Characterization of a capillary driven flow in microgravity by means of
optical technique D. Fiorini, L. Carbonnelle, A. Simonini, J. Steelant, D.
Seveno and M. A. Mendez [1,2]Domenico Fiorini<EMAIL_ADDRESS>von
Karman Institute for Fluid Dynamics, Waterloosesteenweg 72, Sint-Genesius-
Rode, 1640, Belgium

02/20/2023 mm/dd/yyyy

# Characterization of a capillary driven flow in microgravity by means of
optical technique

Louis Carbonnelle  Alessia Simonini  Johan Steelant  David Seveno  Miguel
Alfonso Mendez  von Karman Institute for Fluid Dynamics, Waterloosesteenweg
72, Sint-Genesius-Rode, 1640, Belgium  KU Leuven, Dept. of Materials
Engineering, Leuven 3001, Belgium  ESTEC-ESA, Keplerlaan 1, Noordwijk, The
Netherlands

###### Abstract

The motion of a gas-liquid interface along a solid wall is influenced by the
capillary forces resulting from the interface’s shape and its interaction with
the solid, where it forms a dynamic contact angle. Capillary models play a
significant role in the management of cryogenic propellants in space, where
surface tension dominates the behaviour of gas-liquid interfaces. Yet, most
empirical models have been derived in configurations dominated by viscous
forces. In this study, we experimentally investigate the wetting of a low-
viscosity, highly wetting fluid in a reduced gravity environment. Our setup
consisted of a transparent and diverging U-tube in which capillary forces
sustain the liquid motion. Combining Particle Image Velocimetry (PIV) and
high-speed backlighting visualization, the experimental campaign allowed for
measuring the interface evolution and the velocity field within the liquid
under varying gravity levels. This work reports on the preliminary results
from the image velocimetry and shows that the velocity profile within the tube
is close to parabolic until a short distance from the interface. Nevertheless,
classic 1D models for capillary rise face difficulties reproducing the
interface dynamics, suggesting that the treatment of the surface tension in
these problems must be reviewed.

###### keywords:

Dynamic Wetting in Microgravity, Time-Resolved Particle Image Velocimetry,
Interface dynamics in capillary tubes

††volume: Volume x, Issue x, 2023

##  1 Introduction

Modelling the dynamics of a gas-liquid interface along a solid surface is
fundamental in wetting and dewetting (e.g. inkjet printing), coating flows
(e.g. slot die coating), and capillary-driven flows (e.g. capillary tubes).
These applications have spurred much of the research on the topic and have
often been analyzed using semi-empirical or theoretical correlations that link
the contact angle at the wall to the contact-line dynamics.

Classic empirical laws relate the contact angle to the velocity of the contact
line through the Capillary number (e.g. Voinov (1976); Jiang et al. (1979);
Bracke et al. (1989); Kistler (1993)), while other authors proposed more
complex correlations, in the form of differential equations, to take into
account the time history of the contact-line dynamics (e.g. Ting and Perlin
(1995); Bian et al. (2003)).

Dynamic contact angle correlations are essential to predict interface
dynamics. However, Fiorini et al. (2022) recently showed that classic quasi-
stationary models of the interface shape fail in situations where inertia
plays a significant role. This is the case in many applications in
microgravity, such as the sloshing of space propellants. A critical case study
for cryogenic propellant management in microgravity is the axial sloshing of a
highly wetting liquid in axisymmetrical containers. This test case was
analyzed by Utsumi (2000) and Luppes et al. (2009). Their results demonstrated
that the modelling of surface tension strongly impacts the prediction of
liquid displacement. Similarly, Li et al. (2020, 2018) conducted a CFD
analysis of the oscillations of the gas-liquid interface in a cylindrical tank
upon a step reduction in gravity and compared the results with the
experimental findings of Li et al. (2018). Their results showed that the
dynamic contact angle models significantly impact the rising velocity of the
contact line.

