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This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries (LIBs) as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler–Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in LIBs are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of upscaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo-2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed. [ABSTRACT FROM AUTHOR]

Heat transport in granular and porous media occurs through conduction in the solid and radiation through the voids. By exploiting the separation of length scales between the small typical particles or voids and the large size of whole region, the method of multiple scales can be applied. For a purely diffusive system, this yields a problem on the macroscale with an effective conductivity, deduced by solving a 'cell problem' on the microscale. Here, we apply the method when radiation and conduction are both present; however, care must be taken to correctly handle the integral nature of the radiative boundary condition. Again, an effective conductivity is found by solving a 'cell problem' which, because of the non-linearity of radiative transfer, to be solved for each temperature value. We also incorporate modifications to the basic theory of multiple scales in order to deal with the non-local nature of the radiative boundary condition. We derive the multiple scales formulation of the problem and report on numerical comparisons between the homogenised problem and direct solution of the problem. We also compare the effective conductivity to that derived using Maxwell models and effective medium theory. [ABSTRACT FROM AUTHOR]

This work presents a methodology for coupling two open-source modelling frameworks in a highly parallel fashion across multiple length scales to solve an electrical current and heat transport problem for commercial cylindrical lithium-ion batteries. The global current and heat transfer problems are formulated as resistor networks and solved using a finite difference method on a network extracted from an X-ray tomogram of an MJ1 18650 battery. The electrochemistry governing the heat generation is solved at the local level using a physically parameterized model. Electrochemical models are solved for different regions of a spirally wound cylindrical cell in parallel, coupled via charge conservation at the current collectors in a “battery of batteries” fashion, similar to the concept of modelling a pack. Thermal connections between layers in the spiral winding are established and heat transport is solved globally in a two-dimensional fashion, allowing for the subsequent extension to three dimensions. Great heterogeneity in local current density is predicted by the model which is also found to have some temperature dependence with ramifications for battery degradation. [ABSTRACT FROM AUTHOR]

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct. [ABSTRACT FROM AUTHOR]

A drift-diffusion model for charge transport in an organic bulk heterojunction solar cell, formed by conjoined acceptor and donor materials sandwiched between two electrodes, is formulated. The model accounts for (i) bulk photogeneration of excitons, (ii) exciton drift and recombination, (iii) exciton dissociation (into polarons) on the acceptor–donor interface, (iv) polaron recombination, (v) polaron dissociation into a free electron (in the acceptor) and a hole (in the donor), (vi) electron/hole transport and (vii) electron–hole recombination on the acceptor–donor interface. A finite element method is employed to solve the model in a cell with a highly convoluted acceptor/donor interface. The solutions show that, with physically realistic parameters, and in the power generating regime, the solution varies little on the scale of the micro-structure. This motivates us to homogenise over the micro-structure; a process that yields a far simpler one-dimensional effective medium model on the cell scale. The comparison between the solution of the full model and the effective medium (homogenised) model is very favourable for applied voltages less than the built-in voltage (the power generating regime) but breaks down as the applied voltages increases above it. Furthermore, it is noted that the homogenisation technique provides a systematic way to relate effective medium modelling of bulk heterojunctions [19, 25, 36, 37, 42, 59] to a more fundamental approach that explicitly models the full micro-structure [8, 38, 39, 58] and that it allows the parameters in the effective medium model to be derived in terms of the geometry of the micro-structure. Finally, the effective medium model is used to investigate the effects of modifying the micro-structure geometry, of a device with an interdigitated acceptor/donor interface, on its current–voltage curve. [ABSTRACT FROM PUBLISHER]

Descemet membrane detachment (DMD) is a rare but potentially serious surgical complication that arises most often during cataract surgery. A recent study (Couch, S. M. & Baratz, K. H. (2009) Cornea, 28, 1160–1163) cited the case of a patient with DMDs in both eyes, noting that though one detachment was surgically repaired, the other spontaneously reattached and needed no further treatment. A fluid mechanical model of buoyancy-driven aqueous humour flow in the anterior chamber around a DMD is developed to explain this phenomenon. The analytical model is based on the lubrication theory limit of the Navier–Stokes equations. The flow is determined for a fixed geometry and the possible motion of the DMD is then analysed. Numerical calculations are also carried out (using COMSOL© Multiphysics) to confirm the lubrication theory results. The analytical and numerical results both show that, under the correct conditions, either spontaneous reattachment or worsening of the tear may occur. We conclude that it is possible that clinical outcomes for DMDs may be controlled by adjusting the temperature difference across the eye and/or the orientation of the patient. [ABSTRACT FROM PUBLISHER]

