In some heat exchangers and steam generators, the flow is predominantly axial, and the external fluid flows between baffled compartments through enlarged holes in the baffles around the heat exchanger tubes. Thus, the tube is subjected to relatively high flow velocities over small portions of its length, in the baffle locations. In this paper, the dynamics of such an idealized system is investigated, involving a cylindrical beam with pinned ends in axial flow, going through a baffle plate of finite thickness at some intermediate point, with small radial clearance. The fluid forces along the tube are formulated in a manner reminiscent of the transfer-matrix technique, since the character of these forces changes drastically along the tube. The fluid forces are determined approximately by means of potential flow theory, and viscous effects are taken into account only in a global sense. It was found that if the flow passage through the baffle plate is diffuser-shaped, negative fluid-dynamic damping is generated therein, destabilizing the system and leading to flutter at relatively low flow velocities. The instability depends critically on the shape of the hole through the baffle and on the clearance; thus a convergent-type flow passage does not lead to instability. The negative fluid-dynamic damping is linearly proportional to the flow velocity through the baffle.