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Variational methods for steady‐state Darcy/Fick flow in swollen and poroelastic solids
Author(s) -
Roubíček Tomáš
Publication year - 2017
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201600269
Subject(s) - poromechanics , darcy's law , saddle point , variational principle , mechanics , saddle , flow (mathematics) , diffusion , porous medium , heat transfer , mathematics , mathematical analysis , thermodynamics , physics , porosity , materials science , mathematical optimization , geometry , composite material
Existence of steady states in elastic media at small strains with diffusion of a solvent or fluid due to Fick's or Darcy's laws is proved by combining usage of variational methods inspired from static situations with Schauder's fixed‐point arguments. In the plain variant, the problem consists in the force equilibrium coupled with the continuity equation, and the underlying operator is non‐potential and non‐pseudomonotone so that conventional methods are not applicable. In advanced variants, electrically‐charged multi‐component flows through an electrically charged elastic solid are treated, employing critical points of the saddle‐point type. Eventually, anisothermal variants involving heat‐transfer equation are treated, too.