Trajectories of fluid particles around spheres and in a water wave

F7.largeHisashi Okamoto gave a talk on Tuesday 22 July on the dynamics of fluid particles.  The first half of the talk was on particle motions induced by the motion of a sphere along a straight line in a fluid.  The starting point was the 1870 paper of James Clerk Maxwell which showed that the particle motion was integrable for 2D irrotational flow, giving rise to the curves of the elastica.  This work was then generalised to the equation of Brinkman (1949).  This equation has a parameter which includes irrotational flow as one limit and Stokes flow as the other.  A range of examples of non-integrable particle paths were shown.  These results can be found in the paper Shoji, Okamoto & Ooura (2010).  The second half of the talk was on particle motion in water waves.  First a new proof of global Stokes drift, using the conformal mapping equations of Levi-Civita, was presented.  Then results of particle paths for various parameters and wave amplitude were presented.  The most novel particle paths were those obtained in Crapper waves and near a Wilton resonance.  These results can be found summarised in Okamoto & Shoji (2012). A video of the talk is available here.

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