The influence of weak convection, caused by surface tension forces, on radial dopant segregation occurring in crystals grown under microgravity conditions is studied numerically. The geometry considered corresponds to a floating-zone configuration with partially coated melt surfaces consisting of small evenly distributed spots of free surfaces. In order to distinguish dopant distribution due to weak convection clearly from distribution due to diffusion the spots only cover one quarter of the periphery. Thus, surface tension-driven convection is allowed only over one quarter of the floating-zone configuration resulting in an asymmetric dopant distribution. The percentage of free surfaces present is varied in order to alter the Marangoni flow rates. The maximum dopant concentration due to radial segregation is plotted as a function of a certain convection level. The results of the present numerical study are supposed to be used to design corresponding space experiments launched at the end of the year 2000.