arXiv subjects

Andrew L. Hazel

Publications and source records attributed to Andrew L. Hazel.

13 recordsLinked to original sources

arXiv2025

Dynamics of evaporating, interconnected droplets

We report on the dynamics of a pair of sessile droplets that are connected by a microchannel, yet open to the atmosphere and hence free to evaporate. Our results reveal that fluid exchange between droplets occurs via a pumping flow driven by differences in hydrostatic and Laplace pressure between the two droplets. Evaporation causes the droplets to slowly lose volume and change shape, which subsequently affects the fluid transport between them. We observe that, for equal contact areas, a larger droplet typically feeds a smaller droplet during evaporation and the flow in the connecting channel is unidirectional. However, for unequal contact areas, the flow can reverse in the connecting channel following a sudden switch in dropletshape that occurs during evaporation. A stability,analysis reveals that the dynamics of the exchange flow are underpinned by a supercritical pitchfork bifurcation. Evaporative volume loss permits the droplet pair to step through a sequence of quasi-stationary states determined by the instantaneous volume of the system. Enforcing unequal contact area unfolds the bifurcation such that droplet-shape switching and the associated flow reversal can be understood in terms of a jump from the disconnected to the connected branch of the bifurcation. This establishes symmetry breaking as a mechanism to induce evaporation-driven flow reversal in connected droplets.

physics.flu-dyn↗

arXiv2025

On the oscillatory dynamics of a Saffman--Taylor finger with a bubble at its tip

The complex behaviour of air-liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet, both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder \textit{et. al.} (\textit{Phys. Rev. A}, vol. 34, 1986, p. 5175) to present a systematic investigation of the dynamics that result when a small air bubble is placed at the tip of a steadily propagating air finger in a Hele-Shaw channel. The system can exhibit steady and oscillatory behaviour, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger's tip. We demonstrate that at a fixed flow rate, the oscillations commence when the bubble size is increased sufficiently so that the decreased in-plane curvature of the bubble tip matches the in-plane curvature of the finger tip. The equality between the two in-plane curvatures causes the axial pressure gradient across the bubble, which drives the finger, to vanish, thus rendering the finger susceptible to lateral perturbations. Differing timescales for finger and bubble restoral under perturbation allow sustained oscillations to develop in the finger-bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a compound finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie the observed disordered dynamics in viscous fingering.

physics.flu-dyn↗

arXiv2024

Periodic dynamics in viscous fingering

The displacement of a viscous liquid by air in the narrow gap between two parallel plates - a Hele-Shaw channel - is an exemplar of complex pattern formation. Typically, bubbles or fingers of air propagate steadily at low values of the driving parameter. However, as the driving parameter increases, they can exhibit disordered pattern-forming dynamics. In this paper, we demonstrate experimentally that a remote perturbation of the bubble's tip can drive time-periodic bubble propagation: a fundamental building block of complex unsteady dynamics. We exploit the propensity of a group of bubbles to self-organise into a fixed spatial arrangement in a Hele-Shaw channel with a centralised depth-reduction in order to apply a sustained perturbation to a bubble's shape as it propagates. We find that the bubble with a perturbed shape begins to oscillate after the system undergoes a supercritical Hopf bifurcation upon variation of the tip perturbation and dimensionless flow rate. The oscillation cycle features the splitting of the bubble's tip and advection of the resulting finger-like protrusion along the bubble's length until it is absorbed by the bubble's advancing rear. The restoral of the bubble's tip follows naturally because the system is driven by a fixed flow rate and the perturbed bubble is attracted to the weakly unstable, steadily propagating state that is set by the ratio of imposed viscous and capillary forces. Our results suggest a generic mechanism for time-periodic dynamics of propagating curved fronts subject to a steady shape perturbation.

physics.flu-dyn↗

arXiv2023

The interplay between bulk flow and boundary conditions on the distribution of micro-swimmers in channel flow