In this study, a previously derived two-dimensional model is used to describe the slow spreading of viscous films on the surface of a quiescent deep viscous pool due to gravity. It is assumed that the densities and viscosities of the fluids in the films and pool are comparable, but may be different. It is also assumed that surface tension effects are negligible. The fluid in the films and in the pool are both modelled using the Stokes flow equations. By exploiting the slenderness of the spreading films, asymptotic techniques are used to analyse the flow. It is shown that the dominant forces controlling the spreading are gravity and the tangential stress induced in the films by the underlying pool. As a consequence, the rate of spreading of the films is independent of their viscosity. For the case special of a symmetric configuration of films on the surface of the pool, the flow is studied by assuming that the solution becomes self-similar and hence the problem is recast in a self-similar coordinate system. Stokeslet analysis is then used to derive a singular integral equation for the stresses on the interfaces between the films and the pool. The form of this integral equation depends on the configuration of spreading films that are to be considered. A number of different cases are then studied, namely, a single film, two films, and an infinite periodic array of films. Finally, some results are derived that apply to a general symmetric configuration of films. It is shown that the profile of a spreading film close to its front (where the film thickness becomes zero) is proportional to x1/4. It is also shown that fronts move, and hence, the distance between adjacent fronts increases proportional to t1/3. [ABSTRACT FROM AUTHOR]

Analytical and numerical modelling of the fabrication of optical waveguides in lithium niobate (LiNbO3) through direct writing with a continuous wave ultraviolet (UV) laser has been performed. It is proposed that the UV illumination induces heating of the surface and subsequent thermal diffusion of lithium which alters the refractive index in accordance with the lithium concentration dependence of the Sellmeier equations. [ABSTRACT FROM AUTHOR]

This paper investigates the deposition of the tear film on the cornea of the human eye. The tear film is laid down by the motion of the upper eyelid and then subsequently flows and thins. Of particular interest is the stability of the tear layer and the development of dry patches on the cornea. While there has been significant research on the behaviour of tear films between blinks, this paper focuses on understanding the mechanisms which control the shape and thickness of the deposited film and how this affects the subsequent film behaviour. Numerical and analytical methods are applied to a lubrication model which includes the effects of surface tension, viscosity, gravity and evaporation. The model reveals the importance of the eyelid velocity, motion of the surface lipid layer and the storage of tear film between blinks. [ABSTRACT FROM AUTHOR]

A model is developed for the motion of cells within a multicell spherical tumour. The model allows either for the intercellular forces to be in compression and cells to be compacted to a fixed number density, or for the cell number density to fall and cells to become isolated from each other. The model develops necrotic regions naturally due to force balances rather than being directly attributable to a critical oxygen concentration. These necrotic regions may result in a gradual reduction in local cell density rather than jump to a completely dead region. [ABSTRACT FROM PUBLISHER]

We consider the dynamic growth of a tumour, concentrating on the possible development of a necrotic region and examine some simple tumour geometries in detail. The growth and death rates of the cells in the viable rim of the tumour are taken to be determined by the local oxygen concentration. Crucially the cell motion is determined by the forces generated by cell affinity, by cell interaction and by the need to get the waste products of cell death, primarily water, out of the tumour and products for cell growth, again primarily water, into the tumour. A consolidation type model with surface tension on the cells, slow viscous flow of the cells and porous media flow of the extracellular water is derived. The dynamic behaviour of this model is examined. Considering the very simple case where resistance to extracellular water flow dominates the problem, the model accounts naturally for the formation of a necrotic region. In regions where the extracellular water pressure gets too large, the cells are assumed to be ripped from the extracellular matrix and die. This model contrasts significantly from previous models which typically assume a necrotic region exists and that its behaviour is primarily governed directly by oxygen concentration. Here, the stress determines the necrotic region behaviour and this is affected by the oxygen only indirectly through the cell growth and death rates. The predicted time-dependent growth of one-dimensional, and spherical tumours are illustrated by numerical calculations. [ABSTRACT FROM AUTHOR]

The problem of fluid flow in a two-dimensional pleated filter is considered. Of particular interest is the change in the flow due to cake build-up on the surface of the filter material. The flow is taken to be Darcy flow in the cake and the filter material, with Stokes' flow outside the cake. The particles in the flow are taken to be transported with the flow and to stick to the cake without slippage or resuspension, and the cake is taken to be incompressible. The flow is considered in various geometries, particularly long thin filters and corners. The main parameter in the problem is the ratio of the filter-material resistance to the cake resistance, and limiting cases are considered. Travelling waves of cake build-up are found for arbitrary time-dependent variations in the inflow conditions. The time taken for the filter to become clogged by the cake is also considered. [ABSTRACT FROM PUBLISHER]

We develop a mathematical model to describe the regeneration of a hydrogel inserted into an ex vivo osteochondral explant. Specifically we use partial differential equations to describe the evolution of two populations of cells that migrate from the tissue surrounding the defect, proliferate, and compete for space and resources within the hydrogel. The two cell populations are chondrocytes and cells that infiltrate from the subchondral bone. Model simulations are used to investigate how different seeding strategies and growth factor placement within the hydrogel affect the spatial distribution of both cell types. Since chondrocyte migration is extremely slow, we conclude that the hydrogel should be seeded with chondrocytes prior to culture in order to obtain zonal chondrocyte distributions typical of those associated with healthy cartilage., (Copyright © 2017 Elsevier Inc. All rights reserved.)

