Dynamics and rheology of a dilute suspension of vesicles: Higher-order theory

Vesicles under shear flow exhibit various dynamics: tank treading (TT), tumbling (TB), and vacillating breathing (VB). The 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 TT to TB if Cₐ≡τ ̇ \.γ is small enough---typically below 0.5, but this value is sensitive to the available excess area from a sphere (τ=vesicle relaxation time towards equilibrium shape, ̇ \.γ=shear rate). At larger Cₐ the TB is preceded by the VB mode. For Cₐ⪢1 we recover the leading order original calculation, where the VB mode coexists with 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 TT-TB and TT-VB bifurcation points. At small Cₐ the minimum corresponds to a cusp singularity and is at the TT-TB threshold, while at high enough Cₐ the cusp is smeared out, and is located in the vicinity of the VB mode but in the TT regime.