Pattern Transition on Inertial Focusing of Neutrally Buoyant Particles Suspended in Rectangular Duct Flows

The continuous separation and filtration of particles immersed in fluid flows are important interests in various applications. Although the inertial focusing of particles suspended in a duct flow is promising in microfluidics, predicting the focusing positions depending on the parameters, such as the shape of the duct cross-section and the Reynolds number (Re) has not been achieved owing to the diversity of the inertial-focusing phenomena. In this study, we aimed to elucidate the variation of the inertial focusing depending on Re in rectangular duct flows. We performed a numerical simulation of the lift force exerted on a spherical particle flowing in a rectangular duct and determined the lift-force map within the duct cross-section over a wide range of Re. We estimated the particle trajectories based on the lift map and Stokes drag, and identified the particle-focusing points appeared in the cross-section. For an aspect ratio of the duct cross-section of 2, we found that the blockage ratio changes transition structure of particle focusing. For blockage ratios smaller than 0.3, particles focus near the centres of the long sides of the cross-section at low Re and near the centres of both the long and short sides at relatively higher Re. This transition is expressed as a subcritical pitchfork bifurcation. For blockage ratio larger than 0.3, another focusing pattern appears between these two focusing regimes, where particles are focused on the centres of the long sides and at intermediate positions near the corners. Thus, there are three regimes; the transition between adjacent regimes at lower Re is found to be expressed as a saddle-node bifurcation and the other transition as a supercritical pitchfork bifurcation.

Margination of Platelet-Sized Particles in the Red Blood Cell Suspension Flow through Square Microchannels

In the blood flow through microvessels, platelets show high concentrations near the vessel wall. This phenomenon is called margination of platelets and is closely associated with hemostasis and thrombosis. In the present study, we conducted in vitro experiments using platelet-sized fluorescent particles as platelet substitutes to investigate the cross-sectional distribution of these particles in the red blood cell suspension flowing through microchannels with a square cross section. Fluorescence observations were performed to measure the transverse distribution of particles at various heights from the bottom face with the use of a confocal laser scanning microscope system. In downstream cross sections of the channel, particles showed focusing near the four corners rather than uniform margination along the entire circumference of the cross section. The focusing of particles near the corners was more enhanced for higher hematocrits. On the other hand, particles in circular channel flows showed nearly axisymmetric uniform accumulation adjacent to the channel wall. The present result suggests that the segregation of suspended particles in the flow of multicomponent suspensions could have such heterogeneous 2D features of particle distribution in the cross section of channels, especially for rectangular channels often used in microfluidics.

Axial and Nonaxial Migration of Red Blood Cells in a Microtube

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 μm in a circular microchannel with diameters of 15 μm. 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.

Development of margination of platelet-sized particles in red blood cell suspensions flowing through Y-shaped bifurcating microchannels

BACKGROUND

In the blood flow through microvessels, platelets exhibit enhanced concentrations in the layer free of red blood cells (cell-free layer) adjacent to the vessel wall. The motion of platelets in the cell-free layer plays an essential role in their interaction with the vessel wall, and hence it affects their functions of hemostasis and thrombosis.

OBJECTIVE

We aimed to estimate the diffusivity of platelet-sized particles in the transverse direction (the direction of vorticity) across the channel width in the cell-free layer by in vitro experiments for the microchannel flow of red blood cell (RBC) suspensions containing platelet-sized particles.

METHODS

Fluorescence microscope observations were performed to measure the transverse distribution of spherical particles immersed in RBC suspensions flowing through a Y-shaped bifurcating microchannel. We examined the development of the particle concentration profiles along the flow direction in the daughter channels, starting from asymmetric distributions with low concentrations on the inner side of the bifurcation at the inlet of the daughter channels.

RESULTS

In daughter channels of 40 μm width, reconstruction of particle margination revealed that a symmetric concentration profile was attained in ∼30 mm from the bifurcation, independent of flow rate.

CONCLUSIONS

We presented experimental evidence of particle margination developing in a bifurcating flow channel where the diffusivity of 2.9-μm diameter particles was estimated to be ∼40 μm2/s at a shear rate of 1000 s-1 and hematocrit of 0.2.

