Motion of a spherical microcapsule freely suspended in a linear shear flow

Abstract

The motion of a spherical microcapsule freely suspended in a simple shear flow is studied. The particle consists of a thin elastic spherical membrane enclosing an incompressible Newtonian viscous fluid. The motions of the internal liquid and of the suspending fluid are both described by Stokes equations. On the deformed surface of the membrane, continuity of velocities is imposed together with dynamic equilibrium of viscous and elastic forces. Since this problem is highly nonlinear, a regular perturbation solution is sought in the limiting case where the deviation from sphericity is small. In particular, the nonlinear theory of large deformation of membrane shells is expanded up to second-order terms. The deformation and orientation of the microcapsule are obtained explicitly in terms of the magnitude of the shear rate, the elastic coefficients of the membrane, the ratio of internal to external viscosities. It appears that the very viscous capsules are tilted towards the streamlines, whereas the less viscous particles are oriented at nearly 45° to the streamlines. The tank-treading motion of the membrane around the liquid contents is predicted by the model and appears as the consequence of a solid-body rotation superimposed upon a constant elastic deformation.