A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Dallaston, Michael C.; McCue, Scott W.

doi:10.1098/rspa.2015.0629

Cited by 21 publications

(27 citation statements)

References 42 publications

(71 reference statements)

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“…All of these stability results are analogous to those for contracting bubbles in a Hele-Shaw cell for which the viscous fluid is of a power-law type [48,49] or for which there is a competition between surface tension and a kinetic-type boundary condition [50,51]. For these problems, there are also linear stability results that show a circular interface may be stable or unstable, depending on a parameter value, leading to the existence of noncircular selfsimilar solutions (that evolve to shapes which appear like k-sided polygons with rounded corners).…”

Section: Similarity Solutions With K-fold Symmetrymentioning

confidence: 62%

Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology

McCue

Wang

Moroney

et al. 2019

Physica D: Nonlinear Phenomena

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Continuum mathematical models for collective cell motion normally involve reaction-diffusion equations, such as the Fisher-KPP equation, with a linear diffusion term to describe cell motility and a logistic term to describe cell proliferation. While the Fisher-KPP equation and its generalisations are commonplace, a significant drawback for this family of models is that they are not able to capture the moving fronts that arise in cell invasion applications such as wound healing and tumour growth. An alternative, less common, approach is to include nonlinear degenerate diffusion in the models, such as in the Porous-Fisher equation, since solutions to the corresponding equations have compact support and therefore explicitly allow for moving fronts. We consider here a hole-closing problem for the Porous-Fisher equation whereby there is initially a simply connected region (the hole) with a nonzero population outside of the hole and a zero population inside. We outline how self-similar solutions (of the second kind) describe both circular and non-circular fronts in the hole-closing limit. Further, we present new experimental and theoretical evidence to support the use of nonlinear degenerate diffusion in models for collective cell motion. Our methodology involves setting up a two-dimensional wound healing assay that has the geometry of a hole-closing problem, with cells initially seeded outside of a hole that closes as cells migrate and proliferate. For a particular class of fibroblast cells, the aspect ratio of an initially rectangular wound increases in time, so the wound becomes longer and thinner as it closes; our theoretical analysis shows that this behaviour is consistent with nonlinear degenerate diffusion but is not able to be captured with commonly used linear diffusion. This work is important because it provides a clear test for degenerate diffusion over linear diffusion in cell lines, whereas standard one-dimensional experiments are unfortunately not capable of distinguishing between the two approaches.

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Section: Similarity Solutions With K-fold Symmetrymentioning

confidence: 62%

Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology

McCue

Wang

Moroney

et al. 2019

Physica D: Nonlinear Phenomena

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“…self-similar shrinking interfaces such as those observed in [9,10] when a bubble is driven to extinction by the presence of a sink in a Hele-Shaw cell.…”

Section: 2mentioning

confidence: 83%

“…In particular, for the shrinking interface problem, we choose (1) the space scaling function R( \= t) so that the shrinking interface is always mapped back to its initial size, i.e., the interface does not shrink in the rescaled frame; (2) the time scaling function \rho ( \= t) to slow down the motion of the interface, especially at later times when the interface becomes very small and shrinks extremely rapidly. We note that an alternative time and space rescaling scheme was implemented in [9,10] to accurately simulate vanishing bubbles in a Hele-Shaw cell. Here, we use a semi-implicit, nonstiff time stepping method developed originally in [23] to remove the third order time step constraint.…”

Section: B1207mentioning

confidence: 99%

Computation of a Shrinking Interface in a Hele-Shaw Cell

Zhao

Ying

et al. 2018

SIAM J. Sci. Comput.

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The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman-Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap b(t) is increased more, where τ is the surface tension and C is a function of the interface perturbation mode k. Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behavior of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al., Phys. Fluids, vol. 23, 2011, pp. 123101). When we use the shape invariant gap b(t), our nonlinear results reveal the existence of k-fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost one at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode k of the vanishing interface and the control parameter C.

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“…From a mathematical perspective, these issues are reminiscent of the differences between curve shortening flow and mean curvature flow in differential geometry [35,36]. Given that Hele-Shaw flow may be interpreted as a nonlocal version of curve shortening flow in the plane [21,37,38], a final reason to use (2) is that our three-dimensional moving boundary problem can be considered a nonlocal version of the mean curvature flow. Again, in this context we are interested in studying the effect of surface tension in (2) on the development of axially symmetric curvature singularities.…”

Section: Introductionmentioning

confidence: 99%

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

2019

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We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterised by a blow-up of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a novel numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent α = 1/3 just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.

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A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Cited by 21 publications

References 42 publications

Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology

Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology

Computation of a Shrinking Interface in a Hele-Shaw Cell

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

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