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Diffusion in PoroElastic Media
 Jour. Math. Anal. Appl
, 1998
"... . Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of se ..."
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Cited by 10 (7 self)
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. Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of secondary consolidation and pore fluid exposure on the boundary are included. This quasistatic system is resolved as an application of the theory of linear degenerate evolution equations in Hilbert space, and this leads to a precise description of the dynamics of the system. 1. Introduction We shall consider a system modeling diffusion in an elastic medium in the case for which the inertia effects are negligible. This quasistatic assumption arises naturally in the classical Biot model of consolidation for a linearly elastic and porous solid which is saturated by a slightly compressible viscous fluid. The fluid pressure is denoted by p(x; t) and the displacement of the structure by u(x; t). ...
Macroscale Thermodynamics and the Chemical Potential for Swelling Porous Media
, 1997
"... . The thermodynamical relations for a twophase, Nconstituent, swelling porous medium are derived using a hybridization of averaging and the mixturetheoretic approach of Bowen. Examples of such media include 21 lattice clays and lyophilic polymers. The homogenized field equations are obtained by ..."
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Cited by 6 (5 self)
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. The thermodynamical relations for a twophase, Nconstituent, swelling porous medium are derived using a hybridization of averaging and the mixturetheoretic approach of Bowen. Examples of such media include 21 lattice clays and lyophilic polymers. The homogenized field equations are obtained by volume averaging microscale field equations so that explicit relationships between the macroscale field variables and their microscale counterparts are obtained. The system of equations is closed by assuming the rate of change of the volume fraction is a dependent constitutive variable, resulting in viscoelastic behavior of the porous medium. A novel, scalar definition for the macroscale chemical potential for porous media is introduced, and it is shown how the properties of this chemical potential can be derived by slightly expanding the usual Coleman and Noll approach for exploiting the entropy inequality to obtain nearequilibrium results. Within this approach, we use Lagrange multiplier...
Generalized Forchheimer Equation for TwoPhase Flow Based on Hybrid Mixture Theory
, 1996
"... In this paper, we derive a Forchheimertype equation for twophase flow through an isotropic porous medium using hybrid mixture theory. Hybrid mixture theory consists of classical mixture theory applied to a multiphase system with volume averaged equations. It applies to media in which the charac ..."
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Cited by 2 (0 self)
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In this paper, we derive a Forchheimertype equation for twophase flow through an isotropic porous medium using hybrid mixture theory. Hybrid mixture theory consists of classical mixture theory applied to a multiphase system with volume averaged equations. It applies to media in which the characteristic length of each phase is "small" relative to the extent of the mixture. The derivation of a Forchheimer equation for single phase flow has been obtained elsewhere. These results are extended to include multiphase swelling materials which have nonnegligible interfacial thermodynamic properties. Key words. Porous media, swelling porous media, high velocity flow, nonDarcy flow, twophase flow, multiphase flow, mixture theory, Forchheimer equation. 1 Introduction Darcytype equations are used to describe the flow of a singlephase fluid through porous media in a number of situations. The classical Darcy equation, first derived experimentally in 1856, states that the flux is pro...
Diffusion in Deforming Porous Media
"... We report on some recent progress in the mathematical theory of nonlinear fluid transport and poromechanics, specifically, the design, analysis and application of mathematical models for the flow of fluids driven by the coupled pressure and stress distributions within a deforming heterogeneous p ..."
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Cited by 2 (0 self)
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We report on some recent progress in the mathematical theory of nonlinear fluid transport and poromechanics, specifically, the design, analysis and application of mathematical models for the flow of fluids driven by the coupled pressure and stress distributions within a deforming heterogeneous porous structure. The goal of this work is to develop a set of mathematical models of coupled flow and deformation processes as a basis for fundamental research on the theoretical and numerical modeling and simulation of flow in deforming heterogeneous porous media.
Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: I. Macroscale Field Equations
, 2001
"... A systematic development of the macroscopic field equations (conservation of mass, linear and angular momentum, energy, and Maxwell’s equations) for a multiphase, multicomponent medium is presented. It is assumed that speeds involved are much slower than the speed of light and that the magnitude of ..."
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Cited by 2 (1 self)
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A systematic development of the macroscopic field equations (conservation of mass, linear and angular momentum, energy, and Maxwell’s equations) for a multiphase, multicomponent medium is presented. It is assumed that speeds involved are much slower than the speed of light and that the magnitude of the electric field significantly dominates over the magnetic field so that the electroquasistatic form of Maxwell’s equations applies. A mixture formulation is presented for each phase and then averaged to obtain the macroscopic formulation. Species electric fields are considered, however it is assumed that it is the total electric field which contributes to the electrically induced forces and energy. The relationships between species and bulk phase variables and the macroscopic and microscopic variables are given explicitly. The resulting field equations are of relevance to many practical applications including, but not limited to, swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, and chromatography.
Distributive Smoothers in Multigrid for Problems with Dominating GradDiv Operators
"... In this paper we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating graddiv operator. In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the ..."
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In this paper we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating graddiv operator. In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the mesh sizes in space and time. The applications range from model problems to secondary consolidation Biot’s model. We focus on the smoothing issue, and mainly solve academic problems on Cartesian staggered grids. Copyright c○