1,168 research outputs found
Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation
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
Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations
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. Tapering
plates in either converging or diverging directions 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.
Simulations compare well with previous experimental results for various
injection rates and geometric configurations. Further, we demonstrate how the
fully nonlinear dynamics of the problem affect the number of fingers that
emerge and how well this number agrees with predictions from linear stability
analysis
A novel model for one-dimensional morphoelasticity. Part I - Theoretical foundations
While classical continuum theories of elasticity and viscoelasticity have long been used to describe the mechanical behaviour of solid biological tissues, they are of limited use for the description of biological tissues that undergo continuous remodelling. The structural changes to a soft tissue associated with growth and remodelling require a mathematical theory of ‘morphoelasticity’ that is more akin to plasticity than elasticity. However, previously-derived mathematical models for plasticity are difficult to apply and interpret in the context of growth and remodelling: many important concepts from the theory of plasticity do not have simple analogues in biomechanics.\ud
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In this work, we describe a novel mathematical model that combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. While our focus here is on one-dimensional problems, our model builds on earlier work based on the multiplicative decomposition of the deformation gradient and can be adapted to develop a three-dimensional theory. The foundation of this work is the concept of ‘effective strain’, a measure of the difference between the current state and a hypothetical state where the tissue is mechanically relaxed. We develop one-dimensional equations for the evolution of effective strain, and discuss a number of potential applications of this theory. One significant application is the description of a contracting fibroblast-populated collagen lattice, which we further investigate in Part II
A novel model for one-dimensional morphoelasticity. Part II - Application to the contraction of fibroblast-populated collagen lattices
Fibroblast-populated collagen lattices are commonly used in experiments to study the interplay between fibroblasts and their pliable environment. Depending on the method by which\ud
they are set, these lattices can contract significantly, in some cases contracting to as little as 10% of their initial lateral (or vertical) extent. When the reorganisation of such lattices by fibroblasts is interrupted, it has been observed that the gels re-expand slightly but do not return to their original size. In order to describe these phenomena, we apply our theory of one-dimensional morphoelasticity derived in Part I to obtain a system of coupled ordinary differential equations, which we use to describe the behaviour of a fibroblast-populated collagen lattice that is tethered by a spring of known stiffness. We obtain approximate solutions that describe the behaviour of the system at short times as well as those that are valid for long times. We also obtain an exact description of the behaviour of the system in the case where the lattice reorganisation is interrupted. In addition, we perform a perturbation analysis in the limit of large spring stiffness to obtain inner and outer asymptotic expansions for the solution, and examine the relation between force and traction stress in this limit. Finally, we compare predicted numerical values for the initial stiffness and viscosity of the gel with corresponding values for previously obtained sets of experimental data and also compare the qualitative behaviour with that of our model in each case. We find that our model captures many features of the observed behaviour of fibroblast-populated collagen lattices
Saffman-Taylor fingers with kinetic undercooling
The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.Comment: 10 pages, 6 figures, accepted for publication by Physical Review
A two-compartment mechanochemical model of the roles of\ud transforming growth factor β and tissue tension in dermal wound healing
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
Differential Synchronization in Default and Task-Specific Networks of the Human Brain
On a regional scale the brain is organized into dynamic functional networks. The activity within one of these, the default network, can be dissociated from that in other task-specific networks. All brain networks are connected structurally but evidently are only transiently connected functionally. One hypothesis as to how such transient functional coupling occurs is that network formation and dissolution is mediated by increases and decreases in oscillatory synchronization between constituent brain regions. If so, then we should be able to find transient differences in intra-network synchronization between the default network and a task-specific network. In order to investigate this hypothesis we conducted two experiments in which subjects engaged in a Sustained Attention to Response Task while having brain activity recorded via high-density electroencephalography (EEG). We found that during periods when attention was focused internally (mind wandering) there was significantly more neural phase synchronization between brain regions associated with the default network, whereas during periods when subjects were focused on performing the visual task there was significantly more neural phase synchrony within a task-specific brain network that shared some of the same brain regions. These differences in network synchrony occurred in each of theta, alpha, and gamma frequency bands. A similar pattern of differential oscillatory power changes, indicating modulation of local synchronization by attention state, was also found. These results provide further evidence that the human brain is intrinsically organized into default and task-specific brain networks, and confirm that oscillatory synchronization is a potential mechanism for functional coupling within these networks
A fibrocontractive mechanochemical model of dermal wound\ud closure incorporating realistic growth factor kinetics
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
Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow
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
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.Comment: 16 pages, 16 figure
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