Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell

Numerical simulation of transport and adhesion of thermogenic nano-carriers in microvessels

Externally triggered thermogenic nanoparticles (NPs) are potential drug carriers and heating agents for drug delivery and hyperthermia. A good understanding of the transport and adhesion behaviors of NPs in microvessels is conducive to improving the efficiency of NP-mediated treatment. Given the thermogenesis of NPs and interactions of NP-blood flow, NP-NP, NP-red blood cell (RBC) and ligand-receptor, the movement of NPs in blood flow was modeled using a hybrid immersed boundary and coupled double-distribution-function lattice Boltzmann method. Results show that the margination probability of NPs toward the vessel wall was evidently increased by NP thermogenesis owing to the noticeable variation in blood flow velocity distribution, thereby enhancing their adhesion to the target region. NP-RBC collision can promote NP movement to the acellular layer in microvessels to increase the NP adhesion rate. The number of adhered smaller NPs was larger than that of the larger NPs having the same ligand density due to the enhancement of Brownian force although their adhesion was relatively less firm. Compared with the NPs with a regular shape, the irregularly shaped NPs can adhere to the vessel wall more readily and strongly as a result of the higher turbulence levels caused by NP-blood flow interaction and relatively higher ligand density, which led to a higher rate of NP adhesion.

Research on Human Erythrocyte's Threshold Free Energy for Hemolysis and Damage from Coupling Effect of Shear and Impact Based on Immersed Boundary-Lattice Boltzmann Method

Researches on the principle of human red blood cell's (RBC) injuring and judgment basis play an important role in decreasing the hemolysis in a blood pump. In the current study, the judgment of hemolysis in a blood pump study was through some experiment data and empirical formula. The paper forms a criterion of RBC's mechanical injury in the aspect of RBC's free energy. First, the paper introduces the nonlinear spring network model of RBC in the frame of immersed boundary-lattice Boltzmann method (IB-LBM). Then, the shape, free energy, and time needed for erythrocyte to be shorn in different shear flow and impacted in different impact flow are simulated. Combining existing research on RBC's threshold limit for hemolysis in shear and impact flow with this paper's, the RBC's free energy of the threshold limit for hemolysis is found to be 3.46 × 10⁻¹⁵ J. The threshold impact velocity of RBC for hemolysis is 8.68 m/s. The threshold value of RBC can be used for judgment of RBC's damage when the RBC is having a complicated flow of blood pumps such as coupling effect of shear and impact flow. According to the change law of RBC's free energy in the process of being shorn and impacted, this paper proposed a judging criterion for hemolysis when the RBC is under the coupling effect of shear and impact based on the increased free energy of RBC.

Shear stress in the microvasculature: influence of red blood cell morphology and endothelial wall undulation

The effect of red blood cells and the undulation of the endothelium on the shear stress in the microvasculature is studied numerically using the lattice Boltzmann-immersed boundary method. The results demonstrate a significant effect of both the undulation of the endothelium and red blood cells on wall shear stress. Our results also reveal that morphological alterations of red blood cells, as occur in certain pathologies, can significantly affect the values of wall shear stress. The resulting fluctuations in wall shear stress greatly exceed the nominal values, emphasizing the importance of the particulate nature of blood as well as a more realistic description of vessel wall geometry in the study of hemodynamic forces. We find that within the channel widths investigated, which correspond to those found in the microvasculature, the inverse minimum distance normalized to the channel width between the red blood cell and the wall is predictive of the maximum wall shear stress observed in straight channels with a flowing red blood cell. We find that the maximum wall shear stress varies several factors more over a range of capillary numbers (dimensionless number relating strength of flow to membrane elasticity) and reduced areas (measure of deflation of the red blood cell) than the minimum wall shear stress. We see that waviness reduces variation in minimum and maximum shear stresses among different capillary and reduced areas.

Transport mechanism of deformable micro-gel particle through micropores with mechanical properties characterized by AFM

Deformable micro-gel particles (DMP) have been used to enhanced oil recovery (EOR) in reservoirs with unfavourable conditions. Direct pore-scale understanding of the DMP transport mechanism is important for further improvements of its EOR performance. To consider the interaction between soft particle and fluid in complex pore-throat geometries, we perform an Immersed Boundary-Lattice Boltzmann (IB-LB) simulation of DMP passing through a throat. A spring-network model is used to capture the deformation of DMP. In order to obtain appropriate simulation parameters that represent the real mechanical properties of DMP, we propose a procedure via fitting the DMP elastic modulus data measured by the nano-indentation experiments using Atomic Force Microscope (AFM). The pore-scale modelling obtains the critical pressure of the DMP for different particle-throat diameter ratios and elastic modulus. It is found that two-clog particle transport mode is observed in a contracted throat, the relationship between the critical pressure and the elastic modulus/particle-throat diameter ratio follows a power law. The particle-throat diameter ratio shows a greater impact on the critical pressure difference than the elastic modulus of particles.

