14th International Symposium on Particle Image Velocimetry – ISPIV 2021 August 1–5, 2021 Time Resolved PIV of the Flow Field underneath an accelerating meniscus M. Ratz1,2, D. Fiorini2, A. Simonini2, C. Cierpka1, M. A. Mendez2∗ 1 Technische Universität Ilmenau, Institute of Thermodynamics and Fluid Mechanics, Ilmenau, Germany 2 von Karman Institute for Fluid Dynamics, Enviromental and Applied Fluid Mechanics Department, Brussels, Belgium ∗ mendez@vki.ac.be Abstract We present an experimental analysis of the flow field near an accelerating contact line using time-resolved Particle Image Velocimetry (TR-PIV). Both advancing and receding contact lines are investigated. The analyzed configuration consists of a liquid column that moves along a vertical 2D channel, open to the atmosphere and driven by a controlled pressure head. Large counter-rotating vortices were observed and an- alyzed in terms of the maximum intensity of the Q-field. To compute smooth spatial derivatives and improve the measurement resolution in the post-processing stage, we propose a combination of Proper Orthogonal Decomposition (POD) and Radial Basis Functions (RBF). The RBFs are used to regress the spatial and tem- poral structures of the leading POD modes, so that “high-resolution” modes are obtained. These can then be combined to reconstruct high-resolution fields that are smooth and robust against measurement noise and amenable to analytic differentiation. The results show significant differences in the flow topology between the advancing and the receding cases despite velocity and acceleration of contact lines are comparable in absolute values. This suggests that the flow dynamics are tightly linked to the shape of the interface, which significantly differs in the two cases. 1 Introduction The dynamics of a gas-liquid interface moving along a solid wall plays an essential role in many wet- ting/dewetting processes (e.g. inkjet printing), coating applications (e.g. slot die coating) and capillary driven flows (e.g. liquid absorption in porous media or liquid pumping in microgravity). The capillary force plays a major role in these processes. This force depends on the curvature of the interface, which is linked to the angle formed at the contact line, known as contact angle. Two main chal- lenges exists in the modelling of these flows. First, the value of the contact angle cannot be predicted in the continuous fluid mechanics framework: on the contrary, this is a necessary boundary condition that must be imposed to solve for the interface shape, and hence for the fluid flows on both the gas and the liquid sides. The most common solution to the problem is the implementation of empirical and semi empirical correlations for the (macroscopic) contact angle as a function of the contact line velocity (Hoffman, 1975; Kistler, 1993). The limitations of these formulations are described in Fiorini et al. (2021). The second challenge is that the very notion of moving contact line is at odds with the notion of no-slip boundary conditions that viscosity imposes to the liquid close to the contact line. This is the well known hydrodynamic paradox, which has motivated many theoretical and experimental works on the flow field near moving contact lines. A solution to this paradox has been proposed by Huh and Scriven (1971), who described the flow motion sufficiently close to the contact line as a creeping flow obeying the biharmonic equation. Within this theory, the velocity fields compatible with advancing or receding contact lines for the case of wetting fluid configuration (θ < 90o) are characterised by a splitting streamline as sketched in Figure 1. In an advancing contact line (see Figure 1(a)), the fluid required to displace the interface is injected from the bulk (split injection configuration) while the opposite occurs in a receding contact line (split ejection configuration, see Figure 1(b)). Only few experimental investigation of these flow patterns have been presented (Fuentes and Cerro (2005); Nasarek et al. (2008); Zimmerman et al. (2010)) in the literature, mostly focused on quasi steady conditions. (a) (b) Figure 1: Schematic sketch of the stream lines in proximity of the contact line for a wetting configuration (θ < 90o) in case of an advancing contact line (left) and a receding contact line (right). The flow pattern on the left is known as stream line split injection, the flow pattern on the right is known as stream line split ejection. Images are taken from Fuentes and Cerro (2005). While sufficiently close to the contact line the dominance of viscosity gives a sound