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.

scholarly journals Multiple steadily translating bubbles in a Hele-Shaw channel

2014 ◽  
Vol 470 (2163) ◽  
pp. 20130698 ◽  
Author(s):  
Christopher C. Green ◽  
Giovani L. Vasconcelos
Keyword(s):  
Steady Motion ◽  
Flow Region ◽  
Circular Domain ◽  
Multiply Connected Domains ◽  
Moving Frames ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Region Exterior ◽  
Prime Function ◽  
Connected Circular Domain

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.

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.

The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

2005 ◽  
Vol 461 (2061) ◽  
pp. 2653-2678 ◽  
Author(s):  
Darren Crowdy
Keyword(s):  
Mapping Function ◽  
Circular Domain ◽  
Schottky Group ◽  
Polygonal Domain ◽  
Schottky Groups ◽  
Image Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Connected Circular Domain ◽  
Natural Way

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.

Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

2007 ◽  
Vol 142 (2) ◽  
pp. 319-339 ◽  
Author(s):  
DARREN CROWDY
Keyword(s):  
Real Axis ◽  
Mapping Function ◽  
Circular Domain ◽  
Connected Region ◽  
Polygonal Domain ◽  
The Real ◽  
Multiply Connected ◽  
Region Exterior ◽  
New Formula ◽  
Connected Circular Domain

AbstractA 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.

scholarly journals Equation of motion for point vortices in multiply connected circular domains

2009 ◽  
Vol 465 (2108) ◽  
pp. 2589-2611 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Connected Domain ◽  
Single Point ◽  
Point Vortex ◽  
Equation Of Motion ◽  
Circular Domain ◽  
Point Vortices ◽  
Multiply Connected Domain ◽  
Multiply Connected ◽  
Circular Domains ◽  
Prime Function

The paper gives the equation of motion for N point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles, called the circular domain. The velocity field induced by the point vortices is described in terms of the Schottky–Klein prime function associated with the circular domain. The explicit representation of the equation enables us not only to solve the Euler equations through the point-vortex approximation numerically, but also to investigate the interactions between localized vortex structures in the circular domain. As an application of the equation, we consider the motion of two point vortices with unit strength and of opposite signs. When the multiply connected domain is symmetric with respect to the real axis, the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle, which we investigate in detail.

scholarly journals Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions

2015 ◽  
Vol 780 ◽  
pp. 299-326 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Surface Tension ◽  
Conformal Mapping ◽  
Circular Domain ◽  
Multiply Connected ◽  
Transcendental Functions ◽  
Region Exterior ◽  
Complete Set ◽  
Multiple Bubbles ◽  
Steady Solutions ◽  
Connected Circular Domain

Analytical solutions for both a finite assembly and a periodic array of bubbles steadily moving in a Hele-Shaw channel are presented. The particular case of multiple fingers penetrating into the channel and moving jointly with an assembly of bubbles is also analysed. The solutions are given by a conformal mapping from a multiply connected circular domain in an auxiliary complex plane to the fluid region exterior to the bubbles. In all cases the desired mapping is written explicitly in terms of certain special transcendental functions, known as the secondary Schottky–Klein prime functions. Taken together, the solutions reported here represent the complete set of solutions for steady bubbles and fingers in a horizontal Hele-Shaw channel when surface tension is neglected. All previous solutions under these assumptions are particular cases of the general solutions reported here. Other possible applications of the formalism described here are also discussed.

scholarly journals The Prime Function, the Fay Trisecant Identity, and the van der Pauw Method

2021 ◽  
Author(s):  
Hiroyuki Miyoshi ◽  
Darren Crowdy ◽  
Rhodri Nelson
Keyword(s):  
Applied Sciences ◽  
Science Literature ◽  
Multiply Connected ◽  
Schottky Double ◽  
Van Der Pauw ◽  
Simply Connected ◽  
Genus One ◽  
Physical Quantities ◽  
Prime Function ◽  
Van Der Pauw Method

AbstractThe van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

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