Introduction
Fluid flows propelled by undulating boundaries are ubiquitous in nature, spanning various length scales. Despite evolving under different driving mechanisms in diverse environments, they serve a common purpose: the directed movement of liquid. Engineered devices have emulated nature-inspired strategies to control and guide flow. In this study, we illustrate how undulating boundaries induce significant pumping of a thin liquid layer near the liquid-air interface. Surprisingly, two-dimensional traveling waves on the undulating surface, a conventional method for fluid transport at low Reynolds numbers, result in flow rates that exhibit a non-monotonic dependence on wave speed. By employing an asymptotic analysis of thin-film equations incorporating gravity and surface tension, we predict the optimal speed that maximizes pumping. Our discoveries illuminate how proximity to free surfaces, which minimize energy dissipation, can be harnessed to achieve directional liquid transport.
The manipulation of fluid flow and liquid transport is fundamental to numerous biophysical processes, including embryonic growth and development, mucus transport in the bronchial tree, food movement within the intestine, and animal drinking. Engineered systems also rely on efficient liquid transport, such as in heat sinks and exchangers for integrated circuits, micropumps, and lab-on-a-chip devices. At small scales, transporting liquids requires non-reciprocal motion to overcome the time reversibility of low Reynolds number flows. Nature employs deformable boundaries, such as rhythmic undulation of cilia beds and peristaltic waves, to achieve directional liquid transport. While peristaltic pumps are integral to biomedical devices, artificial ciliary metasurfaces capable of actuating, pumping, and mixing flow have only recently been developed.
The design principle of valveless micropumps operates similarly to cilia-lined walls; sequential actuation of a channel wall by electrical or magnetic fields creates a traveling wave that propels the liquid. While micropumps primarily focus on transporting liquids within channels, many technological applications require handling liquids near fluid–fluid interfaces. Processes like self-assembly, encapsulation, and emulsification involving micron-sized particles critically depend on liquid flow near interfaces. Liquid–air interfaces also play a vital role for organisms like Neuston, which inhabit at and below the water surface. For instance, the underwater apple snail Pomacea canaliculata utilizes the water surface to generate a large-scale surface flow and capture floating food particles from a distance in a process known as pedal surface collection. Understanding the physics behind these natural phenomena could inspire new bio-inspired strategies for flow manipulation and sensing at interfaces.
In this study, we demonstrate how a rhythmically deforming solid boundary pumps viscous liquid at the interface and transports floating objects from distances much larger than its size. Inspired in part by the underwater snails’ ability to create flow through undulations on their flexible foot, our design produces traveling waves on an undulating surface. While traveling boundaries are a conventional method to drive flow within enclosed spaces near a liquid–air interface, the undulator leads to unexpected observations. Pumping does not increase proportionally to the wave speed, and we observe non-monotonic behavior in the average motion of surface floaters as the wave speed increases. Through detailed velocity field measurements and lubrication theory analysis, we uncover the interfacial hydrodynamics arising from the coupling between capillary, gravity, and viscous forces. The non-monotonic flow is found to result from whether the interface remains flat or conforms to the undulator phase. Theoretical analysis allows us to predict the optimal wave speed that maximizes pumping, which aligns well with experimental results. Finally, we demonstrate that pumping near an interface is a less dissipative strategy for liquid transport compared to pumping near a rigid boundary.
Results
Experimental Setup
To conduct experiments, we utilized a 3D-printed undulator capable of generating traveling waves, affixed to the bottom of an acrylic tank. The tank was filled with a viscous liquid, either silicone oil or a glycerin–water mixture, ensuring that the mean depth of liquid above the undulator (H) remained significantly smaller than the undulator wavelength (λ), satisfying the condition H/λ ≪ 1. The undulator was driven by a servo motor connected to a DC power source. Millimeter-sized styrofoam spheres were dispersed on the liquid surface, and their movement was tracked during the experiment to estimate the large-scale liquid flow. Additionally, we analyzed the flow within the thin liquid film in direct contact with the undulator by conducting 2D particle image velocimetry (PIV) measurements. Our experimental setup essentially represented a mesoscale realization of Taylor’s sheet placed near a free surface; however, unlike in free swimming, the sheet or undulator remained stationary here.
The undulator’s images are depicted in Fig. 1a, b. The primary component of this design consisted of a helical spine surrounded by a series of hollow, rectangular links interconnected through a thin top surface. (See SI and Supplementary Movies 1 and 2 for further details). Together, the links and top surface formed an outer shell that translated the rotation of the helix into a planar traveling wave characterized by the equation . The wavelength λ and amplitude δ of the undulations were determined by the pitch and radius of the helix, respectively. By adjusting the angular frequency of the helix, we could vary the wave speed Vw = ωλ from 15 to 120 mm/s (with λ fixed at 50 mm). We conducted experiments using undulators of lengths λ and 2λ, and the results remained consistent regardless of the undulator size. The shapes of the undulator surface for a single period of oscillation are illustrated in Fig. 1c for a given Vw.
