dc.contributor.author WINTER, C. LARRABEE. * dc.creator WINTER, C. LARRABEE. en_US dc.date.accessioned 2011-10-31T17:17:16Z dc.date.available 2011-10-31T17:17:16Z dc.date.issued 1982 en_US dc.identifier.uri http://hdl.handle.net/10150/184747 dc.description.abstract Suppose C(x,t) is the concentration at position x in Rᵈ and time t > 0 of a solute which is diffusing in some medium. If on a local scale the dispersion of the solute is governed by a constant dispersion matrix, 1/2(δ²), and a random velocity field, V(x), then C satisfies a convection-diffusion equation with random coefficients, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (1). Usually V(x) is taken to be μ + εU(x) where μ ε Rᵈ, U(x) is a given stationary random field with mean zero, and ε > 0 is a dimensionless parameter which measures the variability of V(x). Hydrological experiments suggest that on a regional scale the diffusion is classically Fickian with effective diffusion matrix D(ε) and drift velocity α(ε). Thus for large scales (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (2) is satisfied by the solute concentration. Here τ and χ are respectively time and space measured on large scales. It is natural to investigate the relation of the large scale coefficients D and α to the statistical properties of V(x). To relate (1) to (2)--and thus to approximate D(ε) and α(ε)--it is necessary to rescale t and x and average over the distribution of V. It can then be shown that the transition form (1) to (2) is equivalent to (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (3) where A = (∇•δ²∇)/2 + √nμ• ∇ and B(U) = √nU(√nx) • ∇. By expanding each side of (3) estimates of D(ε) and α(ε) can be obtained. The estimates have the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (4). Both D₂ and α₂ depend on the power spectrum of U. Analysis shows that in at least the case of incompressible fluids D₂ is positive definite. In one dimensional transport α₂ < 0, hence α(k) < μ(k) through second order. dc.language.iso en en_US dc.publisher The University of Arizona. en_US dc.rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. en_US dc.subject Fluid dynamics -- Mathematical models. en_US dc.subject Hydrologic models. en_US dc.subject Groundwater -- Pollution -- Mathematical models. en_US dc.title ASYMPTOTIC PROPERTIES OF MASS TRANSPORT IN RANDOM POROUS MEDIA. en_US dc.type text en_US dc.type Dissertation-Reproduction (electronic) en_US dc.identifier.oclc 682971002 en_US thesis.degree.grantor University of Arizona en_US thesis.degree.level doctoral en_US dc.identifier.proquest 8227377 en_US thesis.degree.discipline Applied Mathematics en_US thesis.degree.discipline Graduate College en_US thesis.degree.name Ph.D. en_US refterms.dateFOA 2018-08-14T04:27:29Z html.description.abstract Suppose C(x,t) is the concentration at position x in Rᵈ and time t > 0 of a solute which is diffusing in some medium. If on a local scale the dispersion of the solute is governed by a constant dispersion matrix, 1/2(δ²), and a random velocity field, V(x), then C satisfies a convection-diffusion equation with random coefficients, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (1). Usually V(x) is taken to be μ + εU(x) where μ ε Rᵈ, U(x) is a given stationary random field with mean zero, and ε > 0 is a dimensionless parameter which measures the variability of V(x). Hydrological experiments suggest that on a regional scale the diffusion is classically Fickian with effective diffusion matrix D(ε) and drift velocity α(ε). Thus for large scales (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (2) is satisfied by the solute concentration. Here τ and χ are respectively time and space measured on large scales. It is natural to investigate the relation of the large scale coefficients D and α to the statistical properties of V(x). To relate (1) to (2)--and thus to approximate D(ε) and α(ε)--it is necessary to rescale t and x and average over the distribution of V. It can then be shown that the transition form (1) to (2) is equivalent to (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (3) where A = (∇•δ²∇)/2 + √nμ• ∇ and B(U) = √nU(√nx) • ∇. By expanding each side of (3) estimates of D(ε) and α(ε) can be obtained. The estimates have the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (4). Both D₂ and α₂ depend on the power spectrum of U. Analysis shows that in at least the case of incompressible fluids D₂ is positive definite. In one dimensional transport α₂ < 0, hence α(k) < μ(k) through second order.
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