Rheology and dynamics of vesicle suspension in comparison with droplet emulsion

Abstract

This paper deals with the study of dynamics and rheology of a dilute suspension of vesicles. The study is analytical and is based on the small deformation theory. Vesicles in the small deformation limit exhibit, under shear flow, rich dynamics in comparison to droplets (in the regime where the droplet maintains its integrity). For example, droplets only assume a fixed orientation with respect to the flow, while vesicles undergo three types of motions: (i) tank-treading (tt), where the vesicle assumes a fixed orientation with respect to the flow, while the membrane makes a tank-treading motion, (ii) tumbling (tb), which occurs as a saddle-node bifurcation from the tank-treading motion for a certain critical viscosity ratio, (iii) vacillating-breathing (vb), where the vesicle long axis undergoes oscillations around the shear direction whereas its shape executes a breathing-like motion. This mode is found to coexist with tumbling in the high shear rate limit (or high capillary number Ca ≡ γ ˙ τ , where γ ˙ is the shear rate and τ is the relaxation time towards equilibrium shape of the vesicle). After analyzing these modes and comparing dynamics to droplets, we study rheology. It is found that the constitutive law, written in the co-moving frame, is nonlinear even to leading order. This markedly contrasts with droplet emulsion where the equation is linear to leading order. We make a link between rheology and the above three dynamical states. It is found that the effective viscosity undergoes a cusp singularity at the tumbling bifurcation (which happens at small enough Ca), while the normal stress differences collapse in the tumbling and vb regimes. At high enough Ca the tb transition is preceded by the vb mode. There we find that the apparent viscosity (defined as the averaged shear stress over the applied shear rate; which is distinct from the shear viscosity defined by the limit of this ratio at zero shear rate) exhibits a minimum at the vicinity of the vb boundary. We also report on shear thinning and the behavior of the normal stress difference as a function of γ ˙ .