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 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.

Construction of complex potentials for multiply connected domain

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pyotr N. Ivanshin
Keyword(s):  
Integral Equation ◽  
Analytic Function ◽  
Linear System ◽  
Unbounded Domain ◽  
Connected Domain ◽  
Complex Potential ◽  
Cauchy Integral ◽  
Multiply Connected Domain ◽  
Multiply Connected ◽  
Impermeable Boundary

AbstractThe method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential – an analytic function in an unbounded multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given 𝑛-connected unbounded domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and 𝑛 logarithmic summands. The method is easily computable.

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