Bounds for systems of coupled advection-diffusion equations with application to the Poisson-Nernst-Plank equations.

We consider coupled systems of advection-diffusion equations with initial and boundary conditions and determine conditions on the advection terms that allow us to obtain solutions that can be explicitly bounded above and below using the initial and boundary conditions. Given the advection terms, using our methodology one can easily check if such bounds can be obtained and then one can construct the necessary nonlinear transformation to allow the bounds to be determined. We apply this technique to determine bounding quantities for a number of examples.

A continuum study of layer analysis for single species ion transport inside double-layered graphene sheets with various separations.

We investigate the formation of thin ionic layers driven by electro-osmotic forces, that are commonly found in micro- and nano-channels. Recently, multi-layers have been reported in the literature. However, the relation between classical Debye layers and multi-layers, which is a practically and fundamentally important question, was previously unexplained.

Dual-Gated Transistor Platform for On-Site Detection of Lead Ions at Trace Levels.

On-site monitoring of heavy metals in drinking water has become crucial because of several high profile instances of contamination. Presently, reliable techniques for trace level heavy metal detection are mostly laboratory based, while the detection limits of contemporary field-based methods are barely meeting the exposure limits set by regulatory bodies such as the World Health Organization (WHO). Here, we show an on-site deployable, Pb sensor on a dual-gated transistor platform whose lower detection limit is 2 orders of magnitude better than the traditional sensor and 1 order of magnitude lower than the exposure limit set by WHO.

Modelling of particle-laden flow inside nanomaterials.

In this paper, we demonstrate the usage of the Nernst-Planck equation in conjunction with mean-field theory to investigate particle-laden flow inside nanomaterials. Most theoretical studies in molecular encapsulation at the nanoscale do not take into account any macroscopic flow fields that are crucial in squeezing molecules into nanostructures. Here, a multi-scale idea is used to address this issue.

Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems.

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained.

Mathematical approaches to modeling of cortical spreading depression.

Migraine with aura (MwA) is a debilitating disease that afflicts about 25%-30% of migraine sufferers. During MwA, a visual illusion propagates in the visual field, then disappears, and is followed by a sustained headache. MwA was conjectured by Lashley to be related to some neurological phenomenon.

A mathematical model of the metabolic and perfusion effects on cortical spreading depression.

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism-aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD.

Periodic orbits of inelastic particles on a ring.

We consider the dynamics of N rigid particles of arbitrary mass that are constrained to move on a frictionless ring. Collisions between particles are inelastic with a constant coefficient of restitution e, and between collisions the particles move with constant velocity. We study sequences of collisions that are self-similar in the sense that the relative positions return to their original relative positions after the collision sequence while the relative velocities are reduced by a constant factor.

Critical role of friction for a single particle falling through a funnel.

We investigate a single frictional, inelastic, spherical particle falling under gravity through a symmetric funnel. A recent study showed that, for a frictionless particle in such a system, several anomalous phenomena occur: The particle can stay longer, lose more energy, and exert more impulsive force in a funnel with steeper walls. For frictionless particles, such phenomena exist for many small ranges of funnel angles and are a consequence of the many possible repeated patterns in particle trajectories.

Restricted diffusion in cellular media: (1+1)-dimensional model.

We consider the diffusion of molecules in a one-dimensional medium consisting of a large number of cells separated from the extra-cellular space by permeable membranes. The extra-cellular space is completely connected and allows unrestricted diffusion of the molecules. Furthermore, the molecules can diffuse within a given cell, i.

Deflection of a dilute stream of particles.

We consider a two-dimensional system in which a dilute stream of particles collides with an oblique planar wall. Both collisions between particles and collisions between particles and the wall are inelastic. We perform numerical simulations in two dimensions and show that the mean force experienced by the wall can be a nonmonotonic function of the angle between the wall and the particle stream.

Degenerate orbit transitions in a one-dimensional inelastic particle system.

Continuous transitions between different periodic orbits in a one-dimensional inelastic particle system are investigated. We show that continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. We give a full mechanical description of the system and explain why this is the case.

Driven inelastic-particle systems with drag.

We study steady states of the motion of a large number of particles in a closed box that are excited by a vibrating boundary and experience a linear drag force from the interstitial fluid. The dissipation in such systems arises from two main sources: Inelasticity in particle collisions and the effects of interstitial fluid on the particles. In many applications, order of magnitude estimates suggest that the dissipation due to interstitial fluid effects may greatly exceed that due to inelasticity and one is naturally led to neglect inelastic effects.

Anomalous behavior of a single particle falling through a funnel.

We show several surprising phenomena that occur in an extremely simple system of a single frictionless, inelastic, spherical particle falling under gravity through a symmetric funnel. One might naively expect that particles would fall through funnels with steeper sides more quickly, exert a smaller total impulse on the funnel walls, and lose less energy. However, we show that there are special ranges of angles of the funnel walls for which exactly the opposite occurs.

Anomalous Richtmyer-Meshkov fingering in dissipative particle systems.

We study the fingering instability induced by a shock that propagates across a perturbed interface that separates two types of discrete particles. If collisions between particles conserve energy, then the relative sizes and growth rates of the fingers are similar to those in the analogous shock-induced fingering instability in fluids. However, we show that energy loss during particle collisions, even when very small, causes the qualitative features of the finger growth to be completely opposite to the fluid case.

Traveling waves in coupled reaction-diffusion models with degenerate sources.

We consider a general system of coupled nonlinear diffusion equations that are characterized by having degenerate source terms and thereby not having isolated rest states. Using a general form of physically relevant source terms, we derive conditions that are required to trigger traveling waves when a stable uniform steady-state solution is perturbed by a highly localized disturbance. We show that the degeneracy in the source terms implies that traveling waves have a number of surprising properties that are not present for systems with nondegenerate source terms.

Collapse of periodic orbits in a driven inelastic particle system.

The dynamical behavior of a one-dimensional inelastic particle system with particles of unequal mass traveling between two walls is investigated. The system is driven by adding energy at one of the walls while the other wall is stationary and does not add energy. By deriving analytic solutions for the periodic orbits of this system, we show that there are a countable infinity of critical mass ratios at which the particle dynamics become highly degenerate in the following sense.