# Formation and Breakup of an Immiscible Compound Jet with Density or Viscosity Stratification
## 2 Formulation  
We consider an axisymmetric compound jet, comprising three viscous immiscible liquids, in a cylindrical tube of length \(L\) and diameter \(2R_{3}\), as shown in Figure 1. The inner fluid (\"\(fluid\) 1\") and intermediate fluid (\"\(fluid\) 2\") are respectively injected from the inner nozzle and the annular region between the inner and outer nozzle into another outer coflowing fluid (\"\(fluid\) 3\") within the cylindrical tube. We denote the inner radius of the inner nozzle by \(R_{1}\), and the inner radius of the outer nozzle by \(R_{2}\). We assume the wall of the cylinder to be rigid, with inlet at \(z=0\) and outlet at \(z=L\). The fluids are assumed to be Newtonian and incompressible, with densities and dynamic viscosities of \(\rho_{1}\), \(\mu_{1}\), \(\rho_{2}\), \(\mu_{2}\) and \(\rho_{3}\), \(\mu_{3}\) respectively. The average velocity of inner, intermediate and outer fluids are denoted by \(U_{1}\), \(U_{2}\) and \(U_{3}\) respectively. \(U_{3}\) = 0 represents the state where the outer fluid is at rest and \(U_{3}>0\) reflects outer fluid coflowing. Both these configurations drastically affect the formation of compound jets and drop [14; 16]. At the inlet, we apply fully developed laminar profiles, as specified in Equation (4). Also, the interfacial tension of the two fluid interfaces is assumed to be constant. Here, we use the cylindrical coordinate system \((r,z)\) to solve the three immiscible fluid flow problem, where \(r\) denotes the radial coordinate and \(z\) denotes the axial coordinate. The calculations are assumed to be axisymmetric along z axis. The flow dynamics are governed by the continuity and Navier-Stokes equations, given by:  
\[\
abla\cdot\mathbf{u}=0 \tag{1}\] \[\rho\left[\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot \
abla)\mathbf{u}\right]=-\
abla p+\
abla\cdot\mu(\
abla\mathbf{u}+\
abla \mathbf{u^{T}})+\int_{f}\sigma\kappa\mathbf{n}_{f}\delta(\mathbf{x}-\mathbf{x} _{f})ds \tag{2}\]  
wherein \(\mathbf{u}=(u,v)\) denotes the velocity in which \(u\) and \(v\) represent the radial and axial velocity components, respectively; \(p\) represents the pressure and \(t\) is the time; \(\sigma\) denotes the interfacial tension and \(f\) denotes the interfaces. The gravity is set to zero. The Dirac delta function \(\delta(\mathbf{x}-\mathbf{x}_{f})\) is zero everywhere except that it provides unit impulse at the interface \(\mathbf{x}_{f}\) and \(S\) defines the inner and outer surfaces of the jets and drops. \(\kappa\) is twice the mean curvature, and \(\mathbf{n}_{f}\) represents the unit normal vector to the interface. Here, we solve Equations (1) and (2) using no-slip and no penetration conditions on the cylindrical tube and nozzle walls. At the symmetric axis \(r=0\), symmetrical B.C is applied. To avoid recirculation in the computing domain, the outflow B.C, \(u=0\) and \(\partial\mathbf{v}/\partial z=0\) are enforced at \(z=L\). At the inlet, a constant flow rate and a fully-developed velocity profile are imposed:  
\[u=0 \tag{3}\]  
\[v=\left\{ \begin{array}{l}2U_{1}\big{\{}1-(r/R_{1})\big{\}},\&for0\leq r\leq R _{1}\\ 2U_{2}\bigg{\{}1-\bigg{(}\frac{r-(R_{1W}+R_{2})/2}{R_{2}-R_{1W}}\bigg{)}^{2} \bigg{\}},\&forR_{1W}\leq r\leq R_{2}\\ 2U_{3}\frac{R_{3}^{2}-r^{2}}{R_{3}^{2}}+\frac{(R_{3}^{2}-R_{2W}^{2})/R_{2}^{2} }{\ln(R_{3}/R_{2W})}\ln(r/R_{3})\\ 1+\frac{R_{3W}}{R_{3}}-\frac{1-(R_{2W}/R_{3})^{2}}{\ln(R_{3}/R_{2W})},\&forR_ {2W}\leq r\leq R_{3}\end{array} \right. \tag{4}\]  
With the thickness of the nozzle denoted by \(W_{n}\), we express \(R_{1W}=R_{1}+W_{n}\) and \(R_{2W}=R_{2}+W_{n}\). We specify the nozzle thickness \(W_{n}\) to be \(0.2R_{1}\), and the length of the nozzle in the axial direction (\(L_{n}\)) as \(2R_{1}\). The radius (\(R_{3}\)) of the cylindrical tube is assumed to be \(8R_{1}\) and its length \(L\) to be \(15R_{3}\). At \(t=0\), we assign a hemispherical shape to the two interfaces forming the nascent compound jet with the outer fluid at resting.