Benjamin Marchetti

Laurence Bergougnoux

Elisabeth Guazzelli

Falling clouds of particles in vortical flows

The coupling between particle-particle and particle-fluid interactions is examined by studying the sedimentation of clouds of spheres in a model cellular flow at a small but finite Reynolds number. The model flow consists of counter-rotating vortices and is aimed at capturing key features of the vortical effects on particles. The dynamics of clouds settling in this vortical flow is investigated through a comparison between experiments and point-particle simulations.

Introduction

In many natural phenomena or industrial applications, heavy particles are transported in complex flows. The flow structures may happen to promote the stirring and dispersion of the particles. A representative example in geophysics is the cloud or plume of ash particles coming from a pyroclastic volcanic eruption that is thrown into the atmosphere and which can spread over extremely large distance depending on wind speed and direction. But the opposite can also take place and the flow configuration may contribute to the focussing and accumulation of particles within specific regions of the flow. A typical example in the natural environment is the observed patchiness of plankton concentration caused by wind-induced Langmuir cells occurring at the surface of lakes and oceans. The objective of this paper is to explore how a collection (i.e. a cloud) of heavy particles evolves under gravity in a complex flow. The key question is whether the cloud maintains a cohesive entity or disintegrates and spreads. Most of the flow structures occurring in the environmental examples mentioned above contain coherent whirls. The present work considers a simple vortical flow, specifically a cellular flow consisting of counter-rotating vortices, and addresses the interaction of these controlled vortices with the settling cloud. The focus here is on flow regimes where inertia is small but can become finite.

A significant body of studies has been dedicated to the dynamics of settling clouds in quiescent fluids (see e.g. Noh & Fernando 1993;Nitsche & Batchelor 1997;Machu et al. 2001;[START_REF] Bosse | Numerical simulation of finite Reynolds number suspension drops settling under gravity[END_REF]Metzger et al. 2007;Subramanian & Koch 2008;Pignatel et al. 2011). Different regimes have been identified depending on the magnitude of the particle and cloud Reynolds numbers (Subramanian & Koch 2008). We define the particle Reynolds number as Re a = ρ f U S a/µ, where U S = 2(ρ p -ρ f )a 2 g/9µ is the Stokes velocity of an individual particle of radius a and density ρ p settling in a fluid of viscosity µ and density ρ f under the gravity acceleration g, and the cloud Reynolds number as Re c = ρ f U c R c /µ, where U c ≈ N 0 U S 6a/5R c is the Stokes velocity of a spherical cloud of radius R c containing N 0 particles settling in the same quiescent fluid. When both Re a and Re c 1, the settling of the cloud lies in the Stokes regime. While falling under gravity, the cloud is seen to undergo an internal toroidal circulation similar to what is found for a spherical drop of heavy fluid settling in an otherwise lighter fluid (Hadamard 1911;[START_REF] Rybczyński | Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium[END_REF]). However, this cloud becomes unstable even in the complete absence of inertia and without the need to perturb its initial shape. It first remains roughly spherical with a leakage of particles in a vertical tail and then evolves into a torus which breaks up into two droplets in a repeating cascade. When inertia is finite, two macro-and a micro-scale inertial regimes are subsequently observed. When inertia is increased, the falling cloud transitions first to a regime dominated by macro-scale inertia when the cloud Reynolds number becomes of order one, i.e. Re c ∼ 1. The subsequent transition toward the micro-scale inertial regime occurs when the individual particle wakes are interacting within the cloud boundaries, i.e. when the inertial length is of the order of the cloud size, e.g. = a/Re a ∼ R c . In both inertial regimes, the cloud deforms into a flat torus that eventually destabilises and breaks up into a number of secondary droplets but particle leakage is much weaker if not null. While this evolution resembles that observed in the Stokes regime, the physical mechanisms involved are qualitatively different. Whereas the inertial cloud evolution is strongly determined by the importance of wake-mediated interactions, the key feature of the Stokes cloud is the chaotic motion of the particles which leads to escapes from the cloud internal circulation and to particle leakage. Simulations using point-particle approaches, which contain the minimal physics of the multibody particle interactions, capture these dynamics (Nitsche & Batchelor 1997;[START_REF] Bosse | Numerical simulation of finite Reynolds number suspension drops settling under gravity[END_REF]Metzger et al. 2007;Pignatel et al. 2011).

If the particle cloud settles now in a flowing fluid instead of a quiescent fluid, there are interactions not only between the particles but also between the particles and the local spatial structures of the flow (e.g. large vortices). The literature on sedimentation of heavy particles in non-uniform flows, and specially in random or turbulent flows, has shown the important effect of these vortical structures on the local particle transport and concentration, in particular through a phenomenon known as preferential sweeping where particle paths accumulate at the periphery of the vortical structures (see e.g. [START_REF] Wang | Settling Velocity and Concentration Distribution of Heavy-Particles in Homogeneous Isotropic Turbulence[END_REF]Aliseda et al. 2002;[START_REF] Toschi | Lagrangian properties of particles in turbulence[END_REF]Balachandar & Eaton 2010). Preferential sweeping is generally considered responsible for the turbulence to increase the settling velocity of inertial particles but other mechanisms, such as vortex trapping and loitering or nonlinear drag effects, have also been proposed for, on the contrary, reducing the settling velocity (see e.g. Nielsen 1993;Mei 1994;[START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF][START_REF] Akutina | Experimental evidence of settling retardation in a turbulence column[END_REF]. These features can be also seen using a model cellular flow of counter-rotating vortices. At low particle Reynolds number, individual particles settle at their Stokes velocity simply augmented by the local fluid velocity (Stommel 1949). Their trajectories can be classified according to the magnitude of the ratio of the Stokes velocity, U S , to the vortex velocity, U 0 , i.e. the parameter W = U S /U 0 . When W > 1, the trajectories are essentially straight vertical line as the particles settle through the vortex array without being mostly influenced by it. For W ≈ 1, the particles are affected by the local flow and the trajectories can exhibit zigzagging motions depending on their position of release within the vortex array. For smaller values of W , the particles can be suspended in the flow. When inertia is increased, the dynamics of single individual particles has been described by the Boussinesq-Basset-Oseen equation (Gatignol 1983;Maxey & Riley 1983). The major influence of inertia on the motion is that the particles cannot be resuspended in the flow and are seen to settle out even when they are trapped momentarily into a vortex (Maxey 1987). This effect has been observed in experiments using a cellular flow field created by electroconvection [START_REF] Bergougnoux | The motion of solid spherical particles falling in a cellular flow field at low Stokes number[END_REF]. The experiments show that, for small values of the Stokes number, St (defined as the ratio of the particle response time to the characteristic time of the flow), added mass and history forces are inconsequential and that the dominant forces are buoyancy and drag provided that the Stokes drag is replaced by a nonlinear drag depending of the particle Reynolds number. Collective effects between the particles may also affect this settling but they do not have been thoroughly addressed. The role of particle clusters and their association with settling velocity enhancement has been pointed in Stokes sedimenting flows [START_REF] Good | Settling regimes of inertial particles in isotropic turbulence[END_REF] but also more recently in the turbulent regime (Huck et al. 2018).