While previous experimental and numerical studies of dilute micro-swimmer suspensions have focused on the behaviours of swimmers in the bulk flow and near boundaries, models typically do not account for the interplay between bulk flow and the choice of boundary conditions imposed in continuum models. In our work, we highlight the effect of boundary conditions on the bulk flow distributions, such as through the development of boundary layers or secondary peaks of cell accumulation in bulk-flow swimmer dynamics. For the case of a dilute swimmer suspension in Poiseuille flow, we compare the distribution (in physical and orientation space) obtained from individual based stochastic models with those from continuum models, and identify mathematically sensible continuum boundary conditions for different physical scenarios (i.e. specular reflection, uniform random reflection and absorbing boundaries). We identify that the spread of preferred cell orientations is dependent on the interplay between rotation driven by the shear flow (Jeffery orbits) and rotational diffusion. We find that in the absence of hydrodynamic wall-interactions, swimmers preferentially approach the walls perpendicular to the surface in the presence of high rotational diffusion, and that the preferential approach of swimmers to the walls is shape-dependent at low rotational diffusion (when suspensions tend towards a fully deterministic case). In the latter case, the preferred orientations are nearly parallel to the surface for elongated swimmers and nearly perpendicular to the surface for near-spherical swimmers. Furthermore, we highlight the effects of swimmer geometries and shear throughout the bulk-flow on swimmer trajectories and show how the full history of bulk-flow dynamics affects the orientation distributions of micro-swimmer wall incidence.

physics.flu-dyn↗

arXiv2023

Stable bubble formations in a Hele-Shaw channel

Deformable bubbles propagated by the flow of a viscous liquid in a planar Hele-Shaw channel of uniform depth tend to travel steadily along the channel's streamwise axis and pairs of neighbouring bubbles will either separate or coalesce because an individual bubble's propagation speed increases monotonically with its size. Thus, any group of bubbles will eventually rearrange itself in order of decreasing size and all of the bubbles will separate. We show that, by introducing a small geometric perturbation to the channel in the form of an axially-uniform depth reduction along its centreline, the system supports a multitude of stable bubble formations and this can disrupt the usual reordering by bubble size. The constituent bubbles of a stable formation lie in alternation on opposite sides of the depth-perturbation, retain fixed shapes and propagate steadily at the same speed. A stable formation is always led by the smallest of its constituent bubbles, which would be the slowest bubble in isolation. The leading bubble propagates as if it were isolated and the trailing bubbles reduce their speeds by adjusting their shapes and overlaps of the depth-perturbation in the perturbation fields of their preceding nearest-neighbours in order to match that of the leading bubble. The trailing bubbles can be arranged in any order and, hence, the number of stable formations increases factorially as the number of bubbles is increased.

physics.flu-dyn↗

arXiv2022

Diffusion in multi-dimensional solids using Forman's combinatorial differential forms

The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar variables. The resulting description is intrinsic, different from the approach known as Discrete Exterior Calculus, because it does not assume the existence of smooth vector fields and forms extrinsic to the discrete complex. In addition, the proposed formulation provides a significant new modelling capability: physical processes may be set to operate differently on cells with different dimensions within a complex. An application of the new method to the heat/diffusion equation is presented to demonstrate how it captures the effect of changing properties of microstructural elements on the macroscopic behavior. The proposed method is applicable to a range of physical problems, including heat, mass and charge diffusion, and flow through porous media.

math-ph↗

arXiv2022

The effect of isolated ridges and grooves on static menisci in rectangular channels

We present theoretical and numerical results that demonstrate the sensitivity of the shape of a static meniscus in a rectangular channel to localised geometric perturbations in the form of narrow ridges and grooves imposed on the channel walls. The Young--Laplace equation is solved for a gas/liquid interface with fixed contact angle using computations, analytical arguments and semi-analytical solutions of a linearised model for small-amplitude perturbations. We find that the local deformation of the meniscus's contact line near a ridge or groove is strongly dependent on the shape of the perturbation. In particular, small-amplitude perturbations that change the channel volume induce a change in the pressure difference across the meniscus, resulting in long-range curvature of its contact line. We derive an explicit expression for this induced pressure difference directly in terms of the boundary data. We show how contact lines can be engineered to assume prescribed patterns using suitable combinations of ridges and grooves.

