Tumbling of vesicles under shear flow within an advected-field approach

We study dynamics of vesicles with a viscosity contrast between the interior and exterior, and subjected to a linear shear flow. It is shown that the vesicle, which assumes a steady orientation with respect to the shear flow for r=eta(in)/eta(out)<r(c) (where eta(in) and eta(out) are the internal and external viscosities, and r(c) is a critical value which depends upon the swelling factor), undergoes a tumbling bifurcation for r >= r(c). Tumbling occurs as a saddle-node bifurcation. We present analytical and numerical results. We develop a powerful method, the advected-field approach. This method allows one to treat several different phenomena with great flexibility.