Axisymmetric unsteady stokes flow past an oscillating finite-length cylinder

Abstract

The flow field generated by axial oscillations of a finite-length cylinder in an incompressible viscous fluid is described by the unsteady Stokes equations and computed with a first-kind boundary-integral formulation. Numerical calculations were conducted for particle oscillation periods comparable with the viscous relaxation time and the results are contrasted to those for an oscillating sphere and spheroid. For high-frequency oscillations, a two-term boundary-layer solution is formulated that involves two, sequentially solved, second-kind integral equations. Good agreement is obtained between the boundary-layer solution and fully numerical calculations at moderate oscillation frequencies. The flow field and traction on the cylinder surface display several features that are qualitatively distinct from those found for smooth particles. At the edges, where the base joins the side of the cylinder, the traction on the cylinder surface exhibits a singular behaviour, characteristic of steady two-dimensional viscous flow. The singular traction is manifested by a sharply varying pressure profile in a near-field region. Instantaneous streamline patterns show the formation of three viscous eddies during the decelerating portion of the oscillation cycle that are attached to the side and bases of the cylinder. As deceleration proceeds, the eddies grow, coalesce at the edges of the particle, and thus form a single eddy that encloses the entire particle. Subsequent instantaneous streamline patterns for the remainder of the oscillation cycle are insensitive to particle geometry: the eddy diffuses outwards and vanishes upon particle reversal; a simple streaming flow pattern occurs during particle acceleration. The evolution of the viscous eddies is most apparent at moderate oscillation frequencies. Qualitative results are obtained for the oscillatory flow field past an arbitrary particle. For moderate oscillation frequencies, pathlines are elliptical orbits that are insensitive to particle geometry; pathlines reduce to streamline segments in constant-phase regions close to and far from the particle surface.