Large-scale flow
The trajectories of floating styrofoam particles, as depicted in Figure 1d, are the result of 30 minutes of continuous oscillations within a 1000 cSt silicone oil environment, housed in an acrylic tank measuring 61 × 46 cm (Supplementary Movie 3 illustrates the motion of surface floaters under different Vw conditions). The directional movement of Vw in Fig. 1d corresponds to the downward motion of traveling waves on the actuator. These undulations induce a large-scale flow that propels the floaters. In the trajectory representation, the color progression from blue to yellow signifies the arrow of time, with blue indicating initial positions and yellow indicating final positions. By situating the undulator close to a tank wall, we observe the motion of floaters over a significant distance range. We quantify the net flow generated near the free surface by the undulator. Unlike flow in closed pipes or channels, this net flow isn’t propelled by an imposed pressure gradient. Instead, it’s influenced by the shape of the free surface, which determines the local pressure within the liquid film. The undulator’s open ends and proximity to the tank walls lead to flow recirculation, as seen in Fig. 1d.
The typical Stokes number, St = ρpRpVw/η, remains quite small, approximately 10^-2 for the styrofoam floaters (calculated with wave speed Vw = 100 mm/s, particle radius Rp = 1 mm, particle density ρp ≃ 50 kg/m^3, and silicone oil viscosity ηs = 0.97 Pa ⋅ s). Consequently, we leverage these particles’ trajectories to estimate fluid motion at the interface. The particle tracks in Fig. 1d reveal some particles being recirculated due to proximity to the tank wall, which we disregard in our analysis. Instead, we focus solely on floaters whose initial positions align directly ahead of the undulator. These trajectories are depicted in Fig. 1e, where black circles represent initial positions. For each Vw, we interpolate 20 trajectories to create a velocity-distance curve, as shown in Fig. 1f (refer to the SI for measurement details). Here, represents the magnitude of velocity at the liquid–air interface, and x signifies the distance from the actuator’s edge. At the undulator’s edge, a sharp boundary change induces a transition region in the flow field, where surface floaters exhibit non-uniform and unsteady motion. To mitigate edge effects, we exclude the first 20 mm of data. The color code on the curves in Fig. 1f denotes the magnitude of Vw. Intriguingly, we observe a non-monotonic response in surface fluid velocity: initially, the fluid parcel speed increases with Vw, but subsequently decreases with further increases in Vw. Plotting at a fixed location (x = − 50 mm) against Vw (inset of Fig. 1f) reveals that maximum surface flow occurs at an intermediate speed, Vw ≃ 80 mm/s. Given that the overall flow in the liquid is driven by hydrodynamics within the thin liquid film atop the undulator, we focus on quantifying the velocity field and flow rate in this region.
Dimensionless Groups
Before delving further into experimental results, let’s identify the relevant dimensionless groups dictating the system’s response. The system comprises eight-dimensional parameters: film thickness H, undulator amplitude (δ), undulator wavelength (λ), velocity scale Vw, gravitational constant g, and fluid properties including surface tension (γ), density (ρ), and dynamic viscosity (η). These parameters give rise to five dimensionless groups: ϵ = δ/H, a = H/λ, Reynolds number Re = ρVwλa^2/η, Capillary number Ca = ηVw/(γa^3), and Bond number Bo = ρgλ^2/γ. Here, both Re and Ca are defined for the thin-film limit, where a ≪ 1. We consider two working liquids: silicone oil (ηs = 0.97 Pa ⋅ s, γs = 0.021 N/m) and a glycerin–water mixture (85% glycerin, 15% water by volume), GW (ηGW = 0.133 Pa ⋅ s, γGW = 0.067 N/m). Across experiments, Re remains below 1 (0.01 − 0.45), indicating subdominant inertial effects. Thus, the problem is fully described by ϵ, Ca, and Bo. We vary Ca over three orders of magnitude, from 7 to 6049, and Bo varies with values of 1133 and 426 for silicone oil and glycerin–water mixture, respectively. For a comprehensive list of experimental parameters and dimensionless numbers, refer to Table S1 in SI. As we will illustrate in subsequent sections, Ca/Bo, representing the ratio of viscous force to gravitational force, emerges as the key governing parameter.
Non-monotonic Flow Rate
The flow field within the thin liquid film above the undulator is characterized by performing PIV at five longitudinal planes along the undulator’s width (see “Methods” section and SI section III for details). We find that the velocity field remains consistent across PIV planes and exhibits low divergence, indicating dominant 2D flow in the thin film. A long-exposure image of illuminated tracer particles in Fig. 2a offers a qualitative view of the flow. These particles oscillate up and down with the actuator but experience net horizontal displacement due to the traveling wave. The presence of an interface plays a crucial role in the transport mechanism; interfacial curvature induces capillary pressure, altering the local flow field. The coupling between the two deforming boundaries dictates the flow within the gap. Snapshots of typical velocity fields for both liquids are depicted in Fig. 2b. The top panel illustrates a silicone oil flow field with Vw = 23 mm/s and H = 5.7 mm (Ca = 714, Bo = 1133), while the bottom panel shows a glycerin–water mixture flow field with Vw = 17 mm/s and H = 6.3 mm (Ca = 17, Bo = 426). For both liquids, the instantaneous Vx profile remains half-parabolic along the thin film’s depth, indicating a shear-free liquid–air interface. Higher Ca leads to greater deformation of the free surface. Colors in the plot represent the magnitude of the horizontal velocity component, Vx, with red indicating liquid following the wave and blue representing liquid moving opposite to the wave. Velocity vectors at a given location switch directions depending on the actuator’s phase. To
