# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 4 Results and Discussion
### Effect of Intermediate-to-Outer Viscosity Ratio (\(\mu_{23}\))  
Finally, we consider variations in \(\mu_{23}\). Figure 11a-c shows snapshots of the compound liquid jets for viscosity ratio \(\mu_{23}=\mu_{2}/\mu_{3}=0.18\), 1 and 5 respectively, with \(\rho_{13}=1\), \(\rho_{23}=1\) and \(\mu_{13}=1\). Initially, for lower values of \(\mu_{23}\), the inner and compound drops form in the dripping mode (Figure 11a). Compared to Figure 9a-c, Figure 11a-c situation is opposite. For \(\mu_{23}=0.18\), the outer and inner fluids are approximately 5.5 times more viscous than the intermediate fluid. Due the lower viscous resistance of the intermediate fluid, the intermediate jet deforms faster [28], whence jet breakup length is shorter. The inner fluid is more viscous than the intermediate fluid, so the inner jet might be expected to break slowly. But due to the early necking of the intermediate jet, which encloses the inner fluid, the inner jet is forced to break up at an early stage, and compound drops form in the dripping mode. As we increase \(\mu_{23}\), both inner and compound drops are seen to form in mixed dripping-jetting mode. Similar to the previous section, an increase in the viscosity of intermediate fluid resists the formation and breakup of the drops, and formation takes place in the jetting mode.  
The transition from jetting to dripping for \(\mu_{23}\) was found in the range of \(1\leq\mu_{23}\leq 1.78\). Compared to Figure 9c, Figure 11c shows more multi-core drops. For \(2\leq\mu_{23}\leq 5\), the inner jet breaks at fairly short breakup length, and the intermediate jet breaks up at longer lengths i.e., in jetting mode (Figure 11d). As the inner drops are formed at early stages, more inner drops are enclosed in the intermediate jet, and many two-core drops are observed at high \(\mu_{23}\) ratios. Figure 12 shows the plot of percent of multi-core inner drops occupied in the compound drop. For less values of \(\mu_{23}\), the compound drops formed are single-core. Whereas, increasing the ratio \(\mu_{23}\) increases the percentage of two-core compound drops.  
Figure 11e,f shows the average drop diameter _vs_\(\mu_{23}\) and average drop formation time vs. \(\mu_{23}\) for the case where \(\mu_{1}=\mu_{3}\). Increasing \(\mu_{23}\) results in larger compound drops (Figure 11e), but with little change in average inner drop diameter. In Figure 11f, drop formation time for various values of \(\mu_{23}\) has been plotted. For \(0.18\leq\mu_{23}\leq 1.78\), it is seen that inner and compound drop formation time decreases and for \(\mu_{23}>2\), only the compound drops formation time increases. The sudden increase of the compound drop formation time is due to more resistive viscous force of the intermediate jet which deforms at later stages. For \(\mu_{23}\leq 1.56\), the difference in drop formation times between inner and compound drops is small, which reflects single-core drop formation. For \(\mu_{23}\geq 3\), the difference in drop formation time increases gradually, which corresponds to multi-core drop formation.  
Figure 10: Percent of multi-core compound drops formed versus inner-to-outer viscosity ratio. The horizontal and slant lines depicted in this plot correspond to 1-core and 2-core compound drops formation respectively.