  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 a novel mode of pumping with less energy
dissipation near a free surface compared to a rigid boundary.
