Vesicles under simple shear flow: elucidating the role of relevant control parameters

The dynamics of vesicles under shear flow are carefully analyzed in the regime of a small vesicle excess area relative to a sphere. This regime corresponds to the quasispherical limit, for which several groups have analytically extracted simple nonlinear differential equations. Under shear flow, vesicles are known to exhibit three types of motion: (i) tank-treading (TT): the vesicle assumes a steady inclination angle with respect to the flow direction, while its membrane undergoes a tank-treading motion, (ii) tumbling (TB), and (iii) vacillating-breathing (VB): the vesicle main axis oscillates about the flow direction, whereas the overall shape undergoes a breathinglike motion. The region of existence for each regime depends on material and control parameters. The whole set of parameters can be cast into three dimensionless control parameters: (i) the viscosity ratio between the internal and external fluid, lambda , (ii) the excess area relative to a sphere (this parameter measures the degree of the vesicle deflation), Delta , and (iii) the capillary number (the ratio between the vesicle relaxation time toward its equilibrium shape after cessation of the flow and the flow time scale, which is the inverse shear rate), Ca. Recent studies [Danker, Phys. Rev. E 76, 041905 (2007)] have focused on the shape of the phase diagram (representing the TT, TB, and VB regimes in the Ca-lambda plane). In this paper, the physical quantities are analyzed in detail and attention is brought to features that are essential for future experimental studies. It is shown that the boundaries delimiting different dynamical regimes (TT, TB, and VB) in parameter space depend on the three dimensionless control parameters, in contrast with a recent study [V. V. Lebedev, Phys. Rev. Lett. 99, 218101 (2007)] where it is claimed that only two parameters are relevant. Consideration of the amplitude of oscillation (of the vesicle orientation angle and its shape deformation) in the VB mode reveals an even more significant dependence on the three parameters. It is also shown that the inclination angle in the TT regime significantly depends on the shear rate (Ca), which runs contrary to common belief. Finally, we show that the TB and VB periods are quite insensitive to Ca, in marked contrast with a recent study [H. Noguchi and G. Gompper, Phys. Rev. Lett. 98, 128103 (2007)].