Dynamics of deformation and pinch-off of a migrating compound droplet in a tube

A computational fluid dynamic investigation has been carried out to study the dynamics of a moving compound droplet inside a tube. The motions associated with such a droplet is uncovered by solving the axisymmetric Navier-Stokes equations in which the spatiotemporal evolution of a pair of twin-deformable interfaces has been tracked employing the volume-of-fluid approach. The deformations at the interfaces and their subsequent dynamics are found to be stimulated by the subtle interplay between the capillary and viscous forces. The simulations uncover that when a compound drop composed of concentric inner and outer interfaces migrates inside a tube, initially in the unsteady domain of evolution, the inner drop shifts away from the concentric position to reach a morphology of constant eccentricity at the steady state. The coupled motions of the droplets in the unsteady regime causes a continuous deformation of the inner and outer interfaces to obtain a configuration with a (an) prolate (oblate) shaped outer (inner) interface. The magnitudes of capillary number and viscosity ratio are found to have significant influence on the temporal evolution of the interfacial deformations as well as the eccentricity of the droplets. Further, the simulations uncover that, following the asymmetric deformation of the interfaces, the migrating compound droplet can undergo an uncommon breakup stimulated by a rather irregular pinch-off of the outer shell. The breakup is found to initiate with the thinning of the outer shell followed by the pinch-off. Interestingly, the kinetics of the thinning of outer shell is found to follow two distinct power-law regimes—a swiftly thinning stage at the onset followed by a rate limiting stage before pinch-off, which eventually leads to the uncommon breakup of the migrating compound droplets.

Figure 1

Schematic diagram showing the computational domain (not to scale) and boundary conditions in cylindrical coordinates system $(r,z,\theta )$.

Figure 2

Comparison of experimental results of Olbritch and Kung [71] with present computations (blue line) for steady shapes of a droplet migrating inside a capillary tube at (a) $\text{Ca}=0.05$, (b) $\text{Ca}=0.10$, and (c) $\text{Ca}=0.16$. The operating parameters are $\eta =1.01$ and $a=0.95$. (d) Temporal evolution of droplet shapes obtained by Tsai and Miksis [29]. (e) The shapes obtained from the present computations using the same configuration of Tsai and Miksis [29]. The corresponding time instants are $t^{*}=0,2,4$, and 6.

Figure 3

Quantitative comparison of drop shapes and average droplet velocity between theoretical solutions of Nadim and Stone [Eq. (6)] and present computations. Panels (a) and (b) show the drop shapes for Ca = 0.28 and 1.4, respectively, while other parameters are fixed at $\eta =1$ and $a=0.3$. Panel (c) demonstrates the variation of $u_{\text{avg}}^{*}$ with $\eta$ keeping Ca and $a$ fixed at 0.2 and 0.3, respectively.

Figure 4

Comparison of the average drop velocity for different grid sizes ($8\times 240,16\times 480,32\times 960$, and $64\times 1920$) as a function of dimensionless time $t^{*}$. The operating parameters are Ca $=0.5,\eta =0.2,a=0.6$, and $k=0.5$.

Figure 5

The computed parameters used to describe the dynamics of compound droplets. (a) Deformation index ($D_i$ and $D_o$) corresponding to inner and outer drop interfaces. (b) Drop eccentricity ($e^{*}$). (c) Minimum shell thickness $d_{\text{min}}^{*}$.

Figure 6

Temporal evolution of the compound droplet under conditions of $\text{Ca}=0.5$ and $\eta =0.2$. Panel (a) represents the pressure field inside the compound droplet, panel (b) shows the streamline patterns inside the drop at steady state, panel (c) shows the variation of deformation index $D^{*}$ with time for the inner and outer drop, and panel (d) presents the evolution of drop eccentricity with time.

Figure 7

Effect of capillary number on the (a) deformation of inner and outer drop and (b) eccentricity of the drop. The viscosity ratio $\eta$ is kept fixed at 0.1.

Figure 8

Contour of pressure (zoomed in the region of the compound drop) at time $t^{*}=8.0$ under condition of (a) $\text{Ca}=0.3$ and (b) $\text{Ca}=0.6$. The viscosity ratio $\eta$ is kept fixed at 0.1.

Figure 9

Effect of viscosity ratio on the (a) deformation of inner and outer drop and (b) eccentricity of the drop. The capillary number is kept fixed at 0.5.

Figure 10

Temporal evolution of inner and outer drop velocities under varying $\eta$. The capillary number is kept fixed at 0.5.

Figure 11

Snapshots showing the temporal evolution of a compound droplet leading to breakup.

Figure 12

Grid resolution test for drop breakup. (a) Comparison of drop profile using grid sizes of $128\times 3840$ (upper half) and $256\times 7680$ (lower half). The profiles are plotted for two values of $d_{\text{min}}^{*},0.05$ and 0.01, as the drop proceeds toward breakup. (b) The variation of the volume of primary and secondary drops formed after breakup with different grid resolutions. The capillary number is kept fixed at 0.5.

Figure 13

(a) The temporal evolution of minimum distance between the inner and outer interfaces $d_{\text{min}}^{*}$ for different drops with $k=0.73,0.76,0.80$, and 0.83. (b) Relation between $d_{\text{min}}^{*}$ and time before rupture $\tau$ near the breakup of the droplet for $k=0.73$ and 0.83. The dotted lines in black, orange, red, and yellow represent $d_{\text{min}}^{*}\sim {\tau }^{1.5},{\tau }^{1.9},{\tau }^{0.2}$, and ${\tau }^{0.1}$, respectively. The other operating parameters are $\text{Ca}=0.5$ and $\eta =1.0$.

Figure 14

(a) The temporal evolution of minimum distance between the inner and outer interfaces $d_{\text{min}}^{*}$ for $\text{Ca}=0.1,0.3,0.5$, and 0.7. (b) Relation between $d_{\text{min}}^{*}$ and time before rupture $\tau$ near the breakup of the droplet for $\text{Ca}=0.1$ and 0.5. The dotted lines represents $d_{\text{min}}^{*}\sim {\tau }^{1.5}$ and $d_{\text{min}}^{*}\sim {\tau }^{0.2}$. The other operating parameters are $\eta =1.0$ and $k=0.83$.