Streaming controlled by meniscus shape
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Streaming controlled by meniscus shape
Y. Huang † C.P. Wolfe J. Zhang J.-Q. Zhong ‡ School of Physics Science and Engineering, Tongji University, Shanghai, China School of Marine and Atmospheric Sciences, Stony Brook University, Stony Brook, U.S. NYU-ECNU Institute of Physics at NYU Shanghai, Shanghai, China(Received xx; revised xx; accepted xx)
Surface waves called meniscus waves often appear in the systems that are close to thecapillary length scale. Since the meniscus shape determines the form of the meniscuswaves, the resulting streaming circulation has a structure distinct from that caused byother capillary-gravity waves recently reported in the literature. In the present study, weproduce symmetric and antisymmetric meniscus shapes by controlling boundary wetta-bility and excite meniscus waves by oscillating the meniscus vertically. The symmetricand antisymmetric configurations produce different surface capillary-gravity wave modesand streaming flow structures. The energy density of the streaming circulation increasesat the rate of the fourth power of the forcing amplitude in both configurations. The flowsymmetry of streaming circulation is retained under the symmetric meniscus, while itis lost under the antisymmetric meniscus. In our experiments, the streaming circulationprimarily originates from the Stokes boundary layer beneath the meniscus and can besuccessfully explained using the existing streaming theory.
Key words:
Meniscus wave, Streaming, Broken Symmetry
1. Introduction
Steady streaming is driven by Reynolds stresses and balanced by viscous stresses in theStokes boundary layer (Batchelor 2000; Riley 2001). The momentum of the streamingflow in the boundary layer can be transferred into the liquid bulk through fluid viscosityto form an Eulerian mean circulation. Streaming circulations play essential roles in fluidmass and passive scalar transport in many environments, such as inside blood veins,cochleae, and the boundary layer over ocean sediments (Lesser & Berkley 1972; Schneck& Walburn 1976; Holmedal & Myrhaug 2009). In recent years, streaming circulationsdriven by the surface waves have also been used to control the distribution of thesuspended particles in the biomedical production and the food manufacturing (Chen et al. et al. et al. et al. l . When such a system is oscillated vertically, a variety of standing surfacewaves can be generated by the capillary effect (Lucassen-Reynders & Lucassen 1970; † [email protected] ‡ [email protected] Y. Huang et al.
Perlin & Schultz 2000). The most well-known type of waves are Faraday waves (Faraday1831; Miles & Henderson 1990), which are excited by parametric instability. However,even when the system is stable to Faraday waves, one transverse surface wave is stillobserved if there is a static meniscus. This meniscus wave (Strickland et al. et al. et al. et al. et al. et al. e k ∼ ( a /g ) ;the single power function indicating a single dynamical regime over the parameter rangestudied. Remarkably, we further discover that for the two meniscus shapes we use,the left-right symmetry of the streaming flow pattern differs markedly under variousforcing amplitudes a /g . For the symmetric case, the streaming pattern remains highlysymmetric in the full range of the applied forcing amplitude. In the antisymmetric case,however, the streaming pattern becomes less symmetric as a /g increases owing to theinteraction between the static meniscus profile and the meniscus wave. To explain theseobservations, we establish a simple analytical model to estimate the meniscus wave atlinear order. We then compute the streaming velocity distribution in the boundary layerfollowing the method of Gordillo & Mujica (2014) and Prinet et al. (2017) and use thisto drive a steady laminar flow model that reproduces the observed streaming velocityfields.
2. Experiment
Experimental setup
In the experiment, we used a rectangular Plexiglas tank divided into three cells bytwo optical glass plates (figure 1a). The middle cell was the working cell, with thedimensions L x = 2 . L y = 5 . L z = 10 . h = 6 . treaming controlled by meniscus shape ◦ . A hydrophobic boundary was made by applying a coating of hydrophobic nano SiO film (CHEMNANO ® NC317) to an optical glass surface, creating a contact angle closeto 135 ◦ . A working cell with two hydrophilic lateral boundaries creates the symmetricsetting while choosing one boundary hydrophobic and the other hydrophilic creates anantisymmetric system. The contact angle of water with the rest Plexiglas boundary wasclose to 90 ◦ .The fluid tank was installed on an electrodynamic shaker (Labworks ® ET-139) thatvibrated sinusoidally in the z direction with an excess gravitation acceleration varyingfrom 0 . g ∼ . g and a frequency of f = 8 Hz. The control signal produced from awave-function generator was amplified using a linear power amplifier to drive the shaker.An accelerometer mounted on the base plate of the fluid tank (not shown in figure 1)measured the instantaneous acceleration that was fed back to the wave-function generatorto regulate the control signal. The closed-loop control system reduced error and ensuredthe stability of the shaker oscillation.We measured the fluid velocity field in the working cell using a Particle ImagingVelocimetry (PIV) system. Neutrally-buoyant particles of diameter 10 µ m were seededin the flow. Because of the effect of surface tension, some of the seeding particles weresuspended on the free surface. We used these particles to track the surface deformation(see section 2.2). We found that these surface particles also played a role in enhancingsurface damping and surface-deformation induced streaming. The PIV particles in thesystem was illuminated from the right using a continuous, diode pump solid-state (DPSS)laser, creating a 1 mm thick light-sheet that passed through a vertical cross-section andthe mid-plane in the y direction of the cell. PIV images that covered a region fromthe surface to a fluid depth of around 3 cm were captured by high-speed camera ata frame rate of 80 Hz, achieving a time resolution of 10 frames per oscillation period.Two-dimensional velocity maps were obtained by cross-correlating either two consecutiveimages (for the primary flow) or two images with the same phase in two consecutiveperiods (for the secondary flow). Each velocity vector was calculated from interrogationwindows spanning 32 ×
32 pixels, with 50% overlap of neighbouring sub-windows. Eachvector covers a region of 16 ×
16 pixels and there are 80 ×
64 velocity vectors per frame,achieving a spatial resolution of 0 .
