Challenges in Nonlinear Systems:

a meeting to celebrate the 65th birthday of Professor Tom Mullin
10th–11th September, 2015


Mr Matthew Arran
Avalanche statistics on erodible beds
Avalanches on erodible beds are important in a variety of industrial and environmental contexts, but are poorly understood, exhibiting a complex nonlinear rheology resulting in the existence of an intermittent flow regime. To investigate the statistics of avalanche events in this regime, experiments were conducted in a channel of length 2m, width 5cm, with an erodible bed of depth 10cm developed on a rough base. Two modes of behaviour were found, with episodes of quasi-periodic behaviour alternating with episodes of irregular avalanches. Characteristics are described for each mode, and for switching between them, and links are drawn to the theory of phase transitions.
Mr James Baker
Segregation-induced fingering instabilities in granular flows
The process of particle size-segregation can have important feedback effects on the bulk flow of granular avalanches. For example, frictional large particles may accumulate at the flow front, causing it to become unstable and split into a number of different finger-like structures. This self-channelization significantly enhances overall run-out distances and is therefore crucial to understand for hazard mitigation purposes. This work presents a fully coupled depth-averaged model for such segregation-mobility feedback effects, which is regularised by incorporating further details of the specific granular rheology.
Mr Thomas Barker
Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow
The stability of the continuum equations of granular flow are studied. It is found that in the high-wavenumber limit, the growth rate of perturbations increases unboundedly as their wavelength is reduced. This ill-posedness precludes transient numerical simulations and suggests important physics is missing from the current model.
Dr Richard Bowman
Deformable Plastic Structures
After a conversation with Tom at a conference, we used a 3D printer to create auxetic cellular materials, with potential applications in shock absorption. We have since gone on to exploit the deformability of 3D printed structures to produce a variety of useful mechanisms, including a translation stage. This 3-axis translation stage achieves micron precision over a range of 8×8×4mm, and is stable over weeks. Combined with a Raspberry Pi camera, we have developed this into a miniature microscope suitable for use in incubators, fume hoods, and myriad other places where small size and low cost are an advantage.
Dr Finn Box
The leaky liquid blister
The formation and propagation of fluid-driven fractures is a common occurrence in a wide variety of systems, from the development of magma-filled laccoliths to the delamination of surfaces bonded by adhesion. We have developed an experiment which contains many of the underlying physical processes of a hydraulic fracture in porous rock. The experiment comprises a hydraulic injection underneath an elastic sheet supported by a rigid, permeable surface. On injection, the fluid spreads outwards radially, delaminating the elastic sheet from the surface and leaking through the pore spaces in the supporting mesh. Steady states are attained for which the input flux balances the fluid that drains away. A combination of experimental, numerical and mathematical modelling of the elemental system has enabled characterisation of the dimensions of the fracture and dynamics of fracture propagation. A discussion is presented of the role of the air gap which exists at the fracture tip, in between the leading fracture front and the fluid front.
Prof. Alan Champneys
The impacting dynamics of pressure relief valves
This is the story of an industrial collaboration with Pentair Valves and Controls from Houston and Budapest University of Technology and Economics which came about by chance. After my Hungarian collaborator Csaba Hos discovered some unexplained oscillations in a hydraulic circuit during his PhD, we wrote a paper together on a simple 3-variable autonomous model for an impacting relief valve. The preprint was noticed by the company who flew to Europe to talk to us. After several years, presentations to the American Petrolium Institute and several sector-specific journal papers, we have now built a complete theory of how acoustic waves in inlet pipes couple with impacting dynamics to produce a new instability mechanism that seems to have secretly plaguing industry for years. This poster tells the story.
Dr Lucie Ducloué
Multiple bubble propagation modes in elasto-rigid models of airway reopening
Understanding the mechanics of airway reopening is essential to design ventilation strategies that quickly reopen the airway while minimizing lung damage. During the propagation of a reopening air finger, the airway changes shape in response to fluid motion inside it as well as external forces (bronchoconstriction). To understand the reopening of a single model airway, previous experiments [1] have investigated the coupling of a tube deformation to a two-phase flow inside it, by studying air injection in a fluid-filled, initially collapsed elastic tube. The results have evidenced the existence of a variety of air propagation modes as the tube reopens, including bubbles that propagate at high speed and relatively low pressure in tubes that are initially highly collapsed. To shed light on the selection of the air finger shape, we study the reopening of a rigid channel topped with an elastic membrane, which decouples the channel geometry from its wall elasticity. This simplified set-up allows us to identify two kinds of reopening modes, dominated by either viscous or elastic forces. We discuss the reopening modes observed in the tube within that framework. We also show that this set-up is of great interest for the study of viscous fingering under an elastic membrane.

