Free-Surface Oscillations in a Slowly Draining Tank

Abstract

The initial behavior of a free surface in a draining (filling) circular tank is analyzed using a linearized model. The withdrawal (injection) of fluid damps (enhances) oscillations which either exist before the withdrawal (filling) or are induced by the withdrawal (injection). The initial growth rate of the drainage-initiated free-surface oscillations strongly depends on the initial behavior of the drain rate function. If the drain is turned on gradually, the drainage-initiated free-surface oscillation is weaker compared to the forced one, so there are no drainage-initiated oscillations. However, if the drain is turned on suddenly, the induced oscillatory motion dominates the forced motion. For periodic drainage, the results show that the strongest resonant oscillation occurs when the drainage frequency w coincides with the first natural frequency of the flow system. All of the nonresonant modes of the oscillations are stable regardless of the initial behavior of the drain rate. If q(t) = sinωt, all of the resonant oscillations are stable. In the case when q(t) = cosωt, the initial jump in the drainage means that the resonance modes can be either unconditionally unstable, unconditionally stable, or conditionally unstable, depending on the various parameters.