The following model problem examines the thermocapillary feedback mechanism important at the edge of weld pools and other materials processes. A pool of liquid with a flat horizontal free surface is bounded on one side by a vertical solid wall, which is maintained at a cold temperature to unit depth, and at a warmer temperature below; far away the fluid is at the warmer temperature. Surface tension is a decreasing function of temperature, so that the surface thermal gradient drives flow toward the corner. When convection is vigorous, the flow compresses the thermal gradient which is driving the flow; this positive feedback results in small local length scales and high velocities near the corner. This problem is examined through a detailed scaling analysis and through numerical simulation for a range of parameters.... Thermocapillary flow.