[1] A. Heap and A. Juel (2008) Anomalous bubble propagation in elastic tubes. Phys. Fluid 20, 081702.
Dr Andrew Edwards
Size segregation in a granular bore
We study a system of bidisperse granular matter with an initial inflow that is completely segregated and normally graded. Particles flow continuously down a smooth chute inclined at an angle of 24 degrees until they reach the closed end. The particles impact with the wall at the end of the chute, which leads to the formation of an upslope propagating granular bore, defined as a normal shock in the flow. The bore propagates with constant speed until coming to rest when all of the material has been released from the hoppers. Images are captured with a high speed camera during the experiment to measure the inflow thickness, the bore thickness and its upslope propagation speed. The deposit is studied to investigate the segregation process in the bore, by inferring the particle concentration profiles from still images. We find a purely small particle layer on the bottom with a nearly linear transition up to a constant concentration of small and large particles on top. This profile is then used as an assumed concentration in the segregation theory, which extends the existing model.
Mr Parmesh Gajjar
Underlying Asymmetry with Particle-Size Segregation
The natural tendency for granular materials to segregate provides serious challenges to both industrial and geophysical environments, where the separation of different sized constituents can degrade the product quality, or increase the flow mobility and run-out. Yet, despite its importance, understanding individual particle motion during segregation has been problematic with conventional techniques such as binning and sidewall observation. Recent experiments have utilised the Refractive Index Matched Scanning (RIMS) technique to study particle scale segregation dynamics during oscillatory shear. Analysis of motion within the interior of the flow revealed an underlying asymmetry between large and small particles that is dependant on the local particle concentration. Small particles were seen to segregate faster through regions of many large particles, whilst large particles rise slower through regions of many small particles. The asymmetry is quantified on both bulk and particle length scales, and it is shown how the behaviour may be captured in a continuum model using a cubic segregation flux.
Mr Michael Gomez
The role of ghosts in elastic ‘snap-through’
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 dynamic process by which snap-through occurs remains much less well understood. Several examples have been studied that show slower dynamics than would be expected based on purely elastic time scales 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 a similar ‘anomalous dynamics’; however, we show that here this dynamics is a consequence of the ‘ghost’ of the snap-through bifurcation.
Mr Edgar Häner
Deformation of ovalbuminalginate capsules in a T-Junction
We study experimentally the flow­ induced deformation of liquid­filled ovalbumin-alginate capsules in a T­junction. In applications, capsules/cells often negotiate branched networks with junctions thus experiencing large deformations. We investigate the constant volume­flux viscous flow of buoyancy­neutral thin­walled capsules close to the centreline of rectangular channels, by comparison to near­rigid gelled beads. The motion of the capsules in straight channels scales with the capillary number — the ratio of viscous to elastic forces. However, the effect of elastic deformation on the motion is sufficiently weak that a rigid sphere model predicts the velocity of capsules with diameters of up to 70% of that of the channel to within 5%. In the T-Junction, systematic selection of daughter channel (right­left) occurs outside a finite region around the channel centreline, by contrast with near­rigid gelled beads, where the actual centreline is the separator. We quantify the behaviour of capsules in terms of their longitudinal stretching (up to a factor of three without rupture). We show the large range of deformations encountered can be applied to the measurement of the elastic properties of capsules as well as to the geometric­ induced sorting and manipulation of capsules.
Prof. Jonathan Healey
On the origin of viscous instability of shear layers
The fact that viscosity, a dissipative effect, can create instability in an otherwise stable flow is one of the most surprising results in hydrodynamic stability theory. It was discovered in the 1920s using asymptotic methods that thin viscous layers adjacent to solid boundaries could be destabilizing, but a simple physical mechanism has proved elusive. Why, for example, does viscosity not destabilize plane Couette flow or circular pipe flow? Benjamin (1963) showed that the lightly damped inviscid mode is a Negative Energy Wave (NEW), i.e. energy must be removed from the flow in order to establish the wave. He argued that dissipation removes energy and so allows the wave to grow, hence the destabilizing role of viscosity. However, we show that the existence of a NEW is not sufficient to produce a viscous instability. We suggest that a better way to understand the instability is via the resonant interaction of a pair of lightly damped modes, and we present a short derivation that does not rely on matching, and produces the classical asymptotic dispersion relation in the form of a coupled oscillator system, in which resonance creates instability.
