Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularisation

We perform an analytic and numerical study of an inviscid contracting bubble in a two-dimensional Hele-Shaw cell, where the effects of both surface tension and kinetic undercooling on the moving bubble boundary are not neglected. In contrast to expanding bubbles, in which both boundary effects regularise the ill-posedness arising from the viscous (Saffman-Taylor) instability, we show that in contracting bubbles the two boundary effects are in competition, with surface tension stabilising the boundary, and kinetic undercooling destabilising it. This competition leads to interesting bifurcation behaviour in the asymptotic shape of the bubble in the limit it approaches extinction. In this limit, the boundary may tend to become either circular, or approach a line or "slit" of zero thickness, depending on the initial condition and the value of a nondimensional surface tension parameter. We show that over a critical range of surface tension values, both these asymptotic shapes are stable. In this regime there exists a third, unstable branch of limiting self-similar bubble shapes, with an asymptotic aspect ratio (dependent on the surface tension) between zero and one. We support our asymptotic analysis with a numerical scheme that utilises the applicability of complex variable theory to Hele-Shaw flow.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Mathematical models of bubble evolution in a Hele-Shaw Cell

This thesis concerns the mathematical model of moving fluid interfaces in a Hele-Shaw cell: an experimental device in which fluid flow is studied by sandwiching the fluid between two closely separated plates. Analytic and numerical methods are developed to gain new insights into interfacial stability and bubble evolution, and the influence of different boundary effects is examined. In particular, the properties of the velocity-dependent kinetic undercooling boundary condition are analysed, with regard to the selection of only discrete possible shapes of travelling fingers of fluid, the formation of corners on the interface, and the interaction of kinetic undercooling with the better known effect of surface tension. Explicit solutions to the problem of an expanding or contracting ring of fluid are also developed.

Bwembya Chikolwa quoted by Mike Taylor for Money Management 7th September 2009, Thaw for listed property

Australian property bond markets are starting to improve, but don’t expect a return to the buoyant days of the past any time soon.

Bubble geometry and chaotic pricing behaviour

This paper is a deductive theoretical enquiry into the flow of effects from the geometry of price bubbles/busts, to price indices, to pricing behaviours of sellers and buyers, and back to price bubbles/busts. The intent of the analysis is to suggest analytical approaches to identify the presence, maturity, and/or sustainability of a price bubble. We present a pricing model to emulate market behaviour, including numeric examples and charts of the interaction of supply and demand. The model extends into dynamic market solutions myopic (single- and multi-period) backward looking rational expectations to demonstrate how buyers and sellers interact to affect supply and demand and to show how capital gain expectations can be a destabilising influence – i.e. the lagged effects of past price gains can drive the market price away from long-run market-worth. Investing based on the outputs of past price-based valuation models appear to be more of a game-of-chance than a sound investment strategy.

The organic fraction of bubble-generated, accumulation mode Sea Spray Aerosol (SSA)

Recent studies have detected a dominant accumulation mode (~100 nm) in the Sea Spray Aerosol (SSA) number distribution. There is evidence to suggest that particles in this mode are composed primarily of organics. To investigate this hypothesis we conducted experiments on NaCl, artificial SSA and natural SSA particles with a Volatility-Hygroscopicity-Tandem-Differential-Mobility-Analyser (VH-TDMA). NaCl particles were atomiser generated and a bubble generator was constructed to produce artificial and natural SSA particles. Natural seawater samples for use in the bubble generator were collected from biologically active, terrestrially-affected coastal water in Moreton Bay, Australia. Differences in the VH-TDMA-measured volatility curves of artificial and natural SSA particles were used to investigate and quantify the organic fraction of natural SSA particles. Hygroscopic Growth Factor (HGF) data, also obtained by the VH-TDMA, were used to confirm the conclusions drawn from the volatility data. Both datasets indicated that the organic fraction of our natural SSA particles evaporated in the VH-TDMA over the temperature range 170–200°C. The organic volume fraction for 71–77 nm natural SSA particles was 8±6%. Organic volume fraction did not vary significantly with varying water residence time (40 secs to 24 hrs) in the bubble generator or SSA particle diameter in the range 38–173 nm. At room temperature we measured shape- and Kelvin-corrected HGF at 90% RH of 2.46±0.02 for NaCl, 2.35±0.02 for artifical SSA and 2.26±0.02 for natural SSA particles. Overall, these results suggest that the natural accumulation mode SSA particles produced in these experiments contained only a minor organic fraction, which had little effect on hygroscopic growth. Our measurement of 8±6% is an order of magnitude below two previous measurements of the organic fraction in SSA particles of comparable sizes. We stress that our results were obtained using coastal seawater and they can’t necessarily be applied on a regional or global ocean scale. Nevertheless, considering the order of magnitude discrepancy between this and previous studies, further research with independent measurement techniques and a variety of different seawaters is required to better quantify how much organic material is present in accumulation mode SSA.

