# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 3 Numerical Method  
We use the Eulerian-Lagrangian method of Front Tracking, which applies the finite difference method to solve Equations (1) and (2) by the marker-and-cell (MAC) method [20] on a fixed, staggered grid. The Lagrangian front points are used to track the interfaces, which are transferred by the flow velocity interpolated from the fixed staggered grid. The momentum equation is discretized by second-order centered spatial differences and an explicit second-order time-integration method. The updated front point locations are used to spread the interfacial jumps of density and viscosity to the nearby grid points by area weighting, then density and viscosity fields are solved by Poisson equations. The interfacial tension force is calculated from the curvature formed by the linked front points which is spread to the Eulerian grid cells within a radius of one grid spacing by the discretized \(\delta\) function. For detailed description of the method used, we refer readers to [PERSON] et al. [21]. This solver has been thoroughly confirmed for a compound jet by [PERSON] et al. [11; 12] by comparing with previous numerical and experimental results. A grid refinement test was conducted with 64 \(\times\) 960, 128 \(\times\) 1920 and 256 \(\times\) 3840 resolutions. Figure 2 shows the results of the interfacial evolution using the 128 \(\times\) 1920 grid compared with the 256 \(\times\) 3840 grid. The results were nearly the same with breakup length changing by less than 0.5-1.5% for the 128 \(\times\) 1920 and 256 \(\times\) 3840 grid resolutions, whereas 64 \(\times\) 960 yielded some differences. Similar grid refinement tests were also conducted by [PERSON] et al. [22] and [PERSON] et al. [13] to numerically investigate the formation of a single and compound jet, by using the Front Tracking Method in both resting and coflowing outer fluid. For the results presented below, we incorporate 1920 grid points in the axial direction and 128 grid points in the radial direction. In the previous works, method validations have been conducted carefully. A validation case is shown in Figure 3a,b where comparison with the non-coflowing experiments of [PERSON] and [PERSON] [3] has been presented. Such validations are satisfactory and confirm the accuracy of the method used in this analysis. We also assume that the drop doesn't merge with the jets so as to obtain the most suitable results that are observed in experiments [16]. Here, the properties of inner, intermediate and outer fluids are kept constant to focus on the effects of density and viscosity difference between the inner and intermediate fluids, for example, compound jets of immiscible liquids such as an aqueous solution of a k-Carrageenan or aqueous solution of polyethylene glycol injected from the inner and outer nozzles and sunflower oil injected from the annular nozzle [23]. Such liquid systems have nearly the same values of density, surface tension and the effect of viscosity can be studied by varying the weight fraction of k-Carrageenan in the aqueous solution. For the effect of density, examples such as immiscible liquids of water/Heptane/water or water/n-Decane/water system having nearly the same viscosity but different densities values may be considered [22]. Some density ratios simulated in the current study lie outside the range achievable by realistic choices of standard liquids. However such density ratios might be produced with bubbly suspensions having high volumetric gas fractions dispersed in a liquids which can produce fluids of rather exceptionally low bulk density and viscosity [24]. We keep the Reynolds number, Weber number, velocity ratios, surface tension ratio and jet radii ratio constant: \(Re=37.5\), \(We=1.5\), \(U_{21}=1\), \(U_{31}=1\), \(\sigma_{21}=1\) and \(R_{21}=2\). The selection of such parametric values i.e., \(Re<50\), \(We<2\), \(U_{21}<1.5\), \(U_{31}<1.5\) and \(\sigma_{21}>0.5\) form compound drops near the nozzle, dripping mode [13; 25].