We have studied 2-D time-dependent convection for a rheology which is both non-Newtonian and temperature-dependent. Strong effects associated with viscous heating are found in the downwelling sheets, which are heated on both sides with an intensity around O(102) times the chondritic value. The magnitude of viscous heating increases strongly with the subduction speed. The slab interior is weakened by viscous heating and slab breakoff then takes place. This process provides a self-regulating mechanism for governing the speed of intact slabs able to reach the deep mantle. Timescales associated with viscous heating are quite short, a few million years. Internal heating by radioactivity decreases the amount of shear heating. ¿ American Geophysical Union 1995