We consider the drawing of a hollow Newtonian fibre with temperature-dependent viscosity. The drawing is affected by surface tension, inertia, hole pressurisation and externally applied cooling. We apply long-wavelength techniques to determine the steady states and examine their stability. In the presence of surface tension but with no cooling or internal hole pressure, we show the counter-intuitive result that the hole radius at the outlet of the device is a non-monotonic function of the hole radius at the inlet. We also show that if the internal hole is pressurised and the hole size at the inlet is sufficiently large, then the exit temperature of the fibre is a non-monotonic function of the applied cooling rate. We have found a number of surprising mechanisms related to how the various physical effects influence the stability of drawing. For the isothermal case, we show that increasing the internal hole pressure has a destabilising effect for non-zero surface tension while the stability is completely independent of the internal hole pressure for zero surface tension. We further show that there is a complicated interplay between internal hole pressure, external cooling and surface tension in determining the stability and that it is possible that increasing the hole size at the inlet can act to destabilise, then stabilise and finally destabilise the flow. We discuss the mechanisms that determine the counter-intuitive steady-state behaviour and stability.

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We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the $n$th power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as $n$ is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects.

Small finite-size particles suspended in fluid flow through an enclosed curved duct can focus to points or periodic orbits in the two-dimensional duct cross-section. This particle focusing is due to a balance between inertial lift forces arising from axial flow and drag forces arising from cross-sectional vortices. The inertial particle focusing phenomenon has been exploited in various industrial and medical applications to passively separate particles by size using purely hydrodynamic effects. A fixed size particle in a circular duct with a uniform rectangular cross-section can have a variety of particle attractors, such as stable nodes/spirals or limit cycles, depending on the radius of curvature of the duct. Bifurcations occur at different radii of curvature, such as pitchfork, saddle-node and saddle-node infinite period (SNIPER), which result in variations in the location, number and nature of these particle attractors. By using a quasi-steady approximation, we extend the theoretical model of Harding et al. (J. Fluid Mech., vol. 875, 2019, pp. 1–43) developed for the particle dynamics in circular ducts to spiral duct geometries with slowly varying curvature, and numerically explore the particle dynamics within. Bifurcations of particle attractors with respect to radius of curvature can be traversed within spiral ducts and give rise to a rich nonlinear particle dynamics and various types of tipping phenomena, such as bifurcation-induced tipping (B-tipping), rate-induced tipping (R-tipping) and a combination of both, which we explore in detail. We discuss implications of these unsteady dynamical behaviours for particle separation and propose novel mechanisms to separate particles by size in a non-equilibrium manner.

Six patents were secured by E. H. Lanier from 1930 to 1933 for aeroplane designs that were intended to be exceptionally stable. A feature of five of these was a flow-induced “vacuum chamber” which was thought to provide superior stability and increased lift compared to typical wing designs. Initially, this chamber was in the fuselage, but later designs placed it in the wing by replacing a section of the upper skin of the wing with a series of angled slats. We report upon an investigation of the Lanier wing design using inviscid aerodynamic theory and viscous numerical simulations. This took place at the 2005 Australia–New Zealand Mathematics-in-Industry Study Group. The evidence from this investigation does not support the claims but, rather, suggests that any improvement in lift and/or stability seen in the few prototypes that were built was, most probably, due to thicker airfoils than were typical at the time.

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We consider the stability of the drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape that evolves due to axial drawing, inertia and surface tension effects. Despite the complicated geometry of the boundaries, we use asymptotic techniques to determine a particularly convenient formulation of the equations of motion that is well-suited to stability calculations. We will determine an explicit asymptotic solution for steady states with (a) large surface tension and negligible inertia, and (b) large inertia. In both cases, we will show that complicated boundary layer structures can occur. We will use linear stability analysis to show that the presence of an axisymmetric hole destabilises the flow for finite capillary number and which answers a question raised in the literature. However, our formulation allows us to go much further and consider arbitrary hole structures or non-axisymmetric shapes, and show that any structure with holes will be less stable than the case of a solid axisymmetric thread. For a solid axisymmetric thread, we will also determine a closed-form expression that delineates the unconditional instability boundary in which case the thread is unstable for all draw ratios. We will determine how the detailed effects of the microstructure affect the stability, and show that they manifest themselves only via a single function that occurs in the stability problem and hence have a surprisingly limited effect on the stability.

