Abstract
For an axisymmetric cellular structure of wavenumberq 0, the amplitude of the first linearly unstable mode is proportional to the Bessel functionJ 0(q 0r). If the radius of the structure is much larger thanq −10 , the first mode has a large relative maximum near the center. The nonlinear saturation of this instability gives an amplitude of order ε1/2, ε being the small growth rate of the linear instability. The saturation amplitude near the center is of order ε1/4. This prediction is confirmed by numerical analysis and numbers are given to make possible a comparison with the Rayleigh-Bénard instability.
Resumé
Pour une structure cellulaire axisymétrique de nombre d'ondeq 0, l'amplitude du premier mode instable est proportioneile á la fonction de BesselJ 0(q 0r). Si le rayon de la structure est très supérieur àq −10 ce premier mode a un grand maximum relatif au centre. La saturation non linéaire de cette instabilité conduit à une amplitude en ε1/2 (ε∼ taux de croissance-petit-de l'instabilité linéaire). L'amplitude de saturation au centre est d'ordre ε1/4. Cette prédiction est confirmée par l'analyse numérique, et les nombres nécessaires à la comparaison avec l'instabilité de Rayleigh-Bénard sont donnés.
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Pomeau, Y., Zaleski, S. & Manneville, P. Axisymmetric cellular structures revisited. Z. angew. Math. Phys. 36, 367–394 (1985). https://doi.org/10.1007/BF00944631
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DOI: https://doi.org/10.1007/BF00944631