Fractal Diffusion Research Articles

THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER

The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.

A fractal model for current generation in porous electrodes

This article aims to investigate the solution of the fractional Troesch's equation that models the mechanism of current generation in porous electrodes by numerical integration of its equivalent power-form representation and by deriving its approximate Taylor series solution. Simulation results compare to experimental data demonstrate that the overpotential distribution and efficiency within a porous electrode, from the bottom to the top of the bed is predicted by using Troesch's fractal overpotential diffusion equation, when the fractal parameter values are α ≥ 1.2618 for a porous electrode that follows the Hausdorff dimension, or α ≥ 1.236 for El Naschie's fractal space-time.

FRACTAL ANALYSIS OF STRESS-DEPENDENT DIFFUSIVITY OF POROUS CEMENTITIOUS MATERIALS

The understanding of the diffusion process and mechanisms of harmful species (e.g. chlorides) in porous cementitious materials is important to control and improve the material durability under harsh environments. In this paper, fractal analysis on the pore structure of porous cementitious materials was conducted and involved in a diffusion model. Macro material geometric parameters were considered in the model to avoid the difficulties in the measurements of microscopic pore parameters. The deformations of porous cementitious materials under the uniaxial elastic loads were considered to correct the diffusion model. The stress-affected diffusivity was displayed in an elegant expression involving some macro material parameters (e.g. total porosity, elastic modulus of solid skeleton, Poisson ratio). Results show that the effective diffusivity is greatly influenced by the porosity and stress ratio. The uniaxial elastic loads decrease the pore areas but increase the lengths of the pore channels for mass diffusion, which eventually causes the decrease of the effective diffusivity. The plots of the relative diffusivity against the stress ratio follow linear forms. The developed fractal diffusion model may help better understand the diffusion process in complex porous cementitious materials under elastic loads. Going beyond this, the fractal diffusion model may provide a new tool to predict the diffusivity of porous building materials under complex mechanical and environmental loads.

The geometry of the toxicity of silver nanoparticles to freshwater mussels

The question about the influence of the geometry of silver nanoparticle (nAg) towards toxicity in aquatic organisms is largely unanswered. The purpose of this study was to examine if different geometries of nAg could initiate biophysical stress in the soft tissues of mussels. Freshwater Dreissenna bugensis mussels were exposed for 48 h at 15 °C to 10 and 50 μg/L of ionic Ag and to 3 forms of polyvinylpyrrolidone (PVP)-coated nAg of similar size: sphere, cube and prism. At the end of the exposure period, mussels were allowed to depurate overnight and the post-mitochondrial fraction of the soft tissues were analyzed for the levels of liquid crystals (LCs), changes in the activity and fractal dimensions of pyruvate kinase-lactate dehydrogenase (PK-LDH), F-actin and protein-ubiquitin (UB) levels. The data revealed that exposure to nAg forms lead to increased formation of LCs in increasing order of intensity: prismatic > cubic > spherical nAg. The activity in PK-LDH was decreased by all forms of nAg but not by ionic Ag+ (as with the following effects). Fractal kinetics of the PK-LDH system revealed that the nAg forms increased the spectral dimension (sD) in increasing order: spherical > cubic > prismatic nAg. A decrease in the fractal diffusion rate (fDR) with small changes in the fractal dimension (fD) was also obtained. The levels of F-actin and protein-UB were significantly affected for most forms of nAg and followed a pattern similar to LCs levels. In conclusion, the geometry of nAg could influence the formation of LCs, alter the fractal kinetics of the PK-LDH system, F-actin levels and protein damage in the soft tissues of freshwater mussels.

Analysis and Calculation of the Probability Selectivity Using the Modern Distributed Algorithms

The calculation of the characteristics of the selectivity in the membrane separation of mixtures, based on basic physicochemical characteristics of the solution, is a difficult task due to the large number of influencing factors. To solve this problem, for a theoretical evaluation of the selectivity of ultrafiltration membranes, modern distributed algorithms are used for a probabilistic approach, in which the mechanism of the separation process differs significantly from the separation mechanism in nanomembranes. As a result of the proposed models and modern distributed algorithms, estimates of the selectivity of nanofiltration membranes for individual ions were obtained. It was found that an important feature of mixing with the mixture flowing through the membrane is the dependence of the effective diffusion coefficient on time. Also, as a result of calculations by the proposed model, it was found that this feature is modeled by the coefficient of anomalous fractal diffusion in time, as well as by the displacement of the effective separation zone in the membrane.

