A. C. King

The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory

We examine a spiralling slender inviscid liquid jet which emerges from a rapidly rotating orifice. The trajectory of this jet is determined using asymptotic methods, and the stability using a multiple scales approach. It is found that the trajectory of the jet becomes more tightly coiled as the Weber number is decreased. Unstable travelling wave modes are found to grow along the jet. The breakup length of the jet is calculated, showing good agreement with experiments.

Free jets spun from a prilling tower

A mathematical model of the dynamics of an inviscid liquid jet, subjected to both gravity and surface tension, which emerges from rotating drum is derived and analysed using asymptotic and computational methods. The trajectory and linear stability of this jet is determined. By use of the stability results, the break up length of the jet is calculated. Such jets arise in the manufacture of pellets (for example, of fertilizer or magnesium) using the prilling process. Here the drum would contain many thousands of holes, and the molten liquid would be pumped into the rotating drum. After the jet has broken up into droplets, these droplets solidify to form pellets. The jets in this prilling process are curved in space by both gravity and surface tension.

Differential Equations: Linear, Nonlinear, Ordinary, Partial

Preface Part I. Linear Equations: 1. Variable coefficient, second order, linear, ordinary differential equations 2. Legendre functions 3. Bessel functions 4. Boundary value problems, Greens functions and Sturm-Liouville theory 5. Fourier series and the Fourier transform 6. Laplace transforms 7. Classification, properties and complex variable methods for second order partial differential equations Part II. Nonlinear Equations and Advanced Techniques: 8. Existence, uniqueness, continuity and comparison of solutions of ordinary differential equations 9. Nonlinear ordinary differential equations: phase plane methods 10. Group theoretical methods 11. Asymptotic methods: basic ideas 12. Asymptotic methods: differential equations 13. Stability, instability and bifurcations 14. Time-optimal control in the phase plane 15. An introduction to chaotic systems Appendix 1. Linear algebra Appendix 2. Continuity and differentiability Appendix 3. Power series Appendix 4. Sequences of functions Appendix 5. Ordinary differential equations Appendix 6. Complex variables Appendix 7. A short introduction to MATLAB Bibliography Index.

The Trajectory and Stability of a Spiralling Liquid Jet

We examine a spiralling slender inviscid liquid jet with a curved centre line which emerges from a rotating orifice. The trajectory of this jet is determined using asymptotic methods, and the stability using a multiple scales approach. Some novel results will be presented and discussed.

The interaction of a moving fluid/fluid interface with a flat plate

A well-known technique for metering a multiphase flow is to use small probes that utilize some measurement principle to detect the presence of different phases surrounding their tips. In almost all cases of relevance to the oil industry, the flow around such local probes is inviscid and driven by surface tension, with negligible gravitational effects. In order to study the features of the flow around a local probe when it meets a droplet, we analyse a model problem: the interaction of an infinite, initially straight, interface between two inviscid fluids, advected in an initially uniform flow towards a semi-infinite thin flat plate oriented at 90° to the interface. This has enabled us to gain some insight into the factors that control the motion of a contact line over a solid surface, for a range of physical parameter values. The potential flows in the two fluids are coupled nonlinearly at the interface, where surface tension is balanced by a pressure difference. In addition, a dynamic contact angle boundary condition is imposed at the three-phase contact line, which moves along the plate. In order to determine how the interface deforms in such a flow, we consider the small- and large-time asymptotic limits of the solution. The small-time and linearized large-time problems are solved analytically, using Mellin transforms, whilst the general large-time problem is solved numerically, using a boundary integral method. The form of the dynamic contact angle as a function of contact line velocity is the most important factor in determining how an interface deforms as it meets and moves over the plate. Depending on this, the three-phase contact line may, at one extreme, hang up on the leading edge of the plate or, at the other extreme, move rapidly along the surface of the plate. At large times, the solution asymptotes to an interface configuration where the contact line moves at the far-field velocity.

The initial development of a jet caused by fluid, body and free-surface interaction. Part 2. An impulsively moved plate

The free surface deformation and flow field caused by the impulsive horizontal motion of a rigid vertical plate into a horizontal strip of inviscid, incompressible fluid, initially at rest, is studied in the small time limit using the method of matched asymptotic expansions. It is found that three different asymptotic regions are necessary to describe the flow. There is a main, O(1) sized, outer region in which the flow is singular at the point where the free surface meets the plate. This leads to an inner region, centered on the point where the free surface initially meets the plate, with size of O(it log t). To resolve the singularities that arise in this inner region, it is necessary to analyse further the flow in an inner-inner region, with size of O(t), again centered upon the wetting point of the nascent rising jet. The solutions of the boundary value problems in the two largest regions are obtained analytically. The solution of the parameter-free nonlinear boundary value problem that arises in the inner-inner region is obtained numerically.

On the initial stages of cement hydration

After the initial mixing of cement, an induction period occurs during which its consistency remains constant. Thickening occurs at the end of this period when the consistency is observed to increase very rapidly. In this paper we propose a reaction-diffusion model for the hydration of tricalcium silicate, a principal constituent of cement, which is believed to be responsible for the initial development of its strength. Our model is based on the assumption that the hydration of cement can be described as a dissolution -precipitation reaction. The mathematical solutions enable us to determine some of the factors that control the length of the induction period and make predictions of the ionic concentrations which are in agreement with experimental data.

Surface-tension-driven flow outside a slender wedge with an application to the inviscid coalescence of drops

We consider the two-dimensional inviscid flow that occurs when a fluid, initially at rest around a slender wedge-shaped void, is allowed to recoil under the action of surface tension. As noted by Keller & Miksis, a similarity scaling is available, with lengths scaling like t 2/3 . We find that an asymptotic balance is possible when the wedge semi-angle, a, is small, in an inner region of O(α 4/9 ), a distance of O(α -2/9 ) from the origin, which leads to a simpler boundary value problem at leading order. Although we are able to reformulate the inner problem in terms of a complex potential and reduce it to a single nonlinear integral equation, we are unable to find a solution numerically. This is because, as noted by Vanden-Broeck, Keller & Milewski, and reproduced here numerically using a boundary integral method, the free surface is self-intersecting for α < α 0 2.87°. Since disconnected solutions are only possible when there is a void inside the initial wedge, we consider the effect of an inviscid low-density fluid inside the wedge. In this case, a solution is available for a slightly smaller range of wedge semi-angles, since the flow of the interior fluid sucks the free surfaces together, with the exterior flow seeing pinch-off at a finite angle

Analysis of a model for a loaded, planar, solid oxide fuel cell

We analyze a model for the combustion of methane in a planar solid oxide fuel cell. The model includes diffusive and advective transport processes, an electrochemical source of oxygen, and the consumption of oxygen and methane through combustion. The effect of the presence of the reaction products and atmospheric nitrogen is also included. Since the combustion takes place in a narrow gap we are able to reduce the problem from three to two dimensions. After assuming that the flow is steady and axisymmetric, we use the method of matched asymptotic expansions to construct solutions in six asymptotic regions. This allows us to model the voltage--current characteristics for a given flowrate of fuel and predict the position of the flame region that separates an oxygen-rich region from a fuel-rich region. Comparisons show that the theory is in reasonable agreement with experiments.

Chemical clock reactions: The effect of precursor consumption

During a clock reaction an initial induction period is observed before a significant change in concentration of one of the chemical species occurs. In this study we develop the results of Billingham and Needham (1993) who studied a particular class of inhibited autocatalytic clock reactions. We obtain modified expressions for the length of the induction period and show that characteristic clock reaction behaviour is only observed within certain parameter limits.