theoretical justifica- tion to these patterns, it remains unclear what is their macroscopic impact on the dynamics of the interface and on the flow field far from it. At large Reynolds number or in the presence of strong acceleration, i.e. when inertia becomes important, the flow field sketched in Figure 1 might exist only very close to the interface but still have a strong impact to the flow at larger distances. The practical implications are numerous, especially considering that flow field and contact angle are intimately linked (see Savelski et al. (1995)). In the classic capillary rise problem, for example, the most common models are based on the assumption that the flow field is fully developed (Washburn, 1921; Levine et al., 1976; Quéré, 1997), but no theoretical nor empirical law can identify the distance from the interface below which this assumption ceases to be valid (see Nasarek et al. (2008)). This work presents an experimental investigation of the flow field near a contact line moving at relatively high velocities and subject to large accelerations. The configuration of interest is a narrow rectangular channel in which a vertical column of liquid is driven into motion by a controlled pressure step. The gas liquid interface in contact with the walls produces two dynamic menisci which forces the interface to form two dynamic contact angles. Both advancing and receding configurations are investigated. The flow field was measured via time-resolved Particle Image Velocimetry (TR-PIV), in a region of interest that follows the interface in its displacement. The measurement reveal the formation of two large counter rotating vortices with time-varying intensity. The estimation of the vortex intensity was carried out in terms of Q-field (Hunt et al., 1988). To accurately compute the required spatial derivatives, the velocity field is interpolated using a com- bination of Proper Orthogonal Decomposition (POD) and Radial Basis Function (RBF) regression. This approach was first proposed by (Karri et al., 2009; Raben et al., 2012) to compute spatial derivatives, and it is used here also for super-resolution. In particular, the RBF is used to smooth, filter and interpolate the spatial and temporal structures of the POD modes. The velocity field obtained from the reconstruction of the high-resolution POD modes enables super-resolution with robust outlier removal and analytic calcula- tion of spatial and temporal derivatives. An adaptation of the tool for computing pressure field is currently in preparation (Sperotto et al., 2021). The method is reminiscent of Kriging enhanced gappy POD (Gunes et al., 2006), but differs in the use of the RBF as opposed to the kernel-based Kriging. Section 2 shows the configuration of the narrow, two-dimensional channel in which the flow beneath the dynamic contact lines is observed. Section 3 presents the data processing for the PIV interrogation as well as the proposed regression tool. This algorithm is applied to a simple test case in Section 4.1 and compared to traditional interpolation methods. The experimental results are discussed in Section 4.2 and conclusions are given in Section 5. (a) (b) Figure 2: Fig (a) Sketch of the channel showing the area of the TR-PIV measurements. Fig (b) Picture of the experimental setup with the camera (1), the channel (2), the reservoir (3), the release valve (4) and the pressure ports (5 and 6). 2 Experimental Setup and Test Cases A sketch of the experimental setup is shown in Figure 2a. This consists of a rectangular channel with a width of δ = 5mm much smaller than its depth L = 250mm. The channel height is H = 150mm and is open to atmosphere. This channel is connected to a pressurized chamber in the bottom, sustaining the flow via a pressure difference ∆p. To reproduce an advancing contact line (i.e. interface moving upwards, along the y-axis), a positive pressure difference ∆p is suddenly introduced in the chamber by the opening of a fast release electronic valve connected to a pressure line. This valve opens linearly in 0.2 s. To reproduce a receding contact line (i.e. interface moving downward), the opposite procedure is followed: the chamber is initially at a ∆p= 1200 Pa and the release valve is suddenly opened to the atmosphere, resulting in ∆p = 0 Pa. A picture of the experimental setup is shown in Figure 2b. The camera is placed perpendicularly to the channel’s flow cross section and illuminated with a laser sheet from the right side. The injection/removal of air during the pressurisation/depressurisation used in the advancing/receding experiments is performed