Darren Crowdy

Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains

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 Greens function in multiply connected circular domains. The Greens 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.

Conformal Mappings between Canonical Multiply Connected Domains

Explicit analytical formulae for the conformal mappings from the canonical class of multiply connected circular domains to canonical classes of multiply connected slit domains are constructed. All the formulae can be expressed in terms of the Schottky-Klein prime function associated with the multiply connected circular domains.

Computing the Schottky-Klein Prime Function on the Schottky Double of Planar Domains

A numerical algorithm is presented for the computation of the Schottky-Klein prime function on the Schottky double of multiply connected circular domains in the plane. While there exist classical formulae for the Schottky-Klein prime function in the form of infinite products over a Schottky group, such products are not convergent for all choices of multiply connected circular domains. The prime function itself, however, is a well-defined function for any multiply connected circular domain. The present algorithm facilitates the evaluation of this prime function when the planar domains are such that the classical infinite product representation is either not convergent or so slowly convergent as to be impracticable.

Exact solutions for rotating vortex arrays with finite-area cores

A class of explicit solutions of the two-dimensional Euler equations consisting of a nite-area patch of uniform vorticity surrounded by a nite distribution of corotating satellite line vortices is constructed. The results generalize the classic study of co-rotating vortex arrays by J. J. Thomson. For N satellite line vortices (N > 3) a continuous one-parameter family of rotating vortical equilibria is derived in which dierent values of the continuous parameter correspond to dierent shapes and areas of the central patch. In an appropriate limit, vortex patch equilibria with cusped boundaries are found. A study of the linear stability is performed and a wide range of the solutions found to be linearly stable. Contour dynamics methods are used to calculate the typical nonlinear evolution of the congurations. The results are believed to provide the only known exact solutions involving rotating vortex patches besides the classical Kirchho ellipse. Since the early work of Lord Kelvin (1878) who studied the case of three vortices arranged in a ring, vortex arrays (or vortex ‘lattices’ or ‘crystals’) have been a problem of perennial interest to fluid dynamicists. Thomson (1882) considered the more general situation in which N line vortices are arranged in a co-rotating conguration equispaced around the circumference of a circle for up to N = 7. This problem was later reappraised by Havelock (1931) who also considered larger values of N and corrected some erroneous conclusions made by Thomson regarding the stability of the congurations. Generalizations of this classical work include that of Morikawa & Swenson (1971) who, in an attempt to model geostrophic vortices in the atmosphere, placed an additional line vortex at the centre of the co-rotating polygonal array of satellites. The central line vortex was a simple model of the polar vortex whereas the satellites modelled vortex cores associated with atmospheric pressure systems in the hemisphere surrounding the pole. Further applications of this model to atmospheric pressure

The Schwarz-Christoffel mapping to bounded multiply connected polygonal 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.

The motion of a point vortex around multiple circular islands

A new constructive method for computing the motion of a single point vortex around an arbitrary finite number of circular islands in the special case when the circulations around all the islands are zero is presented. In this case, explicit representations for the governing Hamiltonians can be found and used to study the vortex trajectories. An example application is to geophysical flows and this study provides a simple model of the interaction of ocean eddies with topography. A wide range of illustrative examples are given, including the case of various multi-island configurations lying off an infinite coastline as well as in an unbounded ocean. The critical trajectories (or separatrices) dividing the flow domain into regions of qualitatively different dynamics of the vortices can be computed in a systematic and unified fashion irrespective of the number of islands present.

Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

Quadrature Domains and Fluid Dynamics

Few physical scientists interested in the mathematical description of fluid flows will know what a quadrature domain is; just as few mathematicians interested in quadrature domain theory would profess to know much about fluid dynamics. And yet, recent research has shown that a surprisingly large number of the by-now classic exact solutions of two-dimensional fluid dynamics can be understood within the context of quadrature domain theory. This article surveys a number of different physical applications of quadrature domain theory arising in the general field of fluid dynamics.

Geometric function theory: a modern view of a classical subject

Geometric function theory is a classical subject. Yet it continues to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, nonlinear integrable systems theory and the theory of partial differential equations. This paper surveys, with a view to modern applications, open problems and challenges in this subject. Here we advocate an approach based on the use of the Schottky–Klein prime function within a Schottky model of compact Riemann surfaces.

Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles

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)].