Forthcoming events in this series
Acrobatics of Liquid Ropes
Abstract
Honey poured from a sufficient height onto toast undergoes the well-known `liquid rope coiling’ instability.
We have studied this instability using a combination of laboratory experiments, theory, and numerics, with the aim of determining phase diagrams and scaling laws for the different coiling modes. Finite-amplitude coiling has four distinct modes - viscous, gravitational, inertio-gravitational, and inertial - depending on how the viscous forces that resist deformation of the rope are balanced. The inertio-gravitational mode is particularly interesting as it involves resonance between the coiling portion of the rope and its long trailing `tail’. Further experiments using less viscous fluids reveal that the rope can exhibit five different morphologies, of which steady coiling is only one. We determine the detailed phase diagram of these morphologies, which includes a novel `liquid supercoiling’
state in which the coiled cylinder formed by the primary coiling instability undergoes in turn its own complex buckling instability. We show that the onset of these different patterns is determined by a non-penetrability condition which takes different forms in the viscous, gravitational and inertial limits. To close, we will briefly evoke two additional related phenomena: spiral waves of bubbles generated by coiling, and the `fluid mechanical sewing machine’ in which the fluid falls onto a moving belt.
Interactions of noise and discontinuities: transitions and qualitative changes
Abstract
While there have been recent advances for analyzing the complex deterministic
behavior of systems with discontinuous dynamics, there are many open questions about
understanding and predicting noise-driven and noise-sensitive phenomena in the
non-smooth context. Stochastic effects can often change the picture dramatically,
particularly if multiple time scales are present. We demonstrate novel approaches
for exploring and explaining surprising phenomena driven by the interplay of
nonlinearities, delays, randomness, in specific applications with piecewise smooth
dynamics - nonlinear models of balance, relay control, and impacting dynamics.
Effective techniques typically depend on the combination of mathematical techniques,
multiple scales techniques, and phenomenological intuition from seemingly unrelated
canonical models of biophysics, mechanics, and chemical dynamics. The appropriate
strategy is not always immediately obvious from the area of application or model
type. This gap may follow from the limited attention that stochastic models with
discontinuous dynamics have received in the past, or it may be the reason for this
limited attention. Combining the geometrical perspective with asymptotic approaches
in physical and phase space appears to be a critical part of developing effective
approaches.
Wave-particle coupling in fluid mechanics: bouncing droplets and flapping swimmers
Abstract
Group Meeting
Abstract
Barbara Mahler: 15+5 min
Thomas Woolley: 15+5 min
Julian A. Garcia Grajales: 15+5 min
Predictive simulations for optimisation of inhaled drug delivery
Abstract
Respiratory illnesses, such as asthma and chronic obstructive pulmonary disease, account for one in five deaths worldwide and cost the UK over £6 billion a year. The main form of treatment is via inhaled drug delivery. Typically, however, a low fraction of the inhaled dose reaches the target areas in the lung. Predictive numerical capabilities have the potential for significant impact in the optimisation of pulmonary drug delivery. However, accurate and efficient prediction is challenging due to the complexity of the airway geometries and of the flow in the airways. In addition, geometric variation of the airways across subjects has a pronounced effect on the aerosol deposition. Therefore, an accurate model of respiratory deposition remains a challenge.
High-fidelity simulations of the flow field and prediction of the deposition patterns motivate the use of direct numerical simulations (DNS) in order to resolve the flow. Due to the high grid resolution requirements, it is desirable to adopt an efficient computational strategy. We employ a robust immersed boundary method developed for curvilinear coordinates, which allows the use of structured grids to model the complex patient-specific airways, and can accommodate the inter-subject geometric variations on the same grid. The proposed approach reduces the errors at the boundary and retains the stability guarantees of the original flow solver.
A Lagrangian particle tracking scheme is adopted to model the transport of aerosol particles. In order to characterise deposition, we propose the use of an instantaneous Stokes number based on the local properties of the flow field. The effective Stokes number is then defined as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability. Our results demonstrate that the effective Stokes number can deviate significantly from the reference value based solely on a characteristic flow velocity and length scale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency.
Group Meeting
Abstract
Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;
John Ockendon: Wave Homogenisation, 10 min + questions;
Hilary Ockendon: Sloshing, 10 min + questions
Sharp interface limit in a phase field model of cell motility
Abstract
We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model. Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.
Attributes and Artifacts of Network Optimization
Abstract
Much of the recent interest in complex networks has been driven by the prospect that network optimization will help us understand the workings of evolutionary pressure in natural systems and the design of efficient engineered systems. In this talk, I will reflect on unanticipated attributes and artifacts in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. Then I will comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes/variables often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. Optimization is a double-edged sword for which desired and adverse effects can be exacerbated in complex network systems due to the high dimensionality of their dynamics.
