Calculations of convection driven by crystal and feed rod rotation and by surface‐tension‐gradients are presented for small scale silicon floating zones operating under microgravity and earthbound conditions. The analyses are based on a thermal‐capillary model of the floating zone process that presents a self‐consistent analysis of convection in the melt, heat transport in the melt, feed rod and growing crystal, the shape of the melt/gas meniscus and the shapes of the solidification and melting interfaces. Results for small‐scale silicon floating zones demonstrate show the intense convection driven by thermocapillary forces caused by the axial temperature gradients that are needed to sustain the molten zone. Sample results illustrate the difference between the shape and flows in floating zones on earth and in microgravity.