# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 4 Results and Discussion
### Effect of Inner-to-Outer Density Ratio \((\rho_{13})\)  
The density ratio \(\rho_{13}\) was varied in the range from 0.1 to 5. Here \(\rho_{13}<1\) corresponds to the situation where the inner fluid is less dense compared to the intermediate and outer fluids. The effect of inner-to-outer density ratio \(\rho_{13}\) on the compound jet is shown in Figure 5a-c for \(\rho_{13}=0.5\), 2 and 4.5, respectively. The rest of the parameters used to generate these figures are \(\rho_{23}=1\), \(\mu_{13}=1\) and \(\mu_{23}=1\). For density ratio \(\rho_{13}=0.5\) (Figure 5a), the compound drops are formed in the dripping mode; both the inner and intermediate drops form near the nozzle exit and the compound drops are single core. As the density ratio is increased (\(\rho_{13}\geq 2\)), the jet breakup length of both inner and intermediate fluids increases, transiting from dripping to jetting mode (Figure 5c). Figure 5: Snapshots of compound liquid jets for **(a)**\(\rho_{13}=0.5\); **(b)**\(\rho_{13}=2\) and **(c)**\(\rho_{13}=4.5\). The rest of the parameters are \(\rho_{23}=1\), \(\mu_{13}=1\), \(\mu_{23}=1\), \(Re=37.5\) and \(We=1.5\). Graphs **(d–f)** represents the average breakup length, drop diameter and drop formation time for variable inner-to-outer density ratios. Readers can refer to Video “S2:Inner-to-outer Density ratio” in the Supplementary Materials for better understanding of the jetting mode and the chaotic drop formation process. To understand the transitional behavior of the jet from dripping to jetting, let us consider the rate of necking of the inner interface when \(\rho_{13}<1\) (Figure 5a), as compared with the baseline case \(\rho_{13}=1\) (Figure 4). Clearly, the faster the necking rate, the shorter the breakup length (Figure 5d). By analogy to the classical analysis of Plateau-Rayleigh instability, we neglect viscosity. Then the necking is driven by the spatial gradients of surface tension, which from Equation (23.31) of [26] is inversely proportional to the radius of curvature at the interface and is resisted by the inertia of the fluid that must escape from the necking region. From consideration of this balance, we expect that fluid can escape faster from the necking region when \(\rho_{13}<1\), as compared with \(\rho_{13}=1\), and correspondingly that the necking proceeds more rapidly. The early instability of the inner jet also appears to influence the jet breakup length (Figure 5d) of the intermediate jet (compound drops). In Figure 5d, it can be seen that for \(\rho_{13}<1\), both the inner and intermediate jet breakup lengths are lower than that of the baseline case (\(\rho_{13}=1\)). On the other hand, for \(\rho_{13}>1\), due to the higher density of the inner fluid, the inertia tends to stabilize the inner jet and thereby increase its jet length. Also, as the inner jet is enclosed by the intermediate jet, the inner jet carries the intermediate jet along with it and the jet length of the intermediate jet increases. In general, increasing \(\rho_{13}\) favors the jetting mode for both inner and intermediate jet. The transition from dripping to jetting, in this case, was found in the range of \(1.5<\rho_{13}<2\), as shown in Figure 5d by a thin dotted line that separates the dripping and jetting regions. Note the drop diameter, formation time and jet breakup length calculated in Figures 5, 7, 9 and 11 are over 5-10 drops on average. The number of drops is increased to more than 15 in the case of chaotic drops. Figure 6 shows the percent of multi-core compound drops formed with varying density ratios. For density ratio \(\rho_{13}<1.5\), the compound drops are single-core. As the density ratio is increased, the compound drops formed are two/three-core, and for higher values, four-core compound drops are formed. The percent of one-core, two-core, three-core and four-core inner drops are different due to which the drop types formed are not periodic. In this paper, the phenomenon of alternating formation of single and multi-core drops is termed \"chaotic drop formation\".