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 $r\equiv {\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}<r_c$ (where ${\eta }_{\mathrm{in}}$ and ${\eta }_{\mathrm{out}}$ 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.

• Received 15 October 2001

DOI:https://doi.org/10.1103/PhysRevE.67.031908