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.

We numerically investigated margination of leukocytes at arteriole shear rate in straight circular channels with diameters ranging from 10 to 22 μ m. Our results demonstrated that passing motion of RBCs effectively induces leukocyte margination not only in small channels but also in large channels. A longer time is needed for margination to occur in a larger channel, but once a leukocyte has marginated, passing motion of RBCs occurs continuously independent of the channel diameter, and leukocyte margination is sustained for a long duration. We also show that leukocytes rarely approach the wall surface to within a microvillus length at arteriole shear rate.

The deformability of a red blood cell (RBC) is one of the most important biological parameters affecting blood flow, both in large arteries and in the microcirculation, and hence it can be used to quantify the cell state. Despite numerous studies on the mechanical properties of RBCs, including cell rigidity, much is still unknown about the relationship between deformability and the configuration of flowing cells, especially in a confined rectangular channel. Recent computer simulation techniques have successfully been used to investigate the detailed behavior of RBCs in a channel, but the dynamics of a translating RBC in a narrow rectangular microchannel have not yet been fully understood. In this study, we numerically investigated the behavior of RBCs flowing at different velocities in a narrow rectangular microchannel that mimicked a microfluidic device. The problem is characterized by the capillary number $Ca$ , which is the ratio between the fluid viscous force and the membrane elastic force. We found that confined RBCs in a narrow rectangular microchannel maintained a nearly unchanged biconcave shape at low $Ca$ , then assumed an asymmetrical slipper shape at moderate $Ca$ , and finally attained a symmetrical parachute shape at high $Ca$ . Once a RBC deformed into one of these shapes, it was maintained as the final stable configurations. Since the slipper shape was only found at moderate $Ca$ , measuring configurations of flowing cells will be helpful to quantify the cell state.

We present a numerical analysis of the lateral movement and equilibrium radial positions of red blood cells (RBCs) with major diameter 8 $\mathrm {\mu }$ m under a Newtonian fluid in a circular channel with 50 $\mathrm {\mu }$ m diameter. Each RBC, modelled as a biconcave capsule whose membrane satisfies strain-hardening characteristics, is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$ , the latter of which indicates the ratio of the fluid viscous force to the membrane elastic force. The effects of initial orientation angles and positions on the equilibrium radial position of an RBC centroid are also investigated. The numerical results show that depending on their initial orientations, RBCs have bistable flow modes, so-called rolling and tumbling motions. Most RBCs have a rolling motion. These stable modes are accompanied by different equilibrium radial positions, where tumbling RBCs are further away from the channel axis than rolling ones. The inertial migration of RBCs is achieved by alternating orientation angles, which are affected primarily by the initial orientation angles. Then the RBCs assume the aforementioned bistable modes during the migration, followed by further migration to the equilibrium radial position at much longer time periods. The power (or energy dissipation) associated with membrane deformations is introduced to quantify the state of membrane loads. The energy expenditures rely on stable flow modes, the equilibrium radial positions of RBC centroids, and the viscosity ratio between the internal and external fluids.

Human red blood cells (RBCs) are subjected to high viscous shear stress, especially during microcirculation, resulting in stable deformed shapes such as parachute or slipper shape. Those unique deformed RBC shapes, accompanied with axial or nonaxial migration, cannot be fully described according to traditional knowledge about lateral movement of deformable spherical particles. Although several experimental and numerical studies have investigated RBC behavior in microchannels with similar diameters as RBCs, the detailed mechanical characteristics of RBC lateral movement—in particular, regarding the relationship between stable deformed shapes, equilibrium radial RBC position, and membrane load—has not yet been fully described. Thus, we numerically investigated the behavior of single RBCs with radii of 4 $\mathsf{\mu }$m in a circular microchannel with diameters of 15 $\mathsf{\mu }$m. Flow was assumed to be almost inertialess. The problem was characterized by the capillary number, which is the ratio between fluid viscous force and membrane elastic force. The power (or energy dissipation) associated with membrane deformations was introduced to quantify the state of membrane loads. Simulations were performed with different capillary numbers, viscosity ratios of the internal to external fluids of RBCs, and initial RBC centroid positions. Our numerical results demonstrated that axial or nonaxial migration of RBC depended on the stable deformed RBC shapes, and the equilibrium radial position of the RBC centroid correlated well with energy expenditure associated with membrane deformations.

We present a numerical analysis of the rheology of a suspension of red blood cells (RBCs) for different volume fractions in a wall-bounded, effectively inertialess, small amplitude oscillatory shear (SAOS) flow for a wide range of applied frequencies. The RBCs are modeled as biconcave capsules, whose membrane is an isotropic and hyperelastic material following the Skalak constitutive law. The frequency-dependent viscoelasticity in the bulk suspension is quantified by the complex viscosity, defined by the amplitude of the particle shear stress and the phase difference between the stress and shear. SAOS flow basically impedes the deformation of individual RBCs as well as the magnitude of fluid-membrane interactions, resulting in a lower specific viscosity and first and second normal stress differences than in steady shear flow. Although it is known that the RBC deformation alone is sufficient to give rise to shear-thinning, our results show that the complex viscosity weakly depends on the frequency-modulated deformations or orientations of individual RBCs but rather depends on combinations of the frequency-dependent amplitude and phase difference. The effect of the viscosity ratio between the cytoplasm and plasma and of the capillary number is also assessed.