A nonlinear stability model for the breakup of annular liquid sheets subject to inner and outer gases of negligible inertia was developed. Using this model, we demonstrate the effect of viscosity and advection velocity on the breakup process. The formulation used is similar to that of Eggers and Dupont (1994) and involves expanding the flow field variables as asymptotic expansions of a "thin" variable and then solving the one-dimensional equations using the Galerkin finite element framework. One-dimensional (paravaricose) disturbances were assumed and the long wave ansatz was employed. We show that a nonzero mean axial velocity allows convection of disturbances downstream and can cause disturbance waves that are linearly stable to grow in time and eventually lead to sheet breakup (if the initial disturbance amplitude is greater than a critical value). We also show that the annular sheet could form satellite rings (analogous to the satellite drops that have been observed when cylindrical jets break up) under certain conditions. Model predictions show that increasing the Ohnesorge number yields the expected stabilization of the sheet. Finally, we show that the effect of the annular sheet thickness is counterintuitive, in that thicker sheets tend to break up faster.