The deformation and breakup processes of single droplets in well-defined fields of flow have been extensively studied in the literature. In spite of the fact that in real mixers the conditions are far from equilibrium, most studies are confined to (Newtonian) systems undergoing almost stationary deformation. The transient character of the dispersive mixing process is much less documented. In a Plexiglas-walled Couette-apparatus, the time-dependent deformation of Newtonian droplets into extended threads has been studied When the shear rate is very slowly increased, allowing for almost equilibrium deformation, the results of the critical capillary number Cacrit as a function of viscosity ratio, as reported in the literature, are reproduced. However, in transient flows at capillary numbers Ca>> Cacrit, droplets are deformed into long slender bodies which continue to extend, until the shear has stopped. They then disintegrate into lines of droplets because of interfacial tension-driven disturbances. The time scale for deformation and breakup is important for a better understanding of the dispersive mixing process, e.g., during polymer blending in screw extruders.