Séminaire de mathématiques appliquées

The isotropic helicoid is a peculiar object imagined by Kelvin in 1871: it has octahedral chiral symmetry, but behaves isotropically when sinking in a viscous fluid. Since then, successive studies of particle dynamics in flow have repeatedly highlighted the same intriguing observation: the symmetry properties of the particle’s hydrodynamic behaviour are distinct from the geometric symmetry of the particle itself. In this work, I propose to characterise precisely the distinction between hydrodynamic and geometric symmetry, using classical tools of representation theory applied to this fluid mechanics problem. Hence, I will consider a smooth bounded particle in a fluid governed by the Stokes equation, with prescribed Dirichlet conditions corresponding to an imposed ambient flow. The resistance problem then consists in determining the hydrodynamic drag force moments exerted on the particle from the flow velocity data. By linearity of the Stokes equations, these quantities are organised into a hierarchy of linear operators (Rk) called resistance tensors, living in finite-dimensional O(3)-representation spaces. Then, one can define invariant subspaces of Rk for each symmetry subgroup of O(3), and in particular distinguish the visible classes at each level and their dimension. This systematic approach shows in particular that polyhedral symmetries are isotropic at low level and become visible in strict order: beyond the first (linear) level k=1, Kelvin’s helicoid is not isotropic anymore. Further, I will show how this framework can be applied to derive normal forms and new equations of motion, with interesting applications to classical problems of Stokes flow dynamics, such as the Jeffery equations and effective viscosity of dilute suspensions.

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