Simulation of Mixing within Drops due to Surface Tension Variations

We present the results of a numerical investigation of the mixing within drops caused by surface tension variations. With microfluidic applications in mind, we simulate drops surrounded by a fluid of equal density and viscosity. We investigate both stationary coalescing drops and steadily flowing drops, and study the influence of drop size ratio, viscous effects, and surface tension variations. We measure the mixing efficiency using the variance of the concentration distribution and find that surface tension variations may result in faster mixing than geometric effects.

Figure 1

(a) Dependence on a log scale (linear in inset) of the dimensionless tangential velocity on the Ohnesorge number with $R_{\sigma }=0.4$. The dashed line has slope $-1$. (b) Tangential velocity dependence on $R_{\sigma }$, with $\mathrm{Oh}=0.02$.

Figure 2

(a) Coalescence of drops with size ratio 0.6, identical surface tensions, and $\mathrm{Oh}=0.02$. (b), (c), and (d) Coalescence of equal size drops at $\mathrm{Oh}=0.02$, with $R_{\sigma }=0.04$ (b), $R_{\sigma }=0.24$ (c), and $R_{\sigma }=0.4$ (d). Images are $\tau =\sqrt{\rho R^3/\sigma }$ apart, and the black lines are one initial radius long.

Figure 3

Simulations of a flowing drop with $R/R_t=2/3$, $\mathrm{Ca}=2.5\times {10}^{-4}$, $\mathrm{Re}=1.5$, and $\mathrm{Pe}=0.15$. In (a) $R_{\sigma }=0$, in (b) $R_{\sigma }=0.05$, and in (c) $R_{\sigma }=-0.05$. Positive values of $R_{\sigma }$ correspond to tangential flow opposing the drop motion. Images are $\Delta t=0.035R/U$ apart.

Figure 4

Standard deviation of $C_1$ within the drop (measure of mixing) as a function of time for various $R_{\sigma }$ and Ca. The cases with $\mathrm{Ca}=2.5\times {10}^{-4}$ are those shown in Fig. 3.