34 mm (one order in magnitude smaller than thecapillary length scale).The laser light sheet may heat up the fluid on the right side of the working cell andgive rise to an overturning circulation. The speed of this overturning flow could be up tothe order of 10 − cm / s, comparable with the speed of the streaming flow we study. Toavoid this laser-induced convection, the two lateral cells were filled with water to reducethe thermal heating of the fluid in the working cell. With this thermal protection, thespeed of the overturning flow was reduced to O (cid:0) − cm / s (cid:1) and was negligible.2.2. Observation of the surface deformation
The time varying acceleration of the fluid tank in the lab frame is a ( t ) = a sin ϕ ( t ),where a is the forcing amplitude and ϕ ( t ) = 2 πf t is the oscillation phase. From Newton’sthird law, the effective gravity felt by the fluid in the tank is g ( t ) = − [ g + a sin ϕ ( t )] , in which g is the gravitational acceleration constant. When ϕ = π/
2, the fluid tank isat its lowest position with maximum super-gravity; when ϕ = 3 π/
2, the tank is at itshighest position with maximum sub-gravity. To force the system without exciting the
Y. Huang et al.
Figure 1. (a) Schematic drawing of the experimental apparatus. A water tank wasoscillated sinusoidally in the z direction with a time-varying gravitational acceleration g ( t ) = − ( g + a sin 2 πft ). The tank was divided into three cells by two SiO glass plates. Thefluid velocity field was measured in the working cell using PIV. A DPSS laser sheet illuminatedthe seeding particles in the working cell and the particle motion was recorded by a high-speedcamera. (b) Schematic of meniscus in symmetric and antisymmetric configurations. Faraday instability, we select f = 8 . . a /g .
34 (see Appendix A).Thus the periodic deformation of the liquid-air interface resonates with the cavity modes,giving rise to standing meniscus waves. We observe that such a surface deformation modeis strongly dependent on the geometry of the static meniscus. To quantify the relationbetween this surface deformation mode and the shape of the static meniscus, we measurethe surface profile in different oscillation phases from the PIV images. It is found thata large proportion of the liquid-air interface is clearly visible due to seeding particlesaccumulated at the interface which are illuminated by the laser sheet (figure 2a and 2c).From these images we extract the surface profiles ζ ( x ) at four oscillatory phases φ = 0, π/ π and 3 π/ f . The response of meniscus wave amplitude ζ ( x ) to the forcing acceleration amplitude a /g is estimated by the horizontally averagedsurface deformation amplitude h ζ i , which is defined by h ζ i = | ζ ϕ = π/ − ζ ϕ =3 π/ | , treaming controlled by meniscus shape SymmetricAnti-symmetric (a)(c)
Figure 2.
Images and surface profiles of meniscus shape deformation at φ = 0 , π/ , π, π/ f = 8 Hz and a = 0 . g . (a) PIV image of the vibrating surface in the symmetricconfiguration. (b) The extracted surface profiles corresponding to (a). (c) PIV image of thevibrating surface in the antisymmetric configuration. (d) The extracted surface profiles ofsymmetric scenario corresponding to (c). The insets in (b) and (d) shows the horizontallyaveraged surface deformation amplitude h ζ i as a function of a /g . The symbols in the insetsare experimental measurements and the solid line is a linear regression. where ζ ϕ = π/ and ζ ϕ =3 π/ are the surface profiles captured at ϕ = π/ ϕ = 3 π/ x direction. The mean amplitude of themeniscus oscillation h ζ i grows linearly with the forcing amplitude (see the insets figure2b, 2d). In the symmetric case, the excited mode is symmetric and has two nodes. Atmaximum super-gravity ( ϕ = π/
2) the liquid surface is bulging from the middle of thecell, analogous to the phenomenon of liquid jet observed before (Antkowiak et al. ϕ = 3 π/ x = L/
2) is instead excited. When the system is at maximum sub-gravity( ϕ = 3 π/ S ’ shape, with the convex part on the hydrophobicside ( x < L/
2) and the concave part on the hydrophilic side ( x > L/ ϕ = π/ y direction. Representing the modenumber as ( l, m ), where l and m are positive integers denoting the mode number on x and y axes respectively, the symmetric and antisymmetric configurations give rise to Y. Huang et al. modes (2 ,
0) and (1 ,
0) waves, respectively. The PIV results indicated that the oscillatorymotion decays exponentially with depth, so these are deep water waves.2.3.
