Category: diffusion

Experimental Implications of Bochner-Levy-Riesz Diffusion

Post author By R Hilfer
Post date August 18, 2014

R. Hilfer

Fractional Calculus and Applied Analysis 18, 333-341 (2015)
https://doi.org/10.1515/fca-2015-0022

submitted on
Monday, August 18, 2014

Fractional Bochner-Levy-Riesz diffusion arises from ordinary diffusion by replacing the Laplacean with a noninteger power of itself. Bochner- Levy-Riesz diffusion as a mathematical model leads to nonlocal boundary value problems. As a model for physical transport processes it seems to predict phenomena that have yet to be observed in experiment.

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Tags Anomalous Diffusion, fractional Laplacian

diffusion Fractional Calculus Porous Media Two-Phase Flow

Modeling Infiltration by Means of a Nonlinear Fractional Diffusion Model

Post author By R Hilfer
Post date May 18, 2006

E. Gerolymatou, I. Vardoulakis, R. Hilfer

Journal of Physics D: Applied Physics 39, 4104 (2006)

submitted on
Thursday, May 18, 2006

The classical Richards equation describes infiltration into porous soil as a nonlinear diffusion process. Recent experiments have suggested that this process exhibits anomalous scaling behaviour. These observations suggest generalizing the classical Richards equation by introducing fractional time derivatives. The resulting fractional Richards equation with appropriate initial and boundary values is solved numerically in this paper. The numerical code is tested against analytical solutions in the linear case. Saturation profiles are calculated for the fully nonlinear fractional Richards equation. Isochrones and isosaturation curves are given. The cumulative moisture intake is found as a function of the order of the fractional derivative. These results are compared against experiment.

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diffusion Fractional Calculus Porous Media Two-Phase Flow

Simulating the Saturation Front Using a Fractional Diffusion Model

Post author By R Hilfer
Post date June 30, 2005

E. Gerolymatou, I. Vardoulakis, R. Hilfer

in: Proceedings of the GRACM05 International Congress on Computational Mechanics, Limassol 2005
edited by: G. Georgiou, P. Papanastasiou, M. Papadrakakis
GRACM, Athens, 653 (2005)

submitted on
Thursday, June 30, 2005

In this paper the possibility of making use of fractional derivatives for the simulation of the flow of water through porous media and in particular through soils is considered. The Richards equation, which is a non-linear diffusion equation, will be taken as a basis and is used for the comparison of results. Fractional derivatives differ from derivatives of integer order in that they entail the whole history of the function in a weighted form and not only its local behavior, meaning that a different numerical approach is required. Previous work on the topic will be examined and a consistent approach based on fractional time evolutions will be presented.

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diffusion Fractional Calculus

On fractional diffusion and continuous time random walks

Post author By R Hilfer
Post date May 22, 2003

R. Hilfer

Physica A 329, 35 (2003)
https://doi.org/10.1016/S0378-4371(03)00583-1

submitted on
Thursday, May 22, 2003

A continuous time random walk model is presented with long-tailed waiting time density that approaches a Gaussian distribution in the continuum limit. This example shows that continuous time random walks with long time tails and di!usion equations with a fractional time derivative are in general not asymptotically equivalent.

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dielectric relaxation diffusion electrical conductivity Heterogeneous Materials Porous Media

Quantitative comparison of meanfield mixing laws for conductivity and dielectric constant of porous media

Post author By R Hilfer
Post date June 4, 2002

R. Hilfer, J. Widjajakusuma, B. Biswal

Physica A 318, 319 (2003)
https://doi.org/10.1016/S0378-4371(02)01197-4

submitted on
Tuesday, June 4, 2002

Exact numerical solution of the electrostatic disordered potential problem is carried out for four fully discretised threedimensional experimental reconstructions of sedimentary rocks. The measured effective macroscopic dielectric constants and electrical conductivities are compared with parameterfree predictions from several mean field type theories. All these theories give agreeable results for low contrast between the media. Predictions from Local porosity theory, however, match for the entire range of contrast.

