Gravity-driven draining of a thin rivulet with constant width down a slowly varying substrate

The locally unidirectional gravity-driven draining of a thin rivulet with constant width but slowly varying contact angle down a slowly varying substrate is considered. Specifically, the flow of a rivulet in the azimuthal direction from the top to the bottom of a large horizontal cylinder is investigated. In particular, it is shown that, despite behaving the same locally, this flow has qualitatively different global behaviour from that of a rivulet with constant contact angle but slowly varying width. For example, whereas in the case of constant contact angle there is always a rivulet that runs all the way from the top to the bottom of the cylinder, in the case of constant width this is possible only for sufficiently narrow rivulets. Wider rivulets with constant width are possible only between the top of the cylinder and a critical azimuthal angle on the lower half of the cylinder. Assuming that the contact lines de-pin at this critical angle (where the contact angle is zero) the rivulet runs from the critical angle to the bottom of the cylinder with zero contact angle, monotonically decreasing width and monotonically increasing maximum thickness. The total mass of fluid on the cylinder is found to be a monotonically increasing function of the value of the constant width

Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder

The steady three-dimensional flow of a thin, slowly varying ring of Newtonian fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder is investigated. Specifically, we study “full-ring” solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all the way around the cylinder. In particular, it is found that there is a critical solution corresponding to either a critical load above which no full-ring solution exists (if the rotation speed is prescribed) or a critical rotation speed below which no full-ring solution exists (if the load is prescribed). We describe the behaviour of the critical solution and, in particular, show that the critical flux, the critical load, the critical semi-width and the critical ring profile are all increasing functions of the rotation speed. In the limit of small rotation speed, the critical flux is small and the critical ring is narrow and thin, leading to a small critical load. In the limit of large rotation speed, the critical flux is large and the critical ring is wide on the upper half of the cylinder and thick on the lower half of the cylinder, leading to a large critical load.\ud \ud We also describe the behaviour of the non-critical full-ring solution, and, in particular, show that the semi-width and the ring profile are increasing functions of the load but, in general, non-monotonic functions of the rotation speed. In the limit of large rotation speed, the ring approaches a limiting non-uniform shape, whereas in the limit of small load, the ring is narrow and thin with a uniform parabolic profile. Finally, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there is a region of parameter space close to the critical solution for sufficiently small rotation speed in which backflow occurs in a small region on the right-hand side of the cylinder

Thermoviscous Coating and Rimming Flow

A comprehensive description is obtained of steady thermoviscous (i.e. with temperature-dependent viscosity) coating and rimming flow on a uniformly rotating horizontal cylinder that is uniformly hotter or colder than the surrounding atmosphere. It is found that, as in the corresponding isothermal problem, there is a critical solution with a corresponding critical load (which depends, in general, on both the Biot number and the thermoviscosity number) above which no ``full-film'' solutions corresponding to a continuous film of fluid covering the entire outside or inside of the cylinder exist. The effect of thermoviscosity on both the critical solution and the full-film solution with a prescribed load is described. In particular, there are no full-film solutions with a prescribed load M for any value of the Biot number when M is greater than or equal to M_{c0} divided by the square root of f for positive thermoviscosity number and when M is greater than M_{c0} for negative thermoviscosity number, where f is a monotonically decreasing function of the thermoviscosity number and M_{c0} = 4.44272 is the critical load in the constant-viscosity case. It is also found that when the prescribed load M is less than 1.50315 there is a narrow region of the Biot number - thermoviscosity number parameter plane in which backflow occurs

Measuring Visual Complexity of Cluster-Based Visualizations

Handling visual complexity is a challenging problem in visualization owing to the subjectiveness of its definition and the difficulty in devising generalizable quantitative metrics. In this paper we address this challenge by measuring the visual complexity of two common forms of cluster-based visualizations: scatter plots and parallel coordinatess. We conceptualize visual complexity as a form of visual uncertainty, which is a measure of the degree of difficulty for humans to interpret a visual representation correctly. We propose an algorithm for estimating visual complexity for the aforementioned visualizations using Allen's interval algebra. We first establish a set of primitive 2-cluster cases in scatter plots and another set for parallel coordinatess based on symmetric isomorphism. We confirm that both are the minimal sets and verify the correctness of their members computationally. We score the uncertainty of each primitive case based on its topological properties, including the existence of overlapping regions, splitting regions and meeting points or edges. We compare a few optional scoring schemes against a set of subjective scores by humans, and identify the one that is the most consistent with the subjective scores. Finally, we extend the 2-cluster measure to k-cluster measure as a general purpose estimator of visual complexity for these two forms of cluster-based visualization

Defining extreme sport: conceptions and misconceptions

One feature in how sport is defined is the distinction between extreme and non-extreme. BASE jumping is an example of an ‘extreme sport’ because it involves a high degree of ‘risk’, whilst swimming is classified as ‘non-extreme’ because the risks are minimal. This broad definition falls short of identifying the extent of risk and ignores the psychological, social-demographic and life style variables associated with engagement in each sport

