D. J. Needham

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.

Mathematical modelling of chemical clock reactions

A clock reaction is a chemical reaction which gives rise to an initial induction period before a significant concentration of one of the chemical species involved is produced. We study four closely related isothermal model reactions schemes which can exhibit clock reaction behaviour in a well-stirred situation. These reaction schemes represent a combination of quadratic or cubic autocatalysis with linear or quadratic inhibition.

A fundamental model exhibiting nonlinear oscillatory dynamics in solid oxide fuel cells

In this paper, we address the phenomenon of temporal, self-sustained oscillations which have been observed under quite general conditions in solid oxide fuel cells. Our objective is to uncover the fundamental mechanisms giving rise to the observed oscillations. To this end, we develop a model based on the fundamental chemical kinetics and transfer processes which take place within the fuel cell. This leads to a three-dimensional dynamical system, which, under typical operating conditions, is rationally reducible to a planar dynamical system. The structural dynamics of the planar dynamical system are studied in detail. Self-sustained oscillations are shown to arise through Hopf bifurcations in this planar dynamical system, and the key parameter ranges for the occurrence of such oscillations are identified.

The evolution to localized and front solutions in a non-Lipschitz reaction–diffusion Cauchy problem with trivial initial data

In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an associated two-dimensional non-Lipschitz non-autonomous dynamical system, for which, we establish the existence of a two-parameter family of homoclinic connections on the origin, and a heteroclinic connection between two equilibrium points. Additionally, we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin.

On a L∞ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

In this paper, we consider a

The initial development of a jet caused by fluid, body and free surface interaction. part 3. an inclined accelerating plate

L^infty

The evolution of travelling wavefronts in a hyperbolic Fisher model. III. The initial-value problem when the initial data has exponential decay rates

functional derivative estimate for the first spatial derivative of bounded classical solutions

The evolution of travelling wave-fronts in a hyperbolic Fisher model. I. The travelling wave theory

u:mathbb{R}times [0,T]tomathbb{R}

Breakdown of the shallow water equations due to growth of the horizontal vorticity

to the Cauchy problem for scalar semi-linear parabolic partial differential equations with a continuous nonlinearity

The cauchy problem for non-lipschitz semi-linear parabolic partial differential equations

f:mathbb{R}tomathbb{R}