Mr Utkarsh Jain
Buckling in an elastic column with holes
Buckling of an Euler column under uniaxial compression is a classic example of buckling instability. We observe a number of new and surprising buckled states if a column contains a regular array of holes. Here we developed a model of the system and study bifurcations in columns of finite length and columns with periodic boundary conditions using in-house software OOMPH-LIB. For samples with periodic boundary conditions we validate a theoretical prediction that we have developed for the types of buckled states in the limit of 'very-thin' side walls. We also show that for a finite column, only two buckled states are preferred at the onset of instability. Furthermore we study secondary bifurcations in the system. This is a subject of high interest as predicted behaviour is robust, scale independent and observable in experiments.
Mr Pallav Kant
Asymmetric Spreading Dynamics during Electrowetting on Conductors
Electrowetting (EW) offers a simple approach to control the wetting characteristics by applying potential difference across the liquid droplet and solid substrate. It has been addressed in many experimental and numerical studies motivated by the potential wide range applications in the industry and interesting fundamental science involved. Berge (1993) successfully demonstrated EW on a dielectric layer (EWOD) while avoiding the electrolytic decomposition of the droplet, observed in earlier attempts using conducting surfaces (Mugele & Baret (2007)). Here we revisit the earlier works on EW on conducting surfaces and demonstrate that it can be successfully achieved on special conducting surfaces with a large enough potential window to achieve very low contact angles. This was performed avoiding the electrolysis of the conducting droplet. Specifically, we focus on the wetting dynamics of the droplet using high speed imaging and try to develop an understanding about the asymmetric spreading dynamics observed in experiments. Experiments indicate that time scales associated with the advancing motion of the contact line depend on the potential difference applied across the droplet and conducting surface. From scaling arguments we show that there are two distinct stages of the advancing motion in which different forces are dominant. In the early stage of the contact line motion, inertial forces resist spreading and at the later stage bulk viscous dissipation governs the dynamics. Also, pure power law relationship for short time spreading proposed in earlier studies does not hold true in the current system. Interestingly, the time scales associated with the receding motion when applied potential difference is reduced to its base state are much faster than of advancing motion. This kind of asymmetric spreading dynamics has not been reported earlier.

[1] Berge, B. Electrocapillarity and wetting of insulator films by water. C R Acad. Sci. Ser. II 317, 157–163 (1993)
[2] F. Mugele and J.-C. Baret, Electrowetting: from basics to applications. J. Phys.: Condens. Matter 19, 375112 (2007)
Dr Laura Kimpton
Homogenisation for fibre-reinforced hydrogels
Articular cartilage can become damaged either through diseases, such as osteoarthritis, or through trauma. In adults, cartilage has poor reparative capacity and has become a popular target for tissue engineering therapies. A key challenge is to develop tissue engineering scaffolds that promote cell viability whilst also being sufficiently stiff to withstand the mechanical loading experienced in a joint. Our experimental collaborators have developed a new way to reinforce hydrogels with fibres produced by melt electrospinning writing (Visser et al. Nat. Commun. 2015). They have demonstrated that the inclusion of a very small amount of fibre material (7% by volume) can have a large synergistic reinforcing effect. Mathematical modelling is required to test hypotheses for the reinforcement mechanism and to understand how the fibre architecture affects the strength of the composite. Ultimately, the goal of this modelling is to design fibre arrangements that result in cartilage-like material properties. We will present a series of increasingly detailed mathematical models of the unconfined compression test of these constructs. We first present a simple model that idealises the fibres as elastic strings and the hydrogel as an incompressible elastic material. Geometrical considerations of the uniaxial compression of the construct then allow us to calculate the extension in the fibres and the associated force. This simple model provides support for the hypothesis that in uniaxial compression, the lateral spreading of the hydrogel puts the fibres under tension. Next we present a more detailed model that exploits the periodic fibre arrangement. We treat the fibres as an elastic material and the hydrogel as a poroelastic material and use techniques from homogenisation theory to develop effective governing equations for the anisotropic composite. This approach will allow us to understand the effect of different fibre sizes and arrangements.
Dr Celia Lozano
Segregation driven by particle interactions
We report the results of an experimental study of particle-particle interaction in an horizontally shaken granular layer which undergoes a second order phase transition from a binary gas to a segregation liquid as the packing fraction C is increased. By focusing on the behavior of individual particles, the effect of C is studied on 1) the process of cluster formation, 2) cluster dynamics, and 3) cluster destruction. The outcomes indicate that the segregation is driven by two mechanisms: attraction between same sized particles and random motion with a characteristic length which is inversely proportional to C. All clusters investigated are found to be transient and the probability distribution functions of the separation times have a power law decay, indicating that the splitting probability decreases with time.
