An open question that has persisted for decades is whether the cytoskeleton of a red blood cell is stress-free or under a stress. This question is important in the context of theoretical modeling of cellular motion under a flowing condition where it is necessary to make an assumption about the stress-free state. Here, we present a 3D numerical study to compare the cell dynamics in a simple shear flow under two different stress-free states, a biconcave discocyte representing the resting shape of the cell, and a nearly spherical oblate shape. We find that whether the stress-free states make a significant difference or not depends on the viscosity of the suspending medium. If the viscosity is close to that of blood plasma, the two stress-free states do not show any significant difference in cell dynamics. However, when the suspending medium viscosity is well above that of the physiological range, as in many in vitro studies, the shear rate separating the tank-treading and tumbling dynamics is observed to be higher for the biconcave stress-free state than the spheroidal state. The former shows a strong shape oscillation with repeated departures from the biconcave shape, while the latter shows a nearly stable biconcave shape. It is found that the cell membrane in the biconcave stress-free state is under a compressive stress and a weaker bending force density, leading to a periodic compression of the cell. The shape oscillation then leads to a higher energy barrier against membrane tank-tread leading to an early transition to tumbling. However, if the cells are released with a large off-shear plane angle, the oscillations can be suppressed due to an azimuthal motion of the membrane along the vorticity direction leading to a redistribution of the membrane points and lowering of the energy barrier, which again results in a nearly similar behavior of the cells under the two different stress-free states. A variety of off-shear plane dynamics is observed, namely, rolling, kayaking, precession, and a new dynamics termed “hovering.” For the physiological viscosity range, the shear-plane tumbling appears to be relatively less common, while the rolling is observed to be more stable.

Many numerical studies have considered the dynamics of capsules and red blood cells in shear flow under the condition that the axis of revolution of such bodies remained aligned in the shear plane. In contrast, several experimental studies have shown that the axis of revolution of red blood cells could drift away from the shear plane in a certain range of controlling parameters. In this article, we present three-dimensional numerical simulations on the orientation dynamics of capsules in simple shear flow with different initial undeformed shapes, namely, prolate, oblate, and biconcave disk. It is observed that unlike rigid ellipsoids in Stokes flow, capsules reorient their axis of revolution either towards the vorticity axis while undergoing a precessing motion or towards the shear plane while undergoing a kayaking-type motion. The specific dynamics are observed to depend on initial shape, capillary number, and the ratio of the internal to external fluid viscosity. Near the physiological values of the viscosity ratio, the biconcave shape performs a rolling motion like a wheel. If the viscosity ratio is reduced below the physiological range, a transition to the kayaking dynamics is observed with increasing capillary number. The critical shear stress at which the rolling-to-kayaking transition occurs is found to be dependent on the viscosity ratio. C 2013 AIP Publishing LLC. [http://dx.

We present the first full-scale computational evidence of intermittent and synchronized dynamics of red blood cells in shear flow. These dynamics are characterized by the coexistence of a tumbling motion in which the cell behaves like a rigid body and a tank-treading motion in which the cell behaves like a liquid drop. In the intermittent dynamics, we observe sequences of tumbling interrupted by swinging, as well as sequences of swinging interrupted by tumbling. In the synchronized dynamics, the tumbling and membrane rotation are observed to occur simultaneously with integer ratios of the rotational frequencies. These dynamics are shown to be dependent on the stress-free state of the cytoskeleton, and are explained based on the cell membrane energy landscape.