While these studies highlight the relevance of capillary forces on the motion
of the liquid interface in the absence of gravity and complex geometries,
other studies have addressed the fundamentals of dynamic wetting in
microgravity using dedicated capillary rise experiments. Stange et al. (2003)
performed a capillary rise experiment for a perfect wetting fluid on a large
cylindrical tube in a drop tower to reproduce microgravity conditions. They
showed that the interface reorientation and capillary rise consist of three
phases. The first is dominated by the inertia of the meniscus acceleration.
Viscous effects dominate the second, while capillary forces dominate the third
phase. The extension of these three phases depends on the duration of the
experiments. Wang et al. (2009) conducted a parametric study of capillary rise
in the same configuration to examine the impact of the contact angle and tube
size. The two parameters proved to be coupled, with the interface in large
tubes producing faster capillary rise at an earlier stage for low contact
angles. This behaviour was interpreted as the result of a competition between
viscous and inertial effects, with the former dominating the capillary rise in
small-diameter tubes and the latter affecting the initial stage of the rise in
large (inertia-dominated) tubes.

The aforementioned studies provide essential data for validating and deriving
dynamic contact angle correlations through 1D models of the capillary rise,
such as the one developed by Stange et al. (2003). However, these experiments
provide limited information on capillary-dominated conditions as they neglect
the role of the flow field beneath the interface, which, in turn, impacts the
dynamic contact angle. The velocity field underneath an advancing meniscus has
been observed by a few experiments in small capillary tubes, where the motion
of the liquid can be assumed to be viscous-dominated. For example, Nasarek et
al. (2008) and Waghmare et al. (2012) performed a micro-PIV experiment in a
vertical channel and analyzed the advancing motion of a meniscus of ethanol
and ethylene glycol, respectively. Both works showed that the common
assumption of Poiseuille flow holds until a certain distance from the
interface, below which the velocity profile flattens, and radial components
appear. Similarly, Ratz et al. (2023) analyzed the flow of a quasi-capillary
water channel in inertia-dominated conditions and showed that rolling motion
under the interface could be produced in advancing and receding conditions.

In this work, we use parabolic flights to study experimentally the capillary
rise and the interface dynamics of highly wetting fluids in microgravity. The
experimental conditions achieved through the flights provide a larger duration
of the capillary-dominated phase compared to previous studies and allow for
characterizing the relationship between the interface shape, its motion, and
the underlying flow field at different gravity levels.

The analyzed configuration consists of a diverging U-tube in a saturated gas
environment. This prevents evaporation, which can have a significant impact on
the wetting process, as shown in Waghmare et al. (2012). We use particle image
velocimetry to measure the velocity field beneath the meniscus and high-speed
visualization in back-lighting, combined with image processing, to measure the
interface evolution. In this article, we report on the PIV measurements of a
representative test case. Moreover, we present a simple 1D modelling of the
interface dynamics on the U tube and compare its prediction with the available
experimental data.

The results highlight the role of the capillary forces near the interface and
their impact on the capillary rise of highly wetting liquids. These results
are particularly interesting to the management of space propellants, which are
also characterized by low (static) contact angles.

##  2 Experimental Facility

The diverging U-tube (DUT) configuration analyzed in this work is shown in
Fig. 1. The experiments were carried out during the 78th ESA parabolic flight
campaign.

In the DUT, the liquid reservoir used in classic capillary rise experiments is
replaced by the larger side of the U-tube. This minimizes the impact of liquid
sloshing occurring during a parabolic flight. Moreover, capillary forces act
on both sides of the liquid column, creating a capillary pressure drop that
pulls the interface in opposite directions. This smooths the acceleration
profile compared to the step response in experiments with a single driving
interface, reducing the inertia-dominated phase and increasing the impact of
capillary forces on the interface motion. Finally, the lack of a tube entrance
reduces viscous losses due to sudden tube constriction, minimizing the effect
of viscous forces and reducing the duration of the viscous-dominated phase
identified by Stange et al. (2003).

Figure 1: The Figure on the left shows a schematic of the DUT test cell with
the characteristic dimensions. The Figure on the right shows a picture of the
DUT mounted on a rigid structure to avoid resonance vibrations in the quartz.

