Extended Stefan problem for the solidification of binary alloys in a sphere

Planella, Ferran Brosa; Please, Colin P.; Gorder, Robert A. Van

doi:10.1017/s095679252000011x

Cited by 17 publications

(12 citation statements)

References 44 publications

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“…Heat maps in Figure 9(a)-(b) compare numerical estimates of κ from the phase plane with the perturbation result, Equation (33). The heat map of δκ = κ − κ p in Figure 9(c) shows that the O(c −8 ) perturbation solutions leads to extremely accurate solutions for κ for c < −2 for all u f .…”

Section: Fast Retreating Travelling Wavesmentioning

confidence: 88%

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Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

El‐Hachem¹,

McCue²,

Simpson³

2021

Preprint

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The Fisher-KPP model, and generalisations thereof, are simple reaction-diffusion models of biological invasion that assume individuals in the population undergo linear diffusion with diffusivity D, and logistic proliferation with rate λ. For the Fisher-KPP model, biologically-relevant initial conditions lead to long-time travelling wave solutions that move with speed c = 2 √ λD. 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 = 2This means that, for biologically-relevant initial data, the Fisher-KPP model can not be used to study invasion with c 2 √ λD, or retreating travelling waves with c < 0. Here, we reformulate the Fisher-KPP model as a moving boundary problem on x < s(t) 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, and can propagate with any wave speed, −∞ < c < ∞. 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.

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Section: Fast Retreating Travelling Wavesmentioning

confidence: 88%

“…given by a classical one-phase Stefan condition ds(t)/dt = −κ∂u(s(t), t)/∂x [25][26][27][28][29][30][31][32][33], has been called the Fisher-…”

Section: Various Mathematical Extensions Have Been Proposed To Overco...mentioning

confidence: 99%

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

El‐Hachem¹,

McCue²,

Simpson³

2021

Preprint

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“…Moving-boundary problems of this type are traditionally used to model physical and industrial processes [26][27][28][29][30]. Indeed, the boundary condition (1.3) is analogous to the classical Stefan condition [31] for a material undergoing phase change, where κ is the inverse Stefan number.…”

Section: Introductionmentioning

confidence: 99%

The Effect of Geometry on Survival and Extinction in a Moving-Boundary Problem Motivated by the Fisher-KPP Equation

Tam,

Simpson

2022

Preprint

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The Fisher-Stefan model involves solving the Fisher-KPP equation on a domain whose boundary evolves according to a Stefan-like condition. The Fisher-Stefan model alleviates two practical limitations of the standard Fisher-KPP model when applied to biological invasion. First, unlike the Fisher-KPP equation, solutions to the Fisher-Stefan model have compact support, enabling one to define the interface between occupied and unoccupied regions unambiguously. Second, the Fisher-Stefan model admits solutions for which the population becomes extinct, which is not possible in the Fisher-KPP equation. Previous research showed that population survival or extinction in the Fisher-Stefan model depends on a critical length in one-dimensional Cartesian or radially-symmetric geometry. However, the survival and extinction behaviour for general two-dimensional regions remains unexplored. We combine analysis and level-set numerical simulations of the Fisher-Stefan model to investigate the survival-extinction conditions for rectangular-shaped initial conditions. We show that it is insufficient to generalise the critical length conditions to critical area in two-dimensions. Instead, knowledge of the region geometry is required to determine whether a population will survive or become extinct.

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“…Moving boundary problems are frequently used to model phenomena in physics, in particular melting/freezing processes where the moving boundary represents a solid/melt interface [1][2][3] or interfacial flow problems where the moving boundary is the interface between two fluids [4]. In diffusion-driven process, a moving boundary can be driven by swelling of one phase, for example a polymer or a porous material [5][6][7] One key motivation from a physics perspective is to use moving boundary problems to describe pattern formation at the interface, such as that which occurs when a crystal forms from a supercooled melt [8,9] or when viscous fingering develops in a Hele-Shaw cell [10,11].…”

Section: Introductionmentioning

confidence: 99%

Travelling waves, blow-up and extinction in the Fisher-Stefan model

McCue,

El-Hachem,

Simpson

2021

Preprint

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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 example, 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 generalisation of the well-known Fisher-KPP model, characterised by a leakage coefficient κ 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 with compact support which is more natural in terms of mathematical modelling. Almost all of the existing analysis of the standard Fisher-Stefan model involves setting κ > 0, which can lead to either invading travelling wave solutions or complete extinction of the population. Here, we demonstrate how setting κ < 0 leads to retreating travelling 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 scaling analysis, leading to new insight into the possibilities of travelling waves, blow-up and extinction for this moving boundary problem. Matlab software used to generate the results in this work are available on Github

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Extended Stefan problem for the solidification of binary alloys in a sphere

Cited by 17 publications

References 44 publications

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

The Effect of Geometry on Survival and Extinction in a Moving-Boundary Problem Motivated by the Fisher-KPP Equation

Travelling waves, blow-up and extinction in the Fisher-Stefan model

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