Several outstanding issues concerning modeling of melt blowing of viscoelastic materials are addressed in this work. Using a slender-jet model for the melt-blown fiber, we probe the effects of rheology by considering Newtonian, upper-convected Maxwell, Phan-Thien and Tanner (PTT), and Giesekus constitutive equations. The effects of a non-uniform shear stress along the fiber length and heat transfer are also addressed. Our results suggest that by combining the slender-jet approach with a Giesekus (or PTT) constitutive equation, useful engineering predictions can be made concerning the final fiber diameter, even when assuming a constant shear stress and neglecting heat transfer. Finally, questions related to linear stability, nonlinear dynamics, and sensitivity are explored. Steady-state fiber profiles are found to be linearly stable, and numerical simulations indicate that the predictions from linear theory can be carried over into the nonlinear regime. Sensitivity analysis reveals that disturbances are likely to become especially amplified at particular frequencies, with elasticity reducing the magnitude of the amplification but broadening the spectrum of frequencies susceptible to large amplification. This suggests an explanation for the narrower fiber diameter distributions that are observed experimentally. (C) 2011 Elsevier Ltd. All rights reserved.