Optimal free-surface pumping by an undulating carpet

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.


INTRODUCTION
The necessity to manipulate flow and transport liquids is primitive to many biophysical processes such as embryonic growth and development [1,2], mucus transport in bronchial tree [3][4][5], motion of food within intestine [6,7], animal drinking [8,9].Engineered systems also rely on efficient liquid transport such as in heat sinks and exchangers for integrated circuits [10,11], micropumps [12,13] and lab-on-a-chip devices [14].Transporting liquids at small scales requires non-reciprocal motion to overcome the time reversibility of low Reynolds number flows.Deformable boundaries in the form of rhythmic undulation of cilia beds and peristaltic waves are nature's resolutions to overrule this reversibility and achieve directional liquid transport.While peristaltic pumps have become an integral component of biomedical devices, artificial ciliary metasurfaces that can actuate, pump, and mix flow have been realized only recently [15][16][17][18][19].
Design strategy of valveless micropumps essentially relies on a similar working principle as cilia-lined walls; sequential actuation of a channel wall by electrical or magnetic fields creates a travelling wave which drags the liquid along with it [20,21].While the primary focus of micropumps has been on the transport of liquids enclosed within a channel, numerous technological applications require handling liquids near fluid-fluid interfaces.In particular, processes such as self-assembly, encapsulation, emulsification involving micron-sized particles critically rely on the liquid flow near interfaces [22,23].Thus the ability to maneuver interfacial flows will open up new avenues for micro-particle sensing and actuating at interfaces.Interestingly, the apple snail Pomacea canaliculata leverages its flexible foot to create large-scale surface flows that fetch floating food particles from afar while feeding underwater in a process called pedal surface collection [24,25]; the physics of which is yet to be fully understood [26,27].
Here we reveal how a rhythmically undulating solid boundary pumps viscous liquid at the interface, and transports floating objects from distances much larger than its size.Surprisingly, pumping does not increase proportionally to the speed of the traveling wave, and we observe non-monotonicity in the average motion of surface floaters as the wave speed is gradually increased.Detailed measurements of the velocity field in combination with an analysis of the lubrication theory unravel the interfacial hydrodynamics of the problem that emerges from a coupling between capillary, gravity, and viscous forces.We find that the non-monotonic flow is a direct consequence of whether the interface remains flat or conforms to the phase of the undulator.Through the theoretical analysis, we are able to predict the optimal wave speed that maximizes pumping, and this prediction is in excellent agreement with experiments.Finally, we show how pumping near an interface is a less dissipative strategy to transport liquid compared to pumping near a rigid boundary.

Experiments
A 3D printed undulator capable of generating travelling waves is attached to the bottom of an acrylic tank.The tank is filled with a viscous liquid (silicone oil or glycerin-water mixture) such that the mean depth of liquid above the undulator (H) remains much smaller the undulator wavelength (λ), i.e.H/λ 1.The undulator is driven by a servo motor attached to a DC power source.Millimetric styrofoam spheres are sprinkled on the liquid surface and their motion is tracked during the experiment to estimate the large scale flow of liquid.Additionally, we characterize the flow within the thin film of liquid directly in contact with the undulator by performing 2D particle image velocimetry (PIV) measurements.Our experimental design is essentially a mesoscale realization of the Taylor's sheet [28] placed near a free surface [29,30]; the crucial difference, however, is that the sheet or undulator is held stationary here, in contrast to free swimming.
Images of the undulator are shown in fig.1a and 1b.The primary component of this design is a helical spine encased with a series of hollow, rectangular links that are interconnected through a thin top surface [31] (see SI and supplementary movies 1 and 2 for details).The links along with the top surface forms an outer shell that transforms the helix rotation to planar travelling wave of the form, δ sin[(x − V w t)/λ].The pitch and radius of the helix determine the wavelength λ and amplitude δ of the undulations respectively.By modulating the angular frequency of the helix, we are able to vary the wave speed V w = ωλ from 15 to 120 mm/s (λ is fixed at 50 mm).We perform experiments with undulators of length λ and 2λ, and the results remain invariant of the undulator size.For given V w , shapes of undulator surface are shown in fig.1c for one period of oscillation.
