# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 4 Results and Discussion
### Effect of Inner-to-Outer Viscosity Ratio (\(\mu_{13}\))  
We now investigate the effect of viscosity ratio \(\mu_{13}=\mu_{1}/\mu_{3}\) on the dynamics of the compound liquid jet where ratio of 0.18 to 5 is varied. As compared to moderate density ratios, such viscosity ratios are more commonly found and studied experimentally [23]. In Figure 9a-c, snapshots of the compound liquid jets are presented for \(\mu_{13}=0.18\), 1 (the \"baseline case\") and 5 respectively. The rest of the parameters used for figures are \(\rho_{13}=1\), \(\rho_{23}=1\) and \(\mu_{23}=1\).  
For \(\mu_{13}=0.18\) (Figure 9a), it can be seen that the inner drop forms in the dripping mode and the compound drops form close to the jetting mode. The lower viscosity of inner fluid reduces the resistance to capillary-driven necking instability, facilitating the breakup of the inner fluid jet, and thus results in decreased inner jet length. Such observations are also seen in viscous fingering phenomena, in which less viscous fluid displacing a more viscous immiscible fluid tends to become unstable [28]. Due to the lower viscosity of inner fluid (\(\mu_{13}=0.18\)) which break up more easily than the intermediate fluid, several drops formed at early stages can occupy the core of an intermediate drop, and hence form multi-core compound drops. On the other hand, with \(\mu_{13}=0.18\), the intermediate fluid is approximately 5.5 times more viscous than the inner fluid. Due to the large viscous resistance of  
Figure 8: Percent of multi-core compound drops formed vs intermediate-to-outer density ratio. The horizontal and slant lines depicted in this plot correspond to 1-core and 2-core compound drops respectively whereas circles denote 3-core compound drops formation.  
the intermediate fluid, which stabilizes the interface and delays necking, the breakup length of the compound drop increases.  
As we increase the viscosity ratio \(\mu_{13}\) from 0.18 to 2, the inner jet breakup length increases while the intermediate jet breakup length slightly decreases, and both jets remain in the dripping mode (Figure 9d). [PERSON] et al. [29] also reported similar outcomes in a two-phase system wherein water-glycerol solution (dispersed phase) of different viscosities was injected in oil (continuous phase). For viscosity ratios, \(2\leq\mu_{13}\leq 5\), the inner and intermediate jets break at the same length forming mostly single-core drops whereas, for \(\mu_{13}<0.56\), multi-core drop formation is observed. Figure 10 shows the plot of percent of multi-core compound drops formed versus the inner-to-outer viscosity ratios. For the case where \(\mu_{13}<1\), the compound drops formed contain a mixture of single-core and double-core drops. Whereas for \(\mu_{13}\geq 1\), the compound drops are single-core. For inner-to-outer viscosity ratio \(\mu_{13}\) less than 5, no transition from dripping to jetting was observed. All the jet breakup lengths fall at or below the value of \(2.5R_{3}\).  
Figure 9e,f shows average drop diameter vs. \(\mu_{13}\) and average drop formation time vs. \(\mu_{13}\) plot. For \(0.18\leq\mu_{13}\leq 5\), there is no drastic change in the drop diameter of inner and compound drops. Such conclusions were also reached by [PERSON] et al. [30] for drop formation in two-phase viscous flow system.  
Figure 9: Snapshots of compound liquid jets for (**a**) \(\mu_{13}=0.18\); (**b**) \(\mu_{13}=1\) and (**c**) \(\mu_{13}=5\). The rest of the parameters are \(\mu_{13}=1\), \(\rho_{23}=1\) and \(\mu_{23}=1\). Graphs (**d–f**) represents the average breakup length, drop diameter and drop formation time for variable inner-to-outer viscosity ratios. Readers can refer to the Video “S3:Inner-to-outer Viscosity ratio” in the Supplementary Materials for better understanding of the multi-core drop formation process in the dripping mode.