Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-31T03:01:27.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic bifurcations and pattern formation in melting-boundary convection

Published online by Cambridge University Press:  23 September 2011

G. M. Vasil  and
M. R. E. Proctor
G. M. Vasil*
Affiliation:
Canadian Institute for Theoretical Astrophysics, 60 St George Street, Toronto, ON M5S 3H8, Canada JILA, University of Colorado, Boulder, CO 80309-0440, USA
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
*
†Email address for correspondence: vasil@cita.utoronto.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider weakly nonlinear convection in a fluid layer with a melting top boundary. This leads us to derive a new set of non-autonomous envelope equations as a dynamic generalization to the well-known Ginzburg–Landau equation. However, this new system possesses a number of interesting properties not found in systems close to a traditional dynamic bifurcation, because it involves the interaction of two destabilizing mechanisms. We investigate the system both analytically and numerically; specifically, we find the robust ‘locking in’ of spatially complex patterns, and show this is a general feature of systems of this nature.

JFM classification

Convection: Buoyancy-driven instability Phase change: Morphological instability Phase change: Solidification/melting

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 686 , 10 November 2011 , pp. 77 - 108
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Benoît, E. 1990 Dynamic Bifurcations (ed. Benoît, E. ). Springer.Google Scholar
2. Busse, F. H. 1967 Non-stationary finite amplitude convection. J. Fluid Mech. 28, 223–239.CrossRefGoogle Scholar
3. Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. International Series of Monographs on Physics , Oxford.Google Scholar
4. Cox, S. M. & Matthews, P. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455.CrossRefGoogle Scholar
5. Cross, M. C. 1980 Derivation of the amplitude equations at the Rayleigh–Bénard instability. Phys. Fluids 23, 1727–1731.CrossRefGoogle Scholar
6. Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1123.CrossRefGoogle Scholar
7. Davis, S. H., Müller, U. & Dietsche, C. 1984 Pattern selection in single-component systems coupling Bénard convection and solidification. J. Fluid Mech. 144, 133–151.CrossRefGoogle Scholar
8. Foster, T. D. 1965 Stability of a homogenous fluid cooled uniformly from above. Phys. Fluids 8, 1249–1257.CrossRefGoogle Scholar
9. Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh–Bénard convection. Physica D 10, 249–276.CrossRefGoogle Scholar
10. Hoyle, R. B. 2006 Pattern Formation. Cambridge University Press.CrossRefGoogle Scholar
11. Huppert, H. E. 2000 Geological fluid mechanics. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G. ), pp. 447–495. Cambridge University Press.Google Scholar
12. Kuznetsov, E. A., Nepomnyashchy, A. A. & Pismen, L. M. 1995 New amplitude equation for Boussinesq convection and nonequilateral hexagonal patterns. Phys. Lett. A 205, 261–265.CrossRefGoogle Scholar
13. Landau, L. D. & Lifshitz, E. M. 1995 Fluid Mechanics. Pergamon.Google Scholar
14. Liu, Y., Ning, L. & Ecke, R. E. 1996 Dynamics of surface patterning in salt-crystal dissolution. Phys. Rev. E 53, R4271–R4274.CrossRefGoogle ScholarPubMed
15. Lythe, G. D. 1996 Domain formation in transitions with noise and a time-dependent bifurcation parameter. Phys. Rev. E 53, R5572–R5575.CrossRefGoogle Scholar
16. Majdalani, J. 1998 A hybrid multiple scale procedure for boundary layers involving several dissimilar scales. Z. Angew. Math. Phys. 49, 849–868.CrossRefGoogle Scholar
17. Mandel, P. & Erneux, T. 1987 The slow passage through a steady bifurcation: delay and memory effects. J. Stat. Phys. 48, 1059–1070.CrossRefGoogle Scholar
18. Moore, D. R. & Weiss, N. O. 1973 Nonlinear penetrative convection. J. Fluid Mech. 61, 563–581.CrossRefGoogle Scholar
19. Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303.CrossRefGoogle Scholar
20. Schlüter, A., Lortz, D. & Busse, F. H. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.CrossRefGoogle Scholar
21. Segel, L. A. 1969 Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224.CrossRefGoogle Scholar
22. Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 13, 442–447.CrossRefGoogle Scholar
23. Sullivan, T. S., Liu, Y. & Ecke, R. E. 1996 Turbulent solutal convection and surface patterning in solid dissolution. Phys. Rev. E 54, 486–495.CrossRefGoogle ScholarPubMed
24. Woods, A. W. 1992 Melting and dissolving. J. Fluid Mech. 239, 429–448.CrossRefGoogle Scholar
25. Worster, M. G. 2000 Solidification of fluids. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G. ), pp. 393–446. Cambridge University Press.Google Scholar