Large-scale flow -Figure 1d shows the trajectories of floating styrofoam particles generated by 30 mins of continuous oscillations in 1000 cSt silicone oil contained in an acrylic tank of dimensions 61 cm × 46 cm (supplementary movie 3 shows motion of surface floaters for different V w ).Traveling waves on the actuator move in the downward direction as shown by the direction of V w in fig.1d.Thus the floaters are dragged towards the undulator by forming the large-scale flow.The color codes on the trajectories represent time: blue and yellow colors represent the initial and final positions, respectively.Placing the undulator near a side wall of the tank, we measure the floaters' motion over a decade in distance.However, some particles are recirculated back due to the nearby wall.We disregard these trajectories in our analysis.
Fluid motion at the interface is traced by the styrofoam floaters because of their low density (ρ p 50 kg/m 3 ), which ensures that the Stokes number, St = ρ p R p V w /η remains very small ( 10 −2 ) (based on typical wave speed of V w = 100 mm/s, particle radius of R p = 1 mm, is the magnitude of the velocity at the liquid-air interface and x is the distance from the edge of the actuator.We disregard the first 20 mm of data to avoid edge effects.The color code on the curves represents the magnitude of V w .Interestingly, we observe a nonmonotonic response with the particle velocity reaching the maximum value at an intermediate V w .Once | V | at a given location (x = 50 mm) is plotted against the wave speeds (inset of fig.1f), it becomes apparent that the maximum surface flow is achieved for V w 80 mm/s.Since the overall flow in the liquid is driven by the hydrodynamics within the thin film of liquid atop the undulator, we focus on quantifying the velocity field and flow rate in this region.
Dimensionless groups -Before we discuss the experimental results further, it is instructive to identify the relevant dimensionless groups which dictate the response of the system.
The system has eight dimensional parameters: three length scales given by film thickness H, amplitude (δ) and wavelength (λ) of the undulator, the velocity scale V w , gravitational constant g, and three fluid properties set by surface tension (γ), density (ρ), dynamic viscosity (η).These parameters lead to 5 dimensionless groups, namely, = δ/H, a = H/λ, Reynolds number Re = ρV w λa 2 /η, Capillary number Ca = ηV w /(γa 3 ), and Bond number Bo = ρgλ 2 /γ.Here both Re and Ca are defined for the thin-film limit, a 1.We choose two working liquids, silicone oil (η s = 0.97 Pa•s) and glycerin water mixture, GW (η GW = 0.133 Pa•s).For each of the liquids, the thickness, H (maintaining a 1) and wave speed, V w are varied independently.Across all experiments, Re remains lower than 1 (0.01 − 0.35).Thus inertial effects are subdominant and the problem is fully described by , Ca, and Bo.We vary Ca, the ratio of viscous to capillary forces, over three orders in magnitude, 1.4 − 1140.The value of Bo, representing the strength of gravitational forces to surface tension, is 1133 and 426 for silicone oil and glycerin water mixture respectively.As we will demonstrate in the next sections, Ca/Bo which represents the ratio of viscous force to gravitational force, turns out to be the key governing parameter.
Non-monotonic flow rate -The flow field within the thin film of liquid above the undulator is characterized by performing 2D PIV at a longitudinal plane in the middle of the undulator (see Materials & Methods section for details).Figure 2a shows a long-exposure image of illuminated tracer particles, giving a qualitative picture of the flow.The particles essentially oscillate up-down with the actuator, but exhibit net horizontal displacement over a period due to the traveling wave.The presence of the shear-free interface is also crucial to the transport mechanism; the interfacial curvature induces a capillary pressure that modifies the local flow field.The coupling between the two deforming boundaries determines the flow within the gap.Snapshots of typical velocity fields for the two liquids are shown in fig.2b.The top panel is a silicone oil flow field with V w = 23 mm/s and H = 10 mm (Ca = 132, Bo = 1133), while the bottom panel represents flow field of glycerin-water mixture with V w = 17 mm/s and H = 11 mm (Ca = 3, Bo = 426).Higher Ca leads to larger deformation of the free surface.Colors in the plot represent the magnitude of horizontal velocity component, V x ; portion of the liquid that follows the wave is shown in red, whereas a blue region represents part of the liquid that moves in the opposite direction to the wave.In fact, the velocity vectors at a given location switch directions depending on the phase of the actuator (see supplementary movies 4 and 5).Thus, to estimate the net horizontal transport of liquid across a section, we first integrate V x across thickness in the middle of the undulator (marked by the black dashed line in fig.2b i) which yields an instantaneous flow rate Here, h a and h f are the positions of the bottom and top boundaries from the reference point, respectively.Figure 3a plots Q as a function of time, measured in silicone oil for three distinct wave speeds.It shows that Q oscillates with the the same time period as the undulator (τ = λ/V w ), but there is a net flow of liquid along the traveling wave.Thus a time-averaged flow rate, gives a measure of liquid transport by the undulator.Figure 3b gives a comprehensive picture of the flow rate measured across all the experiments.Q is plotted against the characteristic flow rate V w H.The geometric prefactor of 2 is a direct consequence of the thin geometry of the flow [32].Two interesting observations are in order.Regardless of the fluid properties, the flow rates at first increase linearly with 2 V w H.All the data sets other than the GW exhibit a non-monotonic behavior with flow rates reaching maximum values at intermediate 2 V w H. Thus, we find that the non-monotonic surface flow observed in fig.1f is a direct consequence of the flow within the thin film above the undulator.It is important to note that these measurements remain invariant of the undulator size as shown in the SI where we compare the time averaged flow rates measured in single and double wave undulators.In the next section, we develop a theoretical model to explain how the geometrical and material parameters combine to give the optimal wave speed that maximizes flow rate.

