The N-vortex problem in a doubly periodic rectangular domain with constant background vorticity

Krishnamurthy, Vikas (2023) The N-vortex problem in a doubly periodic rectangular domain with constant background vorticity. Physica D: Nonlinear Phenomena, 448. p. 133728. ISSN 01672789

Text
1-s2.0-S0167278923000829-main.pdf - Published Version
Download (734kB)

Abstract

We formulate the N point vortex problem in a doubly-periodic rectangle with a constant background vorticity using the hydrodynamic Green's function. We derive an explicit formula for the hydrodynamic Green's function and the velocities of the point vortices in terms of the Schottky–Klein prime function using a conformal mapping approach. The sum of vortex strengths can be arbitrary, and when it is zero we recover previous known results including the integrability of the two and three-vortex problems. A non-zero sum of vortex strengths leads to a constant background vorticity. We derive a Hamiltonian structure for the equations and show that the two-vortex problem is integrable, and classify all possible vortex motions. In addition, for general N, we obtain several fixed lattice equilibrium configurations including single-layered and double-layered equilibria. In the latter case, we also obtain lattice configurations with defects consisting of point vortices with inhomogeneous strengths. We find equilibria arranged in doubly-periodic rectangular and parallelogram lattices consisting of N=1,2,3,4 vortices per fundamental lattice.

[error in script]

IITH Creators:

IITH Creators	ORCiD
Krishnamurthy, Vikas	http://www.orcid.org/0000-0002-1518-0994

Item Type:

Article

Uncontrolled Keywords:

Doubly-periodic domain; Hydrodynamic Green's function; Lattice equilibria; Point vortex dynamics; Schottky–Klein prime function; Conformal mapping; Hamiltonians; Hydrodynamics; Vortex flow; Doubly periodic; Doubly-periodic domain; Greens function; Hydrodynamic green function; Lattice equilibrium; Periodic domains; Point vortex dynamic; Point vortices; Schottky; Schottky–klein prime function; Vortex dynamics; Vorticity

Subjects:

Mathematics
Mathematics > General principles of mathematics

Divisions:

Department of Mathematics

Depositing User:

Mr Nigam Prasad Bisoyi

Date Deposited:

04 Jan 2024 14:17

Last Modified:

04 Jan 2024 14:17

URI:

http://raiithold.iith.ac.in/id/eprint/11767