Dr. Roland Glowinski

In this lecture, we return, in some sense, to one of the Magnus Lectures given some years ago by Tom Mullin, and offer a computational science perspective to this former speaker exciting presentation. Indeed, in this lecture, we investigate computationally the clustering of rigid solid particles in rotating cylinders containing an incompressible viscous fluid, for particle populations ranging from 10 to more than 100.  We study in particular the influence of the angular velocity on these clustering phenomena. The presentation will be illustrated by animations visualizing these truly three-dimensional phenomena, which to the best of our knowledge are not fully understood, as of today. The presentation will include a description of the numerical methodology retained for the solution of the differential system coupling the Navier-Stokes equations modeling the flow, with the Newton-Euler equations describing the particle motion.

Motivated by the numerical simulation of particulate flow with slip-boundary  conditions at the interface fluid-particles, we are going to address in this lecture the  solution of the following elliptic problem 

         (1)
     (2)

by a fictitious domain method (new to the best of our knowledge); in (1), (2),  Ω denotes  a bounded domain of Rd and  ω a sub-domain of Ω. Our approach relies essentially on the   transformation of (1), (2) in a (virtual) control problem (in the sense of J.L. Lions),   involving an extension of (1) (completed by u =g0 on ∂Ω) on the whole Ω, the restriction of the extended solution to  being the solution of (1), (2).  From an algorithmic point of view, one solves the control problem by a least-squares/conjugate gradient algorithm whose finite element implementation is rather easy, even if the mesh associated with Ω does not match the geometry of ω. 
Numerical experiments, including the generalization to the solution of parabolic equations with moving ω, suggest optimal orders of convergence.