EFFECT OF A HIGH-FREQUENCY VIBRATION BOUNDARY ON RBC

The vibrations in blood pumps were often caused by high speed, suspension structure, viscoelastic implantation environment and other factors in practical application. Red blood cell (RBC) was modeled using a nonlinear spring network model. The immersed boundary-lattice Boltzmann method (IB-LBM) was used to investigate the impact of high-frequency vibration boundary on RBC. To confirm the RBC model, the simulation results of RBC stretching were compared with experimental results. We examined the force acting on RBC membrane nodes; moreover, we determined whether RBC energy was affected by different frequencies, amplitudes, and vibration models of the boundary. Furthermore, we examined whether RBC energy was affected by the distance between the top and bottom boundaries. The energy of RBCs in shear flow disturbed by the vibration boundary was also investigated. The results indicate that larger amplitude (Am), frequency (Fr), and opposite vibration velocity of top and bottom boundary produced a larger force that acted on RBC membrane nodes and larger energy changes in RBCs. The vibration boundary may cause turbulence and alter RBC energy. When the blood pump was designed and optimized, the vibration frequency and amplitude of the blood pump body and impeller should be reduced, the phase of the blood pump body and impeller vibration velocity should be close. To alleviate the free energy of RBCs and to reduce RBC injury in the blood pump, the distance between RBCs and the boundary should not be less than 20[Formula: see text][Formula: see text]m.

ATP Release by Red Blood Cells under Flow: Model and Simulations

ATP is a major player as a signaling molecule in blood microcirculation. It is released by red blood cells (RBCs) when they are subjected to shear stresses large enough to induce a sufficient shape deformation. This prominent feature of chemical response to shear stress and RBC deformation constitutes an important link between vessel geometry, flow conditions, and the mechanical properties of RBCs, which are all contributing factors affecting the chemical signals in the process of vasomotor modulation of the precapillary vessel networks. Several in vitro experiments have reported on ATP release by RBCs due to mechanical stress. These studies have considered both intact RBCs as well as cells within which suspected pathways of ATP release have been inhibited. This has provided profound insights to help elucidate the basic governing key elements, yet how the ATP release process takes place in the (intermediate) microcirculation zone is not well understood. We propose here an analytical model of ATP release. The ATP concentration is coupled in a consistent way to RBC dynamics. The release of ATP, or the lack thereof, is assumed to depend on both the local shear stress and the shape change of the membrane. The full chemo-mechanical coupling problem is written in a lattice-Boltzmann formulation and solved numerically in different geometries (straight channels and bifurcations mimicking vessel networks) and under two kinds of imposed flows (shear and Poiseuille flows). Our model remarkably reproduces existing experimental results. It also pinpoints the major contribution of ATP release when cells traverse network bifurcations. This study may aid in further identifying the interplay between mechanical properties and chemical signaling processes involved in blood microcirculation.

Investigation of red blood cell mechanical properties using AFM indentation and coarse-grained particle method

Background

Red blood cells (RBCs) deform significantly and repeatedly when passing through narrow capillaries and delivering dioxygen throughout the body. Deformability of RBCs is a key characteristic, largely governed by the mechanical properties of the cell membrane. This study investigated RBC mechanical properties using atomic force microscopy (AFM) with the aim to develop a coarse-grained particle method model to study for the first time RBC indentation in both 2D and 3D. This new model has the potential to be applied to further investigate the local deformability of RBCs, with accurate control over adhesion, probe geometry and position of applied force.

Results

The model considers the linear stretch capacity of the cytoskeleton, bending resistance and areal incompressibility of the bilayer, and volumetric incompressibility of the internal fluid. The model’s performance was validated against force–deformation experiments performed on RBCs under spherical AFM indentation. The model was then used to investigate the mechanisms which absorbed energy through the indentation stroke, and the impact of varying stiffness coefficients on the measured deformability. This study found the membrane’s bending stiffness was most influential in controlling RBC physical behaviour for indentations of up to 200 nm.

