Kinematics of a simple reciprocal model swimmer at intermediate Reynolds numbers

We computationally studied the kinematics of a simple reciprocal model swimmer (asymmetric dumbbell) in a Newtonian fluid as a function of the Reynolds number (Re), and investigated how the onset and gradual increase of inertia impacts swimming behavior: a reversal in the swim direction, flow field, and the swim stroke. We divided the swim stroke into the increase and decrease in the distance between the two spheres (expansion and compression respectively) and related them to power and recovery strokes. We found that the switch in swim direction also corresponds to a switch in power and recovery strokes. We obtained expressions for the mean swimming velocity by collapsing the net displacement during expansion and compression under power-law relationships with respect to Re, the swimmer's amplitude, and the distance between the two spheres. Analyzing the fluid flows, we saw that the averaged flow field during expansion always resembles a pusher and during compression it always resembles a puller, but when averaged over the whole cycle, the flow that dominates is the one that occurs during the power stroke. We also related the power and recovery strokes to the swimming efficiency during times of expansion and compression, and found that the power stroke is, surprisingly, not always more efficient than the recovery stroke. Our results may have important implications in biology and ultimately the design of artificial swimmers.

Figure 1

(a) Reciprocal oscillation of the spherobot swimmer over one cycle. The large sphere (orange) with radius $R$ always oscillates in the opposite direction of the small sphere (blue) with radius $r$. The distance between the spheres $d\left(t\right)=d_0+Asin\left(2\pi ft\right)$ is prescribed to be of a simple harmonic oscillator with frequency $f$, where $d_0$ is the equilibrium distance between the spheres, and $A=A_r+A_R$ is the amplitude of the spherobot. The amplitudes of the small and large spheres are $A_r$ and $A_R$, respectively. When absent of fluid, the spherobot's center of mass (CM), shown in purple, does not move throughout the oscillation. (b) Simulation specifications and parameters.

Figure 2

Kinematic quantities plotted as functions of time over one cycle of oscillation after steady state had been reached. (a) Displacement of spherobot, represented by the net displacement of its center of mass $\mathrm{\Delta }{\widehat{y}}_{\text{CM}}$, (b) velocity of spherobot represented by ${\widehat{\text{v}}}_{\text{CM}}$, and (c) displacement of individual spheres of radius $R$ and $r$. Net swimming direction is indicated by the colors of the curves (black, Stokes flow; pink to yellow, small-sphere-leading; and green to blue, large-sphere-leading). The Reynolds number is represented by the shading of the curves; see the legend.

Figure 3

We define slip to be the region of movement where both spheres move in the same direction. As a result, the entire spherobot moves in the same direction as its spheres. Here, we show the nondimensional velocities ($v$/$fr$) of the spherobot ${\widehat{v}}_{\text{CM}}$ (purple), its large sphere ${\widehat{v}}_R$ (orange), and its small sphere ${\widehat{\text{v}}}_r$ (blue) when it is (a) small-sphere-leading and (b) large-sphere-leading. We identify the regions of slip observed during each spherobot's oscillation with black circles, and the region is magnified to the right. For the (a) small-sphere-leading spherobot, slip regions (1) and (2) are shown. Region (1) displays a region where slip is small-sphere-leading. It occurs at the end of compression and at the start of expansion. Region (2) shows a region where slip is large-sphere-leading. It occurs at the end of expansion and the start of compression. For the (b) large-sphere-leading spherobot, slip regions (3) and (4) are identified. Regions (3) and (4) show regions where the spherobot slips large-sphere-leading. Region (3) occurs at the end of compression and the start of expansion. Region (4) occurs at the end of expansion and the start of compression.

Figure 4

(a) Net displacement of the spherobot during expansion $\mathrm{\Delta }{\widehat{y}}_{\text{exp}}$ as a function of Re on a log-log scale for all $\epsilon$ and ${\widehat{d}}_0$ simulated; see the legend. Curves show a constant negative slope followed by a positive slope except for when both $\epsilon =0.8$ (pink) and ${\widehat{d}}_0=9.0,10.0$. The turning point and the positive slope change for different amplitudes $\epsilon$ (color). (b) Net displacement of the spherobot during compression $\mathrm{\Delta }{\widehat{y}}_{\text{com}}$ as a function of Re on a log-log scale. There are three distinct positive slope trends with Re. We fit the expansion and compression displacements with respect to Re, $\epsilon$, and $d_0$ using a multiple variable linear regression. (c) $\mathrm{\Delta }{\widehat{y}}_{\text{exp}}$ vs Re on a log-log scale collapsed into a negative slope (black dashed) and a positive slope region (red dashed). (d) $\mathrm{\Delta }{\widehat{y}}_{\text{com}}$ vs Re on a log-log scale collapsed into three distinct positive slope regions. The corresponding relationships with Re, $\epsilon$, and $d_0$ are also shown in (c) and (d). The dotted gray line represents the critical Reynolds number where the swimming direction switches from small-sphere-leading to large-sphere-leading.

