The overall objective of the present paper is to develop an accurate and robust numerical modeling of the instability of an interface separating two fluids (liquid-gas) due to buoyancy and capillary effects. The governing unsteady Navier-Stokes along with the stress balance and kinematic conditions at the interface are solved separately in each fluid using the finite-volume approach. The present numerical model interprets the surface and the body forces as a boundary value conditions on the interface. Thus enables accurate modeling of fluid flows driven by either body or surface forces. The position of the interface is captured implicitly on the Eulerian grid by the zero level set function, while appropriate interpolations at the interface are used to enforce the associated jump conditions. To asses the developed numerical model and its versatility, a selection of different unsteady tests are examined: oscillation of a capillary wave, sloshing in a rectangular tank, and the broken-dam problem involving different density fluids. The computational results demonstrate a remarkable capability of the developed numerical model to predict the dynamical characteristics of the two-phase flows, which is of great importance in many industrial and engineering applications.