David Hicks and Cathie Holden (eds.) (2007). Teaching The Global Dimension. London: Routledge, Taylor & Francis Group, 212 pages.

Teaching The Global Dimension (2007) is intended for primary and secondary teachers, pre-service teachers and educators interested in fostering global concerns in the education system. It aims at linking theory and practice and is structured as follows. Part 1, the global dimension, proposes an educational framework for understanding global concerns. Individual chapters in this section deal with some educational responses to global issues and the ways in which young people might become, in Hick’s terms, more “world-minded”. In the first two chapters, Hicks presents first, some educational responses to global issues that have emerged in recent decades, and second, an outline of the evolution of global education as a specific field. As with all the chapters in this book, most of the examples are drawn from the United Kingdom. Young people’s concerns, student teachers’ views and the teaching of controversial issues, comprise the other chapters in this section. Taken collectively, the chapters in Part 2 articulate the conceptual framework for developing, teaching and evaluating a global dimension across the curriculum. Individual chapters in this section, written by a range of authors, explore eight key concepts considered necessary to underpin appropriate learning experiences in the classroom. These are conflict, social justice, values and perceptions, sustainability, interdependence, human rights, diversity and citizenship. These chapters are engaging and well structured. Their common format consists of a succinct introduction, reference to positive action for change, and examples of recent effective classroom practice. Two chapters comprise the final section of this book and suggest different ways in which the global dimension can be achieved in the primary and the secondary classroom.

Bubble rise velocity and trajectory in xanthan gum crystal suspension

An experimental set-up was used to visually observe the characteristics of bubbles as they moved up a column holding xanthan gum crystal suspensions. The bubble rise characteristics in xanthan gum solutions with crystal suspension are presented in this paper. The suspensions were made by using different concentrations of xanthan gum solutions with 0.23 mm mean diameter polystyrene crystal particles. The influence of the dimensionless quantities; namely the Reynolds number, Re, the Weber number, We, and the drag co-efficient, cd, are identified for the determination of the bubble rise velocity. The effect of these dimensionless groups together with the Eötvös number, Eo, the Froude number, Fr, and the bubble deformation parameter, D, on the bubble rise velocity and bubble trajectory are analysed. The experimental results show that the average bubble velocity increases with the increase in bubble volume for xanthan gum crystal suspensions. At high We, Eo and Re, bubbles are spherical-capped and their velocities are found to be very high. At low We and Eo, the surface tension force is significant compared to the inertia force. The viscous forces were shown to have no substantial effect on the bubble rise velocity for 45 < Re < 299. The results show that the drag co-efficient decreases with the increase in bubble velocity and Re. The trajectory analysis showed that small bubbles followed a zigzag motion while larger bubbles followed a spiral motion. The smaller bubbles experienced less horizontal motion in crystal suspended xanthan gum solutions while larger bubbles exhibited a greater degree of spiral motion than those seen in the previous studies on the bubble rise in xanthan gum solutions without crystal.

Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation

The Saffman-Taylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a Hele-Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem suffciently diffcult that asymptotic or numerical methods must be used. In this paper we adapt numerical methods used to solve the Saffman-Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele-Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as with the surface tension problem.

