Affine motion of 2d incompressible fluids and flows in ${\rm SL}(2,{\mathbb R})$
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Affine motion of 2d incompressible fluids and flows in ${\rm SL}(2,{\mathbb R})$

  • Author(s): Roberts, J
  • Shkoller, S
  • Sideris, TC
  • et al.
Abstract

The affine motion of two-dimensional (2d) incompressible fluids can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A complete description of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $t\to\pm\infty$. For MHD, solutions are bounded and generically multiply periodic and recurrent.

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