OCIAM Group Meeting - New singularities for Stokes waves
Abstract
Inferring the large-scale structure of networks
Abstract
Networks form the backbones of a wide variety of complex systems,
ranging from food webs, gene regulation and social networks to
transportation networks and the internet. Due to the sheer size and
complexity of many of theses systems, it remains an open challenge to
formulate general descriptions of their large-scale structures.
Although many methods have been proposed to achieve this, many of them
yield diverging descriptions of the same network, making both the
comparison and understanding of their results very
difficult. Furthermore, very few methods attempt to gauge the
statistical significance of the uncovered structures, and hence the
majority cannot reliably separate actual structure from stochastic
fluctuations. In this talk, I will show how these issues can be tackled
in a principled fashion by formulating appropriate generative models of
network structure that can have their parameters inferred from data. I
will also consider the comparison between a variety of generative
models, including different structural features such as degree
correction, where nodes with arbitrary degrees can belong to the same
group, and community overlap, where nodes are allowed to belong to more
than one group. Because such model variants possess an increased number
of parameters, they become prone to overfitting. We demonstrate how
model selection based on the minimum description length criterion and
posterior odds ratios can fully account for the increased degrees of
freedom of the larger models, and selects the most appropriate trade-off
between model complexity and quality of fit based on the statistical
evidence present in the data.
Throughout the talk I will illustrate the application of the methods
with many empirical networks such as the internet at the autonomous
systems level, the global airport network, the network of actors and
films, social networks, citations among websites, co-occurrence of
disease-causing genes and many others.
Acoustic liners in aircraft engines
Abstract
Noise limits are one of the major constraints when designing
aircraft engines. Acoustic liners are fitted in almost all civilian
turbofan engine intakes, and are being considered for use elsewhere in a
bid to further reduce noise. Despite this, models for acoustic liners
in flow have been rather poor until recently, with discrepancies of 10dB
or more. This talk will show why, and what is being done to model them
better. In the process, as well as mathematical modelling using
asymptotics, we will show that state of the art Computational
AeroAcoustics simulations leave a lot to be desired, particularly when
using optimized finite difference stencils.
Group Meeting
Abstract
Michael Gomez:
Title: The role of ghosts in elastic snap-through
Abstract: Elastic `snap-through' buckling is a striking instability of many elastic systems with natural curvature and bistable states. The conditions under which bistability exists have been reasonably well studied, not least because a number of engineering applications make use of the rapid transitions between states. However, the dynamics of the transition itself remains much less well understood. Several examples have been studied that show slower dynamics than would be expected based on purely elastic timescales of motion, with the natural conclusion drawn that some other effect, such as viscoelasticity, must play a role. I will present analysis (and hopefully experiments) of a purely elastic system that shows similar `anomalous dynamics'; however, we show that here this dynamics is a consequence of the ‘ghost’ of the snap-through bifurcation.
Andrew Krause:
Title: Fluid-Growth Interactions in Bioactive Porous Media
Abstract: Recent models in Tissue Engineering have considered pore blocking by cells in a porous tissue scaffold, as well as fluid shear effects on cell growth. We implement a suite of models to better understand these interactions between cell growth and fluid flow in an active porous medium. We modify some existing models in the literature that are spatially continuous (e.g. Darcy's law with a cell density dependent porosity). However, this type of model is based on assumptions that we argue are not good at describing geometric and topological properties of a heterogeneous pore network, and show how such a network can emerge in this system. Therefore we propose a different modelling paradigm to directly describe the mesoscopic pore networks of a tissue scaffold. We investigate a deterministic network model that can reproduce behaviour of the continuum models found in the literature, but can also exhibit finite-scale effects of the pore network. We also consider simpler stochastic models which compare well with near-critical Percolation behaviour, and show how this kind of behaviour can arise from our deterministic network model.
Abstract:We study an evolving network where the nodes are considered as represent particles with a corresponding state vector. Edges between nodes are created and destroyed as a Poisson process, and new nodes enter the system. We define the concept of a “local state degree distribution” (LSDD) as a degree distribution that is local to a particular point in phase space. We then derive a differential equation that is satisfied approximately by the LSDD under a mean field assumption; this allows us to calculate the degree distribution. We examine the validity of our derived differential equation using numerical simulations, and we find a close match in LSDD when comparing theory and simulation. Using the differential equation derived, we also propose a continuum model for osteocyte network formation within bone. The structure of this network has implications regarding bone quality. Furthermore, osteocyte network structure can be disrupted within cancerous microenvironments. Evidence suggests that cancerous osteocyte networks either have dendritic overgrowth or underdeveloped dendrites. This model allows us to probe the density and degree distribution of the dendritic network. We consider a traveling wave solution of the osteocyte LSDD profile which is of relevance to osteoblastic bone cancer (which induces net bone formation). We then hypothesise that increased rates of differentiation would lead to higher densities of osteocytes but with a lower quantity of dendrites.