Observation of the streaming motion
The streaming circulation is observed when the PIV images are broadcast stroboscopi-cally. The streaming circulation is one order of magnitude weaker than the instantaneousprimary flow. Once the circulation forms, it persists as long as the forcing oscillationcontinues. During the experiment, we collected the PIV images one minute after theforcing was turned on, when a steady streaming circulation had been established. Theflow velocity and the vertical scale of the circulations, for both wettability configurations,appear to increase if the oscillation amplitude a increases. The streaming circulation hasdistinct structures in symmetric and antisymmetric cases.In the symmetric case (figure 3), the secondary circulation has a stable four-vortexstructure, with clockwise (counterclockwise) rotation near the right (left) lateral bound-ary. When a ∼ . g , the central vortices are not clearly visible in the PIV image.As the forcing gets stronger, the central vortices become apparent with approximatelycircular streamlines. At the strongest forcing amplitude, the boundary vortices are welldeveloped, whereas the central vortices become suppressed and asymmetric.In the antisymmetric case, the secondary flow field has a dipole circulation structurewith a weaker counterclockwise circulation near the hydrophobic side and a strongerclockwise circulation near the hydrophilic side (figure 4). When the forcing is weak, a ∼ . g , the counterclockwise circulation is only partially visible in the PIV image,but a strong leftward velocity is found below the free surface. As the forcing amplitude in-creases, the counterclockwise circulation becomes more dominant and downward motionbecomes more prominent near the hydrophobic side.To quantify the dynamical behavior of the secondary flow, we integrate the streamingflow velocity on the observed section to determine the kinetic energy density at y -direction: e k = 12 ρ Z Z S ( u + v ) dx dz. (2.1)As shown in figure 5a, the kinetic energy density for fully-established secondary flowstructure e k scales like e k ∼ ( a /g ) β , (2.2)where β ≈ et al. (2017). Note thatthe flow velocity near the boundary is not resolvable by PIV, so the integration in (2.1)excludes the region within 0 .
034 cm of the lateral boundaries. Further, the camera doesnot capture the whole water bulk, so the integration covers the region z > − .
79 cm inthe symmetric case and z > − .
65 cm in the antisymmetric case.The spatial symmetry of the secondary circulation pattern is quantified by the sym-metry parameter γ = RR x
Velocity fields of the streaming flows overlying the raw PIV image in the symmetriccase for three values of a . For each case, the driving frequency is f = 8 Hz. The liquid-airinterface has been highlighted manually. Y. Huang et al.
Figure 4.
Velocity fields of the streaming flows overlying the raw PIV image in theantisymmetric case for four values of a . For each case, the driving frequency is f = 8 Hz.The liquid-air interface has been highlighted manually. vorticity, A x>L/ and A x Response of the system to changes in forcing amplitude a /g . a /g is increasedfrom 0 . . 34 for symmetric case; from 0 . 04 to 0 . 34 for antisymmetric case from experiment.(a) The response of energy density e k . The fitted trends give e k ∼ ( a /g ) and e k ∼ ( a /g ) . forthe symmetric and antisymmetric cases, respectively. (b) The response of symmetry parameter γ , normalized by γ = γ | a /g =0 . . rapidly as the forcing amplitude increases then saturates at γ ≈ . γ for large forcingamplitudes. 3. Theory In this section, we develop a theoretical description of the meniscus wave and resultingsecondary circulation. We start by deriving the static shape of the meniscus and solvefor the wave modes on this background state in section 3.1. The secondary circulation isobtained in section 3.2 by determining the streaming circulation along the free surface andlateral boundaries and using these as boundary conditions for a steady hydrodynamicssolver. 3.1. Meniscus wave The meniscus wave is solved by linearization of the ideal fluid equations with a freesurface (e.g., Batchelor 2000). The nonlinear fluid equations are ∇ φ = 0 , (3.1) ∂ t φ | z = ζ + [ g + a sin( ωt )] ζ = σρ ∂ xx ζ [1 + ( ∂ x ζ ) ] / , (3.2) ∂ t ζ + ∂ x φ | z = ζ ∂ x ζ = ∂ z φ | z = ζ , (3.3) φ | z →−∞ = 0 , (3.4) ∂ x φ | x =0 ,L = 0 , (3.5)where ζ = ζ ( x, t ) is the surface elevation and φ = φ ( x, z, t ) is the velocity potentialfield. The parameter notation is the same as in the previous section. The cell width is L = 2 . g = 980 cm / s is the gravitational acceleration, ρ = 1 g / cm is the densityof water, σ = 72 dyne / cm is the surface tension of water, a is the forcing acceleration,and ω = 2 πf is the forcing angular frequency.The linearization is done in the following way: We use the capillary length l =0 Y. Huang et al. p σ/ ( ρg ) = 0 . 