Brendan Florio

Boundary tracing is a technique whereby exact solutions to boundary value problems in known domains are used to generate alternate domains admitting the same solution. Here we use line-source solutions of Laplace’s equation to produce exact solutions to the conduction–radiation problem in novel domains, such as circular regions with protrusions and teardrop-shaped regions with short or long tails. The shapes and sizes obtained are practical and could be used for novel radiator design.

We analyse a model of the phosphorus cycle in the ocean given by Slomp and Van Cappellen (Biogeosciences 4:155-171, 2007. ). This model contains four distinct oceanic boxes and includes relevant parts of the water, carbon and oxygen cycles. We show that the model can essentially be solved analytically, and its behaviour completely understood without recourse to numerical methods. In particular, we show that, in the model, the carbon and phosphorus concentrations in the different ocean reservoirs are all slaved to the concentration of soluble reactive phosphorus in the deep ocean, which relaxes to an equilibrium on a time scale of 180,000 y, and we show that the deep ocean is either oxic or anoxic, depending on a critical parameter which we can determine explicitly. Finally, we examine how the value of this critical parameter depends on the physical parameters contained in the model. The presented methodology is based on tools from applied mathematics and can be used to reduce the complexity of other large, biogeochemical models.

In contexts such as space travel, thermal radiation is the primary mode of heat transfer. The Stefan-Boltzmann law gives rise to a boundary flux which is quartic in temperature, and this nonlinearity renders even the simplest of conduction-radiation problems analytically insur-mountable in more than one dimension. An unconventional approach known as boundary tracing allows for analytical inroads into flux boundary value problems that would otherwise require numer-ical study. In this paper, the method of boundary tracing is used to generate near-exact results for an infinite family of conduction-radiation domains representing radiating fins; realistic lengths and temperatures can be realized.

Complicated geometrical objects like aggregate clusters can be characterised by a simple parameter: the fractal dimension. There are many ways to measure this fractal dimension. For most methods, it is found by the exponent of a relationship between an intrinsic value, such as mass or area, to a characteristic value, such as bulk aggregate length. The perimeter-area method was derived by Mandelbrot to measure the fractal dimension of chips of ore. In this method, there is no distinction of whether the perimeter and area values are characteristic or intrinsic.

Various methods of measuring the fractal dimension are used on different known fractal objects to demonstrate these issues in an idealised setting.

We show that while the distinction is not important for Mandelbrot's ore chips, which are island-type objects, it is very important in the analysis of cluster-type objects. The perimeter-area method is a valid tool in the fractal characterisation of aggregates and clusters, however, researchers must be careful to take the appropriate intrinsic and characteristic measurements.

Unit operations used to achieve solid-liquid separation for fine particle suspensions rely upon efficient aggregate formation. There is considerable potential for predictions from population balance models describing particle aggregation to help optimise full-scale processes. The vast majority of studies in this area make use of the classical coagulation equation of Smoluchowsld, and while developed primarily for coalescence phenomena, it has been adapted and modified extensively to describe particle aggregation for many different substrates and procedures for inducing aggregate formation. This has resulted in a wide variety of mathematical expressions, some of which are highly sophisticated but can only be applied successfully to a limited range of conditions. For this reason, it is necessary for researchers to understand the main aggregation mechanisms involved in the processes (coagulation, bridging flocculation) and how the system conditions (flow regime, particle size, solids concentration) can then influence aggregate growth, breakage and the resulting structures. Such understanding is essential for the appropriate selection of Mathematical equations to then obtain a successful model that can be solved at low computational cost. The main mathematical expressions developed for the different phenomena that occur during particle aggregation (i.e. collision frequency and efficiency, aggregate breakage rate and distribution, the structure formed and their potential for restructuring over time) by different mechanisms are reviewed. The main published studies are critically assessed, indicating their scope and the conditions under which the models can be usefully applied. Important challenges remain towards achieving wider practical applications, particularly in reducing reliance on empiricism. Particular emphasis is placed on research oportunities, focusing mainly on i) the incorporation of interaction forces for colloidal systems at submicron particle; ii) importance of achieving more reliable representations of aggregation behaviour at high solid concentration; and iii) incorporation of PBEs within computational fluid dynamics (CFD) models that describe industrial aggregation processes as a powerful tool for full-scale unit design and process optimisation. (C) 2017 Elsevier B.V. All rights reserved.

