Transition in swimming direction in a model self-propelled inertial swimmer

We propose a reciprocal, self-propelled model swimmer at intermediate Reynolds numbers (Re). Our swimmer consists of two unequal spheres that oscillate in antiphase, generating nonlinear steady streaming (SS) flows. We show computationally that the SS flows enable the swimmer to propel itself, and also switch direction as Re increases. We quantify the transition in the swimming direction by collapsing our data on a critical Re and show that the transition in swimming directions corresponds to the reversal of the SS flows. Based on our findings, we propose that SS can be an important physical mechanism for motility at intermediate Re.

Figure 1

Reciprocal oscillation of the spherobot swimmer over one cycle. Spheres' centers of mass (COM) (blue circles) and the spherobot COM (green circle) are indicated. The distance between the spheres' centers $d\left(t\right)$ is $d_0$ at the equilibrium distance, $d_0-A$ at minimum distance, and $d_0+A$ at maximum distance. The total amplitude is $A=A_R+A_r$.

Figure 2

Velocity of the spherobot as a function of Re in 2D for the range of $A$, $d_0$, $R$, and $r$ shown in the legend. The inset shows the small-sphere-leading regime plotted on a semilogarithmic $x$ scale. Parameters $A$, $d_0$, $R$, and $r$ are nondimensionalized by the length scale, $r_0=0.15\mathrm{m}$, the radius for the small sphere. Line color indicates $A$, line saturation indicates $d_0$, line style indicates $R$, and symbols indicate $r$. Negative velocity indicates swimming in the direction of the small sphere, and positive velocity indicates swimming in the direction of the large sphere. Vertical dashed lines denote critical Re for the transition.

Figure 3

Left column: Small-sphere-leading regime at $\text{Re}=2$. Right column: Large-sphere-leading regime at $\text{Re}=30$. Spherobot velocity field superimposed with the vorticity field and streamlines in (a) the small-sphere-leading regime and (b) the large-sphere-leading regime. The largest dimensionless velocity magnitude in (a), (b) is $|{\mathbf{v}}_{\mathbf{max}}|/fr=0.88$. Schematic diagrams showing the reversal of steady streaming flows for one sphere in the limiting cases (c) $\delta \gg r$, and (e) $\delta \ll r$. Due to symmetry, the lower left quadrant is indicated with a dashed line. Velocity vector plot superimposed with the vorticity field for one sphere at (d) $\text{Re}=2$ and (f) $\text{Re}=30$. The largest dimensionless velocity magnitude in (d), (f) is $|{\mathbf{v}}_{\mathbf{max}}|/fr=1.1$.