Dr Nick McCullen
Experiments on Nonlinear Oscillator Networks
Oscillators are found throughout nature and can be used as models for many physical systems to explain interesting phenomena. Networks of coupled oscillators have been used to describe the synchronisation of fireflies and model many other biological systems including insect hearing and neuronal assemblies in the brain. They are also used to analyse and the oscillating generators in power networks. However, despite much theoretical work, experimental results on such systems remain scarce. Electronic nonlinear oscillators are perfectly suited to such experiments, being easily controlled, as well as modelled by well defined differential equations. They can quickly provide high resolution data, while at the same time having small real imperfections and noise, as they are real physical systems. This poster describes a system of coupled nonlinear oscillators used for such well controlled experiments, which is used to validate theoretical results and reveal new interesting phenomena.
Dr Carlos del Pino
Wingtip vortex in a NACA0012 airfoil and its active control
We conduct experiments in a towing-tank to analyse the flow patterns of wingtip vortices in a NACA 0012 airfoil. In this experimental research, we provide PIV measurements and flow visualisations. Without active control, several parameters are given experimentally as function of the Reynolds number, so we compare these data with the theoretical models of Batchelor, and Moore and Saffman together with DNS. Secondly, we analyse the effect of a continuous injection in the spanwise direction. The continuous jet has a strong influence on the wing-tip vortex formation. We explore this effect at low chord based Reynolds number ranging from 7000 up to 20000. We change the aspect ratio of the injection, R, defined as the ratio of the velocities between the jet (Uj) and free-stream (U). For R=1, we find that the jet strongly affects the wingtip vortex formation with a sudden decrement of the axial vorticity and the azimuthal velocity. This technique is a challenge and a promising tool to reduce the intensity of the vortex core. *Carlos del Pino would like to thank Tom Mullin his wisdom, knowledge and sageness during the short stays in Machester (2002, 2003, 2004, and 2007).
Ms Elze Porte
Experimental investigation into sound emissions from granular flows in a vertical tube.
When large storage silo’s containing granular material are discharged, a loud sound emits from the silo. The noise causes disturbances for people working on site and for nearby residential areas. Insufficient knowledge exists to solve the problem efficiently and adequately. An experimental study using a scaled silo setup shows that the particle flow dynamics and system characteristics are both actors in determining the occurrence of the sound and its frequency. The extensive use of frequency analysis provides new insights into the complexity of the related parameters. The particle flow and tube characteristics are manipulated by changing the outflow rate, bulk material, wall material, wall pressure and tube dimensions. Frequency analysis of the recorded sound shows that the frequency depends on both the externally forced parameter changes and internal changes during flow. The latter indicates that during the flow, characteristic properties such as the packing fraction and sound speed change. As a result, the frequency changes as well. However, the external parameters that are manipulated as an initial condition are equally important in describing the frequency response.
Prof. Alastair Rucklidge
Quasipatterns in two and three dimensions
Quasipatterns (patterns with no translation symmetry but with rotation symmetry on average) can occur in a variety of systems, including Faraday waves, nonlinear optics, polymer micelles and coupled reaction-diffusion systems. This poster will explore how having two interacting wavelengths in the physical system can stabilise quasipatterns in two and three dimensions, or can lead to spatio-temporal chaos.
Dr Anne Skeldon
Equation free analysis of agent-based models
Agent based models (ABM)s are increasingly in social science, economics, mathematics, biology and computer science where a closed form description is difficult. They consist of a set of individual agents and a specification for how each agent moves and interacts with other agents. In spite of their increasing ubiquity, few tools are currently available for the systematic analysis of ABM behaviour. Numerical continuation and bifurcation analysis is a well-established tool for the study of deterministic systems and has been widely used to investigate the behaviour of nonlinear dynamical systems. There are two barriers to the application of standard numerical continuation techniques to ABMs: firstly, ABMs are in general stochastic; secondly, standard continuation methods require an analytic description of the model, which is unknown for ABMs in the vast majority of cases. Recently, equation-free (EF) methods have been developed to perform numerical continuation in systems where the dynamics are described at a microscopic scale and continuation of a macroscopic measure is considered. Essentially these work by replacing equations for macroscopic behaviour with an ensemble of microscopic simulations. To date, this formulation has been applied to a few specific examples. For each case, choices have to be made for algorithmic parameters. These parameters have been chosen largely by trial-and-error and large numbers of micro-simulations are often needed. Here we develop a generic framework for the EF continuation of ABMs which does not require any knowledge of the mathematics of programming implementation involved, enabling numerical continuation to be performed by any ABM user. Our computationally efficient implementation is coded in Java, compatible with the all major operating systems, and able to interface with the NetLogo programming language which is a popular tool for coding ABMs. Our generic framework includes a systematic method for determining algorithmic parameters essential for robust EF continuation. We demonstrate our method with application to several ABM models revealing parameter dependence, bifurcation and stability analysis of these complex systems that are not otherwise easily obtainable.