Cross-sectional distributions of normal and abnormal red blood cells in capillary tubes determined by a new technique

BACKGROUND

In the microcirculation, red blood cells (RBCs) were observed to be confined to an axial stream surrounded by a marginal RBC depleted layer. This axial accumulation of RBCs is considered to arise from the RBC deformability.

OBJECTIVE

To quantitatively evaluate the effect of RBC deformability on their axial accumulation at a flow condition comparable to that in arterioles by developing a new observation system for accurate measurements of radial RBC positions in the cross section of capillary tubes.

METHODS

The cross-sectional distributions of normal and hardened RBCs as well as softened RBCs suspended in capillary tube flows were measured with high spatial resolution. A new observation system was developed in which enface views of the cross-section of the tube were obtained at small distances upstream of the outlet at various longitudinal positions in the tube.

RESULTS

The radial positions of individual RBCs were detected within 1 μm accuracy. It was found that normal and softened RBCs rapidly migrated away from the wall towards the tube axis, whereas glutaraldehyde-hardened RBCs were dispersed widely over the tube cross-section, depending on the concentration of glutaraldehyde solution.

CONCLUSIONS

The newly devised observation system revealed quantitatively the essential role of RBC deformability in their axial accumulation.

Transition of planar Couette flow at infinite Reynolds numbers

An outline of the state space of planar Couette flow at high Reynolds numbers (Re<10^{5}) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of the hairpin vortex state (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of the HVS at high Re belongs to the stability boundary that initiates a transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit.

Charge effects on the hindered transport of macromolecules across the endothelial surface glycocalyx layer

A fluid mechanical and electrostatic model for the transport of solute molecules across the vascular endothelial surface glycocalyx layer (EGL) was developed to study the charge effect on the diffusive and convective transport of the solutes. The solute was assumed to be a spherical particle with a constant surface charge density, and the EGL was represented as an array of periodically arranged circular cylinders of like charge, with a constant surface charge density. By combining the fluid mechanical analyses for the flow around a solute suspended in an electrolyte solution and the electrostatic analyses for the free energy of the interaction between the solute and cylinders based on a mean field theory, we estimated the transport coefficients of the solute across the EGL. Both of diffusive and convective transports are reduced compared to those for an uncharged system, due to the stronger exclusion of the solute that results from the repulsive electrostatic interaction. The model prediction for the reflection coefficient for serum albumin agreed well with experimental observations if the charge density in the EGL is ranged from approximately -10 to -30 mEq/l.

Mechanotransduction in leukocyte activation: a review

We review recent evidence which suggests that leukocytes in the circulation and in the tissue may readily respond to physiological levels of fluid shear stress in the range between about 1 and 10 dyn/cm 2, a range that is below the level to achieve a significant passive, viscoelastic response. The response of activated neutrophilic leukocytes to fluid shear consists of a rapid retraction of lamellipodia with membrane detachment from integrin binding sites. In contrast, a subgroup of non-activated neutrophils may project pseudopods after exposure to fluid shear stress. The evidence suggests that G-protein coupled receptor downregulation by fluid shear with concomitant downregulation of Rac-related small GTPases and depolymerization of F-actin serves to retract the lamellipodia in conjunction with proteolytic cleavage of beta 2 integrin to facilitate membrane detachment. Furthermore, there exists a mechanism to up- and down-regulate the fluid shear-response, which involves nitric oxide and the second messenger cyclic guanosine monophosphate (cGMP). Many physiological activities of circulating leukocytes are under the influence of fluid shear stress, including transendothelial migration of lymphocytes. We describe a disease model with chronic hypertension that suffers from an attenuated fluid shear-response with far reaching implications for microvascular blood flow.

