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.

Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

Idyll number eleven

This is a short horror story formulated in the research process for the novel "That Blackfella Bloodsucka Dance!"

Prandtl number scaling of natural convection of an inclined flat plate due to uniform surface heat flux

A new scaling analysis has been performed for the unsteady natural convection boundary layer under a downward facing inclined plate with uniform heat flux. The development of the thermal or viscous boundary layers may be classified into three distinct stages including a start-up stage, a transitional stage and a steady stage, which can be clearly identified in the analytical as well as numerical results. Earlier scaling shows that the existing scaling laws of the boundary layer thickness, velocity and steady state time scale for the natural convection flow on a heated plate of uniform heat flux provide a very poor prediction of the Prandtl number dependency of the flow. However, those scalings performed very well with Rayleigh number and aspect ratio dependency. In this study, a new Prandtl number scaling has been developed using a triple-layer integral approach for Pr > 1. It is seen that in comparison to the direct numerical simulations, the new scaling performs considerably better than the previous scaling.

Symbolic number : the integration of magnitude and spatial representations in children aged 6 to 8 years

The process of learning symbolic Arabic digits in early childhood requires that magnitude and spatial information integrates with the concept of symbolic digits. Previous research has separately investigated the development of automatic access to magnitude and spatial information from symbolic digits. However, developmental trajectories of symbolic number knowledge cannot be fully understood when considering components in isolation. In view of this, we have synthesized the existing lines of research and tested the use of both magnitude and spatial information with the same sample of British children in Years 1, 2 and 3 (6-8 years of age). The physical judgment task of the numerical Stroop paradigm (NSP) demonstrated that automatic access to magnitude was present from Year 1 and the distance effect signaled that a refined processing of numerical information had developed. Additionally, a parity judgment task showed that the onset of the Spatial-Numerical Association of Response Codes (SNARC) effect occurs in Year 2. These findings uncover the developmental timeline of how magnitude and spatial representations integrate with symbolic number knowledge during early learning of Arabic digits and resolve inconsistencies between previous developmental and experimental research lines.

Representational change and strategy use in children's number line estimation during the first years of primary school

Background: The objective of this study was to scrutinize number line estimation behaviors displayed by children in mathematics classrooms during the first three years of schooling. We extend existing research by not only mapping potential logarithmic-linear shifts but also provide a new perspective by studying in detail the estimation strategies of individual target digits within a number range familiar to children. Methods: Typically developing children (n = 67) from Years 1 – 3 completed a number-to-position numerical estimation task (0-20 number line). Estimation behaviors were first analyzed via logarithmic and linear regression modeling. Subsequently, using an analysis of variance we compared the estimation accuracy of each digit, thus identifying target digits that were estimated with the assistance of arithmetic strategy. Results: Our results further confirm a developmental logarithmic-linear shift when utilizing regression modeling; however, uniquely we have identified that children employ variable strategies when completing numerical estimation, with levels of strategy advancing with development. Conclusion: In terms of the existing cognitive research, this strategy factor highlights the limitations of any regression modeling approach, or alternatively, it could underpin the developmental time course of the logarithmic-linear shift. Future studies need to systematically investigate this relationship and also consider the implications for educational practice.

Trends in policies, programs and practices in the Australasian First Year Experience literature 2000-2010. The First Year in Higher Education Research Series on Evidence-based Practice. Number 1

This publication is the first in a series of scholarly reports on research-based practice related to the First Year Experience in Higher Education. This report synthesises evidence about practice-based initiatives and pragmatic approaches in Aotearoa (New Zealand) and Australia that aim to enhance the experience of commencing students in the higher education sector. Trends in policies, programs and practices ... examines the first year experience literature from 2000-2010. It acknowledges the uniqueness of the Australasian socio-political context and its influence on the interests and output of researchers. The review surveyed almost 400 empirical reports and conceptual discussions produced over the decade that dealt with the stakeholders, institutions and the higher education sector in Australasia. The literature is examined through two theoretical constructs or “lenses”: first, a set of first year curriculum design principles and second, the generational approach to describing the maturation of initiatives. These outcomes and suggested directions for further research provide the challenges and the opportunities for FYE adherents, both scholars and practitioners, to grapple with in the next decade.

Scaling for the Prandtl number of the natural convection boundary layer of an inclined flat plate under uniform surface heat flux

An improved scaling analysis and direct numerical simulations are performed for the unsteady natural convection boundary layer adjacent to a downward facing inclined plate with uniform heat flux. The development of the thermal or viscous boundary layers may be classified into three distinct stages: a start-up stage, a transitional stage and a steady stage, which can be clearly identified in the analytical as well as the numerical results. Previous scaling shows that the existing scaling laws of the boundary layer thickness, velocity and steady state time scale for the natural convection flow on a heated plate of uniform heat flux provide a very poor prediction of the Prandtl number dependency of the flow. However, those scalings perform very well with Rayleigh number and aspect ratio dependency. In this study, a modified Prandtl number scaling is developed using a triple layer integral approach for Pr > 1. It is seen that in comparison to the direct numerical simulations, the modified scaling performs considerably better than the previous scaling.

A comparative study of the number and mass of fine particles emitted with diesel fuel and marine gas oil (MGO)

The current investigation reports on diesel particulate matter emissions, with special interest in fine particles from the combustion of two base fuels. The base fuels selected were diesel fuel and marine gas oil (MGO). The experiments were conducted with a four-stroke, six-cylinder, direct injection diesel engine. The results showed that the fine particle number emissions measured by both SMPS and ELPI were higher with MGO compared to diesel fuel. It was observed that the fine particle number emissions with the two base fuels were quantitatively different but qualitatively similar. The gravimetric (mass basis) measurement also showed higher total particulate matter (TPM) emissions with the MGO. The smoke emissions, which were part of TPM, were also higher for the MGO. No significant changes in the mass flow rate of fuel and the brake-specific fuel consumption (BSFC) were observed between the two base fuels.

Branch-length estimation bias misleads molecular dating for a vertebrate mitochondrial phylogeny

Despite recent methodological advances in inferring the time-scale of biological evolution from molecular data, the fundamental question of whether our substitution models are sufficiently well specified to accurately estimate branch-lengths has received little attention. I examine this implicit assumption of all molecular dating methods, on a vertebrate mitochondrial protein-coding dataset. Comparison with analyses in which the data are RY-coded (AG → R; CT → Y) suggests that even rates-across-sites maximum likelihood greatly under-compensates for multiple substitutions among the standard (ACGT) NT-coded data, which has been subject to greater phylogenetic signal erosion. Accordingly, the fossil record indicates that branch-lengths inferred from the NT-coded data translate into divergence time overestimates when calibrated from deeper in the tree. Intriguingly, RY-coding led to the opposite result. The underlying NT and RY substitution model misspecifications likely relate respectively to “hidden” rate heterogeneity and changes in substitution processes across the tree, for which I provide simulated examples. Given the magnitude of the inferred molecular dating errors, branch-length estimation biases may partly explain current conflicts with some palaeontological dating estimates.