Three-dimensional Marangoni-Benard flows in square and nearly square containers

Physics of Fluids

Volumn 13, Issue 1, 2001, Pages 92-98

  Bergeon, A ^a,b   Henry, D ^c   Knobloch, E ^d  

^a UNIVERSITÉ PAUL SABATIER   (France)

^b IMFT   (France)

^c LABORATOIRE DE MÉCANIQUE DES FLUIDES ET D ACOUSTIQUE   (France)

^d UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ASPECT RATIO; BOUNDARY CONDITIONS; GRAVITATIONAL EFFECTS; SURFACE TENSION; THERMAL EFFECTS;

SQUARE CONTAINERS; SURFACE DEFLECTION; THREE-DIMENSIONAL MARAGONI-BERNARD CONVECTION;

FLUID DYNAMICS;

BIFURCATION; COMPUTATIONAL MODEL; SURFACE TENSION; TEMPERATURE EFFECT; THREE-DIMENSIONAL FLOW; ZERO GRAVITY ENVIRONMENT;

EID: 0034746262     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1329905     Document Type: Article

References (27)

1. S. M. Pimpuktar and S. Ostrach, "Convective effects in crystal growth from melt," J. Cryst. Growth 55, 614 (1981).
2. S. Rosenblat, G. M. Homsy, and S. H. Davis, "Nonlinear Marangoni convection in bounded layers. Part 2. Rectangular cylindrical containers," J. Fluid Mech. 120, 123 (1982).
3. H. A. Dijkstra, "On the structure of cellular solutions in Rayleigh-Bénard-Marangoni flows in small-aspect-ratio containers," J. Fluid Mech. 243, 73 (1992).
4. H. A. Dijkstra, "Surface tension driven cellular patterns in three-dimensional boxes - Linear stability," Microgravity Sci. Technol. 7, 307 (1995); "Part II: A bifurcation study," ibid. 8, 70 (1995).
5. H. A. Dijkstra, "Surface tension driven cellular patterns in three-dimensional boxes - Linear stability," Microgravity Sci. Technol. 7, 307 (1995); "Part II: A bifurcation study," ibid. 8, 70 (1995).
6. P. C. Dauby and G. Lebon, "Bénard-Marangoni instability in rigid rectangular containers," J. Fluid Mech. 329, 25 (1996).
7. A. Bergeon, D. Henry, and H. Ben Hadid, "2D and 3D Marangoni pattern selection in shallow cavities," Adv. Space Res. 8, 1223 (1998).
8. cmc Congrès Francais de Mécanique, Toulouse, France (1999).
9. A. Bergeon, D. Henry, H. BenHadid, and L. S. Tuckerman, "Marangoni convection in binary mixtures with Soret effect," J. Fluid Mech. 375, 143 (1998).
10. A. T. Patera, "A spectral element method for fluid dynamics: Laminar flow in a channel expansion," J. Comput. Phys. 54, 468 (1984).
11. G. Em. Karniadakis, M. Israeli, and S. A. Orszag, "High-order splitting method for the incompressible Navier-Stokes equations," J. Comput. Phys. 97, 414 (1991).
12. L. S. Tuckerman, "Steady-state solving via Stokes preconditioning: Recursion relations for elliptic operators," in Proceedings of the 11th International Conference on Numerical Methods in Fluid Dynamics, edited by D. L. Dwoyer, M. Y. Hussaini, and R. G. Voigt (Springer-Verlag, New York, 1989).
13. C. K. Mamun and L. S. Tuckerman, "Asymmetry and Hopf bifurcation in spherical Couette flow," Phys. Fluids 7, 80 (1995).
14. A. Bergeon and D. Henry, "A continuation method applied to the study of three dimensional Rayleigh-Bénard and Marangoni-Bénard instabilities," in Continuation Methods in Fluid Dynamics, edited by D. Henry and A. Bergeon, Notes on Numerical Fluid Mechanics 74 (Vieweg, Braunschweig/Wiesbaden, 2000).
15. C. Canuto, M. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Mechanics (Springer-Verlag, New York, 1987).
16. This terminology is opposite to that used by S. H. Davis, "Convection in a box: Linear theory," J. Fluid Mech. 30, 465 (1967).
17. J. D. Crawford and E. Knobloch, "Symmetry and symmetry-breaking bifurcations in fluid dynamics," Annu. Rev. Fluid Mech. 23, 341 (1991).
18. 4 symmetry. This observation is due to J. W. Swift, Ph.D. thesis, University of California at Berkeley, 1984.
19. M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory (Springer-Verlag, New York, 1988), Vol. 2.
20. For a discussion of these representations in a fluid dynamical context, see A. M. Rucklidge, N. O. Weiss, D. P. Brownjohn, P. C. Matthews, and M. R. E. Proctor, "Compressible magnetoconvection in three dimensions: Pattern formation in a strongly stratified layer," J. Fluid Mech. 419, 283 (2000).
21. E. Knobloch, "On the degenerate Hopf bifurcation with O(2) symmetry," Contemp. Math. 56, 193 (1986).
22. 2, if these are not saddles (see Ref. 17, proposition 3.2).
23. S. van Gils and J. Mallet-Paret, "Hopf bifurcation and symmetry: Travelling and standing waves on the circle," Proc. R. Soc. Edinburgh, Sect. A: Math. 104, 279 (1986).
24. J. D. Crawford and E. Knobloch, "On degenerate Hopf bifurcation with broken O(2) symmetry," Nonlinearity 1, 617 (1988).
25. M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory (Springer-Verlag, New York, 1985), Vol. 1.
26. E. L. Koschmieder and S. A. Prahl, "Surface-tension-driven Bénard convection in small containers," J. Fluid Mech. 215, 571 (1990).
27. G. B. Mindlin, T. Ondarçuhu, H. L. Mancini, C. Pérez-García, and A. Garcimartín, "Comparison of data from Bénard-Marangoni convection in a square container with a model based on symmetry arguments," Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 1121 (1994).