Pulse dynamics in a three-component system : stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.

A variable diffusivity model for the drying of spherical food particulates

An investigation of the drying of spherical food particles was performed, using peas as the model material. In the development of a mathematical model for drying curves, moisture diffusion was modelled using Fick’s second law for mass transfer. The resulting partial differential equation was solved using a forward-time central-space finite difference approximation, with the assumption of variable effective diffusivity. In order to test the model, experimental data was collected for the drying of green peas in a fluidised bed at three drying temperatures. Through fitting three equation types for effective diffusivity to the data, it was found that a linear equation form, in which diffusivity increased with decreasing moisture content, was most appropriate. The final model accurately described the drying curves of the three experimental temperatures, with an R2 value greater than 98.6% for all temperatures.

Pupil shape as viewed along the horizontal visual field

Purpose: Changes in pupil size and shape are relevant for peripheral imagery by affecting aberrations and how much light enters and/or exits the eye. The purpose of this study is to model the pattern of pupil shape across the complete horizontal visual field and to show how the pattern is influenced by refractive error. Methods: Right eyes of thirty participants were dilated with 1% cyclopentolate and images were captured using a modified COAS-HD aberrometer alignment camera along the horizontal visual field to ±90°. A two lens relay system enabled fixation at targets mounted on the wall 3m from the eye. Participants placed their heads on a rotatable chin rest and eye rotations were kept to less than 30°. Best-fit elliptical dimensions of pupils were determined. Ratios of minimum to maximum axis diameters were plotted against visual field angle. Results: Participants’ data were well fitted by cosine functions, with maxima at (–)1° to (–)9° in the temporal visual field and widths 9% to 15% greater than predicted by the cosine of the field angle . Mean functions were 0.99cos[( + 5.3)/1.121], R2 0.99 for the whole group and 0.99cos[( + 6.2)/1.126], R2 0.99 for the 13 emmetropes. The function peak became less temporal, and the width became smaller, with increase in myopia. Conclusion: Off-axis pupil shape changes are well described by a cosine function which is both decentered by a few degrees and flatter by about 12% than the cosine of the viewing angle, with minor influences of refraction.

Bifurcations to travelling planar spots in a three-component FitzHugh-Nagumo system

In this article, we analyse bifurcations from stationary stable spots to travelling spots in a planar three-component FitzHugh-Nagumo system that was proposed previously as a phenomenological model of gas-discharge systems. By combining formal analyses, center-manifold reductions, and detailed numerical continuation studies, we show that, in the parameter regime under consideration, the stationary spot destabilizes either through its zeroth Fourier mode in a Hopf bifurcation or through its first Fourier mode in a pitchfork or drift bifurcation, whilst the remaining Fourier modes appear to create only secondary bifurcations. Pitchfork bifurcations result in travelling spots, and we derive criteria for the criticality of these bifurcations. Our main finding is that supercritical drift bifurcations, leading to stable travelling spots, arise in this model, which does not seem possible for its two-component version.

A variable diffusivity model for the drying of spherical food particulates

An investigation of the drying of spherical food particles was performed, using peas as the model material. In the development of a mathematical model for drying curves, moisture diffusion was modelled using Fick’s second law for mass transfer. The resulting partial differential equation was solved using a forward-time central-space finite difference approximation, with the assumption of variable effective diffusivity. In order to test the model, experimental data was collected for the drying of green peas in a fluidised bed at three drying temperatures. Through fitting three equation types for effective diffusivity to the data, it was found that a linear equation form, in which diffusivity increased with decreasing moisture content, was most appropriate. The final model accurately described the drying curves of the three experimental temperatures, with an R2 value greater than 98.6% for all temperatures.

Dust charge and ion drag forces in a high-voltage, capacitive radio frequency sheath

The charge of an isolated dust grain and ion drag forces on the grain in a collisionless, high-voltage, capacitive rf sheath are studied theoretically. The studies are carried out assuming that the positive ions are monoenergetic, as well as in more realistic approximation, assuming that the time-averaged energy distribution of ions impinging on the dust grain has a double-peaked hollow profile. For the nonmonoenergetic case, an analytical expression for the ion flux to the dust grain is obtained. It is studied how the dust charge and ion drag forces depend on the rf frequency, electron density at plasma-sheath boundary, electron temperature and ratio of the effective oscillation amplitude of rf current to the electron Debye length. It is shown that the dust charge and ion drag forces obtained in the monoenergetic ion approximation may differ from those calculated assuming that the ions are nonmonoenergetic. The difference increases with increasing the width of the ion energy spread in the ion distribution. © 2009 American Institute of Physics.

