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The force on a slender particle under oscillatory translational motion in unsteady Stokes flow

Published online by Cambridge University Press:  17 December 2019

Jason K. Kabarowski  and
Aditya S. Khair Open the ORCID record for Aditya S. Khair [Opens in a new window]
Jason K. Kabarowski
Affiliation:
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Aditya S. Khair*
Affiliation:
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
*
†Email address for correspondence: akhair@andrew.cmu.edu
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Abstract

Asymptotic approximations are derived for the hydrodynamic force on a rigid, axisymmetric slender particle executing longitudinal or transverse oscillation in unsteady Stokes flow. The slender particle has an aspect ratio $\unicode[STIX]{x1D716}=a/\ell \ll 1$, where $\ell$ is the half-length of the particle, and $a$ is its characteristic cross-sectional width. It is assumed that the particle has zero thickness at its ends. The frequency of oscillation is parameterized by the complex quantity $\unicode[STIX]{x1D706}^{2}=-\text{i}\ell ^{2}\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708}$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity, $\unicode[STIX]{x1D714}$ is the particle angular oscillation frequency (units of radians per second) and $\text{i}=\sqrt{-1}$. Asymptotic approximations for the force are obtained in three distinguished limits for longitudinal oscillations: (i) a ‘low-frequency’ regime with $\unicode[STIX]{x1D716}\rightarrow 0$ and $|\unicode[STIX]{x1D706}|$ fixed; (ii) a ‘moderate-frequency’ regime with $\unicode[STIX]{x1D716}\rightarrow 0$ and $\unicode[STIX]{x1D716}|\unicode[STIX]{x1D706}|=O(1)$; and (iii) a ‘high-frequency’ regime with $\unicode[STIX]{x1D716}\rightarrow 0$ and $\unicode[STIX]{x1D716}|\unicode[STIX]{x1D706}|=O(1/\unicode[STIX]{x1D716}^{2})$. The acceleration reaction is a leading-order contributor to the force in this last regime, whereas it is subdominant at moderate frequency. For transverse oscillation we construct asymptotic approximations in the low and moderate-frequency regimes. Here, the acceleration reaction here plays a leading-order role at moderate frequency; hence, a ‘high frequency’ regime in this case simply corresponds to the limiting behaviour for $\unicode[STIX]{x1D716}|\unicode[STIX]{x1D706}|\gg 1$. Our asymptotic predictions are in good agreement with the numerically computed frequency-dependent force on a prolate spheroid ($\unicode[STIX]{x1D716}=0.1$) for longitudinal and transverse oscillations by Lawrence and Weinbaum (J. Fluid Mech., vol. 189, 1988, pp. 463–489) and Pozrikidis (Phys. Fluids, vol. 1, 1989a, pp. 1508–1520), respectively.

JFM classification

Low-Reynolds-number flows: Slender-body theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 884 , 10 February 2020 , A44
Copyright
© 2019 Cambridge University Press

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