Bubble Growth From First Principles

We consider the simplest relevant problem in the foaming of molten plastics, the growth of a single bubble in a sea of highly viscous Newtonian fluid, and without interference from other bubbles. This simplest problem has defied accurate solution from first principles. Despite plenty of research on foaming, classical approaches from first principles have neglected the temperature rise in the surrounding fluid, and we find that this oversimplification greatly accelerates bubble growth prediction. We use a transport phenomena approach to analyze the growth of a solitary bubble, expanding under its own pressure. We consider a bubble of ideal gas growing without the accelerating contribution from mass transfer into the bubble. We explore the roles of viscous forces, fluid inertia, and viscous dissipation. We find that bubble growth depends upon the nucleus radius and nucleus pressure. We begin with a detailed examination of the classical approaches (thermodynamics without viscous heating). Our failure to fit experimental data with these classical approaches, sets up the second part of our paper, a novel exploration of the essential decelerating role of viscous heating. We explore both isothermal and adiabatic bubble expansion, and also the decelerating role of surface tension. The adiabatic analysis accounts for the slight deceleration due to the cooling of the expanding gas, which depends on gas polyatomicity. We also explore the pressure profile, and the components of the extra stress tensor, in the fluid surrounding the growing bubble. These stresses can eventually be frozen into foamed plastics. We find that our new theory compares well with measured bubble behavior.

Bubble Growth From First Principals

We consider the simplest relevant problem in the foaming of molten plastics, the growth of a single bubble in a sea of highly viscous Newtonian fluid, and without interference from other bubbles. This simplest problem has defied accurate solution from first principles. Despite plenty of research on foaming, classical approaches from first principles have neglected the temperature rise in the surrounding fluid, and we find that this oversimplification greatly accelerates bubble growth prediction. We use a transport phenomena approach to analyze the growth of a solitary bubble, expanding under its own pressure. We consider a bubble of ideal gas growing without the accelerating contribution from mass transfer into the bubble. We explore the roles of viscous forces, fluid inertia, and viscous dissipation. We find that bubble growth depends upon the nucleus radius and nucleus pressure. We begin with a detailed examination of the classical approaches (thermodynamics without viscous heating). Our failure to fit experimental data with these classical approaches, sets up the second part of our paper, a novel exploration of the essential decelerating role of viscous heating. We explore both isothermal and adiabatic bubble expansion, and also the decelerating role of surface tension. The adiabatic analysis accounts for the slight deceleration due to the cooling of the expanding gas, which depends on gas polyatomicity. We also explore the pressure profile, and the components of the extra stress tensor, in the fluid surrounding the growing bubble. These stresses can eventually be frozen into foamed plastics. We find that our new theory compares well with measured bubble behavior.

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.

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.

Some features of spray breakup in effervescent atomizers

The near orifice spray breakup at low GLR (gas to liquid ratio by mass) values in an effervescent atomizer is studied experimentally using water as a simulant and air as atomizing gas. From the visualizations, the near orifice spray structures are classified into three modes: discrete bubble explosions, continuous bubble explosions and annular conical spray. The breakup of the spray is quantified in terms of the mean bubble bursting distance from the orifice. The parametric study indicates that the mean bubble bursting distance mainly depends on airflow rate, jet diameter and mixture velocity. It is also observed that the jet diameter has a dominant effect on the bubble bursting distance when compared to mixture velocity at a given airflow rate. The mean bubble bursting distance is shown to be governed by a nondimensional two-phase flow number consisting of all the aforementioned parameters. The location of bubble bursting is found to be highly unsteady spatially, which is influenced by flow dynamics inside the injector. It is proposed that this unsteadiness in jet breakup length is a consequence of varying degree of bubble expansion caused due to the intermittent occurrence of single phase and two-phase flow inside the orifice.

Optimizing mixing during the sponge cake manufacturing process

Sponge cakes have traditionally been manufactured using multistage mixing methods to enhance potential foam formation by the eggs. Today, use of all-in (single-stage) mixing methods is superseding multistage methods for large-scale batter preparation to reduce costs and production time. In this study, multistage and all-in mixing procedures and three final high-speed mixing times (3, 5, and 15 min) for sponge cake production were tested to optimize a mixing method for pilot-scale research. Mixing for 3 min produced batters with higher relative density values than did longer mixing times. These batters generated well-aerated cakes with high volume and low hardness. In contrast, after 5 and 15 min of high-speed mixing, batters with lower relative density and higher viscosity values were produced. Although higher bubble incorporation and retention were observed, longer mixing times produced better developed gluten networks, which stiffened the batters and inhibited bubble expansion during mixing. As a result, these batters did not expand properly and produced cakes with low volume, dense crumb, and high hardness values. Results for all-in mixing were similar to those for the multistage mixing procedure in terms of the physical properties of batters and cakes (i.e., relative density, elastic moduli, volume, total cell area, hardness, etc.). These results suggest the all-in mixing procedure with a final high-speed mixing time of 3 min is an appropriate mixing method for pilot-scale sponge cake production. The advantages of this method are reduced energy costs and production time.