Conclusions

As the bilayer provides bending resistance, this infers that structural changes within the bilayer are responsible for the deformability changes experienced by deteriorating RBCs. The numerical model presented here established a foundation for future investigations into changes within the membrane that cause differences in stiffness between healthy and deteriorating RBCs, which have already been measured experimentally with AFM.

Electronic supplementary material

The online version of this article (10.1186/s12938-017-0429-5) contains supplementary material, which is available to authorized users.

Numerical simulation of the red blood cell aggregation and deformation behaviors in ultrasonic field

The objective of this paper is to propose an immersed boundary lattice Boltzmann method (IB-LBM) considering the ultrasonic effect to simulate red blood cell (RBC) aggregation and deformation in ultrasonic field. Numerical examples involving the typical streamline, normalized out-of-plane vorticity contours and vector fields in pure plasma under three different ultrasound intensities are presented. Meanwhile, the corresponding transient aggregation behavior of RBCs, with special emphasis on the detailed process of RBC deformation, is shown. The numerical results reveal that the ultrasound wave acted on the pure plasma can lead to recirculation flow, which contributes to the RBCs aggregation and deformation in microvessel. Furthermore, increasing the intensity of the ultrasound wave can significantly enhance the aggregation and deformation of the RBCs. And the formation of the RBCs aggregation leads to the fluctuated and dropped vorticity value of plasma in return.

Total Cavopulmonary Connection is Superior to Atriopulmonary Connection Fontan in Preventing Thrombus Formation: Computer Simulation of Flow-Related Blood Coagulation

The classical Fontan route, namely the atriopulmonary connection (APC), continues to be associated with a risk of thrombus formation in the atrium. A conversion to a total cavopulmonary connection (TCPC) from the APC can ameliorate hemodynamics for the failed Fontan; however, the impact of these surgical operations on thrombus formation remains elusive. This study elucidates the underlying mechanism of thrombus formation in the Fontan route by using a two-dimensional computer hemodynamic simulation based on a simple blood coagulation rule. Hemodynamics in the Fontan route was simulated with Navier-Stokes equations. The blood coagulation and the hemodynamics were combined using a particle method. Three models were created: APC with a square atrium, APC with a round atrium, and TCPC. To examine the effects of the venous blood flow velocity, the velocity at rest and during exercise (0.5 and 1.0 W/kg) was measured. The total area of the thrombi increased over time. The APC square model showed the highest incidence for thrombus formation, followed by the APC round, whereas no thrombus was formed in the TCPC model. Slower blood flow at rest was associated with a higher incidence of thrombus formation. The TCPC was superior to the classical APC in terms of preventing thrombus formation, due to significant blood flow stagnation in the atrium of the APC. Thus, local hemodynamic behavior associated with the complex channel geometry plays a major role in thrombus formation in the Fontan route.

Applications of computational models to better understand microvascular remodelling: a focus on biomechanical integration across scales

Microvascular network remodelling is a common denominator for multiple pathologies and involves both angiogenesis, defined as the sprouting of new capillaries, and network patterning associated with the organization and connectivity of existing vessels. Much of what we know about microvascular remodelling at the network, cellular and molecular scales has been derived from reductionist biological experiments, yet what happens when the experiments provide incomplete (or only qualitative) information? This review will emphasize the value of applying computational approaches to advance our understanding of the underlying mechanisms and effects of microvascular remodelling. Examples of individual computational models applied to each of the scales will highlight the potential of answering specific questions that cannot be answered using typical biological experimentation alone. Looking into the future, we will also identify the needs and challenges associated with integrating computational models across scales.

Hemodynamics in the microcirculation and in microfluidics

Hemodynamics in microcirculation is important for hemorheology and several types of circulatory disease. Although hemodynamics research has a long history, the field continues to expand due to recent advancements in numerical and experimental techniques at the micro-and nano-scales. In this paper, we review recent computational and experimental studies of blood flow in microcirculation and microfluidics. We first focus on the computational studies of red blood cell (RBC) dynamics, from the single cellular level to mesoscopic multiple cellular flows, followed by a review of recent computational adhesion models for white blood cells, platelets, and malaria-infected RBCs, in which the cell adhesion to the vascular wall is essential for cellular function. Recent developments in optical microscopy have enabled the observation of flowing blood cells in microfluidics. Experimental particle image velocimetry and particle tracking velocimetry techniques are described in this article. Advancements in micro total analysis system technologies have facilitated flowing cell separation with microfluidic devices, which can be used for biomedical applications, such as a diagnostic tool for breast cancer or large intestinal tumors. In this paper, cell-separation techniques are reviewed for microfluidic devices, emphasizing recent advances and the potential of this fast-evolving research field in the near future.

