Schottky Klein Prime Function Research Articles

Harmonic-measure distribution functions of simply connected and doubly connected polygonal domains

The h-function encodes aspects of the behaviour of Brownian particles released from a specified fixed point inside the region. In turn, this behaviour is influenced by the shape of the region's boundary and the location of the fixed point. We compute the h-functions of numerous simply connected and doubly connected polygonal regions. The Schottky–Klein prime function plays a key role in computing h-functions of these regions. Also, in the computation, we use the Schwarz–Christoffel mappings, the Cayley-type mappings, and their generalisations, which are expressed in terms of the prime function. We also cross-check the validity of all the h-functions we find by rederiving them via numerical simulation of Brownian motion.

Computation of plane potential flow around multi-element airfoils using the Schottky–Klein prime function

Potential flow is computed around multi-element airfoils in two dimensions using numerical conformal mapping combined with the calculus of the Schottky–Klein prime function. The corners at the trailing edges of the airfoils are successively removed by Kármán–Trefftz maps. The map from the domain exterior to disks to the domain exterior to the smooth images of the airfoils is computed using an extension of a Fourier series method for the disk due to Fornberg. The velocity potential for flow in the circle domain, with circulation calculated to satisfy the Kutta condition in the airfoil domain, is computed using the Schottky–Klein prime function. The computations are compared to examples from the aerospace engineering literature.

The [formula omitted]-vortex problem in a doubly periodic rectangular domain with constant background vorticity

We formulate the N point vortex problem in a doubly-periodic rectangle with a constant background vorticity using the hydrodynamic Green’s function. We derive an explicit formula for the hydrodynamic Green’s function and the velocities of the point vortices in terms of the Schottky–Klein prime function using a conformal mapping approach. The sum of vortex strengths can be arbitrary, and when it is zero we recover previous known results including the integrability of the two and three-vortex problems. A non-zero sum of vortex strengths leads to a constant background vorticity. We derive a Hamiltonian structure for the equations and show that the two-vortex problem is integrable, and classify all possible vortex motions. In addition, for general N, we obtain several fixed lattice equilibrium configurations including single-layered and double-layered equilibria. In the latter case, we also obtain lattice configurations with defects consisting of point vortices with inhomogeneous strengths. We find equilibria arranged in doubly-periodic rectangular and parallelogram lattices consisting of N=1,2,3,4 vortices per fundamental lattice.

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

The harmonic-measure distribution function, or h -function, of a planar domain Ω ⊂ C with respect to a basepoint z 0 ∈ Ω is a signature that profiles the behaviour in Ω of a Brownian particle starting from z 0 . Explicit calculation of h -functions for a wide array of simply connected domains using conformal mapping techniques has allowed many rich connections to be made between the geometry of the domain and the behaviour of its h -function. Until now, almost all h -function computations have been confined to simply connected domains. In this work, we apply the theory of the Schottky–Klein prime function to explicitly compute the h -function of the doubly connected slit domain C ∖ ( [ − 1 / 2 , − 1 / 6 ] ∪ [ 1 / 6 , 1 / 2 ] ) . In view of the connection between the middle-thirds Cantor set and highly multiply connected symmetric slit domains, we then extend our methodology to explicitly construct the h -functions associated with symmetric slit domains of arbitrary even connectivity. To highlight both the versatility and generality of our results, we graph the h -functions associated with quadruply and octuply connected slit domains.

Hollow vortex in a corner

Abstract

A calculus for flows in periodic domains

Purpose: We present a constructive procedure for the calculation of 2-D potential flows in periodic domains with multiple boundaries per period window.Methods: The solution requires two steps: (i) a conformal mapping from a canonical circular domain to the physical target domain, and (ii) the construction of the complex potential inside the circular domain. All singly periodic domains may be classified into three distinct types: unbounded in two directions, unbounded in one direction, and bounded. In each case, we use conformal mappings to relate the target periodic domain to a canonical circular domain with an appropriate branch structure.Results: We then present solutions for a range of potential flow phenomena including flow singularities, moving boundaries, uniform flows, straining flows and circulatory flows.Conclusion: By using the transcendental Schottky-Klein prime function, the ensuing solutions are valid for an arbitrary number of obstacles per period window. Moreover, our solutions are exact and do not require any asymptotic approximations.

