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Research Activities > Programs > Incompressible Flows 2006> John Gibbon

Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

CSIC Building (#406), Seminar Room 4122.
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The 3D Euler Equations and Quaternions

John Gibbon

  Department of Mathematics, Imperial College London

Abstract:  More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It will be shown that they provide a natural way of selecting an appropriate ortho-normal frame -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations will be considered. This work may have some bearing on the issue of whether the Euler equations develop a singularity in a finite time. If time permits some of the literature on this topic will be reviewed, which will include both the BKM theorem and associated work on the direction of vorticity by both Constantin, Fefferman & Majda and Deng, Hou and Yu. It will then shown how the quaternion formulation provides a further direction of vorticity result using the Hessian of the pressure.