ORCID Profile
0000-0001-5304-2384
Current Organisation
Queensland University of Technology
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Research Topics
In Research Link Australia (RLA), "Research Topics" refer to ANZSRC FOR and SEO codes. These topics are either sourced from ANZSRC FOR and SEO codes listed in researchers' related grants or generated by a large language model (LLM) based on their publications.
Top 5 Research Topics
ANZSRC Field of Research (FoR)
Applied Mathematics | Theoretical and Applied Mechanics | Biological Mathematics | Mathematical Software | Numerical Solution of Differential and Integral Equations | Ship and Platform Hydrodynamics | Crop and Pasture Protection (Pests, Diseases and Weeds) | Fluidisation and Fluid Mechanics | Approximation Theory and Asymptotic Methods | Computation Theory and Mathematics | Biological Mathematics | Numerical Computation |
ANZSRC Socio-Economic Objective (SEO)
Expanding Knowledge in the Mathematical Sciences | Horticultural Crops not elsewhere classified | Expanding Knowledge in Engineering | Cancer and Related Disorders | Surgical methods and procedures | Coastal Sea Freight Transport | Environmentally Sustainable Transport not elsewhere classified
Related Links
Publications
Simplified approach for calculating moments of action for linear reaction-diffusion equations
Publisher: American Physical Society (APS)
Date: 15-11-2013
Free Surface Problems for Static Coulomb-Mohr Granular Solids
Publisher: SAGE Publications
Date: 12-2005
Abstract: This paper is concerned with some plane strain and axially symmetric free surface problems which arise in the study of static granular solids that satisfy the Coulomb-Mohr yield condition. Such problems are inherently nonlinear, and hence difficult to attack analytically. Given a Coulomb friction condition holds on a solid boundary, it is shown that the angle a free surface is allowed to attach to the boundary is dependent only on the angle of wall friction, assuming the stresses are all continuous at the attachment point, and assuming also that the coefficient of cohesion is nonzero. As a model problem, the formation of stable cohesive arches in hoppers is considered. This undesirable phenomena is an obstacle to flow, and occurs when the hopper outlet is too small. Typically, engineers are concerned with predicting the critical outlet size for a given hopper and granular solid, so that for hoppers with outlets larger than this critical value, arching cannot occur. This is a topic of considerable practical interest, with most accepted engineering methods being conservative in nature. Here, the governing equations in two limiting cases (small cohesion and high angle of internal friction) are considered directly. No information on the critical outlet size is found however solutions for the shape of the free boundary (the arch) are presented, for both plane and axially symmetric geometries.
The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell
Publisher: The Royal Society
Date: 05-2017
Abstract: New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.
Critical timescales and time intervals for coupled linear processes
Publisher: Cambridge University Press (CUP)
Date: 2013
DOI: 10.1017/S1446181113000059
Abstract: In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. S le and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for ex le the analysis of models describing coupled chemical decay and cell differentiation processes.
Gravity-driven fingering simulations for a thin liquid film flowing down the outside of a vertical cylinder
Publisher: American Physical Society (APS)
Date: 28-05-2013
Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries
Publisher: The Royal Society
Date: 10-2023
Travelling waves for a velocity-jump model of cell migration and proliferation
Publisher: Elsevier BV
Date: 08-2013
DOI: 10.1016/J.MBS.2013.04.010
Abstract: Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher-Kolmogorov equation. These traditional parabolic models cannot be used to represent experimental measurements of in idual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity-jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left-moving cells, L(x,t), and a subpopulation of right-moving cells, R(x,t). This leads to a system of hyperbolic partial differential equations that includes a turning rate, Λ⩾0, describing the rate at which in iduals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where Λ=0 and in the limit that Λ→∞. For intermediate turning rates, 0<Λ<∞, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as Λ decreases through a critical value Λcrit. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small Λ limit produces results that are consistent with experimental observations.
Implicit reconstructions of thin leaf surfaces from large, noisy point clouds
Publisher: Elsevier BV
Date: 10-2021
Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations
Publisher: Cambridge University Press (CUP)
Date: 02-09-2019
DOI: 10.1017/JFM.2019.623
Abstract: Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or erging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.
What is the apparent angle of a Kelvin ship wave pattern?
Publisher: Cambridge University Press (CUP)
Date: 09-10-2014
DOI: 10.1017/JFM.2014.530
Abstract: While the half-angle which encloses a Kelvin ship wave pattern is commonly accepted to be 19.47°, recent observations and calculations for sufficiently fast-moving ships suggest that the apparent wake angle decreases with ship speed. One explanation for this decrease in angle relies on the assumption that a ship cannot generate wavelengths much greater than its hull length. An alternative interpretation is that the wave pattern that is observed in practice is defined by the location of the highest peaks for wakes created by sufficiently fast-moving objects, these highest peaks no longer lie on the outermost ergent waves, resulting in a smaller apparent angle. In this paper, we focus on the problems of free-surface flow past a single submerged point source and past a submerged source doublet. In the linear version of these problems, we measure the apparent wake angle formed by the highest peaks, and observe the following three regimes: a small Froude number pattern, in which the ergent waves are not visible standard wave patterns for which the maximum peaks occur on the outermost ergent waves and a third regime in which the highest peaks form a V-shape with an angle much less than the Kelvin angle. For nonlinear flows, we demonstrate that nonlinearity has the effect of increasing the apparent wake angle so that some highly nonlinear solutions have apparent wake angles that are greater than Kelvin’s angle. For large Froude numbers, the effect on apparent wake angle can be more dramatic, with the possibility of strong nonlinearity shifting the wave pattern from the third regime to the second. We expect that our nonlinear results will translate to other more complicated flow configurations, such as flow due to a steadily moving closed body such as a submarine.
Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow
Publisher: American Physical Society (APS)
Date: 25-11-2019
Coulomb-mohr granular materials: Quasi-static flows and the highly frictional limit
Publisher: ASME International
Date: 07-10-2008
DOI: 10.1115/1.2987874
Abstract: One approach to modeling fully developed shear flow of frictional granular materials is to use a yield condition and a flow rule, in an analogous way to that commonly employed in the fields of metal plasticity and soil mechanics. Typically, the yield condition of choice for granular materials is the Coulomb–Mohr criterion, as this constraint is relatively simple to apply but at the same time is also known to predict stresses that are in good agreement with experimental observations. On the other hand, there is no strong agreement within the engineering and applied mechanics community as to which flow rule is most appropriate, and this subject is still very much open to debate. This paper provides a review of the governing equations used to describe the flow of granular materials subject to the Coulomb–Mohr yield condition, concentrating on the coaxial and double-shearing flow rules in both plane strain and axially symmetric geometries. Emphasis is given to highly frictional materials, which are defined as those granular materials that possess angles of internal friction whose trigonometric sine is close in value to unity. Furthermore, a discussion is provided on the practical problems of determining the stress and velocity distributions in a gravity flow hopper, as well as the stress fields beneath a standing stockpile and within a stable rat-hole.
Examining Go-or-Grow Using Fluorescent Cell-Cycle Indicators and Cell-Cycle-Inhibiting Drugs
Publisher: Elsevier BV
Date: 03-2020
Spectrogram analysis of surface elevation signals due to accelerating ships
Publisher: American Physical Society (APS)
Date: 29-10-2021
Preface to fourth Special Issue on Practical Asymptotics
Publisher: Springer Science and Business Media LLC
Date: 28-11-2008
Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronization in cell proliferation
Publisher: The Royal Society
Date: 08-2019
Abstract: We present a suite of experimental data showing that cell proliferation assays, prepared using standard methods thought to produce asynchronous cell populations, persistently exhibit inherent synchronization. Our experiments use fluorescent cell cycle indicators to reveal the normally hidden cell synchronization, by highlighting oscillatory subpopulations within the total cell population. These oscillatory subpopulations would never be observed without these cell cycle indicators. On the other hand, our experimental data show that the total cell population appears to grow exponentially, as in an asynchronous population. We reconcile these seemingly inconsistent observations by employing a multi-stage mathematical model of cell proliferation that can replicate the oscillatory subpopulations. Our study has important implications for understanding and improving experimental reproducibility. In particular, inherent synchronization may affect the experimental reproducibility of studies aiming to investigate cell cycle-dependent mechanisms, including changes in migration and drug response.
Three-dimensional free-surface flow over arbitrary bottom topography
Publisher: Cambridge University Press (CUP)
Date: 03-05-2018
DOI: 10.1017/JFM.2018.254
Abstract: We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton–Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.
Extending fields in a level set method by solving a biharmonic equation
Publisher: Elsevier BV
Date: 08-2017
A novel mathematical model of heterogeneous cell proliferation
Publisher: Springer Science and Business Media LLC
Date: 12-03-2021
Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection
Publisher: Elsevier BV
Date: 02-2016
DOI: 10.1016/J.JTBI.2015.10.040
Abstract: Scratch assays are difficult to reproduce. Here we identify a previously overlooked source of variability which could partially explain this difficulty. We analyse a suite of scratch assays in which we vary the initial degree of confluence (initial cell density). Our results indicate that the rate of re-colonisation is very sensitive to the initial density. To quantify the relative roles of cell migration and proliferation, we calibrate the solution of the Fisher-Kolmogorov model to cell density profiles to provide estimates of the cell diffusivity, D, and the cell proliferation rate, λ. This procedure indicates that the estimates of D and λ are very sensitive to the initial density. This dependence suggests that the Fisher-Kolmogorov model does not accurately represent the details of the collective cell spreading process, since this model assumes that D and λ are constants that ought to be independent of the initial density. Since higher initial cell density leads to enhanced spreading, we also calibrate the solution of the Porous-Fisher model to the data as this model assumes that the cell flux is an increasing function of the cell density. Estimates of D and λ associated with the Porous-Fisher model are less sensitive to the initial density, suggesting that the Porous-Fisher model provides a better description of the experiments.
Evaporating droplets on inclined plant leaves and synthetic surfaces: Experiments and mathematical models
Publisher: Elsevier BV
Date: 06-2021
Linear stern waves in finite depth channels
Publisher: Oxford University Press (OUP)
Date: 11-2000
Traveling waves, blow‐up, and extinction in the Fisher–Stefan model
Publisher: Wiley
Date: 03-11-2021
DOI: 10.1111/SAPM.12465
Abstract: While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For ex le, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher–Stefan model, a generalization of the well‐known Fisher–KPP model, characterized by a leakage coefficient which relates the speed of the moving boundary to the flux of population there. This Fisher–Stefan model overcomes one of the well‐known limitations of the Fisher–KPP model, since time‐dependent solutions of the Fisher–Stefan model involve a well‐defined front which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting , which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting leads to retreating traveling waves and an interesting transition to finite‐time blow‐up. For certain initial conditions, population extinction is also observed. Our approach involves studying time‐dependent solutions of the governing equations, phase plane, and asymptotic analysis, leading to new insight into the possibilities of traveling waves, blow‐up, and extinction for this moving boundary problem. MATLAB software used to generate the results in this work is available on Github.
Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns
Publisher: Elsevier BV
Date: 07-2014
Modelling water droplet movement on a leaf surface
Publisher: Elsevier BV
Date: 04-2011
Travelling wave analysis of cellular invasion into surrounding tissues
Publisher: Elsevier BV
Date: 12-2200
Spray retention on whole plants: modelling, simulations and experiments
Publisher: Elsevier BV
Date: 10-2016
A sharp-front moving boundary model for malignant invasion
Publisher: Elsevier BV
Date: 11-1011
New exact solutions for Hele-Shaw flow in doubly connected regions
Publisher: AIP Publishing
Date: 05-2012
DOI: 10.1063/1.4711274
Abstract: Radial Hele–Shaw flows are treated analytically using conformal mapping techniques. The geometry of interest has a doubly connected annular region of viscous fluid surrounding an inviscid bubble that is either expanding or contracting due to a pressure difference caused by injection or suction of the inviscid fluid. The zero-surface-tension problem is ill-posed for both bubble expansion and contraction, as both scenarios involve viscous fluid displacing inviscid fluid. Exact solutions are derived by tracking the location of singularities and critical points in the analytic continuation of the mapping function. We show that by treating the critical points, it is easy to observe finite-time blow-up, and the evolution equations may be written in exact form using complex residues. We present solutions that start with cusps on one interface and end with cusps on the other, as well as solutions that have the bubble contracting to a point. For the latter solutions, the bubble approaches an ellipse in shape at extinction.