Placental amino acid transfer is essential for fetal development and its impairment is associated with poor fetal growth. Amino acid transfer is mediated by a broad array of specific plasma membrane transporters with overlapping substrate specificity. However, it is not fully understood how these different transporters work together to mediate net flux across the placenta. Therefore the aim of this study was to develop a new computational model to describe how human placental amino acid transfer functions as an integrated system. Amino acid transfer from mother to fetus requires transport across the two plasma membranes of the placental syncytiotrophoblast, each of which contains a distinct complement of transporter proteins. A compartmental modelling approach was combined with a carrier based modelling framework to represent the kinetics of the individual accumulative, exchange and facilitative classes of transporters on each plasma membrane. The model successfully captured the principal features of transplacental transfer. Modelling results clearly demonstrate how modulating transporter activity and conditions such as phenylketonuria, can increase the transfer of certain groups of amino acids, but that this comes at the cost of decreasing the transfer of others, which has implications for developing clinical treatment options in the placenta and other transporting epithelia., (Copyright © 2016 The Authors. Published by Elsevier B.V. All rights reserved.)

Placental amino acid transport is required for fetal development and impaired transport has been associated with poor fetal growth. It is well known that placental amino acid transport is mediated by a broad array of specific membrane transporters with overlapping substrate specificity. However, it is not fully understood how these transporters function, both individually and as an integrated system. We propose that mathematical modelling could help in further elucidating the underlying mechanisms of how these transporters mediate placental amino acid transport. The aim of this work is to model the sodium independent transport of serine, which has been assumed to follow an obligatory exchange mechanism. However, previous amino acid uptake experiments in human placental microvillous plasma membrane vesicles have persistently produced results that are seemingly incompatible with such a mechanism; i.e. transport has been observed under zero-trans conditions, in the absence of internal substrates inside the vesicles to drive exchange. This observation raises two alternative hypotheses; (i) either exchange is not fully obligatory, or (ii) exchange is indeed obligatory, but an unforeseen initial concentration of amino acid substrate is present within the vesicle which could drive exchange. To investigate these possibilities, a mathematical model for tracer uptake was developed based on carrier mediated transport, which can represent either facilitated diffusion or obligatory exchange (also referred to as uniport and antiport mechanisms, respectively). In vitro measurements of serine uptake by placental microvillous membrane vesicles were carried out and the model applied to interpret the results based on the measured apparent Michaelis-Menten parameters Km and Vmax. In addition, based on model predictions, a new time series experiment was implemented to distinguish the hypothesised transporter mechanisms. Analysis of the results indicated the presence of a facilitated transport component, while based on the model no evidence for substantial levels of endogenous amino acids within the vesicle was found., (Copyright © 2014 The Authors. Published by Elsevier Ltd.. All rights reserved.)

Amino acid transfer to the fetus is dependent on several different factors. While these factors can be understood in isolation, it is still not possible to predict the function of the system as a whole. In order to do this an integrated approach is required which incorporates the interactions between the different determinants of amino acid transfer. Computational modelling of amino acid transfer in the term human placenta provides a mechanism by which this integrated approach can be delivered. Such a model would be invaluable for understanding amino acid transfer in both normal and pathological pregnancies. In order to develop a computational model it is necessary to determine all the biological factors which are important contributors to net amino acid transfer and the ways in which they interact. For instance, how different classes of amino acid transporter must interact to transfer amino acids across the placenta. Mathematically, the kinetics of each type of transporter can be represented by separate equations that describe their transfer rate as a non-linear function of amino acid concentrations. These equations can then be combined in the model to predict the overall system behaviour. Testing these predictions experimentally will demonstrate the strengths and weaknesses of the model, which can then be refined with increasing complexity and retested in an iterative fashion. In this way we hope to develop a functional computational model which will allow exploration of the factors that determine amino acid transfer across the placenta. This model may also allow the development of strategies to optimise placental transfer in pathologies associated with impaired amino acid transfer such as fetal growth restriction., (Copyright © 2012. Published by Elsevier Ltd.)