27 cm as the characteristic length scale in the x - and z -directions. The scaleof the surface displacement due to the forced oscillation is a /ω ≈ . . 13 cm, so theratio between these two length scales is a small dimensionless number, ǫ = a / ( lω ) ≈ . . a /g has the magnitude range of 0 . . 32, which is O ( ǫ ). After scalingthe velocity potential, φ , by g a /ω , the solution of the ideal fluid equations (3.1)–(3.5)can be expanded in an asymptotic series in powers of ǫ : ζ = ζ + ζ , (3.6) φ = φ + φ , (3.7)The quantities with the subscript 0 are O (1), which represents a situation when theforcing is neglected and depicts the static balance in the fluid bulk. Thus, both φ and ζ are independent of time. The quantities with subscript 1—that is, O ( ǫ )—represent thelinear wave forced by the vertical oscillation. As the motion decays exponentially fromthe free surface and the depth is 3 times larger than cell width, we apply an infinite-depthbottom boundary condition.3.1.1. Zeroth order solution At O (1), (3.1)–(3.2) become the Young-Laplace equation: ∂ xx ζ = ρg σ ( ζ − z ∗ ) . (3.8)Since the left plate and right plate completely prevent the liquid mass exchange, theexperiment is done with total fluid volume conserved. The reference elevation, z ∗ = − σρg L ( ∂ x ζ | x = L − ∂ x ζ | x =0 ) = − σρg L [tan( θ r ) − tan( π/ − θ l )] . (3.9)is determined by integrating equation (3.8) in x , applying the lateral boundary conditions,and requiring that the horizontal integral of ζ vanish. The quantities θ l and θ r is thecontact angles at the left and right boundaries, respectively. Following the experimentalsetup, in the symmetric case θ l = θ r = 45 ◦ and θ l = 135 ◦ and θ r = 45 ◦ in the anti-symmetric case. The corresponding boundary conditions for equation (3.8) are writtenas ∂ x ζ | x =0 = − s, (3.10) ∂ x ζ | x = L = 1 . (3.11)Here, s controls the symmetry of the meniscus; it is 1 in the symmetric case and − ζ = (1 + s cosh( mL )) m sinh( mL ) cosh( mx ) − sm sinh( mx ) + z ∗ , (3.12)where m = l − . In figure (6), the solutions are compared to the static surface profile mea-surements for both symmetric and antisymmetric cases—the theoretical result capturesthe shape of the free surface well.3.1.2. First order solution According to the surface pressure balance condition (3.2), a change in the gravitationalacceleration g results in horizontally varying pressure differences along the meniscus.Taking the mean surface elevation to be z = 0, a rising of gravitational acceleration treaming controlled by meniscus shape Figure 6. The zeroth-order meniscus solution ζ (solid line) compared to the experimentalmeasurement (dotted line) for the (a) symmetric and (b) antisymmetric cases. The + and − signs represent the pressure surplus and deficit, respectively, that arises when the accelerationis changed. The pressure anomaly distribution out of (in) the bracket is for the situation when g increases (decreases). produces a positive pressure difference in the region where ζ > ζ < 0. The pressure difference distribution is inverted for fallinggravitational acceleration. The distribution of pressure anomalies illustrated in figure 6lead to a wave with two nodes [a (2 , 0) wave] in the symmetric case and a wave with asingle node [a (1 , 0) wave] in the antisymmetric case.At O ( ǫ ), the governing equations and boundary conditions become ∇ φ = 0 , (3.13) ∂ t φ | z =0 + g ζ − σρ ∂ xx ζ = − a ζ sin( ωt ) , (3.14) ∂ z φ | z =0 − ∂ t ζ = 0 , (3.15) φ | z = −∞ = 0 , (3.16) ∂ x φ | x =0 ,L = 0 . (3.17)The solution that satisfies (3.13)–(3.17) takes the form φ = ˆ φ cos( ωt ) = ∞ X n =1 A n e k n z cos( k n x ) cos( ωt ) , (3.18) ζ = ˆ ζ sin( ωt ) = ∞ X n =1 A n k n ω cos( k n x ) sin( ωt ) , (3.19)where the eigen-wavenumber k n = nπ/L and coefficients A n are given by A n = 2 ωa ω − ω n ( − n + s ( m + k n ) L , (3.20)where ω n = g k n + σρ k n (3.21)2 Y. Huang et al. Figure 7. The first order meniscus solutions ˆ ζ (solid line) compared to the experimentalmeasurements (dotted line) when a = 0 . g and f = 8 Hz for the (a) symmetric and(b) antisymmetric cases. The experimental ˆ ζ are produced by subtracting the static surfaceprofiles from the oscillating surface profiles captured during the slack phase. The insets showthe combination of ζ and ˆ ζ solutions and the correspondent experiment measurements. is the (squared) natural frequency of a gravity-capillary wave with