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Unconditional Nonlinear Stability in Temperature‐Dependent Viscosity Flow in a Porous Medium
Author(s) -
Payne L. E.,
SStraughan B.
Publication year - 2000
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00142
Subject(s) - porous medium , nonlinear system , viscosity , stability (learning theory) , convection , mechanics , work (physics) , flow (mathematics) , quadratic equation , mathematics , thermodynamics , thermal , temperature dependence of liquid viscosity , mathematical analysis , physics , materials science , porosity , geometry , computer science , relative viscosity , quantum mechanics , machine learning , composite material
The equations of flow in porous media attributable to Forchheimer are considered. In particular, the problem of thermal convection in such a medium is addressed when the viscosity varies with temperature. It is shown that nonlinear stability may be achieved naturally for all initial data by working with L 3 or L 4 norms. It is also shown that L 2 theory is not sufficient for such unconditional stability. Previous work has established nonlinear stability for vanishingly small initial data thresholds, but we believe this is the first analysis that addresses the important physical problem of unconditional stability. It is shown how to extend the nonlinear analysis for a viscosity linear in temperature to the cases when the viscosity may be quadratic or when penetrative convection is allowed in the layer.