The objective of the present work is to tackle the interplay between the multibody particle interactions and the interaction between the particles and the spatial structures of the flow. This coupling is examined for a cloud settling in a cellular flow field which is a simple model flow capturing key features of vortical effects on the particles. In the experiments described in § 2, electroconvection is used to generate a two-dimensional array of controlled vortices and clouds of particles are released and tracked in this vortical flow. The observed cloud dynamics is compared in § 4 against point-particle simulations described in § 3. Conclusion are drawn in § 5.

Experiments

The experimental apparatus consists of a cell made of Plexiglas (of 50 cm height, 38 cm width, and 4 cm depth) filled with an aqueous mixture of citric acid (a certain amount of UCON R oil can be added to increase the fluid viscosity), as depicted in figure 1. The vortical flow is created by electromagnetic convection. The magnetic field is produced by a checkerboard of permanent square magnets (2 × 2 cm 2 ) placed against the back wall of the cell. An electrical current is generated between two carbon electrodes placed on the opposite small sides of the cell. The coupling between the magnetic field and the uniform electric current gives rise to an electromagnetic force in the fluid. This induces a flow of counter-rotating vortices having the same size as the magnets and an intensity controlled by the magnetic field of the permanent magnets, the intensity of the electric current, and the viscosity of the fluid. Further details can be found in [START_REF] Bergougnoux | The motion of solid spherical particles falling in a cellular flow field at low Stokes number[END_REF] and Lopez & Guazzelli (2017).

The flow generated by electroconvection is characterised by particle image velocimetry (PIV). For this purpose, the fluid is seeded by tracer particles (hollow spheres with diameter ≈ 15 µm and density ≈ 1.4 g.cm -3 from Dantec Measurement Technology) and is illuminated by a green laser sheet which can be positioned in the vertical or horizontal planes. A camera is focused on the illuminated particles which scatter the light and records two images separated in time by typically 1/7 to 1/15 s. These images are processed to find the velocity-vector map of the flow field using the Matlab PIV software DPIVsoft (Meunier & Leweke 2003). The resulting three-dimensional flow field possesses a quasi two-dimensional periodic structure of counter-rotating vortices of size L = 2 cm (i.e. with a spatial wavenumber k = 2π/2L) in the vertical plane having an intensity which reaches a maximum at x ≈ 5 mm away from the back wall and then decays rapidly. This cellular flow is close to a Taylor-Green cellular flow when the flow Reynolds number is small, i.e. for Re k = U 0 k -1 /(µ/ρ f ) < 1, see figure 2 (a). For larger values of Re k , the spatial structure can be slightly distorted and three-dimensional, see figure 2 (b). It eventually becomes unstable for Re k > 15.

Clouds are prepared by mixing a weighted amount of spherical particles with the same fluid as that filling the cell, see tables 1 and 2 describing the characteristics of the different fluids and particles which have been used. This suspension mixture is dropped from the top of the cell using a custom-made syringe positioned at x ≈ 5 mm, i.e. at the time-evolution of the position of the centre of mass of the cloud, X, Y , and Z, and of some other quantities such as its projected surface in the horizontal plane, Σ. The detection of the cloud dynamics is undertaken until the cloud begins to break, i. e. when the cloud starts to bend to break into pieces. The time-evolutions of the clouds have been examined for different combinations of particles and fluid mixtures listed in table 3 which also indicates different dimensionless parameters different values of vortex velocities, U 0 . Note also that the cloud size is always smaller (by a factor ten) than the size of the vortices. For each experimental condition listed in table 3, typically 4-5 experimental runs have been performed but only trajectories which are not too close to the walls and do not have a large x-extension are retained. Typical three-dimensional trajectories of centres of mass of clouds are depicted in figure 3.

(P = ka, W , Re a , Re k , Re c , N 0 , /R c , Q = kR c ).

Numerical modelling

We consider N 0 spherical particles initially randomly distributed in a sphere of radius R c (or alternatively in a prolate spheroid of same volume to mimic the initial shape of the cloud obtained in the experiments) settling under gravity in a vortical flow field u vortex characterised by a spatial wavenumber k and a vortex intensity U 0 . We adopt the simplest model containing the minimum physics to describe the problem at low Reynolds number in which particles are represented by identical point forces. Within this approximation, the velocity, ṙα i , of a particle α located at a position r α i is its own Stokes velocity U S incremented by the local fluid velocity at r α i (Stommel 1949) and the sum of the velocity disturbances (only represented by its far-field portion, i.e. the Stokeslet) coming from the other particles β located at a position r β i (see e.g. Metzger et al. 2007). Using k -1 as the length scale and U 0 as the velocity scale, the dimensionless equations of motion (with dimensionless quantities denoted with a hat) are written as

rα i = ûvortex i (r α i ) + W δ i3 + 3 4 P W β =α δ i3 rαβ + rαβ i rαβ 3 (r αβ ) 3 , (3.1)
where

r αβ i = r α i -r β i and r αβ = |r α i -r β i |.
We have used the dimensionless parameters P = ka and W = U S /U 0 .

