Withdrawal of a fluid through an isolated line sink (for planar flow) or a point sink (for axi-symmetric flow in three dimensions) is considered. A linearized solution is presented in both cases, under the assumption that the sink strength is small and the sink is turned on gradually. The results show that the behaviour for small times is as if an image source were present above the surface. Asymptotic results indicate that, in both cases, the flow rapidly develops an effective image sink above the free surface.