Models of collective cell spreading with variable cell aspect ratio : a motivation for degenerate diffusion models

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multi-scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (pme). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

A review of natural convection and heat transfer in attic-shaped space

This paper presents a review of studies on natural convection heat transfer in the triangular enclosure namely, in attic-shaped space. Much research activity has been devoted to this topic over the last three decades with a view to providing thermal comfort to the occupants in attic-shaped buildings and to minimising the energy costs associated with heating and air-conditioning. Two basic thermal boundary conditions of attic are considered to represent hot and cold climates or day and night time. This paper also reports on a significant number of studies which have been performed recently on other topics related to the attic space, for example, attics subject to localized heating and attics filled with porous media.

Active regulation of the epidermal calcium profile

A distinct calcium profile is strongly implicated in regulating the multi-layered structure of the epidermis. However, the mechanisms that govern the regulation of this calcium profile are currently unclear. It clearly depends on the relatively impermeable barrier of the stratum corneum (passive regulation) but may also depend on calcium exchanges between keratinocytes and extracellular fluid (active regulation). Using a mathematical model that treats the viable sublayers of unwounded human and murine epidermis as porous media and assumes that their calcium profiles are passively regulated, we demonstrate that these profiles are also actively regulated. To obtain this result, we found that diffusion governs extracellular calcium motion in the viable epidermis and hence intracellular calcium is the main source of the epidermal calcium profile. Then, by comparison with experimental calcium profiles and combination with a hypothesised cell velocity distribution in the viable epidermis, we found that the net influx of calcium ions into keratinocytes from extracellular fluid may be constant and positive throughout the stratum basale and stratum spinosum, and that there is a net outflux of these ions in the stratum granulosum. Hence the calcium exchange between keratinocytes and extracellular fluid differs distinctly between the stratum granulosum and the underlying sublayers, and these differences actively regulate the epidermal calcium profile. Our results also indicate that plasma membrane dysfunction may be an early event during keratinocyte disintegration in the stratum granulosum.

Exponential integrators and a dual-scale model for wood drying

For the timber industry, the ability to simulate the drying of wood is invaluable for manufacturing high quality wood products. Mathematically, however, modelling the drying of a wet porous material, such as wood, is a diffcult task due to its heterogeneous and anisotropic nature, and the complex geometry of the underlying pore structure. The well{ developed macroscopic modelling approach involves writing down classical conservation equations at a length scale where physical quantities (e.g., porosity) can be interpreted as averaged values over a small volume (typically containing hundreds or thousands of pores). This averaging procedure produces balance equations that resemble those of a continuum with the exception that effective coeffcients appear in their deffnitions. Exponential integrators are numerical schemes for initial value problems involving a system of ordinary differential equations. These methods differ from popular Newton{Krylov implicit methods (i.e., those based on the backward differentiation formulae (BDF)) in that they do not require the solution of a system of nonlinear equations at each time step but rather they require computation of matrix{vector products involving the exponential of the Jacobian matrix. Although originally appearing in the 1960s, exponential integrators have recently experienced a resurgence in interest due to a greater undertaking of research in Krylov subspace methods for matrix function approximation. One of the simplest examples of an exponential integrator is the exponential Euler method (EEM), which requires, at each time step, approximation of φ(A)b, where φ(z) = (ez - 1)/z, A E Rnxn and b E Rn. For drying in porous media, the most comprehensive macroscopic formulation is TransPore [Perre and Turner, Chem. Eng. J., 86: 117-131, 2002], which features three coupled, nonlinear partial differential equations. The focus of the first part of this thesis is the use of the exponential Euler method (EEM) for performing the time integration of the macroscopic set of equations featured in TransPore. In particular, a new variable{ stepsize algorithm for EEM is presented within a Krylov subspace framework, which allows control of the error during the integration process. The performance of the new algorithm highlights the great potential of exponential integrators not only for drying applications but across all disciplines of transport phenomena. For example, when applied to well{ known benchmark problems involving single{phase liquid ow in heterogeneous soils, the proposed algorithm requires half the number of function evaluations than that required for an equivalent (sophisticated) Newton{Krylov BDF implementation. Furthermore for all drying configurations tested, the new algorithm always produces, in less computational time, a solution of higher accuracy than the existing backward Euler module featured in TransPore. Some new results relating to Krylov subspace approximation of '(A)b are also developed in this thesis. Most notably, an alternative derivation of the approximation error estimate of Hochbruck, Lubich and Selhofer [SIAM J. Sci. Comput., 19(5): 1552{1574, 1998] is provided, which reveals why it performs well in the error control procedure. Two of the main drawbacks of the macroscopic approach outlined above include the effective coefficients must be supplied to the model, and it fails for some drying configurations, where typical dual{scale mechanisms occur. In the second part of this thesis, a new dual{scale approach for simulating wood drying is proposed that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of softwood at low temperatures and is valid in the so{called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradient on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic ux to be defined as an average of the microscopic ux over the unit cell. This formulation provides a first step for moving from the macroscopic formulation featured in TransPore to a comprehensive dual{scale formulation capable of addressing any drying configuration. Simulation results reported for a sample of spruce highlight the potential and flexibility of the new dual{scale approach. In particular, for a given unit cell configuration it is not necessary to supply the effective coefficients prior to each simulation.