To test the effect of a vertical wall on the cloud dynamics, we have followed [START_REF] Blake | A note on the image system for a stokeslet in a no-slip boundary[END_REF] and implemented a mirror image system of the point forces enabling to satisfy the no-slip condition on the wall boundary. The typical evolution with or without the wall is essentially identical as pointed by My lyk et al. (2011) for a cloud falling in a quiescent fluid. We have verified that, for a distance to the wall larger than a few cloud radii as achieved in the experiments, the effect happens to be insignificant.

When inertia becomes finite, equation (3.1) is no longer valid. However, if the Stokes number (i.e. the ratio of the particle response time, ∝ ρ p a 2 /µ, to the flow time scale, ∝ 1/kU 0 , St ∝ a 2 ρ p kU 0 /µ) is still small (St ≈ 2 10 -3 for case C2a and ≈ 4 10 -3 for case C2b), the assumption of linearity can be maintained but instead of a Stokes drag a nonlinear drag may be used, e.g. a Schiller-Naumann drag [START_REF] Bergougnoux | The motion of solid spherical particles falling in a cellular flow field at low Stokes number[END_REF].

As a first estimate, the far-field Stokes interactions can also be replaced by the steady far-field Oseen interactions (i.e. the Oseenlet) in equation (3.1) to encompass the inertial wake interactions between particles (Subramanian & Koch 2008;Pignatel et al. 2011). This is however a crude approximation as assuming steady Oseen wake interactions is far from being justified (see e.g. Lovalenti & Brady 1993). The new dimensionless equation of motion then reads

rα i = ûvortex i (r α i ) + W δ i3 + 3 4 P W β =α rαβ i (r αβ ) 2 2 ˆ rαβ (1 -Ê) -Ê + Ê rαβ δ i3 , (3.2)
where the inertial screening length is = a/Re p and E = exp -(1

+ r αβ 3 r ) r αβ 2
. Knowing the initial positions of the N 0 particles at time t = 0, the sets of equations (3.1) or (3.2) represent a close system of 3N 0 coupled ordinary differential equations which can be numerically integrated for a cellular flow u vortex inferred from interpolating the three-dimensional PIV measurements of the flow field described in § 2. The integration has been performed using an explicit Runge-Kutta method of order (4)5 (the 'dopri5' integrator of the 'ode' solver in Python). To optimise and run the Python code with a large number of point-particles, we use the Numba package to translate the mobilitymatrix function written in Python into a fast native machine code. The numerical trajectory of each particles of the initial spherical cloud can then be obtained and analysed. The generated images of the clouds can also be treated in a similar manner as in the experiments to give the trajectory and shape of the numerical clouds for a close comparison examination.

Results and comparison

In the following, we present the experimental results of the time-evolution of the cloud and compare them to the numerical predictions obtained using the point-particle approaches introduced in § 3. This latter numerical modelling also enables access to some dynamical features which are not amenable in the experiments. We investigate the Stokes regime but also the effect of increasing finite inertia. Before embarking in these analyses, it is important to consider the various dimensionless parameters involved in the problem. For Stokes clouds settling in a quiescent fluid, the only relevant parameter is the initial number of particles, N 0 . For Stokes clouds settling in a vortical flow of intensity U 0 and size k -1 , two additional dimensionless numbers come into play, the velocity ratio W = U S /U 0 and the size ratio P = ka or alternatively Q = kR c . When inertia is increased, it is important to consider additionally the cloud Reynolds number, Re c , and the particle Reynolds number, Re a , or alternatively the dimensionless inertial length k or /R c . It is important to mention that the Stokes number, St ∝ a 2 ρ p kU 0 /µ (≈ 2 -4 10 -3 for the most inertial cases C2a and C2b), and size ratio, P = ka (≈ 0.01 -0.03), of the individual particles are always kept small in the present study. Of course, the Stokes number of the cloud itself is larger, typically (R c /a) 2 larger, leading to values ≈ 1 for the most inertial cases C2a and C2b. The cloud radius is always smaller than the spatial wavelength of the vortical structure (Q = kR c ≈ 0.2 -0.6) because of the small xextension wherein the cloud can be dropped and for which the vortex intensity can be kept constant away from the back wall (see figure 2).

