Convection with a Newtonian temperature-dependent rheology leads to little or no surface velocity unless zones of weakness are introduced. “Plate-like” features are observed in calculations both with Newtonian rheology, employing imposed weak zones, and with power-law (non-Newtonian) rheology, where high stresses at the trench reduce the effective viscosity. Since deformation at subduction zones involves faulting, both of these parameterizations should be treated with some skepticism. It is important to understand how the parameterizations affect the model results. We study the relationship between trench viscosity and plate velocity using a Newtonian rheology by varying the viscosity at the trench. The plate velocity is a function of the trench viscosity for fixed Rayleigh number and plate/slab viscosity. Slab velocities for non-Newtonian rheology calculations are significantly different from slab velocities from Newtonian rheology calculations at the same effective Rayleigh number. Both models give reasonable strain-rates for the slab when compared with estimates of seismic strain-rate. Non-Newtonian rheology eliminates the need for imposed weak zones and provides a self-consistent fluid dynamical mechanism for subduction in numerical convection models.