Theoretical framework
Thin-film equation -We consider the two dimensional geometry depicted in fig.4a for the theoretical model.An infinite train of periodic undulations of the form h a = δ sin(x − V w t)/λ propagates on the actuator located at a mean depth of H from the free surface.We analyze the flow in the thin-film limit, such that a = H/λ 1.A key aspect of the problem is that the shape of the interface, h f is unknown along with the flow field.The explicit time dependence in this problem is a direct manifestation of the traveling wave on the boundary.Thus in a coordinate system (X, Z) moving with the wave, the flow becomes steady.A simple Galilean transformation relates these coordinates to the laboratory coordinates (x, z): X = x − V w t, and Z = z.Thus, we first solve the problem in the wave frame and then transform the solution to the lab frame.Leaving the details of the derivation in Materials & Methods, here we present the key results.
In the thin-film limit, the separation of vertical and horizontal scales leads to a predominantly horizontal flow-field, and both mass and momentum conservation equations are integrated across the film thickness to reach an ordinary differential equation involving the free surface shape, h f and volume flow rate q.Introducing dimensionless variables X = X/λ, hf = h f /H, ha = h a /H, and q = q/V w H we get q = 1 3 1 Ca The thin film geometry with relevant quantities.We consider an infinite train of traveling undulations of amplitude δ and wavelength λ moving at a speed of V w .The coordinate frame (X, Z) travels with the undulations.The red curve represents the bottom boundary in motion.A liquid layer of mean thickness H reside on top the deformable bottom boundary.Shape of the free surface is given by h f , while the bottom surface is given by h a .b) Numerical solution of the thin-film equation is plotted in terms of the flow rate as a function of Ca, for different Bo.The two largest Bond numbers correspond to the experimental values.c) The rescaled experimental data of fig.3b are in excellent agreement with the theoretical prediction of ( 7), plotted as the solid black line.The small V w (Ca/Bo 1) limit is given by Q = 3 2 /2, while large V w (Ca/Bo 1) limit is given by Q = 2 (Ca/Bo) −2 /6.
where both q and hf are unknowns, and ha = sin X is known.We close the problem by imposing the following constraint on hf : which states that the mean film thickness over one wavelength does not change due to deformation.Along with periodic boundary conditions, equations ( 3) and ( 4) form a set of nonlinear coupled equations whose solutions depend on the three parameters, Ca, Bo, and .For chosen Bo and , these equations are solved by a shooting method for a wide range of Ca.To be able to compare the numerical results with the experimental data of fig.3, we transform the results to the lab frame using the relation Q = q + hf (x, t) − ha (x, t) .Owing to the periodic nature of hf and ha , the time-averaged flow rate simplifies to Q = q + 1. Figure 4b shows the numerical solution of Q as a function of Ca for different Bo.All curves exhibit the same qualitative behavior; at low Ca, the scaled flow rate reaches a constant value as Q ∼ V w H, which is analogous to what we observe in fig.3b.At large Ca, however, we recover a decreasing flow rate as Q ∼ (V w H) −α with α > 0. The transition between the two regimes scales with the Bo.Thus the thin-film equation captures the qualitative behavior found in the experiments.
