Generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

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The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1480 ◽

2005 ◽

Vol 461 (2061) ◽

pp. 2653-2678 ◽

Cited By ~ 50

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.

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.469 ◽

2015 ◽

Vol 780 ◽

pp. 299-326 ◽

Cited By ~ 6

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.

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Multiple steadily translating bubbles in a Hele-Shaw channel

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0698 ◽

2014 ◽

Vol 470 (2163) ◽

pp. 20130698 ◽

Cited By ~ 4

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.

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