J. Billingham

The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves

We study the isothermal autocatalytic system , A + nB → (n + 1)B , where n = 1 or 2 for quadratic or cubic autocatalysis respectively. In addition, we allow the chemical species, A and B, to have unequal diffusion rates. The propagating reaction-diffusion waves that may develop from a local initial input of the autocatalyst, B, are considered in one spatial dimension. We find that travelling wave solutions exist for all propagation speeds v ≥ v*n,where v*n is a function of the ratio of the diffusion rates of the species A and B and represents the minimum propagation speed. It is also shown that the concentration of the autocatalyst, B, decays exponentially ahead of the wavefront for quadratic autocatalysis. However, for cubic autocatalysis, although the concentration of the autocatalyst decays exponentially ahead of the wavefront for travelling waves which propagate at speed v = v*2, this rate of decay is only algebraic for faster travelling waves with v > v*2. This difference in decay rate has implications for the selection of the long time wave speed when such travelling waves are generated from an initial-value problem.

A note on the properties of a family of travelling-wave solutions arising in cubic autocatalysis

The propagating reaction–diffusion waves that may develop in the isothermal, autocatalytic system A+2B→3B, from a local initial input of the autocatalyst B, are considered. We find that travelling-wave solutions exist for all propagation speeds . It is shown that the travelling-wave solution with minimum propagation speed , exhibits exponential decay in the concentration of the autocatalyst B ahead of the wavefront, whilst, for faster travelling waves, with , the autocatalyst concentration decays only algebraically. This difference in behaviour between the minimum-speed and faster-speed waves has implications concerning the selection of the long-time wave speed when such travelling waves are generated from an initial-value problem

Dynamics of a strongly nonlocal reaction-diffusion population model

We study the development of travelling waves in a population that competes with itself for resources in a spatially nonlocal manner. We model this situation as an initial value problem for the integro-differential reaction?diffusion equation with g an even function that satisfies g(y) ? 0 as y ? ? ?, , ? > 0, 0 0. We concentrate on the limit of highly nonlocal interactions, ? 1, focusing on the particular case g(y) = ? e?|y|, which is equivalent to the reaction?diffusion system Using numerical and asymptotic methods, we show that in different, well-defined regions of parameter space, steady travelling waves, unsteady travelling waves and periodic travelling waves develop from localized initial conditions. A key feature of the system for ? 1 is the local existence of travelling wave solutions that propagate with speed c < 2, and which, although they cannot exist globally, attract the solution of the initial value problem for an asymptotically long time. By using a Cole?Hopf transformation, we derive a first order hyperbolic equation for the gradient of log u ahead of the wavefront, where u is exponentially small. An analysis of this equation in terms of its characteristics, allowing for the formation of shocks where necessary, explains the dynamics of each of the different types of travelling wave. Moreover, we are able to show that the techniques that we develop for this particular case can be used for more general kernels g(y) and that we expect the same range of different types of travelling wave to be solutions of the initial value problem for appropriate parameter values. As another example, we briefly consider the case , for which the system cannot be simplified to a pair of partial differential equations.

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.

Laminar, unidirectional flow of a thixotropic fluid in a circular pipe

Abstract In this paper we study the unidirectional, axisymmetric flow of a bentonite mud in a circular pipe. Bentonite mud is an inelastic, thixotropic, generalised-Newtonian fluid. We use a rheological model that characterises this behaviour in terms of a single parameter Λ which is a measure of the amount of structure in the fluid. The behaviour of Λ is determined by a single rate equation which models the tendency of fluid structure to increase whilst being limited by the imposed shear rate. We find that, for certain parameter ranges, the model is not structurally stable, but that this problem can be eliminated by including diffusion of fluid structure. A graph of the equilibrium shear stress for a given shear rate (the rheogram) is not monotonic, yet no mechanical instability occurs in pipe flow. We contrast this with recent work on the pipe flow of a Johnson-Segalman-Oldroyd fluid which displays spurting and oscillatory behaviour. The difference lies in the relative magnitude of normal stress effects in the two fluids. There appear to be no grounds for discarding the constitutive model studied here simply because of the non-monotonicity of the equilibrium rheogram.

Three-dimensional flow due to a microcantilever oscillating near a wall: an unsteady slender-body analysis

We compute the drag on a slender rigid cylinder, of uniform circular cross-section, oscillating in a viscous fluid at small amplitude near a horizontal wall. The cylinders axis lies at an angle α to the horizontal and the cylinder oscillates in a vertical plane normal to either the wall or its own axis. The flow is described using an unsteady slender-body approximation, which we treat both numerically and using an iterative scheme that extends resistive-force theory to account for the leading-order effects of unsteady inertia and the wall. When α is small, two independent screening mechanisms are identified which suppress end-effects and produce approximately two-dimensional flow along the majority of the cylinder; however, three-dimensional effects influence the drag at larger tilt angles.

The Development of Travelling Waves in Quadratic and Cubic Autocatalysis with Unequal Diffusion Rates. II. An Initial-Value Problem with an Immobilized or Nearly Immobilized Autocatalyst

We study the isothermal autocatalytic reaction schemes, A + B -> 2B (quadratic autocatalysis), and A + 2B → 3B (cubic autocatalysis), where A is a reactant and B is an autocatalyst. We consider the situation when a quantity of B is introduced locally into a uniform expanse of A, in one-dimensional slab geometry. In addition, we allow the chemical species A and B to have unequal diffusion rates DA and DB respectively, and study the two closely related cases, (DB/ DA) = 0 and 0 < (DB/ DA) < 1. When (Db/Da) = 0 a spike forms in the concentration of B, which grows indefinitely, and we can obtain both large and small time asymptotic solutions. For 0 < (DB/ DA) < 1 there is a long induction period during which a large spike forms in the concentration of B, before a minimum speed travelling wave is generated. We can relate the results for case (DB/DA) = 0 to the solution when 0 < (DB/ DA) < 1 to obtain detailed information about its behaviour.

Simple chemical clock reactions: application to cement hydration

In the first part of this paper, we review some recent mathematical work which shows how the duration of induction periods for simple non-linear chemical kinetic models can be related directly to initial reactant concentrations and reaction rate constants in these schemes. These results should be of interest to both theorists and experimentalists.Chemical clock behaviour has obvious parallels with the induction period observed during the hydration and setting of cements: a slurry remains liquid for a characteristic time before rapidly hardening into a porous solid. Cement chemistry is not completely understood, but our approach concentrates on what we believe to be the dominant kinetic steps which control the overall setting process. Specifically, the rate-determining process is taken to be an inhibited autocatalytic nucleation mechanism for which we propose a connection with one of our model clock reaction schemes. The model is in at least qualitative agreement with experimental observations.

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.