A multi-scaling approach to predict hydraulic damage of poromaterials

The purpose of this paper is to introduce the concept of hydraulic damage and its numerical integration. Unlike the common phenomenological continuum damage mechanics approaches, the procedure introduced in this paper relies on mature concepts of homogenization, linear fracture mechanics, and thermodynamics. The model is applied to the problem of fault reactivation within resource reservoirs. The results show that propagation of weaknesses is highly driven by the contrasts of properties in porous media. In particular, it is affected by the fracture toughness of host rocks. Hydraulic damage is diffused when it takes place within extended geological units and localized at interfaces and faults.

Accessibility of pores in coal to methane and carbon dioxide

Fluid–solid interactions in natural and engineered porous solids underlie a variety of technological processes, including geological storage of anthropogenic greenhouse gases, enhanced coal bed methane recovery, membrane separation, and heterogeneous catalysis. The size, distribution and interconnectivity of pores, the chemical and physical properties of the solid and fluid phases collectively dictate how fluid molecules migrate into and through the micro- and meso-porous media, adsorb and ultimately react with the solid surfaces. Due to the high penetration power and relatively short wavelength of neutrons, smallangle neutron scattering (SANS) as well as ultra small-angle scattering (USANS) techniques are ideally suited for assessing the phase behavior of confined fluids under pressure as well as for evaluating the total porosity in engineered and natural porous systems including coal. Here we demonstrate that SANS and USANS can be also used for determining the fraction of the pore volume that is actually accessible to fluids as a function of pore sizes and study the fraction of inaccessible pores as a function of pore size in three coals from the Illinois Basin (USA) and Bowen Basin (Australia). Experiments were performed at CO2 and methane pressures up to 780 bar, including pressures corresponding to zero average contrast condition (ZAC), which is the pressure where no scattering from the accessible pores occurs. Scattering curves at the ZAC were compared with the scattering from same coals under vacuum and analysed using a newly developed approach that shows that the volume fraction of accessible pores in these coals varies between �90% in the macropore region to �30% in the mesopore region and the variation is distinctive for each of the examined coals. The developed methodology may be also applied for assessing the volume of accessible pores in other natural underground formations of interest for CO2 sequestration, such as saline aquifers as well as for estimating closed porosity in engineered porous solids of technological importance.

Dynamic Micromapping of CO2 Sorption in Coal

We have applied X-ray and neutron small-angle scattering techniques (SAXS, SANS, and USANS) to study the interaction between fluids and porous media in the particular case of subcritical CO2 sorption in coal. These techniques are demonstrated to give unique, pore-size-specific insights into the kinetics of CO2 sorption in a wide range of coal pores (nano to meso) and to provide data that may be used to determine the density of the sorbed CO2. We observed densification of the adsorbed CO2 by a factor up to five compared to the free fluid at the same (p, T) conditions. Our results indicate that details of CO2 sorption into coal pores differ greatly between different coals and depend on the amount of mineral matter dispersed in the coal matrix: a purely organic matrix absorbs more CO2 per unit volume than one containing mineral matter, but mineral matter markedly accelerates the sorption kinetics. Small pores are filled preferentially by the invading CO2 fluid and the apparent diffusion coefficients have been estimated to vary in the range from 5 × 10-7 cm2/min to more than 10-4 cm2/min, depending on the CO2 pressure and location on the sample.

Characterisation of the micro-architecture of direct writing melt electrospun tissue engineering scaffolds using diffusion tensor and computed tomography microimaging

This article describes the first steps toward comprehensive characterization of molecular transport within scaffolds for tissue engineering. The scaffolds were fabricated using a novel melt electrospinning technique capable of constructing 3D lattices of layered polymer fibers with well - defined internal microarchitectures. The general morphology and structure order was then determined using T 2 - weighted magnetic resonance imaging and X - ray microcomputed tomography. Diffusion tensor microimaging was used to measure the time - dependent diffusivity and diffusion anisotropy within the scaffolds. The measured diffusion tensors were anisotropic and consistent with the cross - hatched geometry of the scaffolds: diffusion was least restricted in the direction perpendicular to the fiber layers. The results demonstrate that the cross - hatched scaffold structure preferentially promotes molecular transport vertically through the layers ( z - axis), with more restricted diffusion in the directions of the fiber layers ( x – y plane). Diffusivity in the x – y plane was observed to be invariant to the fiber thickness. The characteristic pore size of the fiber scaffolds can be probed by sampling the diffusion tensor at multiple diffusion times. Prospective application of diffusion tensor imaging for the real - time monitoring of tissue maturation and nutrient transport pathways within tissue engineering scaffolds is discussed.

