A. D. Fitt

Modeling the fabrication of hollow fibers: capillary drawing

A method for modeling the fabrication of small-scale hollow glass capillaries is developed. The model is based on an asymptotic analysis of the Navier-Stokes equations, which yields a simple closed-form solution for this problem. We demonstrate the validity of this approach using experimental data and use it to make predictions for a range of regimes of interest for the development of microstructured optical fiber technology.

The mathematical modelling of capillary drawing for holey fibre manufacture

Microstructured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years, and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for the particularly simple case of annular capillaries. A fluid-mechanics model is constructed using asymptotic analysis based on the small aspect ratio of the capillary. The leading-order equations are then examined in a number of asymptotic limits, many of which give valuable practical information about the control parameters that influence the drawing process. Finally, some comparisons with experiment are performed. For a limited set of experiments where the internal hole is pressurised, the theoretical predictions give qualitatively accurate results. For a much more detailed set of experiments carried out with a high-grade silica glass where no hole pressurisation is used, the relevant asymptotic solution to the governing equations is shown to give predictions that agree remarkably well with experiment.

Aerodynamics of slot-film cooling: theory and experiment

A simple model is proposed for the two-dimensional injection of irrotational inviscid fluid from a slot into a free stream. In a certain range of values of the ratio of free-stream to injection total heads, the film thickness satisfies a nonlinear integral equation whose solution enables the mass flow in the film to be found. Some experiments are described which both agree with this theory when it is relevant and indicate its deficiencies at other values of the total head ratio.

The unsteady motion of two-dimensional flags with bending stiffness

The motion of a two-dimensional flag at a time-dependent angle of incidence to an irrotational flow of an inviscid, incompressible fluid is examined. The flag is modelled as a thin, flexible, impermeable membrane of finite mass with bending stiffness. The flag is fixed at the leading edge where it is assumed to be either freely hinged or clamped with zero gradient. The angle of incidence to the outer flow is assumed to be small and thin aerofoil theory and simple beam theory are employed to obtain a partial singular integro-differential equation for the flag shape. Steady solutions to the problem are calculated analytically for various limiting cases and numerically for order one values of a non-dimensional parameter that measures the relative importance of outer flow momentum flux and flexural rigidity. For the unsteady problem, the stability of steady solutions depends only upon two non-dimensional parameters. Stability analysis is performed in order to identify the regions of instability. The resulting quadratic eigenvalue problem is solved numerically and the marginal stability curves for both the hinged and the clamped flags are constructed. These curves show that both stable and unstable solutions may exist for various values of the mass and flexural rigidity of the membrane and for both methods of attachment at the leading edge. In order to confirm the results of the linear stability analysis, the full unsteady flag equation is solved numerically using an explicit method. The numerical solutions agree with the predictions of the linear stability analysis and also identify the shapes that the flag adopts according to the magnitude of the flexural rigidity and mass.

Asymptotic analysis of the flow of shear-thinning foodstuffs in annular scraped heat exchangers

The problem of isothermal flow of a shear-thinning (pseudoplastic) fluid in the gap between two concentric cylinders is considered. A pump provides an axial pressure gradient which causes flow down the device. The outer cylinder is fixed and has ‘scrapers’ attached to it to cause flow mixing, whilst the inner cylinder rotates about its axis to provide shear and thus thin the fluid. The goal is to determine the optimal distribution of power between rotation and pumping. Although ostensibly the flow is nonlinear and three-dimensional we show that judicious use of fairly straightforward asymptotic methods can yield a great deal of information about the device, including cross-sectional flow predictions and throughput results. Furthermore, these results are derived for a variety of different flow conditions. Some numerical calculations are carried out using a commercial CFD code. These show good agreement with the asymptotic analysis.

Mathematical Modeling as an Accurate Predictive Tool in Capillary and Microstructured Fiber Manufacture: The Effects of Preform Rotation

A method for modeling the fabrication of capillary tubes is developed that includes the effects of preform rotation, and is used to reduce or remove polarization mode dispersion and fiber birefringence. The model is solved numerically, making use of extensive experimental investigations into furnace temperature profiles and silica glass viscosities, without the use of fitting parameters. Accurate predictions of the geometry of spun capillary tubes are made and compared directly with experimental results, showing remarkable agreement and demonstrating that the mathematical modeling of fiber drawing promises to be an accurate predictive tool for experimenters. Finally, a discussion of how this model impacts on the rotation of more general microstructured optical fiber preforms is given.

Models for high-Reynolds-number flow down a step

We consider inviscid, incompressible flow down a backward-facing step. Using thin-aerofoil theory, a model is proposed in which the separated region downstream of the back face of the step consists of a constant-pressure zone immediately behind the step, followed by a Prandtl-Batchelor constant-vorticity region

Mathematical Modeling of the Self-Pressurizing Mechanism for Microstructured Fiber Drawing

A method is proposed for modeling the self-pressurization of optical fibers that are sealed before drawing. The model is solved numerically and the results compared with experimental results. An explanation of the mechanism is presented and a numerical investigation is undertaken to optimize the choice of experimental parameters to minimize the transient effects of sealed preform drawing.

Mathematical modeling: case studies from industry

There is a growing need in todays world to support the study of industrial problems with an appropriate analysis anchored in the underlying basic physical principles that are involved and their description in mathematical terms. With the increasing availability of ever-improving computational facilities, it is all the more important to ensure that a balanced understanding of the problem is based on sound physical principles that concentrate on effects of leading importance and to ensure that secondary effects are appropriately considered. This process requires a proper consideration of mathematical techniques that often concentrate on partial differential equations to formulate the equations of motion and to achieve their solution. This involves an appreciation of a wide range of techniques, both analytical as well as numerical, ranging from underlying symmetries to asymptotic techniques. The purpose of this book is to address a selection of a wide-ranging set of typical industrial problems highlighting just these very features in a balanced way. The book achieves these objectives in a clear way, ensuring a high quality of presentation as well as content. The wide-ranging scope of the problems considered does not sacrifice the depth to which they are treated. In summary, it achieves its objectives in a very useful, clear and satisfactory way and constitutes a very good example of how practical problems of industry can be helped by mathematical modelling. The material is very well laid out in a clear and accessible way with maximum simplicity in mind. Appropriate diagrams and a very suitable set of references to support further study of the problem are also included. The book meets the high standards of production characteristic of Cambridge University Press. Its readership should comprise third-year undergraduate students as well as postgraduate MSc and PhD students, extending to include all those in industry whose interests lie in the province of practical mathematical modelling with applications in mind. The book is available in hardback as well as in economical paperback copies that could be of particular benefit to students. Phiroze Kapadia

The numerical and analytical solution of ill-posed systems of conservation laws

Abstract Recently, much attention has been given to the study of mixed systems of conservation laws, in which evolutionary systems of partial differential equations have the property that some of their eigenvalues are complex. This has led to some confusion, particularly in the field of two-phase flow, in which the correct form of the governing equations for different flow regimes is not clear. In this study we consider two mixed systems, one being a 2 x 2 system in which the analytic solution is known if certain special waves are defined and the other a prototype system of equations for modelling single-pressure two-phase flow. By using these examples it is shown both analytically and by numerical experiment that solving such sets of equations is far from an easy matter. The results have implications for the modelling of two-phase flows and other mixed systems, suggesting that although in some cases it might be possible to calculate solutions successfully, great care is generally needed in interpreting numerical results. This emphasizes the continuing requirement for more detailed mathematical modelling of two-phase flows.