Information processing in feedforward neuronal networks
Abstract
Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum.
Localized Patterns & Spatial Heterogeneitie
Abstract
We consider the impact of spatial heterogeneities on the dynamics of
localized patterns in systems of partial differential equations (in one
spatial dimension). We will mostly focus on the most simple possible
heterogeneity: a small jump-like defect that appears in models in which
some parameters change in value as the spatial variable x crosses
through a critical value -- which can be due to natural inhomogeneities,
as is typically the case in ecological models, or can be imposed on the
model for engineering purposes, as in Josephson junctions. Even such a
small, simplified heterogeneity may have a crucial impact on the
dynamics of the PDE. We will especially consider the effect of the
heterogeneity on the existence of defect solutions, which boils down to
finding heteroclinic (or homoclinic) orbits in an n-dimensional
dynamical system in `time' x, for which the vector field for x > 0
differs slightly from that for x < 0 (under the assumption that there is
such an orbit in the homogeneous problem). Both the dimension of the
problem and the nature of the linearized system near the limit points
have a remarkably rich impact on the defect solutions. We complement the
general approach by considering two explicit examples: a heterogeneous
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous
generalized FitzHugh–Nagumo system (n = 6).
Spatial Efficiency of Complex Networks
Abstract
Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.
16:00
Swarming Models with Repulsive-Attractive Effects: Pattern Stability
Abstract
I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.
Evaporation of droplets with moving contact lines
Abstract
Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.
Some non-local problems arising in mathematical biology
Abstract
This talk covers two topics: (1) Phenotype change, where we consider the steady-fitness states, in a model developed by Korobeinikov and Dempsey (2014), in which the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. This enables thresholds for survival/extinction to be established in terms of fitness.
Topic (2) looks at the steady-size distribution of an evolving cohort of cells, such as tumour cells in vitro, and therein establishes thresholds for growth or decay of the cohort. This is established using a new class of non-local (but linear) singular eigenvalue problems which have point spectra, like the traditional Sturm-Liouville problems. The first eigenvalue gives the threshold required. But these problems are first order unless dispersion is added to incorporate random perturbations. But the same idea will apply here also. Current work involves binary asymmetrical division of cells, simultaneous with growth. It has implications to cancer biology, helping biologists to conceptualise non-local effects and the part they may play in cancer. This is developed in Zaidi et al (2015).
Acknowledgement. The support of Gravida (NCGD) is gratefully acknowledged.
References
Korobeinikov A & Dempsey C. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences and Engineering 11, (2014), 919-927.
Zaidi AA, van-Brunt B, & Wake GC. A model for asymmetrical cell division Mathematical Biosciences and Engineering (June 2015).
Complex Solutions of the Navier-Stokes Equations
Abstract
It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.
We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane.
We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.
Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.
by Jonathan Mestel and Florencia Boshier
Epidemic processes in temporal networks
Abstract
In today's interconnected world, the dissemination of an idea, a trend, a rumor through social networks, as well as the propagation of information or cyber-viruses through digital networks are all common phenomena. They are conceptually similar to the spread of infectious diseases among hosts, as common to all these phenomena is the dissemination of a spreading agent on a networked system. A large body of research has been produced in recent years to characterize the spread of epidemics on static connectivity patterns in a wide range of biological and socio-technical systems. In particular, understanding the mechanisms and conditions for widespread dissemination represents a crucial step for its prevention and control (e.g. in the case of diseases) or for its enhancement (e.g. in the case of viral marketing). This task is however further hindered by the temporal nature characterizing the activation of the connections shaping the networked system, for which data has recently become available. As an example, in networks of proximity contacts among individuals, connections represent sequences of contacts that are active for given periods of time. The time variation of contacts in a networked system may fundamentally alter the properties of spreading processes occurring on it, with respect to static networks, and affect the condition at which epidemics become possible. In this talk I will present a novel theoretical framework adopting a multi-layer perspective for the analytical understanding of the interplay between temporal networks and spreading dynamics. The framework is tested on a set of time-varying network models and empirical networks.