The Pennyland Project

This is the final report on the performance of 177 low-energy houses at Pennyland, Milton Keynes, monitored by the Open University Energy Research Group (ERG), for the Milton Keynes Development Corporation (MKDC), under contract to the Energy Technology Support Unit (ETSU) at Harwell.[The smaller file contains the Executive Summary; the larger file contains the main report, excluding the Executive Summary]

Stochastic modelling of reaction-diffusion processes:\ud algorithms for bimolecular reactions

Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A+B → C, or A+A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site)

Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations

Time scale of random sequential adsorption

A simple multiscale approach to the diffusion-driven adsorption from a solution to a solid surface is presented. The model combines two important features of the adsorption process: (i) the kinetics of the chemical reaction between adsorbing molecules and the surface; and (ii) geometrical constraints on the surface made by molecules which are already adsorbed. The process (i) is modelled in a diffusion-driven context, i.e. the conditional probability of adsorbing a molecule provided that the molecule hits the surface is related to the macroscopic surface reaction rate. The geometrical constraint (ii) is modelled using random sequential adsorption (RSA), which is the sequential addition of molecules at random positions on a surface; one attempt to attach a molecule is made per one RSA simulation time step. By coupling RSA with the diffusion of molecules in the solution above the surface the RSA simulation time step is related to the real physical time. The method is illustrated on a model of chemisorption of reactive polymers to a virus surface

A hierarchy of models for superconducting thin films

A hierarchy of models for type-II superconducting thin films is presented. Through appropriate asymptotic limits this hierarchy passes from the mesoscopic Ginzburg--Landau model to the London model with isolated vortices as $\delta$-function singularities to vortex-density models and finally to macroscopic critical-state models. At each stage it is found that a key nondimensional parameter is $\Lambda = \lambda^2/d L$, where $\lambda$ is the penetration depth of the magnetic field, a material parameter, and d and L are a typical thickness and lateral dimension of the film,respectively. The models simplify greatly if this parameter is large or small

Continuum and discrete models of dislocation pile-ups. I Pile-up at a lock

A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip pane under the action of an external force in the direction of a locked dislocation in that plane is considered. As $n \rightarrow \infty$ there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away

Asymptotic analysis of a system of algebraic equations arising in dislocation theory

The system of algebraic equations given by\ud \ud $\sum_{j=0, j \neq i}^n sgn(x_i - x_j) / |x_i - x_j|^a = 1, i = 1, 2, \ldots n, x_0 = 0,$\ud \ud appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole.\ud \ud We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n -> ∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment but, up to corrections of logarithmic order, it also leads to a differential equation.\ud \ud The continuum approximation is only valid for i not too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem

Ray theory for high-Péclet-number convection-diffusion

Asymptotic methods based on those of geometrical optics are applied to some steady convection-diffusion streamed flows at a high Péclet number. Even with the assumption of inviscid, irrotational flow past a body with uniform ambient conditions, the rays from which the solution is constructed can only be found after local analyses have been carried out near the stagnation points. In simple cases, the temperature away from the body is the sum of contributions from each stagnation point