physics.flu-dyn↗

arXiv2021

A microstructural model of tendon failure

Collagen fibrils are the most important structural component of tendons. Their crimped structure and parallel arrangement within the tendon lead to a distinctive non-linear stress-strain curve when a tendon is stretched. Microstructural models can be used to relate microscale collagen fibril mechanics to macroscale tendon mechanics, allowing us to identify the mechanisms behind each feature present in the stress-strain curve. Most models in the literature focus on the elastic behaviour of the tendon, and there are few which model beyond the elastic limit without introducing phenomenological parameters. We develop a model, built upon a collagen recruitment approach, that only contains microstructural parameters. We split the stress in the fibrils into elastic and plastic parts, and assume that the fibril yield stretch and rupture stretch are each described by a distribution function, rather than being single-valued. By changing the shapes of the distributions and their regions of overlap, we can produce macroscale tendon stress-strain curves that generate the full range of features observed experimentally, including those that could not be explained using existing models. These features include second linear regions occurring after the tendon has yielded, and step-like failure behaviour present after the stress has peaked. When we compare with an existing model, we find that our model reduces the average root mean squared error from 4.15MPa to 1.61MPa, and the resulting parameter values are closer to those found experimentally. Since our model contains only parameters that have a direct physical interpretation, it can be used to predict how processes such as ageing, disease, and injury affect the mechanical behaviour of tendons, provided we can quantify the effects of these processes on the microstructure.

q-bio.TO↗

arXiv2020

The life and fate of a bubble in a geometrically-perturbed Hele-Shaw channel

Motivated by the desire to understand complex transient behaviour in fluid flows, we study the dynamics of an air bubble driven by the steady motion of a suspending viscous fluid within a Hele-Shaw channel with a centred depth perturbation. Using both experiments and numerical simulations of a depth-averaged model, we investigate the evolution of an initially centred bubble of prescribed volume as a function of flow rate and initial shape. The experiments exhibit a rich variety of organised transient dynamics, involving bubble break up as well as aggregation and coalescence of interacting neighbouring bubbles. The long-term outcome is either a single bubble or multiple separating bubbles, positioned along the channel in order of increasing velocity. Up to moderate flow rates, the life and fate of the bubble are reproducible and can be categorised by a small number of characteristic behaviours that occur in simply-connected regions of the parameter plane. Increasing the flow rate leads to less reproducible time evolutions with increasing sensitivity to initial conditions and perturbations in the channel. Time-dependent numerical simulations that allow for break up and coalescence are found to reproduce most of the dynamical behaviour observed experimentally including enhanced sensitivity at high flow rate. An unusual feature of this system is that the set of steady and periodic solutions can change during temporal evolution because both the number of bubbles and their size distribution evolve due to break up and coalescence events. Calculation of stable and unstable solutions in the single- and two-bubble cases reveals that the transient dynamics are orchestrated by weakly-unstable solutions of the system termed edge states that can appear and disappear as the number of bubbles changes.

physics.flu-dyn↗

arXiv2017

Bubble propagation in Hele-Shaw channels with centred constrictions

We study the propagation of finite bubbles in a Hele-Shaw channel, where a centred occlusion (termed a rail) is introduced to provide a small axially-uniform depth constriction. For bubbles wide enough to span the channel, the system's behaviour is similar to that of semi-infinite fingers and a symmetric static solution is stable. Here, we focus on smaller bubbles, in which case the symmetric static solution is unstable and the static bubble is displaced towards one of the deeper regions of the channel on either side of the rail. Using a combination of experiments and numerical simulations of a depth-averaged model, we show that a bubble propagating axially due to a small imposed flow rate can be stabilised in a steady symmetric mode centred on the rail through a subtle interaction between stabilising viscous forces and destabilising surface tension forces. However, for sufficiently large capillary numbers Ca, the ratio of viscous to surface tension forces, viscous forces in turn become destabilising thus returning the bubble to an off-centred propagation regime. With decreasing bubble size, the range of Ca for which steady centred propagation is stable decreases, and eventually vanishes through the coalescence of two supercritical pitchfork bifurcations. The depth-averaged model is found to accurately predict all the steady modes of propagation observed experimentally, and provides a comprehensive picture of the underlying steady bifurcation structure. However, for sufficiently large imposed flow rates, we find that initially centred bubbles do not converge onto a steady mode of propagation. Instead they transiently explore weakly unstable steady modes, an evolution which results in their break-up and eventual settling into a steady propagating state of changed topology.

