A note on a reaction-diffusion model describing the bone morphogen protein gradient in Drosophila embryonic patterning

In this article, we consider the Eldar model [3] from embryology in which a bone morphogenic protein, a short gastrulation protein, and their compound react and diffuse. We carry out a perturbation analysis in the limit of small diffusivity of the bone morphogenic protein. This analysis establishes conditions under which some elementary results of [3] are valid.

Near-Wall Turbulent Pressure Diffusion Modeling and Influence in Three-Dimensional Secondary Flows

The purpose of this paper is to develop a second-moment closure with a near-wall turbulent pressure diffusion model for three-dimensional complex flows, and to evaluate the influence of the turbulent diffusion term on the prediction of detached and secondary flows. A complete turbulent diffusion model including a near-wall turbulent pressure diffusion closure for the slow part was developed based on the tensorial form of Lumley and included in a re-calibrated wall-normal-free Reynolds-stress model developed by Gerolymos and Vallet. The proposed model was validated against several one-, two, and three-dimensional complex flows.

'Savoir fare': are cooking skills a new morality

There has been a recent surge of interest in cooking skills in a diverse range of fields, such as health, education and public policy. There appears to be an assumption that cooking skills are in decline and that this is having an adverse impact on individual health and well-being, and family wholesomeness. The problematisation of cooking skills is not new, and can be seen in a number of historical developments that have specified particular pedagogies about food and eating. The purpose of this paper is to examine pedagogies on cooking skills and the importance accorded them. The paper draws on Foucault’s work on governmentality. By using examples from the USA, UK and Australia, the paper demonstrates the ways that authoritative discourses on the know how and the know what about food and cooking – called here ‘savoir fare’ – are developed and promulgated. These discourses, and the moral panics in which they are embedded, require individuals to make choices about what to cook and how to cook, and in doing so establish moral pedagogies concerning good and bad cooking. The development of food literacy programmes, which see cooking skills as life skills, further extends the obligations to ‘cook properly’ to wider populations. The emphasis on cooking knowledge and skills has ushered in new forms of government, firstly, through a relationship between expertise and politics which is readily visible through the authority that underpins the need to develop skills in food provisioning and preparation; secondly, through a new pluralisation of ‘social’ technologies which invites a range of private-public interest through, for example, television cooking programmes featuring cooking skills, albeit it set in a particular milieu of entertainment; and lastly, through a new specification of the subject can be seen in the formation of a choosing subject, one which has to problematise food choice in relation to expert advice and guidance. A governmentality focus shows that as discourses develop about what is the correct level of ‘savoir fare’, new discursive subject positions are opened up. Armed with the understanding of what is considered expert-endorsed acceptable food knowledge, subjects judge themselves through self-surveillance. The result is a powerful food and family morality that is both disciplined and disciplinary.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Metallic and Carbon Nanotube-Catalyzed Coupling of Hydrogenation in Magnesium

Synergistic effect of metallic couple and carbon nanotubes on Mg results in an ultrafast kinetics of hydrogenation that overcome a critical barrier of practical use of Mg as hydrogen storage materials. The ultrafast kinetics is attributed to the metal−H atomic interaction at the Mg surface and in the bulk (energy for bonding and releasing) and atomic hydrogen diffusion along the grain boundaries (aggregation of carbon nanotubes) and inside the grains. Hence, a hydrogenation mechanism is presented.

Hydrogen spillover mechanism on a pd-doped mg surface as revealed by ab initio density functional calculation

The hydrogenation kinetics of Mg is slow, impeding its application for mobile hydrogen storage. We demonstrate by ab initio density functional theory (DFT) calculations that the reaction path can be greatly modified by adding transition metal catalysts. Contrasting with Ti doping, a Pd dopant will result in a very small activation barrier for both dissociation of molecular hydrogen and diffusion of atomic H on the Mg surface. This new computational finding supports for the first time by ab initio simulationthe proposed hydrogen spillover mechanism for rationalizing experimentally observed fast hydrogenation kinetics for Pd-capped Mg materials.

Modelling carbon membranes for gas and isotope separation

Molecular modelling has become a useful and widely applied tool to investigate separation and diffusion behavior of gas molecules through nano-porous low dimensional carbon materials, including quasi-1D carbon nanotubes and 2D graphene-like carbon allotropes. These simulations provide detailed, molecular level information about the carbon framework structure as well as dynamics and mechanistic insights, i.e. size sieving, quantum sieving, and chemical affinity sieving. In this perspective, we revisit recent advances in this field and summarize separation mechanisms for multicomponent systems from kinetic and equilibrium molecular simulations, elucidating also anomalous diffusion effects induced by the confining pore structure and outlining perspectives for future directions in this field.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

Deformation with coupled chemical diffusion

The deformation of rocks is commonly intimately associated with metamorphic reactions. This paper is a step towards understanding the behaviour of fully coupled, deforming, chemically reacting systems by considering a simple example of the problem comprising a single layer system with elastic-power law viscous constitutive behaviour where the deformation is controlled by the diffusion of a single chemical component that is produced during a metamorphic reaction. Analysis of the problem using the principles of non-equilibrium thermodynamics allows the energy dissipated by the chemical reaction-diffusion processes to be coupled with the energy dissipated during deformation of the layers. This leads to strain-rate softening behaviour and the resultant development of localised deformation which in turn nucleates buckles in the layer. All such diffusion processes, in leading to Herring-Nabarro, Coble or “pressure solution” behaviour, are capable of producing mechanical weakening through the development of a “chemical viscosity”, with the potential for instability in the deformation. For geologically realistic strain rates these chemical feed-back instabilities occur at the centimetre to micron scales, and so produce structures at these scales, as opposed to thermal feed-back instabilities that become important at the 100–1000 m scales.

