Inertial effects on free surface pumping with an undulating surface

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 Pandey et al. (Nat. Commun., vol. 14, no. 1, 2023, p. 7735) with the lubrication model. By contrast, in the regime of finite inertia (Re ∼ 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 h₂. Going beyond the experimental data, we derive analytical expressions of Q and h₂, 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.