Numerical simulation of red blood cell distributions in three-dimensional microvascular bifurcations

We constructed three-dimensional microvascular bifurcation models using a parent vessel of diameter 10μm and investigated the flow behavior of the red blood cells (RBCs) through bifurcations. We considered symmetric and asymmetric model types. Two cases of equal daughter vessel diameter were employed for the asymmetric models, where the first was 10μm, which is the same as the parent vessel and the second was 7.94μm, which satisfies Murray's law. Simulated blood flow was computed using the lattice Boltzmann method in conjunction with the immersed boundary method for incorporating fluid-membrane interactions between the flow field and deformable RBCs. First, we investigated the flow behavior of a single RBC through microvascular bifurcations. In the case of the symmetric bifurcation, the turning point of the fractional plasma flow wherein the RBC flow changed from one daughter vessel to the other was 0.50. This turning point was however different for asymmetric bifurcations. Additionally, we varied the initial offset of RBCs from the centerline of the parent vessel. The simulation results indicated that the RBCs preferentially flow through the branch of a larger flow ratio. Next, we investigated the distribution characteristics of multiple RBCs. Simulations indicated that the results of the symmetric model were similar to those predicted by a previously published empirical model. On the other hand, results of asymmetric models deviated from those of the symmetric and empirical models. These results suggest that the distribution of RBCs varies according to the bifurcation angle and daughter vessel diameter in a microvascular bifurcation of the size considered.

Elastic behavior of a red blood cell with the membrane's nonuniform natural state: equilibrium shape, motion transition under shear flow, and elongation during tank-treading motion

Direct numerical simulations of the mechanics of a single red blood cell (RBC) were performed by considering the nonuniform natural state of the elastic membrane. A RBC was modeled as an incompressible viscous fluid encapsulated by an elastic membrane. The in-plane shear and area dilatation deformations of the membrane were modeled by Skalak constitutive equation, while out-of-plane bending deformation was formulated by the spring model. The natural state of the membrane with respect to in-plane shear deformation was modeled as a sphere ([Formula: see text]), biconcave disk shape ([Formula: see text]) and their intermediate shapes ([Formula: see text]) with the nonuniformity parameter [Formula: see text], while the natural state with respect to out-of-plane bending deformation was modeled as a flat plane. According to the numerical simulations, at an experimentally measured in-plane shear modulus of [Formula: see text] and an out-of-plane bending rigidity of [Formula: see text] of the cell membrane, the following results were obtained. (i) The RBC shape at equilibrium was biconcave discoid for [Formula: see text] and cupped otherwise; (ii) the experimentally measured fluid shear stress at the transition between tumbling and tank-treading motions under shear flow was reproduced for [Formula: see text]; (iii) the elongation deformation of the RBC during tank-treading motion from the simulation was consistent with that from in vitro experiments, irrespective of the [Formula: see text] value. Based on our RBC modeling, the three phenomena (i), (ii), and (iii) were mechanically consistent for [Formula: see text]. The condition [Formula: see text] precludes a biconcave discoid shape at equilibrium (i); however, it gives appropriate fluid shear stress at the motion transition under shear flow (ii), suggesting that a combined effect of [Formula: see text] and the natural state with respect to out-of-plane bending deformation is necessary for understanding details of the RBC mechanics at equilibrium. Our numerical results demonstrate that moderate nonuniformity in a membrane's natural state with respect to in-plane shear deformation plays a key role in RBC mechanics.

Erythrocyte hemodynamics in stenotic microvessels: a numerical investigation

This paper presents a two-dimensional numerical investigation of deformation and motion of erythrocytes in stenotic microvessels using the immersed boundary-fictitious domain method. The erythrocytes were modeled as biconcave-shaped closed membranes filled with cytoplasm. We studied the biophysical characteristics of human erythrocytes traversing constricted microchannels with the narrowest cross-sectional diameter as small as 3 μm. The effects of essential parameters, namely, stenosis severity, shape of the erythrocytes, and erythrocyte membrane stiffness, were simulated and analyzed in this study. Moreover, simulations were performed to discuss conditions associated with the shape transitions of the cells along with the relative effects of radial position and initial orientation of erythrocytes, membrane stiffness, and plasma environments. The simulation results were compared with existing experiment findings whenever possible, and the physical insights obtained were discussed. The proposed model successfully simulated rheological behaviors of erythrocytes in microscale flow and thus is applicable to a large class of problems involving fluid flow with complex geometry and fluid-cell interactions. Our study would be helpful for further understanding of pathology of malaria and some other blood disorders.

