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SUMMARY:New conservation laws of helical flows - Oberlack\, M (TU Darmstad
 t)
DTSTART:20120723T130000Z
DTEND:20120723T134000Z
UID:TALK39016@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Conservation laws in incompressible fluid dynamics\, in partic
 ular inviscid motion\, constitute an axiomatic basis for fluid mechanics. 
 In 3D\, mass and momentum conservation forms the fundamental basis\, which
  is further extended by energy\, vorticity and helicity conservation. Inte
 resting enough considering reduced dimensions a much broader set of conser
 ved quantities is observed in particularly for 2D/planar and axisymmetric 
 flows. For the planar case it is well known that any once differential fun
 ction of the vorticity is a materially conserved quantity and hence an inf
 inite number of additional conservation laws exist. Further\, the most sim
 ple one\, the square of the vorticity\, is named enstrophy\, and is weakly
  conserved in the viscous case and constitutes a fundamental invariant for
  2D turbulence. Recently we have shown that the known set of additional co
 nservation laws may be considerably extended for helical flows which const
 itute a Lie symmetry induced concaten ation of planar and axisymmetric flo
 ws living on the (r\, a z + b phi\; t) spatially reduced system with a^2 +
  b^2 > 0 and r\, z\, phi are the classical coordinates in a cylinder coord
 inate system. Various infinity dimensional new conservations laws have bee
 n established including e.g. a generalized helicity. Even for the 2D/plana
 r and axisymmetric flows new conservation laws have been derived not repor
 ted in the literature before. The construction of the new results is based
  on the direct method. It relies on two key theorems: (i) the Euler operat
 or applied to a term is always zero if and only if the term is in divergen
 ce form\; (ii) any non-trivial conservation law of a given set of differen
 tial equations can only be constructed by a linear combination of the give
 n equations with some multipliers to be determined by theorem (i). This is
  a necessary and sufficient condition. The process of finding the new cons
 ervation laws was aided by the computer algebra system Maple employing the
  package GeM by A. Cheviakov.\n
LOCATION:Seminar Room 1\, Newton Institute
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