Haemorheology in dilute, semi-dilute and dense suspensions of red blood cells

Abstract

We present a numerical analysis of the rheology of a suspension of red blood cells (RBCs) in a wall-bounded shear flow. The flow is assumed as almost inertialess. The suspension of RBCs, modelled as biconcave capsules whose membrane follows the Skalak constitutive law, is simulated for a wide range of viscosity ratios between the cytoplasm and plasma,$\unicode[STIX]{x1D706}=0.1$–10, for volume fractions up to$\unicode[STIX]{x1D719}=0.41$and for different capillary numbers ($Ca$). Our numerical results show that an RBC at low$Ca$tends to orient to the shear plane and exhibits so-called rolling motion, a stable mode with higher intrinsic viscosity than the so-called tumbling motion. As$Ca$increases, the mode shifts from the rolling to the swinging motion. Hydrodynamic interactions (higher volume fraction) also allow RBCs to exhibit tumbling or swinging motions resulting in a drop of the intrinsic viscosity for dilute and semi-dilute suspensions. Because of this mode change, conventional ways of modelling the relative viscosity as a polynomial function of$\unicode[STIX]{x1D719}$cannot be simply applied in suspensions of RBCs at low volume fractions. The relative viscosity for high volume fractions, however, can be well described as a function of an effective volume fraction, defined by the volume of spheres of radius equal to the semi-middle axis of a deformed RBC. We find that the relative viscosity successfully collapses on a single nonlinear curve independently of$\unicode[STIX]{x1D706}$except for the case with$Ca\geqslant 0.4$, where the fit works only in the case of low/moderate volume fraction, and fails in the case of a fully dense suspension.