ASYMMETRIC, OSCILLATORY MOTION OF A FINITE-LENGTH CYLINDER - THE MACROSCOPIC EFFECT OF PARTICLE EDGES

The oscillatory motion of a finite-length, circular cylinder perpendic ular to its symmetry axis in an incompressible, viscous fluid is descr ibed by the unsteady Stokes equations. Numerical calculations are perf ormed using a first-kind, boundary-integral formulation for particle o scillation periods comparable to the viscous relaxation time. For high -frequency oscillations, a two-term, boundary layer solution is implem ented that involves two, sequentially solved, second-kind integral equ ations. Good agreement is obtained between the boundary layer solution and fully numerical solutions at moderate oscillation frequencies. At the edges, where the base joins the side of the cylinder, the pressur e and both components of tangential stress exhibit distinct, singular behaviors that are characteristic of steady, two-dimensional, viscous flow. Numerical calculations accurately capture the theoretically pred icted singular behavior. The unsteady flow reversal process is initiat ed by a complex near-field how reversal process that is inferred from the tangential stress distribution. A qualitative picture is construct ed that involves the formation of three viscous eddies during the dece lerating portion of the oscillation cycle: two attached to the ends of a finite-length cylinder, and a third that wraps around the cylinder centerline; the picture is similar to the results for axisymmetric how . As deceleration proceeds, the eddies grow and coalesce at the cylind er edges to form a single eddy that encloses the entire particle. The remainder of the oscillatory flow cycle is insensitive to particle geo metry and orientation. The macroscopic effect of the sharp edges is il lustrated by considering ultrasonic, viscous dissipation in a dilute s uspension. For a fixed particle-to-fluid density ratio, four different frequency regimes are identified. Four distinct viscous dissipation s pectra are shown for different particle-to-fluid density ratios. The r esults indicate that particle geometry is important only for particles considerably less dense than the suspending fluid. The effect of edge s is most apparent for disk- and rod-shaped particles.