The Fluid Shear Stress Distribution on the Membrane of Leukocytes in the Microcirculation

Recent in-vivo and in-vitro evidence indicates that fluid shear stress on the membrane of leukocytes has a powerful control over several aspects of their cell function. This evidence raises a question about the magnitude of the fluid shear stress on leukocytes in the circulation. The flow of plasma on the surface of a leukocyte at a very low Reynolds number is governed by the Stokes equation for the motion of a Newtonian fluid. We numerically estimated the distribution of fluid shear stress on a leukocyte membrane in a microvessel for the cases when the leukocyte is freely suspended, as well as rolling along or attached to a microvessel wall. The results indicate that the fluid shear stress distribution on the leukocyte membrane is nonuniform with a sharp increase when the leukocyte makes membrane attachment to the microvessel wall. In a microvessel (10 μm diameter), the fluid shear stress on the membrane of a freely suspended leukocyte (8 μm diameter) is estimated to be several times larger than the wall shear stress exerted by the undisturbed Poiseuille flow, and increases on an adherent leukocyte up to ten times. High temporal stress gradients are present in freely suspended leukocytes in shear flow due to cell rotation, which are proportional to the local shear rate. In comparison, the temporal stress gradients are reduced on the membrane of leukocytes that are rolling or firmly adhered to the endothelium. High temporal gradients of shear stress are also present on the endothelial wall. At a plasma viscosity of 1 cPoise, the peak shear stresses for suspended and adherent leukocytes are of the order of 10 dyn/cm2 and 100 dyn/cm2, respectively.

Flow around cells adhered to a microvessel wall II: comparison to flow around adherent cells in channel flow

The flow around adherent cells in a parallel-plate channel and that in a circular cylindrical tube are numerically analyzed, and their effects on the adherent cells are compared. The cells are modeled as rigid spherical particles and they are assumed to be attached to a wall of a 2D channel uniformly in a square array, or a wall of a circular tube regularly in a line along the tube axis. It is found that, when the size ratios of the particle-to-channel height and the particle-to-tube diameter are smaller than approximately 0.2, the distributions of the shear stress and the pressure exerted on the surface of an adherent particle as well as the drag force and torque acting on it compare favorably in the 2D channel flow and tube flow. As the size ratios increase from 0.2, the differences between the 2D channel and the tube increase drastically, especially when separation distances between neighboring particles are large.

Flow around cells adhered to a microvessel wall. I. Fluid stresses and forces acting on the cells

To evaluate the fluid forces acting on cells adhered to a microvessel wall, we numerically studied the flow field around adherent cells and the distribution of the stresses on their surfaces. For simplicity, the cells were modeled as rigid particles attached to a wall of a circular cylindrical tube regularly in the flow direction, in a row or two rows. It was found that not the detailed shape of the model cells but their height from the vessel wall is a key determinant of the fluid forces and torque acting on them. In both arrangements of one row and two rows, the axial spacing between neighboring adherent cells significantly affects the distributions of the stresses on them, which results in drastic variations of the fluid forces with the axial spacing and the relative positions with respect to their neighboring cells. The drag force acting on an adherent cell in the vessel was evaluated to be larger than the value in the 2D chamber flow at the same wall shear stress, mainly due to much larger variations of the pressure distribution on the cell surface in the vessel flow.

A mechanism for erythrocyte-mediated elevation of apparent viscosity by leukocytes in vivo without adhesion to the endothelium

In spite of the relatively small number of leukocytes in the circulation, they have a significant influence on the perfusion of such organs as skeletal muscle or kidney. However, the underlying mechanisms are incompletely understood. In the current study a combined in vivo and computational approach is presented in which the interaction of individual freely flowing leukocytes with erythrocytes and its effect on apparent blood viscosity are explored. The skeletal muscle microcirculation was perfused with different cell suspensions with and without leukocytes or erythrocytes. We examined a three-dimensional numerical model of low Reynolds number flow in a capillary with a train of erythrocytes (small spheres) in off-axis positions and single larger leukocytes in axisymmetric positions. The results indicate that in order to match the slower axial velocity of leukocytes in capillaries, erythrocytes need to position themselves into an off-axis position in the capillary. In such off-axis positions at constant mean capillary velocity, erythrocyte axial velocity matches on average the axial velocity of the leukocytes, but the apparent viscosity is elevated, in agreement with the whole organ perfusion observations. Thus, leukocytes influence the whole organ resistance in skeletal muscle to a significant degree only in the presence of erythrocytes.

