Hydrodynamic slender-body theory for local rotation at zero Reynolds number

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Hydrodynamic slender-body theory for local rotation at zero Reynolds number

Benjamin J. Walker, Kenta Ishimoto, and Eamonn A. Gaffney

Phys. Rev. Fluids 8, 034101 – Published 2 March 2023

Abstract

Slender objects are commonplace in microscale flow problems, from soft deformable sensors to biological filaments such as flagella and cilia. While much research has focused on the local translational motion of these slender bodies, relatively little attention has been given to local rotation, even though it can be the dominant component of motion. In this study, we explore a classically motivated ansatz for the Stokes flow around a rotating slender body via superposed rotlet singularities, which leads us to pose an alternative ansatz that accounts for both translation and rotation. Through an asymptotic analysis that is supported by numerical examples, we determine the suitability of these flow ansatzes for capturing the fluid velocity at the surface of a slender body, assuming local axisymmetry of the object but allowing the cross-sectional radius to vary with arclength. In addition to formally justifying the presented slender-body ansatzes, this analysis reveals a markedly simple relation between the local angular velocity and the torque exerted on the body, which we term resistive torque theory. Though reminiscent of classical resistive force theories, this local relation is found to be algebraically accurate in the slender-body aspect ratio, even when translation is present, and is valid and required whenever local rotation contributes to the surface velocity at leading asymptotic order.

Received 3 November 2022
Accepted 1 February 2023

DOI:https://doi.org/10.1103/PhysRevFluids.8.034101

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Slender body theory

Stokes equations

Fluid Dynamics

Authors & Affiliations

Benjamin J. Walker ^1,2,*, Kenta Ishimoto^3,†, and Eamonn A. Gaffney^4,‡

¹Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
²Department of Mathematics, University College London, London, WC1H 0AY, United Kingdom
³Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
⁴Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom

^*Corresponding author: bjw43@bath.ac.uk
^†ishimoto@kurims.kyoto-u.ac.jp
^‡gaffney@maths.ox.ac.uk

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Issue

Vol. 8, Iss. 3 — March 2023

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Images

Figure 1
A slender body with circular cross section of arclength-dependent radius $ε η (s)$ , centerline $ξ (s)$ , and cross-sectional angle $ϕ$ . Points on the surface of the body are parameterized as $X (s, ϕ)$ , where $s \in [- 1, 1]$ is an arclength parameter. Figure adapted from Walker et al. [11] with permission.
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Figure 2
Approximation of $I_{1} (s)$ . We illustrate $I_{1} (s)$ and a leading-order approximation to it, each normalized by $1 / ε^{2} η^{2} (s)$ . The exact value of $I_{1} (s)$ is shown as a solid black curve, with the approximation of Eq. (24) shown as a gray dashed line. We observe excellent agreement between the approximation of Eq. (24) and the true value of $I_{1} (s)$ for the vast majority of the slender body, with the discrepancy towards the ends highlighted in the inset. Here we have taken $η (s) = {(1 - s^{2})}^{1 / 2}$ and $ε = 10^{- 2}$ , corresponding to a slender prolate ellipsoid, though we note that the qualitative features of this figure are not sensitive to this choice.
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Figure 3
Maximum absolute error in the no-slip boundary condition for axially rotating slender bodies. We report the error $E$ in the surface velocity generated by Eq. (9), making use of numerically computed torque densities that result from Eq. (12) and Eq. (28) in turn, denoting the corresponding errors by $E^{rotlet}$ and $E^{RTT}$ , respectively. which correspond to the full rotlet boundary condition and the derived resistive torque theory, respectively. Having taken $η (s) = \sqrt{1 - s^{2}}$ and discretizing the centerline into $N = 100$ segments, we report the maximum absolute errors, as measured in the infinity norm and defined in Eq. (53), for $ε \in {0.01, 0.005}$ , prescribing $V (s) = 0$ and $Ω (s) = e_{t}$ . In (b)–(e) the numerically computed errors are seen to be scaling approximately with $ε^{2}$ , with $E_{ε = 0.01} \approx 4 E_{ε = 0.005}$ , in line with the analysis of Sec. 3, while the errors in (a) are dictated by the centerline discretization and reduce upon refinement (see Sec. 5d). The slender bodies with the largest errors correspond to those with the largest curvatures, with the largest errors being localized to regions with higher curvature. These plots have aspect ratio 1:1 and are independently scaled for visual clarity. Parameterizations of these centerlines are given in Appendix pp3.
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Figure 4
Off-axis rotation of a prolate ellipsoid. Imposing the angular velocity $Ω (s) = e_{n} (s)$ on a straight prolate ellipsoid, taking $V (s) = 0$ and $ε = 0.01$ , we numerically solve for the forces and/or torques in the pure rotlet of Sec. 3 and the combined ansatz of Sec. 4. We sample the induced velocity at 612 points on the surface of the slender body, sampling at 12 equispaced angles at 51 equally spaced arclengths. The components of velocity at the sample points are shown for the rotlet ansatz, the prescribed boundary condition, and the combined ansatz in panels (a), (b), and (c), respectively, with different colors corresponding to different components. The $e_{t}$ component, shown lightest, is captured well by both ansatzes, while the $e_{n}$ and $e_{b}$ components, shown as dark gray and black curves, respectively, deviate significantly from the exact solution when using the pure rotlet ansatz. The combined ansatz of Sec. 4 accurately captures all velocity components, with errors on the order of $10^{- 6}$ . Sample points are in ascending order by arclength, with perceived oscillations corresponding to sampling at different points around the circumference of each cross section.
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Figure 5
Relative error in using the generalized resistive torque theory. Numerically computing the torques associated with the resistive torque theory relation of Eq. (51), we report the maximum relative error in torque when using the resistive torque theory, denoted by $E^{m}$ . This is measured relative to the torque computed from enforcing the full boundary condition of Eq. (41) and is shown as a solid curve and as a function of the ratio $ε | Ω | / | V |$ . The analogous error in the surface velocity, denoted by $E^{u}$ and evaluated at the discrete surface points described in Sec. 5a, is shown dot-dashed. In line with the analysis of Sec. 4d, the errors in both the torque and the surface velocity are approximately on the order of $ε$ , here $10^{- 2}$ , when $ε | Ω | / | V | > 1$ . The apparent spike in $E^{u}$ is an artifact of the discrete approximation to the maximum relative error. Here we have taken $N = 100, V (s) = e_{x}$ and $Ω (s) ∥ e_{t}$ , adopting the parabolic centerline of Fig. 3 and fixing $η (s) = \sqrt{1 - s^{2}}$ .
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