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Abstract Examples of fluid flows driven by undulating boundaries are found in nature across many different length scales. Even though different driving mechanisms have evolved in distinct environments, they perform essentially the same function: directional transport of liquid. Nature-inspired strategies have been adopted in engineered devices to manipulate and direct flow. Here, we demonstrate how an undulating boundary generates large-scale pumping of a thin liquid near the liquid-air interface. Two dimensional traveling waves on the undulator, a canonical strategy to transport fluid at low Reynolds numbers, surprisingly lead to flow rates that depend non-monotonically on the wave speed. Through an asymptotic analysis of the thin-film equations that account for gravity and surface tension, we predict the observed optimal speed that maximizes pumping. Our findings reveal how proximity to free surfaces, which ensure lower energy dissipation, can be leveraged to achieve directional transport of liquids. 

Free surface flows driven by boundary undulations are observed in many biological phenomena, including the feeding and locomotion of water snails. To simulate the feeding strategy of apple snails, we develop a centimetric robotic undulator that drives a thin viscous film of liquid with the wave speed$$V_w$$. Our experimental results demonstrate that the behaviour of the net fluid flux$$Q$$strongly depends on the Reynolds number$$Re$$. Specifically, in the limit of vanishing$$Re$$, we observe that$$Q$$varies non-monotonically with$$V_w$$, which has been successfully rationalised by Pandeyet al.(Nat. Commun., vol. 14, no. 1, 2023, p. 7735) with the lubrication model. By contrast, in the regime of finite inertia ($${Re} \sim O(1)$$), the fluid flux continues to increase with$$V_w$$and completely deviates from the prediction of lubrication theory. To explain the inertia-enhanced pumping rate, we build a thin-film, two-dimensional model via the asymptotic expansion in which we linearise the effects of inertia. Our model results match the experimental data with no fitting parameters and also show the connection to the corresponding free surface shapes$$h_2$$. Going beyond the experimental data, we derive analytical expressions of$$Q$$and$$h_2$$, which allow us to decouple the effects of inertia, gravity, viscosity and surface tension on free surface pumping over a wide range of parameter space.