One dimensional drift-diffusion between two absorbing boundaries: application to granular segregation

Motivated by a novel method for granular segregation, we analyze the one dimensional drift-diffusion between two absorbing boundaries. The time evolution of the probability distribution and the rate of absorption are given by explicit formulae, the splitting probability and the mean first passage time are also calculated. Applying the results we find optimal parameters for segregating binary granular mixtures.Comment: RevTeX, 5 pages, 6 figure

Spreading of a density front in the K\"untz-Lavall\'ee model of porous media

We analyze spreading of a density front in the K\"untz-Lavall\'ee model of porous media. In contrast to previous studies, where unusual properties of the front were attributed to anomalous diffusion, we find that the front evolution is controlled by normal diffusion and hydrodynamic flow, the latter being responsible for apparent enhancement of the front propagation speed. Our finding suggests that results of several recent experiments on porous media, where anomalous diffusion was reported based on the density front propagation analysis, should be reconsidered to verify the role of a fluid flow

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

Velocity profile of granular flows inside silos and hoppers

We measure the flow of granular materials inside a quasi-two dimensional silo as it drains and compare the data with some existing models. The particles inside the silo are imaged and tracked with unprecedented resolution in both space and time to obtain their velocity and diffusion properties. The data obtained by varying the orifice width and the hopper angle allows us to thoroughly test models of gravity driven flows inside these geometries. All of our measured velocity profiles are smooth and free of the shock-like discontinuities ("rupture zones") predicted by critical state soil mechanics. On the other hand, we find that the simple Kinematic Model accurately captures the mean velocity profile near the orifice, although it fails to describe the rapid transition to plug flow far away from the orifice. The measured diffusion length $b$, the only free parameter in the model, is not constant as usually assumed, but increases with both the height above the orifice and the angle of the hopper. We discuss improvements to the model to account for the differences. From our data, we also directly measure the diffusion of the particles and find it to be significantly less than predicted by the Void Model, which provides the classical microscopic derivation of the Kinematic Model in terms of diffusing voids in the packing. However, the experimental data is consistent with the recently proposed Spot Model, based on a simple mechanism for cooperative diffusion. Finally, we discuss the flow rate as a function of the orifice width and hopper angles. We find that the flow rate scales with the orifice size to the power of 1.5, consistent with dimensional analysis. Interestingly, the flow rate increases when the funnel angle is increased.Comment: 17 pages, 8 figure

Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models

Modelling topical photodynamic therapy treatment including the continuous production of Protoporphyrin IX

C L Campbell acknowledges financial support from an UK EPSRC PhD studentship (EP/K503162/1) and the Alfred Stewart Trust.Most existing theoretical models of photodynamic therapy (PDT) assume a uniform initial distribution of the photosensitive molecule, Protoporphyrin IX (PpIX). This is an adequate assumption when the prodrug is systematically administered; however for topical PDT this is no longer a valid assumption. Topical application and subsequent diffusion of the prodrug results in an inhomogeneous distribution of PpIX, especially after short incubation times, prior to light illumination. In this work a theoretical simulation of PDT where the PpIX distribution depends on the incubation time and the treatment modality is described. Three steps of the PpIX production are considered. The first is the distribution of the topically applied prodrug, the second in the conversion from the prodrug to PpIX and the third is the light distribution which affects the PpIX distribution through photobleaching. The light distribution is modelled using a Monte Carlo radiation transfer model and indicates treatment depths of around 2 mm during daylight PDT and approximately 3 mm during conventional PDT. The results suggest that treatment depths are not only limited by the light penetration but also by the PpIX distributionPostprintPeer reviewe

Driven diffusion in a periodically compartmentalized tube: homogeneity versus intermittency of particle motion

We study the effect of a driving force F on drift and diffusion of a point Brownian particle in a tube formed by identical ylindrical compartments, which create periodic entropy barriers for the particle motion along the tube axis. The particle transport exhibits striking features: the effective mobility monotonically decreases with increasing F, and the effective diffusivity diverges as F → ∞, which indicates that the entropic effects in diffusive transport are enhanced by the driving force. Our consideration is based on two different scenarios of the particle motion at small and large F, homogeneous and intermittent, respectively. The scenarios are deduced from the careful analysis of statistics of the particle transition times between neighboring openings. From this qualitative picture, the limiting small-F and large-F behaviors of the effective mobility and diffusivity are derived analytically. Brownian dynamics simulations are used to find these quantities at intermediate values of the driving force for various compartment lengths and opening radii. This work shows that the driving force may lead to qualitatively different anomalous transport features, depending on the geometry design

Behavior of Metallic Inclusions in Uranium Dioxide

The mobility of micron-size powders of refractory and noble metals in UO{sub 2} was investigated under isothermal and temperature gradient conditions, The metal particles were initially placed between two polished surfaces of UO{sub 2} and any movement which occurred during high temperature annealing was determined microscopically. Tungsten and molybdenum particles 1 to 10 {micro}m in diameter were immobile in UO{sub 2} at 2500°C in a temperature gradient of 1400°C/cm. Ruthenium, however, dissolved into and spread through hypostoichiometric, polycrystalline urania and was found after isothermal annealing as the U-Ru intermetallic compound in the grain boundaries of the oxide. The mechanism does not involve bodily motion of the metal particles. Rather, ruthenium dissolves in the grain bmmdaries of the oxide, migrates as atoms via the same pathway, and reacts while migrating to form URu{sub 3}, This product grows as layers in the grain boundaries. Isothermal ruthenium spreading followed simple diffusion theory, and apparent solubilities and effective diffusivities were obtained from the data for the temperature nmge 2000 to 2300°C. In a temperature gradient, ruthenium moves to the hot zones of UO{sub 2}; the mechanism appears to be the same as found for isothermal spreading, but the extent of movement up the temperature gradient cannot be explained by simple diffusion theory, even with an appreciable Soret effect