Motion of a capsule in a cylindrical tube: effect of membrane pre-stress

We present a numerical model of the axisymmetric flow of an initially spherical capsule in a co-axial cylindrical tube. The capsule consists of a liquid droplet enclosed by a thin hyper-elastic membrane that is assumed to obey different membrane constitutive equations such as Mooney–Rivlin, Skalak et al. (1973) or Evans & Skalak (1980) laws. It is further assumed that the capsule may be subjected to some isotropic pre-stress due to initial swelling. We compute the steady flow of the capsule inside the tube as a function of the size ratio between the capsule and tube radii, the amount of pre-swelling and the membrane constitutive law. We thus determine the deformed profile geometry and specifically the onset of the curvature inversion at the back of the particle. We show that for a given size ratio, the critical flow rate at which the back curvature changes is strongly dependent on pre-inflation. The elastic tension level in the membrane as well as the additional pressure drop created by the presence of the particle are also computed. The numerical results are then compared to experimental observations of capsules with alginate membranes as they flow in small tubes (Risso. et al. 2006). It is found that the experimental capsules were probably pre-inflated by about 3% and that their membrane is best modelled by the Skalak et al. law.