New exact solutions for Hele-Shaw flow in doubly connected regions

Radial Hele-Shaw flows are treated analytically using conformal mapping techniques. The geometry of interest has a doubly-connected annular region of viscous fluid surrounding an inviscid bubble that is either expanding or contracting due to a pressure difference caused by injection or suction of the inviscid fluid. The zero-surface-tension problem is ill-posed for both bubble expansion and contraction, as both scenarios involve viscous fluid displacing inviscid fluid. Exact solutions are derived by tracking the location of singularities and critical points in the analytic continuation of the mapping function. We show that by treating the critical points, it is easy to observe finite-time blow-up, and the evolution equations may be written in exact form using complex residues. We present solutions that start with cusps on one interface and end with cusps on the other, as well as solutions that have the bubble contracting to a point. For the latter solutions, the bubble approaches an ellipse in shape at extinction.

Rapid synthesis of porous honeycomb Cu/Pd through a hydrogen-bubble templating method

A rapid electrochemical method based on using a clean hydrogen-bubble template to form a bimetallic porous honeycomb Cu/Pd structure has been investigated. The addition of palladium salt to a copper-plating bath under conditions of vigorous hydrogen evolution was found to influence the pore size and bulk concentration of copper and palladium in the honeycomb bimetallic structure. The surface was characterised by X-ray photoelectron spectroscopy, which revealed that the surface of honeycomb Cu/Pd was found to be rich with a Cu/Pd alloy. The inclusion of palladium in the bimetallic structure not only influenced the pore size, but also modified the dendritic nature of the internal wall structure of the parent copper material into small nanometre-sized crystallites. The chemical composition of the bimetallic structure and substantial morphology changes were found to significantly influence the surface-enhanced Raman spectroscopic response for immobilised rhodamine B and the hydrogen-evolution reaction. The ability to create free-standing films of this honeycomb material may also have many advantages in the areas of gas- and liquid-phase heterogeneous catalysis.

The effect of electrode material on the electrochemical formation of porous copper surfaces using hydrogen bubble templating

The electrodeposition of copper onto copper, gold, palladium and glassy carbon (GC) electrodes via a hydrogen bubble templating method is reported. It is found that the composition of the underlying electrode material significantly influences the morphology of the copper electrodeposit. Highly ordered porous structures are achieved with Cu and Au electrodes, however on Pd this order is disrupted and a rough randomly oriented surface is formed whereas on GC a bubble templating effect is not observed. Chronopotentiograms recorded during the electrodeposition process allows bubble formation and detachment from the surface to be monitored where distinctly different potential versus time profiles are observed at the different electrodes. The porous Cu surfaces are characterised with scanning electron microscopy, X-ray diffraction and cyclic voltammetric measurements recorded under alkaline conditions. The latter demonstrates that there are active sites present on electrodeposited copper whose coverage and reactivity depend on the underlying electrode material. The most active Cu surface is achieved at a Pd substrate for both the hydrogen evolution reaction and the catalytic reduction of ferricyanide ions with thiosulphate ions. This demonstrates that the highly ordered porous structure on the micron scale which typifies the morphology that can be achieved with the hydrogen bubbling template method is not required in producing the most effective material.

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

Reducing bubble defect on the outsole production process in the footwear manufacturing industry

The purpose of this study is to discover the significant factors causing the bubble defect on the outsoles manufactured by the Case Company. The bubble defect occurs approximately 1.5 per cent of the time or in 36 pairs per day. To understand this problem, experimental studies are undertaken to identify various factors such as injector temperature, mould temperature; that affects the production of waste. The work presented in this paper comprises a review of the relevant literature on the Six Sigma DMAIC improvement process, quality control tools, and the design of the experiments. After the experimentation following the Six Sigma process, the results showed that the defect occurred in approximately 0.5 per cent of the products or in 12 pairs per day; this decreased the production cost from 6,120 AUD per month to 2,040 AUD per month. This research aimed to reduce the amount of waste in men’s flat outsoles. Hence, the outcome of research presented in this paper should be used as a guide for applying the appropriate process for each type of outsole.