Dynamics and Rheology of highly deflated vesicles

We study the dynamics and rheology of a single two-dimensional vesicle embedded in a linear shear flow by means of numerical simulations based on the boundary integral method. The viscosities inside and outside the vesicle are supposed to be identical. We explore the rheology by varying the reduced area, i.e. we consider more and more deflated vesicles. Effective viscosity and normal stress differences are computed and discussed in detail, together with the inclination angle and the lateral membrane velocity (tank-treading velocity). The angle is found, surprisingly, to reach a zero value (flow alignment) at a critical reduced area even in the absence of viscosity contrast. A Fast Multipole Method is presented that enables to run efficiently simulations with a large number of vesicles. This method prevails over the direct summation for a number of mesh points beyond a value of about 103. This offers an interesting perspective for simulation of semi-dilute and concentrated suspensions