Figure 1 shows the installation of the DUT quartz tube on its robust structure
in aluminium. The structure consists of two plates joined by four cylindrical
beams via M8 screws. A resin is poured into two pockets of the structure to
clamp the tube. The system is designed to allow optical access while having
sufficient rigidity to avoid vibration, which could result in the cracking of
the quartz. The complete structure with the U tube and the resin weighs five
kg.

The two sides of the tube are connected with a flexible junction. This
prevents leakages and creates a closed environment in which saturated
conditions are reached before the experiments begin. The flexible is connected
to the plate with two fast connectors on the top. The assembly is then
installed on a larger breadboard containing the entire experimental equipment,
as shown in Fig. 2. A separate electronic rack contains all the electronic
devices necessary for the experiment. These include acquisition cards for the
acceleration and pressure signals connected to a laptop, accelerometer
conditioner, laser controller and camera controller connected to a desktop.

Figure 2: The figures on the left show a schematic of the optical set-up for
the PIV measurements. The back-lighting is performed using a LED screen behind
the U-tube’s aluminium frames. The Figure on the right shows the installation
on the parabolic flight breadboard: two experiments were carried out
simultaneously.

Two experiments were performed simultaneously, with a camera acquiring images
on each side of the two tubes. Both experiments used HFE7200
($\text{C}_{\text{4}}\text{F}_{\text{9}}\text{O}\text{C}_{\text{2}}\text{H}_{\text{5}}$)
produced by 3M Novec as test fluid. The fluid physical properties have been
characterized by Rausch et al. (2015) and evaluated at the experiment
temperature (298K). The gas phase is constituted by air saturated with HFE7200
vapor. The motion of each interface is recorded with a high-speed camera
(model JAY SP-12000-CXP4), acquiring grey-scale images at 300 fps. The active
region of all cameras is restricted to the central region of size $4096\times
n_{px}$ pixels where the lateral size $n_{px}$ is optimized according to the
tube geometry to achieve the highest acquisition frequency allowed by the
camera. All cameras mount objectives with 105mm focal lenses and are
positioned to acquire the full motion of the interface while spanning the
largest possible tube length. The final pixel size corresponds to $0.013\mu
m$.

As shown in Fig. 2 on the left, the PIV is carried out using a planar laser
sheet produced on one side of the tube. A laser stopper is introduced to avoid
optical disturbances on the other side of the tube. The high speed in
backlighting conditions is carried out using a LED screen behind the DUTs.
However, in this work, we solely focus on the PIV campaign. The fluid was
seeded with FMR-1.3 1-5 $\mu m$ Red Fluorescent Miscospheres from Cospheric.
The particles were illuminated with a continuous laser source (LMX-532 from
Oxxius) with a wavelength of 532 nm and a maximum power of 800 mW.

The experiment recordings start with the Pilot’s announcement that the plane
reached 40 degrees inclination. Shortly after the announcement, the gravity
levels drop from approximately 1.8$g$ to approximately 0$g$, and the two
liquid interfaces begin moving along the axis of the tube due to the
unbalanced capillary action at the interfaces. The image processing consists
of several stages. First, the images are compensated for the vibrations of the
camera objective, which appears as displacement of the image in the vertical
(z) and lateral (x) direction (max 4 px.) and without significant image
rotation. To compensate for the vibrations, we detect the outer edges of the
tube in each image via standard canny edge detection and compute the necessary
x and y correction by iteratively minimizing the difference of the global
overlap of the image with a reference image acquired before the experiment
start. In the second step, the PIV images are pre-processed using the Proper
Orthogonal Decomposition (POD)-based background removal introduced by Mendez
et al. (2017). The optical distortion due to the tube’s curved surface was
corrected using the method proposed by Darzi and Park (2017). This approach
corrects the radial distortion given the distance of the camera from the tube
center, the tube radius and the refraction index of the materials involved.
The method was validated in Fiorini et al. (2022) in similar size cylindrical
tube. The undistorted image is then mapped at the original camera resolution
using bicubic image interpolation.

