# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 1 Introduction  
Compound liquid jet formation and breakup has been studied extensively due to its attractive industrial applications such as atomization, microencapsulation, and drug delivery [1; 2; 3; 4; 5; 6]. For example, magnetic drug targeting applies compound droplets that are formed by coating a layer of ferrofluid around the core region of the drug [7].  
A compound jet consists of two fluids, namely an \"intermediate fluid\" enclosing an \"inner fluid\", extruded through a coaxial nozzle into the surrounding \"outer fluid\" which is coaxially resting or flowing. The resulting compound jet then decomposes into droplets due to the effects of surface tension forces. This phenomenon is known as capillary instability [8]. Several researchers have numerically and experimentally studied the formation of a compound jet in the dripping and jetting modes [3; 4; 6; 8; 9; 10; 11; 12; 13]. In the dripping mode, breakup of the drops occurs near the nozzle exit whereas in jetting mode, breakup takes place farther downstream. These two modes are essential in the application of compound jets [6; 14]. Recently, the effect of outer coflowing fluid on a compound jet was experimentally examined by [PERSON] et al. [15] and [PERSON] et al. [16]. They demonstrated that increasing the volumetric flow rate of the external coflowing fluid not only changes the jet's transition mode, but also affects the compound drop's diameter. [PERSON] et al. [12; 13] numerically investigated the breakup modes inlaminar compound jets in an external coflowing fluid by varying parameters such as Reynolds number, Weber number, surface tension, velocity, and nozzle size. However, the effect of differences of density and viscosity on the dynamics of a compound jet has received far less attention, especially when the outer fluid is coflowing.  
[PERSON] et al. [17] experimentally examined the effect of flow rate and surfactants on the compound jet instabilities. They described drop formation with special attention to the conditions yielding concentricity. [PERSON] et al. [9] conducted linear stability analysis and found absolute instability under certain parametric ranges. [PERSON] et al. [13] considered the effect of outer fluid properties neglected by [PERSON] et al. [9], and varied parameters such as Reynolds number, Weber number, interfacial tension ratio, velocity ratio, and jet radii ratio, when the outer fluid is either at rest or coflowing. Experimental investigations by [PERSON] et al. [14] and [PERSON] et al. [16] confirm that when a compound jet forms with an outer coflowing fluid, it can breakup into simple, one-core and multi-core drops types.  
The effect of inner to outer fluid viscosity ratios on the dynamics of a compound jet in a flow focusing device was studied by [PERSON] et al. [18]. By using the diffuse-interface model, they noticed that with increase in the viscosity ratio, multi-core compound drops form. On the other hand, [PERSON] et al. [19] studied the density and viscosity ratios effect on the breakup time of the inner fluid and encapsulated volume percentage of the inner fluid after drop breakup by using a Galerkin/finite element method. However, their calculations stopped when the compound jet accomplished stable shape and the domain used was half of the wavelength of the perturbation in the axial direction. Hence, later development of the jet and drop motion from dripping to jetting was not seen in their results.  
As the above review shows, little consideration has been paid on the transition of a compound jet from dripping to jetting in a density-stratified and viscosity-stratified systems. Motivated by the importance of the compound jet in several industrial applications [1; 2; 3; 4; 5; 6], the present study extends the work of [PERSON] et al. [13] to numerically compute the breakup of a compound jet at fairly moderate density and viscosity ratios, when the outer fluid is coflowing.