Fractal analysis of shape factor for matrix-fracture transfer function in fractured reservoirs

As the core function of dual-porosity model in fluids flow simulation of fractured reservoirs, matrix-fracture transfer function is affected by several key parameters, such as shape factor. However, modeling the shape factor based on Euclidean geometry theory is hard to characterize the complexity of pore structures. Microscopic pore structures could be well characterized by fractal geometry theory. In this study, the separation variable method and Bessel function are applied to solve the single-phase fractal pressure diffusion equation, and then the obtained analytical solution is used to deduce one-dimensional, two-dimensional and three-dimensional fractal shape factors. The proposed fractal shape factor can be used to explain the influence of microstructure of matrix on the fluid exchange rate between matrix and fracture, and is verified by numerical simulation. Results of sensitivity analysis indicate that shape factor decreases with tortuosity fractal dimension and characteristic length of matrix, increases with maximum pore diameter of matrix. Furthermore, the proposed fractal shape factor is effective in the condition that tortuosity fractal dimension of matrix is roughly between 1 and 1.25. This study shows that microscopic pore structures have an important effect on fluid transfer between matrix and fracture, which further improves the study on flow characteristics in fractured systems.

Method of Fractal Reservoir with Irregular Crack Bedrock System and Its Application for Well Test Analysis

In this paper the fractal theory is used to derive formulas for calculating seepage velocity, permeability, and porosity for the flow of/kids in porous media with dual fractal dimensions (capillary diameter and capillary distortion fractal dimension). The dual fractal model is applicable for most limestone reservoirs. It is shown that the percolation equation of micro-compressible fluids, or the double fractal pressure diffusion equation, is a partial differential equation. The equation is solved for the pressure drop and pressure recovery conditions. The solutions for the actual wellbore consider wellbore storage and skin Wen parameters, and a model curve of an infinite reservior is drawn. Analysis of the double fractal media shows that the traditional well test analysis methods, like the pressure drop semi-log linear curve method, pressure recovery Horner method, and Gringarten-Bourdet diagram method will bring great errors, especially when the distortion fractal dimension is large. Using the established well test model, the parameters of the limestone oil reservoir and the unstable pressure data are measured, and the accurate oil reservoir effective permeability, fractal length, and fractal dimensions are calculated.

Application of fractal geometry to describe reservoirs with complex structures

Abstract This study provides basic instantaneous source functions for different source-reservoir geometries in infinite fractal reservoirs (FRs) and infinite slab FRs. The concept of fractal geometry (FG) has already been adopted to analyse and model well-reservoir systems with complex structures. All these complexities have been modelled through a partial differential equation called fractal diffusivity equation (FDE). In this paper, by use of the source/Green’s function technique, we analytically solve the FDE for different boundary conditions, i.e., different source-reservoir geometries. The wellbore pressure response for horizontal wells in an infinite slab FR (synthetic example) is derived and analysed to illustrate the applications of the provided analytical source solutions and how to use these solutions.

Trap-controlled fractal diffusion model of the Havriliak-Negami dielectric relaxation

A new model of non-Debye dielectric relaxation, which leads to the popular in broadband dielectric spectroscopy Havriliak-Negami expression for two-power-law dielectric response, is proposed. The new model is based on the multiple trapping model and the fractal diffusion model. Explicit analytical expressions for experimentally measured values such as dielectric strength, characteristic relaxation time, and power-law exponents, which include the microscopic parameters of the medium, are obtain. The comparison of the expression for the relaxation time with the experiment is demonstrated. A clear physical interpretation of the power-law exponents of the Havriliak-Negami relaxation function is found, one makes it possible to extend the area application of this function and to use it for the fitting of dielectric data, which exhibit atypical relaxation behavior.

Parameter identification for fractional fractal diffusion model based on experimental data.

This paper studies the techniques of parameter estimation and their application in determining parameters of the fractional fractal diffusion model. On account of the basic structural characteristics of the porous coal matrix, the fractional fractal diffusion model is established to express the gas transport mechanism in the heterogeneous coal matrix. A L1 finite difference method in the temporal direction while spectral collocation method in the spatial direction is proposed to solve the model numerically. Then, by means of the gas adsorption and desorption experiments in coal samples, attempts have been made by the BFGS method, nonlinear conjugate gradient method, and Bayesian method to compare and contrast to obtain the physical parameters of the model. Furthermore, advantages and limitations of different estimation methods are discussed.

