Stephen O’Brien

Acid polishing of lead glass

PurposeThe polishing of cut lead glass crystal is effected through the dowsing of the glass in a mixture of two separate acids, which between them etch the surface and as a result cause it to be become smooth. In order to characterise the resultant polishing the rate of surface etching must be known, but when this involves multicomponent surface reactions it becomes unclear what this rate actually is.MethodsWe develop a differential equation based discrete model to determine the effective etching rate by means of an atomic scale model of the etching process.ResultsWe calculate the etching rate numerically and provide an approximate asymptotic estimate.ConclusionsThe natural extension of this work would be to develop a continuum advection-diffusion model.

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.

Improving the Industrial/Mathematics Interface

Mathematics is ubiquitous in the sciences and engineering. Arguably a science is considered to have come of age when it has become sufficiently mathematical as illustrated by the burgeoning areas of mathematical biology and mathematical finance. Despite all the potential applications, some mathematicians have moved away from industry and from real applications. From the late 1960s under the influence of the Oxford group [Alan Tayler and Leslie Fox, see Ockendon (1998), Tayler (1990)], interest in modeling real industrial problems has steadily grown. Industrial mathematics and its near synonyms is problem-driven mathematics for the sake of the sciences while pure mathematics may be regarded as being mathematics for its own sake. In this context ‘industry’ is interpreted in a very broad sense: the remit of these groups includes more than collaboration with industry (problems may come from anywhere in the sciences, e.g. mathematical biology, mathematical finance, the environment).

Mathematical Modelling of Waves in Guinness

We provide a simple two-dimensional model of bubbly two-phase flow which can be used to investigate why waves form and propagate downward while a pint of Guinness is settling. We start out with the basic equations of the two-phase flow and use the large timescale difference of beer convection and rising bubbles in order to treat the convection flow as quasistatic. Using this argument we further simplify the two-phase mixture equations to that of a single liquid whose density varies with bubble concentration. A stability analysis shows that waves can occur through an instability analogous to the Kelvin-Helmholtz instability which forms in parallel shear flow. 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 shear flows.

Decoupling the Interaction of Solid and Fluid Mechanics in the Modelling of Continuous Casting Processes

The modelling of the continuous casting of metals is known to involve the complex interaction of non-isothermal fluid and solid mechanics. However, using asymptotic methods and an earlier numerical result obtained via computational fluid dynamics, we demonstrate how the motion of the liquid metal can be systematically decoupled from the stresses induced in the solidified shell. The resulting asymptotically reduced model can then serve as a computationally efficient module for stress mechanics models that aim to predict segregation and crack formation in the solid metal.

Mathematical Modelling of the Coffee Brewing Process

The drip filter coffee market is a multi-billion euro industry. Despite this, although the chemistry of coffee brewing has been investigated in great detail, the physics of the process has received relatively little attention. In order to explain in scientific terms correlations between the coffee quality and the process variables, a physical model is required. In this study, flow through a static, saturated coffee bed, under the influence of a pressure gradient, is described using a double porosity model. The model is parametrised using experimentally obtained data from a cylindrical flow-through cell containing a coffee bed. Mass transfer from the coffee grains to the interstitial water is modelled using two mechanisms; mass transfer from the surface of the grains and mass transfer from the interior (bulk) of the grains. Mass transfer resistances are estimated by fitting experimental data. Initially coffee extraction is dominated by mass transfer from the grain surface, while transfer from the kernel of the grain is the rate limiting mechanism once the surface coffee has been exhausted.

Study Groups in Ireland: A Reflection

Study groups were first introduced to Ireland in 2008 by MACSI. We present an overview of MACSI study groups focusing on the role study groups play in initiating longer term interactions between industry and academia.

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.