Asymmetric, oscillatory motion of a finite-length cylinder: The macroscopic effect of particle edges

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Abstract
The oscillatory motion of a finite-length, circular cylinder perpendicular to its symmetry axis in an incompressible, viscous fluid is described by the unsteady Stokes equations. Numerical calculations are performed using a first-kind, boundary-integral formulation for particle oscillation periods comparable to the viscous relaxation time. For high-frequency oscillations, a two-term, boundary layer solution is implemented that involves two, sequentially solved, second-kind integral equations. 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 pressure 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 predicted singular behavior. The unsteady flow reversal process is initiated by a complex near-field flow reversal process that is inferred from the tangential stress distribution. A qualitative picture is constructed that involves the formation of three viscous eddies during the decelerating 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 flow. As deceleration proceeds, the eddies grow and coalesce at the cylinder edges to form a single eddy that encloses the entire particle. The remainder of the oscillatory flow cycle is insensitive to particle geometry and orientation. The macroscopic effect of the sharp edges is illustrated by considering ultrasonic, viscous dissipation in a dilute suspension. For a fixed particle-to-fluid density ratio, four different frequency regimes are identified. Four distinct viscous dissipation spectra are shown for different particle-to-fluid density ratios. The results indicate that particle geometry is important only for particles considerably less dense than the suspending fluid. The effect of edges is most apparent for disk- and rod-shaped particles.

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