Superpropulsion of Droplets and Soft Elastic Solids

We investigate the behavior of droplets and soft elastic objects propelled with a catapult. Experiments show that the ejection velocity depends on both the projectile deformation and the catapult acceleration dynamics. With a subtle matching given by a peculiar value of the projectile/catapult frequency ratio, a 250% kinetic energy gain is obtained as compared to the propulsion of a rigid projectile with the same engine. This superpropulsion has strong potentialities: actuation of droplets, sorting of objects according to their elastic properties, and energy saving for propulsion engines.

Figure 1

(a) Image sequences of a typical droplet propulsion standing on a SHS initially at rest at $t=0$ (see movie M1 [19]). In the initial stage of the droplet propulsion ($t_1$), the droplet deformation is not homogeneous but concentrated in its lower part. In an intermediate stage the deformation reaches the top of the droplet. Later on ($t_e$) the droplet leaves the catapult with a complex and deformed shape. During its flight ($t_3$), we clearly see the oscillation modes of frequency $f_0$. (b) A space-time diagram is built along the dashed line drawn in image $t=0$. It illustrates the oscillatory motion of the catapult plate and the difference between the maximum plate velocity and the ejection velocity of the droplet.

Figure 2

The figure displays snapshots of three experiments performed with several drop sizes and the same catapult characteristics ($f=37\mathrm{Hz}$ and $A=1.4\mathrm{mm}$), see movie M1 [19]. Left: Position before the motion initiation. Right: Vertical position reached after ejection once all the kinetic energy is transferred into gravitational potential energy. The dashed line indicates the position that would reach the center of mass of a rigid object.

Figure 3

The figure summarizes measurements performed with both droplets (circles) and hydrogel balls (squares). Top: Energy transfer factor $\alpha$ plotted as a function of the frequency ratio $f_0/f$. The dashed line represents $\alpha =1$, the value expected for a rigid projectile, while the model based on the projectile deformation dynamics is represented by the solid line. Bottom: Same data and analysis for $t_e/T$ as a function of $f_0/f$. The dashed line represents $t_e/T=0.25$, the value expected for a rigid projectile.

Figure 4

The figure displays the simultaneous propulsion of soft elastic balls for several catapult frequencies (see movies M2, M3, and M4 [19]). The two first projectiles are hydrogel balls of eigenfrequency $f_0=90$ and 150 Hz, respectively. The right-most ball is rigid.

Figure 5

(a) Plate velocity $V_p(t)$ (dotted line), center of mass velocity in the reference frame of the plate $V_{\mathrm{c}\mathrm{m}}(t)$ (dashed lines), and ejection velocity $V_e(t)$ (solid lines) as functions of time for three frequency ratios. (b) $V_p$, $V_{\mathrm{c}\mathrm{m}}$, and $V_e$ at the ejection time $t_e$ as functions of $f_0/f$. All velocities are rescaled by the maximal plate velocity $V_p^{*}=2\pi fA$.