Finally, the PIV interrogation is carried out using OpenPIV processing tool
(Liberzon et al. (2020)) and windows with a 2:1 aspect ratio, given that
velocity component v (along y) is much larger than the velocity component u
(along x). We use a Dynamic Region of Interest (DROI) to follow the gas-liquid
interface across the available Field of View (FOV). This allows focusing the
analysis always in the region below the interface. The DROI is identified by
performing standard gradient-based edge detection on the POD-filtered images
to detect the element of interest in the image (e.g. the interface). Spurious
points and outliers are eliminated through both masking of the tube solid
surfaces and unsupervised outlier detection using the Local Outlier Factor
(LOF) by Breunig et al. (2000). While this approach allows identifying the
region of the image where PIV can be applied, the light refraction at the
interface of the PIV images prevents extracting the interface contact angle
with high accuracy, for which the backlight technique is necessary. The
results of the contact angle analysis will be presented in the extended
version of this work. The PIV images are processed using adaptive iterative
multigrid interrogation (Scarano and Riethmuller (2000)) to retrieve the
velocity fields. The initial window size is set to $128\times 64$ px and
reduced over two steps to $32\times 16$ px. Outliers are removed based on the
signal-to-noise ratio measured in terms of peak-to-peak in the cross-
correlation map.

##  3 Modeling of the capillary rise

We describe the motion of the liquid’s center of mass inside the DUT with a 1D
integral model balancing the acceleration of the center of mass with the
friction, gravity and the interface capillary pressure drop. We derive the
model assuming a gentle curvature of the tube and a cylindrical coordinate
system $(\zeta,\eta)$ with $\zeta$ aligned with the tube axis and $\eta$
aligned along the tube radius as illustrated in Figure 3.

Figure 3: Equivalent cylindrical coordinate system for DUT where gentle
curvature of the tube is assumed.

We integrate the contributions of all forces along the liquid domain to obtain
the following second-order non-linear ODE:

$\rho V_{l}\ddot{Y}=-F_{g}(Y,a(t))+F_{v}(\dot{Y},Y)+F_{c}(\dot{Y},Y)\,,$ (1)

where $Y$, $\dot{Y}$ and $\ddot{Y}$ are respectively the position, the
velocity and the acceleration of the liquid column’s center of mass along the
tube axis $\zeta\in[-L/2,L/2]$. On the right-hand side, $F_{g}$ denotes the
gravitational force acting to the center of mass of the liquid column, $F_{v}$
denotes the viscous shear applied to the liquid column, and $F_{c}$ denotes
the net result of the capillary pressure due to the two gas-liquid interfaces
on the opposite sides of the liquid column. $V_{l}$ is the volume of liquid
within the tube and $a(t)=[a_{x}(t),a_{y}(t),g(t)]$ is the local acceleration
of the tube.

We denote with $z_{A}(Y)$ and $z_{B}(Y)$ the position of the two interfaces
along the curvilinear coordinate $\zeta$. The gravity term in (1) results from
the contribution of both the height difference $|z_{A}(Y)|-|z_{B}(Y)|$ and the
local acceleration $a(t)$.

The viscous force term in (1) is computed from two contributions: the first is
the integration of the viscous term in the Navier Stokes equations, here
denoted as $\tau$, along the liquid column $z_{A}(Y)-z_{B}(Y)$ and the radial
direction $\eta$; the second is an empirical correction to account for the
viscous losses at the tube’s bending. These two terms reads as follows:

$F_{v}(\dot{Y},Y)=\int_{z_{A}(Y)}^{z_{B}(Y)}\int_{0}^{R(\zeta)}\tau\,2\pi\eta\,d\eta\,d\zeta+C_{U}\,.$
(2)

where value of $C_{U}$ was computed from the correction proposed by Itō
(1969):

$C_{U}=8\pi\mu V_{0}\left(\pi
D_{U}\right)\left(0.1033De^{0.5}\left(\left(1+\frac{1.729}{De}\right)^{0.5}-\left(\frac{1.729}{De}\right)^{0.5}\right)^{-3}\right)\,,$
(3)

with $De$ the Dean number computed as $De=Re(V_{0})\sqrt{{2R}/{D_{U}}}$, with
$V_{0}=V(\zeta=0,t)$ the velocity at the centerline, $Re(V_{0})$ the local
Reynolds number based on the centerline velocity and $D_{U}$ the distance
between the tube axes.

