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

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1175
Author(s):  
El Mostafa Kalmoun ◽  
Mohamed M. S. Nasser ◽  
Khalifa A. Hazaa
Keyword(s):  
Cross Sections ◽  
Single Point ◽  
Vortex Motion ◽  
Point Vortex ◽  
Multiply Connected Domains ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Contour Lines ◽  
Circular Domains ◽  
Prime Function

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.

Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains

2005 ◽  
Vol 461 (2060) ◽  
pp. 2477-2501 ◽  
Author(s):  
Darren Crowdy ◽  
Jonathan Marshall
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Vortex Motion ◽  
Multiply Connected Domains ◽  
Single Vortex ◽  
Multiply Connected ◽  
Analytical Formulae ◽  
Circular Domains ◽  
Explicit Formulae ◽  
Prime Function

Explicit formulae for the Kirchhoff–Routh path functions (or Hamiltonians) governing the motion of N -point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero. The method uses the Schottky–Klein prime function to find representations of the hydrodynamic Green's function in multiply connected circular domains. The Green's function is then used to construct the associated Kirchhoff–Routh path function. The path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains. Illustrative examples are presented for the case of single vortex motion in multiply connected domains.

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 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 CONFORMAL SLIT MAPS IN APPLIED MATHEMATICS

2012 ◽  
Vol 53 (3) ◽  
pp. 171-189 ◽  
Author(s):  
DARREN CROWDY
Keyword(s):  
Special Function ◽  
Function Theory ◽  
Applied Mathematics ◽  
Building Blocks ◽  
Review Article ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Mathematical Problems ◽  
Explicit Formulae ◽  
Prime Function

AbstractConformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky–Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.

scholarly journals Secondary Schottky–Klein prime functions associated with multiply connected planar domains

2015 ◽  
Vol 471 (2173) ◽  
pp. 20140688 ◽  
Author(s):  
Giovani L. Vasconcelos ◽  
Jonathan S. Marshall ◽  
Darren G. Crowdy
Keyword(s):  
Mixed Type ◽  
Compact Riemann Surface ◽  
Mathematical Framework ◽  
Multiply Connected Domains ◽  
Additional Function ◽  
Multiply Connected ◽  
Planar Domains ◽  
Recent Interest ◽  
Schottky Double ◽  
Prime Function

In recent years, a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the Schottky–Klein (S–K) prime function of an underlying compact Riemann surface known as the Schottky double of the domain. In this paper, we describe additional function-theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary S–K prime functions. The basic idea develops, and extends, an observation of Burnside dating back to 1892. Applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given. Other possible applications are also surveyed.

Conformal and Quasiconformal Mappings of Close Multiply-Connected Domains

2002 ◽  
Vol 9 (2) ◽  
pp. 367-382
Author(s):  
Z. Samsonia ◽  
L. Zivzivadze
Keyword(s):  
Integral Equations ◽  
Quasiconformal Mappings ◽  
Multiply Connected Domains ◽  
Multiply Connected

Abstract Doubly-connected and triply-connected domains close to each other in a certain sense are considered. Some questions connected with conformal and quasiconformal mappings of such domains are studied using integral equations.

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