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Table of Contents
Intro
Preface
Contents
1 Kirchhoff's Insufficiently-Celebrated Equations of Motion
1.1 Introduction
1.2 Kirchhoff's Equations
1.3 The Legacy of Kirchhoff's Equations
1.4 The Geometric Mechanics of Kirchhoff's Equations
1.4.1 The Euler-Lagrange and Hamilton's Equations in the Spatially-Fixed Frame
1.5 Extending Kirchhoff's Model
1.5.1 The Sum Poisson Bracket
2 The Addition of Vortices
2.1 The Importance of Vorticity
2.2 Singular Vortex Models
2.2.1 The N-Point-Vortex Model
2.2.2 The N Vortex Ring Model
3 Dynamically Coupled Rigid Body+Point Vortices in R2
3.1 N-Point-Vortices and Stationary Rigid Boundaries: C.C. Lin's Problem
3.2 N-Point-Vortices Dynamically Coupled with a Single Rigid Contour of Arbitrary Shape
3.2.1 The Euler-Lagrange Equations in a Spatially-FixedFrame
3.2.2 The Vortical Momenta and Reciprocity Relations
3.3 N-Point-Vortices Dynamically Coupled with a Single Rigid Circular Contour
3.3.1 The Half-Space Model
4 Dynamically Coupled Rigid Body+Vortex Rings in R3
4.1 N Vortex Rings and a Single Stationary Rigid Boundary
4.2 N Vortex Rings Dynamically Coupled with a Single Rigid Body of Arbitrary Shape
4.2.1 The Euler-Lagrange Equations in a Spatially-FixedFrame
4.2.2 The Vortical Momenta and Reciprocity Relations
4.3 N Vortex Rings Dynamically Coupled with a Rigid Sphere
4.3.1 The Axisymmetric Model of a Sphere and N CircularRings
5 Viscous Effects and Their Modeling
5.1 System Momentum Balance Laws in the Viscous Setting
5.2 Some Experimental and Numerical Work of Vortex Rings Colliding with Rigid Bodies
6 Miscellaneous Extensions
6.1 Dynamically Coupled Rigid Body+free Surface
6.1.1 A Free Surface Dynamically Coupled with a Completely Submerged Single Rigid Body of Arbitrary Shape
6.1.1.1 Phase Space and Hamiltonian Formalism
6.2 Dynamically Coupled N Rigid Bodies in the Absenceof Vorticity
6.2.1 The Euler-Lagrange Equations in a Spatially-FixedFrame
6.3 A Single Buoyant Rigid Body Above an Impermeable FlatBoundary
A Brief Introduction to Geometric Mechanics
B Leading Order Behavior of Velocity and Vector Potential Fields of a Curved Vortex Filament
C Hamiltonian Function and Vector Field in the Half-space Model for Np=2 Sh2006
References
Preface
Contents
1 Kirchhoff's Insufficiently-Celebrated Equations of Motion
1.1 Introduction
1.2 Kirchhoff's Equations
1.3 The Legacy of Kirchhoff's Equations
1.4 The Geometric Mechanics of Kirchhoff's Equations
1.4.1 The Euler-Lagrange and Hamilton's Equations in the Spatially-Fixed Frame
1.5 Extending Kirchhoff's Model
1.5.1 The Sum Poisson Bracket
2 The Addition of Vortices
2.1 The Importance of Vorticity
2.2 Singular Vortex Models
2.2.1 The N-Point-Vortex Model
2.2.2 The N Vortex Ring Model
3 Dynamically Coupled Rigid Body+Point Vortices in R2
3.1 N-Point-Vortices and Stationary Rigid Boundaries: C.C. Lin's Problem
3.2 N-Point-Vortices Dynamically Coupled with a Single Rigid Contour of Arbitrary Shape
3.2.1 The Euler-Lagrange Equations in a Spatially-FixedFrame
3.2.2 The Vortical Momenta and Reciprocity Relations
3.3 N-Point-Vortices Dynamically Coupled with a Single Rigid Circular Contour
3.3.1 The Half-Space Model
4 Dynamically Coupled Rigid Body+Vortex Rings in R3
4.1 N Vortex Rings and a Single Stationary Rigid Boundary
4.2 N Vortex Rings Dynamically Coupled with a Single Rigid Body of Arbitrary Shape
4.2.1 The Euler-Lagrange Equations in a Spatially-FixedFrame
4.2.2 The Vortical Momenta and Reciprocity Relations
4.3 N Vortex Rings Dynamically Coupled with a Rigid Sphere
4.3.1 The Axisymmetric Model of a Sphere and N CircularRings
5 Viscous Effects and Their Modeling
5.1 System Momentum Balance Laws in the Viscous Setting
5.2 Some Experimental and Numerical Work of Vortex Rings Colliding with Rigid Bodies
6 Miscellaneous Extensions
6.1 Dynamically Coupled Rigid Body+free Surface
6.1.1 A Free Surface Dynamically Coupled with a Completely Submerged Single Rigid Body of Arbitrary Shape
6.1.1.1 Phase Space and Hamiltonian Formalism
6.2 Dynamically Coupled N Rigid Bodies in the Absenceof Vorticity
6.2.1 The Euler-Lagrange Equations in a Spatially-FixedFrame
6.3 A Single Buoyant Rigid Body Above an Impermeable FlatBoundary
A Brief Introduction to Geometric Mechanics
B Leading Order Behavior of Velocity and Vector Potential Fields of a Curved Vortex Filament
C Hamiltonian Function and Vector Field in the Half-space Model for Np=2 Sh2006
References