Diffusion enhancement in a levitated droplet via oscillatory deformation

Recent experimental results indicate that mixing is enhanced by a reciprocal flow induced inside a levitated droplet with an oscillatory deformation [T. Watanabe et al., Sci. Rep. 8, 10221 (2018)]. Generally, reciprocal flow cannot convect the solutes in time average, and agitation cannot take place. In the present paper, we focus on the diffusion process coupled with the reciprocal flow. We theoretically derive that the diffusion process can be enhanced by the reciprocal flow, and the results are confirmed via numerical calculation of the over-damped Langevin equation with a reciprocal flow.

Figure 1

Time series of the droplet shape and flow field. The gray regions illustrate the cross section of the droplet at $xy$ and $xz$ planes. The black arrowheads represent the flow fields.

Figure 2

Trajectories of 400 tracer particles for one cycle (from $t=0$ to $T$) on $xy$ (top) and $xz$ (bottom) planes. The particles move back and forth with the reciprocal flow. However, they do not follow the streamlines due to the thermal noise. As the initial condition, the particles are randomly located on each plane. The oscillation amplitude $\varepsilon$ is set as 0.1. The gray regions indicate the area of the droplet, where the time-dependent shapes are all superimposed.

Figure 3

Numerical results of the normalized effective diffusion coefficient ${\overline{D}}^{\mathrm{eff}}/D$ dependent on the initial distance $\rho$ from the center (left) and the oscillation amplitude $\varepsilon$ (right) on the $xy$ plane for $n=2,3$, and 4. The effective diffusion coefficient was estimated from the mean square displacement of the particles. With respect to the left panel, $\varepsilon$ is fixed at $\varepsilon =0.1$, and for the right panel, the initial radius is fixed at $\rho =0.5$ [red (gray)] and $\rho =0.9$ [blue (light gray)]. The black thin curves show the theoretical prediction in Eq. (45).

Figure 4

Numerical results on the components of the normalized effective diffusion coefficient dependent on $\phi$ for $n=2,3$, and 4. $D_{\rho \rho }^{\mathrm{eff}}/D$ [red (gray)], $D_{\phi \phi }^{\mathrm{eff}}/D$ [blue (light gray)], $D_{zz}^{\mathrm{eff}}/D$ [green (dark gray)] are shown in the left panel, while $D_{\rho \phi }^{\mathrm{eff}}/D$ is plotted on the right panel. The initial distance from the center is $\rho =0.5$, and $\varepsilon =0.1$. The black thin curves show the theoretical predictions in Eqs. (47) to (50).