Concerning the first term in (2), assuming axisymmetric flow, the diffusion
term $\tau$ can be written as

$\tau=\mu\left[\frac{1}{\eta}\frac{\partial
v}{\partial\eta}+\frac{\partial^{2}v}{\partial\eta^{2}}+\frac{\partial^{2}v}{\partial\zeta^{2}}\right]$
(4)

where $v=v(\zeta,\eta,t)$ is the instantaneous axial velocity at the
coordinate $\zeta,\eta$. In normal gravity conditions ($g=9.8m/s^{2}$) and for
small diameters, classic capillary rise models assume a Poiseuille velocity
profile everywhere in the channel (see for example Zhong et al. (2019)). With
this simplification, tested in this work for the DUT configuration, the
velocity profile can be written as

$v=v_{p}(\zeta,\eta,V(\zeta,t))=2V(\zeta,t)\left(1-\frac{\eta^{2}}{R(\zeta)^{2}}\right)\,.$
(5)

where $V$ denotes the mean velocity and $R(\zeta)$ the radius of the channel
as a function of the channel axial coordinate.

The capillary term is modelled as the net sum of the pressure drop on the two
interfaces, which in the analyzed wetting configuration pulls the liquid
towards opposite sides. The pressure drop results from the integration of
Laplace equation for each interface and reads

$F_{c}(\dot{Y},Y)=\int_{0}^{R(z_{A}(Y))}\Delta P_{A}(\eta)2\pi\eta
d\eta-\int_{0}^{R(z_{B}(Y))}\Delta P_{B}(\eta)2\pi\eta d\eta\,.$ (6)

In the absence of mass transfer and Marangoni effects, the capillary pressure
drops $\Delta P(\eta,t)_{A|B}$ become

$\Delta P(\eta,t)=\sigma\nabla\cdot\mathbf{n}\,$ (7)

where $\nabla\cdot\mathbf{n}$ is the interface curvature, with $\nabla$ the
nabla operator and $\mathbf{n}$ the local normal vector. If the interface is
treated as spherical, the curvature becomes $\nabla\cdot\mathbf{n}=2/R_{s}$,
with $R_{s}$ the sphere’s radius. With this assumption, introducing the
dependency on the apparent and dynamic contact angle $\theta$, one has
$R_{s}(\zeta)\approx R(\zeta)/cos(\theta)$ and thus the capillary pressure
drops on each side reads:

$\Delta
P_{A}=\frac{2\sigma}{R(z_{A}(Y))}cos(\theta_{A})\quad\mbox{and}\quad\Delta
P_{B}=\frac{2\sigma}{R(z_{B}(Y))}cos(\theta_{B})$ (8)

##  4 Results and Discussion

Fig. 4 shows the acceleration profiles for the presented experiments during
the $\approx 38$s constituting a parabola. The figure on the top shows the
envelope of (vertical) gravitational acceleration while the figure on the
bottom report on the lateral accelerations (see Fig. 2).

Figure 4: Acceleration profile for the selected experiment

The micro-gravity phase begins approximately at $t=10$ s and ends at $t=30$s.
This is preceded and followed by a hypergravity phase. The lateral
accelerations are non-negligible during the hypergravity phases and can have
an important effect also during the microgravity phase. Figure 5a shows a
sample image acquired during the experiment. All images are pre-processed
using the POD background removal; the result for the snapshot in 5a is shown
in 5b. The velocity field is computed using the multigrid approach with final-
size windows (32, 16). The result is shown in Fig 6 for an extracted axial
location near the interface.

Figure 5:  From top to bottom, the Figure shows the raw PIV images and the
result from the background removal. All figures are rotated for plotting
purposes.

