We consider a simple experimental setup, based on a harmonic confinement, where a Bose-Einstein condensate and a thermal cloud of weakly interacting alkali-metal atoms are trapped in two different vessels connected by a narrow channel. Using the classical field approximation, we theoretically investigate the analog of the celebrated superfluid-helium fountain effect. We show that this thermomechanical effect might indeed be observed in this system. By analyzing the dynamics of the system, we are able to identify the superfluid and normal components of the flow as well as to distinguish the condensate fraction from the superfluid component. We show that the superfluid component can easily flow from the colder vessel to the hotter one while the normal component is practically blocked in the latter. In the long-time limit, the superfluid component exhibits periodic oscillations reminiscent of the ac Josephson effect obtained in superfluid weak-link experiments.