General evolution of the clouds

The typical evolution of a cloud settling in the vortical flow in the Stokes regime (case A1b) is shown in the chronophotography of figure 4 (a). The cloud is seen to settle along the downstream flow of the successive vortices and to display zigzagging motions due to the modulations caused by the periodic cellular flow. It does not maintain a spherical shape as it is slightly stretched in the elongational region of the flow, in particular close to the stagnation points surrounding the purely rotational regions at the centres of the vortices. A significant leakage of particles is observed at its rear and the released particles move with the vortical flow and thus can be trapped into the vortices as the ratio of their Stokes velocity to the flow velocity, W (= 0.0017), and their Reynolds number, Re a (= 10 -4 ), are both very small. No break-up is observed as the travel distance of the cloud is not long enough for this phenomenon to happen. The cloud falls across 12 vortices (≈ 100 times its radius) and it would necessitate for the cloud to fall at least 500 times its radius for break-up to occur in the Stokes regime (Metzger et al. 2007). Similar qualitative evolution is found between the point-particle simulation using the Stokeslet approximation and the experiment in this regime as evidenced by comparing figure 4 (a) and (b). However, the experiments show a mostly rounded shape of the cloud while the simulations give more like a mushroom shape. This discrepancy is observed for case A1b having a cloud with a large number of particles (see table 3) and may be due to excludedvolume effects not accounted for in the point-particle simulations. The behaviour of the cloud for case B1b is displayed in figure 5 and presents similar features as for case A1b since inertia is still very small. The cloud deformation by the flow structure is however This is even more significant when inertia becomes finite for case C2b shown in 6. The cloud is highly deformed and eventually widens in the upward elongational of the flow and breaks into multiple pieces. Point-particle simulations using an Oseenlet (instead of Stokeslet) approximation can capture this phenomenon. 4.2. Cloud trajectory more quantitative comparison between experiments and simulations can be obtained by examining the time evolution of the vertical, Z, and horizontal, Y , coordinates of the centre of mass, its vertical velocity, U z and the cloud projected surface in the plane normalised by its initial value, Σ/Σ 0 . a typical Stokes case (B1a) depicted in figure 7, Z(t) is approximatively a straight while Y (t) exhibits oscillations due to the modulation induced by the vortical flow. The mean settling velocity of the cloud is similar to that observed when falling in a quiescent fluid, U c ≈ N 0 U S 6a/5R c as shown in figure 7 (a). Its vertical velocity U z (t) and the normalised projected surface Σ(t)/Σ 0 experience periodic oscillations. The cloud down (accelerates, respectively) while simultaneously expanding (shrinking, respectively) when it falls through the successive upward (downward, respectively) elongational portions of the flow. Numerical simulations using the Stokeslet approximation are also presented in figure 7 for N 0 2500 corresponding to the experimental value. The different curves correspond to simulations having different initial positions for the cloud centre of mass which are chosen within the experimental accuracy of the cloud initial location. More precisely, the initial position is shifted along the vertical or horizontal directions by the experimental error ∆ on the initial location, as indicated in the legend of graph (a). The set of obtained curves encloses the experimental data, and in particular, the amplitude and phase of the oscillations are captured in graphs (b), (c), and (d) of figure 7. Dispersion and phase shift between the different curves are seen to increase slightly with time because of the sensitivity to the local velocity field, in particular at the stagnation saddle points, as observed previously for individual particles [START_REF] Bergougnoux | The motion of solid spherical particles falling in a cellular flow field at low Stokes number[END_REF]).

(Y0, Z0 -∆) (Y0, Z0 + ∆) (Y0 + ∆, Z0) (Y0 -∆, Z0) k Y Σ/Σ0 Uz/U0 (b) (c) (d) g Y Z 1 Uc/U0
In the finite-inertia case (C2b), the trajectories can be followed only over a distance of ≈ 4 -5 vortices since break-ups (indicated by the symbols) quickly occur, as shown in figure 8 for a typical example. The vertical position Z(t) again follows approximately a straight trajectory with a slope ≈ U c while the horizontal position Y (t) presents oscillations. The normalised projected surface Σ(t)/Σ 0 experiences similar periodic oscillations. Conversely, the vertical velocity U z (t) shows oscillations but with a doubled period compared to those of Y (t). This is a sign of the successive slowdowns and accelerations during the fall of the cloud through a vortical structure. This period doubling was not distinctly seen in the Stokes case of figure 7 (c). Numerical simulations using the Oseenlet approximation (with N 0 = 500 to match the experimental condition) capture these behaviours. Some discrepancy is observed for Σ(t)/Σ 0 . It may be due to experimental error in the determination of Σ coming from difficulties to discriminate between the particles within or without the cloud (this is enhanced after some settling distance when the tail behind the cloud due to particle leakage becomes important).

Comparison between behaviours in the viscous and inertial cases is shown in figure 9 where snapshots of the numerical clouds are shown during settling for cases (a) B1a (N 0 = 2500) and (b) C2b (N 0 = 500). The perturbed streamlines are represented together with the original PIV flow fields. The vortices are pushed apart when the cloud is falling in a downward portion of the flow between the two vortices while they are pulled closer when the cloud sweeps around the vortical structure when there is an upward flow between the two vortices. The oscillations between the expansions and contractions of the cloud when settling in the successive upward and downward, respectively, portions of the flow are clearly seen. This phenomenon seems to be enhanced in the Oseenlet simulation wherein the upward elongational flow favours the cloud expansion.

Particle leakage from the cloud

Since the evaluation of particle leakage from the cloud is difficult to achieve in the experiments, we have chosen to rely on numerical simulations. To identify the particles located inside the cloud, a connected-component-labeling algorithm has been used to detect connected regions in the binary digital images and to identify blob regions, in particular the cloud entity. First, we construct a boolean matrix containing the coordinates of the particles with an element resolution corresponding to the interparticle distance. Second, we use the connected-component-labeling algorithm (function measure.label in Python skimage library) which identifies blobs of adjacent elements and labels the different blobs. Then, the cloud entity is determined as the blob having the highest coordinates along z; the other blobs of particles are considered as leaking from the cloud. Figure 10 shows the percentage (N 0 -N )/N 0 of particles that have leaked away from the cloud as a function of time. The data correspond to averages over 10 numerical simulations having different initial particle distributions using the Stokelet (cases A1a and B1b) and the Oseenlet (case C2b) models.

Figures 10 (a obtained for Stokes clouds settling in quiescent fluids (Metzger et al. 2007). This contrasts with the scaling as tU c /R c which seems to be obtained in the inertial case. This again evidences the strong increase of leakage by the vortical flow when inertia becomes finite.

Figures 10 (e) and (f) show that increasing the Reynolds number of the flow intensifies the leakage, in both viscous (for N 0 = 2500) and inertial (for N 0 = 500) cases. It is worth noticing that the percentage of leakage presents oscillations induced by the modulation of the cellular flows in graphs (b), (c), (d) (e), and (f). This is likely to be linked to the modulation of the size of the cloud when falling through the vortical structure as shown in figure 9. The cloud seems to loose particles when it is shrinking, i.e. when falling in a downward elongational portion of the flow, see figure 9(b) at t/(kU 0 ) -1 = 3.8.