Asymptotic solution -For Bo 1, the third-order term in Eq. ( 3) can be neglected, which simplifies the governing equation to q = − 1 3 Bo Ca Indeed Bo values in experiments are large (433 and 1132) justifying the above simplification.Furthermore, we assume that the amplitude of the wave, δ is much smaller than H, 1. Interestingly Ca/Bo, the single parameter dictating the solution of eq. ( 5), does not contain surface tension.This ratio is reciprocal to the Galileo number which plays a crucial role in the stability of thin films driven by gravity [33].Here we look for asymptotic solutions of the form, hf = 1 + hf1 + 2h f 2 + O( 3) and q = q 0 + q1 + 2 q2 + O( 3 ) [34].We insert these expansions in eqns.( 5) and ( 4), and solve the equations in orders of .Leaving the solution of hf in the SI, here we present the solution of q which becomes q Thus the time averaged flow rate in the lab frame is given by This is the key result of the theoretical model.It demonstrates that the flow rate is quadratic in amplitude of the traveling wave, which is why we incorporated 2 in the horizontal scale of fig.3b.Importantly, eq. ( 7) captures the non-monotonic behavior of the experiments.Once the data in fig.3b are rescaled, all the different cases collapse onto a master curve which is in excellent agreement with the black solid line representing eq. ( 7), as shown in fig.4c.
Optimal wave speed -The physical picture behind the nonmonotonic nature of the flow rate becomes clear once the free surface shapes are found.For a given Bo with a low Ca, the liquid-air interface behaves as an infinitely taut membrane with minimal deformations.Thus, a liquid parcel moves primarily in the horizontal direction, and the flow rate is given purely by the kinematics (V w , H, δ).Indeed for Ca/Bo 1 eq.( 7) simplifies to give Q / 2 = 3/2.In the dimensional form, this relation explains the increase in the flow rate with the wave speed, Q = 3δ 2 V w /2H.Thus, the flow is independent of the liquid properties which we have noted in fig.3b.As Ca increases, the interface starts to deform up and down by conforming with the undulating actuator.In this limit, the translational velocity of tracer particles decreases thereby lowering the flow rate.Indeed, in the limit of Ca/Bo 1, we find a decreasing flow rate given by Q = 2 (Ca/Bo) −2 /6.These two asymptotic limits are shown as dashed lines in fig.4c.The flow rate attains a maximum at the intersection of these two lines where Ca/Bo = 1/3.In the dimensional form, this particular value of Ca/Bo gives the optimal wave speed at which flow rate peaks, The optimal wave speed emerges from a competition between hydrostatic pressure (∼ ρgH) and lubrication pressure (∼ ηV w λ/H 2 ); surface tension drops out in the above expression.Eq. ( 8) gives the optimal speed at which the undulator maximizes pumping.Now we are in a position to examine whether eq. ( 8) captures the peak surface velocities observed in fig.1f.Plugging in the density (ρ = 970 kg/m 3 ), viscosity (η s = .97Pa•s), H = 10.8 mm, λ = 50 mm, we find V (max) w = 82.3mm/s which matches very well with the observation (shown as the gray line in fig.1f inset).
Pumping Efficacy -The flow rate achieved by this mechanism comes at the expense of the power needed to drive the undulator.This power expenditure equals the viscous dissipation within the flow.To this end, we estimate the efficacy of the mechanism by comparing the output, Q to the input, viscous dissipation Ē (see Materials & Methods for a derivation of Ē).To demonstrate the benefit of having an interface on the pumping capability of this mechanism, we compare with the flux and FIG.5: Pumping efficacy of the undulator.The dimensionless flow rate is plotted against the dimensionless dissipation for a wide range of Ca/Bo values.The data points represent numerical results, which are obtained for = 0.3.The dashed line is the asymptotic prediction of eq. ( 9).The solid line is the result for a top rigid boundary and represents eq.(10).dissipation for a rigid top boundary.These results are shown in fig. 5.The data points represent dimensionless flux plotted against dissipation for a wide range of Ca/Bo.The 1 asymptotic result, shown as the black dashed line in fig.5, captures these results perfectly, giving the following algebraic relation between the two Importance of the free surface becomes apparent when the above result is compared to the scenario of the thin film bounded by a rigid, solid wall on top.As shown in the SI, for a rigid top boundary, both the flow rate and dissipation is given purely by the ratio of δ/H.We find that the flow dissipates 4 times more energy to achieve the same amount of flow, This is plotted as the solid black line in fig. 5. Thus it is clear that the liquid-air interface facilitates pumping by promoting horizontal transport of fluid parcels at a lower power consumption.

