Steven I. Barry

Finite difference schemes for multilayer diffusion

Although numerical methods have been developed for diffusion through single layer materials, few have been developed for multiple layers. Diffusion processes through a multilayered material are of interest for a wide range of applications, including industrial, biological, electrical, and environmental areas. We present finite difference schemes for multilayered materials with a range of matching conditions between the layers, in particular for a jump matching condition. We show the finite difference methods are flexible, simple to implement, and help illustrate interesting behaviour in multilayered diffusion.

Modelling the effect of seasonal influenza vaccination on the risk of pandemic influenza infection

BackgroundRecent studies have suggested that vaccination with seasonal influenza vaccine resulted in an apparent higher risk of infection with pandemic influenza H1N1 2009. A simple mathematical model incorporating strain competition and a hypothesised temporary strain-transcending immunity is constructed to investigate this observation. The model assumes that seasonal vaccine has no effect on the risk of infection with pandemic influenza.ResultsResults of the model over a range of reproduction numbers and effective vaccination coverage confirm this apparent increased risk in the Northern, but not the Southern, hemisphere. This is due to unvaccinated individuals being more likely to be infected with seasonal influenza (if it is circulating) and developing hypothesised temporary immunity to the pandemic strain. Because vaccinated individuals are less likely to have been infected with seasonal influenza, they are less likely to have developed the hypothesised temporary immunity and are therefore more likely to be infected with pandemic influenza. If the reproduction number for pandemic influenza is increased, as it is for children, an increase in the apparent risk of seasonal vaccination is observed. The maximum apparent risk effect is found when seasonal vaccination coverage is in the range 20-40%.ConclusionsOnly when pandemic influenza is recently preceded by seasonal influenza circulation is there a modelled increased risk of pandemic influenza infection associated with prior receipt of seasonal vaccine.

DYNAMICS OF HUMAN MILK EXTRACTION: A COMPARATIVE STUDY OF BREAST FEEDING AND BREAST PUMPING

We describe a mathematical model of the flow and deformation in a human teat. Our aim is to compare the theoretical milk yield during infant breast feeding with that obtained through the use of a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk, whereas breast pumps use a cyclic pattern of suction only. Our model is based on quasi-linear poroelasticity whereby the teat is modelled as a cylindrical porous elastic material saturated with fluid. We impose a cyclic axial suction pressure difference across the teat and impose a radial compressive force moving along the teat which mimics infant suckling. This is compared to the case of cyclic and steady pumping only which models the action of breast pumps. The results illustrate that there is an optimal time to apply the compressive force during the suction cycle that will increase the flow rate in our theoretical teat. The model and results may be of use in the future design of effective breast pumps.

Obtaining the concentration-dependent diffusion coefficient from ultrafiltration experiments: Application to hyaluronate

A method for determining the concentration-dependent mutual diffusion coefficient D(C) of a macromolecule-solvent combination over a wide macromolecular concentration range is presented. All necessary data are gathered from a single experiment, in which polymer concentration profiles are measured during one-dimensional dead-end ultrafiltration. Based on these profiles, the convection-diffusion equation is used to deduce the dependence of D on C. To demonstrate the utility of the approach, studies using sodium hyaluronate dissolved in either 10 mM NaCl or a phosphate buffer were carried out. For hyaluronate (HA) in 10 mM NaCl, the mutual diffusion coefficient varies approximately linearly with concentration according to D(C) = 4.1 × 10−6 C0.96 cm2/s in the range 0 ≤ C ≤ 0.6 mass %, for C expressed in mass %. However, transition points (slope changes) in the D(C) curve are present at C ≅ 0.7 mass % and C ≅ 1.4 mass %. For HA in the phosphate buffer, the mutual diffusion coefficient is well described by D(C) = 1.9 × 10−6 C0.825, for 0 ≤ C ≤ 1.8 mass %. These values agree well with previously published data. The technique is robust, and permits reasonably high polymer concentrations to be easily studied.

