Hostname: page-component-7b9c58cd5d-g9frx Total loading time: 0 Render date: 2025-03-15T19:34:25.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Pattern selection in single-component systems coupling Bénard convection and solidification

Published online by Cambridge University Press:  20 April 2006

S. H. Davis ,
U. Müller  and
C. Dietsche
S. H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60201, USA
U. Müller
Affiliation:
Kernforschungszentrum Karlsruhe GMBH, Institüt fur Reaktorbauelemente, Karlsruhe, West Germany
C. Dietsche
Affiliation:
Kernforschungszentrum Karlsruhe GMBH, Institüt fur Reaktorbauelemente, Karlsruhe, West Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

A horizontal layer is heated from below and cooled from above so that the enclosed single-component liquid is frozen in the upper part of the layer. When the imposed temperature difference is such that the Rayleigh number across the liquid is supercritical, there is Bénard convection coupled with the dynamics of the solidification interface. An experiment is presented which shows that the interfacial corrugations that result are two-dimensional when this solid is thin but hexagonal when the solid is thick. A weakly nonlinear convective instability theory is presented which explains this behaviour, and isolates this ‘purely thermal’ mechanism of pattern selection. Jump behaviour is seen in the liquid-layer thickness at the onset of hexagonal convection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 144 , July 1984 , pp. 133 - 151
Copyright
© 1984 Cambridge University Press

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.)

References

Busse, F. 1967 The stability of finite amplitude convection and its relation to an extreme principle. J. Fluid Mech. 30, 625–649.Google Scholar
Coriell, S. R., Cordes, M. R., Boettinger, W. J. & Sekerka, R. F. 1980 Convective and interfacial instabilities during unidirectional solidification of a binary alloy. J. Crystal Growth 49, 13–28.Google Scholar
Coriell, S. R. & Sekerka, R. F. 1982 Effect of convective flow on morphological stability. Physico-Chem. Hydrodyn. 2, 281–293.Google Scholar
Davis, S. H. & Segel, L. A. 1968 Effects of surface curvature and property variation on cellular convection. Phys. Fluids 11, 470–476.Google Scholar
Farhadieh, R. & Tankin, R. S. 1975 A study of the freezing of sea water. J. Fluid Mech. 71, 293–304.Google Scholar
Fischer, K. M. 1981 The effects of fluid flow on the solidification of industrial castings and ingots. Physico-Chem. Hydrodyn. 2, 311–326.Google Scholar
Foster, T. D. 1969 Experiments on haline convection induced by the freezing of sea water. J. Geophys. Res. 74, 6967–6974.Google Scholar
Hurle, D. T. J. & Jakeman, E. 1981 Introduction to the techniques of crystal growth. Physico-Chem. Hydrodyn. 2, 237–244.Google Scholar
Hurle, D. T. J., Jakeman, E. & Wheeler, A. A. 1982 Effect of solutal convection on the morphological stability of a binary alloy. J. Crystal Growth 58, 163–179.Google Scholar
Hurle, D. T. J., Jakeman, E. & Wheeler, A. A. 1983 Hydrodynamic stability of the melt during solidification of a binary alloy. Phys. Fluids 23, 624–626.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.Google Scholar
Marshall, R. 1981 Experimental experience with the ASHRAE/NBS procedures for testing a phase change thermal storage device. In Proc. Int. Conf. on Energy Storage, Brighton, vol. 1, pp. 129–143.
Mihaljan, J. M. 1962 A rigorous exposition of the Boussinesq approximation applicable to a thin layer of fluid. Astrophys. J. 136, 1126–1133.Google Scholar
Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35, 444–451.Google Scholar
Nield, D. A. 1968 The Rayleigh—Jeffreys problem with boundary slab of finite conductivity. J. Fluid Mech. 32, 393–398.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Pantaloni, J., Velarde, M. G., Bailleux, R. & Guyon, E. 1977 Sur la convection cellulaire dans les sels fondus pres de leur point de solidification. C.R. Acad. Sci. Paris B285, 275–278.Google Scholar
Saitoh, T. & Hirose, K. 1980 Thermal instability of natural convection flow over a horizontal ice cylinder encompassing a maximum density point. Trans. ASME C: J. Heat Transfer 102, 261–266.Google Scholar
Saitoh, T. & Hirose, K. 1982 High Rayleigh number solutions to problems of latent heat thermal energy storage in a horizontal cylinder capsule. Trans. ASME C: J. Heat Transfer 104, 545–553.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Segel, L. A. 1965 The nonlinear interaction of a finite number of disturbances to a layer of fluid heated from below. J. Fluid Mech. 21, 359–384.Google Scholar
Segel, L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13, 289–306.Google Scholar
Seki, N., Fukusako, S. & Sugawara, M. 1977 A criterion of onset of free convection in a horizontal melted water layer with a free surface. Trans ASME C: J. Heat Transfer 99, 92–98.Google Scholar
Iranganathan, R., Wollkind, D. J. & Oulton, D. B. 1983 A theoretical investigation of the development of interfacial cells during the solidification of a dilute binary alloy: comparison with the experiments of Morris and Winegard. J. Crystal Growth 62, 265–283.Google Scholar
Wollkind, D. J. & Raissi, S. 1974 A nonlinear stability analysis of the melting of a dilute binary alloy. J. Crystal Growth 26, 277–293.Google Scholar
Wollkind, D. J. & Segel, L. A. 1970 A nonlinear stability analysis of the freezing of a dilute binary alloy. Phil. Trans. R. Soc. Lond. A 268, 351–380.Google Scholar
Yen, Y.-C. 1968 Onset of convection in a layer of water formed by melting ice from below. Phys. Fluids 11, 1263–1270.Google Scholar
Yen, Y.-C. 1980 Free convection heat transfer characteristics in a melt water layer. Trans. ASME C: J. Heat Transfer 102, 550–556.Google Scholar