We examine the effect of Dean number on the inertial focusing of spherical particles suspended in flow through curved microfluidic ducts. Previous modelling of particle migration in curved ducts assumed the flow rate was small enough that a leading-order approximation of the background flow with respect to the Dean number produces a reasonable model. Herein, we extend our model to situations involving a moderate Dean number (in the microfluidics context) while the particle Reynolds number remains small. Variations in the Dean number cause a change in the axial velocity profile of the background flow which influences the inertial lift force on a particle. Simultaneously, changes in the cross-sectional velocity components of the background flow directly affect the secondary flow induced drag. In keeping the particle Reynolds number small, we continue to approximate the inertial lift force using a regular perturbation while capturing the subtle effects from the modified background flow. This approach pushes the limits at which a regular perturbation is applicable to provide some insights into how variations in the Dean number influence particle focusing. Our results illustrate that, as the extrema in the background flow move towards the outside of edge of the cross-section with increasing Dean number, we observe a similar shift in the stable equilibria of some, but not all, particle sizes. This might be exploited to enhance the lateral separation of particles by size in a number of practical scenarios.

When a party or candidate loses the popular vote but still wins the election, do voters view the winner as legitimate? This scenario, known as an electoral inversion, describes the winners of two of the last six presidential elections in the United States. We report results from two experiments testing the effect of inversions on democratic legitimacy in the US context. Our results indicate that inversions significantly decrease the perceived legitimacy of winning candidates. Strikingly, this effect does not vary with the margin by which the winner loses the popular vote, nor by whether the candidate benefiting from the inversion is a co-partisan. The effect is driven by Democrats, who punish inversions regardless of candidate partisanship; few effects are observed among Republicans. These results suggest that the experience of inversions increases sensitivity to such outcomes among supporters of the losing party.

The collapse under surface tension of a long axisymmetric capillary, held at both ends and softened by a travelling heater, is used to determine the viscosity or surface tension of silica glasses. Capillary collapse is also used in the manufacture of some optical fibre preforms. Typically, a one-dimensional (1-D) model of the closure of a concentric fluid annulus is used to relate a measure of the change in the cross-sectional geometry, for example the external radius, to the desired information. We here show that a two-dimensional (2-D) asymptotic model developed for drawing of optical fibres, but with a unit draw ratio, may be used and yields analytic formulae involving a single dimensionless parameter, the scaled heater speed $V$, equivalently a capillary number. For a capillary fixed at both ends, this 2-D model agrees with the 1-D model and offers the significant benefit that it enables determination of both the surface tension and viscosity from a single capillary-collapse experiment, provided the pulling tension in the capillary during collapse is measured. The 2-D model also enables our investigation of the situation where both ends of the capillary are not fixed, so that the capillary cannot sustain a pulling tension. Then the collapse of the capillary is markedly different from that predicted by the 1-D model and the ability to determine both surface tension and viscosity is lost.

Ice shelves restrain flow from the Greenland and Antarctic ice sheets. Climate-ocean warming could force thinning or collapse of floating ice shelves and subsequently accelerate flow, increase ice discharge and raise global mean sea levels. Petermann Glacier (PG), northwest Greenland, recently lost large sections of its ice shelf, but its response to total ice shelf loss in the future remains uncertain. Here, we use the ice flow model Úa to assess the sensitivity of PG to changes in ice shelf extent, and to estimate the resultant loss of grounded ice and contribution to sea level rise. Our results have shown that under several scenarios of ice shelf thinning and retreat, removal of the shelf will not contribute substantially to global mean sea level (<1 mm). We hypothesize that grounded ice loss was limited by the stabilization of the grounding line at a topographic high ~12 km inland of its current grounding line position. Further inland, the likelihood of a narrow fjord that slopes seawards suggests that PG is likely to remain insensitive to terminus changes in the near future.

