Applications of knot theory in fluid mechanics

Renzo Ricca

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 321-346
  • ISSN: 0137-6934

Abstract

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In this paper we present an overview of some recent results on applications of knot theory in fluid mechanics, as part of a new discipline called `topological fluid mechanics' (TFM). The choice of the topics covered here is deliberately restricted to those areas that involve mainly a combination of ideal fluid mechanics techniques and knot theory concepts, complemented with a brief description of some other concepts that have important applications in fluid systems. We begin with the concept of topological equivalence of fluid flow maps, giving a definition of knotted and linked flux-tubes. In the fluid mechanics context Reidemeister's moves are interpreted in terms of local actions of fluid flows performed on fluid structures. An old theorem of Lichtenstein (1925) concerning the isotopic evolution of vortex structures in the context of the Euler equations is re-proposed and discussed in the TFM context for the first time. Then, we review the relationship between helicity and linking numbers and we present some recent results on magnetic relaxation of linked, knotted and braided structures in magnetohydrodynamics. In the context of the Euler equations (and under certain approximations given by the so-called `localized induction' for vortex structures) we briefly examine some interesting relationships between integrability and existence and stability of vortex filaments in the shape of torus knots. We conclude with an overview of some new results concerning electrically charged knots embedded in a viscous fluid, elastic relaxation of strings and braids and relationships between energy levels and topological information. Some simple bounds on elastic energy levels given by global geometric quantities and topological quantities are presented and discussed for the first time.

How to cite

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Ricca, Renzo. "Applications of knot theory in fluid mechanics." Banach Center Publications 42.1 (1998): 321-346. <http://eudml.org/doc/208816>.

@article{Ricca1998,
abstract = {In this paper we present an overview of some recent results on applications of knot theory in fluid mechanics, as part of a new discipline called `topological fluid mechanics' (TFM). The choice of the topics covered here is deliberately restricted to those areas that involve mainly a combination of ideal fluid mechanics techniques and knot theory concepts, complemented with a brief description of some other concepts that have important applications in fluid systems. We begin with the concept of topological equivalence of fluid flow maps, giving a definition of knotted and linked flux-tubes. In the fluid mechanics context Reidemeister's moves are interpreted in terms of local actions of fluid flows performed on fluid structures. An old theorem of Lichtenstein (1925) concerning the isotopic evolution of vortex structures in the context of the Euler equations is re-proposed and discussed in the TFM context for the first time. Then, we review the relationship between helicity and linking numbers and we present some recent results on magnetic relaxation of linked, knotted and braided structures in magnetohydrodynamics. In the context of the Euler equations (and under certain approximations given by the so-called `localized induction' for vortex structures) we briefly examine some interesting relationships between integrability and existence and stability of vortex filaments in the shape of torus knots. We conclude with an overview of some new results concerning electrically charged knots embedded in a viscous fluid, elastic relaxation of strings and braids and relationships between energy levels and topological information. Some simple bounds on elastic energy levels given by global geometric quantities and topological quantities are presented and discussed for the first time.},
author = {Ricca, Renzo},
journal = {Banach Center Publications},
keywords = {localized induction for vortex structures; stability of vortex filaments; topological fluid mechanics; topological equivalence of fluid flow maps; linked flux-tubes; Reidemeister's moves; theorem of Lichtenstein; isotopic evolution of vortex structures; Euler equations; helicity; magnetic relaxation; structures in magnetohydrodynamics},
language = {eng},
number = {1},
pages = {321-346},
title = {Applications of knot theory in fluid mechanics},
url = {http://eudml.org/doc/208816},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Ricca, Renzo
TI - Applications of knot theory in fluid mechanics
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 321
EP - 346
AB - In this paper we present an overview of some recent results on applications of knot theory in fluid mechanics, as part of a new discipline called `topological fluid mechanics' (TFM). The choice of the topics covered here is deliberately restricted to those areas that involve mainly a combination of ideal fluid mechanics techniques and knot theory concepts, complemented with a brief description of some other concepts that have important applications in fluid systems. We begin with the concept of topological equivalence of fluid flow maps, giving a definition of knotted and linked flux-tubes. In the fluid mechanics context Reidemeister's moves are interpreted in terms of local actions of fluid flows performed on fluid structures. An old theorem of Lichtenstein (1925) concerning the isotopic evolution of vortex structures in the context of the Euler equations is re-proposed and discussed in the TFM context for the first time. Then, we review the relationship between helicity and linking numbers and we present some recent results on magnetic relaxation of linked, knotted and braided structures in magnetohydrodynamics. In the context of the Euler equations (and under certain approximations given by the so-called `localized induction' for vortex structures) we briefly examine some interesting relationships between integrability and existence and stability of vortex filaments in the shape of torus knots. We conclude with an overview of some new results concerning electrically charged knots embedded in a viscous fluid, elastic relaxation of strings and braids and relationships between energy levels and topological information. Some simple bounds on elastic energy levels given by global geometric quantities and topological quantities are presented and discussed for the first time.
LA - eng
KW - localized induction for vortex structures; stability of vortex filaments; topological fluid mechanics; topological equivalence of fluid flow maps; linked flux-tubes; Reidemeister's moves; theorem of Lichtenstein; isotopic evolution of vortex structures; Euler equations; helicity; magnetic relaxation; structures in magnetohydrodynamics
UR - http://eudml.org/doc/208816
ER -

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