The time-varying flow in which fluid is withdrawn from or added to a reservoir of infinite or arbitrary finite depth through a point sink or source of variable strength beneath a free surface is considered. Backed up by some analytic work, a numerical method is used, and the results are compared with previous work on steady and unsteady flows. In the case of withdrawal for an impulsively started flow, it is found that the critical flow rate increases with reservoir depth, although it changes little as the depth increases beyond double the sink submergence depth. The largest flow rate at which steady solutions can evolve in source flows follows a similar pattern although at a considerably higher value. Simulations indicate that some of the previously calculated steady state solutions at higher flow rates may be unstable, if they exist at all.

In the low-Froude number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Previous studies have considered linearized steady flow past a submerged source in infinite-depth fluids, in which exponential asymptotics were used to determine the behaviour of downstream longitudinal and transverse free-surface gravity waves. Here, unsteady flow past a submerged source in an infinite-depth fluid is investigated, with the free surface taken to be initially waveless. The source is taken to be weak, and the flow is linearized about the undisturbed solution. Exponential asymptotics are applied to determine the wave behaviour on the free surface in terms of the two-dimensional plan-view, in order to show how the free surface waves evolve over time and eventually tend to the steady solution.

Two-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.