A two-dimensional, quasi-steady-state, thermal-capillary model is developed for a micro-pulling-down (mu-PD) system to study limitations to steady growth of sapphire. The model incorporates mass, energy, and momentum conservation equations, and also accounts for the physics of the melt meniscus, the solidification front, and the crystal radius. Limit points with respect to pull rate are found under higher-gradient thermal conditions but are shown to unfold with changes in die heating and ambient temperature. Limit points related to crystal size and capillary effects are also found with respect to static head (melt height); however, classical criteria of capillary instability are shown to be invalid. Thus, a more fundamental understanding is obtained for mu-PD operating limits, their origins, and their possible avoidance. (C) 2011 Elsevier B.V. All rights reserved.