Vertically-aligned carbon nanotube membranes for hydrogen separation

Vertically-aligned carbon nanotube membranes have been fabricated and characterized and the corresponding gas permeability and hydrogen separation were measured. The carbon nanotube diameter and areal density were adjusted by varying the catalyst vapour concentration (Fe/C ratio) in the mixed precursor. The permeances are one to two magnitudes higher than the Knudsen prediction, while the gas selectivities are still in the Knudsen range. The diameter and areal density effects were studied and compared, the temperature dependence of permeation is also discussed. The results confirm the existence of non-Knudsen transport and that surface adsorption diffusion may affect the total permeance at relative low temperature. The permeance of aligned carbon nanotube membranes can be improved by increasing areal density and operating at an optimum temperature.

Mathematical modelling of controlled drug release from polymer micro-spheres : incorporating the effects of swelling, diffusion and dissolution via moving boundary problems

Controlled drug delivery is a key topic in modern pharmacotherapy, where controlled drug delivery devices are required to prolong the period of release, maintain a constant release rate, or release the drug with a predetermined release profile. In the pharmaceutical industry, the development process of a controlled drug delivery device may be facilitated enormously by the mathematical modelling of drug release mechanisms, directly decreasing the number of necessary experiments. Such mathematical modelling is difficult because several mechanisms are involved during the drug release process. The main drug release mechanisms of a controlled release device are based on the device’s physiochemical properties, and include diffusion, swelling and erosion. In this thesis, four controlled drug delivery models are investigated. These four models selectively involve the solvent penetration into the polymeric device, the swelling of the polymer, the polymer erosion and the drug diffusion out of the device but all share two common key features. The first is that the solvent penetration into the polymer causes the transition of the polymer from a glassy state into a rubbery state. The interface between the two states of the polymer is modelled as a moving boundary and the speed of this interface is governed by a kinetic law. The second feature is that drug diffusion only happens in the rubbery region of the polymer, with a nonlinear diffusion coefficient which is dependent on the concentration of solvent. These models are analysed by using both formal asymptotics and numerical computation, where front-fixing methods and the method of lines with finite difference approximations are used to solve these models numerically. This numerical scheme is conservative, accurate and easily implemented to the moving boundary problems and is thoroughly explained in Section 3.2. From the small time asymptotic analysis in Sections 5.3.1, 6.3.1 and 7.2.1, these models exhibit the non-Fickian behaviour referred to as Case II diffusion, and an initial constant rate of drug release which is appealing to the pharmaceutical industry because this indicates zeroorder release. The numerical results of the models qualitatively confirms the experimental behaviour identified in the literature. The knowledge obtained from investigating these models can help to develop more complex multi-layered drug delivery devices in order to achieve sophisticated drug release profiles. A multi-layer matrix tablet, which consists of a number of polymer layers designed to provide sustainable and constant drug release or bimodal drug release, is also discussed in this research. The moving boundary problem describing the solvent penetration into the polymer also arises in melting and freezing problems which have been modelled as the classical onephase Stefan problem. The classical one-phase Stefan problem has unrealistic singularities existed in the problem at the complete melting time. Hence we investigate the effect of including the kinetic undercooling to the melting problem and this problem is called the one-phase Stefan problem with kinetic undercooling. Interestingly we discover the unrealistic singularities existed in the classical one-phase Stefan problem at the complete melting time are regularised and also find out the small time behaviour of the one-phase Stefan problem with kinetic undercooling is different to the classical one-phase Stefan problem from the small time asymptotic analysis in Section 3.3. In the case of melting very small particles, it is known that surface tension effects are important. The effect of including the surface tension to the melting problem for nanoparticles (no kinetic undercooling) has been investigated in the past, however the one-phase Stefan problem with surface tension exhibits finite-time blow-up. Therefore we investigate the effect of including both the surface tension and kinetic undercooling to the melting problem for nanoparticles and find out the the solution continues to exist until complete melting. The investigation of including kinetic undercooling and surface tension to the melting problems reveals more insight into the regularisations of unphysical singularities in the classical one-phase Stefan problem. This investigation gives a better understanding of melting a particle, and contributes to the current body of knowledge related to melting and freezing due to heat conduction.

Comment on "Local accumulation times for source, diffusion, and degradation models in two and three dimensions" [J. Chem. Phys. 138, 104121 (2013)]

In a recent paper, Gordon, Muratov, and Shvartsman studied a partial differential equation (PDE) model describing radially symmetric diffusion and degradation in two and three dimensions. They paid particular attention to the local accumulation time (LAT), also known in the literature as the mean action time, which is a spatially dependent timescale that can be used to provide an estimate of the time required for the transient solution to effectively reach steady state. They presented exact results for three-dimensional applications and gave approximate results for the two-dimensional analogue. Here we make two generalizations of Gordon, Muratov, and Shvartsman’s work: (i) we present an exact expression for the LAT in any dimension and (ii) we present an exact expression for the variance of the distribution. The variance provides useful information regarding the spread about the mean that is not captured by the LAT. We conclude by describing further extensions of the model that were not considered by Gordon,Muratov, and Shvartsman. We have found that exact expressions for the LAT can also be derived for these important extensions...