Designing a Green Roof for Ireland

A model is presented for the gravity-driven flow of rainwater descending through the soil layer of a green roof, treated as a porous medium on a at permeable surface representing an efficient drainage layer. A fully saturated zone is shown to occur. It is typically a thin layer, relative to the total soil thickness, and lies at the bottom of the soil layer. This provides a bottom boundary condition for the partially saturated upper zone. It is shown that after the onset of rainfall, well-defined fronts of water can descend through the soil layer. Also the rainwater flow is relatively quick compared with the moisture uptake by the roots of the plants in the roof. In a separate model the exchanges of water are described between the (smaller-scale) porous granules of soil, the roots and the rainwater in the inter-granule pores

Ink Drying in Inkjet Printers

The first problem put to the Study Group for Maths in Industry by Domino UK Ltd concerns ink drying and blocking nozzles in a printer. The goals were as follows: 1. To propose mechanisms for the growth of a plug of dried ink in the open end of a Drop-on-Demand drop generator, 2. To suggest cures to this problem, 3. To consider why oscillating the meniscus appears to alleviate the problem

Electron self-trapping on a nano-circle

We study the self-trapping of quasiparticles (electrons, holes, excitons, etc) in a molecular chain with the structure of a ring, taking into account the electron-phonon interaction and the radial and tangential deformations of the chain. A discrete system of equations is obtained and solved numerically. The analytical solutions for the wave function of a quasiparticle and for the molecule displacements that determine the distortion of the ring, are also obtained and solved in the continuum approximation. The numerical solutions of the system of discrete nonlinear equations reveals several regimes of quasiparticle localisation in the chain which depend on the values of the parameters of the system. It is shown that the transversal deformation of the chain favours the formation of a soliton.Comment: 43 pages 9 figure

Mobile radio propagation prediction using ray tracing methods

The basic problem is to solve the two-dimensional scalar Helmholtz equation for a point source (the antenna) situated in the vicinity of an array of scatterers (such as the houses and any other relevant objects in 1 square km of urban environment). The wavelength is a few centimeters and the houses a few metres across, so there are three disparate length scales in the problem. The question posed by BT concerned ray counting on the assumptions that: (i) rays were subject to a reflection coefficient of about 0.5 when bouncing off a house wall and (ii) that diffraction at corners reduced their energy by 90%. The quantity of particular interest was the number of rays that need to be accounted for at any particular point in order for those neglected to only contribute 10% of the field at that point; a secondary question concerned the use of rays to predict regions where the field was less than 1% of that in the region directly illuminated by the antenna. The progress made in answering these two questions is described in the next two sections and possibly useful representations of the solution of the Helmholtz equations in terms other than rays are given in the final section

Electro-painting

A mathematical model for electro-painting is developed and analysed. Several sub-problems are studied analytically and numerically

Electric arc problem

This problem was concerned with the behaviour of an electric arc as the current supplied to the arc passed through zero. We sought to establish a model that predicts the conditions under which the arc was extinguished. During the study group we attempted to show how an equation of state could be derived by proposing that the arc should be in thermal equilibrium with the surrounding gas. The behaviour of the system as the current goes to zero was examined but no firm conclusions could be draw

Homeless population

The aim was to derive and analyze a model for numbers of homeless and non-homeless people in a borough, in particular to see how these figures might be affected by different policies regarding housing various categories of people. Most attention was focused on steady populations although the stability of these and possible timescales of dynamic problems were also discussed. The main outcome of this brief study is the identification of the key role played by the constant k_1 - the constant which fixes the speed at which the homeless are rehoused in permanent council property. Reducing this constant, i.e. making the system "fairer" with less priority to accommodating homeless families, appears to have little effect on the sizes of other categories on the waiting list but there is a marked increase in the number of households in temporary accommodation. The model, indicated by the size of its longest time-scale, should be modified to allow for births etc. It could be varied by allowing people to remove themselves from the register or by allowing the rates at which registered and unregistered people become homeless to differ, but these modifications are unlikely to substantially change the main result. The inclusion of movement from the homeless to the general population would have the effect of limiting the numbers in temporary accommodation. However, it is thought this effect is very small so a great reduction in k_1 would be needed for this flow to become significant