DISCUSSION
In summary, we have demonstrated the pumping capability of a sub-surface undulating carpet; the travelling wave triggers a large-scale flow beyond its body size.A direct observation of the liquid motion above the undulator in combination with a quantitative analysis of the thin film equations, yields the optimal speed at which this device transports the maximum amount of liquid for given geometric and fluid properties.This optimal wave speed scales inversely with the wavelength of the undulations and linearly with the cube of the film thickness.It is interesting to note that the key governing parameter, Ca/Bo can be interpreted as a ratio of two velocitieswave speed (V w ) to a characteristic relaxation or leveling speed (V r = ρgH 3 /ηλ) at which surface undulations flatten out.This leveling process is dominated by gravity since the scale of undulations (∼ λ) is much larger than the capillary length ( γ/ρg).Thus for V r V w , the undulator essentially works against a relaxed, flat interface, and liquid parcels primarily exhibit horizontal displacement over a period.In the other limit of V r V w , the free surface tends to beat in phase with the travelling boundary amplifying the vertical displacement, and subsequently reducing the net transport.
Our study demonstrates that the large-scale surface flow is a direct manifestation of the thin-film hydrodynamics above the undulator by showing how the optimal pumping speed captures the peak velocities in surface floaters.However, a quantitative analysis connecting the above two aspects of the flow field is necessary to exploit the full potential of this mechanism.Additionally, in the unexplored inertial regime, we expect the mechanism to showcase interesting dynamics due to the coupling between surface waves and finite-size particles [35].We believe that this work opens up new pathways for self-assembly and patterning at the mesoscale [36,37], bio-inspired strategies for remote sensing and actuation within liquids [38,39], and control of interfacial flows using active boundaries [40,41].

MATERIALS & METHODS
Modeling & printing of the undulator -The models are designed in Fusion 360 (Autodesk).The helix is 3D printed in a Formlab Form 2 SLA printer by photo-crosslinking resin, whereas the outer shell comprising the top surface and rectangular links is printed in a Ultimaker S5 (Ultimaker Ltd.) using a blue TPE (thermoplastic elastomer).Due to the relative flexibility of TPE, the outer shell conforms to the helix.The helix is connected to a mini servo motor which is driven by DC power supply.All other parts (Base, Undulator holders, etc.) are printed using PLA (Polylactic acid) filaments on an Ultimaker S5 (Ultimaker Ltd.) printer.
Measurement of the flow-field -We perform particle image velocimetry measurements on the thin liquid layer above the undulator.The viscous liquid is seeded with 10 µm glass microspheres (LaVision).A 520 nm 1W laser sheet (Laserland) illuminates a longitudinal plane in the middle of the undulator.Images are recorded by a Photron Fastcam SAZ camera at 500 frames per second.Image cross-corelation is performed in the open source PIVlab [42] to construct the velocity field.
Theoretical Modeling -The separation of scales (H λ) in the thin film geometry leads to a set of reduced momentum equations and a flow field that is predominantly horizontal.Thus integration of the X-momentum equation with no slip boundary condition on the undulator (Z = ha) and no shear stress condition at the free surface (Z = h f ) results in Similarly, we integrate the Z-momentum equation and apply the Young-Laplace equation at the free surface, which yields the following expression for the pressure p, p = −γh f + ρg(h f − Z).
Integration of the continuity equation gives the volume flow rate, q = h f ha vX dZ, (per unit depth in this two dimensional case).Plugging eqs.(11) and (12) into the expression of flow rate gives the ODE, This equation relates the yet unknown constant q and the unknown free surface shape h f .We close the problem by imposing the following additional constraint on h f : which states that the mean film thickness over one wavelength remains the same as that of the unperturbed interface, H.In dimensionless form the above set of equations take the form of eqs.( 3) and ( 4) of the main text.For a direct comparison with experiments, we transform the flow rate, q back to the lab frame which is an explicit function of time, Q(x, t) = q + Vw(h f (x, t) − δ sin(x − Vwt)/λ).We seek for periodic free surface shapes, such that h f = H + periodic terms.The time averaged flow rate thus simplifies to where the integration is performed at a fixed spatial location.In dimensionless form this equation becomes, Q = q + 1, as mentioned in the main text.
To estimate the efficacy of the pumping mechanism, we compare the output, flow rate to the energy dissipation within the flow, which is given by, Using the expression for velocity field of eq. ( 11), we integrate on Z to find that the free surface shape h f fully determine the amount of dissipation in the flow.In dimensionless form, dissipation becomes Bo Ca where Ē = EH/(ηλV 2 w ).

