Inertial oscillations in the impulsively started flow between a porous injector and an impermeable substrate

The dynamic behaviour of impulsively started axisymmetric flow between a porous injector and an impermeable substrate is considered for the case where fluid is injected at a constant rate. Flow in the separate boundary layer regions near each disk are matched with the evolving inviscid core flow to provide approximate analytical solutions to the short-time dynamics of the flow field. The short-time analytical solutions are valid for all Reynolds numbers where the flow remains laminar. The intermediate and long-time dynamics of the fluid motion are analysed using numerical methods. It is shown that this simple injection flow can exhibit a wide range of interesting dynamic behaviours, the nature of which depend upon the size of the Reynolds number. In the vicinity of the impermeable substrate, under-damped inertial oscillations are present in large Reynolds number flows, whereas over-damped inertial oscillations are present in low Reynolds number flows. Critical-damping of flow near the impermeable substrate occurs at Re ≈ 9. The implications of inertial oscillation for electrochemical materials processing are discussed.