Strombolian activity is generally assumed to be driven by overpressurized gas slugs that rise through the magma-filled volcanic conduit and burst at the surface. We develop an analytical model for this process that incorporates a generic, depth-dependent, and non-Newtonian magma rheology. Our model also describes the film draining after the burst and allows for the computation of the stresses exerted on the conduit walls using a new analytical solution for the thickness of an annular film flow. Using Stromboli volcano, Italy, as reference, it was evaluated with a specifically designed, non-Newtonian rheological model based on petrochemical data. The results show the importance of using a realistic magma viscosity model when modeling the slug ascent: A 100 kg gas slug attains its maximum overpressure of 2.1 bars about 40 s before it rapidly expands and finally bursts at only 1.4 bars. This preburst pressure drop is not reproduced by constant viscosity models. The normal stresses on the conduit wall are generally dilatational above and adjacent to the slug and constrictive below it and after the burst. In both cases their maximum values exceed 3 bars, whereas the shear stresses stay negligible. Furthermore, it was found that the slug needs to start deeper than 25 m to build up its full burst pressure. At greater depth, the burst pressure decreases because of the adiabatic slug gas. Finally, a viscous plug at the top of the conduit is shown to increase the explosive potential of the volcano significantly.