Force acting on spheres adhered to a vessel wall

To evaluate the force and torque acting on leukocytes attached to the vessel wall, we numerically study the flow field around the leukocytes by using rigid spherical particles adhered to the wall of a circular cylindrical tube as a model of adherent leukocytes. The adherent particles are assumed to be placed regularly in the flow direction with equal spacings, in one row or two rows. The flow field of the suspending fluid is analyzed by a finite element method applied to the Stokes equations, and the drag force and torque acting on each particle, as well as the apparent viscosity, are evaluated as a function of the particle to tube diameter ratio and the particle arrangements. For two-row arrangements of adhered particles where neighboring particles are placed alternately on opposite sides of the vessel, the drag and the torque exerted on each particle are higher than those for single-row arrangements, for constant particle to tube diameter ratio and axial spacing between neighboring particles. This is enhanced for larger particles and smaller axial spacings. The apparent viscosity of the flow through vessels with adhered particles is found to be significantly higher than that without adhered particles or when the particles are freely floating through the vessels.

Asymmetric flows of spherical particles in a cylindrical tube

To study the rheological behavior of blood cells in various flow patterns through narrow vessels, we analyzed numerically the motion of blood cells arranged in one row or two rows in tube flow, at low Reynolds numbers. The particles are assumed to be identical rigid spheres placed periodically along the vessel axis at off-axis positions with equal spacings. The flow field of the suspending fluid in a circular cylindrical tube is analyzed by a finite element method applied to the Stokes equations, and the motion of each particle is simultaneously determined by a force-free and torque-free condition. In both cases of single- and two-file arrangements of the particles, their longitudinal and angular velocities are largely affected by the radial position and the axial spacing between neighboring particles. The apparent viscosity of the asymmetric flows in higher than that of the symmetric flow where particles are located on the tube centerline, and this is more pronounced when particles are placed farther from the tube centerline and when the axial distance between neighboring particles is reduced.

The motion of two cylinders in contact in channel flow

The motion of a doublet consisting of two rigidly connected circular cylinders freely suspended in an incompressible Newtonian fluid in a narrow channel bounded by two parallel planes is studied numerically at zero Reynolds number. The Stokes equations are solved by a finite element method for various positions and orientations of the particle, and the trajectories of the particle are determined for a number of initial configurations. A doublet of equal-sized cylinders is found either to tumble or oscillate in rotation, while a doublet of unequal-sized cylinders possesses a stable, steady equilibrium configuration, at which the particle is located close to the channel centerline with a constant angle of inclination. Thus, doublets of unequal diameters tumble, oscillate in rotation or approach the equilibrium configuration asymptotically, depending on the radius ratio of cylinders, particle size-channel width ratio and initial conditions.

Motion of a doublet of two cylinders in contact in a narrow channel flow

The motion of two rigid circular cylinders in contact immersed in an incompressible Newtonian fluid in a channel is examined numerically in the zero Reynolds number limit, for various values of the cylinder radius/channel width ratio. Analyses of the time courses of the lateral position and the orientation of the doublet showed that, depending on the initial condition and the doublet/channel size ratio, the doublet exhibit one of the three types of motion: a continuous rotation in the same direction during a period, and a rotation changing its direction at every half period with a large or a small variation of the orientation.

Two-dimensional analysis of two-file flow of red cells along capillaries

The effect of a tank-treading motion of the red cell membrane on a staggered "zipper"-type red cell flow in capillaries is examined, using a two-dimensional theoretical model. An approximately triangular cell with a thin flexible membrane enclosing a viscous fluid is adopted as a model of the red cell. The motion of model red cells arranged periodically along a two-dimensional channel in an idealized zipper-type arrangement is analyzed numerically by a finite element method applied to the Stokes equations for the flow both inside and outside the model cells. It is shown that, if the viscosity ratio of the internal fluid to the suspending fluid is lower than a critical value, there exists a stable zipper-type arrangement of cells. In that arrangement, the cells remain stationary relative to each other with the membrane tank-treading. In contrast, inhibiting tank-treading by increasing the viscosity ratio above the critical value induces a cyclic oscillatory motion of red cells. The critical viscosity ratio increases if the channel is narrowed or if the spacing between cells is reduced. The present results suggest that the membrane tank-treading tends to stabilize zipper-type arrangements of red cells in capillaries at high hematocrit.