We examine the effect of gravity and (rotational) inertia on the inertial focusing of spherical non-neutrally buoyant particles suspended in flow through curved microfluidic ducts. In the neutrally buoyant case, examined in Harding et al. (J. Fluid Mech., vol. 875, 2019, pp. 1–43), the gravitational contribution to the force on the particle is exactly zero and the net effect of centrifugal and centripetal forces (due to the motion around the curved duct) is negligible. Inertial lift force and drag from the secondary fluid flow vortices interact and lead to focusing behaviour which is sensitive to the bend radius of the device and the particle size (each measured relative to the height of the cross-section). In the case of non-neutrally buoyant particles the behaviour becomes more complex with the two additional perturbing forces. The gravitational force, relative to the inertial lift force, scales with the inverse square of the flow velocity, making it a potentially important factor for devices operating at low flow rates with a suspension of non-neutrally buoyant particles. In contrast, the net centripetal/centrifugal force scales with the inverse of the bend radius, similar to the drag force from the secondary flow. We examine how these forces perturb the stable equilibria within the cross-sectional plane to which neutrally buoyant particles ultimately migrate.

Small mountain glaciers are an important part of the cryosphere and tend to respond rapidly to climate warming. Historically, mapping very small glaciers (generally considered to be <0.5 km2) using satellite imagery has often been subjective due to the difficulty in differentiating them from perennial snowpatches. For this reason, most scientists implement minimum size-thresholds (typically 0.01–0.05 km2). Here, we compare the ability of different remote-sensing approaches to identify and map very small glaciers on imagery of varying spatial resolutions (30–0.25 m) and investigate how operator subjectivity influences the results. Based on this analysis, we support the use of a minimum size-threshold of 0.01 km2 for imagery with coarse to medium spatial resolution (30–10 m). However, when mapping on high-resolution imagery (<1 m) with minimal seasonal snow cover, glaciers <0.05 km2 and even <0.01 km2 are readily identifiable and using a minimum threshold may be inappropriate. For these cases, we develop a set of criteria to enable the identification of very small glaciers and classify them as certain, probable or possible. This should facilitate a more consistent approach to identifying and mapping very small glaciers on high-resolution imagery, helping to produce more comprehensive and accurate glacier inventories.

We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrate. The Navier–Stokes equations determine the hydrodynamic forces acting on a particle. A leading-order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales: specifically, the particle radius, duct height and duct bend radius. Additionally, a trapezoidal-shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles.

We consider the role of heating and cooling in the steady drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape. The internal holes and the external boundary evolve as a result of the axial drawing and surface-tension effects. The heating and cooling affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We use asymptotic techniques to show that, under a suitable transformation, this complicated three-dimensional free boundary problem can be formulated in such a way that the transverse aspect of the flow can be reduced to the solution of a standard Stokes flow problem in the absence of axial stretching. The solution of this standard problem can then be substituted into a system of three ordinary differential equations that completely determine the flow. We use this approach to develop a very simple numerical method that can determine the way that thermal effects impact on the drawing of threads by devices that either specify the fibre tension or the draw ratio. We also develop a numerical method to solve the inverse problem of determining the initial cross-sectional geometry, draw tension and, importantly, heater temperature to obtain a desired cross-sectional shape and change in cross-sectional area at the device exit. This precisely allows one to determine the pattern of air holes in the preform that will achieve the desired hole pattern in the stretched fibre.

Is American democracy under threat? The question is more prominent in political debate now than at any time in recent memory. However, it is also too blunt; there is widespread recognition that democracy is multifaceted and that backsliding, when it occurs, tends to be piecemeal. To address these concerns, we provide original data from surveys of political science experts and the public measuring the perceived importance and performance of U.S. democracy on a number of dimensions during the first year-and-a-half of the Trump presidency. We draw on a theory of how politicians may transgress limits on their authority and the conditions under which constraints are self-enforcing. We connect this theory to our survey data in an effort to identify potential areas of agreement—bright lines—among experts and the public about the most important democratic principles and whether they have been violated. Public and expert perceptions often differ on the importance of specific democratic principles. In addition, though our experts perceive substantial democratic erosion, particularly in areas related to checks and balances, polarization between Trump supporters and opponents undermines any social consensus recognizing these violations.