The various problems tackled during the week are described below.
British Gas This problem concerned the flow of a gas jet that was directed downwards, through a crossflow, towards a planar surface; it is an idealised model for a gas leak from a ruptured pipe. One of the most noticeable features of the flow, from the experimental results provided by British Gas, is the presence of `horseshoe' vortices which curve around the front of the descending jet and travel downstream, entraining air. A flow dominated by these vortices, which originate from a point source where the jet strikes the wall, was proposed and from the conditions that the position of the vortices remain fixed and that the core radius of the line vortex remained small, a nonlinear differential equation was derived for the position of the horseshoe vortex system. The extent of the gas jet upstream was estimated using a simple conservation argument, with the assumption that the mass of air entrained was proportional to the difference between the velocity of the jet and the external flow. This argument provided estimates of the extent of the flow which were in reasonably good agreement with experiment. An independent two-dimensional analysis of the flow, using dimensional arguments, suggested a parabolic shape for the gas jet as it spreads out along the wall.
Domino Printing The objective was to understand the mechanism by which a solid plug forms in the end of an ink jet nozzle and so help in formulating a strategy to prevent this. It had been noted that while such a plug forms rapidly in the confines of a small nozzle the surface of the same ink in a beaker only produced a crust near the container wall.
The principal cause for solidification was thought to be the reduction in solvent concentration due to evaporation, resulting in higher polymer concentration. Other related effects considered were (i) phase separation - similar three-phase (e.g. solvent, polymer, pigment) mixtures are known to exhibit this phenomena - together with nucleation of the solid phase at the ink/container boundary; (ii) higher concentrations of polymer will occur near a contact line if the contact angle is acute because of the difficulty of replenishing such regions with solvent; (iii) effective lack of mechanical strength of any crust on a large surface can allow it to break up; (iv) evaporation causes cooling, thermal contraction, and, for a large expanse of ink (with air above), e.g. in a beaker, natural convection and resultant mixing - thereby reducing the localised polymer concentration.
It was apparent that polymer build-up near the free surface of the ink would be reduced by mixing, for example using forced convection.
A second problem brought by Domino involved designing a valve by moulding an aperture in a sheet of rubber and closing it by compression. To avoid wear due to excessive stresses sharp edges were to be avoided in the aperture. Also, the valve should not buckle and would close in from the sides in order to prevent breakup of an ink droplet flowing through the valve.
Contact problem theory indicates that a thin line crack with the initial shape of an ellipse will close instantaneously along its length on the application of a large enough pressure. It would be preferable to zip the aperture in from the sides, hence an initial shape of superimposed ellipses was considered. A change in pressure causes the crack to close from the sides inwards in a continuous fashion as each ellipse snaps shut. This shape however would have cusps at the edges and hence would be difficult to mould. However by superimposing an outer ellipse on the obtained shape, a shape which is possible to mould is formed. On the application of a pressure this outer ellipse can be shut to obtain the cusped edge shape and then the valve can be operated between that pressure and the pressure needed for closure.
Elkem This problem concerned the process of calcining anthracite. In particular Elkem wished to understand the stability of the calcining process and the temperature variation caused by uneven flow at the bottom of the calciner.
A number of different aspects of this problem were considered. Analysis of the thermal boundary layer near the top electrode shed light on the heated wake and showed where the electrode preheats the anthracite encouraging the calcining process. A similarity solution modelled the reaction plume and a simple ODE system was developed allowing the stability to be considered.
MAFF The problem brought by MAFF was to devise a method to predict the freezing times for foods in an industrial freezer to within 10% accuracy whilst still being simple enough to use on a PC.
Many foods such as pizzas or hamburgers have one length scale sufficiently less than the others to allow them to be treated as one-dimensional. The Study Group derived conditions to be satisfied by the food and the freezer for this to be justified. The transfer of heat from the surface of the food was modelled by Newton cooling. Numerical experiments showed that the constant of proportionality for the cooling law is a function of the conditions in the freezer and does not vary significantly for different foods.
The 1-dimensional model can simulate the layered structure of foods by allowing the physical parameters (thermal conductivity, specific heat and density) to vary with the spatial coordinate. It is believed that modelling the distinct layers will be significantly more accurate than the current practice of simply averaging the physical parameters and simulating a homogeneous food.
Pilkingtons An issue raised during the OCIAM presentation on problems in glass flow concerned bubbles bursting at a free surface. Pilkingtons find that most bubbles produced in molten glass will float to the surface and quickly burst through, yet a small number of bubbles can sit underneath the surface for up to an hour.
