Keywords

Vortex Filament, Self-Advection, Local Induction Approximation, Slipping motion, OHAM

Abstract

This thesis explores the self advection of a thin vortex filament in an incompressible fluid. The effect of slipping motion on the filament caused by viscosity is investigated. This problem is addressed by working with three models developed for this purpose.

The Da Rios-Betchov equations are obtained from the expression for the velocity of the vortex filament under the assumptions of local induction approximation. They describe the evolution of the intrinsic geometric quantities, namely, curvature and torsion of the filament. The invariants associated with these equations are deduced. The linear instability of the vortex filament motion obtained by Betchov is identified to be the linear modulational instability. We then proceed to address the nonlinear evolution of this modulational instability. We give the Lagrangian and the Hamiltonian formulations for the Da Rios-Betchov equations. The interpretation of curvature and torsion of the filament as the density and velocity respectively of a fictitious fluid is extended to the case where the slipping motion of the vortex filament is constant.

The Hasimoto model describes the motion of the vortex filament in terms of a wave function that depends on the curvature and torsion of the filament using the nonlinear Schrodinger equation. The invariants of this equation are deduced and the effects of an exact solution on these invariants are noted. A similarity reduction of the nonlinear Schrodinger equation is performed and a compatibility condition required for this reduction deduced. The Optimal Homotopy Analysis Method (OHAM) is used to obtain a truncated power series solution to the evolution equation. We then introduce a slipping motion of the vortex filament and investigate its effects on the filament dynamics by deriving a new evolution equation describing its motion. The new invariants associated with this equation are deduced and their physical interpretations are given. We investigate the conservation of physical quantities such as the length of the vortex filament, total torsion, kinetic energy, linear momentum and angular momentum. We give the Lagrangian and the Hamiltonian formulations for the new evolution equation for the case when the slipping motion is constant.

Shivamoggi and van Heijst posed the problem of self advection of a vortex filament in terms of extrinsic filament coordinates and derived the Shivamoggi-van Heijst equation for this model. We determine the evolution equations for the amplitude and phase of the vortex motion using a Madelung transformation. We investigate the linear modulational instability of the vortex motion using these evolution equations. We deduce invariants for the Shivamoggi-van Heijst equation and its fully nonlinear version. We construct steady state solutions for special cases stipulating restrictions on the amplitude and phase of the vortex motion. We perform a singular point analysis to investigate whether the Shivamoggi-van Heijst equation possesses the Painleve property. The Optimal Homotopy Analysis method (OHAM) is used to obtain a truncated power series solution to the Shivamoggi-van Heijst equation. We perform a similarity reduction of the Shivamoggi-van Heijst equation and deduce a compatibility condition required for this reduction. We introduce a slipping motion of the vortex filament and investigate its effects on the filament dynamics.

Completion Date

2025

Semester

Summer

Committee Chair

Shivamoggi, Bhimsen K.

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Department of Mathematics

Format

PDF

Identifier

DP0029572

Language

English

Document Type

Thesis

Campus Location

Orlando (Main) Campus

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