Comparing Breastfeeding and Breast Pumps Using a Computer Model

There is a role for computer models in increasing the understanding of milk extraction from the human teat. A computer model can be used to investigate aspects of extracting milk from the human teat which are not feasible using clinical experiments. In this paper, the behavior of the human teat during an infant suckling and with the use of a breast pump is modeled. The model is used to (1) identify the role of suction and the peristaltic motion of the tongue during suckling and (2) compare the volume of milk extracted by an infant breastfeeding with that obtained using a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk. Breast pumps use a cyclic pattern of suction only. In the model, the human teat is represented as a cylindrical porous elastic material saturated with fluid. We mimic an infant suckling by imposing both suction and a peristaltic force in the computer model of the human teat. This is compared to the effect of suction only, which models the action of breast pumps. The results demonstrate that there is an optimal time to apply the peristaltic force during the suction cycle which will increase the milk volume. The model and results may be of use in the future design of effective breast pumps.

Flow and deformation in poroelasticity-II numerical method

A numerical method based on finite difference approximations is presented for the calculation of the time dependent deformation, pressure and flow within a finite two-dimensional poroelastic medium. The method allows for sources or sinks of fluid within the medium, slip or no slip boundaries, a permeable or impermeable free surface and applied pressures or forces at the boundaries. The applications of this method are varied and include filtration problems, soil consolidation, and the indentation of biological tissue.

Flow and deformation in poroelasticity-I unusual exact solutions

We briefly discuss some unusual solutions to the poroelastic equations governing flow and deformation within an elastic porous material. These solutions are ideal for testing computer modelling codes used in poroelasticity especially for irregular geometries and nonconstant permeabilities.

Deformation and fluid flow due to a source in a poro-elastic layer

Abstract The flow of fluid from a point source or sink at some arbitrary height in a layer of deformable porous material is considered. This problem is applicable to filtration through beds of sand and flow in soils. The porous material is assumed to be an isotropic, homogeneous, linear elastic solid. The equations governing the behavior of the medium and fluid are derived for an axisymmetric geometry using linear poro-elasticity theory and are solved using the Hankel transform with the Hankel inversion integral evaluated numerically. The upper surface is stress free and permeable, with the lower surface impermeable to fluid flow. Two different boundary conditions are applied to the lower surface; stress free and tethered. Results are given for the pressure contours, surface fluid velocity, and the displacement of the solid matrix for a variety of source heights and boundary conditions. These results provide an indication of the amount of swelling of the medium and subsequent deformation of the free surface as a function of the location of the point source and boundary conditions.

Critical times in one- and two-layered diffusion

Summary The study of diffusion is commonplace in engineering mathematics courses as a classic example of partial differential equations and separation of variables. However, many textbooks stop at a derived solution without going further to explore what the solution means. The analysis of the critical diffusion time is used here to demonstrate how the solutions obtained can be used to explore additional useful results for diffusion through a single layer of material. We also show how consideration of diffusion through two layers gives rise to some surprising new results. This problem was motivated by analysing the annealing of steel coils, where knowledge of the time to heat a system of air and steel layers is critical in the manufacturing process.

Heat Transfer During Annealing of Steel Coils

Steel becomes brittle during the cold rolling process which is used to produce sheet metal. Heat treatment (annealing) is required to release stresses and reform the crystalline structure. The 2008 Mathematics-and Statistics-in- Industry Study Group in Wollongong (MISG08) modelled the approach used by New Zealand Steel for which steel coils are heated in a batch annealing furnace. Determining the temperature within each coil is complicated by height-dependent gaps within the coils. Deciding on suitable boundary conditions for the outside of the coils provides a further challenge. This is explored with two alternative models. Having made reasonable assumptions, a linear model is found to be sufficient for modelling the heating process and allows the cold point in the steel coil to be established.