Spectrograms of ship wakes: identifying linear and nonlinear wave signals
Publisher: Cambridge University Press (CUP)
Date: 06-12-2017
DOI: 10.1017/JFM.2016.753
Abstract: A spectrogram is a useful way of using short-time discrete Fourier transforms to visualise surface height measurements taken of ship wakes in real-world conditions. For a steadily moving ship that leaves behind small- litude waves, the spectrogram is known to have two clear linear components, a sliding-frequency mode caused by the ergent waves and a constant-frequency mode for the transverse waves. However, recent observations of high-speed ferry data have identified additional components of the spectrograms that are not yet explained. We use computer simulations of linear and nonlinear ship wave patterns and apply time–frequency analysis to generate spectrograms for an idealised ship. We clarify the role of the linear dispersion relation and ship speed on the two linear components. We use a simple weakly nonlinear theory to identify higher-order effects in a spectrogram and, while the high-speed ferry data are very noisy, we propose that certain additional features in the experimental data are caused by nonlinearity. Finally, we provide a possible explanation for a further discrepancy between the high-speed ferry spectrograms and linear theory by accounting for ship acceleration.
Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions
Publisher: IOP Publishing
Date: 07-10-2016
DOI: 10.1088/1478-3975/13/5/056003
Abstract: Two-dimensional collective cell migration assays are used to study cancer and tissue repair. These assays involve combined cell migration and cell proliferation processes, both of which are modulated by cell-to-cell crowding. Previous discrete models of collective cell migration assays involve a nearest-neighbour proliferation mechanism where crowding effects are incorporated by aborting potential proliferation events if the randomly chosen target site is occupied. There are two limitations of this traditional approach: (i) it seems unreasonable to abort a potential proliferation event based on the occupancy of a single, randomly chosen target site and, (ii) the continuum limit description of this mechanism leads to the standard logistic growth function, but some experimental evidence suggests that cells do not always proliferate logistically. Motivated by these observations, we introduce a generalised proliferation mechanism which allows non-nearest neighbour proliferation events to take place over a template of [Formula: see text] concentric rings of lattice sites. Further, the decision to abort potential proliferation events is made using a crowding function, f(C), which accounts for the density of agents within a group of sites rather than dealing with the occupancy of a single randomly chosen site. Analysing the continuum limit description of the stochastic model shows that the standard logistic source term, [Formula: see text], where λ is the proliferation rate, is generalised to a universal growth function, [Formula: see text]. Comparing the solution of the continuum description with averaged simulation data indicates that the continuum model performs well for many choices of f(C) and r. For nonlinear f(C), the quality of the continuum-discrete match increases with r.
Time-frequency analysis of ship wave patterns in shallow water: modelling and experiments
Publisher: Elsevier BV
Date: 06-2018
Asymptotic and Numerical Results for a Model of Solvent-Dependent Drug Diffusion through Polymeric Spheres
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2011
DOI: 10.1137/110821688
Single phase limit for melting nanoparticles
Publisher: Elsevier BV
Date: 05-2009
Comment on Local accumulation times for source, diffusion, and degradation models in two and three dimensions [J. Chem. Phys. 138, 104121 (2013)]
Publisher: AIP Publishing
Date: 03-07-2013
DOI: 10.1063/1.4811832
Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics
Publisher: Elsevier BV
Date: 2023
Clinical strategies for the alleviation of contractures from a predictive mathematical model of dermal repair
Publisher: Wiley
Date: 07-02-2012
Velocity-jump processes with proliferation
Publisher: IOP Publishing
Date: 05-12-2013
Contracting bubbles in Hele-Shaw cells with a power-law fluid
Publisher: IOP Publishing
Date: 11-1111
Exact sharp-fronted travelling wave solutions of the Fisher–KPP equation
Publisher: Elsevier BV
Date: 04-2021
Velocity-jump models with crowding effects
Publisher: American Physical Society (APS)
Date: 28-12-2011
Discrete families of Saffman–Taylor fingers with exotic shapes
Publisher: Elsevier BV
Date: 2015
Application of a continuum theory to vertical vibrations of a layer of granular material
Publisher: Elsevier BV
Date: 11-1999
Sensitivity and robustness of Lagrangian coherent structures in coastal water systems
Publisher: Springer Science and Business Media LLC
Date: 22-04-2021
Modelling the interaction of keratinocytes and fibroblasts during normal and abnormal wound healing processes
Publisher: The Royal Society
Date: 23-05-2012
Abstract: The crosstalk between fibroblasts and keratinocytes is a vital component of the wound healing process, and involves the activity of a number of growth factors and cytokines. In this work, we develop a mathematical model of this crosstalk in order to elucidate the effects of these interactions on the regeneration of collagen in a wound that heals by second intention. We consider the role of four components that strongly affect this process: transforming growth factor- β , platelet-derived growth factor, interleukin-1 and keratinocyte growth factor. The impact of this network of interactions on the degradation of an initial fibrin clot, as well as its subsequent replacement by a matrix that is mainly composed of collagen, is described through an eight-component system of nonlinear partial differential equations. Numerical results, obtained in a two-dimensional domain, highlight key aspects of this multifarious process, such as re-epithelialization. The model is shown to reproduce many of the important features of normal wound healing. In addition, we use the model to simulate the treatment of two pathological cases: chronic hypoxia, which can lead to chronic wounds and prolonged inflammation, which has been shown to lead to hypertrophic scarring. We find that our model predictions are qualitatively in agreement with previously reported observations and provide an alternative pathway for gaining insight into this complex biological process.