Vascular resistance of arterioles with nonuniform diameters

Arterioles in various vascular beds have been often observed to have nonuniform diameters along the vessel axis. The effect of nonuniformity of arteriolar diameter on vascular resistance was investigated using a theoretical model of blood flow in arterioles. Viscous flow of a Newtonian fluid in tubes with periodically changing diameters along the tube axis was analyzed by a finite element method based on the Stokes equations. Vascular resistance for the nonuniform tube was computed over a spatial period of the variation in diameter and was compared to resistance for a uniform tube with a constant diameter equal to the mean diameter of the nonuniform tube. In all cases, resistance for a tube with a nonuniform diameter was larger than resistance for a uniform tube with a diameter equal to the mean diameter of the nonuniform tube. Increases in the amplitude of the variation in diameter resulted in a rapid increase in resistance when the period of the variation remained constant. On the other hand, as the period of the diameter variation increased when amplitude remained constant, resistance decreased and approached the values obtained under the assumption of a Poiseuille flow at each cross section of the tube in the limit of an infinite period. Our theoretical model was applied to our previous in vivo studies of vessel diameter nonuniformity for rabbit mesentery arterioles in a contracted state. It was shown that vascular resistance calculated by our model was 2 to 11% higher than resistance obtained for a uniform tube with a diameter equal to the mean diameter of the arteriole.

Stability of particle motions in a narrow channel flow

Single rows or two rows of identical circular cylinders spaced regularly in a narrow channel flow have been shown to be capable of steady flow provided the cylinders are located at equal lateral positions and with equal spacings in the flow direction. The stability of such steady flows of circular cylinders is studied for periodic perturbations of the particle positions, assuming that every other cylinder is equally perturbed in lateral position and spacing along the channel. This results in two rows which are not symmetrically placed. The suspending fluid is assumed to be an incompressible Newtonian fluid. It is assumed that no external forces or moments act on the cylinders and the effects of inertia forces on the motion of the fluid and the cylinders are negligible. The velocity field of the suspending fluid and the instantaneous velocities of the cylinders are computed by the finite element method. The translational velocities of the cylinders are obtained for a large number of particle positions, from which the trajectories of their relative motion are determined for various initial positions near the regular single-file and two-file arrangements. It is shown that when the initial arrangements of the cylinders are slightly perturbed from the regular (alternating) two-file flows, the trajectories of their relative motions become small closed loops around the regular two-file arrangements. In contrast, such small closed trajectories are not obtained when they start from the arrangements near the regular single-file flows or regular (symmetric) double-file flows, suggesting that these flows are unstable under the conditions examined.

Numerical study of asymmetric flows of red blood cells in capillaries

Experimental studies have shown that red blood cells in capillaries may flow in single-file or multifile arrangements. To model multifile rheological behavior, the asymmetric flows of rigid circular cylinders in a two-dimensional channel are studied by numerical analysis. The rigid circular cylinders are arranged off-center in a channel in a row or two rows with equal spacings. The motion of the suspending fluid is analyzed by the finite element method applied to the Stokes equation, and the motions of the particles are simultaneously determined under the zero force and zero moment conditions appropriate to neutorally buoyant particles. The velocity difference between the particles and the bulk flow is significantly affected by the arrangement of the particles. The particle velocity is reduced as the particles are moved away from the centerline of the channel. At a constant concentration of the particles, the relative apparent viscosity of an off-center arrangement is considerably higher than that of a single-file flow of the particles located on the centerline of the channel. The present results suggest that changes of the radial distribution of red cells flowing through narrow vessels may lead to alterations of the Fahraeus and Fahraeus-Lindqvist effects.

Transient relative motion of two cells in a channel flow

The hydrodynamic interaction of a red blood cell and a white blood cell in microvessels is studied, by use of a two-dimensional numerical model. The red blood cell, modeled as a small rigid circular cylinder, and the white blood cell, modeled as a larger rigid circular cylinder, are immersed in an incompressible Newtonian fluid in a two-dimensional channel. It is assumed that no external force or moment acts on the model cells, and the effect of inertia forces on the motion of the fluid and the cells is neglected. The velocity field of the suspending fluid and the instantaneous velocities of the two model cells are computed by the finite element method. Using the translational velocities of the model cells obtained, the trajectories of their relative motion are determined, for various initial positions. It is shown that the cells may or may not pass each other or separate, depending on the initial positions. The present results compare well to the experimental results.