AN EFFICIENT RED BLOOD CELL MODEL IN THE FRAME OF IB-LBM AND ITS APPLICATION

A two-dimensional red blood cell (RBC) membrane model based on elastic and Euler–Bernoulli beam theories is introduced in the frame of immersed boundary-lattice Boltzmann method (IB-LBM). The effect of the flexible membrane is handled by the immersed boundary method in which the stress exerted by the RBC on the ambient fluid is spread onto the collocated grid points near the boundary. The fluid dynamics is obtained by solving the discrete lattice Boltzmann equation. A "ghost shape", to which the RBC returns when restoring, is introduced by prescribing a bending force along the boundary. Numerical examples involving tumbling, tank-treading and RBC aggregation in shear flow and deformation and restoration in poiseuille flow are presented to verify the method and illustrate its efficiency. As an application of the present method, a ten-RBC colony being compressed through a stenotic microvessel is studied focusing the cell–cell interaction strength. Quantitative comparisons of the pressure and velocity on specified microvessel interfaces are made between each aggregation case. It reveals that the stronger aggregation may lead to more resistance against blood flow and result in higher pressure difference at the stenosis.

Tank-treading of swollen erythrocytes in shear flows

In this paper, we investigate computationally the oscillatory tank-treading motion of healthy swollen human erythrocytes (owing to lower than physiological plasma osmolarity) in shear flows with capillary number Ca=O(1) and small to moderate viscosity ratios 0.01≤λ≤2.75. Swollen cells show similar shear flow dynamics with normal cells but with significantly higher inclination and tank-treading speed owing to the higher cell thickness. For a given viscosity ratio, as the flow rate increases, the steady-state erythrocyte length L (in the shear plane) increases logarithmically while its depth W (normal to the shear plane) decreases logarithmically; increase of the viscosity ratio results in lower cell deformation. The erythrocyte width S, which exists in the shear plane, is practically invariant in time, flow rate, and viscosity ratio and corresponds to a real cell thickness of about 2.5 μm at physiological osmolarity (300 mO) and 3.4 μm at an osmolarity of 217 mO. The erythrocyte inclination decreases as the flow rate increases or as the surrounding fluid viscosity decreases, owing to the increased inner rotational flow which tends to align the cell toward the flow direction. The ektacytometry deformation of swollen cells increases logarithmically with the shear stress but with a slower slope than that for normal cells owing mainly to the higher orientation of the more swollen cells. As the cell swelling increases, the tank-treading period decreases owing to the higher thickness of the actual cell which overcomes the opposite action of the reduced shape-memory effects (i.e., the more spherical-like erythrocyte's reference shape of shearing resistance). The local area incompressibility tensions from the lipid bilayer increase with the cell swelling and cause a higher cytoskeleton prestress; this increased prestress results in smaller, but still measurable, local area changes on the spectrin skeleton of the more swollen erythrocytes. Our work provides insight on the effects of clinical syndromes and biophysical processes associated with lowered plasma osmolarity (and thus higher cell swelling) such as inappropriate antidiuretic hormone secretion and diuretic therapy.

Tank treading of optically trapped red blood cells in shear flow

Tank-treading (TT) motion is established in optically trapped, live red blood cells (RBCs) held in shear flow and is systematically investigated under varying shear rates, temperature (affecting membrane viscosity), osmolarity (resulting in changes in RBC shape and cytoplasmic viscosity), and viscosity of the suspending medium. TT frequency is measured as a function of membrane and cytoplasmic viscosity, the former being four times more effective in altering TT frequency. TT frequency increases as membrane viscosity decreases, by as much as 10% over temperature changes of only 4°C at a shear rate of ∼43 s(-1). A threshold shear rate (1.5 ± 0.3 s(-1)) is observed below which the TT frequency drops to zero. TT motion is also observed in shape-engineered (spherical) RBCs and those with cholesterol-depleted membranes. The TT threshold is less evident in both cases but the TT frequency increases in the latter cells. Our findings indicate that TT motion is pervasive even in folded and deformed erythrocytes, conditions that occur when such erythrocytes flow through narrow capillaries.

