Swimming droplets driven by a surface wave

Self-propelling motion is ubiquitous for soft active objects such as crawling cells, active filaments, and liquid droplets moving on surfaces. Deformation and energy dissipation are required for self-propulsion of both living and non-living matter. From the perspective of physics, searching for universal laws of self-propelled motions in a dissipative environment is worthwhile, regardless of the objects' details. In this article, we propose a simple experimental system that demonstrates spontaneous migration of a droplet under uniform mechanical agitation. As we vary control parameters, spontaneous symmetry breaking occurs sequentially, and cascades of bifurcations of the motion arise. Equations describing deformable particles and hydrodynamic simulations successfully describe all of the observed motions. This system should enable us to improve our understanding of spontaneous motions of self-propelled objects.


B. Traveling wave of squirming motion
The traveling wave is generated at the water-oil-air triple line. Thus, a squirming droplet swims due to the traveling wave. A spatio-temporal plot of the droplet with the traveling wave is shown in Fig. S1 (a). The color indicates the magnitude of deformation. Red and blue represent the peak and bottom of the traveling wave, respectively. As shown in Fig. S1 (a), the traveling wave periodically propagates from the "head" to the "tail" of the droplet at nearly the same frequency as the Faraday wave (typically 50Hz). In addition, the Faraday wave slowly propagates from the "tail" to the "head" along the long axis of the droplet (typically 3Hz), and nodes and anti-nodes are sequentially generated at the "tail" and disappear at the "head" (Fig. S1 (b)).

C. Model of the deformable particle
The time-evolution equation for deformable particles is derived in [S1 -S3].
Equivalent equations can also be derived from symmetry arguments [S4]. By using a Fourier series, the deformation of a 2D particle around a circular shape can be written as where we set the radius to be unity. Complex amplitude z n can be expressed as z n =ρ n exp(inφ n ). ρ n and φ n are identical to those in the Methods. Here, we define the following complex velocity: where V and φ v are the speed of the centroid and the direction of migration, respectively.
For convenience, we set z 1 = v 1 and zn as the complex conjugate of z n . The general time-evolution equations for z n (n is a positive integer) are derived by considering possible coupling terms up to the third order [S4].
where i, j, and k are integers excluding 0, and A i , B ij , and C ijk are real constant coefficients.
Here, we found that all of the observed motions in the experiment can be described by model equations of deformable particles by considering a 4 th order rank tensor [S3]. For simplicity, we only consider the terms that seem to be important for mode bifurcation.
First, we show the numerical results of the equations for v 1 , z 2 and z 4 , while ignoring z 3 .
Thus, we only consider Eqs. (S1), (S2) and (S4) with α2 = β2 =0. The phase diagram obtained from the numerical calculation is shown in Fig Around the bifurcation from spinning motion to rotational motion, V 2 becomes an almost linear function of ρ 2 cos2ψ ( Fig. S2 (b)). As long as ρ 2 is not so large, z 3 and z 4 are negligible. If we omit the terms z 3 and z 4 , Eq. (S1) and (S2) can be transformed into Eqs. (1) and (2)  Next, we show the numerical results of the equations for v 1 , z 2 and z 3 , while ignoring z 4 .

D. Simulation of the 2D Navier-Stokes equation
To determine the relation between oscillatory flow and steady flow in the silicone oil, we simulate the 2D Navier-Stokes equation. Before the simulation, we measure the oscillatory flow near the droplet-silicone oil interface of the stationary elongated droplet.
The flow near the droplet is measured by particle-tracking using tracers. Next, we focus on the flow along the droplet -oil interface. We measure the velocity 1.2mm from the droplet interface. The velocity is decomposed into a tangential component V t (t,θ) and a normal component V n (t,θ) with respect to the droplet-oil interface (see schematic illustration, Fig.S4 (c)). Here, θ is defined around the centroid and is the angle with respect to the long axis. We found that V n (t,θ) is several times larger than V t (t,θ), and the spatio-temporal plot of V n (t,θ) is shown in Fig. S4 (d). Figure  S4 (d) indicates that strong oscillatory flow is localized around the long axis, and the flow almost becomes a standing wave.
By using the experimental data on oscillatory flow, we simulate the two-dimensional flow of the Navier-Stokes equation. As a boundary condition on the droplet, we use the oscillatory flow V t (t,θ) and V n (t,θ) which are obtained in the experiment. Next, we check whether the steady flow near the droplet is reconstructed or not. In the simulation, we ignore deformation of the droplet and flow in the droplet. The parameter values are chosen to be the same as those in the experiment.
We calculate the 2D Navier Stokes equation with polar coordinates: The equation of continuity is At the surface of the droplet r=R, where R and σ are the radius of the droplet and the surface tension, respectively. At the boundary far from the droplet r=r F , where we set a constant pressure to eliminate the mean flow.
In the simulation, we obtained the four vortices of steady streaming as seen in the experiment (Fig.S4 (e)). Compared to the experiment, the speed of the vortices is slightly small near the droplet and large at a distance from the droplet. This difference could be due to the two-dimensionality of the simulation. We introduce fitting parameter α and substitute αV t (t,θ) and αV n (t,θ) at the boundary condition. Then, we found that we could fit the experimental results well (Fig. S4 (f)). These results indicate that steady streaming is generated by oscillation of the triple line.