# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 4 Results and Discussion  
### Baseline Case  
We investigate the effect of density ratios \(\rho_{13}\), \(\rho_{23}\) and viscosity ratios \(\mu_{13}\), \(\mu_{23}\) on the breakup modes of the compound jet, focusing on the transition of a compound jet from dripping to jetting. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\rho_{13}\) & \(\rho_{23}\) & \(\mu_{13}\) & \(mu_{23}\) \\ \hline
0.1–5 & 1–5 & 0.18–5 & 0.18–5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Investigated ranges of parameters in the present study. Figure 3: A comparison between experimental jets of [PERSON] and Hermanrud [3] (top section) and the computed jet (bottom section) profiles at computational time: (**a**) \(\tau=251\) and (**b**) \(\tau=253\). The parameter used to generate the figures, calculated based on the inner most fluid, are \(Re=75\), \(We=6\), \(Fr=5442\), \(\rho_{21}=1\), \(\mu_{21}=1\), \(\sigma_{21}=0.38\), \(U_{21}=1\), \(U_{31}=0\), \(R_{21}=2\), for both experimental and computed jets. \(\rho_{31}=\mu_{31}=0.05\) is for the computed jet; \(\rho_{31}=0.001\) for the experimental jet. The figure is taken from [PERSON] et al. [11]  
Figure 2: Grid refinement test for \(\rho_{13}=1\), \(\rho_{23}=1\), \(\mu_{13}=1\), \(\mu_{23}=1\) at non-dimensional times of: (**a**) \(\tau=0\); (**b**) \(\tau=6.42\) and (**c**) \(\tau=8.72\), with the rest of the parameters as \(Re=37.5\) and \(We=1.5\). We first consider the \"baseline case\" with all fluids having the same density and viscosity, \(\rho_{13}=\rho_{23}\) = \(\mu_{13}=\mu_{23}\) = 1, \(U_{21}=1\), \(U_{31}=1\), \(\sigma_{21}=1\) and \(R_{21}=2\), for which results are illustrated in Figure 4. In such conditions, the only destabilizing forces are the surface tension forces. Herein we set the criterion for \"dripping\" to be when the drops detach at a length less than \(10R_{2}\), i.e., 5 non-dimensional units herein. For jet breakup above this length, we refer to such drop formation as being in the \"jetting mode\". In Figure 4a-e, the inner and intermediate jets are ejected from the coaxial nozzle into the outer co-flowing fluid and proceed to form compound drops. In the initial stages, a vortex forms inside the inner drop as shown in Figure 4a. The right sides of the snapshots in Figure 4 show the streamlines, whence the vortex formation process can be clearly seen. The vortex in the inner drop starts to weaken when the diameter of the growing inner drop reaches 0.75 times its ultimate value. This weakening of the vortex formation in the inner drop(Figure 2b) is basically due to more fluid accumulation in the growing inner drop which later results in pinch-off [13]. This growing drop detaches from the inner jet and then is carried away by the intermediate fluid motion (Figure 4d). Similarly, in the initial stages, vortex formation starts to occur on the inner surface of the intermediate fluid (Figure 4a,b). The vortex in the intermediate drop starts to weaken after the diameter ratio of the growing intermediate drop to the fully developed intermediate drop reaches 0.68 (Figure 4c). The motion of the inner drop further weakens the vortex formation inside the nascent intermediate drop (Figure 4d), which results in necking and detachment of the intermediate drop (Figure 4e). After the pinch-off of the inner drop, the interface of the inner jet retracts due to surface tension. This causes a collision in the inner fluid streams, coming from opposite directions within the inner jet which gives rise to new vortex formation in the inner jet (Figure 4d). Similarly, after the detachment of the intermediate drop, surface tension causes the interface of the intermediate jet to retract. The retraction of the intermediate fluid gives rise to a new vortex formation due to the fluid collision from the opposite directions within the jet, leading in most cases to periodic compound drop formation (Figure 4e). Apart from compound drops, small inner and intermediate \"satellite drops\" are also formed due to capillary instability, is seen in Figure 4b.