Abstract.
A study of dynamic regimes in Bénard-Marangoni convection was carried out for various Prandtl and Marangoni numbers in small aspect ratio geometries (Γ = 2.2 and 2.8). Experiments in a small hexagonal vessel, for a large range of the Marangoni number (from 148 to 3636), were carried out. Fourier spectra and an auto-correlation function were used to recognize the various dynamic regimes. For given values of the Prandtl number (Pr = 440) and aspect ratio (Γ = 2.2), mono-periodic, bi-periodic and chaotic states were successively observed as the Marangoni number was increased. The correlation dimensions of strange attractors corresponding to the chaotic regimes were calculated. The dimensions were found to be larger than those obtained by other authors for Rayleigh-Bénard convection in aspect ratio geometries of the same order. The transition from temporal chaos to spatio-temporal chaos was also observed. For Γ = 2.2, when larger values of the Marangoni number were imposed (Ma = 1581 for Pr = 160 and Ma = 740 for Pr = 440), spatial modes were involved through the convective pattern dynamics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.C. Cross, P.E. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)
E. Bodenschatz, W. Pesch, G. Ahlers, Ann. Rev. Fluid Mech. 32, 709 (2000)
M.F. Schatz, G.P. Neitzel, Ann. Rev. Fluid Mech. 33(6), 93 (2001)
E.A. Ramadan, J.M. Hay, R.E. Khayat, J. Non-Newtonian Fluid Mech. 115, 79 (2003)
F.T. Akyildiz, H. Bellout, Applied Mathematics and Computation 162, 1103 (2005)
M. Bestehorn, Phys. Rev. E 48, 3622 (1993)
F. Sain, H. Riecke, Physica D 144, 124 (2000)
W.V.R. Malkus, Proc. Roy. Soc. A 225, 196 (1954)
G.E. Willis, J.W. Deardorff, Phys. Fluids 10, 931 (1967)
R. Krishnamurti, J. Fluid Mech. 42, 309 (1970)
G. Ahlers G., Phys. Rev. Lett. 33, 1185 (1974)
J.P. Gollub, S.V. Benson, J. Fluid Mech. 100, 449 (1980)
J.P. Gollub, S.V. Benson, J. Steinman, in Nonlinear Dynamics, edited by R.H.G. Helleman (Annals of the New York Academy of Sciences, 1980), Vol. 357, p. 22
M. Giglio, S. Musazzi, U. Perini, Phys. Rev. Lett. 47, 243 (1981)
A. Libchaber, J. Maurer, J. Phys. Colloq. France 41, C3-51 (1980)
A. Libchaber, C. Laroche, S. Fauve, J. Phys. Lett. 43, 211 (1982)
P. Bergé, M. Dubois, P. Manneville, Y. Pomeau, J. Phys. Lett. 41, L 341 (1980)
M. Dubois, M.A. Rubio, P. Bergé, Phys. Rev. Lett. 51, 1446 (1983)
M. Sano, Y. Sawada, Chaos and Statistical Methods, edited by Y. Kuramoto (Springer-Verlag, Berlin, 1984)
M. Dubois, P. Bergé, Physica Scripta 33, 159 (1986)
S. Sato, M. Sano, Y. Sawada, Phys. Rev. A 37, 1679 (1988)
T. Boeck, N.K. Vitanov, Phys. Rev. E 65, 3720 (2002)
P. Bergé, Y. Pomeau, C. Vidal, Order within Chaos (Wiley, New York, 1984)
P. Bergé, Le Chaos: théorie et expériences (Eyrolles, 1988)
P. Cerisier, C. Jamond, J. Pantaloni, J.C. Charmet, J. Phys. France 45, 405 (1984)
J.C. Berg, A. Acrivos, Chemical Engineering Science 20, 737(1965)
M.F. Schatz, S.J. VanHook, W.D. McCormick, J.B. Swift, H.L. Swinney, Phys. Fluids 11, 2577 (1999)
P. Cerisier, S. Rahal, B. Billia, Phys. Rev. E 54, 3508 (1996)
O. Ozen, E. Theisen, D.T. Johnson, P.C. Dauby, R. Narayanan, Journal of Colloid and Interface Science 289, 271 (2005)
M. Dubois, P. Bergé, Phys. Lett. 93, 365 (1983)
F. Takens, in Dynamical Systems and Turbulence, edited by D.A. Rand, L.S. Young (Springer-Verlag, Berlin, 1981)
F. Takens, in Dynamical Systems and Bifurcations, edited by H.W. Broer, F. Takens (Springer-Verlag, New York, 1985)
N.H. Packard, J.P. Crutchfield, J.D. Farmer, R.S. Shaw, Phys. Rev. Lett. 45, 712 (1980)
P. Grassberger, I. Procaccia, Phys. Rev. Lett. 50, 346 (1983)
P. Grassberger, I. Procaccia, Physica D 9, 189 (1983)
C. Lausberg, Ph.D. thesis, INPG, Grenoble, France, 1987
H. Abarbanel, R. Brown, J.J. Sidorowich, L.S. Tsimiring, Rev. Mod. Phys. 65, 1331 (1993)
J.C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, 2003)
P. Cvitanovic, G. Gunaratne, I. Procaccia, Phys. Rev. A 38, 1503 (1988)
S. Chang, J. McCown, Phys. Rev. A 30, 1149 (1984)
A. Thess, S.A. Orszag, J. Fluid Mech. 283, 201 (1995)
P. Cerisier, R. Occelli, J. Pantaloni, Physico-Chemical Hydrodynamics 7(4), 191 (1986)
C. Perez-Garcia, J. Pantaloni, R. Occelli, P. Cerisier, J. Phys. France 46, 2047 (1985)
A. Libchaber, J. Maurer, J. Phys. Lett. France 39, L 369 (1978)
B. Malraison, P. Atten, P. Bergé, M. Dubois, C.R. Acad. Sc. Paris 297, 209 (1983)
J.P. Eckmann, S. Oliffson - Kamphorst, D. Ruelle, S. Ciliberto, Phys. Rev. A 34, 4971 (1986)
F.E. Udwadia, H.F. Von Bremen, Appl. Math. Comput. 121, 219 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rahal, S., Cerisier, P. & Abid, C. Transition to chaos via the quasi-periodicity and characterization of attractors in confined Bénard-Marangoni convection . Eur. Phys. J. B 59, 509–518 (2007). https://doi.org/10.1140/epjb/e2007-00307-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2007-00307-0