Exact and asymptotic low Reynolds, time-varying solutions for spinning rods with a comparison to experiments on the micro and macroscale

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  • March 20, 2019
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  • Leiterman, Terry Jo
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • An exact mathematical solution for the low Reynolds number, quasi steady, hydrodynamic motion induced by a rod in the form of a prolate spheroid sweeping a double cone is developed and the influence of the ensuing fluid motion upon passive particles is studied. The resulting fluid motion is fully three-dimensional and time-varying. The advected particles are observed to admit slow orbits around the rotating rods and a fast epicyclic motion roughly commensurate with the rod rotation rate. The epicycle amplitudes, vertical fluctuations, arclengths, and angle traveled per rotation are mapped as functions of their initial coordinates and rod geometry. These trajectories exhibit a rich spatial structure with greatly varying trajectory properties. We examine these complexities via an auxiliary flow in the rotating frame which provides a generator that defines the epicycles. Further, an additional spin around the major spheroidal axis is included in the exact hydrodynamic solution. This exact solution is compared to an asymptotic solution for a slender body sweeping a double cone by carefully assessing how slenderness affects the domain in which trajectories and flow properties can be accurately captured. By utilizing slender body theory and a family of singularities developed for Stokes flow past a no-slip plane, an asymptotic solution for a slender body attached to a plane rotating about its base sweeping out an upright cone is constructed. Trajectory and flow properties are examined with special attention paid to iii the case study on slenderness between the exact and asymptotic free space solutions. Far field asymptotic analysis is presented for both the exact free space and the asymptotic no-slip plane velocity solutions. The present study is of direct use to nano-scale actuated fluidics where similar epicyclical behavior has been observed. On such scales, thermal fluctuations and Brownian motion are observed and the proper interpretation of experimental measurements relies on the ability for accurate predictions of the deterministic component of the hydrodynamics. Through dynamic similarity, analysis of a table top experiment performed by the UNC RMX group validates the mathematical theory and allows for direct comparison to the microscale.
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  • In Copyright
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  • McLaughlin, Richard
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