Low-Reynolds-number mixing of two immiscible viscous fluids in a lid-driven rectangular cavity -a problem of interest in the processing of polymers-is studied computationally using a modified marker and cell method. Two prototypical configurations-a round blob of fluid within the other fluid and a layered configuration-are considered. It is found that the length of the interface between the two fluids -a good indicator of the degree of intermingling between initially segregated materials-is a strong function of the viscosity ratios of the two fluids, the aspect ratio of the cavity, and the initial configuration of the fluids. The Reynolds number is taken to be small and the capillary number is considered large enough for surface tension effects to be ignored. The aspect ratios range from 0.80 to 2.00, while viscosity ratios range in value from 0.01 to 5. The bulk of the results pertain to steady lid forcing; however, time-periodic forcing leading to chaotic motion is considered as well. The maximum computed length stretch is 65. The results suggest that care should be exercised in the placing of feeding locations in mixing devices.