In what follows, we focus on the velocity field close to the interface up to 5
mm below the interface. The measured velocity fields in this area are shown in
Fig. 6 using a moving reference frame with rising velocity $\dot{Y}$. The red
curve in this plot shows the result of the interface detection described in
the previous section. These results show that the liquid accelerates towards
the contact line as it approaches the interface. This lateral acceleration is
due to mass conservation: since the interface shape remains unvaried and hence
all its points move at a constant velocity, the flow must adapt from a nearly
parabolic profile to a flat one. A similar result was reported by Waghmare et
al. (2012) for purely capillary-dominated flow with no evaporation. The
classic counter-rotating vortices shown in similar configurations by Davis et
al. (1974) and Ratz et al. (2023) are not visible in this case.

At the scale of interest, namely at about the half-width of the channel, the
flow appears to be different from the classic wedge flow postulated by Voinov
(1976) in the derivation of its classic theoretical model for the dynamic
contact angle. This result might thus question the validity of such a model
for the investigated test case.

Concerning the stream-wise variation of the velocity profile, at a distance of
approximately 3 mm, the velocity profile appears nearly parabolic in all
snapshots. At larger acceleration (see snapshot e) the flow inertia appears to
push the validity of the parabolic profile closer to the interface (up to
nearly 1 mm from it).

Figure 6: From left to right, the plot shows the axial velocity profile in a
moving reference frame with rising velocity $\dot{Y}$ and at extracted
locations in the axis of the tube. Each plot also shows the contour of the
velocity magnitude beneath the advancing meniscus.

Figure 7 compares the velocity profiles at various distances from the
interface in this latter snapshot, characterized by larger inertia. The
validity of the self-similar assumption is illustrated by comparing the
extracted velocity profiles with the classic parabolic profile from the
assumption of Poiseuille flow. The progressive flattening of the velocity
profile starts appearing at approximately 1 mm from the interface and the
parabolic shape of the velocity profile increasingly deteriorates at closer
distances from the interface. Fig. 8 use contour plots to provide an
indication of a lack of similarity with the Poiseuille flow, measured as

$P_{\%}=\frac{|v_{P}-v|}{{v}(0,t)}\times 100\,,$ (9)

where $v(\eta,\zeta)$ is the measured velocity and
$v_{P}(\zeta,\eta,V(\zeta,t))$ is the local velocity under the assumption of
parabolic profile in (5). This parameter is defined up to the lowest point of
the interface, above which the wetting meniscus begins. The figure shows that
at the latest stages of the capillary rise, the velocity field is close to
that of a Poiseuille flow until a close distance from the interface.

Figure 7: Comparison of experimental radial velocity profiles, sampled at
various axial positions along the channel, with theoretical Poiseuille
profile. Figure 8: From left to right, the contour plots show the difference
of the velocity profile with e corresponding Poiseuille profile at each axial
location of the channel. The line plots show the increase of disagreement with
the velocity profile moving from the bulk flow toward the meniscus. Figure 9:
The Figure shows the comparison of the experimental interface position with
the prediction achieved by solving (1). The gravity profile corresponding to
the time frame examined by the experiment is also shown in the back.

To conclude the analysis of the velocity field, it appears that the assumption
of fully developed Poiseuille flow holds until a short distance from the
interface, especially when the flow is strongly dominated by inertia as in the
last snapshots of Figure 8. These results validate the assumption of a self-
similar profile in the theoretical model presented in (1), particularly in the
computation of the viscous term. Figure 9 compares the model prediction and
the experimental data. The axis on the left reports the interface position
while the one on the right reports the instantaneous vertical acceleration.

The figure illustrates on the right axis the examined gravity profile and on
the left the comparison of the model solution with experimental data on the
interface position during the experiment. The model correctly shows the
oscillation of the interface velocity due to minor oscillation in the
gravitational level at about $t\in[6-8]$s. Since all terms in equation 1 have
a comparable order of magnitude during the interface evolution, the accurate
modelling of all terms is relevant to achieve a satisfactory prediction of the
interface history. Considering that the assumption of self-similarity appears
reasonable, the sources of the discrepancy are found either in modelling the
pressure losses due to the bend or the treatment of the capillary forces. The
first appears less plausible, given that the employed correlation has been
well-calibrated in ground experiments. Therefore, the mismatch is addressed
here to the modelling of the capillary term and the impact of the different
gravity levels on the interface shape.