on two sides of the chamber in order to ensure a symmetric flow and to minimise the entry effects. Together with the camera (1), the picture shows the channel (2) and the pressurized chamber (3), the fast opening vale (4), and the two pressure ports (5 and 6). Water is used as a fluid and the walls are made of plexiglass. The fluid is seeded with Red Fluorescent Polymer Microspheres from Thermo Fisher Scientific with a diameter of 12 µs. The flow is observed with a TR-PIV system from Dantec dynamics. The particles are illuminated with the laser (DM40-527-DH Nd: YLF from Photonics Industries) with a wavelength of 527 µm and a maximum pulse energy of 40 mJ at a frequency of 1 kHz. Images are recorded with a SpeedSense Ethernet M310 camera having a resolution of 1280×800 px at a maximum frame rate of 3260 Hz. An objective lens with a focal length of 105 mm is used to get an optical magnification of 60 px/mm. The images are acquired in single frame mode using the Dantec’s software Dynamic Studio. The expo- sure time of the camera is set to 250 s and the frame rate is set to 1200 Hz. The time resolved configuration allows to capture about 60 snapshots during the rising/advancing experiment and about 100 during the de- scending/receding experiments For the rising interface, the flow is observed at the very top of the channel, close to the maximum height. In the first case, the reservoir is at an initial pressure of 0 Pa and the pressure differce is 1500 Pa. This is referred to as ‘Test Cases 1’. The second case, referred to as ‘Test Case 2’, has an initial pressure of 200 Pa, the pressure difference is 1200 Pa. These different initial conditions result in the same final height of the meniscus. The descending interface is observed at multiple positions inside the channel by changing the height of the camera between measurements. Additionally, the release speed of the liquid is varied between cases to investigate the effects of different accelerations. The release speed is controlled by opening the release valve to the surrounding air at different speeds. 3 Data Processing: Smoothing, Super-Resolution and Q-Fields The PIV images are pre-processed using the POD-based background removal introduced by Mendez et al. (2017). The PIV interrogation is carried out using an in-house adaptation of OpenPIV (Liberzon et al., 2020). In particular, due to the nature of the flow, two additional features were implemented. First, because of the large aspect ratio of the flow (the velocity component v, along y, is much larger than the velocity component u, along x), rectangular interrogation windows were implemented. This allows for a better sampling of the flow in the direction where the most substantial gradients are expected without deteriorating the signal-to-noise ratio in the cross-correlation maps. The second implementation is a dynamic Region of Interest (ROI) to follow the gas-liquid interface across the available Field of View (FOV), as schematically illustrated in Figure 2(a). At each image pair k at time tk, the velocity profile along the edge 1-2 (denoted as y1,2) is used to compute the average velocity vk = ∫ δ/2 −δ/2 v(x,y1,2, tk)dx and displace vertically the FOV in the following image pair by v∆t. This allows focusing the processing always in the same region below the interface. Adaptive iterative multigrid interrogation (Scarano and Riethmuller, 2000) is used to calculate the ve- locity fields. For the advancing contact line experiments, the initial window size is set to 256×64 px and reduced over two steps to 64×16 px. For the receding contact line experiments, the initial size is 64×64 px, and reduced to 24×24 px in two iterations. Outliers are removed based on the signal-to-noise ratio measured in terms of peak to standard deviation in the cross-correlation map. The post-processing of the PIV fields begins with a POD-based filter using a classic truncation ~u(x0, t0) = R ∑ i=1 σi~φi(x0)ϕi(t0) , (1) where R� nt is the truncation index, σi is the amplitude of the i-th POD mode having spatial structure ~φi(x0) and temporal structure ϕi(t0). Note that x0 ∈Rns×1 and t0 ∈Rnt×1 here denote the spatial and temporal grids in which the PIV field is available, reshaped as column vectors for computational convenience. Because the flow fields here are bidimensional, given nx and ny the number of points along x and y, we have ns = 2nxny. The spatial structures~φ(x0) = (φu(x0),φv(x0)) collects the dominant velocity fields, computed by pro- jecting the velocity field ~u = (u,v) on the POD temporal basis. The details of the POD computation can be found elsewhere (e.g. Mendez et al. (2019, 2020a)) and are thus omitted here. The main idea behind the super-resolution strategy developed in this work is to perform Radial Basis Function (RBFs, see Fornberg and Flyer (2015)) regression of the spatial and temporal structures of the decomposition, and then use a reconstruction of high-resolution POD modes to rebuild the velocity field. The RBF regression consists in writing both of these bases as linear combination of RBFs bases, i.e.: ~φi(x0) = nφ ∑ j=1 wφ i j γ φ j(x0|xi, j,Σφ j) and ϕi(t0) = nϕ ∑ j=1 wϕ i, j γ ϕ j (t0|t j,Σϕ j) , (2) where wφ i, j and wϕ i, j are the set of weights defining the regression functions, γ φ j(x0|x j,Σφ j) are the nφ radial basis functions in space and γ ϕ j (t0|t j,Σϕ j) are the nϕ radial basis functions in time. These have collocation points x j and t j and shape parameters Σφ j and Σϕ j . Note that the coefficients wφ i, j must be defined as vectors, i.e. wφ i, j = (wφ u,i, j,w φ v,i, j) since the same radial basis γ φ j is used for both components (φu,φv). However, because every snapshot is reshaped as a column vector, regardless of whether this collects a vector or a scalar field, we keep the same notation as for the interpolation of the temporal structures. In this work, we consider Gaussian RBF of the form: γ φ j(x0|x j,Σφ j) = exp ( −(x0−x j) T ( 1/2σ2 x 0 0 1/2σ2 y ) (x0−x j) ) ; γ ϕ j (t0|t j,Σϕ j) = exp (−(t0− t j) 2 σ2 t ) , (3) i.e. Σφ j = diag(1/2σ2 x ,1/2σ2 y) and Σϕ j = 1/2σ2 t . For both space and time regressions, the collocation point and the shape parameters are defined a priori and the regression is solved once the weights are identified. Reshaping all bases functions as columns of matrices Γφ(x0) ∈ Rns×nφ and Γϕ(t0) ∈ Rnt×nϕ , collecting all the unknown weights into column vectors wφ ∈ Rnφ×1 , wϕ ∈ Rnφ×1 and collecting all the entries of~φi(x0) and ϕi(t0) into column vectors φ0 ∈ Rns×1 and ϕ0 ∈ Rnt×1, the weights are solution of classic least square problems. Using a standard Tikhonov regularization to mitigate the risks of overfitting (Mendez et al., 2020b), the solution reads: Ax = b → x = ( AT A+αI )−1 AT b, (4) where A = Γφ(x0), x = wφ and b = φi for the regression in space and A = Γϕ(t0), x = wϕ and b = ϕi for the regression in time. The regularization parameter α is taken as ε ||A||F , with ε = 1e− 11 and ||A||F the Frobenious norm of A. Note that the inverse in equation 4 can be precomputed and the system solved for all the modes in a single step. Once the weights are computed, equation (2) can be used on any arbitrary mesh grid x and any time discretization t. Then, the interpolated field reads ~u(x, t) = R ∑ i=1 σi~φi(x)ϕi(t) with { ~φi(x) = ∑ nφ j=1 wφ j γ φ j(x|x j,Σφ j) ϕi(t) = ∑ nϕ j=1 wϕ j γ ϕ j (t|t j,Σϕ j) (5) The derivatives of the flow can now be computed analytically by simply replacing the RBF with their required partial derivatives. For example, the spatial derivative ∂xu can be computed as: ∂xu(x, tk) = ∂x ( R ∑ i=1 σiφu,i(x)ϕi(tk) ) = R ∑ i=1 σi∂xφu,i(x)ϕi(tk) with ∂xφu,i(x) = nφ ∑ j=1 wφ j ∂xγ φ j(x|x j,Σφ j) . (6) This derivative is analytically available because ∂xγ φ j is analytically available by differentiating eq.3. Finally, as all derivatives can be accurately computed, the strength of the vortices detected in the PIV fields can be measured in terms of Q-field (Hunt et al., 1988) Q = 1 2 (||Ω||2F −||S||2F) =− 1 2 ( (∂xu)2 +2∂yu∂xv+(∂yv)2 ) (7) where S = 1/2(∇V +∇V T ) and Ω = 1/2(∇V −∇V T ) the symmetric and anti symmetric portions of the velocity gradient tensor. 4 Results 4.1 Synthetic Test Case Before analysing the experimental data, we consider a simple synthetic test case to test the robustness of the implemented RBF regression against noise. The flow field chosen for the test is ~u = (u,v) = (sin(x) · cos(y),−cos(x) · sin(y)) over a domain x = [0,π]× [0,π]. This field is sampled on a Cartesian grid with nx = ny = 11 points and is shown in Figure 3(a). To simulate different noise levels, we add a white noise field ~w(x) ∈ [0,1] with relative intensity p, i.e. ~uδ(x) =~u(x)(1+ p~w(x)). The perturbed fields with p = 0.5 and p = 1 are shown in Figures 3(b) and 3(c) respectively. The spatial regression is performed using a total of nΦ = 40× 40 Gaussian RBFs (see eq 3) with σx = σy = 2. The collocation points are uniformly distributed over a Cartesian grid in the domain x. The RBF regression is then used to re-sample the velocity field onto a much finer grid, i.e. nx = ny = 22. The results for the three fields in Figures 3(a), 3(b) and 3(c) are shown in Figures 3(d), 3(e) and 3(f) respectively. In all these cases, the