The steady state structure of thermal plumes rising from a small heater is studied in high Prandtl number fluids. We show good agreement between laboratory experiments and numerical simulations. We study the effect of the boundaries on the plume development by numerically simulating the plume rise in very large geometries. The thermal structure of the plume axis is similar for all box sizes considered, but the velocity structure changes strongly as box sizes are increased. We show that the effect of the side boundaries becomes unimportant for large aspect ratio, but that the free-slip top boundary has a strong influence on the velocity structure under all conditions. We show that the use of an outflow boundary condition significantly reduces the influence of the top boundary. Under these conditions we recover to good precision the theoretical predictions for plumes rising in an semi-infinite half-space. The strong influence of the boundaries in high Prandtl number fluids is important in the interpretation of laboratory experiments and numerical simulation for the dynamics of the Earth's mantle.