Break-up of the cloud

As mentioned previously in § 4.1, break-up events have been only observed in the finite-inertia case and thus are only analysed in that regime. The key quantity is the time required to reach the break-up, t b , which characterises the lifetime of the cloud. This time can be inferred as the time for which the cloud starts to bend to break into pieces. Table 4 gives the dimensionless break-up time, t b U c /R c , for clouds of particles C settling in fluid 2 both in the quiescent case and in the vortical cases (C2a and C2b). The experimental data correspond to averages over typically 10 runs and the error bars to the standard deviations over the different runs. Clearly, break-up times are shorter when clouds are settling into vortical flows. Numerical simulations using the Oseenlet approximation have been performed for similar inertial length, /R c ≈ 2 -3, and initial size of the cloud, Q = kR c ≈ 0.5, and for N 0 = 500 (and also 1000 in the quiescent case). The numerical data correspond to averages over 5-10 runs having differing initial distributions but also having different initial positions for the centre of mass as done in figures 7 and 8 of § 4.2. The trend is similar to that observed in experiments although the numerical values are somehow smaller than those found experimentally. This latter discrepancy may be due to the fact that the present numerical model is not accounting for unsteady or excluded-volume effects.

A physical picture of the mechanism leading to break-up can be obtained from a visualisation of the flow field in the cloud reference frame for a typical Oseenlet simulation in case C2b, see figure 11. As mentioned before in discussing figure 9, the cloud successively expands and contracts when falling in the consecutive upward and downward, respectively, elongational portions of the flow. When inertia is finite, the oscillation between the expansion and contraction phases is no longer reversible and the cloud expansion becomes stronger. When the cloud reaches an upward elongational region of the flow wherein there is a slow-down of the velocity, break-up then happens as the widened cloud winds around the vortical structure, see the graphs at t/(kU 0 ) -1 = 7.8,

Concluding remarks

We have performed experimental investigations as well as numerical simulations to examine the dynamics of clouds of particles settling in model cellular flows consisting of counter-rotating vortices where inertia is small but can be increased to become finite. The clouds tend to settle along the downstream flow regions of the vortical structures and present zigzagging motions. They do not maintain their initial spherical shape as they successively expand or shrink when settling through the successive upward or downward elongational portions of the flow. Increasing inertia enhances the cloud deformation. A leakage of particles is observed at the rear of the clouds. It is greatly amplified by the vortical flows by comparison with clouds settling in quiescent fluids. It is also intensified when inertia is increased and becomes finite. The vortical structures also induce a faster break-up of the clouds into multiple shatters in the finite inertia case. Point-particle simulations, which contain the minimal physics to describe the coupling between particle-particle and particle-flow interactions, capture well the cloud dynamics.

A key feature that is highlighted in the present work is the importance of the collective effects between the particles in maintaining the cloud as a single entity before breakup occurs owing to stretching in the vortical flow. This can be further investigated in figure 12 c) and (e), respectively, but wherein the hydrodynamic have been switched off. The difference between the simulations with and collective effects (i. e. with and without two-way coupling) is striking. Without hydrodynamic interactions, the individual particles settle much slowly as they sweep around the vortices since W = 0.108 < and a significant amount of them remains inside the vortex circulations since the Reynolds number based on the individual particle size is Re a 1. interesting output of the present study is the identification of a mechanism of expansion and contraction of the cloud leading to a final disintegration into multiple stretched pieces induced by the elongational portions of the vortical flow when inertia is increased (see figures 11 and 12). This last effect may be related to particle segregation in turbulent flow which has been seen to evolve into a network of filamentous shapes similar caustic patterns (see e. g. Reeks 2014). The generation of such inhomogeneities in the context of particles sinking in oceanic flows has been attributed to stretching mechanisms due to the flow (Drótos et al. More work is certainly needed to assess whether stretching by the elongational part of the flow structure is a generic and robust process that is seen in more complex flow than the model vortical flow studied in the present work.

Figure 1 .Figure 2 .

 12 Figure1. Experimental apparatus. The transparent cell is filled with an aqueous mixture of citric acid (UCON R oil can be added to increase the viscosity). The magnetic field is generated by a checkerboard of magnets placed behind the back wall of the cell. The electric current is created by two electrodes placed on the opposite small sides of the cell. The flow generated by this set-up is a periodic flow of counter-rotating vortices that is depicted in figure2. Clouds are dropped from the top of the cell using a custom made syringe and settle through the cellular flow. Their trajectories are imaged by a set of synchronised cameras located below and in front of the cell.

  -0.6 0.4 -0.6 Table 3. Dimensionless numbers for the different combinations of particles (labelled A, B, and C), fluids (labelled 1 and 2) and current intensities (labelled a and b) used in the experiments.

Figure 3 .

 3 Figure 3. Typical three-dimensional trajectories of centres of mass of clouds. The colour coding (colour online) corresponds to the distance to the back wall, and the trajectories projected in the horizontal plane are shown at the bottom. The trajectories which do not possess a large x-extension are selected.

Figure 4 .

 4 Figure 4. Typical (a) experimental and (b) numerical chronophotographies of a cloud for case A1b. The time between successive photos is kept constant (2 s in the experiments and a corresponding interval in the simulation) in order to indicate the difference in velocities along the trajectory. The flow field measured by PIV is indicated by red arrows. The complete dynamics for the experimental case and for the Stokeslet simulation are shown in supplementary movie 1.

Figure 5 .

 5 Figure 5. Same as figure 4 but for case B1b. The complete dynamics for the experimental case and for the Stokeslet simulation are shown in supplementary movie 2.

Figure 6 .

 6 Figure 6. Same as figure 4 but for case C2b. The numerical simulation uses an Oseenlet (instead of Stokeslet) approximation. The complete dynamics for the experimental case and for the Oseenlet simulation are shown in supplementary movie 3.

Figure 7 .