physics.flu-dyn↗

arXiv2017

Stochastic Dynamics of Resistive Switching: Fluctuations Lead to Optimal Particle Number

Resistive switching is one of the foremost candidates for building novel types of non-volatile random access memories. Any practical implementation of such a memory cell calls for a strong miniaturization, at which point fluctuations start playing a role that cannot be neglected. A detailed understanding of switching mechanisms and reliability is essential. For this reason, we formulate a particle model based on the stochastic motion of oxygen vacancies. It allows us to investigate fluctuations in the resistance states of a switch with two active zones. The vacancies' dynamics is governed by a master equation. Upon the application of a voltage pulse, the vacancies travel collectively through the switch. By deriving a generalized Burgers equation we can interpret this collective motion as nonlinear traveling waves, and numerically verify this result. Further, we define binary logical states by means of the underlying vacancy distributions, and establish a framework of writing and reading such memory element with voltage pulses. Considerations about the dis- criminability of these operations under fluctuations together with the markedness of the resistive switching effect itself lead to the conclusion, that an intermediate vacancy number is optimal for performance.

cond-mat.mtrl-sci↗

arXiv2017

Controlling droplet spreading with topography

We present a novel experimental system that can be used to study the dynamics of picoliter droplet spreading over substrates with topographic variations. We concentrate on spreading of a droplet within a recessed stadium-shaped pixel, with applications to the manufacture of POLED displays, and find that the sloping side wall of the pixel can either locally enhance or hinder spreading depending on whether the topography gradient ahead of the contact line is positive or negative. Locally enhanced spreading occurs via the formation of thin pointed rivulets along the side walls of the pixel through a mechanism similar to capillary rise in sharp corners. We demonstrate that a thin-film model combined with an experimentally measured spreading law, which relates the speed of the contact line to the contact angle, provides excellent predictions of the evolving liquid morphologies. We also show that the spreading can be adequately described by a Cox-Voinov law for the majority of the evolution. The model does not include viscous effects and hence, the timescales for the propagation of the thin pointed rivulets are not captured. Nonetheless, this simple model can be used very effectively to predict the areas covered by the liquid and may serve as a useful design tool for systems that require precise control of liquid on substrates.

physics.flu-dyn↗

arXiv2016

Reopening modes of a collapsed elasto-rigid channel

Motivated by the reopening mechanics of strongly collapsed airways, we study the steady propagation of an air finger through a collapsed oil-filled channel with a single compliant wall. In a previous study using fully-compliant elastic tubes, a `pointed' air finger was found to propagate at high speed and low pressure, which may enable rapid reopening of highly collapsed airways with minimal tissue damage (Heap & Juel 2008). In this paper, we identify the selection mechanism of that pointed finger, which remained unexplained, by conducting an experimental study in a rigid rectangular Hele-Shaw channel with an elastic top boundary. The constitutive behaviour of this elasto-rigid channel is broadly similar to that of an elastic tube, but unlike the tube the channel's cross-section adopts self-similar shapes from the undeformed state to the point of first near wall contact. The simplification of the vessel geometry enables the systematic investigation of the reopening dynamics in terms of initial collapse. We find that for low levels of initial collapse, a single centred symmetric finger propagates in the channel and its shape is set by the tip curvature. As the level of collapse increases, the channel cross-section develops a central region of near opposite wall contact, and the finger shape evolves smoothly towards a `flat-tipped' finger whose geometry is set by the strong depth gradient near the channel walls. We show that the flat-tipped mode of reopening is analogous to the pointed finger observed in tubes. Its propagation is sustained by the vessel's extreme cross-sectional profile at high collapse, while vessel compliance stabilises it. A simple scaling argument based on the dissipated power reveals that this reopening mode is preferred at higher propagation speeds when it becomes favourable to displace the elastic channel wall rather than the viscous fluid.

physics.flu-dyn↗