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Periodic solutions for differential equations of order three, with application to heat‐flux induced convection
Author(s) -
Néel M. C.,
Lyubimov D.
Publication year - 1995
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670181403
Subject(s) - mathematics , mathematical analysis , vector field , rayleigh number , eigenvalues and eigenvectors , filtration (mathematics) , field (mathematics) , heat flux , convection , natural convection , geometry , heat transfer , mechanics , physics , pure mathematics , quantum mechanics
In this paper we show the existence of periodic solutions for a class of non‐linear regular vector fields of ℝ 3 admitting the trivial solution and depending on a real parameter μ. Two main assumptions lead to this conclusion: first, the right hand‐side of each vector field anticommutes with a reflection. Then, its linear part admits the eigenvalue zero for all variations of μ, and takes the form of a three‐dimensional Jordan block at μ = 0. This result allows the description of the bidimensional stationary patterns induced by a uniform heat‐flux imposed on the boundaries of an infinite porous layer, saturated by a fluid. Indeed, when the filtration Rayleigh number comes close to its critical value, the field of seepage velocity and the field of temperature can be related to a partial differential equation, which admits a three‐dimensional centre manifold. The existence of convective patterns follows. Their length depends on the filtration Rayleigh number, and tends towards infinity when this parameter comes close to its critical value (from above).