wavenumber k n . Notethat these wave modes [(3.18) and (3.19)] are not parametric resonance modes (Faradaymodes), but ordinary resonance modes since they have the same frequency as the forcing.The wave energy is concentrated around those n for which ω n is close to ω . If ω = ω n ,for a given n , the capillary-gravity wave can travel integer multiples of the cell lengthduring one forcing cycle. When ω = 16 π rad − as in our experiment, both modes n = 1and n = 2 are close to resonance. In our case, the basin modes are further selected bythe symmetry of the resting meniscus, since A n = 0 for n odd in the symmetric case and A n = 0 for n even in the antisymmetric case. Thus, only the n = 2 ( n = 1) mode isclose to resonance in the symmetric (antisymmetric) case. The amplitude A n decays like O ( n − ), so the series converges quickly. In figure 7, the infinite series is terminated at n = 200 to produce the estimate of the meniscus wave amplitude ˆ ζ for both symmetricand antisymmetric cases.From the comparison in figure 7, we can see that our theory captures about 50%variance of the experiment value at first order. The linearized theory is inadequate toproduce a more accurate result, because our basic scaling approximation ∂ x ( ζ + ζ ) → ∂ x ζ = 1, obviously violates theapproximation. To estimate the magnitude of ∂ x ζ , we use the contact line boundarycondition from Hocking (1987) ∂ t ζ = c s ( ∂ x ζ − ∂ x ζ ) . (3.22)Here, c s is the slipping parameter of contact line. Under this new relationship, thegradient of first order surface elevation ∂ x ζ can be scaled by a / ( ωc s ). According tothe experiment measurement done by Cocciaro et al. (1993), the slipping parameter c s = Dla , in which D = 0 . ± . 05 s / cm is a fitting constant (it is necessary to note thatin their experiment the working fluid is water, solid surface is Plexiglas and the contactangle is at 62 ◦ ). ∂ x ζ ∼ . 15 in our experiment is a small number. So, the solution wegive in this section can be valid near the contact line only when contact angles are both treaming controlled by meniscus shape ◦ , when ∂ x ζ is close to 0. Nevertheless, to get a precise analytical model ofmeniscus wave is not the purpose of this work. In the next section, we will show thatsuch a meniscus wave solution is sufficient to explain the streaming circulation patternformation. 3.2. Streaming circulation In this section, we explain the secondary circulation pattern shown in figures 3 and 4.Firstly, we note that as the dominant mode number of primary flow doubles, the numberof secondary vortices doubles as well. Moreover, the nodes of the meniscus wave are aboveupwelling flow, while the anti-nodes cap downwelling flows. The spatial relationshipsbetween the meniscus wave and the streaming flow indicates that the streaming flowis generated by the meniscus wave itself. There are a large number of seeding particlescaptured by the free surface, forming a membrane (see figure 2). Moisy et al. (2018)has shown that under these conditions, the free surface acts like a no-slip boundary.We thus hypothesize that such conditions generate a Stokes boundary layer at the freesurface which drives a secondary streaming flow in the boundary layer. This streamingboundary layer affects the bulk through viscous momentum transfer, so as to generatethe secondary circulation pattern.3.2.1. Streaming under the free surface Following Prinet et al. (2017), the streaming velocity under a meniscus fully contami-nated by seeding particles can be written as u s | z = ζ = − ω ˆ u k ∂ τ ˆ u k ˆ t (3.23)where u || denotes the primary flow parallel to the meniscus, ˆ t is the tangential unit vectoralong the meniscus. The tangential velocity at the meniscus, u k , is obtained by linearextrapolation from z = 0 (see Appendix B). By substituting the expression for u k into3.23, we find u s | z = ζ = ( − ω X n,m A n A m ( k n + ζ k n )( k m + ζ k m ) h k m − ρg σ ( ζ − z ∗ ) i sin( k n x ) cos( k m x ) + O ( ∂ x ζ ) ) ˆ ı + O ( ∂ x ζ )ˆ , (3.24)where ˆ ı and ˆ are the horizontal and vertical unit vectors, respectively. From this result,we can see that the surface profile ζ modifies the streaming velocity distribution. To beconsistent with the scaling assumptions of the primary flow solution, we have neglectedterms that are O ( ∂ x ζ ), which eliminates the vertical velocity along the meniscus.The horizontal velocity along the meniscus, u s | z = ζ , is shown in figure 8. In bothcases, the gravest mode gives the largest contribution to the streaming velocity. Withthe contribution from the higher order waves, the magnitude decreases in the symmetriccase and increases in the antisymmetric