Elsevier

Journal of Biomechanics

Volume 31, Issue 4, 1 April 1998, Pages 347-354
Journal of Biomechanics

Movement of spherical particles in capillaries using a boundary singularity method

https://doi.org/10.1016/S0021-9290(98)00024-4Get rights and content

Abstract

As a first step to investigate the motion of blood cells in capillaries, we have studied the movement of a spherical particle falling in a vertical cylindrical tube using a boundary singularity method. The tube is filled with a Newtonian viscous fluid which would otherwise be stationary. The Reynolds number of the flow is much less than one. The sphere falls at arbitrary positions in the tube and is free to rotate. Point forces, Stokeslets, with unknown strength and direction, are distributed on the surfaces of the tube and the sphere. By forcing the flow field generated by all of the Stokeslets to satisfy proper boundary conditions, we solve for the strength and direction of each Stokeslet. The velocity, U, and rotation, Ω, of the sphere are then calculated from a force and a torque balance. For a sphere falling on the axis of the tube, our results agree with Bohlin’s approximate solution. When the sphere takes eccentric positions in the tube, it rotates as it translates down the tube, the direction of rotation being opposite to that it would have if the sphere rolled along the nearest side of the tube. This results from the zero net torque on the sphere and facilitates flow passing around the sphere. As the distance between the centre of the sphere and the axis of the tube increases, Ω increases almost linearly, while U changes little. When the radius of the tube increases, U increases and approaches the Stokes velocity, while Ω decreases rapidly. The boundary singularity method is relatively simple compared to other numerical methods and can be extended much more easily to the complex geometries typical of blood cells.

Introduction

This study is motivated by the motion of blood cells in capillaries. There is increasing interest in the effects of blood flow on vascular endothelial cells because of the possible influence of flow on vascular biology and pathobiology (e.g. Davies, 1986; Fung and Liu, 1993; Nerem and Girard, 1990). Many properties of endothelial cells in culture depend upon the fluid dynamic conditions in the culture medium (Diamond, et al., 1989; Eskin, et al., 1984). In an earlier paper, we studied the haemodynamic effects of endothelial glycocalyx in capillary flow using a binary mixture model and lubrication theory (Wang and Parker, 1995). In that paper, as in several other studies, the capillary is small and the particles occupy most of its lumen (Bungay and Brenner, 1973; Chester, 1984; Christopherson and Dowson, 1959; Damiano, et al., 1996; Halpern and Secombe, 1989). When the size of the capillary increases to several times of that of the cell, lubrication theory is no longer valid. Asymptotic solutions have been obtained using reflection methods with small perturbations when the sphere is on or near the axis of the tube (Brenner and Happel, 1958; Bohlin, 1960; Tözeren, 1983). However, when the sphere has a finite eccentricity in the tube, its motion has to be calculated using numerical methods. Previous numerical studies concentrated on the non-Newtonian effects of the flow when the sphere moves along the axis of the tube (e.g. Sugeng and Tanner, 1986; Zheng et al. 1990). Muldowney and Higdon (1995) developed a spectral boundary element method and applied this method to calculate resistance functions for spherical particles in cylindrical tubes (Higdon and Muldowney, 1995). However, they did not give any results of the rotation of the sphere which is also of interest.

In this paper, we use a boundary singularity method to study the translation and rotation of a spherical particle in a cylindrical tube. This method is relatively simple compared to other numerical methods and can be extended much more easily to study more complex geometries. The sphere takes arbitrary positions in the tube and is free to rotate as it falls under gravity. The Reynolds number of the flow based on the radius of the tube is much less than one. A number of point forces, Stokeslets, with unknown strength and direction are distributed on the surfaces of the sphere and the tube where interaction with fluid occurs. The flow field generated by each Stokeslet is known (e.g. Batchelor, 1967; Pozrikidis, 1992). We sum contributions from all of the Stokeslets to calculate the overall flow field. By forcing this field to satisfy no slip conditions at the boundaries, we solve for each Stokeslet. The steady state descending velocity of the sphere, U (hereafter called the velocity) and rotating velocity of the sphere, Ω (hereafter called the rotation), are calculated from a force and a torque balance.

The movement of blood cells in the microcirculation is much more complicated: the elastic membrane of red cells allow them to deform when exposed to shear stresses (e.g. Chien, et al. 1984); blood flow is driven by pressure differences in capillaries; and there are generally a number of red cells in a single capillary, although they behave independently when at more than one diameter apart (Wang and Skalak, 1969). In the current model, we consider the simplest problem of a spherical particle under gravity, which allow us to focus on the development of the boundary singularity method and to compare our results with previous solutions available under restricted conditions. This method can easily be extended to study the motion of a sphere in a capillary driven by a pressure gradient, where parabolic flow will replace the zero flow condition at two ends of the tube and the pressure difference is balanced by the sum of the Stokeslets on the surface of the tube.

Section snippets

Theoretical background

Consider a body of arbitrary shape moving in an incompressible Newtonian viscous fluid. When the Reynolds number is small, the far flow field generated by this body depends only on the resultant force exerted on the fluid by the body, and is not affected by its shape and motion. In spherical polar coordinates (r,θ,φ), with the origin on the particle, the stream function can be expressed asψ=F8πμrsin2θ,where μ is fluid viscosity, F is the magnitude of the resultant force on the fluid, F, and θ=0

Numerical algorithm

Stokeslets and the velocity are linearly related. As shown in Fig. 2a, flow is symmetric in the x and z directions over the xz plane and anti-symmetric in the y direction. Mirror Stokeslets over the xz plane, therefore, have the same vectors in the x and the z directions but the opposite vectors in the y direction. With this relationship, only the Stokeslets and velocity points in the y⩾0 half space need to be considered with each Stokeslet representing a mirror Stokeslet pair over the xz

Results and discussion

When a sphere falls in a tube, its velocity is less than the Stokes velocity because of the presence of the tube. The wall correction factor, K, is defined as the Stokes velocity over the velocity of the sphere in a confined domain and has been calculated by Bohlin (1960) in his study of the motion of a sphere on the axis of a tube, using a reflection method,K=1−2.10443R+2.08877R30.94813R51.372R6+3.87R84.19R10+…-1.

This approximation gives good results when R≫1. Fig. 3 shows the K values from

Conclusions

The boundary singular method using Stokeslets gives satisfactory results in calculating the motion of a spherical particle in a cylindrical tube when the size of the tube is more than twice that of the sphere. When the sphere is on the axis of the tube, reasonable agreement with Bohlin’s approximation (1960) is seen over a wide range of tube radius (2<R<20). When the sphere takes an eccentric positions in the tube, it rotates as it translates down the tube. The direction of Ω is opposite to

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