Dynamics and breakup of an axi-symmetric liquid jet injected upwards into another stationary or co-flowing immiscible liquid is investigated numerically. Simulations are done using an in-house code based on a novel DGLSM (dual grid level set method). Furthermore, a novel procedure - based upon physical interpretation of the various functions in Level Set Method - is demonstrated here as a powerful numerical tool to calculate certain parameters (diameter as well as frequency of drop formation and temporal variation of jet length at the axis), which characterize the unsteady interface-dynamics. Six different combination of the dispersed and continuous fluid are subjected to various injection velocity, resulting in a large variation in the non-dimensional governing parameters such as viscosity-ratio and Weber number. From the temporal variation of jet length and instantaneous interface, three drop formation regimes are proposed: Periodic Uniform Drop formation (P-UD), Quasi-Periodic Non-Uniform Drop formation (QP-NUD) and Chaotic Non-Uniform Drop formation (C-NUD); demarcated in a drop formation regime map for various Weber number and viscosity ratio. Their effect on the mean value of jet breakup length (L-d (m)), detached drop diameter (D-d m) and drop formation frequency (St(m)) is also studied. After a more detailed study on stationary continuous fluid, the effect of co-flowing continuous fluid is studied; and is found to stabilize the drop formation regime and increase the frequency of drop formation. (C) 2013 Elsevier Ltd. All rights reserved.