RBF regression removes the noise and reconstructs the flow field on the much finer grid. We close this subsection by comparing the performances of the RBF regression with two classic interpolants: a linear interpolation and a cubic spline interpolation, as available from the Python’s library Scipy. The performances were evaluated in terms of Mean Absolute Error (MAE), defined as: MAE = 1 nx ny nx·ny ∑ i=1 |vGT,i− vi| (8) (a) (b) (c) (d) (e) (f) Figure 3: Illustrative synthetic test case to analyze the robustness of the RBF regression: a vortex field is sampled over a coarse grid. Fig (a) - (c): velocity field on the coarse grid with noise level (defined in the text) p = 0,0.5,1. Fig (d)-(f): results of the re-sampling of the RBF regression on a much finer grid for the cases (a)-(c). Where vi is the prediction of the interpolant and vGT,i is the analytic flow field. The results are shown in Figure 4 as a function of the noise parameter p. While at low levels of noise the three approaches are comparable, the RBF is significantly more robust as p is increased. 4.2 PIV Fields The PIV velocity fields were processed retaining a total of R = 25 POD modes in equation (1). The temporal structures of the modes were approximated with nϕ = 500 RBFs with σt = 0.01 while nφ = 900 RBFs were used for the spatial structures (see equation (3)). For the regression in space, the RBFs were collocated over a regular grid of nBx×nBy = 60×15 and anisotropic kernels with σx = 10 and σy = 15 were selected. This allows to efficiently account for the high aspect ratio of the flow. No re-sampling was performed in the time domain since the available time resolution was sufficient for the scope of this work. The spatial resolution of the PIV field, i.e. x0 in equation (1)-(3), has a resolution of ∆x = 0.12mm and ∆y = 0.48mm for the rising test cases and a resolution of ∆x = ∆y = 0.22mm for the descending test cases. The resulting RBF regression is used to re-sample the velocity field over a grid (x in equation (5)) with a resolution of ∆x = 0.04mm and ∆y = 0.16mm for the rising test casess, and ∆x = ∆y = 0.07mm for the descending test cases. Figure 5 shows the results of the experiments with the rising interface, i.e. the advancing contact line problem. Figure 5(a) and 5(b) show the acceleration and the maximum Q-field within the ROI as a function of the mean velocity within the channel. The labels correspond to the ones explained in Section 2. The mean velocity is calculated by zero padding the velocity field at the walls and taking the mean of the average velocity of the bottom five rows and the acceleration is computed from the time derivative of the mean 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p 0.0 0.1 0.2 0.3 0.4 M A E RBF Linear Cubic Figure 4: Mean absolute error after interpolation over the level of noise. ’RBF’ refers to the implemented interpolator, ’linear’ and ’cubic’ an interpolator using bivariate splines of first and third order. velocity. A clear trend is visible: the Q-field is increasing with decreasing mean velocity. Three instantaneous velocity fields are shown in Figures 5(c) - 5(e), corresponding to the three points marked in Figure 5(a) and 5(b). For plotting purposes, the velocity fields are slightly high-pass filtered by using a Gaussian filter with σx = σy = 15 and a truncation after 2.5 standard deviations. The quiver plot only shows every second column for the rising interface to avoid overcrowding the image. The Q-field is also shown in each snapshot for quantitative analysis and the axis aspect ratio is set to one. The upper horizontal boundary of the ROI is placed approximately 0.2 mm from the interface at the center of the channel, and this is the case for the three shown snapshots. However, because of light reflections and diffraction some of the rows close to the interface are sometimes considered invalid and hence this distance increases up to 1.5 mm. This might explain the noisy trends in Figure 5(a) and 5(b). In each case, two counter rotating vortices are visible close to the walls. The one on the right is rotating clockwise and the one on the left counterclockwise. The vortices extends considerably along the vertical direction but only over a small distance in the cross-stream direction. At a distance of 1 mm from the wall, these have no appreciable influence on the velocity field. The initial pressure was high enough to produce a strong acceleration, as shown in Figure 5(a). While the rolling motion of the flow is still present, the significant acceleration and high velocities push the stream-line split injection