Natural convection heat transfer in a baffled triangular enclosure

To reduce the natural convection heat loss from enclosures many researchers used convection suppression devices in the past. In this study a single baffle is used under the top tip to investigate numerically the natural convection heat loss in an attic shaped enclosure which is a cost effective approach. The case considered here is one inclined wall of the enclosure is uniformly heated while the other inclined wall is uniformly cooled with adiabatic bottom wall. The finite volume method has been used to discretize the governing equations, with the QUICK scheme approximating the advection term. The diffusion terms are discretized using central-differencing with second order accuracy. A wide range of governing parameters are studied (Rayleigh number, aspect ratio, baffle length etc.). It is observed that the heat transfer due to natural convection in the enclosure reduces when the baffle length is increased. Effects of other parameters on heat transfer and flow field are described in this study.

Butterfly catastrophe for fronts in a three-component reaction–diffusion system

We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically.

Finite volume methods for simulating anomalous transport

In this thesis a new approach for solving a certain class of anomalous diffusion equations was developed. The theory and algorithms arising from this work will pave the way for more efficient and more accurate solutions of these equations, with applications to science, health and industry. The method of finite volumes was applied to discretise the spatial derivatives, and this was shown to outperform existing methods in several key respects. The stability and convergence of the new method were rigorously established.

Segmental bioelectrical impedance in the assessment of body composition of Sri Lankan children

Abstract: Over the years bioelectrical impedance assay (BIA) has gained popularity in the assessment of body composition. However, equations for the prediction of whole body composition use whole body BIA. This study attempts to evaluate the usefulness of segmental BIA in the assessment of whole body composition. A cross sectional descriptive study was conducted at the Professorial Paediatric Unit of Lady Ridgeway Hospital, Colombo, involving 259 (M/F:144/115) 5 to 15 year old healthy children. The height, weight, total and segmental BIA were measured and impedance indices and specific resistivity for the whole body and segments were calculated. Segmental BIA indices showed a significant association with whole body composition measures assessed by total body water (TBW) using the isotope dilution method (D2O). Impedance index was better related to TBW and fat free mass (FFM), while specific resistivity was better related to the fat mass of the body. Regression equations with different combinations of variables showed high predictability of whole body composition. Results of this study showed that segmental BIA can be used as an alternative approach to predict the whole body composition in Sri Lankan children.

A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems

In order to simulate stiff biochemical reaction systems, an explicit exponential Euler scheme is derived for multidimensional, non-commutative stochastic differential equations with a semilinear drift term. The scheme is of strong order one half and A-stable in mean square. The combination with this and the projection method shows good performance in numerical experiments dealing with an alternative formulation of the chemical Langevin equation for a human ether a-go-go related gene ion channel mode

Field performance of in-service cast iron gas reticulation pipe buried in reactive clay

Field instrumentation of an in-service cast iron gas pipe buried in a residential area is detailed in this paper. The aim of the study was to monitor the long-term pipe behavior to understand the mechanisms of pipe bending in relation to ground movement as a result of seasonal fluctuation of soil moisture content. Field data showed that variation of soil temperature, suction, and moisture content are closely related to the prevailing climate. Change of soil temperature is generally related to the ambient air temperature, with a variation of approximately −3°C −3°C per meter depth from the ground surface in summer (decrease with depth) and winter (increase with depth). Seasonal cyclic variation in moisture content was observed with maxima in February and March, and a minimum around September. The pipe top was under tensile strain during summer and subsequently subjected to compressive strain as soil swelling occurred as a result of increase in moisture content. The study suggests that downward pipe bending occurs in summer because of soil shrinkage, while upward pipe bending occurs in winter when the soil swells.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.

Ability of commonly used prediction equations to predict resting energy expenditure in children with inflammatory bowel disease

Background: Paediatric onset inflammatory bowel disease (IBD) may cause alterations in energy requirements and invalidate the use of standard prediction equations. Our aim was to evaluate four commonly used prediction equations for resting energy expenditure (REE) in children with IBD. Methods: Sixty-three children had repeated measurements of REE as part of a longitudinal research study yielding a total of 243 measurements. These were compared with predicted REE from Schofield, Oxford, FAO/WHO/UNU, and Harris-Benedict equations using the Bland-Altman method. Results: Mean (±SD) age of the patients was 14.2 (2.4) years. Mean measured REE was 1566 (336) kcal per day compared with 1491 (236), 1441 (255), 1481 (232), and 1435 (212) kcal per day calculated from Schofield, Oxford, FAO/WHO/UNU, and Harris-Benedict, respectively. While the Schofield equation demonstrated the least difference between measured and predicted REE, it, along with the other equations tested, did not perform uniformly across all subjects, indicating greater errors at either end of the spectrum of energy expenditure. Smaller differences were found for all prediction equations for Crohn's disease compared with ulcerative colitis. Conclusions: Of the commonly used equations, the equation of Schofield should be used in pediatric patients with IBD when measured values are not able to be obtained. (Inflamm Bowel Dis 2010;) Copyright © 2010 Crohn's & Colitis Foundation of America, Inc.