Short, flat-tipped, viscous fingers: novel interfacial patterns in a Hele-Shaw channel with an elastic boundary
Publisher: Cambridge University Press (CUP)
Date: 17-11-2017
DOI: 10.1017/JFM.2017.692
Abstract: Injecting a less viscous fluid into a more viscous fluid in a Hele-Shaw cell triggers two-dimensional viscous fingering patterns which are characterised by increasingly long fingers undergoing tip splitting and branching events. These complex structures are considered to be a paradigm for interfacial pattern formation in porous media flow and other related phenomena. Over the past five years, there has been a flurry of interest in manipulating these interfacial fingering patterns by altering the physical components of the Hele-Shaw apparatus. In this Focus on Fluids article, we summarise some of this work, concentrating on a very recent study in which the alterations include replacing one of the two bounding plates with an elastic membrane (Ducloué et al. , J. Fluid Mech. , vol. 826, 2017, R2). The resulting experimental set-up gives rise to a wide variety of novel interfacial patterns including periodic sideways fingers, dendritic-like patterns and short, flat-tipped viscous fingers that appear to resemble molar teeth. These latter fingers are similar to those observed in the printer’s instability and when peeling off a layer of adhesive tape. This delightful work brings together a number of well-studied themes in interfacial fluid mechanics, including how viscous and surface tension forces compete to drive fingering patterns, how interfaces are affected by fluid–solid interactions and, finally, how novel strategies can be implemented to control interfacial instabilities.
Minimising wave drag for free surface flow past a two-dimensional stern
Publisher: AIP Publishing
Date: 07-2011
DOI: 10.1063/1.3609284
Abstract: Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results obtained by a forced KdV equation are also presented, as are numerical solutions of the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the litude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely
A cellular automata model to investigate immune cell–tumor cell interactions in growing tumors in two spatial dimensions
Publisher: Springer New York
Date: 2014
Non-vanishing sharp-fronted travelling wave solutions of the Fisher–Kolmogorov model
Publisher: Oxford University Press (OUP)
Date: 06-06-2022
Abstract: The Fisher–Kolmogorov–Petrovsky–Piskunov (KPP) model, and generalizations thereof, involves simple reaction–diffusion equations for biological invasion that assume in iduals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $\\lambda $. For the Fisher–KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2\\sqrt {\\lambda D}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher–KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed $c=2\\sqrt {\\lambda D}& 0$. This means that, for biologically relevant initial data, the Fisher–KPP model cannot be used to study invasion with $c \\ne 2\\sqrt {\\lambda D}$, or retreating travelling waves with $c & 0$. Here, we reformulate the Fisher–KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher–KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, $-\\infty & c & \\infty $. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.
Burgers’ equation in the complex plane
Publisher: Elsevier BV
Date: 06-2023
A continuum mathematical model of substrate-mediated tissue growth
Publisher: Springer Science and Business Media LLC
Date: 02-03-2022
DOI: 10.1007/S11538-022-01005-7
Abstract: We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model , involves a partial differential equation describing the density of tissue, $${\\hat{u}}(\\hat{{\\mathbf {x}}},{\\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\\hat{s}}(\\hat{{\\mathbf {x}}},{\\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\\hat{s}}$$ s ^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, $$c = c_{\\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c c_{\\mathrm{min}}$$ c c min . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub .
Including nonequilibrium interface kinetics in a continuum model for melting nanoscaled particles
Publisher: Springer Science and Business Media LLC
Date: 17-11-2014
DOI: 10.1038/SREP07066
Impaction of spray droplets on leaves: influence of formulation and leaf character on shatter, bounce and adhesion
Publisher: Springer Science and Business Media LLC
Date: 30-06-2015
Invading and receding sharp-fronted travelling waves
Publisher: Springer Science and Business Media LLC
Date: 21-02-2021
Lie group symmetry analysis for granular media stress equations
Publisher: Elsevier BV
Date: 2005
Free surface flow into a horizontal slot with zero gravity
Publisher: AIP Publishing
Date: 08-2000
DOI: 10.1063/1.870461
Abstract: The two-dimensional free surface flow of a finite-depth fluid into a horizontal slot is considered. For this study, the effects of viscosity and gravity are ignored. A generalized Schwarz–Christoffel mapping is used to formulate the problem in terms of a linear integral equation, which is solved exactly with the use of a Fourier transform. The resulting free surface profile is given explicitly in closed form.
Extended logistic growth model for heterogeneous populations
Publisher: Elsevier BV
Date: 05-2018
DOI: 10.1016/J.JTBI.2018.02.027
Abstract: Cell proliferation is the most important cellular-level mechanism responsible for regulating cell population dynamics in living tissues. Modern experimental procedures show that the proliferation rates of in idual cells can vary significantly within the same cell line. However, in the mathematical biology literature, cell proliferation is typically modelled using a classical logistic equation which neglects variations in the proliferation rate. In this work, we consider a discrete mathematical model of cell migration and cell proliferation, modulated by volume exclusion (crowding) effects, with variable rates of proliferation across the total population. We refer to this variability as heterogeneity. Constructing the continuum limit of the discrete model leads to a generalisation of the classical logistic growth model. Comparing numerical solutions of the model to averaged data from discrete simulations shows that the new model captures the key features of the discrete process. Applying the extended logistic model to simulate a proliferation assay using rates from recent experimental literature shows that neglecting the role of heterogeneity can, at times, lead to misleading results.