Particle-Based Simulation Reveals Macromolecular Crowding Effects on the Michaelis-Menten Mechanism

Many computational models for analyzing and predicting cell physiology rely on in vitro data collected in dilute and controlled buffer solutions. However, this can mislead models because up to 40% of the intracellular volume—depending on the organism, the physiology, and the cellular compartment—is occupied by a dense mixture of proteins, lipids, polysaccharides, RNA, and DNA. These intracellular macromolecules interfere with the interactions of enzymes and their reactants and thus affect the kinetics of biochemical reactions, making in vivo reactions considerably more complex than the in vitro data indicates. In this work, we present a new, to our knowledge, type of kinetics that captures and quantifies the effect of volume exclusion and other spatial phenomena on the kinetics of elementary reactions. We further developed a framework that allows for the efficient parameterization of these kinetics using particle simulations. Our formulation, entitled generalized elementary kinetics, can be used to analyze and predict the effect of intracellular crowding on enzymatic reactions and was herein applied to investigate the influence of crowding on phosphoglycerate mutase in Escherichia coli, which exhibits prototypical reversible Michaelis-Menten kinetics. Current research indicates that many enzymes are reaction limited and not diffusion limited, and our results suggest that the influence of fractal diffusion is minimal for these reaction-limited enzymes. Instead, increased association rates and decreased dissociation rates lead to a strong decrease in the effective maximal velocities Vmax and the effective Michaelis-Menten constants KM under physiologically relevant volume occupancies. Finally, the effects of crowding were explored in the context of a linear pathway, with the finding that crowding can have a redistributing effect on the effective flux responses in the case of twofold enzyme overexpression. We suggest that this framework, in combination with detailed kinetics models, will improve our understanding of enzyme reaction networks under nonideal conditions.

Schramm–Loewner evolution theory of the asymptotic behaviors of (2+1)-dimensional Das Sarma–Tamborenea model

A mass of theoretical analysis and numerical calculation on (2+1)-dimensional Das Sarma–Tamborenea model have been carried out over the years. A problem about asymptotical behaviors that this model belongs to which universality classes is still interesting and controversial. The Schramm–Loewner evolution theory is an innovative point of view to describe the conformal invariance of contour lines of saturated rough surfaces. In this paper, we used SLE theory to analyze the conformal invariance of the (2+1)-dimensional DT model. A noise reduction technique was also utilized to get better scaling behavior. The results show that the contour lines of (2+1)-dimensional DT model satisfy the property of conformal invariance and belongs to κ=4 universality class. The relation df=1+κ∕8 is also satisfied for the fractal dimension and diffusion coefficient of the contour lines.

Fractal Diffusion Limited Aggregation of Soot Particles Based on Fuzzy Membership Functions

In this paper, the membership function in fuzzy systems is used in the Diffusion Limited Aggregation (DLA) model to investigate the fractal diffusion of soot particles from diesel engine emissions. The transformation of the morphology of soot particle aggregates and the control of fractal diffusion of soot particles are investigated by analyzing the nonlinear relationship between the motion steps and angles of diffusing particles. The simulation results demonstrate that the morphology of the aggregates varies from loose to compact by changing the particles’ motion steps and angles in membership functions. Meanwhile, the Ballistic Aggregation (BA)-like aggregates are obtained. Furthermore, the control of the morphology of soot particle aggregates is realized, which makes the settlement of the aggregates become easier. This will provide a reference for further understanding the growth mechanism of soot particle diffusion and enhancing the purification technology of the soot particles.

A Semianalytic Solution for Temporal Pressure and Production Rate in a Shale Reservoir With Nonuniform Distribution of Induced Fractures

Summary The network of induced fractures and their properties control pressure propagation and fluid flow in hydraulically fractured shale reservoirs. We present a novel fully fractal model in which both the spacing and the porosity/permeability of induced fractures are distributed according to fractal dimensions (i.e., fractal decay of fracture density and the associated porosity/permeability away from the main induced fracture). The fractal fracture distribution is general, and handles exponential, linear, power, and uniform distributions. We also developed a new fully fractal diffusivity equation (FDE) using the fractal distribution of fractures and their properties. We then used, for the first time, the semianalytic Bessel spline scheme to solve the developed diffusivity equation. Our proposed model is general and can capture any form of induced-fracture distribution for better analysis of pressure response and production rates at transient- and pseudosteady-state conditions. We compared the unsteady-state and pseudosteady-state pressure responses calculated by our fully fractal model with former models of limited cases: uniform fracture spacing and uniform porosity/permeability [conventional diffusivity equation (CDE)]; variable fracture spacing and uniform porosity/permeability [modified CDE (MCDE)]; and uniform fracture spacing and fractal porosity/permeability distribution (FPPD). We used these models to match and predict the production data of a multifractured horizontal gas well in the Barnett Shale. Our results showed that the fractal distribution of fracture networks and their associated properties better matches the field data. Uniform distribution of induced-fracture networks underestimates production rate, especially at early time.

