Anomalous diffusion in fractal porous medium

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The velocity of sand particles relative to water and the variation of fluctuation velocity due to the concentration of suspended sand particles

In this paper, we discuss three fundamental problems of the theory of suspended sand particles based on the theory of fluid dynamics. They are: (1) the relative velocity of sand particles to the surrounding fluid in the turbulent flow, (2) the variation of the velocity fluctuation of the fluid when the suspended sand particles exist, (3) the profile of the vertical distribution of the concentration of the suspended sand particles in two-dimensional uniform steady channel flow and the modification of usual diffusion theory. The results of our theoretical analysis are somewhat different from the corresponding expressions in the textbooks of hydraulics and river mechanics.     

Non-Markovian diffusion process in polymers and stretched exponential relaxation

In this contribution, we model the long-time behaviour of the desorption from an LDPE sheet, using non-Markovian random walks. It is shown that the mass of penetrant in the final stage of desorption decays as t ^–m , where m is proportional to the exponent of the probability distribution (t) t ^–(1+u), 0 < v < 1. Furthermore, it is shown that this model may lead to the so-called mechanical stretched exponential relaxation, and that Wagner's memory function can be obtained as a special case.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field

Based on the finite series method, the Gaussian standing or quasi-standing beam is expressed in terms of spherical wave functions and a weighting parameter, which describe the beam shape and location relative to the particle. An expression is derived for the radiation force on a sphere centered on the axis of a Gaussian standing or quasi-standing wave propagating in an ideal fluid. Rigid, fluid, elastic, and viscoelastic spheres immersed in water are treated as examples. In addition, a method is proposed to compute the axial acoustic radiation force when the sphere is translated axially. Results indicate the capability of the proposed method to manipulate and separate spheres based on their mechanical and acoustical properties. The interaction of a Gaussian quasi-standing beam with a sphere can result in periodic axial force under specific operating conditions. The results presented here may provide a theoretical basis for the development of acoustical tweezers in a Gaussian standing beam, which would be useful in micro-fluidic lab-on-chip applications.     

Tracer Dispersion in Rough Open Cracks

Tracer dispersion is studied in an open crack where the two rough crack faces have been translated with respect to each other. The different dispersion regimes encountered in rough-wall Hele-Shaw cell are first introduced, and the geometric dispersion regime in the case of self-affine crack surfaces is treated in detail through perturbation analysis. It is shown that a line of tracer is progressively wrinkled into a self-affine curve with an exponent equal to that of the crack surface. This leads to a global dispersion coefficient which depends on the distance from the tracer inlet, but which is still proportional to the mean advection velocity. Besides, the tracer front is subjected to a local dispersion (as could be revealed by point measurements or echo experiments) very different from the global one. The expression of this anomalous local dispersion coefficient is also obtained.     

Diffusion in pore fractals: A review of linear response models

A major aspect of describing transport in heterogeneous media has been that of relating effective diffusivities to the topological properties of the medium. While such effective transport coefficients may be useful for mass fractals or under steady state conditions, they are not adequate under transient conditions for self-similar pore fractal media. In porous formations without scale, diffusion is anomalous with the mean-squared displacement of a particle proportional to time raised to a fractional exponent less than unity. The objective of this review is to investigate the nature of the laws of diffusion in fractal media using the framework of linear response theory of nonequilibrium statistical mechanics. A Langevin/Fokker-Planck approach reveals that the particle diffusivity depends on its age defined as the time spent by the particle since its entry into the medium. An analysis via generalized hydrodynamics describes fractal diffusion with a frequency and wave number dependent diffusivity.     

Time Fractional Diffusion: A Discrete Random Walk Approach

The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.     

Reduced‐order modeling for nonlocal diffusion problems

In several settings, diffusive behavior is observed to not follow the rate of spread predicted by parabolic partial differential equations (PDEs) such as the heat equation. Such behaviors, often referred to as anomalous diffusion, can be modeled using nonlocal equations for which points at a finite distance apart can interact. An example of such models is provided by fractional derivative equations. Because of the nonlocal interactions, discretized nonlocal systems have less sparsity, often significantly less, compared with corresponding discretized PDE systems. As such, the need for reduced‐order surrogates that can be used to cheaply determine approximate solutions is much more acute for nonlocal models compared with that for PDEs. In this paper, we consider the construction, application, and testing of proper orthogonal decomposition (POD) reduced models for an integral equation model for nonlocal diffusion. For certain modeling parameters, the model we consider allows for discontinuous solutions and includes fractional Laplacian kernels as a special case. Preliminary computational results illustrate the potential of using POD to obtain accurate approximations of solutions of nonlocal diffusion equations at much lower costs compared with, for example, standard finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Unsteady MHD mixed convection flow over an impulsively stretched permeable vertical surface in a quiescent fluid

The unsteady mixed convection flow of an incompressible laminar electrically conducting fluid over an impulsively stretched permeable vertical surface in an unbounded quiescent fluid in the presence of a transverse magnetic field has been investigated. At the same time, the surface temperature is suddenly increased from the surrounding fluid temperature or a constant heat flux is suddenly imposed on the surface. The problem is formulated in such a way that for small time it is governed by Rayleigh type of equation and for large time by Crane type of equation. The non-linear coupled parabolic partial differential equations governing the unsteady mixed convection flow under boundary layer approximations have been solved analytically by using the homotopy analysis method as well as numerically by an implicit finite difference scheme. The local skin friction coefficient and the local Nusselt number are found to decrease rapidly with time in a small time interval and they tend to steady-state values for t*≥5. They also increase with the buoyancy force and suction, but decrease with injection rate. The local skin friction coefficient increases with the magnetic field, but the local Nusselt number decreases. There is a smooth transition from the unsteady state to the steady state.     

Concentration distribution of fractional anomalous diffusion caused by an instantaneous point source

The Fox function expression and the analytic expression for the concentration distribution of fractional anomalous diffusion caused by an instantaneous point source in n-dimensional space (n= 1, 2 or 3) are derived by means of the condition of mass conservation , the time-space similarity of the solution , Mellin transform and the properties of the Fox function . And the asymptotic behaviors for the solutions are also given .