Personality differences amongst drag racers and archers: implications for sport injury rehabilitation

Personality trait of an athlete is a significant factor in sports injury rehabilitation. The aim of the present study is to investigate whether there are differences in personality traits between male and female, professional and amateur athletes from sports representing two ends of extreme to traditional namely; drag racing and archery. Overall 189 male and female, professional and amateur drag racers (n=144) and archers (n=45) took part in this study. Participants completed the personality traits of extroversion and neuroticism as measured by Eysenck’s classic Personality Inventory dimensions and thrill and adventure seeking (TAS), experience seeking (ES), disinhibition (DIS), boredom susceptibility (BS), and sensation seeking (SS) as measured by Zuckerman’s Sensation Seeking Scale. The results showed that professionals scored significantly lower on neuroticism compared to amateurs. Drag racers scored significantly higher on TAS, DIS, and SS compared to archers and there were gender differences amongst archers on TAS and SS with males scoring higher than females. Such differences in personality factors and the readiness to take risks, lack of caution, and adventurous spirit can influence the risk of injury in athletes and indeed may influence the outcome of rehabilitation. Practitioners would need to recognise difference in personality traits associated with the type of sport and the choice of interventions strategies

Porous squeeze-film flow

The squeeze-film flow of a thin layer of Newtonian fluid filling the gap between a flat impermeable surface moving under a prescribed constant load and a flat thin porous bed coating a stationary flat impermeable surface is considered. Unlike in the classical case of an impermeable bed, in which an infinite time is required for the two surfaces to touch, for a porous bed contact occurs in a finite contact time. Using a lubrication approximation an implicit expression for the fluid layer thickness and an explicit expression for the contact time are obtained and analysed. In addition, the fluid particle paths are calculated, and the penetration depths of fluid particles into the porous bed are determined. In particular, the behaviour in the asymptotic limit of small permeability, in which the contact time is large but finite, is investigated. Finally, the results are interpreted in the context of lubrication in the human knee joint, and some conclusions are drawn about the contact time of the cartilage-coated femoral condyles and tibial plateau and the penetration of nutrients into the cartilage

A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow

We use the lubrication approximation to analyse three closely related problems involving a thin rivulet or ridge (i.e. a two-dimensional droplet) of fluid subject to a prescribed uniform transverse shear stress at its free surface due to an external airflow, namely a rivulet draining under gravity down a vertical substrate, a rivulet driven by a longitudinal shear stress at its free surface, and a ridge on a horizontal substrate, and find qualitatively similar behaviour for all three problems. We show that, in agreement with previous numerical studies, the free surface profile of an equilibrium rivulet/ridge with pinned contact lines is skewed as the shear stress is increased from zero, and that there is a maximum value of the shear stress beyond which no solution with prescribed semi-width is possible. In practice, one or both of the contact lines will de-pin before this maximum value of the shear stress is reached, and so we consider situations in which the rivulet/ridge de-pins at one or both contact lines. In the case of de-pinning only at the advancing contact line, the rivulet/ridge is flattened and widened as the shear stress is increased from its critical value, and there is a second maximum value of the shear stress beyond which no solution with a prescribed advancing contact angle is possible. In contrast, in the case of de-pinning only at the receding contact line, the rivulet/ridge is thickened and narrowed as the shear stress is increased from its critical value, and there is a solution with a prescribed receding contact angle for all values of the shear stress. In general, in the case of de-pinning at both contact lines there is a critical “yield” value of the shear stress beyond which no equilibrium solution is possible and the rivulet/ridge will evolve unsteadily. In an Appendix we show that an equilibrium rivulet/ridge with prescribed flux/area is quasi-statically stable to two-dimensional perturbations

Squeeze-Film Flow in the Presence of a Thin Porous Bed, with Application to the Human Knee Joint

Motivated by the desire for a better understanding of the lubrication of the human knee joint, the squeeze-film flow of a thin layer of Newtonian fluid (representing the synovial fluid) filling the gap between a flat impermeable surface (representing the femoral condyles) and a flat thin porous bed (representing the articular cartilage) coating a stationary flat impermeable surface (representing the tibial plateau) is considered. As the impermeable surface approaches the porous bed under a prescribed constant load all of the fluid is squeezed out of the gap in a finite contact time. In the context of the knee, the size of this contact time suggests that when a person stands still for a short period of time their knees may be fluid lubricated, but that when they stand still for a longer period of time contact between the cartilage-coated surfaces may occur. The fluid particle paths are calculated, and the penetration depths of fluid particles into the porous bed are determined. In the context of the knee, these penetration depths provide a measure of how far into the cartilage nutrients are carried by the synovial fluid, and suggest that when a person stands still nutrients initially in the fluid layer penetrate only a relatively small distance into the cartilage. However, the model also suggests that the cumulative effect of repeated loading and unloading of the knees during physical activity such as walking or running may be sufficient to carry nutrients deep into the cartilage