Computational modeling of blood flow in microvessels with internal diameter 20-500 microm is a major challenge. It is because blood in such vessels behaves as a multiphase suspension of deformable particles. A continuum model of blood is not adequate if the motion of individual red blood cells in the suspension is of interest. At the same time, multiple cells, often a few thousands in number, must also be considered to account for cell-cell hydrodynamic interaction. Moreover, the red blood cells (RBCs) are highly deformable. Deformation of the cells must also be considered in the model, as it is a major determinant of many physiologically significant phenomena, such as formation of a cell-free layer, and the Fahraeus-Lindqvist effect. In this article, we present two-dimensional computational simulation of blood flow in vessels of size 20-300 microm at discharge hematocrit of 10-60%, taking into consideration the particulate nature of blood and cell deformation. The numerical model is based on the immersed boundary method, and the red blood cells are modeled as liquid capsules. A large RBC population comprising of as many as 2500 cells are simulated. Migration of the cells normal to the wall of the vessel and the formation of the cell-free layer are studied. Results on the trajectory and velocity traces of the RBCs, and their fluctuations are presented. Also presented are the results on the plug-flow velocity profile of blood, the apparent viscosity, and the Fahraeus-Lindqvist effect. The numerical results also allow us to investigate the variation of apparent blood viscosity along the cross-section of a vessel. The computational results are compared with the experimental results. To the best of our knowledge, this article presents the first simulation to simultaneously consider a large ensemble of red blood cells and the cell deformation.
The effect of free rotation on the drag and lift forces on a solid sphere in unbounded linear shear flow is investigated. The sphere Reynolds number, Re=|ur|d/ν, is in the range 0.5–200, where ur is the slip velocity. Direct numerical simulations of three-dimensional flow past an isolated sphere are performed using spectral methods. The sphere is allowed to rotate and translate freely in the shear flow in response to the hydrodynamic forces and torque acting on it. The effect of free rotation is studied in a systematic way by considering three sets of simulations. In the first set of simulations, we study how fast a pure rotational or translational motion of the sphere approaches steady state. The “history” effect of rotational and translational motions are compared. Results at high Re are found to be significantly different from the analytical prediction based on low Re theory. In steady simulations, the sphere is allowed to rotate in a torque-free condition. The torque-free rotation rate and the drag and lift forces under such a condition are reported. Comparisons are drawn with the forces on a nonrotating sphere. The effect of rotation is observed to be high in the range 5≲Re≲100. The total lift force is shown to be the sum of the lift force on a nonrotating sphere in shear flow and the lift force on a sphere that is forced to spin at the torque-free rotation rate in a uniform stream (Magnus effect). Finally, we consider the effect of combined free rotation and translation. It is observed that under such combined motion, the sphere achieves translational equilibrium with the local fluid much earlier than it can achieve the zero torque state. The sphere rotates at a rate much lower than the torque-free rotation rate. Free rotation is shown to have a negligible effect on the unsteady drag and lift forces.
A direct numerical simulation ͑DNS͒ is used to study the effect of a freestream isotropic turbulent flow on the drag and lift forces on a spherical particle. The particle diameter is about 1.5-10 times the Kolmogorov scale, the particle Reynolds number is about 60-600, and the freestream turbulence intensity is about 10%-25%. The isotropic turbulent field considered here is stationary, i.e., frozen in time. It is shown that the freestream turbulence does not have a substantial and systematic effect on the time-averaged mean drag. The standard drag correlation based on the instantaneous or mean relative velocity results in a reasonably accurate prediction of the mean drag obtained from the DNS. However, the accuracy of prediction of the instantaneous drag decreases with increasing particle size. For the smaller particles, the low frequency oscillations in the DNS drag are well captured by the standard drag, but for the larger particles significant differences exist even for the low frequency components. Inclusion of the added-mass and history forces, computed based on the fluid velocity at the center of the particle, does not improve the prediction. Different estimates of the fluid velocity seen by the particle are examined. It is shown that the mean drag is insensitive to the fluid velocity measured at the particle center, or obtained by averaging over a fluid volume of the order of the particle size. The fluctuations diminish as the size of the averaging volume increases. The effect of increasing freestream turbulence intensity for the same particle size is studied. Fluctuations in the drag and lift forces are shown to scale with the mean drag and freestream intensity. The standard drag without the added-mass and history forces provides the best approximation to the DNS result.
We present computational fluid dynamic (CFD) simulation of aggregation of two deformable cells in a shearflow. This work is motivated by an attempt to develop computational models of aggregation of red blood cells (RBCs). Aggregation of RBCs is a major determinant of blood viscosity in microcirculation under physiological and pathological conditions. Deformability of the RBCs plays a major role in determining their aggregability. Deformability depends on the viscosity of the cytoplasmic fluid and on the rigidity of the cell membrane, in a macroscopic sense. This paper presents a computational study of RBC aggregation that takes into account the rheology of the cells as well as cell-cell adhesion kinetics. The simulation technique considered here is two dimensional and based on the front tracking/immersed boundary method for multiple fluids. Results presented here are on the dynamic events of aggregate formation between two cells, and its subsequent motion, rolling, deformation, and breakage. We show that the rheological properties of the cells have significant effects on the dynamics of the aggregate. A stable aggregate is formed at higher cytoplasmic viscosity and membrane rigidity. We also show that the bonds formed between the cells change in a cyclic manner as the aggregate rolls in a shearflow. The cyclic behavior is related to the rolling orientation of the aggregate. The frequency and amplitude of oscillation in the number of bonds also depend on the rheological properties.
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