Red blood cells (RBCs) undergo remarkably large deformations when subjected to external forces but return to their biconcave discoid resting shape as the forces are withdrawn. In many experiments, such as when RBCs are subjected to a shear flow and undergo the tank-treading motion, the membrane elements are also displaced from their original (resting) locations along the cell surface with respect to the cell axis, in addition to the cell being deformed. A shape memory is said to exist if after the flow is stopped the RBC regains its biconcave shape and the membrane elements also return to their original locations. The shape memory of RBCs was demonstrated by Fischer [“Shape memory of human red blood cells,” Biophys. J. 86, 3304–3313 (2004)] using shear flow go-and-stop experiments. Optical tweezer and micropipette based stretch-relaxation experiments do not reveal the complete shape memory because while the RBC may be deformed, the membrane elements are not significantly displaced from their original locations with respect to the cell axis. Here we present the first three-dimensional computational study predicting the complete shape memory of RBCs using shear flow go-and-stop simulations. The influence of different parameters, namely, membrane shear elasticity and bending rigidity, membrane viscosity, cytoplasmic and suspending fluid viscosity, as well as different stress-free states of the RBC is studied. For all cases, the RBCs always exhibit shape memory. The complete recovery of the RBC in shear flow go-and-stop simulations occurs over a time that is orders of magnitude longer than that for optical tweezer and micropipette based relaxations. The response is also observed to be more complex and composed of widely disparate time scales as opposed to only one time scale that characterizes the optical tweezer and micropipette based relaxations. We observe that the recovery occurs in three phases: a rapid compression of the RBC immediately after the flow is stopped, followed by a slow recovery to the biconcave shape combined with membrane rotation, and a final rotational return of the membrane elements back to their original locations. A fast time scale on the order of a few hundred milliseconds characterizes the initial compression phase while a slow time scale on the order of tens of seconds is associated with the rotational phase. We observe that the response is strongly dependent on the stress-free state of the cells, that is, the relaxation time decreases significantly and the mode of recovery changes from rotation-driven to deformation-driven as the stress-free state becomes more non-spherical. We show that while membrane shear elasticity and non-spherical stress-free shape are necessary and sufficient for the membrane elements to return to their original locations, bending rigidity is needed for the “global” recovery of the biconcave shape. We also perform a novel relaxation simulation in which the cell axis of revolution is not aligned with the shear plane and show that the shape memory is exhibited even when the membrane elements are displaced normal to the imposed flow direction. The results presented here could motivate new experiments to determine the exact stress-free state of the RBC and also to clearly identify different tank-treading modes.

We present a three-dimensional computational study of fully deformable red blood cells of the biconcave resting shape subject to sinusoidally oscillating shear flow. A comprehensive analysis of the cell dynamics and deformation response is considered over a wide range of flow frequency, shear rate amplitude and viscosity ratio. We observe that the cell exhibits either a periodic motion or a chaotic motion. In the periodic motion, the cell reverses its orientation either by passing through the flow direction (horizontal axis) or by passing through the flow gradient (vertical axis). The chaotic dynamics is characterized by a non-periodic sequence of horizontal and vertical reversals. The study provides the first conclusive evidence of the chaotic dynamics of fully deformable cells in oscillating flow using a deterministic numerical model without the introduction of any stochastic noise. In certain regimes of the periodic motion, the initial conditions are completely forgotten and the cells become entrained in the same sequence of horizontal reversals. We show that chaos is only possible in certain frequency bands when the cell membrane can rotate by a certain amount, allowing the cells to swing near the maximum shear rate. As such, the bifurcation between the horizontal and vertical attractors in phase space always occurs via a swinging inflection. While the reversal sequence evolves in an unpredictable way in the chaotic regime, we find a novel result that there exists a critical inclination angle at the instant of flow reversal which determines whether a vertical or horizontal reversal takes place, and is independent of the flow frequency. The chaotic dynamics, however, occurs at a viscosity ratio less than the physiological values. We further show that the cell shape in oscillatory shear at large amplitude exhibits a remarkable departure from the biconcave shape, and that the deformation is significantly greater than that in steady shear flow. A large compression of the cells occurs during the reversals which leads to over/undershoots in the deformation parameter. We show that due to the large deformation experienced by the cells, the regions of chaos in parameter space diminish and eventually disappear at high shear rate, in contradiction to the prediction of reduced-order models. While the findings bolster support for reduced-order models at low shear rate, they also underscore the important role that the cell deformation plays in large-amplitude oscillatory flows.