Rheology of a dilute suspension of vesicles

From the hydrodynamical equations of vesicle dynamics under shear flow, we extract a rheological law for a dilute suspension. This is made analytically in the small excess area limit. In contrast to droplets and capsules, the rheological law (written in the comoving frame) is nonlinear even to the first leading order. We exploit it by evaluating the effective viscosity eta(eff) and the normal stress differences N-1 and N-2. We make a link between rheology and microscopic dynamics. For example, eta(eff) is found to exhibit a cusp singularity at the tumbling threshold, while N-1,N-2 undergoes a collapse.