Sensitivity Analysis of Key Parameters for Population Balance Based Soot Model for Low-Speed Diffusion Flames

In this article, the evolution of in-flame soot species in a slow speed, buoyancy-driven diffusion flame is thoroughly studied with the implementation of the population balance approach in association with computational fluid dynamics (CFD) techniques. This model incorporates interactive fire phenomena, including combustion, radiation, turbulent mixing, and all key chemical and physical formation and destruction processes, such as particle inception, surface growth, oxidation, and aggregation. The in-house length-based Direct Quadrature Method of Moments (DQMOM) soot model is fully coupled with all essential fire sub-modelling components and it is specifically constructed for low-speed flames. Additionally, to better describe the combustion process of the parental fuel, ethylene, the strained laminar flamelet model, which considers detailed chemical reaction mechanisms, is adopted. Numerical simulation is validated against a self-conducted co-flow slot burner experimental measurement. A comprehensive assessment of the effect of adopting different nucleation laws, oxidation laws, and various fractal dimension and diffusivity values is performed. The results suggest the model employing Moss law of nucleation, modified NSC law of oxidation, and adopting a fractal dimension value of 2.0 and Schmidt number of 0.9 yields the simulation result that best agreed with experimental data.

Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface

This article studies the ion release from an unsmooth inner surface of a hollow fiber. A fractal diffusion model is established using the fractal derivative, and the effect of the fractal dimension on the ion release is elucidated. The present theory provides a theoretical basis for the optimization of a hollow fiber contained silver ions for practical applications.

Controlling Fractal Diffusion of Differently-Sized Soot Particles

Fractal diffusion of soot particles is widespread in nature. We propose a mathematical model of interference control for diffusion systems of differently-sized soot particles that can be used to predict and control fractal diffusion. The effect of the source of the interference on diffusion is investigated by analyzing the numerical relationship between the growth probability and the different interferences, a nonlinear logarithm interference and an internal source. Simulation demonstrates the effective control of fractal diffusion for differently-sized soot particles. Meanwhile, analyzing the multifractal spectra of soot particle aggregates under different controlling parameters lends to a quantitative description of the impact on the morphology and distribution of aggregated soot particles due to environmental interferences. This is useful not only for selecting controlling parameters and regions, but also for further developing studies into the kinetics of soot particle diffusion and the improvement of aggregation mechanisms.

Fractal diffusion in high temperature polymer electrolyte fuel cell membranes.

The performance of fuel cells depends largely on the proton diffusion in the proton conducting membrane, the core of a fuel cell. High temperature polymer electrolyte fuel cells are based on a polymer membrane swollen with phosphoric acid as the electrolyte, where proton conduction takes place. We studied the proton diffusion in such membranes with neutron scattering techniques which are especially sensitive to the proton contribution. Time of flight spectroscopy and backscattering spectroscopy have been combined to cover a broad dynamic range. In order to selectively observe the diffusion of protons potentially contributing to the ion conductivity, two samples were prepared, where in one of the samples the phosphoric acid was used with hydrogen replaced by deuterium. The scattering data from the two samples were subtracted in a suitable way after measurement. Thereby subdiffusive behavior of the proton diffusion has been observed and interpreted in terms of a model of fractal diffusion. For this purpose, a scattering function for fractal diffusion has been developed. The fractal diffusion dimension dw and the Hausdorff dimension df have been determined on the length scales covered in the neutron scattering experiments.

Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media

The anomalous diffusion in fractal isotropic/anisotropic porous media is characterized by the Hausdorff derivative diffusion model with the varying fractal orders representing the fractal structures in different directions. This paper presents a comprehensive understanding of the Hausdorff derivative diffusion model on the basis of the physical interpretation, the Hausdorff fractal distance and the fundamental solution. The concept of the Hausdorff fractal distance is introduced, which converges to the classical Euclidean distance with the varying orders tending to 1. The fundamental solution of the 3-D Hausdorff fractal derivative diffusion equation is proposed on the basis of the Hausdorff fractal distance. With the help of the properties of the Hausdorff derivative, the Huasdorff diffusion model is also found to be a kind of time-space dependent convection-diffusion equation underlying the anomalous diffusion behavior.

Point Transformations and Relationships among Linear Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

We present new connections among linear anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Anomalous Diffusion equations of O’Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we can tell) be expressed as examples of the O’Shaughnessy and Procaccia equations. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. We illustrate the ideas both analytically and with a detailed computational example for a non-trivial choice of point transformation. Finally, we summarize our results.