pattern and its vortices towards the wall. It is worth noticing that the velocity fields look similar even though Figure 6(c) and 6(d) have largely different accelerations. Referring to the expected theoretical flow field from Figure 1(a), it appears that the stream-line split injection forms a small angle with the walls. As a result, the rolling motion expected by the viscous-capillary balance is confined to a narrow region of the flow. Figure 6 shows the same results for the cases with the descending interface. The velocities and accel- erations in Figure 6(a) and 6(b) have a different sign because the interface moves downward. A similar trend as for the rising interface can be observed, with the magnitude of the Q-field increasing as the velocity decreases. However, the reader should note that a decreasing velocity in this case means that the absolute value of the velocity is increasing. Another observation is the increased strength of the Q-field, with the maximum value being almost an order of magnitude above the maximum value for the rising interface. The vortices in Figure 6(c) - 6(e) show a different behaviour than the ones for the rising interface. Besides the reversed rotation, as expected from the theoretical flow topology in Figure 1, their centre is much closer to the centre of the channel, and the rolling motion extends much further in both the x and y-direction. In the case of the receding contact line, the stream-line split ejection (see Figure 1(b)) forms a larger angle with the wall than for the advancing contact line. The large difference cannot be justified by the velocity of the contact line, which in the slowest rising test cases is comparable to the fastest falling test cases, nor by its acceleration, which are also comparable in the cases in 6(c) and the one in Figure 5(e). 5 Conclusions and Perspectives Time-Resolved PIV measurements of the flow field underneath a dynamic meniscus have been carried out in a two-dimensional channel. Both advancing and receding contact lines were investigated. In both cases, two counter-rotating vortices were observed below the meniscus. In addition, super-resolution of the velocity 240 260 280 300 320 v [mm/s] −2500 −2250 −2000 −1750 −1500 −1250 a [m m /s 2 ] (c) (e) (d) Test Case 1 Test Case 2 (a) 240 260 280 300 320 v [mm/s] 0 200 400 600 Q m a x [1 /s 2 ] (c) (e) (d) Test Case 1 Test Case 2 (b) −2.50 −1.25 0.00 1.25 2.50 x[mm] 0 100 200 (c) −2.50 −1.25 0.00 1.25 2.50 x[mm] 0 100 200 300 (d) −2.50 −1.25 0.00 1.25 2.50 x[mm] 0 100 200 300 (e) Figure 5: Results for the rising interface. Fig (a) - (b) Acceleration and maximum Q-field over the mean velocity for different test cases. The blue case corresponds to an initial pressure of 1400 Pa and the orange one to 1500 Pa. Fig (c) - (e) Q-field contour with the high-pass filtered velocity field. The aspect ratio of the plot is set to one. fields was achieved using a regression scheme that combines an analytic approximation of the spatial and temporal structures of the leading POD modes using RBFs. This allowed analytical differentiation and accurate computation of the Q-fields. In both advancing and receding cases, the strength of the vortices depends on the mean velocity and the acceleration of the flow, and whether the contact line is advancing or receding. The latter is especially important because the splitting stream-line forms a different angle with the wall and has the largest impact on the flow field. Significant differences were observed between advancing/receding wetting configurations even when the modulus of velocity and acceleration of the contact lines were comparable. The main difference be- tween the two cases was found in the interface’s shape (not shown here): this appeared nearly flat in the advancing cases and characterized by large menisci, with small contact angles, during the receding test cases. Future work will complement the presented investigation with an analysis of the interface shape in both experiments. Acknowledgments This project was funded by Arcelormittal Maizières Research SA in the framework of the ‘Minerva Project’, and the authors thank Jean-Michel Mataigne for many fruitful discussions on dynamic wetting. D. Fiorini is supported by Fonds Wetenschappelijk Onderzoek (FWO), Project number 1S96120N. −200 −180 −160 −140 −120 −100 −80 −60 −40 v [mm/s] 0 200 400 600 800 1000 1200 a [m m /s 2 ] (c) (d) (e) 0 mm 40 mm 120 mm (a) −200 −180 −160 −140 −120 −100 −80 −60 −40 v [mm/s] 0 500 1000 1500 2000 2500 3000 Q m a x [1 /s 2 ] (c) (d) (e) 0 mm 40 mm 120 mm (b) −2.50 −1.25 0.00 1.25 2.50 x[mm] −500 0 500 