Rotational motion of a thin axisymmetric disk in a low Reynolds number linear flow

Abstract

The motion of thin, torque-free axisymmetric rigid particles with fore-aft symmetry in low Reynolds number linear flows is investigated. The rotational motion of such a particle is fully determined by the effective aspect ratio (κe), defined as the aspect ratio of a spheroid having the same period of rotation as that of the particle. We determine the effective aspect ratio for a family of shapes given by, y(ρ) = κ(1 − ρ2)α where α is a positive parameter, ρ is the radial distance from the particle center in polar coordinates, y is half the thickness of the particle, κ is the aspect ratio of the particle defined as the ratio of the thickness (L) of the particle parallel to the axis of symmetry to its diameter (d) perpendicular to the axis of symmetry. This family includes an oblate spheroid (\documentclass[12pt]{minimal}\begin{document}$\alpha =\frac{1}{2}$\end{document}α=12) and the shape approaches a blunt circular cylinder shape as α → 0. For a thin particle, the effective aspect ratio scales like \documentclass[12pt]{minimal}\begin{document}$\smash{G_{A}^{\frac{1}{2}}}$\end{document}GA12, where GA is the torque non-dimensionalized by μγd3 acting on a particle held in a fixed alignment in a simple shear flow with its longer dimensions in the flow-vorticity plane. Here, μ is the fluid viscosity and γ is the shear rate. Starting with the integral representation of the Stokes flow, an analysis based on a matched asymptotic expansions approach is performed to determine the scaling of the torque acting on a stationary particle in simple shear flow with κ as the small parameter. Using boundary element method simulations, the exact torques are calculated and the scaling obtained from the analysis is verified. We find that there are two regions of interest that contribute to the torque, a flat outer region covering most of the disk area and a boundary layer region of large slope at the edge. For \documentclass[12pt]{minimal}\begin{document}$\alpha >\frac{1}{4}$\end{document}α>14, the torque is dominated by the stresses acting on the flat surface of the particle and is O(κ2) to the leading order. For these shapes, the effective aspect ratio scales like the aspect ratio of the particle. On the other hand, for \documentclass[12pt]{minimal}\begin{document}$\alpha \le \frac{1}{4}$\end{document}α≤14, the leading order torque contribution comes from the boundary layer region. To the leading order, the torque scales as O(κ2logκ) for \documentclass[12pt]{minimal}\begin{document}$\alpha =\frac{1}{4}$\end{document}α=14 and as \documentclass[12pt]{minimal}\begin{document}$O(\kappa ^\frac{3}{2(1-\alpha )})$\end{document}O(κ32(1−α)) for \documentclass[12pt]{minimal}\begin{document}$\alpha <\frac{1}{4}$\end{document}α<14. In general, a particle with a blunt edge is found to rotate faster than a particle of the same aspect ratio for which the slope of the edge varies slowly. The fastest rotating particle is identified to be a cylindrical disk for which κe = 1.12κ3/4.