scholarly journals Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI

2018 ◽  
Vol 474 (2216) ◽  
pp. 20180080 ◽  
Author(s):  
Tiago Anselmo ◽  
Rhodri Nelson ◽  
Bruno Carneiro da Cunha ◽  
Darren G. Crowdy
Keyword(s):  
Conformal Mapping ◽  
Asymptotic Expansions ◽  
Numerical Approach ◽  
Young Diagrams ◽  
Target Domain ◽  
Conformal Maps ◽  
Circular Arc ◽  
Tau Function ◽  
Accessory Parameter ◽  
Parameter Problem

We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.

Analogue-numerical approach to conformal mapping

1976 ◽  
Vol 123 (3) ◽  
pp. 212 ◽  
Author(s):  
K.J. Binns ◽  
G. Rowlands Rees ◽  
R. Anderson
Keyword(s):  
Conformal Mapping ◽  
Numerical Approach

scholarly journals A calculus for flows in periodic domains

2020 ◽  
Author(s):  
Peter J. Baddoo ◽  
Lorna J. Ayton
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Moving Boundaries ◽  
Asymptotic Approximations ◽  
Periodic Domain ◽  
Target Domain ◽  
Potential Flows ◽  
Flow Phenomena ◽  
Periodic Domains ◽  
Prime Function

AbstractPurpose: 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.

A uniformly asymptotic solution for incompressible flow past thin sharp-edged aerofoils at zero incidence

1971 ◽  
Vol 70 (1) ◽  
pp. 135-155
Author(s):  
A. F. Sheer
Keyword(s):  
Asymptotic Solution ◽  
Incompressible Flow ◽  
Asymptotic Expansions ◽  
Matched Asymptotic Expansions ◽  
Proper Choice ◽  
Circular Arc ◽  
Flow Past ◽  
Sharp Edges

Expansions obtained from classical subsonic thin-aerofoil theory break down in the neighbourhood of the aerofoil edges. At sharp edges the method of matched asymptotic expansions fails to remedy this. Here this failure is explained, and in the case of incompressible flow past a symmetric aerofoil at zero incidence it is shown that by proper choice of the dependent variable an expansion may be obtained which is uniformly asymptotic. Finally, the case of a circular-arc aerofoil is considered in more detail.

Numerical Conformal Mapping for Undergraduates

1981 ◽  
Vol 18 (4) ◽  
pp. 359-372 ◽  
Author(s):  
G. Warner ◽  
R. Anderson
Keyword(s):  
Conformal Mapping ◽  
Undergraduate Student ◽  
Elementary Mathematics ◽  
Numerical Approach ◽  
Numerical Conformal Mapping ◽  
Sufficient Detail

A numerical approach to conformal mapping is discussed which is conceptually simple and requires only elementary mathematics and simple programming. The technique is illustrated with practical problems well within the ability of an undergraduate student. Sufficient detail is given for the examples to be of immediate in-course use.

scholarly journals Periodic Schwarz–Christoffel mappings with multiple boundaries per period

2019 ◽  
Vol 475 (2228) ◽  
pp. 20190225 ◽  
Author(s):  
Peter J. Baddoo ◽  
Darren G. Crowdy
Keyword(s):  
Circular Domain ◽  
Target Domain ◽  
Multiply Connected ◽  
Single Period ◽  
Straight Line ◽  
Finite Set ◽  
Periodic Configuration ◽  
Parameter Problem ◽  
Prime Function ◽  
Connected Circular Domain

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.

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