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Non‐linear stability in the Bénard problem for a double‐diffusive mixture in a porous medium
Author(s) -
Lombardo S.,
Mulone G.,
Straughan B.
Publication year - 2001
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.263
Subject(s) - mathematics , stability (learning theory) , porous medium , instability , linear stability , thermodynamics , mathematical analysis , porosity , mechanics , physics , chemistry , machine learning , computer science , organic chemistry
The linear and non‐linear stability of a horizontal layer of a binary fluid mixture in a porous medium heated and salted from below is studied, in the Oberbeck–Boussinesq–Darcy scheme, through the Lyapunov direct method. This is an interesting geophysical case because the salt gradient is stabilizing while heating from below provides a destabilizing effect. The competing effects make an instability analysis difficult. Unconditional non‐linear exponential stability is found in the case where the normalized porosity ϵ is equal to one. For other values of ϵ a conditional stability theorem is proved. In both cases we demonstrate the optimum result that the linear and non‐linear critical stability parameters are the same whenever the Principle of Exchange of Stabilities holds. Copyright © 2001 John Wiley & Sons, Ltd.