Vesicles under shear flow exhibit various dynamics: tank treading $\left(\mathit{TT}\right)$, tumbling $\left(\mathit{TB}\right)$, and vacillating breathing $\left(\mathit{VB}\right)$. The $\mathit{VB}$ mode consists in a motion where the long axis of the vesicle oscillates about the flow direction, while the shape undergoes a breathing dynamics. We extend here the original small deformation theory [C. Misbah, Phys. Rev. Lett. 96, 028104 (2006)] to the next order in a consistent manner. The consistent higher order theory reveals a direct bifurcation from $\mathit{TT}$ to $\mathit{TB}$ if $C_a\equiv \tau \dot{\gamma }$ is small enough—typically below 0.5, but this value is sensitive to the available excess area from a sphere ($\tau =\text{vesicle}$ relaxation time towards equilibrium shape, $\dot{\gamma }=\text{shear}$ rate). At larger $C_a$ the $\mathit{TB}$ is preceded by the $\mathit{VB}$ mode. For $C_a⪢1$ we recover the leading order original calculation, where the $\mathit{VB}$ mode coexists with $\mathit{TB}$. The consistent calculation reveals several quantitative discrepancies with recent works, and points to new features. We briefly analyze rheology and find that the effective viscosity exhibits a minimum in the vicinity of the $\mathit{TT}\text{−}\mathit{TB}$ and $\mathit{TT}\text{−}\mathit{VB}$ bifurcation points. At small $C_a$ the minimum corresponds to a cusp singularity and is at the $\mathit{TT}\text{−}\mathit{TB}$ threshold, while at high enough $C_a$ the cusp is smeared out, and is located in the vicinity of the $\mathit{VB}$ mode but in the $\mathit{TT}$ regime.

• Received 26 March 2007

DOI:https://doi.org/10.1103/PhysRevE.76.041905