Convection can occur in a confined saturated porous box when the associated Rayleigh number exceeds a threshold critical value: the identity of the preferred onset convection mode depends sensitively on the geometry of the box. Here we discuss examples for which the box dimensions are such that four modes share a common critical Rayleigh number. Perturbation theory is used to derive a system of coupled ordinary differential equations that governs the nonlinear interaction of the four modes and an analysis of this set is undertaken. In particular, it is demonstrated that as the Rayleigh number is increased beyond critical so a series of pitchfork bifurcations occur and multiple stable states are identified that correspond to the survival of just one of the four modes. The basins of attraction for each mode in the 4D phase space are described by a reduction to a suitable 3D counterpart.

The formation of an air gap in continuous casting systems is detrimental to the process efficiency as it acts to thermally insulate the cast from the water-cooled mould. By tapering the mould wall, the thermal contraction of the cooling cast can be accommodated so that the thickness of the air gap is decreased. We consider a coupled thermomechanical model to investigate the effect of mould tapering on the formation and thickness of the air gap in an axisymmetric mould. Using asymptotic techniques, the model is reduced to allow analytic and inexpensive numerical investigations while maintaining the essential characteristics of the thermomechanical process. This work improves on previous models by including superheat, where the incoming molten metal is at a higher temperature than its melting point. The degree of superheating also affects the formation and thickness of the air gap and presents a viable alternative for control of the system. The efficacy of mould tapering in the presence of superheat is examined.

A multitude of convection modes may occur within a confined rectangular box of saturated porous medium when the associated dimensionless Rayleigh number R is above some critical value. For particular sizes of box, however, it is possible for multiple modes (typically three) to share a common critical Rayleigh number. For box shapes close to these geometries, modes can interact and compete nonlinearly near the onset of convection. The generic examples of this phenomenon can be conveniently classified as belonging to one of two distinctive classes distinguished by whether or not fixed points are possible with all three modes having a non-zero amplitude. It transpires that this classification is not quite exhaustive for there is one particular case which falls outside this pattern. This last case, which is described by a system of evolution equations that is structurally different from that applying in the generic situation, is explored in this paper. Some rich dynamical behaviours are uncovered and, in particular, it is shown that at sufficiently large R stable states arise in which all three modes persist. This contrasts to the typical generic behaviour where a single mode is preferred. The bifurcation sequence that develops as the Rayleigh number grows is mapped out from primary through to tertiary stages. In addition, it is found that a cusp bifurcation is present and that the details of the ordering of the various bifurcations are shown to be sensitive to the precise geometry of the box.

Asymptotic methods are employed to revisit an earlier model for oscillation-mark formation in the continuous casting of steel. A systematic non-dimensionalization of the governing equations, which was not carried out previously, leads to a model with 12 dimensionless parameters. Analysis is provided in the same parameter regime as for the earlier model, and surprisingly simple analytical solutions are found for the oscillation-mark profiles; these are found to agree reasonably well with the numerical solution in the earlier model and very well with fold-type oscillation marks that have been obtained in more recent experimental work. The benefits of this approach, when compared with time-consuming numerical simulations, are discussed in the context of auxiliary models for macrosegregation and thermomechanical stresses and strains.

A mathematical model is developed to describe the melting of nanowires. The first section of the paper deals with a standard theoretical situation, where the wire melts due to a fixed boundary temperature. This analysis allows us to compare with existing results for the phase change of nanospheres. The equivalent solidification problem is also examined. This shows that solidification is a faster process than melting; this is because the energy transfer occurs primarily through the solid rather than the liquid which is a poorer conductor of heat. This effect competes with the energy required to create new solid surface which acts to slow down the process, but overall conduction dominates. In the second section, we consider a more physically realistic boundary condition, where the phase change occurs due to a heat flux from surrounding material. This removes the singularity in initial melt velocity predicted in previous models of nanoparticle melting. It is shown that even with the highest possible flux the melting time is significantly slower than with a fixed boundary temperature condition.