The stationary, periodic solution to the problem of oscillating flow of a conducting fluid, in a duct closed at one end and with periodical mass flow rate at the other end, is known to exhibit a singularity in the mean temperature at the closed end when transverse conduction in the fluid and the duct wall is taken into account, but longitudinal conduction is neglected. For instance, a solution of that type was originally suggested by Gifford and Longsworth as a prototype for pulse-tube refrigeration. Whether these stationary singular solutions are physical or mere theoretical curiosities depend upon the existence of a scenario leading to such a limit cycle. To address that question, a transient theory is formulated, using the narrow duct approximation. The results show that at least for constant fluid thermal conductivity, all singular profiles generated by the mechanism under study are linearly stable. For round tubes, the temperature profile is shown to evolve, from an arbitrary initial value, toward an equilibrium profile which results in balanced energy fluxes. ©1998 American Institute of Physics.