Stop-and-go locomotion of superwalking droplets

Vertically vibrating a liquid bath at two frequencies, $f$ and $f/2$, having a constant relative phase difference can give rise to self-propelled superwalking droplets on the liquid surface. We have numerically investigated such superwalking droplets in the regime where the phase difference varies slowly with time. We predict the emergence of stop-and-go motion of droplets, consistent with experimental observations [Valani et al. Phys. Rev. Lett. 123, 024503 (2019)]. Our simulations in the parameter space spanned by the droplet size and the rate of traversal of the phase difference uncover three different types of droplet motion: back-and-forth, forth-and-forth, and irregular stop-and-go motion, which we explore in detail. Our findings lay a foundation for further studies of dynamically driven droplets, whereby the droplet's motion may be guided by engineering arbitrary time-dependent phase difference functions.

Figure 1

(a) Snapshots of a superwalker undergoing stop-and-go motion (SGM) showing the passive bouncing (stop) phase and the active self-propulsion (go) phase as observed in experiments. (b) The theoretical setup has a droplet located at a vertical position $z_d$ with an underlying wave field $h(\mathbf{x},t)$ that is walking horizontally with velocity ${\dot{x}}_d$ in the frame of reference of the driven bath. The notation is defined in the main text. The vertical amplitude of the wave field is exaggerated.

Figure 2

Droplet dynamics as a function of the relative phase difference. (a) Steady walking speed $u_d$ of a superwalker as a function of a fixed phase difference $\mathrm{\Delta }{\varphi }_0$ for a droplet of radius $R=0.60$ mm from experiments of Valani et al. [30] (markers) and simulations of Valani et al. [31] (multicolored curve). The experimental data to the right of the vertical dashed line at $\mathrm{\Delta }{\varphi }_0={180}^{\circ }$ is repeated. The style of marker indicates the bouncing modes observed in experiments: (1,2,2) are black circles $•$, (1,2,1) are black triangles $▴$, and transition between a (1,2,1) and a (1,2,2) mode are gray squares $\blacksquare$. The colors on the solid curve indicate different vertical bouncing modes observed in simulations: ${(1,2,1)}^{\text{L}}$ in navy blue, (1,2,2) in red, and chaotic modes that arise at the transition between ${(1,2,1)}^{\text{L}}$ and (1,2,2) are shown in gray. The light-purple shaded region indicates the bouncing regime where pure vertical bouncing with no horizontal walking motion is observed. The dotted multicolored horizontal line in the superwalking regime indicates pure bouncing states that are unstable to horizontal perturbations. In this panel, a time-varying phase difference $\mathrm{\Delta }\varphi \left(t\right)=2\pi \epsilon t$ would correspond to traversing the $\mathrm{\Delta }{\varphi }_0$ axis from either left to right $(\epsilon >0)$ or right to left $(\epsilon <0)$. Panel (b) shows the vertical periodic motion of the bath, $\mathcal{B}\left(t\right)=-\left[{\gamma }_f/{\left(2\pi f\right)}^2\right]sin\left(2\pi ft\right)-\left[{\gamma }_{f/2}/{\left(\pi f\right)}^2\right]sin(\pi ft+\mathrm{\Delta }{\varphi }_0)$, in the laboratory frame at different constant phase differences $\mathrm{\Delta }{\varphi }_0$. Panel (c) shows vertical bouncing modes obtained for different values of $\mathrm{\Delta }{\varphi }_0$ that are indicated by a black line segment at the top of panel (a). In panel (c), the solid black curves indicate the bath motion, $\mathcal{B}\left(t\right)$, the multicolored curves represent the vertical motion of the south pole of the droplet, $z_d\left(t\right)+\mathcal{B}\left(t\right)$, and the filled blue regions illustrate the vertical motion of the liquid surface directly beneath the droplet, $h(x_d,t)+\mathcal{B}\left(t\right)$, all in the laboratory frame. The gray regions indicate times at which the droplet is in contact with the underlying waves.

Figure 3

Various types of SGM observed from simulations in the $(\epsilon ,R)$ parameter space. Four distinct types of dynamics are observed: (i) No walking ($\times$, beige region) where the droplet bounces vertically but does not propel horizontally, (ii) irregular SGM ($\circ$, purple region) where the droplet erratically switches its walking direction and/or traverses unequal distances after every half SGM cycle, (iii) back-and-forth SGM ($\mathrm{\Delta }$, green region) where the droplet switches direction after every half SGM cycle, and (iv) forth-and-forth SGM ($+$, maroon region) where the droplet moves in the same direction after every half SGM cycle. The markers indicate the parameter values where simulations were performed while the background colors represent the interpolated region.

