Dynamic Modes of Red Blood Cells in Oscillatory Shear Flow

Abstract

The dynamics of red blood cells (RBCs) in oscillatory shear flow was studied using differential equations of three variables: a shape parameter, the inclination angle $\theta$, and phase angle $\phi$ of the membrane rotation. In steady shear flow, three types of dynamics occur depending on the shear rate and viscosity ratio. i) tank-treading (TT): $\phi$ rotates while the shape and $\theta$ oscillate. ii) tumbling (TB): $\theta$ rotates while the shape and $\phi$ oscillate. iii) intermediate motion: both $\phi$ and $\theta$ rotate synchronously or intermittently. In oscillatory shear flow, RBCs show various dynamics based on these three motions. For a low shear frequency with zero mean shear rate, a limit-cycle oscillation occurs, based on the TT or TB rotation at a high or low shear amplitude, respectively. This TT-based oscillation well explains recent experiments. In the middle shear amplitude, RBCs show an intermittent or synchronized oscillation. As shear frequency increases, the vesicle oscillation becomes delayed with respect to the shear oscillation. At a high frequency, multiple limit-cycle oscillations coexist. For a high mean shear rate with small shear oscillation, the shape and $\theta$ oscillate in the TT motion but only one attractor exists even at high shear frequencies. The measurement of these oscillatory modes is a promising tool for quantifying the viscoelasticity of RBCs and synthetic capsules.