S. J. Chapman

Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models.

An integrative computational model for intestinal tissue renewal.

Objectives:  The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation.

On the Theory of Complex Rays

The article surveys the application of complex-ray theory to the scalar Helmholtz equation in two dimensions. The first objective is to motivate a framework within which complex rays may be used to make predictions about wavefields in a wide variety of geometrical configurations. A crucial ingredient in this framework is the role played by Sp{} in determining the regions of existence of complex rays. The identification of the Stokes surfaces emerges as a key step in the approximation procedure, and this leads to the consideration of the many characterizations of Stokes surfaces, including the adaptation and application of recent developments in exponential asymptotics to the complex Wentzel--Kramers--Brilbuin expansion of these wavefields. Examples are given for several cases of physical importance.

Predicting Tumor Location by Modeling the Deformation of the Breast

Breast cancer is one of the biggest killers in the western world, and early diagnosis is essential for improved prognosis. The shape of the breast varies hugely between the scenarios of magnetic resonance (MR) imaging (patient lies prone, breast hanging down under gravity), X-ray mammography (breast strongly compressed) and ultrasound or biopsy/surgery (patient lies supine), rendering image fusion an extremely difficult task. This paper is concerned with the use of the finite-element method and nonlinear elasticity to build a 3-D, patient-specific, anatomically accurate model of the breast. The model is constructed from MR images and can be deformed to simulate breast shape and predict tumor location during mammography or biopsy/surgery. Two extensions of the standard elasticity problem need to be solved: an inverse elasticity problem (arising from the fact that only a deformed, stressed, state is known initially), and the contact problem of modeling compression. The model is used for craniocaudal mediolateral oblique mammographic image matching, and a number of numerical experiments are performed.

Nucleation of superconductivity in decreasing fields. II

The bifurcation from a normally conducting to a superconducting state as an external magnetic field is lowered is examined using the Ginzburg-Landau theory. The results for three specific examples are reviewed, extended and unified in the framework of a systematic perturbation theory introduced in [1].

A Hierarchy of Models for Type-II Superconductors

A hierarchy of models for type-II superconductors is presented. Through appropriate asymptotic limits we pass from the mesoscopic Ginzburg--Landau model to the London model with isolated superconducting vortices as line singularities, to vortex-density models, and finally to macroscopic critical-state models.

A model for variable thickness superconducting thin films

A model for superconductivity in thin films having variable thickness is derived through an averaging process across the film. When the film is of uniform thickness the model is identical to a model for superconducting cylinders as the Ginzburg-Landau parameter tends to ∞. This means that all superconducting materials, whether type I or type II in bulk, behave as type-II superconductors when made into sufficiently thin films. When the film is of non-uniform thickness the variations in thickness appear as spatially varying coefficients in the thin-film differential equations. After providing a formal derivation of the model, some results about solutions of the variable thickness model are given. In particular, it is shown that solutions obtained from the new model are an appropriate limit of a sequence of averages of solutions of the three-dimensional Ginzburg-Landau model as the thickness of the film tends to zero. An application of the variable thickness thin film model to flux pinning is then provided. In particular, the results of a numerical calculation are given that show that the vortex-like structures present in superconductors are attracted to relatively thin regions.

Vortex pinning by inhomogeneities in type-II superconductors

The methods of formal matched asymptotics are used to examine the motion of a curvilinear vortex in an inhomogeneous type-II superconducting material in the limit as the vortex core radius tends to zero. The resulting law of motion indicates that the logarithm of the equilibrium density of the superconducting electrons acts as a pinning potential for the vortex, so that vortices will be attracted by impurities in the superconducting material.

Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions.

On the rôle of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension

Small surface tension is known to select a discrete family from the continuum of Saffman–Taylor finger solutions with zero surface tension. Here a method developed recently in Chapman et al . [2] is employed in which a naive perturbation expansion in powers of the surface tension parameter is optimally truncated to show exponentially small terms being switched on across Stokes lines. These terms are responsible for finger selection.