Journal ArticleDOI
On the integrability and perturbation of three-dimensional fluid flows with symmetry
Igor Mezic,Stephen Wiggins +1 more
TLDR
In this article, the authors studied three-dimensional volume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving.Abstract:
The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.read more
Citations
More filters
Journal ArticleDOI
Distinguished material surfaces and coherent structures in three-dimensional fluid flows
TL;DR: In this article, the authors prove the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows and characterize coherent structures as local maximizers of the largest finite time Lyapunov exponent field computed directly from particle paths.
Journal ArticleDOI
Foundations of chaotic mixing.
Stephen Wiggins,Julio M. Ottino +1 more
TL;DR: Most importantly for the design process, it is shown how the so–called linked twist maps function as a minimal picture of mixing, provide a mathematical structure for understanding the type of 2D flows that arise in many micromixers already built, and give conditions guaranteeing the best quality mixing.
Journal ArticleDOI
Finite time transport in aperiodic flows
George Haller,Andrew C. Poje +1 more
TL;DR: In this paper, the authors studied the transport of particles in a general, two-dimensional, incompressible flow in the presence of a transient eddy, i.e., a bounded set of closed streamlines with a finite time of existence.
Journal ArticleDOI
Frontiers of chaotic advection
Hassan Aref,John Blake,Marko Budišić,Silvana S. S. Cardoso,Julyan H. E. Cartwright,Herman Clercx,Kamal El Omari,Ulrike Feudel,Ramin Golestanian,Emmanuelle Gouillart,GertJan van Heijst,Tatyana S. Krasnopolskaya,Yves Le Guer,Robert S. MacKay,Vyacheslav V. Meleshko,Guy Metcalfe,Igor Mezic,Alessandro P. S. de Moura,Oreste Piro,Michel F.M. Speetjens,Rob Sturman,Jean-Luc Thiffeault,Idan Tuval +22 more
TL;DR: In this article, the present position of and survey future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers.
Journal ArticleDOI
Magnetic field lines, Hamiltonian dynamics, and nontwist systems
TL;DR: In this paper, a new theory for invariant tori breakup in symplectic twist maps is proposed, which describes magnetic field lines in tokamaks and zonal flows in geophysical fluid dynamics, and comments about renormalization are made.
References
More filters
Book
Table of Integrals, Series, and Products
TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Book
Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Book
Introduction to Applied Nonlinear Dynamical Systems and Chaos
TL;DR: The Poincare-Bendixson Theorem as mentioned in this paper describes the existence, uniqueness, differentiability, and flow properties of vector fields, and is used to prove that a dynamical system is Chaotic.