Feedback control system simulator for the control of biological cells in microfluidic cross slots and integrated microfluidic systems

Control systems for lab on chip devices require careful characterisation and design for optimal performance. Traditionally, this involves either extremely computationally expensive simulations or lengthy iteration of laboratory experiments, prototype design, and manufacture. In this paper, an efficient control simulation technique, valid for typical microchannels, Computed Interpolated Flow Hydrodynamics (CIFH), is described that is over 500 times faster than conventional time integration techniques. CIFH is a hybrid approach, utilising a combination of pre-computed flows and hydrodynamic equations and allows the efficient simulation of dynamic control systems for the transport of cells through micro-fluidic devices. The speed-ups achieved by using pre-computed CFD solutions mapped to an n-dimensional control parameter space, significantly accelerate the evaluation and improvement of control strategies and chip design. Here, control strategies for a naturally unstable device geometry, the microfluidic cross-slot, have been simulated and optimal parameters have been found for proposed devices capable of trapping and sorting cells.

Oscillatory tank-treading motion of erythrocytes in shear flows

In this paper, we investigate the oscillatory dynamics of the tank-treading motion of healthy human erythrocytes in shear flows with capillary number Ca = O(1) and small to moderate viscosity ratios 0.01 ≤ λ ≤ 1.5. These conditions correspond to a wide range of surrounding medium viscosities (4-600 m Pa s) and shear flow rates (2-560 s(-1)), and match those used in ektacytometry systems. For a given viscosity ratio, as the flow rate increases, the steady-state erythrocyte length L (in the shear plane) increases logarithmically while its depth W (normal to the shear plane) decreases logarithmically. In addition, the flow rate increase dampens the oscillatory erythrocyte inclination but not its length oscillations (which show relative variations of about 5-8%). For a given flow rate, as the viscosity ratio increases, the erythrocyte length L contracts while its depth W increases (i.e., the cell becomes less deformed) with a small decrease in the length variations. The average orientation angle of the erythrocyte shows a significant decrease with the viscosity ratio as does the angle oscillation while the oscillation period increases. These trends continue in higher viscosity ratios resulting eventually in the transition from a (weakly oscillatory) tank-treading motion to a tumbling motion. Our computations show that the erythrocyte width S, which exists in the shear plane, is practically invariant in time, capillary number, and viscosity ratio, and corresponds to a real cell thickness of about 2.5 μm. Comparison of our computational results with the predictions of (low degree-of-freedom) theoretical models and experimental findings, suggests that the energy dissipation due to the shape-memory effects is more significant than the energy dissipation due to the membrane viscosity. Our work shows that the oscillatory tank-treading motion can account for more than 50% of the variations found in ektacytometry systems; thus, researchers who wish to study inherent differences between erythrocytes within a population must devise a way of monitoring individual cells over time so that they can remove the oscillation effects.

Dynamic modes of red blood cells in oscillatory shear flow

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 θ, and phase angle ϕ 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): ϕ rotates while the shape and θ oscillate. (ii) tumbling (TB): θ rotates while the shape and ϕ oscillate. (iii) intermediate motion: both ϕ and θ 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. The thermal fluctuations can induce transitions between two orbits at very low shear amplitudes. For a high mean shear rate with small shear oscillation, the shape and θ 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, synthetic capsules, and lipid vesicles.

Dynamic modes of microcapsules in steady shear flow: effects of bending and shear elasticities

The dynamics of microcapsules in steady shear flow were studied using a theoretical approach based on three variables: the Taylor deformation parameter αD , the inclination angle θ , and the phase angle ϕ of the membrane rotation. It is found that the dynamic phase diagram shows a remarkable change with an increase in the ratio of the membrane shear and bending elasticities. A fluid vesicle (no shear elasticity) exhibits three dynamic modes: (i) tank treading at low viscosity ηin of internal fluid (αD and θ relaxes to constant values), (ii) tumbling (TB) at high ηin (θ rotates), and (iii) swinging (SW) at middle ηin and high shear rates γ (θ oscillates). All of three modes are accompanied by a membrane (ϕ) rotation. For microcapsules with low shear elasticity, the TB phase with no ϕ rotation and the coexistence phase of SW and TB motions are induced by the energy barrier of ϕ rotation. Synchronization of ϕ rotation with TB rotation or SW oscillation occurs with integer ratios of rotational frequencies. At high shear elasticity, where a saddle point in the energy potential disappears, intermediate phases vanish and either ϕ or θ rotation occurs. This phase behavior agrees with recent simulation results of microcapsules with low bending elasticity.