The dynamics of vesicles under a shear flow are analyzed analytically in the small deformation regime. We derive two coupled nonlinear equations which describe the vesicle orientation in the flow and its shape evolution. A new type of motion is found, namely, a "vacillating-breathing" mode: the vesicle orientation undergoes an oscillation around the flow direction, while the shape executes breathing dynamics. This solution coexists with tumbling. Moreover, we provide an explicit expression for the tumbling threshold. A rheological law for a dilute vesicle suspension is outlined.

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We investigate the dynamics of a vesicle in a shear flow on the basis of the newly proposed advected field (AF) method [T. Biben and C. Misbah, Eur. Phys. J. E 67, 031908 (2003)]. We also solve the same problem with the boundary integral formulation for the sake of comparison. We find that the AF results presented previously overestimated the tumbling threshold due to the finite size of the membrane, inherent to the AF model. A comparison between the two methods shows that only in the sharp interface limit (extrapolating the results to a vanishing width) the AF method leads to accurate quantitative results. We extensively investigate the tank-treading to tumbling transition, and compare our numerical results to the theory of Keller and Skalak which assumes a fixed ellipsoidal shape for the vesicle. We show that this theory describes correctly the two regimes, at least in two dimensions, even for the quite elongated non-convex shapes corresponding to red blood cells (and therefore far from ellipsoidal), This theory is, however, not fully quantitative. Finally we investigate the effect of a confinement on the tank-treading to tumbling transition, and show that the tumbling regime becomes unfavorable in a capillary vessel, which should have strong effects on blood rheology in confined geometries.