 7 Figure 7. Typical experimental (•) and numerical (solid lines) trajectories in the Stokes case B1a: time evolution of the (a) vertical, Z, and (b) horizontal, Y , coordinates of the cloud centre of mass, (c) its vertical velocity, Uz, and (d) the cloud projected surface in the horizontal plane normalised by its initial value, Σ/Σ0. In graph (a), the black triangle indicates the slope Uc = N0US6a/5Rc and the inset shows the initial positions of the cloud within experimental accuracy ∆ (with k∆ = 7 10 -2 ). The horizontal dotted lines correspond (i) to the initial location kY = 0 in graph (b), (ii) to the value Uc/U0 in graph (c), and (iii) to the initial projected surface Σ/Σ0 = 1.

Figure 8 .

 8 Figure 8. Same figure 7 but for finite-inertia case C2b. The symbols indicate break-up events.

Figure 9 .

 9 Figure 9. Snapshots of the settling numerical cloud for cases (a) B1a (N0 = 2500) and (b) C2b (N0 = 500). The original flow field measured by PIV is indicated by black arrows while the perturbed flow field is represented by blue streamlines. The corresponding movies are available in the supplementary material (movie 4 for the Stokeslet simulation and movie 5 for the Oseenlet simulation).

Figure 10 .

 10 Figure 10. Percentage (N0 )/N0 of particles that have leaked away from the cloud as a function of dimensionless time tUc/Rc: in (a) quiescent fluids and (b) vortical flows for cases A1a, B1b, and C2b; in vortical flows for different N0 for cases (c) B1b and (d) C2b; in vortical flows for two different Re k for cases (e) B1b and (f) The insets of graph (c) and (d) show the same leakage N0 -N but normalised by N 1/3 0 instead of N0.

Figure 11 .

 11 Figure 11. Flow field computed at successive times in the vertical plane in the cloud reference frame for a typical Oseenlet simulation in case C2b. High (low) velocity is indicated in white (dark). The corresponding movie 6 is available in the supplementary material.

  8.3, and 8.8. The remaining pieces of the cloud spread in the flow structures and can undergo the same break-up process, see the graph at t/(kU 0 ) -1 = 10.1.

Figure 12 .

 12 Figure 12. Snapshots of numerical clouds (using the Oseenlet approximation) corresponding to case C2a (P = 0.03, W = 0.108, /Rc = 2) settling in Taylor-Green vortices at five different times t = t/(kU0) -1 = 0, 8, 16, 24, 32 (each time t corresponds to a colour) with (a) Q = 0.5 and N0 = 500, (c) Q = 2 and N0 = 500, and (e) Q = 2 and N0 = 2000. Conditions in (d), (f) are the same as in (a), (c), (e), respectively, but hydrodynamic interactions between particles have been switched off. The corresponding movie 7 is available in the supplementary material.

  by going beyond the experimental constraint of having an initial cloud smaller the vortex size, i. e. by investigating a cloud with Q > 1. When the size of the initial cloud is larger than that of the vortices, the cloud can break into pieces which can be further stretched by the elongational regions of the flow. This is shown by comparing figures 12 (a) and (c) having the same N 0 = 500 but with Q = 0.5 and 2, respectively. Increasing the number of particles the cohesion of the cloud which is eventually stretched into elongated pieces by the vortical structure as seen in figure12 (e) having the same Q = 2 as figure 12 (c) but a different N 0 = 2000. The cloud in figure 12 (e) falls at a faster pace than that in figure 12 (c) since the cloud velocity is U c ∼ N 0 . As a we also show simulations in figures 12 (b), (d), (f) corresponding to the same initial cloud as in figures 12 (a), (

Table 2 .

 2 Fluid characteristics: viscosity µ and density ρ f .

Acknowledgments

article has been written under the auspices of the ANR project "Collective Dynamics of Settling Particles In Turbulence" (ANR-12-BS09-0017-01), the 'Laboratoire d'Excellence Mécanique et Complexité' (ANR-11-LABX-0092), the Excellence Initiative of Aix-Marseille University -A * (ANR-11-IDEX-0001-02) funded by the French Government "Investissements d'Avenir programme", and COST Action MP1305 'Flowing Matter'. We thank D. Lopez, J. Massoni, and J. Vicente for help in data acquisition and processing and E. Climent and A. Soldati for discussions and comments.

Declaration of Interests

authors report no conflict of interest.


Sanjeev Kumar

Marc Medale

Paolo Di

David Brutin

Sessile volatile drop evaporation under microgravity

The evaporation of sessile drops of various volatile and non-volatile liquids and their internal flow patterns with or without instabilities have been the subject of many investigations. The current experiment is a preparatory one for a space experiment planned to be installed in the European Drawer Rack 2 (EDR-2) of the International Space Station (ISS) to investigate drop evaporation in weightlessness. In this work, we concentrate on preliminary experimental results for the evaporation of hydrofluoroether (HFE-7100) sessile drops in a sounding rocket that has been performed in the frame of the MASER-14 Sounding Rocket Campaign, providing the science team with the opportunity to test the module and perform the experiment in microgravity for 6 consecutive minutes. The focus is on the evaporation rate, experimentally observed thermo-capillary instabilities, and the de-pinning process. The experimental results provide evidence for the relationship between thermo-capillary instabilities and the measured critical height of the sessile drop interface. There is also evidence of the effects of microgravity and Earth conditions on the sessile drop evaporation rate and of the shape of the sessile drop interface and its influence on the de-pinning process.