Figure 5

The power and recovery stroke of the spherobot is determined by the movement of its appendage, the small sphere, represented by ${\widehat{v}}_r$. We define the power stroke to be when the small sphere moves opposite to the direction of net motion, ${\widehat{v}}_r\langle {\widehat{v}}_{\mathrm{CM}}\rangle <0$. Vice versa, the recovery stroke is defined to be when the small sphere moves in the same direction as the net motion, ${\widehat{v}}_r\langle {\widehat{v}}_{\mathrm{CM}}\rangle >0$. In this figure, the nondimensional velocity of the spherobot, ${\widehat{v}}_{\text{CM}}$, is represented by the purple curves. The mean swim direction $\langle {\widehat{v}}_{\mathrm{CM}}\rangle$ is indicated by the purple arrow in the accompanying spherobot schematics. The shaded areas under the ${\widehat{v}}_{\text{CM}}$ curve represent the displacements: in the mean swimming direction $\mathrm{\Delta }{\widehat{y}}_{+}$ (green) and opposite to it $\mathrm{\Delta }{\widehat{y}}_{-}$ (red). The power and recovery strokes for each swimmer are labeled by $P$ and $R$, respectively. (a) The spherobot in Stokes flow. Here, there is zero net displacement. Therefore, there is no power or recovery stroke observed. (b) and (c) The spherobot swims net small-sphere-leading at $\text{Re}=2.5$. Its power stroke is during compression, and its recovery stroke is during expansion. (d) The spherobot does not swim and its net displacement is very small and approximated to be zero. Like Stokes flow, we do not observe a power or recovery stroke. (e) and (f) The spherobot swims large-sphere-leading at $\text{Re}=33.0$ and 70.0. Its power and recovery stroke are opposite of those observed for the small-sphere-leading spherobot. The power stroke occurs during expansion, and the recovery stroke occurs during compression.

Figure 6

The average velocity field of a small-sphere-leading (top row) and large-sphere-leading (bottom row) spherobot averaged over (left) expansion, (middle) compression, and (right) an oscillation cycle. Flow magnitudes are represented by the heat map, and the flow direction is indicated by the black arrows. (a) Flow field of small-sphere-leading spherobot averaged over expansion. The fluid flows outward along the swimming axis and inward perpendicular. (b) Flow field of small-sphere-leading spherobot averaged over compression. The flow is opposite to that of expansion, inward along the swimmer's axis and outward perpendicular. (c) Averaged over a whole cycle small-sphere-leading spherobot flow. The net flow is pullerlike. (d) Flow field of large-sphere-leading spherobot averaged over expansion. The fluid flows outward along the swimming axis and inward perpendicular. (e) Flow field of large-sphere-leading spherobot averaged over compression. The flow is opposite to that of expansion, inward along the swimmer's axis, and outward perpendicular. (f) Averaged over a whole cycle large-sphere-leading spherobot flow. The net flow is pusherlike.

Figure 7

Efficiency of a spherobot with parameters $\epsilon =1.2$ and ${\widehat{d}}_0=6.5$ as a function of Re where ${\eta }_{\text{exp}}$ (solid green and red), ${\eta }_{\text{com}}$ (dashed green and red), and ${\eta }_{\text{cyc}}$ (black) are depicted. Also shown are the efficiencies of the power (green) and recovery (red) strokes. The inset shows a closeup of the efficiencies in the small-sphere-leading regime. Here, the power stroke occurs during compression, the recovery stroke occurs during expansion, and ${\eta }_{\text{com}}>{\eta }_{\text{exp}}$. Because there is a switch in swimming direction at $\text{Re}\approx 20$, the power and recovery strokes also switch. Now, ${\eta }_{\text{exp}}>{\eta }_{\text{com}}$. As expected, the power stroke is more efficient than the recovery, but up until $\text{Re}>110$ for this configuration. There is also a trend in the spherobot's cycle efficiency where swimming large-sphere-leading is generally more efficient than swimming small-sphere-leading.

Figure 8

The fluid flow around the spherobot continuously evolves across Re. We provide four flows averaged over a cycle, from a spherobot with ${\widehat{d}}_0=5.0$ and $\epsilon =1.2$, across both swimming regimes to highlight its evolution. Vortices of interest are identified with circles (O) and stagnation points are shown with a ($+$). Their colors are chosen to contrast background vorticity. (a) At $\text{Re}=2.0$, the spherobot swims small-sphere-leading. We observe one pair of vortices from each sphere, and the small sphere's vortices dominate the surrounding flow. (b) At $\text{Re}=12.0$, an outer vortex forms below the small sphere, which rotates counter to the inner vortex. The flow direction change below the small sphere is shown to occur at the specified stagnation point. (c) At $\text{Re}=20.0$, another outer vortex forms above the large sphere with an accompanying stagnation point. There is a competition between pushing and pulling the fluid both above and below the spheres, and the spherobot does not swim. (d) The spherobot now swims large-sphere-leading at $\text{Re}=30.0$. The outer vortex above the large sphere disappears, and the flow merges with the outer vortex above the small sphere, pink circle. The outer vortex below the small sphere (green circle) remains and moves closer to the spherobot. The outer vortices generated from the small sphere movement dominate the surrounding fluid flow.