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Explicit formulae for the Kirchhoff–Routh path functions (or Hamiltonians) governing the motion of N -point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero. The method uses the Schottky–Klein prime function to find representations of the hydrodynamic Green's function in multiply connected circular domains. The Green's function is then used to construct the associated Kirchhoff–Routh path function. The path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains. Illustrative examples are presented for the case of single vortex motion in multiply connected domains.

A formula for the generalized Schwarz–Christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived. The theory of classical Schottky groups is employed. The formula for the derivative of the mapping function contains a product of powers of Schottky–Klein prime functions associated with a Schottky group relevant to the circular pre-image domain. The formula generalizes, in a natural way, the known mapping formulae for simply and doubly connected polygonal domains.

A class of exact solutions to the steady Euler equations representing finite area patches of nonuniform vorticity is presented. It is demonstrated that the solutions constitute a special class of steady multipolar vortical structures and have many qualitative similarities with the multipolar equilibria observed in two-dimensional flows at high Reynolds numbers. The results provide insights into the mathematical structure of the two-dimensional Euler equation that, it is argued, underlies the occurrence of such multipolar coherent structures in real physical flows. Moreover, the new solutions possess the interesting feature of being completely ''invisible'' in that their presence cannot be detected anywhere outside the support of the vorticity.

A general mathematical framework is presented for modelling the pulling of optical glass fibres in a draw tower. The only modelling assumption is that the fibres are slender; cross-sections along the fibre can have general shape, including the possibility of multiple holes or channels. A key result is to demonstrate how a so-called reduced time variable τ serves as a natural parameter in describing how an axial-stretching problem interacts with the evolution of a general surface-tension-driven transverse flow via a single important function of τ , herein denoted by H(τ ), derived from the total rescaled cross-plane perimeter. For any given preform geometry, this function H(τ ) may be used to calculate the tension required to produce a given fibre geometry, assuming only that the surface tension is known. Of principal practical interest in applications is the 'inverse problem' of determining the initial cross-sectional geometry, and experimental draw parameters, necessary to draw a desired final cross-section. Two case studies involving annular tubes are presented in detail: one involves a cross-section comprising an annular concatenation of sintering near-circular discs, the cross-section of the other is a concentric annulus. These two examples allow us to exemplify and explore two features of the general inverse problem. One is the question of the uniqueness of solutions for a given set of experimental parameters, the other concerns the inherent ill-posedness of the inverse problem. Based on these examples we also give an experimental validation of the general model and discuss some experimental matters, such as buckling and stability. The ramifications for modelling the drawing of fibres with more complicated geometries, and multiple channels, are discussed.

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An analytical formula for the frictional slip length associated with transverse shear flow over a bubble mattress comprising a dilute periodic array of parallel circular-arc grooves protruding into the fluid has recently been presented by Davis and Lauga [Phys. Fluids 21, 011701 (2009)]. This letter derives an analytical formula for the slip length associated with longitudinal shear flow over the same surface. The formula is in excellent agreement with a phenomenological result based on finite element simulations given by Teo and Khoo [Microfluid. Nanofluid. 9, 499 (2010)].