Modelling and simulation of 2D stokesian Squirmers^*

¹ Université Paris-Sud 11, Département de Mathématiques, Bâtiment 425, 91405 Orsay cedex ( France )
² Max-Planck Institut fur Dynamik und Selbstorganisation, Am Fassberg 17, 37077 Gottingen ( Germany )

Abstract

Direct numerical simulations of the individual and collective dynamics of neutral squirmers are presented. “Squirmer” refers to a class of swimmers driven by prescribed tangential deformations at their surface, and “cneutral” means that the swimmer does not apply a force dipole on the fluid. The squirmer model is used in this article to describe self-propelled liquid droplets. Each swimmer is a fluid sphere in Stokes flow without radial velocity and with a prescribed tangential velocity, which is constant in time in the swimmer frame. The interaction between two or more swimmers is taken into account through the relaxation of their translational and angular velocities. The algorithm presented for solving the fluid flow and the motion of the liquid particles is based on a variational formulation written on the whole domain (including the external fluid and the liquid particles) and on a fictitious domain approach. The constraint on the tangential velocity of swimmers can be enforced using two different methods: penalty approach of the strain rate tensor on the particles domain, or a saddle-point formulation involving a Lagrange multiplier associated to the constraint. This leads to a minimization problem over unconstrained functional spaces that can be implemented straightforwardly in a finiteelement multi-purpose solver. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. Two-dimensional numerical simulations implemented with FreeFem++ are presented.