# Publications

My publications to date are given below together with their respective abstracts.

## Harmonic measure distribution functions for a class of multiply connected symmetrical slit domains

**C. C. Green**, M. A. Snipes & L. A. Ward. In final preparation for Proc. R. Soc. A.

## The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

**C. C. Green**, C. J. Lustri & S. W. McCue. 2017. Proc. R. Soc. A, 473, 20170050.

*New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.*

## A fast numerical method for ideal fluid flow in domains with multiple stirrers

M. M. S. Nasser & **C. C. Green**. 2017. Nonlinearity.

*A collection of arbitrarily-shaped solid objects, each moving at a constant speed, can be used to mix or stir ideal fluid, and can give rise to interesting flow patterns. Assuming these systems of fluid stirrers are two-dimensional, the mathematical problem of resolving the flow field - given a particular distribution of any finite number of stirrers of specified shape and speed - can be formulated as a Riemann-Hilbert problem. We show that this Riemann-Hilbert problem can be solved numerically using a fast and accurate algorithm for any finite number of stirrers based around a boundary integral equation with the generalized Neumann kernel. Various systems of fluid stirrers are considered, and our numerical scheme is shown to handle highly multiply connected domains (i.e. systems of many fluid stirrers) with minimal computational expense.*

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

D. G. Crowdy, E. H. Kropf, **C. C. Green** & M. M. S, Nasser. 2016. IMA J. App. Math, 81:589-628.

*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.*

## Analytical solutions for two hollow vortex configurations in an infinite channel

**C. C. Green.** 2015. Eur. J. Mech. B/Fluids, 54:69-81.

*New analytical solutions for a co-travelling hollow vortex pair and a single row of hollow vortices in an infinite
channel are presented. These new solutions generalise several known classical solutions for hollow vortices. The mathematical problems to be solved in each geometry are particular types of free boundary problem over a multiply connected domain. We present concise formulae for the conformal mapping determining the shape of the boundaries of the hollow vortices in both channel geometries by employing free streamline theory in combination with the function theory of the Schottky–Klein prime function. Various properties of the solutions are also presented.*

## Multiple steadily translating bubbles in a Hele-Shaw channel

**C. C. Green,** G. L. Vasconcelos. 2014. Proc. R. Soc. A, 470:20130698.

*Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions—corresponding to the complex potentials in the laboratory and co-moving frames—that map the circular domain onto respective degenerate polygonal domains. These functions are obtained using the generalized Schwarz–Christoffel formula for multiply connected domains in terms of the Schottky–Klein prime function. Our solutions are very general in that no symmetry assumption concerning the geometrical disposition of the bubbles is made. Several examples for various bubble configurations are discussed.*

## Green’s function for the Laplace-Beltrami operator on a toroidal surface

**C. C. Green**, J. S. Marshall. 2013. Proc. R. Soc. A, 469:20120479.

*Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.*

## Analytical solutions for von Karman streets of hollow vortices

D. G. Crowdy,** C. C. Green.** 2012. Phys. Fluids, 23:126602.

*New analytical solutions are presented for steadily translating von Karman vortex streets made up of two infinite rows of hollow vortices. First, the solution for a single row of hollow vortices due to Baker, Saffman & Sheffield (1976) is rederived, in a modified form, and using a new mathematical approach. This approach is then generalized to find relative equilibria for both unstaggered and staggered double hollow vortex streets. The method employs a combination of free streamline theory and conformal mapping ideas. The staggered hollow vortex streets are compared with analogous numerical solutions for double streets of vortex patches due to Saffman & Schatzman (1981) and several common features are found. In particular, within the two different inviscid vortex models, the same street aspect ratio of approximately 0.34–0.36 is found to have special significance for the equilibria.*

## Conformal mappings to multiply connected polycircular arc domains

D. G. Crowdy, A. S. Fokas, **C. C. Green.** 2011. Comp. Meth. Funct. Theory, 11(2):685-706.

*There has been much recent interest in finding analytical formulae for conformal mappings from canonical multiply connected circular regions to multiply connected polygonal regions. Such formulae are the multiply connected generalizations of the Schwarz-Christoffel formula of classical function theory. A natural generalization of polygonal domains is the class of polycircular arc domains whose boundaries are a union of circular arc segments. This paper describes a theoretical method for the construction of conformal mappings from multiply connected circular domains, of arbitrary finite connectivity, to conformally equivalent polycircular arc domains. This work generalizes results on the doubly connected case by Crowdy & Fokas (2007).*