Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-01T17:27:46.093Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure of cellular solutions in Rayleigh–Bénard–Marangoni flows in small-aspect-ratio containers

Published online by Cambridge University Press:  26 April 2006

Henk A. Dijkstra
Henk A. Dijkstra
Affiliation:
Mathematical Sciences Institute, Cornell University, Ithaca, NY 14853, USA Present address: Institute of Meteorology and Oceanography, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

Multiple steady flow patterns occur in surface-tension/buoyancy-driven convection in a liquid layer heated from below (Rayleigh–Bénard–Marangoni flows). Techniques of numerical bifurcation theory are used to study the multiplicity and stability of two-dimensional steady flow patterns (rolls) in rectangular small-aspect-ratio containers as the aspect ratio is varied. For pure Marangoni flows at moderate Biot and Prandtl number, the transitions occurring when paths of codimension 1 singularities intersect determine to a large extent the multiplicity of stable patterns. These transitions also lead, for example, to Hopf bifurcations and stable periodic flows for a small range in aspect ratio. The influence of the type of lateral walls on the multiplicity of steady states is considered. ‘No-slip’ lateral walls lead to hysteresis effects and typically restrict the number of stable flow patterns (with respect to ‘slippery’ sidewalls) through the occurrence of saddle node bifurcations. In this way ‘no-slip’ sidewalls induce a selection of certain patterns, which typically have the largest Nusselt number, through secondary bifurcation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 73 - 102
Copyright
© 1992 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

Ahlers, G., Cannell, D. S. & Steinberg, V. 1985 Time dependence of flow patterns near the convective threshold in a cylindrical container. Phys. Rev. Lett. 54, 1373–1375.Google Scholar
Bensimon, D. 1988 Pattern selection in thermal convection: experimental results in an annulus.. Phys. Rev. A 37, 200–206.Google Scholar
Berg, J. C. 1972 In Recent Developments in Separation Science, vol. 2, pp. 1–31. CRC.
Bergéa, P. 1975 Rayleigh—Bénard instability: experimental findings obtained by light scattering and other optical methods. In Fluctuations, Instabilities and Phase Transitions (ed. T., Riste), pp. 323–352. Plenum.
Block, M. J. 1956 Nature 178, 650.
Christodoulou. K. N. & Scriven, L. E. 1988 Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3, 355–405.Google Scholar
Ciliberto, S., Simonelli, F. & Arecchi, F. T. 1986 Measurement of temperature distribution in thermocapillary instability. Il Nuovo Cimente 7D, 195–202.Google Scholar
Cliffe, K. A. 1983 Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233.Google Scholar
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D. 1983 Phase winding solutions in a finite container above convective threshold. J. Fluid Mech. 127. 155–183.Google Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Ann. Rev. Fluid Mech. 19. 403–435.Google Scholar
Dijkstra, H. A. 1988 Mass transfer induced convection near gas—liquid interfaces. Ph.D thesis, Department of Mathematics, University of Groningen, The Netherlands.
Dijkstra, H. A. & Van De Vooren, A. I. 1989 Multiplicity and stability of steady solutions for Marangoni convection in a two-dimensional rectangular container with rigid sidewalls.. Numer. Heat Transfer A 16, 59–75.Google Scholar
Drazin, P. G. 1975 On the effects of sidewalls on Bénard convection. Z. angew Math. Phys. 26, 239–243.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Goldhirsch, I., Orszag, S. A. & Maulik, B. K. 1987 An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. J. Sci. Comput. 2, 33–58.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Golubitsky, M. & Schaeffer, D. G. 1985 Singularities and Groups in Bifurcation Theory, part 1. Springer.
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz). Academic.
Koschmieder, E. L. 1974 Bénard convection. Adv. Chem. Phys. 26, 177–212.Google Scholar
Koschmieder, E. L. & Prahl, S. A. 1990 Surface-tension-driven Bénard convection in small containers. J. Fluid Mech. 215, 571–583.Google Scholar
Metzener, P. 1986 The effect of rigid sidewalls on nonlinear two-dimensional Bénard convection. Phys. Fluids 29, 1373–1377.Google Scholar
Moore, G. & Spence, A. 1980 The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17, 567–576.Google Scholar
Nield, D. A. 1964 Surface tension and buoyancy effects in cellular convection. J. Fluid Mech. 19, 341–352.Google Scholar
Patberg, W. B., Koers, A., Steenge, W. D. E. & Drinkenburg, A. A. H. 1983 Effectiveness of mass transfer in a packed distillation column in relation to surface tension gradients. Chem. Engng Sci. 38, 917–923.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Riley, D. S. & Winters, K. H. 1989 Modal exchange mechanisms in Lapwood convection. J. Fluid Mech. 204, 325–358.Google Scholar
Rosenblat, S., Homsy, G. M. & Davis, S. H. 1982 Nonlinear Marangoni convection in bounded layers. Part 2. Rectangular cylindrical containers. J. Fluid Mech. 120, 123–138.Google Scholar
Steen, P. H. & Aidun, C. K. 1988 Time periodic convection in porous media: transition mechanism. J. Fluid Mech. 196, 263–290.Google Scholar
Steward, W. J. & Jennings, A. 1981 A simultaneous iteration algorithm for real matrices. ACM Trans. Math. Software 7, 184–198.Google Scholar
Stork, K. & Müller, U. 1972 Convection in boxes: experiments. J. Fluid Mech. 54, 599–611.Google Scholar
Thomson, J. 1855 Phil. Mag. 10, 330.
Van De Vooren, A. I. & Dijkstra, H. A. 1989 A finite-element stability analysis of the Marangoni problem in a two-dimensional container with rigid sidewalls. Comput. Fluids 17, 467–485.Google Scholar
Vidal, A. & Acrivos, A. 1966 Effect of nonlinear temperature profiles on the onset of convection driven by surface tension gradients. I & EC Fundamentals 7, 53–58.Google Scholar
Werner, B. & Spence, A. 1984 The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399.Google Scholar
Winters, K. H., Plesser, Th. & Cliffe, K. A. 1988 The onset of convection in a finite container due to surface tension and buoyancy.. Physica D 29, 387–401.Google Scholar
Zuiderweg, F. J. & Harmens, A. 1958 The influence of surface phenomena on the performance of distillation columns. Chem. Engng Sci. 9, 89–103.Google Scholar