case. The streaming velocity field is completelyantisymmetric in the symmetric case, but lacks any definite symmetry in the antisymmet-ric case. This asymmetry can be understood by considering the fact that the velocity ofdeep-water waves decreases exponentially from the mean water depth (i.e., from z = 0).Since the steady liquid surface is higher on the hydrophilic side than on the hydrophobicside, the velocity induced by the meniscus wave is also higher on the hydrophilic side.From (3.23), the wave-induced secondary flow inherits this strength distribution (see theschematic inset in figure 11), leading to an asymmetric streaming velocity distribution.4 Y. Huang et al. Figure 8. The streaming flow velocity along the free surface when a = 0 . g and f = 8 Hz forthe (a) symmetric and (b) antisymmetric cases. The solid line is the result (3.24) truncated at m, n = 200 while the dashed line contains contributions from the gravest mode only. Streaming near the lateral boundary Following Batchelor (2000), the streaming velocity for 2D flow near a flat verticalboundary is w s | x =0 ,L = − ω X n,m ˆ w n ∂ z ˆ w m (3.25)where ˆ w n is the vertical velocity amplitude of the n th mode at the lateral boundary.From (3.18), this is ˆ w n = A n k n e k n z ( x = 0 , ( − n x = L. (3.26)Substituting this into (3.25) and applying linear extrapolation (see Appendix B), we findthat w s | x =0 ,L = − ω P n,m A n A m k n k m (1 + zk n )(1 + zk m ) z > , − ω P n,m A n A m k n k m e ( k n + k m ) z z < . (3.27)This expression is valid for both symmetric and anti-symmetric cases.The vertical velocity distribution near the lateral boundary is shown in figure 9. Thecontribution from the gravest mode is similar in both the symmetric and antisymmetriccases. The two cases become distinct once higher order contributions are included.The streaming velocity becomes stronger, but remains monotonic and negative in thesymmetric case. In the antisymmetric case, the higher order contributions create a profilewhich is positive above z = 0 . cm and has a negative extremum about 0 . A n .All the A n is positive in the symmetric case, so the components add constructively.In the antisymmetric case, A is negative and the other A n are positive, so interactionsbetween the gravest and higher order modes interfere destructively with the contributionsfrom the wave self-interactions. Thus, in contrast to the monochromatic wave situation treaming controlled by meniscus shape Figure 9. The streaming flow velocity distribution along the lateral boundary when a = 0 . g , f = 8 Hz for the (a) symmetric and (b) antisymmetric cases. The solid line is the result (3.25)with the sum terminated at m, n = 200 while the dashed line contains contributions from thegravest mode only. The upper limit of the plot is the position of the liquid surface at hydrophilicboundary. The dash-dotted line in (b) shows the liquid surface position at the hydrophobicboundary. discussed by Prinet et al. (2017), the lateral boundary streaming velocity profile is affectedby the relative phases of the component waves. In this experiment, the relative phasesare controlled by the symmetry of static meniscus. Note that the streaming velocity isleft-right symmetric in the symmetric case, but asymmetric in the antisymmetric casebecause the free surface on the hydrophobic side is lower than that at hydrophilic side(see figure 9b).3.2.3. Numerical simulation of streaming circulation Following Prinet et al. (2017), we obtain the secondary flow in the bulk by usingthe steady streaming velocity profile as boundary conditions for the laminar flow solverin COMSOL Multiphysics ® using a fixed (i.e., not deforming) domain. Taking thecharacteristic velocity U to be the root mean square velocity of the surface streamingvelocity distribution, and the characteristic length to be the cell width L , the maximumReynolds number is 24 for streaming circulation. We simulate the steady secondarycirculation at 7 different oscillation amplitudes with the driving frequency f = 8 Hz.The steady secondary circulation induced by the oscillation at a = 0 . g is shown infigure 10. The numerical simulation successfully reproduces the four-vortex circulationstructure in the symmetric case and dipole circulation structure in the antisymmetriccase. In the symmetric case, the generated circulation structure is nearly symmetric. Inthe antisymmetric case, the circulation is deeper than in the symmetric case and strongernear the hydrophilic side—both in agreement with the experiments.The numerical solutions follow the same scaling law, e k ∼ ( a /g ) , as the experiments,but the value of the energy is approximately