Applications of the Homogenization Method to Flow and Transport in Porous Media

Hornung, Ulrich

doi:10.1142/9789814368438_0002

Cited by 71 publications

(113 citation statements)

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“…Our approach was based on homogenization theory (21,22) applied to planar diffusion through the ECS, regarded as a porous medium composed of repeating obstacles in the form of cellular elements. Cellular elements were defined as sections of cell bodies, dendrites, axons, or glial processes.…”

Section: Lattice Arrangements and Governing Equationmentioning

confidence: 99%

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Changes in brain cell shape create residual extracellular space volume and explain tortuosity behavior during osmotic challenge

Chen

Nicholson

2000

Proc. Natl. Acad. Sci. U.S.A.

123

115

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Diffusion of molecules in brain extracellular space is constrained by two macroscopic parameters, tortuosity factor and volume fraction ␣. Recent studies in brain slices show that when osmolarity is reduced, increases while ␣ decreases. In contrast, with increased osmolarity, ␣ increases, but attains a plateau. Using homogenization theory and a variety of lattice models, we found that the plateau behavior of can be explained if the shape of brain cells changes nonuniformly during the shrinking or swelling induced by osmotic challenge. The nonuniform cellular shrinkage creates residual extracellular space that temporarily traps diffusing molecules, thus impeding the macroscopic diffusion. The paper also discusses the definition of tortuosity and its independence of the measurement frame of reference.diffusion ͉ lattice ͉ volume transmission ͉ cell swelling and shrinkage ͉ numerical simulation M any biological processes involve diffusion of substances through a disordered heterogeneous medium. Diverse examples are intracellular signaling (1), intercellular signaling (2), volume transmission (3, 4), and drug delivery (5, 6). The rapid increase in the use of diffusion-weighted MRI has raised further issues about diffusion in tissues (7). In approaching such problems, two essential parameters are the extracellular space (ECS) volume fraction ␣, the volume ratio of the ECS compared to the whole tissue, and the tortuosity , a measure of how diffusing molecules are hindered by cellular obstructions. Recent studies by using osmotic challenge in brain tissue (8, 9) have shown a complex relation between these two parameters. In this paper, we apply the mathematical technique of homogenization to show that changes in cell shape provide an explanation for osmotic data and are likely to be applicable in a much wider context.To quantify diffusion in a complex biological medium, it is necessary to apply the diffusion equation through the use of some macroscopically effective properties, such as the local average concentration, ͗C͘, and the apparent (or effective) diffusion coefficient, D*. Such macroscopic formulations have been applied successfully in the ECS of brain tissues (8-15). When discussing the ECS, 2 is frequently defined as the ratio between the diffusion coefficient, D, of a given molecule in a bulk liquid and the apparent diffusion coefficient D* of the same molecule in the ECS,although other definitions have been used. Furthermore, some authors (16-18) include terms for viscosity of the ECS and hydrodynamic interaction of the diffusing molecule in their definition of ; we exclude such factors here, to focus on the relationship between geometry and tortuosity, but briefly discuss them later in the paper and in the Appendix, which also comments on the independence of tortuosity from its measurement frame of reference.The question of whether and ␣ are related has been explored in many contexts, but because it depends on the structure of the heterogeneous medium, no universal relationship is known. It is expected tha...

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Section: Lattice Arrangements and Governing Equationmentioning

confidence: 99%

“…Across the impermeable boundaries, the gradient for is specified by (16,22) n⅐ٌ ϭ n⅐ê, [5] where n designates the outward normal at the solid boundaries. If ê aligns with the y axis, Eqs.…”

Section: Lattice Arrangements and Governing Equationmentioning

confidence: 99%

Changes in brain cell shape create residual extracellular space volume and explain tortuosity behavior during osmotic challenge

Chen

Nicholson

2000

Proc. Natl. Acad. Sci. U.S.A.

123

115

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“…In our knowledge, this is the first time that convergence of the homogenization procedure is proved for problems whith nonlinear reactive terms and nonlinear transmission conditions. For a survey on homogenization applied to flow, diffusion, convection, and reactions in porous media see the papers [18]. Problems of related type were investigated in various papers.…”

Section: Introductionmentioning

confidence: 99%

Reactive transport through an array of cells with semi-permeable membranes

Hornung¹,

Jäger²,

Mikelić³

1994

ESAIM: M2AN

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“…This is a well-known technique for qualitatively assessing the structure of the upscaled equations. For a detailed explanation of this method we refer to the books Hornung (2012) and Cioranescu and Donato (2000). For an accurate physical model, the values of the homogenized coefficients should be confirmed by experiments, as the asymptotic expansion method only provides formulas for these in the case of simple microscale geometries.…”

Section: Introductionmentioning

confidence: 99%

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

et al. 2018

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We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.

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Applications of the Homogenization Method to Flow and Transport in Porous Media

Cited by 71 publications

References 0 publications

Changes in brain cell shape create residual extracellular space volume and explain tortuosity behavior during osmotic challenge

Changes in brain cell shape create residual extracellular space volume and explain tortuosity behavior during osmotic challenge

Reactive transport through an array of cells with semi-permeable membranes

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

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