FIG. 1 :
FIG. 1: Large-scale transport of floaters by the undulating carpet.The actuator, shown in panels a) and b), is comprised of a helix rotating inside a blue shell.Rotation of the helix causes an oscillatory motion of the shell forming a traveling wave on the surface.It is placed at a mean depth H below the liquid surface.c) Shape of the undulations over a period of oscillation.These shapes are captured by a traveling sine wave of δ sin(x − V w t)/λ.d) Trajectories of floating styrofoam particles after 30 mins of continuous oscillation in 1000 cSt silicone oil for a fixed actuation speed V w .This panel is a top view image with the actuator position marked at the bottom of the frame.The color coding of dark to light indicates the arrow of time.e) Magnified trajectories of particles located straight ahead of the actuator.The filled circles represent initial positions of the styrofoam particles.f) Particle velocity as a function of distance for increasing wave speeds (V w ).Different wave speeds are marked by the color coding.Distances are measured from the edge of actuator, as shown in panel e).Each of the curves is an average over 20 trajectories.Particle velocity exhibits a non-monotonic behavior with V w with maximum velocities measured at intermediate wave speeds.The inset confirms this behavior by showing particle velocity at a fixed location, x = 50 mm for different V w .Error bars in this plot represent standard deviation in velocity magnitude.The gray line is prediction from eq. (8).

FIG. 2 :
FIG. 2: Thin-film flow atop the undulator.a) A sketch of the actuator and a long exposure image of a typical flow-field measurement, showing motion of the tracer particles in the thin film.The free surface deforms in response to the flow.b) Results of PIV for two different capillary numbers, Ca = 132 (top panel), Ca = 3 (bottom panel).In both these panels the bottom boundary is the actuator surface, while the top boundary is the liquid interface.The color coding represents the horizontal component of the velocity field, V x ; red signifies flow along V w while blue signifies flow opposite to V w .and viscosity of silicone oil of η = 10 −3 Pa•s).To this end, we focus on the floaters that are initially located straight ahead of the actuator to analyze the variation of velocity with distance.These trajectories are shown in fig.1e with black circles representing the initial positions.For a given V w , we interpolate 20 trajectories to construct a velocity-distance curve which is shown in fig.1f (see SI for details of these measurements).Here,| V | = (V 2 x + V y ) 1/2is the magnitude of the velocity at the liquid-air interface and x is the distance from the edge of the actuator.We disregard the first 20 mm of data to avoid edge effects.The color code on the curves represents the magnitude of V w .Interestingly, we observe a nonmonotonic response with the particle velocity reaching the maximum value at an intermediate V w .Once | V | at a given location (x = 50 mm) is plotted against the wave speeds (inset of fig.1f), it becomes apparent that the maximum surface flow is achieved for V w 80 mm/s.Since the overall flow in the liquid is driven by the hydrodynamics within the thin film of liquid atop the undulator, we focus on quantifying the velocity field and flow rate in this region.

FIG. 3 :
FIG.3: Non-monotonic flow rate.a) Instantaneous flow rate in silicone oil over multiple periods of oscillation.The data sets represent increasing V w , from white to black.These measurements are taken at a cross section marked by the dashed line in fig.2bi.b) Time-averaged flow rate, Q is plotted against the flux scale 2 V w H of the problem.The circles represent the experiment in silicone oil with bigger markers denoting larger H: 7.5 mm (red), 9.5 mm (green), 11.5 mm (blue), 14 mm (orange).The squares represent GW experiments with H = 11 mm.The dashed line is the theoretical prediction, given by Q = 3 2 V w H/2.

FIG. 4 :
FIG.4: Theoretical & numerical solutions of thin-film flow.a) The thin film geometry with relevant quantities.We consider an infinite train of traveling undulations of amplitude δ and wavelength λ moving at a speed of V w .The coordinate frame (X, Z) travels with the undulations.The red curve represents the bottom boundary in motion.A liquid layer of mean thickness H reside on top the deformable bottom boundary.Shape of the free surface is given by h f , while the bottom surface is given by h a .b) Numerical solution of the thin-film equation is plotted in terms of the flow rate as a function of Ca, for different Bo.The two largest Bond numbers correspond to the experimental values.c) The rescaled experimental data of fig.3bare in excellent agreement with the theoretical prediction of (7), plotted as the solid black line.The small V w (Ca/Bo 1) limit is given by Q = 3 2 /2, while large V w (Ca/Bo 1) limit is given by Q = 2 (Ca/Bo) −2 /6.