A two-compartment mechanochemical model of the roles of transforming growth factor β
Publisher:
Elsevier BV
Date:
03-2011
DOI:
10.1016/J.JTBI.2010.12.011
Abstract: The repair of dermal tissue is a complex process of interconnected phenomena, where cellular, chemical and mechanical aspects all play a role, both in an autocrine and in a paracrine fashion. Recent experimental results have shown that transforming growth factor -β (TGFβ) and tissue mechanics play roles in regulating cell proliferation, differentiation and the production of extracellular materials. We have developed a 1D mathematical model that considers the interaction between the cellular, chemical and mechanical phenomena, allowing the combination of TGFβ and tissue stress to inform the activation of fibroblasts to myofibroblasts. Additionally, our model incorporates the observed feature of residual stress by considering the changing zero-stress state in the formulation for effective strain. Using this model, we predict that the continued presence of TGFβ in dermal wounds will produce contractures due to the persistence of myofibroblasts in contrast, early elimination of TGFβ significantly reduces the myofibroblast numbers resulting in an increase in wound size. Similar results were obtained by varying the rate at which fibroblasts differentiate to myofibroblasts and by changing the myofibroblast apoptotic rate. Taken together, the implication is that elevated levels of myofibroblasts is the key factor behind wounds healing with excessive contraction, suggesting that clinical strategies which aim to reduce the myofibroblast density may reduce the appearance of contractures.
Publisher: Elsevier BV
Date: 03-2011
DOI: 10.1016/J.JTBI.2010.12.011
Abstract: The repair of dermal tissue is a complex process of interconnected phenomena, where cellular, chemical and mechanical aspects all play a role, both in an autocrine and in a paracrine fashion. Recent experimental results have shown that transforming growth factor -β (TGFβ) and tissue mechanics play roles in regulating cell proliferation, differentiation and the production of extracellular materials. We have developed a 1D mathematical model that considers the interaction between the cellular, chemical and mechanical phenomena, allowing the combination of TGFβ and tissue stress to inform the activation of fibroblasts to myofibroblasts. Additionally, our model incorporates the observed feature of residual stress by considering the changing zero-stress state in the formulation for effective strain. Using this model, we predict that the continued presence of TGFβ in dermal wounds will produce contractures due to the persistence of myofibroblasts in contrast, early elimination of TGFβ significantly reduces the myofibroblast numbers resulting in an increase in wound size. Similar results were obtained by varying the rate at which fibroblasts differentiate to myofibroblasts and by changing the myofibroblast apoptotic rate. Taken together, the implication is that elevated levels of myofibroblasts is the key factor behind wounds healing with excessive contraction, suggesting that clinical strategies which aim to reduce the myofibroblast density may reduce the appearance of contractures.
Extinction Behaviour of Contracting Bubbles in Porous Media
Publisher: Oxford University Press (OUP)
Date: 08-2003
A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices
Publisher: Springer Science and Business Media LLC
Date: 18-05-2017
DOI: 10.1007/S10237-017-0917-3
Abstract: The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.
New stress and velocity fields for highly frictional granular materials
Publisher: Oxford University Press (OUP)
Date: 17-12-2004
Extinction behaviour for two–dimensional inward-solidification problems
Publisher: The Royal Society
Date: 08-04-2003
Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology
Publisher: Elsevier BV
Date: 11-2019
The Extinction Problem for Three-dimensional Inward Solidification
Publisher: Springer Science and Business Media LLC
Date: 08-2005
Stochastic models of cell invasion with fluorescent cell cycle indicators
Publisher: Elsevier BV
Date: 11-2018
Classical two-phase Stefan problem for spheres
Publisher: The Royal Society
Date: 10-04-2008
Abstract: The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one phase only. Even in this idealized case there is no (known) exact solution, and the only way to obtain meaningful results is through numerical or approximate means. In this study, the full two-phase problem is considered, and in particular, attention is given to the large Stefan number limit. By applying the method of matched asymptotic expansions, the temperature in both the phases is shown to depend algebraically on the inverse Stefan number on the first time scale, but at later times the two phases essentially decouple, with the inner core contributing only exponentially small terms to the location of the solid–melt interface. This analysis is complemented by applying a small-time perturbation scheme and by presenting numerical results calculated using an enthalpy method. The limits of zero Stefan number and slow diffusion in the inner core are also noted.
The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem
Publisher: Elsevier BV
Date: 02-0005
Spray droplet impaction outcomes for different plant species and spray formulations
Publisher: Elsevier BV
Date: 09-0011
Logistic proliferation of cells in scratch assays is delayed
Publisher: Springer Science and Business Media LLC
Date: 23-03-2017
DOI: 10.1007/S11538-017-0267-4
Abstract: Scratch assays are used to study how a population of cells re-colonises a vacant region on a two-dimensional substrate after a cell monolayer is scratched. These experiments are used in many applications including drug design for the treatment of cancer and chronic wounds. To provide insights into the mechanisms that drive scratch assays, solutions of continuum reaction-diffusion models have been calibrated to data from scratch assays. These models typically include a logistic source term to describe carrying capacity-limited proliferation however, the choice of using a logistic source term is often made without examining whether it is valid. Here we study the proliferation of PC-3 prostate cancer cells in a scratch assay. All experimental results for the scratch assay are compared with equivalent results from a proliferation assay where the cell monolayer is not scratched. Visual inspection of the time evolution of the cell density away from the location of the scratch reveals a series of sigmoid curves that could be naively calibrated to the solution of the logistic growth model. However, careful analysis of the per capita growth rate as a function of density reveals several key differences between the proliferation of cells in scratch and proliferation assays. Our findings suggest that the logistic growth model is valid for the entire duration of the proliferation assay. On the other hand, guided by data, we suggest that there are two phases of proliferation in a scratch assay at short time, we have a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. These two phases are observed across a large number of experiments performed at different initial cell densities. Overall our study shows that simply calibrating the solution of a continuum model to a scratch assay might produce misleading parameter estimates, and this issue can be resolved by making a distinction between the disturbance and growth phases. Repeating our procedure for other scratch assays will provide insight into the roles of the disturbance and growth phases for different cell lines and scratch assays performed on different substrates.
Free surface flow past topography: A beyond-all-orders approach
Publisher: Cambridge University Press (CUP)
Date: 09-02-2012
DOI: 10.1017/S0956792512000022
Abstract: The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567 , 299–326, the ergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.
