Surface-tension-driven Stokes flow: a numerical method based on conformal geometry

AbstractA novel numerical scheme is presented for solving the problem of two dimensional Stokes flows with free boundaries whose evolution is driven by surface tension. The formulation is based on a complex variable formulation of Stokes flow and use of conformal mapping to track the free boundaries. The method is motivated by applications to modelling the fabrication process for microstructured optical fibres (MOFs), also known as “holey fibres”, and is therefore tailored for the computation of multiple interacting free boundaries. We give evidence of the efficacy of the method and discuss its performance

An integral equation for conformal mapping of multiply connected regions onto a circular region

Abstract. An integral equation is presented for the conformal mapping of multiply connected regions of connectivity m+1 onto a circular region. The circular region is bounded by a unit circle, with centre at the origin, and m number of circles inside the unit circle. The development of theoretical part is based on the boundary integral equation related to a non-homogeneous boundary relationship. An example for verification purpose is given in this paper for the conformal mapping from an annulus onto a doubly connected circular region with centres and radii are assumed to be known

Numerical conformal mapping methods based on Faber series

Methods are presented for approximating the conformal map from the interior of various regions to the interior of simply-connected target regions with a smooth boundary. The methods for the disk due to Fornberg (1980) and the ellipse due to DeLillo and Elcrat (1993) are reformulated so that they may be extended to other new computational regions. The case of a cross-shaped region is introduced and developed. These methods are used to circumvent the severe ill-conditioning due to the crowding phenomenon suffered by conformal maps from the unit disk to target regions with elongated sections while preserving the fast Fourier methods available on the disk. The methods are based on expanding the mapping function in the Faber series for the regions. All of these methods proceed by approximating the boundary correspondence of the map with a Newton-like iteration. At each Newton step, a system of linear equations is solved using the conjugate gradient method. The matrix-vector multiplication in this inner iteration can be implemented with fast Fourier transforms at a cost of O(N log N). It is shown that the linear systems are discretizations of the identity plus a compact operator and so the conjugate gradient method converges superlinearly. Several computational examples are given along with a discussion of the accuracy of the methods

Conformal mapping for multiple terminals

abstract: Conformal mapping is an important mathematical tool that can be used to solve various physical and engineering problems in many fields, including electrostatics, fluid mechanics, classical mechanics, and transformation optics. It is an accurate and convenient way to solve problems involving two terminals. However, when faced with problems involving three or more terminals, which are more common in practical applications, existing conformal mapping methods apply assumptions or approximations. A general exact method does not exist for a structure with an arbitrary number of terminals. This study presents a conformal mapping method for multiple terminals. Through an accurate analysis of boundary conditions, additional terminals or boundaries are folded into the inner part of a mapped region. The method is applied to several typical situations, and the calculation process is described for two examples of an electrostatic actuator with three electrodes and of a light beam splitter with three ports. Compared with previously reported results, the solutions for the two examples based on our method are more precise and general. The proposed method is helpful in promoting the application of conformal mapping in analysis of practical problems.The final version of this article, as published in Scientific Reports, can be viewed online at: https://www.nature.com/articles/srep3691

Mathematical techniques for free boundary problems with multiple boundaries

In this thesis, we study six different free boundary problems arising in the field of fluid mechanics, and the mathematical methods used to solve them. The free boundary problems are all characterised by having more than one boundary and the solution of these problems requires special mathematical treatment. The challenge in each of these problems is to determine the shape of the multiple fluid interfaces making up the particular system under consideration. In each of the free boundary problems we employ aspects of complex function theory, conformal mapping between multiply connected domains, and specialist techniques devised in recent years by Crowdy and collaborators. At the heart of these techniques lies a special transcendental function known as the Schottky-Klein prime function. This thesis makes use of this function in a variety of novel contexts. We first examine a single row of so-called hollow vortices in free space. This problem has been solved before but we present a new methodology which is convenient in being extendible to the case of a double row, or von Karman vortex street, of hollow vortices. We find a concise formula for the conformal mapping describing the shapes of the free boundaries of two hollow vortices in a typical period window in the vortex street and thereby solve the free boundary problem. We next focus on the problem of a pair of hollow vortices in an infinite channel. This free boundary problem exhibits similar mathematical features to the vortex street problem but now involves the new ingredient of solid impenetrable walls. Again we solve the free boundary problem by finding a concise formula for the conformal mapping governing the hollow vortex shapes. We then extend this analysis to a single row of hollow vortices occupying the channel. The problem of a pair of hollow vortices of equal and opposite circulation positioned behind a circular cylinder, superposed with a uniform flow, is then considered. This system is a desingularisation of the so-called Foppl point vortex equilibrium. For this free boundary problem, we employ a hybrid analytical-numerical scheme and we are able to offer a Fourier-Laurent series expansion for the conformal mapping determining the shape of the hollow vortex boundaries. Finally, we investigate an asymmetric assembly of steadily translating bubbles in a Hele- Shaw channel. This free boundary problem can be formulated as a special Riemann-Hilbert problem solvable in terms of the Schottky-Klein prime function. Our method of solution can be used to determine the shapes of any finite number of bubbles in a given assembly

A new transform approach to biharmonic boundary value problems in circular domains with applications to Stokes flows

In this thesis, we present a new transform approach for solving biharmonic boundary value problems in two-dimensional polygonal and circular domains. Our approach provides a unified general approach to finding quasi-analytical solutions to a wide range of problems in Stokes flows and plane elasticity. We have chosen to analyze various Stokes flow problems in different geometries which have been solved using other techniques and present our transform approach to solve them. Our approach adapts mathematical ideas underlying the Unified transform method, also known as the Fokas method, due to Fokas and collaborators in recent years. We first consider Stokes flow problems in polygonal domains whose boundaries consist of straight line edges. We show how to solve problems in the half-plane subject to different boundary conditions along the real axis and we are able to retrieve analytical results found using other techniques. Next, we present our transform approach to solve for a flow past a periodic array of semi-infinite plates and for a periodic array of point singularities in a channel, followed by a brief discussion on how to systematically solve problems in more complex channel geometries. Next, we show how to solve problems in circular domains whose boundaries consist of a combination of straight line and circular edges. We analyze the problems of a flow past a semicircular ridge in the half-plane, a translating and rotating cylinder above a wall and a translating and rotating cylinder in a channel.Open Acces