Propulsion of a Two-Sphere Swimmer

Daphne Klotsa, Kyle A. Baldwin, Richard J. A. Hill, R. M. Bowley, and Michael R. Swift

Phys. Rev. Lett. 115, 248102 – Published 7 December 2015

Abstract

We describe experiments and simulations demonstrating the propulsion of a neutrally buoyant swimmer that consists of a pair of spheres attached by a spring, immersed in a vibrating fluid. The vibration of the fluid induces relative motion of the spheres which, for sufficiently large amplitudes, can lead to motion of the center of mass of the two spheres. We find that the swimming speed obtained from both experiment and simulation agree and collapse onto a single curve if plotted as a function of the streaming Reynolds number, suggesting that the propulsion is related to streaming flows. There appears to be a critical onset value of the streaming Reynolds number for swimming to occur. We observe a change in the streaming flows as the Reynolds number increases, from that generated by two independent oscillating spheres to a collective flow pattern around the swimmer as a whole. The mechanism for swimming is traced to a strengthening of a jet of fluid in the wake of the swimmer.

Received 13 November 2014

DOI:https://doi.org/10.1103/PhysRevLett.115.248102

Authors & Affiliations

Daphne Klotsa^1,2,3, Kyle A. Baldwin¹, Richard J. A. Hill¹, R. M. Bowley¹, and Michael R. Swift¹

¹School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
²Department of Chemistry, Lensfield Road, University of Cambridge, Cambridge CB2 1EW, United Kingdom
³Department of Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290, USA

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Vol. 115, Iss. 24 — 11 December 2015

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Images

Figure 1
Main panel shows the experimental data collapse for the unequal-sized swimmer that swims upwards. The driving conditions are the following: blue stars $f = 65 Hz$ , light blue triangle down $f = 75 Hz$ , pink hexagons $f = 85 Hz$ , cyan circles $f = 95 Hz$ , yellow triangle up $f = 105 Hz$ , green triangles right $f = 125 Hz$ , green diamond $f = 135 Hz$ . In all cases the viscosity was $1.2 {mm}^{2} / s$ except for one data set (turquoise diamond $f = 135 Hz$ ) where the viscosity was $2.5 {mm}^{2} / s$ . Simulations for $Γ$ between 2 and 20 and frequencies $f = 65$ , 75, 100, 125 Hz are shown by the red plus symbols for comparison. The lower inset shows a close-up of the experimental data ( $f = 125 Hz$ ) shown in the main panel near the apparent onset. The upper inset shows a photograph of the swimmer when stationary.
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Figure 2
Main panel shows the experimental data collapse for the equal-sized swimmer. The driving conditions are the following: blue stars $f = 30 Hz$ and green triangles right $f = 35 Hz$ . Simulations for $Γ$ between 2 and 12 and frequency $f = 30 Hz$ are shown by the red plus symbols for comparison. The inset shows a photograph of the swimmer when stationary.
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Figure 3
Illustrations of the fluid flows generated by the vibration of the spheres from experiment and simulation. Panel (a) shows an image taken from experiment showing the flow around the spheres. The arrow illustrates the direction of a jet of fluid evident from the movies (Supplemental Material [24]). Panels (b) and (c) show the direction of the time-averaged velocity field (i.e. the normalized velocity vectors) in the plane of the spheres. In (b) the swimmer is stationary ( ${Re}_{s} = 15$ ) while in (c) it is swimming ( ${Re}_{s} = 60$ ). These figures illustrate the change in topology of the flows as the amplitude of vibration increases. Note that the magnitude of the flows is much greater around the smaller sphere than around the larger sphere, as seen in (a).
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Figure 4
Data collapse from simulations confirming the scaling behaviour for different viscosities (red $1.2 \times 10^{- 6} m^{2} / s$ , blue $2 \times 10^{- 6} m^{2} / s$ , green $3 \times 10^{- 6} m^{2} / s$ ). Each data set includes simulations for $Γ$ between 2–20 and $f = 65$ , 75, 100, 125 Hz. The lower inset shows typical trajectories of a swimmer that stays stationary for $Γ = 2$ , $f = 65 Hz$ (red line) and one that swims for $Γ = 12$ , $f = 75 Hz$ (blue line). The upper left inset is a snapshot from simulations showing the swimmer and the simulated cell.
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