Movement of spherical particles in capillaries using a boundary singularity method

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.