Non-axisymmetric motion of rigid closely fitting particles in fluid-filled tubes

Abstract

We consider non-axisymmetric motion of a rigid particle in a cylindrical fluid-filled tube, with negligible inertial effects. The particle is assumed to fit closely in the tube, and lubrication theory is used to describe the fluid flow in the narrow gap between the particle and the tube wall. The solution to the Reynolds lubrication equation and the components of the resistance matrix are expressed in terms of a Green's function. For the case in which the gap is almost uniform, the Green's function is expanded as a power series in a small parameter δ, characteristic of the variations in gap width, and the first two terms are obtained. The velocity of a freely suspended axisymmetric particle driven by a pressure difference along the tube is deduced from the resistance matrix. According to the results at first order in δ, in general the particle moves transversely with a constant velocity. In the absence of higher-order effects, it would eventually collide with the wall. Motion along the tube axis is a neutrally stable solution to the equations of motion at first order. However, if effects at second order in δ are included, motion of an axisymmetric particle along the tube axis is stable or unstable depending on its shape. Generally, if the particle is narrower near the front than near the rear, and the width near the middle is at least as large as the mean of the widths near the front and rear, then its motion is stable. Numerical calculations (not restricted to small δ) confirm these results for axisymmetric particles, and show that a non-axisymmetric shape similar to a red blood cell has a stable equilibrium position in the tube.