Effect of viscosity and density ratios on two drops rising side by side

We study the dynamics of a pair of initially spherical drops rising side by side in a surrounding, denser, fluid. Our primary focus is on liquid-liquid systems, and a range of viscosity and density ratios is explored by three-dimensional numerical simulations. Interesting dynamics are reported, which cannot be extrapolated from previously known dynamics of gas-liquid systems. Similar to two air bubbles though, we find that two liquid drops move away from each other as they rise, in cases where a single drop would rise vertically. A pair of light drops always remains in two-dimensional motion, and higher drop viscosity increases the tendency of wobbling. This is in contrast with the dynamics of a single drop that follows a highly three-dimensional trajectory at very low drop viscosity, but is restricted to two-dimensional motion at higher drop viscosity. On the other hand, a pair of heavier drops displays three-dimensional behavior at low drop viscosity and two-dimensional behavior at high viscosity. We find that a pair of drops is far less sensitive to viscosity contrast than a single drop is, in our parameter range. In contrast to gas-liquid systems, where the shape change of the bubble was tied to nonlinear dynamics of the trajectory, we find in liquid-liquid systems that interesting drop trajectories can occur without corresponding large shape changes. It is found that the separation distance between the drops exhibits a nonmonotonic trend with an increase in the density ratio. The physical reason for this nonmonotonic trend is simple and explained by inspecting the velocity components.

Figure 1

(a) Density (${\rho }_r$) and viscosity (${\mu }_r$) ratios of about 1650 pairs of fluids. Blue open and red closed symbols represent liquid-liquid and liquid-gas systems, respectively. The list of the viscosity and density ratios of these pairs of fluids is provided in the Supplemental Material [37]. (b) Some of the fluid pairs considered in the present study.

Figure 2

Schematic diagram showing the initial configuration of two initially spherical drops (fluid $B$) rising in a surrounding medium (fluid $A$). A cubic computational domain of size $H\times H\times H$ is used and the drops are placed at $(q_0/2,0,z_0)$ and $(-q_0/2,0,z_0)$. In the present study, $H=120R$ and $z_0=7R$. The acceleration due to gravity $g$ acts in the $-z$ direction.

Figure 3

Summary of earlier work [11], restricted to very light bubbles of very low viscosity. The top row shows the evolution of the separation distance $q$. The second row shows the aspect ratio $A_r$ of bubble 1 and also that of a single bubble versus $z$, as well as the shapes of bubble 1 (on the left) ($x-z$ view) and the shapes of a single bubble (on the right) at different time instances and the corresponding $z$ locations. The third row shows the top view of the trajectories of the two bubbles and the corresponding trajectory of the single bubble. The bottom row shows the contours of the $z$ component of the vorticity (${\omega }_z=\pm 0.02$) for the single- and two-bubble cases at $t=40$. The parameters are $\text{Eo}=3.996, q_0=3, {\rho }_r={10}^{-3}, {\mu }_r={10}^{-2}$, and (a) $\text{Ga}=20$ and (b) $\text{Ga}=40$.

Figure 4

Variations of (a) and (b) the separation distance between the drops $q$ and (c) and (d) the aspect ratio $A_r$ of drop 1 with $z$ for different values of the density ratio ${\rho }_r$. The parameters are $\text{Ga}=20, \text{Eo}=3.996, q_0=3$, and (a) and (c) ${\mu }_r=0.1$ and (b) and (d) ${\mu }_r=10$.

Figure 5

Variations of velocity of the centroid of drop 1 in (a) and (b) the $x$ direction ($u_{CG}$) and (c) and (d) the $z$ direction ($w_{CG}$) and variation of (e) and (f) $u_{CG}/w_{CG}$ of drop 1 with time for different values of the density ratio ${\rho }_r$ The parameters are $\text{Ga}=20, \text{Eo}=3.996, q_0=3$, and (a), (c), and (e) ${\mu }_r=0.1$ and (b), (d), and (f) ${\mu }_r=10$.

Figure 6

Variations of the separation distance $q$ with $z$ for different values of the density ratio ${\rho }_r$. The parameters are $\text{Ga}=40, \text{Eo}=3.996, q_0=3$, and (a) ${\mu }_r=0.1$ and (b) ${\mu }_r=10$.

Figure 7

(a) Variations of the separation distance $q$ versus $z$ for different values of ${\mu }_r$. The top view of the trajectories of the two drops and the corresponding trajectory of the single drop are shown for (b) ${\mu }_r={10}^{-5}$ and (c) ${\mu }_r={10}^{-1}$. The rest of the parameters are $\text{Ga}=20, \text{Eo}=3.996, q_0=3$, and ${\rho }_r={10}^{-3}$.

Figure 8

Variations of the aspect ratio $A_r$ of drop 1 versus $z$ for different values of ${\mu }_r$. The other parameters are the same as in Fig. 7.

Figure 9

(a) Variations of the separation distance $q$ versus $z$ for different values of ${\mu }_r$. The top view of the trajectories of the two drops and the corresponding trajectory of the single drop are shown for (b) ${\mu }_r={10}^{-3}$ and (c) ${\mu }_r={10}^3$. The other parameters are $\text{Ga}=20, \text{Eo}=3.996, q_0=3$, and ${\rho }_r=0.5$.

Figure 10

Variations of the aspect ratio $A_r$ of drop 1 versus $z$ for different values of ${\mu }_r$. The other parameters are the same as in Fig. 9.

Figure 11

Grid convergence test showing the bubble shapes at $t=9$ for ${\mu }_r={10}^{-5}$ with the rest of the parameters the same as those used to generate Fig. 7. The values of the separation distance $q$ and the aspect ratios of drops 1 and 2 (designated below the bubbles) obtained for (a) $\mathrm{\Delta }=0.058$ and (b) $\mathrm{\Delta }=0.029$ are given in each panel.

Figure 12

Grid convergence test showing the drop shapes at $t=9$ for ${\mu }_r={10}^3$ with the rest of the parameters the same as those used to generate Fig. 9. The values of the separation distance $q$ and the aspect ratios of drops 1 and 2 (designated below the drops) obtained for (a) $\mathrm{\Delta }=0.058$ and (b) $\mathrm{\Delta }=0.029$ are given in each panel.