##  5 Conclusion

In this work, we investigated the capillary rise dynamics in a divergent
U-tube setup under micro-gravity conditions using time-resolved Particle Image
Velocimetry (PIV). The configuration produces a flow driven by the difference
in capillary forces between the two sides of the tube. We followed both the
interface position and the velocity field during the capillary rise and
compared the results with a simple 1D integral model based on self-similarity.
The objective was to verify the validity of the self-similar velocity profile
assumption and test the accuracy of a simplified 1D formulation of the
capillary-driven flow.

The PIV measurements showed the assumption of a parabolic velocity profile
until a distance of $\approx 1$ mm from the interface, thus validating one of
the main simplifying assumptions in these flows. The flow profile near the
meniscus change from parabolic to nearly flat, with a strong acceleration near
the contact line. This suggests that the flow field significantly differs from
the wedge flow postulated in the derivation of theoretical models of dynamic
wetting, which leads to contact angle correlations of the form
$\theta_{D}=f(Ca)$.

###### Acknowledgements.

D. Fiorini is supported by Fonds Wetenschappelijk Onderzoek (FWO), Project
number 1S96120N. This work was supported by the ESA and BELSPO via the Prodex
program and by the ESA-GSTP "Physical and Numerical Modeling of cryogenic
sloshing for Space applications" (Contract No. 4000129315/19/NL/MG). We would
like to thank also Novaspace for their support and the fruitful discussions.
Finally, we acknowledge Michel Bavier for the support in the experiment
preparation and execution.

