The variation of radial impurity distribution induced by surface tension driven flow increases as the zone length decreases in silicon crystals grown by floating zone melting. In combined buoyancy driven and surface tension driven convection at the gravity of earth, the buoyancy contribution becomes relatively smaller as the zone diameter decreases and eventually convection is dominated by the surface tension driven flow (in the case of silicon, for zones of less than about 0.8 cm in diameter). Preliminary calculations for sapphire suggest the presence of an oscillatory surface tension driven convection as a result of an unstable melt surface temperature that results when the zone is heated by a radiation heater.