J. Fluid Mech. (2007), vol. 573, pp. 149–169. c  2007 Cambridge University Press doi:10.1017/S0022112006003739 Printed in the United Kingdom 149 Hydrodynamic interaction between two identical capsules in simple shear flow ETIENNE LAC, ARNAUD MOREL AND DOMINIQUE BARTH ` ES-BIESEL UMR CNRS 6600, Biom´ ecanique et G´ enie Biom´ edical, Universit´ e de Technologie de Compi` egne, France (Received 7 December 2005 and in revised form 1 August 2006) We present a numerical model of the hydrodynamic interactions between two capsules freely suspended in a simple shear flow. The capsules are identical and each consists of a liquid droplet enclosed by a thin hyperelastic membrane, devoid of bending resistance and obeying a neo-Hookean constitutive law. The two capsules are slightly prestressed with a given inflation ratio in order to avoid the small deformation instability due to compression observed for a single capsule in simple shear flow. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping flow conditions are assumed to prevail. The boundary- element method is used with bi-cubic B-splines as basis functions on a structured mesh in order to discretize the capsule surface. A new method using two grids with initially orthogonal pole axes is developed to eliminate polar singularities in the load calculation and to allow for long computation times. Two capsules suspended in simple shear flow usually have different velocities and thus eventually pass each other. We study this crossing process as a function of flow strength and initial particle separation. We find that hydrodynamic interactions during crossing lead to large shape alterations, elevated elastic tensions in the membrane and result in an irreversible trajectory shift of the capsules. Furthermore, a tendency towards buckling is observed, particularly during the separation phase where large pressure differences occur. Our results are in qualitative agreement with those obtained for a pair of interacting liquid droplets but show the specific role played by the membrane of capsules. 1. Introduction Artificial capsules consisting of a liquid droplet enclosed by a thin elastic membrane are widely used in many industrial processes (cosmetics, pharmaceuticals, food industry). The role of the membrane is to protect the internal contents, to deliver them through capsule breakup under specific conditions, or to control mass transfer between the internal and external media. We consider here initially spherical liquid- filled capsules designed to be used in suspension in another liquid. As compared to a simple liquid droplet, the capsule has a more complex dynamic behaviour owing to the presence of the membrane that may buckle under compression or burst when the elastic tensions exceed a critical level. There have been several numerical, experimental and theoretical studies of the deformation of a single capsule suspended in a flow. For example, the deformation of an isolated capsule freely suspended in a simple shear flow has been measured by Chang & Olbricht (1993) and by Walter, Rehage & Leonhard (2000, 2001) for different