## References

  * Bian et al. (2003) Bian, X., Schultz, W.W., and Perlin, M., Liquid slug motion and contact lines in an oscillatory capillary tube, Journal of Fluid Dynamics (under revision), http://wwwpersonal. umich. edu/schultz/Manuscripts/BianSchultzPerlin. pdf, vol. 146, p. 151, 2003\.
  * Bracke et al. (1989) Bracke, M., De Voeght, F., and Joos, P., 1989. The kinetics of wetting: the dynamic contact angle. Trends in Colloid and Interface Science III. Springer, pp. 142–149.
  * Breunig et al. (2000) Breunig, M.M., Kriegel, H.P., Ng, R.T., and Sander, J., Lof: identifying density-based local outliers, Proceedings of the 2000 ACM SIGMOD international conference on Management of data, pp. 93–104, 2000.
  * Darzi and Park (2017) Darzi, M. and Park, C., Optical distortion correction of a liquid-gas interface and contact angle in cylindrical tubes, Physics of Fluids, vol. 29, no. 5, p. 052004, 2017.
  * Davis et al. (1974) Davis, S.H. , On the motion of a fluid-fluid interface along a solid surface, Journal of Fluid Mechanics, vol. 65, no. 1, pp. 71–95, 1974.
  * Fiorini et al. (2022) Fiorini, D., Mendez, M.A., Simonini, A., Steelant, J., and Seveno, D., Effect of inertia on the dynamic contact angle in oscillating menisci, Physics of Fluids, vol. 34, no. 10, p. 102116, 2022.
  * Itō (1969) Itō, H., Laminar flow in curved pipes, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 49, no. 11, pp. 653–663, 1969.
  * Jiang et al. (1979) Jiang, T.S., Soo-Gun, O., and Slattery, J.C., Correlation for dynamic contact angle, Journal of Colloid and Interface Science, vol. 69, no. 1, pp. 74–77, 1979.
  * Kistler (1993) Kistler, S.F., 1993. Hydrodynamics of wetting, Wettability. Berg, J. (Ed.). Marcel Dekker, New York, New York, Ch. 6, pp. 311–430.
  * Li et al. (2020) Li, J.C., Lin, H., Li, K., Zhao, J.F., and Hu, W.R., Dynamic behavior in a storage tank in reduced gravity using dynamic contact angle method, Microgravity Science and Technology, vol. 32, no. 6, pp. 1039–1048, 2020\.
  * Li et al. (2018) Li, J.C., Lin, H., Zhao, J.F., Li, K., and Hu, W.R., Dynamic behaviors of liquid in partially filled tank in short-term microgravity, Microgravity Science and Technology, vol. 30, no. 6, pp. 849–856, 2018.
  * Liberzon et al. (2020) Liberzon, A., Lasagna, D., Aubert, M., Bachant, P., Kaufer, T., Jakirkham, Andreas, B., Boyko, V., Cameron, D., Joe, B., Tomerast, and Ranleu, Openpiv/openpiv-python: Open-piv-python (v0. 22.2) with a new extended search piv grid option, 2020.
  * Luppes et al. (2009) Luppes, R., Helder, J.A., and Veldman, A.E., 2009. The numerical simulation of liquid sloshing in microgravity. Computational Fluid Dynamics 2006. Springer, pp. 607–612.
  * Mendez et al. (2017) Mendez, M., Raiola, M., Masullo, A., Discetti, S., Ianiro, A., Theunissen, R., and Buchlin, J.M., Pod-based background removal for particle image velocimetry, Experimental Thermal and Fluid Science, vol. 80, pp. 181–192, 2017.
  * Nasarek et al. (2008) Nasarek, R., Wereley, S., and Stephan, P., Flow field measurements near a moving meniscus of a capillary flow with micro particle image velocimetry ($\mu$piv), International Conference on Nanochannels, Microchannels, and Minichannels, Vol. 48345, pp. 863–871, 2008.
  * Ratz et al. (2023) Ratz, M., Fiorini, D., Simonini, A., Cierpka, C., and Mendez, M.A., Analysis of an unsteady quasi-capillary channel flow with time-resolved piv and rbf-based super-resolution, Journal of Coatings Technology and Research, vol. 20, no. 1, pp. 27–40, 2023.
  * Rausch et al. (2015) Rausch, M.H., Kretschmer, L., Will, S., Leipertz, A., and Froba, A.P., Density, surface tension, and kinematic viscosity of hydrofluoroethers hfe-7000, hfe-7100, hfe-7200, hfe-7300, and hfe-7500, Journal of chemical & engineering data, vol. 60, no. 12, pp. 3759–3765, 2015.
  * Scarano and Riethmuller (2000) Scarano, F. and Riethmuller, M.L., Advances in iterative multigrid piv image processing, Experiments in fluids, vol. 29, no. 1, pp. S051–S060, 2000.
  * Stange et al. (2003) Stange, M., Dreyer, M.E., and Rath, H.J., Capillary driven flow in circular cylindrical tubes, Physics of fluids, vol. 15, no. 9, pp. 2587–2601, 2003.
  * Ting and Perlin (1995) Ting, C.L. and Perlin, M., Boundary conditions in the vicinity of the contact line at a vertically oscillating upright plate: an experimental investigation, Journal of Fluid Mechanics, vol. 295, pp. 263–300, 1995.
  * Utsumi (2000) Utsumi, M., Low-gravity sloshing in an axisymmetrical container excited in the axial direction, J. Appl. Mech., vol. 67, no. 2, pp. 344–354, 2000\.
  * Voinov (1976) Voinov, O.V., Hydrodynamics of wetting, Fluid dynamics, vol. 11, no. 5, pp. 714–721, 1976.
  * Waghmare et al. (2012) Waghmare, P.R., Sen, D., Nobes, D.S., and Mitra, S.K., Experimental investigation of the flow front behind a liquid-air interface for capillary flow, International Conference on Nanochannels, Microchannels, and Minichannels, Vol. 44793, American Society of Mechanical Engineers, pp. 797–806, 2012.
  * Wang et al. (2009) Wang, C., Xu, S., Sun, Z., and Hu, W., Influence of contact angle and tube size on capillary-driven flow under microgravity, AIAA journal, vol. 47, no. 11, pp. 2642–2648, 2009.
  * Zhong et al. (2019) Zhong, X., Sun, B., and Liao, S., Analytic solutions of the rise dynamics of liquid in a vertical cylindrical capillary, European Journal of Mechanics-B/Fluids, vol. 78, pp. 1–10, 2019.