INTRODUCTION

Drops have been fascinating researchers for centuries [START_REF] Leidenfrost | De aquae communis nonnullis qualitatibus tractatus[END_REF][START_REF] Young | an essay on the cohesion of fluids[END_REF][START_REF] Fuchs | Evaporation and Droplet Growth in Gaseous Media[END_REF]. Topics of interest include water falling onto a hot cooking plate, which is a typical example of Leidenfrost drops [START_REF] Leidenfrost | De aquae communis nonnullis qualitatibus tractatus[END_REF], the evaporation of sessile drops with nanoparticle deposition in coffee rings [START_REF] Deegan | Capillary flow as the cause of ring stains from dried liquid drops[END_REF], inkjet printing [START_REF] Ko | Metal nanoparticle direct inkjet printing for low-temperature 3d micro metal structure fabrication[END_REF][START_REF] Schirmer | On the principles of printing sub-micrometer 3d structures from dielectric-liquid-based colloids[END_REF], pesticides sprayed onto leaves [START_REF] Salyani | Droplet size effect on spray deposition efficiency of citrus leaves[END_REF], and blood analysis [START_REF] Sobac | Desiccation of a sessile drop of blood: Cracks, folds formation and delamination[END_REF][START_REF] Smith | Wetting and spreading of human blood: Recent advances and applications[END_REF]. Although sessile drops are simple in geometry, the physics involved in the evaporation process is complex due to the numerous intricate interactions with the substrate and ambient environment and the fluid nature of the sessile drop itself. An accurate quantitative model of the evaporation process can lead to greater understanding of the evaporation rate and control over the pattern formation or the deposition of particles after the evaporation of a sessile drop. This knowledge can then enhance the efficiency of several applications. The physically rich and complex evaporation of sessile drops is thus of interest to both the academic and industry communities.

Parabolic flight experiments on drops of various fluids have been performed multiple times by The National Centre for Space Studies (CNES), France and The European Space Agency (ESA) parabolic flight campaigns [START_REF] Brutin | Sessile drop in microgravity: Creation, contact angle and interface[END_REF][START_REF] Carle | Contribution of convective transport to evaporation of sessile droplets: Empirical model[END_REF][START_REF] Saccone | Effects of force fields on interface dynamics, in view of two-phase heat transfer enhancement and phase management for space applications[END_REF][START_REF] Carle | Hydrothermal waves on ethanol droplets evaporating under terrestrial and reduced gravity levels[END_REF][START_REF] Sobac | Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate[END_REF][START_REF] Dehaeck | Vapor-based interferometric measurement of local evaporation rate and interfacial temperature of evaporating droplets[END_REF]. The existence of thermo-capillary instabilities [START_REF] Sobac | Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate[END_REF][START_REF] Sefiane | Self-excited hydrothermal waves in evaporating sessile drops[END_REF] and the effect of the reduced gravity environment on evaporation [START_REF] Carle | Contribution of convective transport to evaporation of sessile droplets: Empirical model[END_REF][START_REF] Carle | Experimental evidence of the atmospheric convective transport contribution to sessile droplet evaporation[END_REF] and the drop interface [START_REF] Brutin | Sessile drop in microgravity: Creation, contact angle and interface[END_REF][START_REF] Kabov | The effect of wetting hysteresis on drop spreading under gravity[END_REF][START_REF] Diana | Sessile drop wettability in normal and reduced gravity[END_REF] have already been demonstrated. Parabolic flights have enabled these observations, but such flights are not sufficient in terms of duration or residual acceleration for accurate measurements to be taken. Furthermore, the drop interface is highly sensitive to aircraft vibrations. A better level of microgravity and a longer evaporation time are therefore needed.

The Advanced Research on Liquid Evaporation in Space (ARLES) experiment module (see Fig. 1 and2) was designed to support the investigation of the evaporation process in a controlled environment. ARLES was part of the payload in a SubOrbital Express rocket (MASER 14), and it successfully took place on Monday, June 24, 2019 from the Esrange Space Center in northern Sweden under the collaboration of the ESA and SSC (Swedish Space Corporation). The ARLES experiment was conducted as preparation for an experiment that is to be performed in the near future at the European Drawer Rack 2 (EDR-2) of the International Space Station (ISS) under the EVAPORATION project of the ESA. The intent is to study evaporating drops of pure fluids as well as drops of fluids that contain a low concentration of metallic nanoparticles. The influence of an electric field is also of interest. The application of an external electrostatic field induces electric stress at the vapor-liquid interface, deforming it and altering the contact angle. The resulting electric forces press the drop against the surface and elongate it in the vertical direction; additionally, electroconvection is induced in the liquid and in the surrounding vapor atmosphere, resulting in a possible enhancement of evaporation rate, which may result useful when gravity-driven convection is suppressed. The scientific objectives include dealing with the flow motion and the thermocapillary instabilities occurring in the drop, at the drop interface, and in the vapor phase and investigating the pattern formation on the substrate after the evaporation phase.

ARLES was a collaborative experiment among various teams. Each team focused on different aspects of the experiment to contribute to the overall scientific objectives of the experiment, such as flow motion and thermo-capillary instabilities occurring in the drop, at the drop interface, and in the vapor phase, the pattern formation on the substrate after evaporation of the volatile phase, the deposition of nano-particles, and the eventual heat transfer enhancement. Our team primarily focused on the analysis of the flow motion and thermo-capillary instabilities occurring in the drop using data from the infrared (IR) (top view) camera and on the evaporation rate and interface evolution of the sessile drop using data from the side view camera. The experimental results presented here address the effect of microgravity and Earth conditions on the evaporation, thermo-capillary instabilities, drop interface, and de-pinning of a forced sessile drop of hydrofluoroether (HFE-7100) liquid on a heated substrate. The experimental results allow for a comparison of data from both ground and space experiments, thereby providing firm conclusions.

RESULTS

Experimental setup and conditions

In Figure 1a a complete setup of the ALRES experiment has been shown. It consists of two parts namely: Main evaporating cell (top) and Multi evaporating cells (bottom). Our current focus is on the Main evaporating cell (MEC) experiment. The detailed schematic of the MEC is presented in the Figure 2 (left) along with its chamber shown in Figure 1b (top view) and 1c (cut view) with injection system, substrate and electric field electrode (substrate is connected to the negative (-) terminal and the electrode to the positive (+) terminal). Figure 2 (right) shows the electric field distribution around the sessile drop for the axisymmetric case. For more details, please refer to MEC schematic in Figure 2.