Gravity acts to drain the film separating the bubble from the air. The mechanism proposed for stabilising this film is surface tension variation which acts to pull fluid in the opposite direction to gravity. Estimates for the time scales showed that surface tension effects can increase the draining time from a few minutes to a few hours.
Riskcare Riskcare wished to price American-style warrants. These are call options which can be exercised at any time to create new shares. Of particular interest was the situation where the number of shares and warrants issued is a significant proportion of the original number of shares. This has been observed to lead to ``dilution effects'', because the company value does not increase whilst the number of shares does.
First the pricing of American warrants in a classical setting was considered. This showed that given optimal exercise of the warrants no dilution of the share price should occur. The dilution effect observed in practice must therefore be caused by other factors. One possibility is that the price drop is due to large sales of the shares obtained on exercise of the warrants. A pricing equation was also developed for the warrants which would require numerical solution.
Schlumberger Schlumberger Cambridge Research presented the problem of redesigning the valve in a positive displacement pump in order to reduce the wear on its urethane seal. The erosion, or ``pitting'', of the urethane is due to the presence of small proppant particles which constitute about 35 % by volume of the slurry that is being pumped. Consideration of the dimensionless parameters in the problem lead to the conclusion that it is not possible to redesign the valve in such a way that particles are expelled preferentially over the fluid; they are certain to be crapped and trussed as the valve closes (A.D. Fitt, (not very) private communication). However, the problem of pitting could be reduced by redesigning the shape of the seal in order to minimise the stresses imposed on the urethane by trapped particles; several alternative designs were suggested.
UBS UBS were interested in calculating risk/reward profiles for a portfolio problem. Two aspects were discussed: firstly the location of the optimum reward portfolio (or portfolios) for a given level of risk, secondly some issues to do with three-dimensional graphical representations of the risk-reward diagram as sample portfolio parameters varied.
It was a thoroughly enjoyable and well organised week. Thanks to all those people in Cambridge who took the trouble to arrange the meeting and make it a success. The next study group will be held in Oxford and supported by the Smith Institute.
International Conference on Mathematical Modelling (physical, biological engineering and social systems) University Brunei Darassalam, May 29 - June 1, 1995, contact: Organizing Secretary, ICMM 1995, Dept. of Maths, University Brunei Darussalam, Gadong 3186, Negara Brunei Darassalam or contact Prof. G. Wake at G.Wake@massey.ac.nz.
3rd International Applied Statistics in Industry Conference Dallas, Texas, 5-7 June 1995. Contact email@example.com.
Workshop on Mathematical Problems in Industry in association with the RPI study group, University of New Mexico, June 12-16 1995. Contact firstname.lastname@example.org.
International Conference on Industrial and Applied Mathematics Hamburg, July 3-7 1995, contact: GAMM-Office, University Regensburg, NWF I-Mathematik, D-93053 Regensburg, Germay. Fax +49 941 943 4005, e-mail email@example.com.
ECMI Modelling Week University of Strathclyde, Sept. 3-10, 1995. A modelling week for postgraduate students.
Applied Mathematics and Industrial Problems Bucharest, contact: Dr. D. Tiba/Prof. A. Georgescu, firstname.lastname@example.org.
European Coating Symposium Leeds University, Sept. 19 - 22, 1995 contact: The Secretary, European Coating Symposium 1995, Dept. of Mechanical Engineering, Leeds University, Leeds LS2 9JT or Dr J.L. Summers at email@example.com.
ANZIAM '96 The 32nd Australian and New Zealand Applied Mathematics Conference, Masterton, New Zealand, Feb. 4-8 1996, including a half-day mini-symposium on mathematics in agriculture. Contact: Prof. G. Wake, fax (64) 6 3505611 or G.Wake@massey.ac.uk.
Dr G. Sander from Griffith university is here until the end of June. Professors Babich and Kissilev from St. Petersburg will be here May 1 - 14. They are experts on ray theory; Prof. Babich will be talking about his work on May 4th. Professor Xu from McGill University will be making a welcome return for a week at the end of June. Prof. Z.S. Olesiak from Warsaw is visiting in early June.
Density wave oscillations in two-phase flow. C. Aldridge, supervised by Dr. A.C. Fowler.
Two phase flow of water and steam in a liquid metal fast breeder reactor pipe. S. Kane, supervised by Dr. A.D. Fitt.
If you have any suggestions or contributions for the next newsletter please contact Dr Tim Myers, OCIAM, Maths Institute, 24-29 St. Giles', Oxford, OX1 3LB, firstname.lastname@example.org.