Nearly spherical vesicles in an external flow

We theoretically analyze a vesicle with small excess area, which is immersed in an external flow. A dynamical equation for the vesicle evolution is obtained by solving the Stokes equation with suitable boundary conditions imposed on the membrane. The equation has solutions corresponding to different types of motion, such as tank-treading, tumbling and trembling. A phase diagram reflecting the regimes is constructed in terms of two dimensionless parameters that depend on the vesicle excess area, the fluid viscosities, the membrane viscosity and bending modulus, the strength of the flow, and the ratio of the elongational and rotational components of the flow. We investigate the peculiarities of the vesicle dynamics near the tank-treading to tumbling and the tank-treading to trembling transitions, which occur via a saddle-node bifurcation and a Hopf bifurcation, respectively. We examine the slowdown of the vesicle dynamics near the merging point and also predict the existence of a novel dynamic regime, which we call spinning.