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diffusion Fractional Calculus

Fractional Diffusion based on Riemann-Liouville Fractional Derivatives

Post author By R Hilfer
Post date October 12, 1999

R. Hilfer

The Journal of Physical Chemistry B 104, 3914-3917 (2000)
DOI: 10.1021/jp9936289

submitted on
Tuesday, October 12, 1999

A fractional diffusion equation based on Riemann−Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of H-functions. It differs from the known solution of fractional diffusion equations based on fractional integrals. The solution of fractional diffusion based on a Riemann−Liouville fractional time derivative does not admit a probabilistic interpretation in contrast with fractional diffusion based on fractional integrals. While the fractional initial value problem is well defined and the solution finite at all times, its values for t → 0 are divergent.

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Tags Anomalous Diffusion

dielectric relaxation diffusion electrical conductivity fluid flow Porous Media

Effective Physical Properties of Sandstones

Post author By R Hilfer
Post date October 6, 1999

J. Widjajakusuma, R. Hilfer

in: IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials
edited by: W. Ehlers
Solid Mechanics and Its Applications, vol. 87,Kluwer, Dordrecht, 113 (2001)
10.1007/0-306-46953-7
ISBN: 978-0-7923-6766-6

submitted on
Wednesday, October 6, 1999

In this paper we continue the investigation of the effective transport parameters of a digitized sample of Fontainebleau sandstone and three reconstruction models discussed previously in Biswal et. al., Physica A 273, 452 (1999). The effective transport parameters are computed directly by solving the disordered Laplace equation via a finite-volume method. We find that the transport properties of two stochastic models differ significantly from the real sandstone. Moreover, the effective transport parameters are predicted by employing local porosity theory and various traditional mixing-laws (such as effective medium approximation or Maxwell-Garnet theory). The prediction of local porosity theory is in good agreement with the exact result.

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Tags effective medium, sand stone

dielectric relaxation diffusion electrical conductivity fluid flow Heterogeneous Materials Porous Media

Quantitative Prediction of Effective Material Properties of Heterogeneous Media

Post author By R Hilfer
Post date October 8, 1998

J. Widjajakusuma, B. Biswal, R. Hilfer

Computational Materials Science 16, 70 (1999)
https://doi.org/10.1016/S0927-0256(99)00047-6

submitted on
Thursday, October 8, 1998

Effective electrical conductivity and electrical permittivity of water-saturated natural sandstones are evaluated on the basis of local porosity theory (LPT). In contrast to earlier methods, which characterize the underlying microstructure only through the volume fraction, LPT incorporates geometric information about the stochastic microstructure in terms of local porosity distribution and local percolation probabilities. We compare the prediction of LPT and of traditional effective medium theory with the exact results. The exact results for the conductivity and permittivity are obtained by solving the microscopic mixed boundary value problem for the Maxwell equations in the quasistatic approximation. Contrary to the predictions from effective medium theory, the predictions of LPT are in better quantitative agreement with the exact results.

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Tags effective medium, local porosity theory

diffusion Fractional Calculus Random Walks

On Fractional Diffusion and its Relation with Continuous Time Random Walks

Post author By R Hilfer
Post date May 22, 1998

R. Hilfer

in: Anomalous Diffusion: From Basis to Applications
edited by: R. Kutner, A. Pekalski and K. Sznajd-Weron
Lecture Notes in Physics, vol. 519,Springer, Berlin, 77 (1999)
10.1007/BFb0106828
978-3-662-14242-4

submitted on
Friday, May 22, 1998

Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is established between the fractional master equation and a separable continuous time random walk of the Montroll-Weiss type. The waiting time density can be expressed using a generalized Mittag-Leffier function. The first moment of the waiting density does not exist.

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Tags CTRW

diffusion fluid flow Nonequilibrium Pattern Formation

Pattern Formation at Liquid Interfaces

Post author By R Hilfer
Post date January 11, 1988

B. Heidel, C. Knobler, R. Hilfer, R. Bruinsma

Physical Review Letters 60, 2492 (1988)
10.1103/PhysRevLett.60.2492

submitted on
Monday, January 11, 1988

Although many examples of pattern formation resulting from chemical reactions at liquid interfaces are known, few have been studied in detail. We report a quantitative study of patterns formed by the photoproduction of Fe++ and its subsequent reaction to form Turnbull’s Blue. The experiment leads to the postulation of a mechanism in which autocatalysis is enhanced by double diffusion. The phase diagram contains a line of phase transitions whose critical behavior is discussed.

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Tags growth