Rheology and shape transitions of vesicles under capillary flow

We present an analytical description of the rheology and shape of axisymmetric vesicles flowing down narrow capillaries. The vesicle surface is described by the Helfrich bending energy. We find that the rheological properties of the vesicle are independent of the Helfrich bending energy. The classical Bretherton theory for tense drops can be applied provided we replace the drop tension with a “dynamical tension” discussed in the text. Darcyʹs Law is obeyed with an effective permeability which depends on the filling fraction and the dimensions of the vesicle and the pore. For vesicles with tension, there are two rheological regimes. At low applied pressure heads, the vesicle moves very slowly and violates Darcyʹs Law. With increasing-pressure gradient, there is a singular point beyond which the rear of the vesicle becomes tensionless and Darcyʹs Law is obeyed. This singular point marks a whole sequence of shapes transitions of the vesicle, starting from a sphero-cylinder and ending in a Bell shape, similar to those reported for red blood cells in the physiological literature.