Figure 4

Back-and-forth SGM: (a) Horizontal walking speed $u_d$ (green curve) as a function of the evolving phase difference $\mathrm{\Delta }\varphi \left(t\right)=2\pi \epsilon t$ for one cycle of the SGM. The walking speed from constant phase difference superwalker simulations from Fig. 2 is shown as a multicolored curve. The light-purple shaded region indicates the pure bouncing regime. Panel (b) shows the vertical velocity $v_d$ of the droplet at impact (lower frame) and lift-off (upper frame) from constant phase difference superwalker simulations (background navy blue and red multicolored markers with $•$ indicating the superwalking solution branch and $\times$ indicating the nonwalking, pure bouncing solution branch in the superwalking regime) and the back-and-forth SGM (green $•$ indicating impact and green $▴$ indicating lift-off). Panel (c) shows the vertical motion of the bath (black curve), wave height beneath the droplet (blue shaded region), and the vertical location of the droplet's “south pole” (green curve) during back-and-forth SGM. The gray shaded region represents contact between the droplet and the underlying waves. Panel (d) shows the evolution of the horizontal wave force $F_W=-F_N{\left(t\right)\partial h/\partial x|}_{x=x_d}$ (blue), horizontal drag force $F_D=-D_{\text{tot}}\left(t\right){\dot{x}}_d$ (red), and the total horizontal force $F_H=F_W+F_D$ (black) during the back-and-forth SGM. The magenta circular markers represent the integrated total horizontal force over the duration of a single bounce. Panel (e) shows the evolution of the gradient of the wave beneath the droplet ${\partial h/\partial x|}_{x=x_d}$ (black solid curve) with the dashed line indicating zero gradient. The droplet size is fixed to $R=0.60$ mm and the detuning is fixed to $\epsilon =-0.5$ Hz.

Figure 5

Forth-and-forth SGM: (a) Horizontal walking speed $u_d$ (maroon curve) as a function of the evolving phase difference $\mathrm{\Delta }\varphi \left(t\right)=2\pi \epsilon t$ for one cycle of the SGM. The walking speed from constant phase difference superwalker simulations from Fig. 2 is shown as a multicolored curve. The light-purple shaded region indicates the bouncing regime. Panel (b) shows the vertical velocity $v_d$ of the droplet at impact (lower frame) and lift-off (upper frame) from constant phase difference superwalker simulations (background navy blue and red multicolored markers with $•$ indicating the superwalking solution branch and $\times$ indicating the nonwalking, pure bouncing solution branch in the superwalking regime) and the forth-and-forth SGM (maroon $•$ indicating impact and maroon $▴$ indicating lift-off). Panel (c) shows the vertical motion of the bath (black curve), wave height beneath the droplet (blue shaded region), and the vertical location of the droplet's “south pole” (maroon curve) during forth-and-forth SGM. The gray shaded region represents contact between the droplet and the underlying waves. Panel (d) shows the evolution of the horizontal wave force $F_W=-F_N{\left(t\right)\partial h/\partial x|}_{x=x_d}$ (blue), horizontal drag force $F_D=-D_{\text{tot}}\left(t\right){\dot{x}}_d$ (red), and the total horizontal force $F_H=F_W+F_D$ (black) during the forth-and-forth SGM. The magenta circular markers represent the integrated total horizontal force over the duration of a single bounce. Panel (e) shows the evolution of the gradient of the wave beneath the droplet ${\partial h/\partial x|}_{x=x_d}$ (black solid curve) with the dashed line indicating zero gradient. The droplet size is fixed to $R=0.60$ mm and the detuning is fixed to $\epsilon =0.5$ Hz.

Figure 6

Back-and-forth and forth-and-forth SGM for different detunings. Horizontal walking speed $u_d$ as a function of the evolving phase difference $\mathrm{\Delta }\varphi \left(t\right)$ for one cycle of the SGM showing (a) back-and-forth SGM at different negative detunings and (b) forth-and-forth SGM at different positive detunings. Panels (c) and (d) show the vertical velocity of the droplet $v_d$ at impact (lower frame) and lift-off (upper frame) from the back-and-forth and the forth-and-forth SGM corresponding to panels (a) and (b), respectively. The constant phase difference superwalker simulation results are shown as multicolored curves in the background in these panels.

Figure 7

Back-and-forth and forth-and-forth SGM for small detunings for a droplet of radius $R=0.60$ mm. (a) Walking velocity $u_d$ as a function of time for $\epsilon =-0.10$ Hz showing forth-and-forth SGM without noise (black) and SGM with erratic direction reversals due to added noise (gray). Panel (b) shows a zoomed-in section of panel (a) with the vertical axis stretched and the green background highlighting that the droplet continues to walk in the same direction as the previous cycle when no noise is added. (c) Walking velocity $u_d$ as a function of time for $\epsilon =0.08$ Hz showing back-and-forth SGM without noise (black) and SGM with erratic direction reversals due to added noise (gray). Panel (d) shows a zoomed-in section of panel (c) with the vertical axis stretched and the pink and green regions highlighting that the droplet reverses its walking direction compared to the previous cycle when no noise is added. The multicolored curves represent the corresponding results for constant phase difference simulations.

Figure 8

No walking for very small detunings $\left(\right|\epsilon |<0.04)$ and the effect of added noise for a droplet of radius $R=0.60$ mm. Panels (a) and (c) show the walking velocity $u_d$ as a function of time without noise (black) and with noise (gray) for $\epsilon =-0.03$ Hz and $\epsilon =0.03$ Hz, respectively. Panels (b) and (d) show the vertical velocity of the droplet $v_d$ at impact and lift-off without noise (black) and with noise (gray) corresponding to panels (a) and (c), respectively. The constant phase superwalker simulation results are shown as multicolored curves in these panels.