Results 1  10
of
10
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 ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
. 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 ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
. 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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract
 Add to MetaCart
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○
Modified Darcy's Law, Fick's Law, and Terzaghi's Effective Stress Principle for Swelling Clay Soils
, 1997
"... Governing equations often used in soil mechanics and hydrology include the classical Darcy's law, Terzaghi's effective stress principle, and the classical Fick's first law. It is known that the classical forms of these relations apply only to nonswelling, granular materials. In this ..."
Abstract
 Add to MetaCart
Governing equations often used in soil mechanics and hydrology include the classical Darcy's law, Terzaghi's effective stress principle, and the classical Fick's first law. It is known that the classical forms of these relations apply only to nonswelling, granular materials. In this paper we summarize recent generalizations of these results for swelling porous media obtained using hybrid mixture theory by the authors. Hybrid mixture theory (HMT) is a methodical procedure for obtaining macroscopic constitutive restrictions which are thermodynamically admissible by exploiting the entropy inequality in the spirit of Coleman and Noll for spatially averaged properties. HMT applied to the modeling of swelling clay particles, viewed as clusters of adsorbed water and clay minerals, produces additional terms necessary to account for the physicochemical forces between the adsorbed water and clay minerals, or more generally, for swelling colloids. New directions for modeling consolidation of sw...
Contents
"... Abstract. 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 heteroge ..."
Abstract
 Add to MetaCart
Abstract. 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
Moisture and Heat Transport in Unsaturated Swelling Porous Media: A ParallelFlow, MultiScale, MixtureTheoretic Approach
"... . A threescale theory of both moisture and heat transport is presented for an unsaturated swelling porous media. At the microscale the system consists of macromolecules and an adsorbed fluid, both of which are considered as distinct nonoverlapping continua. At the mesoscale the macromolecule, and ..."
Abstract
 Add to MetaCart
. A threescale theory of both moisture and heat transport is presented for an unsaturated swelling porous media. At the microscale the system consists of macromolecules and an adsorbed fluid, both of which are considered as distinct nonoverlapping continua. At the mesoscale the macromolecule, and adsorbed fluid are homogenized to form particles. At this scale each phase within the particle is considered as an overlaying continua. The mesoscale particles coexist with a bulk fluid which may be either liquid or vapor; the particles and bulk fluid exist as distinct nonoverlaying continua. At the macroscale, the mesoscale particles and bulk fluid are homogenized to form overlaying continua at each point in space. The homogenization procedure is modified mixture theoretic. A single energy equation is used at the macroscale where the constitutive theory is developed. The model has one time scale so it may be thought of as a parallel flow model. The particles are saturated, but the mesoscal...