Invited and contributed talks presented at conferences, seminars, workshops, and research meetings.
Dana Knox Student Research Showcase New Jersey Institute of Technology, Newark, New Jersey
A global, singularity-free phase-plane analysis of the three-vortex problem with arbitrary circulations and its associated bifurcation structure.
SIAM-NNP-2024 Gosnell Hall
We investigate two problems in point-vortex dynamics in a two-dimensional, inviscid, incompressible fluid. We derive a novel reduction of a system involv- ing three vortices, initially employing Jacobi coordinates followed by Nambu brackets. First, we conduct a global phase analysis of a three-vortex problem with arbitrary circulations. Second, we generalize the reduction method to study the dynamics of four vortices with vanishing total circulation. The novel reduction method eliminates coordinate singularities that made understanding the dynamics challenging.
SIAM Conference on Nonlinear Waves and Coherent Structures Lord Baltimore Hotel
The motion of three point-vortices in a 2D inviscid, incompressible fluid has been widely studied. Gröbli (1877) derived a closed system for the evolution of the side lengths of the triangle formed by the vortices. This system has been the basis of most studies of this problem. These coordinates have a few disadvantages. First, the coordinates must satisfy the triangle inequality, so not all points in \(\mathbb{R}^{3}\) are physical. Second, the system introduced non-physical singularities because collinear arrangements lie on the boundary of the triangle inequality. Third, these coordinates break the useful Hamiltonian structure and make phase-plane reasoning difficult. We introduce a coordinate system for this problem that overcomes these disadvantages using a sequence of systematic and standard reductions. We first use Jacobi coordinates, a common technique in n-body systems, and further apply a Nambu Bracket reduction. The method avoids creating non-physical singularities and makes the system’s phase-space geometry and topology plain. A previous Nambu-bracket formulation based on Gröbli’s reduction inherited its triangle-inequality-related disadvantages. Depending on the circulations, the phase space may be a sphere or one sheet of a two-sheeted hyperboloid. Prior attempts to classify the dynamics focused solely on the stability of the relative fixed points; this approach allows us to explain the results more clearly using the entire phase space.
SIAM-NNP-2023 NJIT Campus Center
We investigate three problems in point-vortices dynamics within a two-dimensional, inviscid, incompressible fluid. We derive a new reduction of a system of three vortices. The integrable reduced system has an easily visualized phase plane that illuminates the dynamics. We apply it to explain the scattering of the point-vortex dipole with a third vortex in two cases. We then add a fourth vortex and use the reduced dynamics of the three-vortex system as the basis for the perturbative study of dipole-dipole scattering.
NJIT-MATH-SUMMER-TALK-2023 NJIT Cullimore Hall
Point vortex motion arises in the study of concentrated vorticity in an ideal, incompressible fluid described by Euler’s equations. The two-dimensional Euler equations of fluid mechanics, a partial differential equation (PDE) system, support a solution where the vorticity is concentrated at a single point. Helmholtz derived a system of ordinary differential equations (ODEs) that describe the motion of a set of interacting vortices that behave as discrete particles, which approximates the fluid motion in the case that the vorticity is concentrated in very small regions. This system of equations has continued to provide interesting questions for over 150 years. We will discuss the special class of relative periodic orbits known as the leapfrogging orbits. The relative periodic orbit of a four-point vortex problem, consisting of two positive and two negative point vortices of equal absolute circulation arranged as coaxial vortex pairs, is known as the leapfrogging orbit. This dissertation presents generalizations of the leapfrogging motion of point vortices and vortex rings, including their stability and dynamics. More specifically, we study the leapfrogging motion of \(2N\) vortices, with half having positive circulation and half having negative circulation.