This paper presents a numerical study of the dynamics of a viscous liquid drop that is being formed directly at the tip of a vertical tube into ambient air. A model is developed to predict the evolution of the drop shape and its breakup based on RIPPLE, which is a solution algorithm for computing transient, two-dimensional, incompressible fluid flow with surface tension on free surfaces of general topology (D.B. Kothe and R. C. Mjolsness, AIAA J. 30, 2694 (1992)). The full Navier-Stokes system is solved by using finite-difference formulation on a Eulerian mesh. The mesh is fixed in space, with the how and surface moving through it to ensure accurate calculations of complex free surface hows and topology, including surface breakup and coalescence. The novel feature of the numerical algorithm is the use of a Eulerian volume-tracking approach which allows the calculations to pass the breaking point during formation of a drop continuously without interruption or numerical modification and, therefore, to explore the features of generation of satellite droplets. The effects of physical and geometric parameters on the nonlinear dynamics of drop growth and breakup are investigated. The focus here is on drop breakup and subsequent formation of satellite droplets. The effects of finite inertial, capillary, viscous, and gravitational forces are all accounted for to classify different formation dynamics and to elucidate features of satellite droplet generation. The numerical predictions are compared with experimental measurements for water drops, and the results show good agreement.