Exact solutions for rotating vortex arrays with finite-area cores

Published versio

Compressible bubbles in Stokes flow

Published versio

Stress fields around two pores in an elastic body: exact quadrature domain solutions

Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores

Stuart vortices on a sphere

Published versio

Analytical formulas for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci

This paper reports new analytical formulas for the longitudinal slip lengths for simple shear over a superhydrophobic surface, or bubble mattress, comprising a periodic array of unidirectional circular menisci, or bubbles, protruding into, or out of, the fluid. The accuracy of the formulas is tested against results from full numerical simulations; they are found to give small relative errors even at large no-shear fractions. In the dilute limit the formulas reduce to an earlier result by the author [Phys. Fluids, 22, 121703, (2011)]. They also extend analytical results of Sbragaglia & Prosperetti [Phys Fluids, 19, 043603, (2007)] beyond a small protrusion angle limit

An elliptical-pore model for late-stage planar viscous sintering

Published versio

The Schottky-Klein prime function: a theoretical and computational tool for applications

This article surveys the important role, in a variety of applied mathematical contexts, played by the so-called Schottky–Klein (S–K) prime function. While it is a classical special function, introduced by 19th century investigators, its theoretical significance for applications has only been realized in the last decade or so, especially with respect to solving problems defined in multiply connected, or ‘holey’, domains. It is shown here that, in terms of it, many well-known results pertaining only to the simply connected case (no holes) can be generalized, in a natural way, to the multiply connected case, thereby contextualizing those well-known results within a more general framework of much broader applicability. Given the wide-ranging usefulness of the S–K prime function it is important to be able to compute it efficiently. Here we introduce both a new theoretical formulation for its computation, as well as two distinct numerical methods to implement the construction. The combination of these theoretical and computational developments renders the S–K prime function a powerful new tool in applied mathematics