# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 4 Results and Discussion
### Effect of Intermediate-to-Outer Density Ratio (\(\rho_{23}\))  
The density ratio of \(\rho_{23}\) is varied in the range from 1 to 5. In Figure 7a-c, snapshots of the compound liquid jets are presented for \(\rho_{23}=1.5\), 2.5 and 4.5 respectively, with \(\rho_{13}=1\), \(\mu_{13}=1\) and \(\mu_{23}=1\). For \(\rho_{23}=1.5\) (Figure 7a), it can be seen that both the jets breakup in the dripping mode. Increasing \(\rho_{23}\) increases the breakup length of the intermediate jet, which accordingly transitions from dripping to jetting mode (Figure 7b,c). Trends similar to the previous section were observed for this case. As \(\rho_{23}\) increases, due to higher inertia of the intermediate fluid that delays necking of the intermediate jet, the length of the intermediate jet increases, and thus the drops are formed in the jetting mode. Similar results were also observed by [PERSON] et al. [27] for a compound jet, where increase in density ratio of intermediate to outer fluid resulted in longer breakup lengths. By contrast, increasing \(\rho_{23}\) barely affects the breakup length of the inner jet (Figure 7d) yielding the inner-dripping and intermediate-jetting mode (mixed dripping-jetting). Although motion of the intermediate fluid is stabilized by the inertia, we infer that the density of the inner fluid dominates the resistance to necking that is driven by the surface tension force. Despite increase in \(\rho_{2}\), this \"mixed\" mode often results in two or more drops of inner fluid occupying compound drops and hence at high \(\rho_{23}\) ratio, multi-core drop formation is seen. As compared with Figure 5c, Figure 7c shows less multi-core drop formation and the effect of \(\rho_{23}\) is lesser in this case. Figure 8 illustrates the percentage of multi-core compound drops versus intermediate-to-outer density ratios plot. For values of \(\rho_{23}=1\), the compound drops formed are single-core. Whereas for higher values of \(\rho_{23}\), formation of two-core and three-core compound drops is seen. In other words, for high values of \(\rho_{23}\), the compound jet formation falls in the inner-dripping and outer-jetting mode forming more multi-core compound drops  
Figure 7: Snapshots of compound liquid jets for (**a**) \(\rho_{23}=1.5\); (**b**) \(\rho_{23}=2.5\) and (**c**) \(\rho_{23}=4.5\). The rest of the parameters are \(\rho_{13}=1\), \(\mu_{13}=1\) and \(\mu_{23}=1\). Graphs (**d**–**f**) represents the average breakup length, drop diameter and drop formation time for variable intermediate-to-outer density ratios. Figure 7e,f shows the averaged drop diameter _vs_\(\rho_{23}\) and drop formation time vs. \(\rho_{23}\) i.e., as the intermediate fluid density is varied, with inner and outer fluid densities kept constant. Figure 7e shows that as \(\rho_{23}\) is increased, the average drop diameter of the inner drop decreases slightly. Slight decreasing behavior is observed in case of inner drop diameter whereas the compound drop diameter has increasing-decreasing behavior which is mainly due to the multi-core compound drops formation which sometimes encloses more than one drop leading to accumulation of large volume of the inner fluid. The transition from dripping to jetting, in this case, was found at density ratio \(\rho_{23}>2\), shown in Figure 7d by a thin dotted line which separates the dripping and jetting regions. Like the previous case, this case too shows that the system is more sensitive to \(\rho_{23}\) parameter. The effect of \(\rho_{23}\), on the time when the inner and compound drops form is shown in Figure 7f. \(\rho_{23}<2.5\), the difference in the time of inner and compound drops formed is small, which corresponds to single-core drop formation. Whereas, for \(\rho_{23}>2.5\), this difference in drop formation time is much greater, which reflects multi-core drop formation.