one order of magnitude smaller (figure 11a).Since the secondary circulation is due to the quadratic nonlinearity, the energy of thesecondary circulation scales with the amplitude of the meniscus wave to the fourth power.Thus, small errors in the amplitude of the meniscus wave produce large errors in theenergy of the secondary circulation. Figure 7 shows that the ratio of the amplitude ofthe meniscus wave derived from linear theory to the experimentally realized wave is6 Y. Huang et al. Figure 10. The secondary flow velocity (areas) and streamfunction (contours) beneath theoscillating meniscus from numerical simulation for the (a) symmetric and (b) antisymmetriccases. The forcing amplitude is a = 0 . g and the driving frequency is f = 8 Hz. Figure 11. Variation of (a) kinetic energy density, e k , and (b) symmetry parameter, γ/γ , with a /g from the numerical simulations of the symmetric and antisymmetric cases. The lines in(a) give the scaling e k ∼ ( a /g ) . Note that the energy integration is performed over the samespatial region as in the experiments. The insets in panel (b) are two schematic plots illustratingthe streaming velocity distribution under the free surface. treaming controlled by meniscus shape y direction at the observational plane near the hydrophobic angle (not shown in thepaper). So, the convergence induced downwelling on the observational plane will alsocontribute to a portion of e k .As with the experiments, the numerical simulations retains a high degree of symmetryin the symmetric case, but looses symmetry as the forcing amplitude increases in theantisymmetric case (figure 11). However, the loss of symmetry is less dramatic in thenumerical simulations than in the experiments. This is also due to momentum injectioncaused by convergence of the flow in the y -direction, which will accelerates the 2Drotational motion on the hydrophilic side.The numerical experiments were repeated with the surface streaming condition re-placed by a free-slip boundary. The four-vortex structure is lost in the symmetric caseand the kinetic energy drops by an order of magnitude in both cases. These resultsdemonstrate that the free surface is a significant source of momentum for the secondarycirculation and is important for determining its qualitative structure. 4. Conclusion We observe the streaming circulation excited by standing capillary-gravity waves usingPIV. It is found that the structure of the streaming circulation is controlled by the staticmeniscus shape. A semi-analytical theory is developed which explains the streamingpattern.The experiment is designed to inhibit Faraday waves, so the streaming circulation isdriven by resonant meniscus waves, rather than by Faraday waves as reported previously(Gordillo & Mujica 2014; Prinet et al. e k , scales like the forcing amplitude to the fourth power[i.e., e k ∼ ( a /g ) ]. In the symmetric case, the secondary flow retains a high degreeof left-right symmetry as forcing amplitude is increased. For the anti-symmetric case,the flow symmetry is broken, such an asymmetry comes from the exponential decay ofthe velocity from z = 0, so the places with higher mean elevation has greater velocityunder it. As a result, the anti-symmetric surface profile gives rise to the asymmetry ofstreaming circulation.8 Y. Huang et al. Using the strategy suggested by (Prinet et al. ∂ x ζ ∼ 5. Acknowledgement We are grateful to Jie Yu, Robert Wilson and Dongping Wang for stimulating dis-cussions. We also thank Jun-Qiang Shi and Ze Chen for their participation in the earlystage of the experiment. This work was supported by the National Science Foundationof China through Grant No. 11572230 and 11772235, and a NSFC/RGC Joint Grant No.11561161004. 6. Supplemental Materials Supplementary movies that show the streaming circulations produced by oscillatingmeniscus (symmetric/antisymmetric, a = 0 . g , f = 8 Hz) are available at .... Appendix A. Faraday threshold This experiment focuses on streaming driven by a natural cavity mode excited throughordinary resonance, rather than a Faraday wave excited by parametric instability. Theexperiment is therefore conducted in a stable zone of the Faraday instability. Here, wedemonstrate that the experiment is indeed within a stable zone.The wavenumbers, k lm , of linear modes in the working cell are k lm = π ( l L x + m L y ) , (A 1)where l and m are the mode numbers in the x - and y -directions, respectively. From theBenjamin & Ursell (1954), the mode amplitude a lm changes according to the differentialequation d a lm dT + [ p lm − q lm cos(2 T )] a lm = 0 , (A 2)which is known as Mathieu’s equation (Abramowitz & Stegun 1965). On this