The squeezing function on doubly-connected domains via the Loewner differential equation

For any bounded domains varOmega in {mathbb {C}}^{n}, Deng, Guan and Zhang introduced the squeezing function S_varOmega (z) which is a biholomorphic invariant of bounded domains. We show that for n=1, the squeezing function on an annulus A_r = lbrace z in {mathbb {C}} : r<|z| <1 rbrace is given by S_{A_r}(z)= max left{ |z| ,frac{r}{|z|} right} for all 0<r<1. This disproves the conjectured formula for the squeezing function proposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) the squeezing function for all doubly-connected domains in {mathbb {C}} other than the punctured plane. It provides the first non-trivial formula for the squeezing function for a wide class of plane domains and answers a question of Wold. Our main tools used to prove this result are the Schottky–Klein prime function (following the work of Crowdy) and a version of the Loewner differential equation on annuli due to Komatu. We also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in {mathbb {C}}^{n}.

The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

Periodic Schwarz-Christoffel mappings with multiple boundaries per period.

We present an extension to the theory of Schwarz-Christoffel (S-C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the x-direction in an (x, y)-plane, three cases are considered; these differ in whether the period window extends off to infinity as y → ± ∞, or extends off to infinity in only one direction (y → + ∞ or y → - ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S-C mapping formulae are shown to be expressible in terms of the Schottky-Klein prime function associated with the circular preimage domains. As usual for an S-C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results.

Applying improved analytical methods for modelling flood displacement fronts in bounded reservoirs (Quitman field, east Texas)

We apply an improved potential flow model based on conformal mapping to study the sweep pattern of waterfloods in bounded reservoirs. Solutions for streamlines and flood advancement obtained with the new model are validated using an independent but more intricate numerical streamline simulation method. Subsequently, the use of the benchmarked model is demonstrated in a review of the flow patterns in the Quitman field, a tabular reservoir comprising an 18 ft thick payzone of Harris sand interfingering with shales of the Eagle Ford formation. The reservoir is bound by impervious faults modelled by the method of images in earlier studies of the 1970's. In the present study, Riemann's mapping theorem in combination with the Schottky-Klein prime function is applied to find complex potentials that describe the flow in the bounded reservoir. Such an approach can model waterflooding in marginal fields like the Quitman oil field more accurately than previous potential flow methods and can visualize the sweep pattern and compute time-of-flight contours in a simpler and faster fashion than numerical streamline simulators.

Fast evaluation of the fundamental singularities of two-dimensional doubly periodic Stokes flow

Analytical representations of the flows associated with doubly periodic arrangements of point singularities of two-dimensional Stokes flow are derived. The analysis makes use of a conformal mapping from a concentric annulus to a rectangle. A natural mathematical object known as the Schottky–Klein prime function is used to derive the solutions avoiding the theory of elliptic functions. The new expressions have the advantage of being immediately amenable to fast evaluation without the need for Ewald or other fast summation methods or acceleration techniques for lattice sums.

The effect of core size on the speed of compressible hollow vortex streets

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy &amp; Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

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

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.

Finite Gap Jacobi Matrices and the Schottky\u2013Klein Prime Function

A covering map formalism for studying the spectral curves associated with finite gap Jacobi matrices is presented. We advocate a constructive function theoretic framework based on use of the Schottky–Klein prime function. The single gap, or genus-one, case is studied in explicit detail.

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

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

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.

Analytical solutions for two hollow vortex configurations in an infinite channel

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.

Vortex patch equilibria of the Euler equation and random normal matrices

A new analogy is drawn between vortex patch, or V-state, equilibria of the Euler equations for ideal fluids and the planar limit of random normal matrix models. Physically the former is a quite different fluid dynamical problem to Hele-Shaw flow, or Laplacian growth, to which an analogy with matrix models has become well known in recent years. The connection of random matrices with vortex dynamics is made via the so-called modified Schwarz potential. This theoretical link, while interesting in itself, has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics. Hence for multi-support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a Schottky model of the underlying algebraic curves and use of the Schottky–Klein prime function. Two-support eigenvalue distributions in a quartic potential are given as an illustrative example.

On the Construction of Multiply Connected Arc Integral Quadrature Domains

In this paper, we construct an explicit parameterisation describing quadrature domains of finite area and arbitrary finite connectivity admitting quadrature identities with integrals along interior arcs. The formulation is in terms of conformal maps from circular domains, where these maps are expressed in terms of the associated Schottky–Klein prime function. An important role in the derivation is played by the Bergman kernel function. To illustrate the results, several examples are presented.

Multiple steadily translating bubbles in a Hele-Shaw channel

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.