Unsteady free-surface flow induced by a line sink

The unsteady flow of fluid from a deep reservoir through a line sink beneath a free surface with surface tension is considered. Two different initial conditions are discussed; the first effectively represents impulsive withdrawal from rest, and the second can be regarded as a disturbance to an existing steady flow. Small-time expansions and numerical methods are used to investigate both the movement to steady states and the critical drawdown of the free surface in the two situations. It is shown that there are several different critical values of flow parameters at which the flow changes its nature. In the zero-surface-tension case, the situation is not fully resolved, but the addition of surface tension clarifies the flow behaviour greatly, and drawdown or movement to a steady state becomes evident. For the second class of initial conditions, it appears that either movement to a steady state or drawdown are the only subcritical possibilities.