AXISYMMETRICAL UNSTEADY STOKES-FLOW PAST AN OSCILLATING FINITE-LENGTHCYLINDER

The flow field generated by axial oscillations of a finite-length cyli nder in an incompressible viscous fluid is described by the unsteady S tokes equations and computed with a first-kind boundary-integral formu lation. Numerical calculations were conducted for particle oscillation periods comparable with the viscous relaxation time and the results a re contrasted to those for an oscillating sphere and spheroid. For hig h-frequency oscillations, a two-term boundary-layer solution is formul ated that involves two, sequentially solved, second-kind integral equa tions. 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 fe atures that are qualitatively distinct from those found for smooth par ticles. At the edges, where the base joins the side of the cylinder, t he traction on the cylinder surface exhibits a singular behaviour, cha racteristic of steady two-dimensional viscous flow. The singular tract ion is manifested by a sharply varying pressure profile in a near-fiel d region. Instantaneous streamline patterns show the formation of thre e viscous eddies during the decelerating portion of the oscillation cy cle that are attached to the side and bases of the cylinder. As decele ration proceeds, the eddies grow, coalesce at the edges of the particl e, and thus form a single eddy that encloses the entire particle. Subs equent instantaneous streamline patterns for the remainder of the osci llation 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 visc ous eddies is most apparent at moderate oscillation frequencies. Quali tative results are obtained for the oscillatory flow field past an arb itrary particle. For moderate oscillation frequencies, pathlines are e lliptical orbits that are insensitive to particle geometry; pathlines reduce to streamline segments in constant-phase regions close to and f ar from the particle surface.