Analysis of the housing market in the autonomous community of the Basque country: the adjustment after the burst of the housing bubble

In this project study the characteristic and dynamics of the residential housing market in the Basque country. When strong expansion and colapsing emerged in 2007,studied the differents adjustment.

Replacing fat and sugar with inulin in cakes: bubble size distribution, physical and sensory properties

The replacement of fat and sugar in cakes is a challenge as they have an important effect on the structural and sensory properties. Moreover, there is the possibility to incorporate an additional value using novel replacers. In this work, inulin and oligofructose were used as fat and sugar replacers, respectively. Different combinations of replacement levels were investigated: fat replacement (0 and 50 %) and sugar replacement (0, 20, 30, 40 and 50 %). Simulated microbaking was carried out to study bubble size distribution during baking. Batter viscosity and weight loss during baking were also analysed. Cake characteristics were studied in terms of cell crumb structure, height, texture and sensory properties. Fat and sugar replacement gave place to batters with low apparent viscosity values. During heating, bubbles underwent a marked expansion in replaced cakes if compared to the control cake. The low batter stability in fat-replaced samples increased bubble movement, giving place to cakes with bigger cells and less height than the control. Sugar-replaced samples had smaller and fewer cells and lower height than the control. Moreover, sugar replacement decreased hardness and cohesiveness and in- creased springiness, which could be related with a denser crumb and an easily crumbled product. Regarding the sensory analysis, a replacement up to 50 % of fat and 30 % of sugar, separately and simultaneously, did not change remarkably the overall acceptability of the cakes. However, the sponginess and the sweetness could be improved in all the replaced cakes, according to the Just About Right scales.

Economic growth in Sweden, 2000-2010 : The dot-com bubble and the financial crisis

    Economic growth is the increase in the inflation-adjusted market value of the goods and services produced by an economy over time. The total output is the quantity of goods or servicesproduced in a given time period within a country. Sweden was affected by two crises during the period 2000-2010: a dot-com bubble and a financial crisis. How did these two crises affect the economic growth?     The changes of domestic output can be separated into four parts: changes in intermediate demand, final domestic demand, export demand and import substitution. The main purpose of this article is to analyze the economic growth during the period 2000-2010, with focus on the dot-com bubble in the beginning of the period 2000-2005, and the financial crisis at the end of the period 2005-2010. The methodology to be used is the structural decomposition method.     This investigation shows that the main contributions to the Swedish total domestic output increase in both the period 2000-2005 and the period 2005-2010 were the effect of domestic demand. In the period 2005-2010, financial crisis weakened the effect of export. The output of the primary sector went from a negative change into a positive, explained mainly by strong export expansion. In the secondary sector, export had most effect in the period 2000-2005. Nevertheless, domestic demand and import ratio had more effect during the financial crisis period. Lastly, in the tertiary sector, domestic demand can mainly explain the output growth in the whole period 2000-2010.

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.

Improvement to Chinese information retrieval by incorporating word segmentation and query expansion

The increasing diversity of the Internet has created a vast number of multilingual resources on the Web. A huge number of these documents are written in various languages other than English. Consequently, the demand for searching in non-English languages is growing exponentially. It is desirable that a search engine can search for information over collections of documents in other languages. This research investigates the techniques for developing high-quality Chinese information retrieval systems. A distinctive feature of Chinese text is that a Chinese document is a sequence of Chinese characters with no space or boundary between Chinese words. This feature makes Chinese information retrieval more difficult since a retrieved document which contains the query term as a sequence of Chinese characters may not be really relevant to the query since the query term (as a sequence Chinese characters) may not be a valid Chinese word in that documents. On the other hand, a document that is actually relevant may not be retrieved because it does not contain the query sequence but contains other relevant words. In this research, we propose two approaches to deal with the problems. In the first approach, we propose a hybrid Chinese information retrieval model by incorporating word-based techniques with the traditional character-based techniques. The aim of this approach is to investigate the influence of Chinese segmentation on the performance of Chinese information retrieval. Two ranking methods are proposed to rank retrieved documents based on the relevancy to the query calculated by combining character-based ranking and word-based ranking. Our experimental results show that Chinese segmentation can improve the performance of Chinese information retrieval, but the improvement is not significant if it incorporates only Chinese segmentation with the traditional character-based approach. In the second approach, we propose a novel query expansion method which applies text mining techniques in order to find the most relevant words to extend the query. Unlike most existing query expansion methods, which generally select the highly frequent indexing terms from the retrieved documents to expand the query. In our approach, we utilize text mining techniques to find patterns from the retrieved documents that highly correlate with the query term and then use the relevant words in the patterns to expand the original query. This research project develops and implements a Chinese information retrieval system for evaluating the proposed approaches. There are two stages in the experiments. The first stage is to investigate if high accuracy segmentation can make an improvement to Chinese information retrieval. In the second stage, a text mining based query expansion approach is implemented and a further experiment has been done to compare its performance with the standard Rocchio approach with the proposed text mining based query expansion method. The NTCIR5 Chinese collections are used in the experiments. The experiment results show that by incorporating the text mining based query expansion with the hybrid model, significant improvement has been achieved in both precision and recall assessments.