The motion of an elliptical cylinder in channel flow at low Reynolds numbers

Abstract

The motion of an elliptical cylindrical particle immersed in an incompressible Newtonian fluid in a narrow channel is examined numerically in the zero-Reynolds-number limit. It is assumed that no external forces or torques act on the elliptical cylinder, and the effects of inertia forces on the motion of the fluid and the particle are neglected. The Stokes equations are solved by a finite-element method for various positions and orientations of the cylinder, yielding the instantaneous velocities of the particle that satisfy the conditions of zero force and zero torque on the particle. Using the computed longitudinal, lateral, and angular velocities of the particle, the evolution of the particle's position and orientation is determined for various initial configurations. An elliptical cylinder is found to either tumble or oscillate in rotation, depending on the particle-channel size ratio, the axis ratio of the elliptical cylinder, and the initial conditions. In the first case, the particle rotates continuously in one direction that is well approximated by Jeffery's solution for an elliptical cylinder in unbounded shear flow with a so-called equivalent axis ratio; in the second case, the particle changes its direction of rotation during part of each period. In both cases, the particle translates with a periodically varying longitudinal velocity, accompanied by a considerable side drift due to the walls. The oscillatory motion is more likely to occur when the particle-channel size ratio or axis ratio is increased. The tumbling motion is inhibited for elliptic cylinders whose size ratios are larger than threshold values that depend on the axis ratio.