Quadrature Domains and p-Laplacian Growth
Publisher: Springer Science and Business Media LLC
Date: 29-11-2008
Viscous fingering patterns for Hele-Shaw flow in a doubly connected geometry driven by a pressure differential or rotation
Publisher: American Physical Society (APS)
Date: 10-01-2023
Perturbation solutions for flow through symmetrical hoppers with inserts and asymmetrical wedge hoppers
Publisher: Springer Science and Business Media LLC
Date: 07-2005
Wake angle for surface gravity waves on a finite depth fluid
Publisher: AIP Publishing
Date: 06-2015
DOI: 10.1063/1.4921918
Abstract: Linear water wave theory suggests that wave patterns caused by a steadily moving disturbance are contained within a wedge whose half-angle depends on the depth-based Froude number FH. For the problem of flow past an axisymmetric pressure distribution in a finite-depth channel, we report on the apparent angle of the wake, which is the angle of maximum peaks. For moderately deep channels, the dependence of the apparent wake angle on the Froude number is very different to the wedge angle and varies smoothly as FH passes through the critical value FH = 1. For shallow water, the two angles tend to follow each other more closely, which leads to very large apparent wake angles for certain regimes.
A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS
Publisher: Cambridge University Press (CUP)
Date: 07-2021
DOI: 10.1017/S144618112100033X
Abstract: The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.
Mathematical models for cell migration with real-time cell cycle dynamics
Publisher: Elsevier BV
Date: 03-0055
A Bayesian computational approach to explore the optimal duration of a cell proliferation assay
Publisher: Springer Science and Business Media LLC
Date: 28-06-2017
DOI: 10.1007/S11538-017-0311-4
Abstract: Cell proliferation assays are routinely used to explore how a low-density monolayer of cells grows with time. For a typical cell line with a doubling time of 12 h (or longer), a standard cell proliferation assay conducted over 24 h provides excellent information about the low-density exponential growth rate, but limited information about crowding effects that occur at higher densities. To explore how we can best detect and quantify crowding effects, we present a suite of in silico proliferation assays where cells proliferate according to a generalised logistic growth model. Using approximate Bayesian computation we show that data from a standard cell proliferation assay cannot reliably distinguish between classical logistic growth and more general non-logistic growth models. We then explore, and quantify, the trade-off between increasing the duration of the experiment and the associated decrease in uncertainty in the crowding mechanism.
Surface reconstruction of wheat leaf morphology from three-dimensional scanned data
Publisher: CSIRO Publishing
Date: 2015
DOI: 10.1071/FP14058
Abstract: Realistic virtual models of leaf surfaces are important for several applications in the plant sciences, such as modelling agrichemical spray droplet movement and spreading on the surface. In this context, the virtual surfaces are required to be smooth enough to facilitate the use of the mathematical equations that govern the motion of the droplet. Although an effective approach is to apply discrete smoothing D2-spline algorithms to reconstruct the leaf surfaces from three-dimensional scanned data, difficulties arise when dealing with wheat (Triticum aestivum L.) leaves, which tend to twist and bend. To overcome this topological difficulty, we develop a parameterisation technique that rotates and translates the original data, allowing the surface to be fitted using the discrete smoothing D2-spline methods in the new parameter space. Our algorithm uses finite element methods to represent the surface as a linear combination of compactly supported shape functions. Numerical results confirm that the parameterisation, along with the use of discrete smoothing D2-spline techniques, produces realistic virtual representations of wheat leaves.
Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models
Publisher: American Physical Society (APS)
Date: 02-02-2011
A computational modelling framework to quantify the effects of passaging cell lines
Publisher: Public Library of Science (PLoS)
Date: 27-07-2017
Constructive development of the solutions of linear equations in introductory ordinary differential equations
Publisher: Informa UK Limited
Date: 15-07-2009
Characterizing transport through a crowded environment with different obstacle sizes
Publisher: AIP Publishing
Date: 04-02-2014
DOI: 10.1063/1.4864000
Abstract: Transport through crowded environments is often classified as anomalous, rather than classical, Fickian diffusion. Several studies have sought to describe such transport processes using either a continuous time random walk or fractional order differential equation. For both these models the transport is characterized by a parameter α, where α = 1 is associated with Fickian diffusion and α & 1 is associated with anomalous subdiffusion. Here, we simulate a single agent migrating through a crowded environment populated by impenetrable, immobile obstacles and estimate α from mean squared displacement data. We also simulate the transport of a population of such agents through a similar crowded environment and match averaged agent density profiles to the solution of a related fractional order differential equation to obtain an alternative estimate of α. We examine the relationship between our estimate of α and the properties of the obstacle field for both a single agent and a population of agents we show that in both cases, α decreases as the obstacle density increases, and that the rate of decrease is greater for smaller obstacles. Our work suggests that it may be inappropriate to model transport through a crowded environment using widely reported approaches including power laws to describe the mean squared displacement and fractional order differential equations to represent the averaged agent density profiles.
Kelvin wake pattern at small Froude numbers
Publisher: Cambridge University Press (CUP)
Date: 31-03-2021
DOI: 10.1017/JFM.2021.193
Exact time-dependent solutions of a Fisher–KPP-like equation obtained with nonclassical symmetry analysis
Publisher: Elsevier BV
Date: 10-2022
Moments of action provide insight into critical times for advection-diffusion-reaction processes
Publisher: American Physical Society (APS)
Date: 25-09-2012
A Fibrocontractive Mechanochemical Model of Dermal Wound Closure Incorporating Realistic Growth Factor Kinetics
Publisher: Springer Science and Business Media LLC
Date: 13-01-2012
DOI: 10.1007/S11538-011-9712-Y
Abstract: Fibroblasts and their activated phenotype, myofibroblasts, are the primary cell types involved in the contraction associated with dermal wound healing. Recent experimental evidence indicates that the transformation from fibroblasts to myofibroblasts involves two distinct processes: The cells are stimulated to change phenotype by the combined actions of transforming growth factor β (TGFβ) and mechanical tension. This observation indicates a need for a detailed exploration of the effect of the strong interactions between the mechanical changes and growth factors in dermal wound healing. We review the experimental findings in detail and develop a model of dermal wound healing that incorporates these phenomena. Our model includes the interactions between TGFβ and collagenase, providing a more biologically realistic form for the growth factor kinetics than those included in previous mechanochemical descriptions. A comparison is made between the model predictions and experimental data on human dermal wound healing and all the essential features are well matched.
