Asymptotics of a small liquid drop on a cone and plate rheometer

A cone and a plate rheometer is a laboratory apparatus used to measure the viscosity and other related parameters of a non-Newtonian liquid subject to an applied force. A small drop, of order millimetres, of the liquid is located between the horizontal plate and the shallow cone of the rheometer. Rotation of the cone ensues, the liquid begins to flow and the plate starts to rotate. Liquid parameters are inferred based on the difference in the applied rotational force and the resulting rotational force of the plate. To describe the flow of the drop, the initial drop configuration, before rotation commences, must be determined. The equilibrium drop profile is given by the solution to the well-known nonlinear Young-Laplace equation. We formulate asymptotic solutions for the drop profile based on the small Bond number. The modelling of the drop exhibits a rich asymptotic structure consisting of five distinct scalings which are resolved via the method matched asymptotics

Stability of thin liquid curtains

We investigate the stability of thin liquid curtains with respect to two-dimensional perturbations. The dynamics of perturbations with wavelengths exceeding (or comparable to) the curtain's thickness are examined using the lubrication approximation (or a kind of geometric optics). It is shown that, contrary to the previous theoretical results, but in agreement with the experimental ones, all curtains are stable with respect to small perturbations. Large perturbations can still be unstable, however, but only if they propagate upstream and, thus, disrupt the curtain at its outlet. This circumstance enables us to obtain an effective stability criterion by deriving an existence condition for upstream propagating perturbations

Waves in Guinness

We describe a simple model of a bubbly two-phase flow which is able to explain why waves propagate downward when a pint of Guinness is poured, and also how the waves are generated. Our theory involves a physically based regularization of the basic equations of the two-phase flow, using interphasic pressure difference and virtual mass terms, together with bulk or eddy viscosity terms. We show that waves can occur through an instability analogous to that which forms roll waves in inclined fluid flows, and we provide a description of the form of these waves, and compare them to observations. Our theory provides a platform for the description of waves in more general bubbly two-phase flows, and the way in which the flow breaks down to form slug flow. (C) 2008 American Institute of Physics

A validated asymptotic thermomechanical model for air-gap formation in tapered moulds in the continuous casting of steel

A recent asymptotics-based thermomechanical model is adapted and applied to the mould region in the continuous casting of round steel billets, with a view to describing the complex interplay between airgap formation, mould taper, cooling channel width and cooling water velocity. Although the situation is steady state, the analysis leads to what is mathematically a dual moving-boundary problem for the solid–melt and solid–air interfaces, where the distance from the top of the mould region is the time-like variable in the problem. Moreover, the two interfaces are initiated at different locations. In addition, the thermal and mechanical problems are found to decouple and it is possible to solve the first ahead of the second. The model equations are solved numerically using a finite-difference method, and the approach is subsequently successfully validated against a previous finite-element model and experimental data from temperature measurements taken within the mould

Coffee extraction kinetics in a well mixed system

The extraction of coffee solubles from roasted and ground coffee is a complex operation, the understanding of which is key to the brewing of high quality coffee. This complexity stems from the fact that brewing of coffee is achieved through a wide variety of techniques each of which depends on a large number of process variables. In this paper, we consider a recent, experimentally validated model of coffee extraction, which describes extraction from a coffee bed using a double porosity model. The model incorporates dissolution and transport of coffee in the coffee bed. The model was shown to accurately describe extraction of coffee solubles from grains in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. The full model equations can only be solved numerically. In this work we consider asymptotic solutions, based on the dominant mechanisms, in the case of coffee extraction from a dilute suspension of coffee grains. Extraction in this well mixed system, can be described by a set of ordinary differential equations. This allows analysis of the extraction kinetics from the coffee grains independent of transport processes associated with flow through packed coffee beds. Coffee extraction for an individual grain is controlled by two processes: a rapid dissolution of coffee from the grain surfaces in conjunction with a much slower diffusion of coffee through the tortuous intragranular pore network to the grain surfaces. Utilising a small parameter resulting from the ratio of these two timescales, we construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared with numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters, so the solutions facilitate quick investigation of the influence of various process parameters on the coffee extraction curves

Detecting heart rate while jogging: blind source separation of gait and heartbeat

A blind source signal separation problem that was brought to a Study Group in Limerick in 2013 required a way to prevent the gait of a jogger from masking the heartbeat, when detected by a simple photodiode that measures light transmission through a jogger’s wrist tissues. The group was successful in discovering a singular value decomposition (SVD) approach, which not only allows accurate detection of heart rate but also allows recovery of a good facsimile of the entire blood pressure time series from the mixed photodiode signal

Developing 'good' post-primary teachers and teaching in a reform era: cultural dynamics in a programme level study of the PDE

Most discussions about the quality of schooling quickly turn to the quality of teachers, reflections and memories of individual teachers who ‘made a difference’, whether good or not so, in a person’s school biography. The quest for the ‘good teacher’ is important to parents, interleaves itself into a community’s conversations about its schools, animates children’s and adolescents’ reflections on a central feature of their lives and increasingly is the protagonist in policy debates on teacher education