluid. Proc. Roy. Soc. London Ser. A 359 (1978), 1–26, 27–43.Google Scholar
[5]Schaeffer, D. G.. Qualitative analysis of a model for boundary effects in the Taylor problem. Proc. Cambridge Philos. Soc. 87 (1980), 307–337.CrossRefGoogle Scholar
[6]Impey, M. D., Riley, D. S. and Winters, K. H.. The effect of sidewall imperfections on pattern formation in Lapwood convection. Nonlinearity 3 (1990), 197–230.CrossRefGoogle Scholar
[7]Kidachi, H.. Side wall effects on the pattern formation of the Rayleigh-Bénard convection. Prog. Theoret. Phys. 68 (1982), 49–63.CrossRefGoogle Scholar
[8]Metzener, P.. The effect of rigid sidewalls on nonlinear two-dimensional Bénard convection. Phys. Fluids 29 (1986), 1373–1377.CrossRefGoogle Scholar
[9]Daniels, P. G.. The onset of Bénard convection in a shallow sloping container. Quart. J. Mech. Appl. Math. 35 (1982), 49–67.CrossRefGoogle Scholar
[10]Golubitsky, M. and Schaeffer, D. G.. A theory for imperfect bifurcation via singularity theory. Gomm. Pure App. Math. 32 (1979), 21–98.CrossRefGoogle Scholar
[11]Golubitsky, M. and Schaeffer, D. G.. Imperfect bifurcation in the presence of symmetry. Comm. Math. Phys. 67 (1979), 205–232.CrossRefGoogle Scholar
[12]Golubitsky, M. and Schaeffer, D. G.. Singularities and Groups in Bifurcation Theory, vol. 1 (Springer-Verlag, 1984).Google Scholar
[13]Sutton, F.. Onset of convection in a porous channel with net through flow. Phys. Fluids 13 (1970), 1931–1934.CrossRefGoogle Scholar
[14]Knobloch, E. and Guckenheimer, J.. Convective transitions induced by a varying aspect ratio. Phys. Rev. A 27 (1983), 408–417.CrossRefGoogle Scholar
[15]Impey, M. D.. Bifurcation in Lapwood convection. Ph.D. thesis, University of Bristol (1988).Google Scholar
[16]Impey, M. D.. A qualitative model for the effect of sidewall imperfections on modal exchanges in Lapwood convection. Submitted (1991).Google Scholar