Dr Priya Subramanian
Tailoring thermoacoustic instabilities
Thermoacoustic systems consist of a source of unsteady heat release enclosed in an acoustic field. Instabilities arise in such thermoacoustic systems due to positive feedback between fluctuations in acoustic pressure and unsteady heat release rate. During such instabilities, thermoacoustic systems display a wide variety of asymptotic dynamical behaviour; from steady states to chaotic trajectories. The ability to control the nature and characteristics of thermoacoustic oscillations is a desirable capability, which is achieved by altering the bifurcation behaviour of the uncontrolled system - using washout aided feedback control. Suitable choice of feedback control can be employed to delay or advance the instability or to even change the nature of the primary instability from being sub-critical to supercritical Hopf bifurcation for a horizontal Rijke tube.
Dr Alice Thompson
Stabilising falling liquid films
Falling liquid films become unstable to propagating waves if the fluid layer is too thick, leading to two-dimensional waves and eventually three-dimensional chaos. Here we discuss the use of feedback controls to stabilise the system, based on observations of the interface height, and supplied to the system by blowing and suction. We show that these controls can be used to stabilise a uniform film state, and also drive the system into non-uniform steady and travelling waves. We show that the stabilising effect persists through three different nonlinear models for the system, and discuss how the controls can be constructed based on only a few observations of the system state.
Dr Sylvain Viroulet
Particles size segregation and roll waves in dense granular flows
Geophysical granular flows, such as landslides, snow avalanches and pyroclastic flows commonly involve particles with different sizes which are prone to segregate during the flow. This particle-size segregation may lead to the formation of regions with different frictional properties which can have a feedback on the flow. This study aims to understand this effect in the context of bi-disperse roll waves in shallow granular free-surface flows. Experiments have been performed in a 3 meter long chute using several mixtures of spherical glass beads of diameter 75-150 and 400-600 microns flowing on a rough bed. These show that the waves propagate at constant speed that depends on the initial mixture composition. In addition, during their propagation, a higher concentration of large particles is localized at the front of the waves. A theoretical and numerical approach is presented using depth-averaged equations for the conservation of mass, momentum and depth-averaged small particle concentration. Results without frictional feedback are investigated and compared to those that include the enhanced frictional resistance to motion of the large grains.
Dr Nathalie Vriend
Avalanches in a V-shape
In this work, we create avalanches in a V-shaped channel at different apertures and flowrates. For deep flows (high apertures and flowrates), roll waves are triggered that have surprising features due to geometry. For shallower flows (low to medium apertures), the effects of roll waves are reduced and/or eliminated and the base flow appears. This background base flow is characterized by a curved free surface and recirculation cells whose structure and position is a strong function of the flowrate. An alternative rheology is proposed which accounts for this background base flow in terms of second-order stress differences.
Dr Feng Xu
Drop spreading with random viscosity
Airway mucus acts as a barrier that protects the lung from infection. However as a biological material, its physical properties are known imperfectly and can be spatially heterogeneous. In this study we assess the impact of these uncertainties on the rate of spreading of a drop (representing an inhaled aerosol) over a mucus film. We model the film as Newtonian, having a viscosity that depends linearly on the concentration of a passive solute (a crude proxy for mucin proteins). Given an initial random solute (and hence viscosity) distribution, we seek to quantify the variability in outcomes as the drop spreads. We use lubrication theory to describe the spreading of the drop in two dimensions, assuming a thin precursor film ahead of the drop. We consider a regime in which diffusion is sufficient to suppress solute concentration gradients across but not along the film and derive a system of coupled nonlinear PDEs governing the evolution of film height and the vertically-averaged solute concentration. The initial solute distribution is described as a Gaussian random field with a given correlation structure. Exploiting the small ratio of the precursor film thickness to drop height, we derive and validate a system of ODEs describing the coupled motion of the drop's effective contact lines. In this limit, the initial solute concentration at each contact line has a long-lived influence on the subsequent dynamics, whereas heterogeneity in the film ahead of the drop has surprisingly limited influence. To examine the uncertainties introduced by the initial random solute field, we perform Monte Carlo simulations to predict the variability in the drop centre location and width. We show how simulation results are well described (at much lower computational cost) using a weak disorder expansion, involving solution of a small set of ODEs. Our results show for example how variability in the drop width increases (and in the drop location decreases) as the solute correlation length increases.