Mathematical modelling of fumigant transport in stored grain

Computational fluid dynamics, analytical solutions, and mathematical modelling approaches are used to gain insights into the distribution of fumigant gas within farm-scale, grain storage silos. Both fan-forced and tablet fumigation are considered in this work, which develops new models for use by researchers, primary producers and silo manufacturers to assist in the eradication grain storage pests.

Fouling of waste heat recovery : numerical and experimental results

Numerical results are presented to investigate the performance of a partly-filled porous heat exchanger for waste heat recovery units. A parametric study was conducted to investigate the effects of inlet velocity and porous block height on the pressure drop of the heat exchanger. The focus of this work is on modelling the interface of a porous and non-porous region. As such, numerical simulation of the problem is conducted along with hot-wire measurements to better understand the physics of the problem. Results from the two sources are then compared to existing theoretical predictions available in the literature which are unable to predict the existence of two separation regions before and after the porous block. More interestingly, a non-uniform interface velocity was observed along the streamwise direction based on both numerical and experimental data.

A micro-level investigation of the solid displacement method for porosity determination of dried food

Porosity is one of the key parameters of the macroscopic structure of porous media, generally defined as the ratio of the free spaces occupied (by the volume of air) within the material to the total volume of the material. Porosity is determined by measuring skeletal volume and the envelope volume. Solid displacement method is one of the inexpensive and easy methods to determine the envelope volume of a sample with an irregular shape. In this method, generally glass beads are used as a solid due to their uniform size, compactness and fluidity properties. The smaller size of the glass beads means that they enter into the open pores which have a larger diameter than the glass beads. Although extensive research has been carried out on porosity determination using displacement method, no study exists which adequately reports micro-level observation of the sample during measurement. This study set out with the aim of assessing the accuracy of solid displacement method of bulk density measurement of dried foods by micro-level observation. Solid displacement method of porosity determination was conducted using a cylindrical vial (cylindrical plastic container) and 57 µm glass beads in order to measure the bulk density of apple slices at different moisture contents. A scanning electron microscope (SEM), a profilometer and ImageJ software were used to investigate the penetration of glass beads into the surface pores during the determination of the porosity of dried food. A helium pycnometer was used to measure the particle density of the sample. Results show that a significant number of pores were large enough to allow the glass beads to enter into the pores, thereby causing some erroneous results. It was also found that coating the dried sample with appropriate coating material prior to measurement can resolve this problem.

Modelling intermittent microwave convective drying (IMCD) of food materials

This thesis develops comprehensive mathematical models for an advanced drying technology Intermittent Microwave Convective Drying (IMCD). The models provide an improved physical understanding of the heat and mass transport during the drying process, which will help to improve the quality of dried food and energy efficiency of the process, as well as will increase the ability of automation and optimization. The final model in this thesis represents the most comprehensive fundamental multiphase model for IMCD that considers 3D electromagnetics coupled with multiphase porous media transport processes. The 3D electromagnetics considers Maxwell's equation and multiphase transport model considers three different phases: solid matrix, liquid water and gas consisting water vapour and air. The multiphase transport includes pressure-driven flow, capillary diffusion, binary diffusion, and evaporation. The models developed in this thesis were validated with extensive experimental investigations.

Mathematical modelling of gas production and compositional shift of a CSG (coal seam gas) field: Local model development

In this work we discuss the development of a mathematical model to predict the shift in gas composition observed over time from a producing CSG (coal seam gas) well, and investigate the effect that physical properties of the coal seam have on gas production. A detailed (local) one-dimensional, two-scale mathematical model of a coal seam has been developed. The model describes the competitive adsorption and desorption of three gas species (CH4, CO2 and N2) within a microscopic, porous coal matrix structure. The (diffusive) flux of these gases between the coal matrices (microscale) and a cleat network (macroscale) is accounted for in the model. The cleat network is modelled as a one-dimensional, volume averaged, porous domain that extends radially from a central well. Diffusive and advective transport of the gases occurs within the cleat network, which also contains liquid water that can be advectively transported. The water and gas phases are assumed to be immiscible. The driving force for the advection in the gas and liquid phases is taken to be a pressure gradient with capillarity also accounted for. In addition, the relative permeabilities of the water and gas phases are considered as functions of the degree of water saturation.