Assimilation of GPS-tracked drifter data to improve the Eulerian velocity fields in an estuary
Publisher: Elsevier BV
Date: 11-2021
Free-surface flows emerging from beneath a semi-infinite plate with constant vorticity
Publisher: Cambridge University Press (CUP)
Date: 25-06-2002
DOI: 10.1017/S0022112002008510
Abstract: The free-surface flow past a semi-infinite horizontal plate in a finite-depth fluid is considered. It is assumed that the fluid is incompressible and inviscid and that the flow approaches a uniform shear flow downstream. Exact relations are derived using conservation of mass and momentum for the case where the downstream free surface is flat. The complete nonlinear problem is solved numerically using a boundary-integral method and these waveless solutions are shown to exist only when the height of the plate above the bottom is greater than the height of the uniform shear flow. Interesting results are found for various values of the constant vorticity. Solutions with downstream surface waves are also considered, and nonlinear results of this type are compared with linear results found previously. These solutions can be used to model the flow near the stern of a (two-dimensional) ship.
Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularization
Publisher: IOP Publishing
Date: 29-04-0001
Mean exit time for diffusion on irregular domains
Publisher: IOP Publishing
Date: 09-1923
Abstract: Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first passage time, and in particular the mean first passage time, is an active area of research relevant to many disciplines. Exact results for the mean first passage time for diffusion on simple geometries, such as lines, discs and spheres, are well-known. In contrast, computational methods are often used to study the first passage time for diffusion on more realistic geometries where closed-form solutions of the governing elliptic boundary value problem are not available. Here, we develop a perturbation solution to calculate the mean first passage time on irregular domains formed by perturbing the boundary of a disc or an ellipse. Classical perturbation expansion solutions are then constructed using the exact solutions available on a disc and an ellipse. We apply the perturbation solutions to compute the mean first exit time on two naturally-occurring irregular domains: a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing the perturbation solutions with numerical solutions of the elliptic boundary value problem on these irregular domains confirms that we obtain a very accurate solution with a few terms in the series only. MATLAB software to implement all calculations is available at github.com/ProfMJSimpson/Exit_time .
Modeling transport through an environment crowded by a mixture of obstacles of different shapes and sizes
Publisher: Elsevier BV
Date: 05-2016
Image analysis of shatter and pinning events on hard‐to‐wet leaf surfaces by drops containing surfactant
Publisher: Wiley
Date: 03-03-2020
DOI: 10.1002/PS.5796
Accurate series solutions for gravity-driven Stokes waves
Publisher: AIP Publishing
Date: 08-2010
DOI: 10.1063/1.3480394
Abstract: In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.
Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy
Publisher: The Royal Society
Date: 09-2019
Abstract: The Fisher–Kolmogorov–Petrovsky–Piskunov model, also known as the Fisher–KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher–KPP model cannot replicate, such as the extinction of invasive populations. The Fisher–Stefan model is an adaptation of the Fisher–KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher–Stefan model also supports travelling wave solutions however, a key additional feature of the Fisher–Stefan model is that it is able to simulate population extinction, giving rise to a spreading–extinction dichotomy . In this work, we revisit travelling wave solutions of the Fisher–KPP model and show that these results provide new insight into travelling wave solutions of the Fisher–Stefan model and the spreading–extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher–Stefan model and often-neglected travelling wave solutions of the Fisher–KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher–Stefan model in the limit of slow travelling wave speeds, c ≪1.
Optimal fluid injection strategies for in situ mineral leaching in two-dimensions
Publisher: Springer Science and Business Media LLC
Date: 1999
The role of initial geometry in experimental models of wound closing
Publisher: Elsevier BV
Date: 04-2018
Axially symmetric continuum mechanical model for the oscillon
Publisher: SAGE Publications
Date: 23-07-2022
DOI: 10.1177/10812865221107413
Abstract: The oscillon is a highly localized dynamical phenomenon occurring in a thin horizontal layer of granular material, which rests on a rigid plate oscillating in the vertical direction. The geometry is axially symmetric and physically resembles a splash of liquid due to a falling drop, except that it continually perpetuates itself and does not generate a spreading wave, as is the case for a liquid splash. The oscillon moves from “peak” to “crater” and “crater” to “peak” such that the time from “peak” to “peak” or “crater” to “crater” is twice the period of the oscillating plate. The physics of granular phenomena is not properly understood, and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behaviour. Here for a free-flowing (cohesion-less) granular material, under axially symmetric conditions, we present a partial continuum mechanical analysis assuming the Coulomb–Mohr yield function and non-dilatant double-shearing theory. We examine small perturbations superimposed upon a purely vertical vibration, and make the assumption that throughout the motion, the lower surface of the layer remains in contact with the rigid metal plate. We show how the temporal dependence, which decouples from the spatial structure, is governed by Mathieu’s equation for the physically relevant case of the rigid plate oscillating sinusoidally, and therefore stability is determined by certain key parameters. We explore a variety of possible forms for spatial dependence. The present axially symmetric analysis complements that presented by the authors for plane strain conditions, and we find, quite remarkably, that apart from constants, both flows are governed by similar fourth-order systems of ordinary differential equations. This means that for both plane strain and axially symmetry, analogous pattern forming conditions can operate.
Perturbation solutions for flow through symmetrical hoppers with inserts and asymmetrical wedge hoppers
Publisher: Springer Netherlands
Date: 2005
A review of one-phase Hele-Shaw flows and a level-set method for nonstandard configurations
Publisher: Australian Mathematical Publishing Association, Inc.
