Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations

Study of the collapse of granular columns using DEM numerical simulation

Numerical simulations of the collapse and spreading of granular columns onto an horizontal plane using the Contact Dynamics method are presented. The final shape of the deposit seems to depend only on the aspect ratio $a$ of the columns; these results are in good agreement with previous experimental work. In particular, the renormalised runout distance shows a power law dependence on the aspect ratio $a$, which is incompatible with a simple friction model. The dynamics of the collapse is shown to be mostly controlled by the free fall of the column. The energy dissipation at the base of the column can be described simply by a coefficient of restitution. Hence the energy available for the sideways flow is proportional to the initial potential energy $E_0$. The dissipation process within the flow is well approximated by basal friction, contrary to the behaviour of the runout distance. The mass ejected sideways is showned to play a determining role in the spreading process. As $a$ increases, the same fraction of initial potential energy $E_0$ drives more mass against friction. This additional dissipation give a possible explanation for power-law dependence of the runout distance on $a$. Beyond the frictional properties of the material, we show that the flow characteristics strongly depend on the early dynamics of the collapse. We propose a new scaling for the runout distance that matches the data well, is compatible with a friction model, and provide a qualitative explanation to the column collapse phenomenology.Comment: 15 pages, 22 Figure

The rheology of a suspension of nearly spherical particles subject to Brownian rotations

A set of constitutive equations, valid for arbitrary linear bulk flows, is derived for a dilute suspension of nearly spherical, rigid particles which are subject to rotary Brownian couples. These constitutive equations are subsequently applied to find the resulting stress patterns for a variety of time-dependent bulk flow fields. The rheological responses are found to exhibit many of the same qualitative features as have been observed in recent experimental investigations of polymeric solutions and other complex materials

Conceptual models of the climate : 2003 program of study, Non-Newtonian geophysical fluid dynamics

Non-Newtonian fluids occur commonly in our world. These fluids, such as toothpaste, saliva, oils, mud and lava, exhibit a number of behaviors that are different from Newtonian fluids and have a number of additional material properties. In general, these differences arise because the fluid has a microstructure that influences the flow. In section 2 we will present a collection of some of the interesting phenomena arising from flow nonlinearities, the inhibition of stretching, elastic effects and normal stresses. In section 3 we will discuss a variety of devices for measuring material properties, a process known as rheometry.Funding was provided by the Office of Naval Research under Contract No. N00014-97-1-0934 and The National Science Foundation under Contract No. OCE 98-10647

Ink Drying in Inkjet Printers

The first problem put to the Study Group for Maths in Industry by Domino UK Ltd concerns ink drying and blocking nozzles in a printer. The goals were as follows: 1. To propose mechanisms for the growth of a plug of dried ink in the open end of a Drop-on-Demand drop generator, 2. To suggest cures to this problem, 3. To consider why oscillating the meniscus appears to alleviate the problem

Modelling an isolated dust grain in a plasma using matched asymptotic expansions

The study of dusty plasmas is of significant practical use and scientific interest. A characteristic feature of dust grains in a plasma is that they are typically smaller than the electron Debye distance, a property which we exploit using the technique of matched asymptotic expansions. We first consider the case of a spherical dust particle in a stationary plasma, employing the Allen–Boyd–Reynolds theory, which assumes cold, collisionless ions. We derive analytical expressions for the electric potential, the ion number density and ion velocity. This requires only one computation that is not specific to a single set of dust–plasma parameters, and sheds new light on the shielding distance of a dust grain. The extension of this calculation to the case of uniform ion streaming past the dust grain, a scenario of interest in many dusty plasmas, is less straightforward. For streaming below a certain threshold we again establish asymptotic solutions but above the streaming threshold there appears to be a fundamental change in the behaviour of the system

Wrinkling of microcapsules in shear flow

Elastic capsules can exhibit short wavelength wrinkling in external shear flow. We analyse this instability of the capsule shape and use the length scale separation between the capsule radius and the wrinkling wavelength to derive analytical results both for the threshold value of the shear rate and for the critical wave-length of the wrinkling. These results can be used to deduce elastic parameters from experiments.Comment: 4 pages, 2 figures, submitted to PR

Self-Diffusion of Drops in a Dilute Sheared Emulsion

Self-diffusion coefficients that describe cross-flow migration of non-Brownian drops in a dilute sheared emulsion were obtained by trajectory calculations. A boundary integral formulation was used to describe pairwise interactions between deformable drops; interactions between undeformed drops were described with mobility functions for spherical drops. The results indicate that drops have large anisotropic self-diffusivities which depend strongly on the drop viscosity and modestly on the shear-rate. Pairwise interactions between drops in shear-flow do not appreciably promote drop breakup