Heating at the closed end of a duct due to oscillating flow-the transient regime

It is known that the stationary, periodic solution to the problem of flow in a tube filled with conducting fluid, closed at one end and with periodically oscillating mass flow rate at the other end, exhibits a singularity in the temperature at the closed end. In this work, the transient regime is studied. Starting from an arbitrary initial temperature profile, the system evolves toward its equilibrium profile, with the singularity at the closed end, on a time scale that depends upon the ratio of the thermal masses, of the wall and of the fluid and upon how effective heat transfer is, and that is typically much longer than the period of the oscillating velocity at the open end. The transient regime can thus be studied by multiple time scale techniques