equation, treaming controlled by meniscus shape T = ωt is dimensionless time and the parameters p lm and q lm are p lm = 4 k lm tanh( k lm h ) ω (cid:18) g + σρ k lm (cid:19) , (A 3) q lm = 2 k lm a tanh( k lm h ) ω . (A 4)The mathematical character of the Mathieu’s equation is such that some modes grow ex-ponentially in time while others—in the stable zones—decay exponentially. The decayingmodes are not observable in this experiment, since their amplitudes rapidly become toosmall to measure. For the forcing frequency f = 8 Hz and amplitude a . g , theparameters in Mathieu’s equation are p ≈ . , p ≈ . ,q . , q . . For these values of the p lm the corresponding instability thresholds are q T ≈ . q T ≈ . q lm realized in the experiment arewell below the Faraday instability threshold and the observed (1 , 0) and (2 , 0) modes aretherefore ordinary resonant modes. Appendix B. Extrapolation method In this section, we describe the extrapolation method used to calculate the streamingvelocity distribution along the meniscus at z = ζ ( x ). Instead of extrapolating thesecondary flow velocity directly, we extrapolate the primary flow (i.e., the linear meniscuswave) velocity to ensure continuity of the first-order velocity. The primary flow velocitiesextrapolated to the static meniscus are u | z = ζ ≈ u | z =0 + ∂u ∂z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 ζ = − cos ωt ∞ X n A n k n [1 + k n ζ ] sin k n x, (B 1) w | z = ζ ≈ w | z =0 + ∂w ∂z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 ζ = cos ωt ∞ X n A n k n [1 + k n ζ ] cos k n x. (B 2)The velocity tangent to the free surface is the projection of (B 1) and (B 2) onto thetangential direction ˆ t : u k = ( u , w ) | z = ζ · ˆ t = − cos ωt ∞ X n A n k n k n ζ [1 + ( ∂ x ζ ) ] / (sin k n x − ∂ x ζ cos k n x ) ≈ − cos ωt ∞ X n A n k n (1 + k n ζ ) sin k n x, (B 3)where we have used the fact that ∂ x ζ ≪ ∂ x ζ ≪ 1, we estimate the tangential derivative of the tangential velocity as ∂ τ u k ≈ ∂ x u k ≈ cos ωt ∞ X n A n k n (1 + k n ζ ) (cid:0) ∂ xx ζ − k n (cid:1) cos k n x. (B 4)0 Y. Huang et al. Substituting (B 3) and (B 4) into (3.23), we find that the streaming velocity along thefree surface is u s = ( u s , w s ) = − ω ˜ u k ∂ τ ˜ u k ˆ t ≈ − ω ˆ ı ∞ X n,m A n A m k m k n (1 + k m ζ )(1 + k n ζ ) (cid:0) k n − ∂ xx ζ (cid:1) sin k m x cos k n x, (B 5)where we have used the facts that ˆ t · ˆ ı ≈ 1, and ˆ t · ˆ k ≈ z 0, thestreaming velocity is the same as for a flat surface. When z > 0, the vertical velocity atarbitrary position z ′ is estimated by w | z = z ′ ≈ w | z =0 + ∂w ∂z | z = z ′ ζ . (B 6)It should be mentioned that the extrapolation for the velocity does not converge forarbitrary order of the Taylor expansion. As A n converges like n − , the extrapolationpolynomial in (B 1) and (B 2) should be no higher than the second order. REFERENCESAbramowitz, M. & Stegun, I. A. Handbook of mathematical functions: with formulas,graphs, and mathematical tables . New York: Dover Publications. Antkowiak, Arnaud, Bremond, Nicolas, Le Dizs, Stphane & Villermaux, Emmanuel J. Fluid Mech. ,241–250. Batchelor, G. K. An introduction to fluid dynamics . Cambridge: Cambridge universitypress. Benjamin, T. B. & Ursell, F. Proc. R. Soc. Lond. A , 505–515. Carrin, L. M., Herrada, M. A., Montanero, J. M. & Vega, J. M. Phys. Rev. E , 033101. Chen, P., Luo, Z., Gven, S., Tasoglu, S., Ganesan, A. V., Weng, A. & Demirci, U. Adv. Mater. , 5936–5941. Cocciaro, B., Faetti, S. & Festa, C. J. Fluid Mech. , 43–66. Douady, S. J. Fluid Mech. , 383–409. Faraday, M. Proc. R. Soc. Lond. , 299–340. Francois, N., Xia, H., Punzmann, H., Fontana, P. W. & Shats, M. Nat. Commun. , 14325. Gordillo, L. & Mujica, N. J. Fluid Mech. , 590–604. Hocking, L. M. J. Fluid Mech. ,267–281. Holmedal, L. E. & Myrhaug, D. Cont. ShelfRes. , 911–926. Lesser, M.B. & Berkley, D. A. J. Fluid Mech. , 497–512. Longuet-Higgins, M. S. Proc. R. Soc. Lond. A ,535–581. Lucassen-Reynders, E. H. & Lucassen, J. Adv. ColloidInterfac. , 347–395. treaming controlled by meniscus shape Martn, E., Martel, C. & Vega, J. M. J.Fluid Mech. , 57–79. Martn, E. & Vega, J. M. J. Fluid Mech. , 203–225. Miles, J. & Henderson, D. Annu. Rev. Fluid Mech. , 143–165. Moisy, F., Bouvard, J. & Herreman, W. Euro. Phys. Lett. , 34002. Perlin, M. & Schultz, W. W. Annu. Rev. FluidMech. , 241–274. Punzmann, H., Francois, N., Xia, H., Falkovich, G. & Shats, M. Nat. Phys. , 658. Prinet, N., Gutirrez, P., Urra, H., Mujica, N. & Gordillo, L. J. Fluid Mech. , 285–310. Riley, N. Annu. Rev. Fluid Mech. , 43–65. Schneck, D. J. & Walburn, F. J. J. Fluids Eng. , 707–713. Strickland, S. L., Shearer, M. & Daniels, K. E. J. Fluid Mech.777