Mode selection on breakup of a droplet falling into a miscible solution

When a droplet with a relatively high density falls into a miscible solution with a relatively low density, the droplet breaks up spontaneously. We investigated the number $m$ of breakup in experiments with several density differences $\mathrm{\Delta }\rho$ between two solutions, viscosities $\mu$, and droplet radii $r$. The mode number $m$ has a distribution even under the same experimental conditions. We propose a simple model of mode selection based on the linear Rayleigh-Taylor instability and the growing radius of a vortex ring deformed from the droplet. The model provides the probability distribution $P\left(m\right)$ and a relationship between the nondimensional parameter $G\propto \mathrm{\Delta }\rho g{r}^3/{\mu }^2$ and the average value of $m$, which are consistent with experimental results.

Figure 1

(a) Viscosity $\mu$ and density ${\rho }_2$ of the base solution for several concentrations $x$ of PEG. (b) Experimental setup.

Figure 2

Droplet deformation with (a) a digon shape, (b) a triangular shape, (c) a square shape, and (d) a pentagonal shape, as captured from the bottom of a beaker. Solid lines in the photos show 5.0 mm scale bars.

Figure 3

Probability distributions $P\left(m\right)$ for mode $m$ obtained from experiments with several droplet radii (a) $r=0.9$ mm, (b) 1.1 mm, (c) 1.3 mm, (d) 1.5 mm, and (e) 2.0 mm when a droplet starts to fall. The density difference between two solutions is 0.06 g/${\mathrm{cm}}^3$, and viscosities of two solutions are 10.2 $\mathrm{mPa}·\mathrm{s}$.

Figure 4

Probability distributions $P\left(m\right)$ for mode $m$ obtained from experiments with several viscosities (a) $\mu =7.2\mathrm{mPa}·\mathrm{s}$, (b) 10.2 $\mathrm{mPa}·\mathrm{s}$, (c) 15.2 $\mathrm{mPa}·\mathrm{s}$, and (d) 18.2 $\mathrm{mPa}·\mathrm{s}$. Density difference 0.06 g/${\mathrm{cm}}^3$ between two solutions and droplet radius 1.5 mm at the time when a droplet started to fall down were kept constant in the measurements.

Figure 5

(a) Phase diagram of breakup mode $m$ obtained from probability distribution in experiments with several density differences, viscosities, and droplet radii. (b) Relationship between $G$ and $\langle m\rangle$ in a logarithmic scale. The parameters for each marker are as follows: $\mathrm{\Delta }\rho ={\rho }_1-{\rho }_2=0.25\mathrm{g}$/${\mathrm{cm}}^3$ and $r=0.9$ mm for closed triangles; $\mathrm{\Delta }\rho =0.06\mathrm{g}$/${\mathrm{cm}}^3$ and $r=1.5$ mm for asterisks; $\mathrm{\Delta }\rho =0.06\mathrm{g}$/${\mathrm{cm}}^3$ and viscosity $\mu =10.2\mathrm{mPa}·\mathrm{s}$ for closed squares; and $\mathrm{\Delta }\rho =0.06\mathrm{g}$/${\mathrm{cm}}^3$ and viscosity $\mu =14.4\mathrm{mPa}·\mathrm{s}$ for inclined closed triangles. Data obtained from experiments with several density differences are shown as closed diamonds.

Figure 6

(a) Linear growth rate $s\left(k\right)$ of the Rayleigh-Taylor instability for $d_c=0.1$, ${\rho }_1=1.2$, ${\rho }_2=1.1$, $g=980$, and $\nu =0.1$. (b) Time evolution of $s\left(k\right)=s(m/R\left(t\right))$ for $m=2$ (dashed line), $m=3$ (solid line), and $m=4$ (dotted line). (c) Time evolutions of perturbation $A_m\left(t\right)$ for $m=2$ (dashed line), $m=3$ (solid line), and $m=4$ (dotted line) in a semilogarithmic scale.

Figure 7

(a) Selected mode $m$ as a function of $\alpha$ for $A_m\left(0\right)={10}^{-6}$ and $R\left(0\right)=0.1$. (b) Probability distribution $P\left(m\right)$ of selected mode when $A_m\left(0\right)$ obeys the Gaussian distribution of standard deviation of ${10}^{-6}$ for $R\left(0\right)=0.1$ and $\alpha =0.04$.

Figure 8

Plots of $R^2\left(t^{\prime }\right)-R^2\left(0\right)$ for the evaluation of the $\alpha$ value, where $t^{\prime }=0$ is a time when the deformation of the vortex ring becomes visible. Data of circles, squares, and triangles are obtained from experiments of $\mu =10.2\mathrm{mPa}·\mathrm{s}$ ($m=4$), 15.2 $\mathrm{mPa}·\mathrm{s}$ ($m=3$), and 20.2 $\mathrm{mPa}·\mathrm{s}$ ($m=2$), respectively. The radius of a droplet 1.5 mm was kept constant in those experiments.

Figure 9

Relationship between $\alpha /\nu$ and $G$ for various values of $r$ for $\mu =10.2\mathrm{mPa}·\mathrm{s}$ (circles) and those of $\mu$ for $r=1.5$ mm (squares). The linear line denotes $\alpha /\nu =a_{1}G+a_2$, where $a_1=4.30\times {10}^{-2}\pm 0.44\times {10}^{-2}$ and $a_2=-15.99\times {10}^{-2}\pm 6.04\times {10}^{-2}$.

Figure 10

Probability distributions $P\left(m\right)$ of selected mode number. The upper row shows numerical results for (a) $\nu =0.0911$ and (b) 0.145. The lower row shows experimental results for (${\mathrm{a}}^{\prime }$) $\nu =0.0911$ and (${\mathrm{b}}^{\prime }$) 0.145 with a droplet radius $r=1.5$ mm.

Figure 11

Relationship between $G$ and $\langle m\rangle$. Experimental results are denoted by marks and the numerical results are denoted by a solid curve [27].