Convection de Bénard–Rayleigh. Cas libre-libre

Gérard Iooss

doi:10.5802/crmeca.216

Convection de Bénard–Rayleigh. Cas libre-libre
[Bénard-Rayleigh convection in the free-free case]

Gérard Iooss ¹

¹ Laboratoire J.A.Dieudonné, I.U.F., Université Côte d’Azur, CNRS, Parc Valrose, 06108 Nice cedex 2, France

Comptes Rendus. Mécanique, Volume 351 (2023), pp. 281-313.

Abstract

La convection de Bénard–Rayleigh est étudiée dans le cas des conditions aux limites libre-libre, qui a l’avantage de permettre des calculs explicites. On présente les solutions convectives bifurquées périodiques sur réseau hexagonal, lorsque le nombre de Rayleigh est voisin de sa valeur critique. On donne explicitement les plages de valeurs du nombre de Prandtl déterminant la stabilité ou non de chaque solution bifurquée : rouleaux, pattern hexagonal, pattern en triangles équilatéraux, et patchwork. On montre que lorsqu’une de ces solutions est stable, c’est soit les rouleaux, soit le pattern en triangles.

Bénard–Rayleigh convection is investigated in the case of free-free plane boundaries. This has the advantage to allow explicit computations. Convective steady periodic bifurcating solutions are presented on a hexagonal lattice, for a Rayleigh number near its critical value. Explicit intervals of values of the Prandtl number are given for which bifurcating solutions are stable or not (rolls, hexagons, triangular pattern and patchwork). It is shown that when one of these solutions is stable, it can only be rolls or a triangular pattern.

Received: 2023-05-11
Accepted: 2023-06-22
Published online: 2023-08-03

DOI: 10.5802/crmeca.216

Author's affiliations:

Gérard Iooss ¹

¹ Laboratoire J.A.Dieudonné, I.U.F., Université Côte d’Azur, CNRS, Parc Valrose, 06108 Nice cedex 2, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {G\'erard Iooss},
     title = {Convection de {B\'enard{\textendash}Rayleigh.} {Cas} libre-libre},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {281--313},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2023},
     doi = {10.5802/crmeca.216},
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Gérard Iooss. Convection de Bénard–Rayleigh. Cas libre-libre. Comptes Rendus. Mécanique, Volume 351 (2023), pp. 281-313. doi : 10.5802/crmeca.216. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.216/

References
Cited by

[1] Lothar Koschmieder Bénard cells and Taylor vortices, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1993 | Zbl

[2] M. R. Ukhovskij; V. I. Yudovich On the equations of steady-state convection, J. Appl. Math. Stochastic Anal., Volume 27 (1963), pp. 432-440 | DOI | MR | Zbl

[3] V. I. Yudovich On the origin of convection, J. Appl. Math. Stochastic Anal., Volume 30 (1966) no. 6, pp. 1193-1199 | DOI

[4] V. I. Yudovich Free convection and bifurcation, Prikl. Mat. Mekh., Volume 31 (1967), pp. 101-111 (Russian) translated as J. Appl. Math. Mech. 31 (1967), 103–114. | Zbl

[5] V. I. Yudovich Stability of convection flows, J. Appl. Math. Stochastic Anal., Volume 31 (1967), pp. 294-303 | Zbl

[6] David H. Sattinger Group Representation Theory, Bifurcation Pattern Formation, J. Funct. Anal., Volume 28 (1978), pp. 58-101 | DOI | MR | Zbl

[7] Martin Golubitsky; Ian Stewart; David G. Schaeffer Singularities and groups in bifurcation theory, Applied Mathematical Sciences, 69, Springer, 1988 | DOI | Zbl

[8] A. Schlüter; D. Lortz; F. Busse On the stability of steady finite amplitude convection, J. Fluid Mech., Volume 23 (1965), pp. 129-144 | DOI | MR | Zbl

[9] Enok Palm Nonlinear thermal convection, Annu. Rev. Fluid Mech., Volume 7 (1975), pp. 39-61 | DOI | Zbl

[10] Klaus Kirchgässner; Hansjörg Kielhöfer Stability and bifurcation in fluid dynamics, Rocky Mt. J. Math., Volume 3 (1973), pp. 275-318 | DOI | MR | Zbl

[11] Mariana Haragus; Gérard Iooss Local Bifurcations, center manifolds, and Normal Forms in Infinite-dimensional dynamical Systems, Universitext, Springer, 2011 | DOI | Zbl

[12] Anne Pellew; Richard V. Southwell On maintained convection motion in a fluid heated from below, Proc. R. Soc. Lond., Ser. A, Volume 176 (1940), pp. 312-343 | DOI | Zbl

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