Figure 6

Simulation of behavior of compressed water-in-oil droplets based on continuum active fluid equations. (a) Simulated time-averaged velocity fields and vorticity maps from the midplanes of compressed droplets (\(r =\) 2.4 mm, \(h =\) 2 mm) immersed in oil layers of different thicknesses (\(\Delta\)). The velocities and vorticities were plotted as in Fig. 1c. Gray curves represent the water–oil interfaces; blue and red curves represent the no-slip boundaries. Droplets immersed in a thicker oil layer (\(\Delta =\) 2.6 mm) developed circulatory flows (left), whereas in those with a thinner oil layer (\(\Delta =\) 1.1 mm) the circulatory flows were suppressed (right). Simulated instantaneous velocity fields and vorticity maps in \(xy\)-, \(yz\)-, and \(xz\)-midplane cross-section for various oil layer thicknesses are available in Supplementary Fig. S6. (b) Circulation order parameter (COP) as a function of the layer thickness of the oil that surrounds a compressed droplet (\(r =\) 2.4 mm, \(h =\) 2 mm). The intradroplet flows were sensitive to oil layer thickness when the thicknesses were thinner than ~ 2.2 mm. Each error bar represents the standard deviation of the time-averaged COP. Inset: Evolution of COPs within the droplets described in panel a. (c) Flow profiles of azimuthal velocities, \(v_{\theta }\), taken at droplet midplanes. The horizontal axis represents the radial axis in cylindrical coordinate with the origin shifted to the droplet interface (Fig. 5c inset). Each curve represents the averaged flow velocity of the data points shaded in the same color in panel b. Dashed lines represent the water–oil interface. These flow profiles demonstrated that intradroplet circulatory flows induced a thin layer (0.3–2 mm) of circulatory flow in the oil near the interface, indicating that flows within and outside the interface were coupled. Inset: Flow speed profiles near the droplet interface.