1000 1500 (c) −2.50 −1.25 0.00 1.25 2.50 x[mm] −200 0 200 400 600 800 (d) −2.50 −1.25 0.00 1.25 2.50 x[mm] −300 −100 100 300 500 700 (e) Figure 6: Results for the descending interface. Fig (a) - (b) Acceleration and maximum Q-field over the mean velocity for different test cases. The labels refer to the vertical position of the camera, where 0 mm refers to the part of the channel where it meets the reservoir. Fig (c) - (e) Q-field contour with the high-pass filtered velocity field. The aspect ratio of the plot is set to one. References Fiorini D, Mendez AM, Simonini A, Seveno D, and Johan S (2021) A study on the effect of inertia on dynamic contact angles using an inverse method. to appear in Journal of colloid and interface science Fornberg B and Flyer N (2015) Solving pdes with radial basis functions. Acta Numerica 24:215–258 Fuentes J and Cerro R (2005) Flow patterns and interfacial velocities near a moving contact line. Experi- ments in Fluids 38:503–510 Gunes H, Sirisup H, and Karniadakis GE (2006) Gappy data: To krig or not to krig?. Journal of Computa- tional Physics 212 212:358–382 Hoffman RL (1975) A study of the advancing interface. i. interface shape in liquid—gas systems. Journal of Colloid and Interface Science 50:228 – 241 Huh C and Scriven L (1971) Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. Journal of Colloid and Interface Science 35:85–101 Hunt J, Wray A, and Moin P (1988) Eddies, streams, and convergence zones in turbulent flows. in Study- ing Turbulence Using Numerical Simulation Databases, 2. Proceedings of the 1988 Summer Program, Stanford, California, USA, June 27 - July 22 Karri S, Charonko J, and Vlachos P (2009) Robust wall gradient estimation using radial basis functions and proper orthogonal decomposition (pod) for particle image velocimetry (piv) measured fields. Measure- ment Science and Technology 20:045401 Kistler SF (1993) Hydrodynamics of Wetting. in Wettability. chapter 6, pages 311 – 430. CRC Press Levine S, Reed P, Watson E, and Neale G (1976) A theory of the rate of rise of a liquid in a capillary. in Colloid and interface science. pages 403–419. Academic Press Liberzon A, Lasagna D, Aubert M, Bachant P, Käufer T, jakirkham, Bauer A, Vodenicharski B, Dallas C, Borg J, tomerast, and ranleu (2020) Openpiv/openpiv-python: added synthetic image generator Mendez M, Raiola M, Masullo A, Discetti S, Ianiro A, Theunissen R, and Buchlin J (2017) Pod-based background removal for particle image velocimetry. Experimental Thermal and Fluid Science 80:723– 729 Mendez MA, Balabane M, and Buchlin JM (2019) Multi-scale proper orthogonal decomposition of complex fluid flows. Journal of Fluid Mechanics 870:988–1036 Mendez MA, Hess D, Watz BB, and Buchlin JM (2020a) Multiscale proper orthogonal decomposition (mPOD) of TR-PIV data—a case study on stationary and transient cylinder wake flows. Measurement Science and Technology 31:094014 Mendez MA, Pino F, and Fiore M (2020b) Machine learning for fluid mechanics: Challenges, opportunities and perspectives. in Optimization Methods for Computational Fluid Dynamics, VKI Lecture Series. von Karman Institute Nasarek R, Wereley S, and Stephan P (2008) Flow Field Measurements Near a Moving Meniscus of a Capillary Flow With Micro Particle Image Velocimetry (µPIV). in Proceedings of the Sixth International ASME Conference on Nanochannels, Microchannels and Minichannels, Darmstadt, Germany, June 23-25 Quéré D (1997) Inertial capillarity. Europhysics Letters 39:533–538 Raben SG, Charonko JJ, and Vlachos PP (2012) Adaptive gappy proper orthogonal decomposition for par- ticle image velocimetry data reconstruction. Measurement Science and Technology 23:025303 Savelski M, Shetty S, Kolb W, and Cerro R (1995) Flow patterns associated with the steady movement of a solid/liquid/fluid contact line. Journal of Colloid and Interface Science 176:117–127 Scarano F and Riethmuller M (2000) Pod-based background removal for particle image velocimetry. Exper- iments in Fluids 29:51–60 Sperotto P, Mendez M, and Pieraccini S (2021) A meshless method to measure pressure fields via radial basis functions. to appear in Experiments in Fluids Washburn EW (1921) The dynamics of capillary flow. Physical Review 17:273–283 Zimmerman J, Weislogel M, and Tretheway D (2010) Micro-PIV Measurements near a Moving Contact Line. in Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition, Vancouver, British Columbia, Canada, November 12-18 Introduction Experimental Setup and Test Cases Data Processing: Smoothing, Super-Resolution and Q-Fields Results Synthetic Test Case PIV Fields Conclusions and Perspectives