Date: 16-11-2021
DOI: 10.21914/ANZIAMJ.V63.16689
Abstract: The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity. doi:10.1017/S144618112100033X
Nanoparticle Melting as a Stefan Moving Boundary Problem
Publisher: American Scientific Publishers
Date: 02-2009
Abstract: The melting of spherical nanoparticles is considered from the perspective of heat flow in a pure material and as a moving boundary (Stefan) problem. The dependence of the melting temperature on both the size of the particle and the interfacial tension is described by the Gibbs-Thomson effect, and the resulting two-phase model is solved numerically using a front-fixing method. Results show that interfacial tension increases the speed of the melting process, and furthermore, the temperature distribution within the solid core of the particle exhibits behaviour that is qualitatively different to that predicted by the classical models without interfacial tension.
Inferring parameters for a lattice-free model of cell migration and proliferation using experimental data
Publisher: Elsevier BV
Date: 2018
DOI: 10.1016/J.JTBI.2017.10.032
Abstract: Collective cell spreading takes place in spatially continuous environments, yet it is often modelled using discrete lattice-based approaches. Here, we use data from a series of cell proliferation assays, with a prostate cancer cell line, to calibrate a spatially continuous in idual based model (IBM) of collective cell migration and proliferation. The IBM explicitly accounts for crowding effects by modifying the rate of movement, direction of movement, and the rate of proliferation by accounting for pair-wise interactions. Taking a Bayesian approach we estimate the free parameters in the IBM using rejection s ling on three separate, independent experimental data sets. Since the posterior distributions for each experiment are similar, we perform simulations with parameters s led from a new posterior distribution generated by combining the three data sets. To explore the predictive power of the calibrated IBM, we forecast the evolution of a fourth experimental data set. Overall, we show how to calibrate a lattice-free IBM to experimental data, and our work highlights the importance of interactions between in iduals. Despite great care taken to distribute cells as uniformly as possible experimentally, we find evidence of significant spatial clustering over short distances, suggesting that standard mean-field models could be inappropriate.
Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation
Publisher: Cambridge University Press (CUP)
Date: 04-08-2014
DOI: 10.1017/S0956792514000230
Abstract: We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.
Critical time scales for advection-diffusion-reaction processes
Publisher: American Physical Society (APS)
Date: 23-04-2012
Simulating droplet motion on virtual leaf surfaces
Publisher: The Royal Society
Date: 05-2015
DOI: 10.1098/RSOS.140528
Abstract: A curvilinear thin film model is used to simulate the motion of droplets on a virtual leaf surface, with a view to better understand the retention of agricultural sprays on plants. The governing model, adapted from Roy et al. (2002 J. Fluid Mech. 454, 235–261 ( doi:10.1017/S0022112001007133 )) with the addition of a disjoining pressure term, describes the gravity- and curvature-driven flow of a small droplet on a complex substrate: a cotton leaf reconstructed from digitized scan data. Coalescence is the key mechanism behind spray coating of foliage, and our simulations demonstrate that various experimentally observed coalescence behaviours can be reproduced qualitatively. By varying the contact angle over the domain, we also demonstrate that the presence of a chemical defect can act as an obstacle to the droplet's path, causing break-up. In simulations on the virtual leaf, it is found that the movement of a typical spray size droplet is driven almost exclusively by substrate curvature gradients. It is not until droplet mass is sufficiently increased via coalescence that gravity becomes the dominating force.
Persistency of debris accumulation in tidal estuaries using Lagrangian coherent structures
Publisher: Elsevier BV
Date: 08-2021
Symmetry analysis for uniaxial compression of a hypoplastic granular material
Publisher: Springer Science and Business Media LLC
Date: 11-2005
Simulating spray droplet impaction outcomes: comparison with experimental data
Publisher: Wiley
Date: 06-02-2020
DOI: 10.1002/PS.5736
Abstract: A suite of plant retention spray models has been developed to simulate spray retention using virtual surfaces (either single leaves or whole plants) and their outputs compared with experimental data for the equivalent spray scenarios. The results for a single formulation (0.1% v/v lecithin mixture in water) and difficult to wet plant species Chenopodium album L (common lambsquarters) are presented. They include experimental observations with single leaves, as well as simulations of virtual impaction events, conducted to provide for the first time estimates of f (the proportion of theoretical impact drop diameter at shatter). With this factor prescribed, multi-plant simulations using a range of nozzle types and droplet sizes (volume mean diameter (VMD) range 241 to 530 μm) are compared with equivalent experimentally determined spray retention by real plants. The simulations demonstrated that impaction resulted predominantly in shatter with the production of daughter droplets, and that retention is mainly due to re-capture of these droplets. Overall the simulations show the same trends as experimental retention results from different nozzle applications, but at best predicted retention results were 68% to 79% of experimental percentage retention, depending on plant spacing. Retention is the result of some primary drop capture but predominantly by recapture of shatter droplets as the modelling illustrates. The value of f affects the droplet shatter outcome and can result in fewer, more energetic daughter droplets, or more droplets but with lower energies. However, this effect alone cannot explain the discrepancy between actual and simulated results. Possible operational influences are discussed. © 2020 Society of Chemical Industry.
A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area
Publisher: The Royal Society
Date: 2016
Abstract: Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.
Exponential asymptotics of free surface flow due to a line source
Publisher: Oxford University Press (OUP)
Date: 12-04-2013
Free-surface flow past arbitrary topography and an inverse approach for wave-free solutions
Publisher: Oxford University Press (OUP)
Date: 23-04-2013
Moving Boundary Problems for Quasi-Steady Conduction Limited Melting
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2019
DOI: 10.1137/18M123445X
Saffman-Taylor fingers with kinetic undercooling
Publisher: American Physical Society (APS)
Date: 23-02-2015
Micro/nanoparticle melting with spherical symmetry and surface tension
Publisher: Oxford University Press (OUP)
Date: 26-12-2008
Selection of a Hele-Shaw bubble via exponential asymptotics
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2020
DOI: 10.1137/18M1220868
Efficient computation of two‐dimensional steady free‐surface flows
Publisher: Wiley
Date: 17-10-2018
DOI: 10.1002/FLD.4469
Towards a model of spray–canopy interactions: Interception, shatter, bounce and retention of droplets on horizontal leaves
Publisher: Elsevier BV
Date: 10-2014
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