The ideal experimental conditions for the MEC are as follows: target theoretical nominal parameters for microgravity and Earth conditions were set to be similar for the purposes of comparison. The injection velocity of liquid HFE-7100 for sessile drop creation on the heated substrate was 4 µL s -1 , and the nominal volume of each sessile drop was set at 6 µL. However, multiple ground experiments have shown that it is difficult to precisely control the injection liquid volume with the current injection system and hardware. Even though the actual injected volume of the drops during the ground experiment is lower than the target theoretical nominal value but the actual injected volume of the drops during the microgravity conditions is higher than the target nominal one (see Fig. 3). The temperature of the main test cell was set at 26 • C, and the temperature of the substrate was set at 28 • C with an imposed electric field 8 kV for all drops with electric field except for drop 8DPµgEF under microgravity, for which the field was set at 5.7 kV. Due to the grooves on the substrate, the base diameter of all the sessile drops remained constant (4 mm) during evaporation until the drops de-pinned.

Experimental results

For Earth gravity, the experimental data from the sensor are as follows: the main cell pressure (inside chamber) P amb , ambient temperature (inside chamber) T amb , substrate center temperature T sc , and difference between the substrate center and ambient temperatures (T sc -T amb ) were in the ranges 1053-1058 mbar, 26.16-25.87 • C, 27.93-28.00 • C, and 1.84-2.14

• C, respectively, for all drops. Furthermore, the substrate edge temperature T se was in the range 28.11-28.24 • C. (T se -T sc ) was thus in the range 0.16-0.22 • C, and (T se -T amb ) was in the range 2.04-2.31 • C for all drops.

Similarly for the microgravity experiment, the sensor data are as follows: the main cell pressure (inside chamber) P amb , ambient temperature (inside chamber) T amb , substrate center temperature T sc , and difference between the substrate center and ambient temperatures (T sc -T amb ) were in the range 1050-1057 mbar, 25.65-25.21 • C, 27.95-28.08 • C, and 2.39-2.79

• C, respectively, for all drops. The substrate edge temperature T se was in the range 28.13-28.18 • C. (T se -T sc ) was thus in the range 0.15-0.21 • C, and (T se -T amb ) was in the range 2.58-2.99 • C for all drops. The data and results from the Earth gravity and microgravity experiment are summarized in Table 1.

A comparison of the sessile drop volume with respect to time during evaporation is presented in Figure 3. We can see that in the Earth's gravity experiment all drops evaporated from the heated substrate before flushing started, whereas in the microgravity experiment flushing started before evaporation was complete (see the sudden fall in the drop volume). In the latter, only drop 6DPµgEF de-pinned, conversely to Earth's gravity experiment, where all drops did.

To compare the evaporation rates of sessile drops measured in microgravity experiment, one can refer to the analytical model for evaporation limited by diffusion, first derived by Picknett and Bexon [START_REF] Picknett | The evaporation of sessile or pendant drops in still air[END_REF] for a constant contact area (up to depinning) and a spherical cap shape. In our experiments the wetted area between the liquid HFE-7100 and heated substrate was constant with a base diameter of 4 mm (owing to the groove in the substrate). The analytical evaporation rate is thus:

dV dt = 2πD ef f C sat LF (θ) (1)
C sat = P sat M l R gas T amb (2)
F (θ) = (8.957 10 -5 + 0.633 θ + 0.116 θ 2 -0.08878 θ 3 + 0.01033 θ 4 )/sin θ for π/18 ≤ θ ≤ π,

where L is the drop base radius, C sat is the saturated vapor concentration, T amb is the ambient temperature in Kelvin, R gas is the universal gas constant, P sat is the saturation pressure based on the ambient temperature T amb in the MEC, M l is the molecular weight of the liquid (HFE-7100) and D ef f is the diffusion coefficient of HFE-7100 in a nitrogen gas environment.

The diffusion coefficient D ef f was calculated according to the Fuller-Schettler-Giddings equation [START_REF] Fuller | Diffusion of halogenated hydrocarbons in helium. the effect of structure on collision cross sections[END_REF] and F (θ) is a function of the contact angle of the sessile drop, derived by Picknett and Bexon [START_REF] Picknett | The evaporation of sessile or pendant drops in still air[END_REF].

A comparison of experimental and theoretical evaporation rates is presented in Figure 4 for drop 7DPµg under microgravity conditions at time t = 30 s (see Fig. To better understand the overall evaporation process, it could be interesting to address the related coupled fluid-flow problem that is induced. For that purpose, Figure 6 displays top view infrared and side-view images of drop 6DPµgEF in the microgravity experiment and drop 4DP1gEF in the ground experiment subjected to an 8-kV electric field as these were the only two drops of similar initial volume (see Fig. 3). The drop evaporation time series is divided in five sections, starting the sequence from the liquid injection to flushing. Next to the injection phase, surface temperature was almost uniform in both experiments, until thermo-capillary instabilities take place for drop 6DPµgEF at t = 18.3 s in the microgravity experiment and drop 4DP1gEF at t = 12 s in the ground experiment. The pattern of thermo-capillary instabilities shows several cells coming from bottom to surface of sessile drop and then moving toward the contact line. It clearly appears that these thermo-capillary instabilities only occur once the drop volume gets below a critical value (see horizontal lines in Figure 3 and detailed values in table 1). It is noteworthy from Figure 3 that these thermo-convective instabilities do not significantly modify the evaporation rates, whatever been under Earth or microgravity conditions. The last two sections of Figure 6 display the initiation stage of de-pinning and that of flushing, respectively.

DISCUSSION

Owing to unrepeatable injection drop volumes, one was faced with very different initial evaporation conditions between Earth and microgravity experiments, see Figure 3. Moreover, the time plot for drop 5DPµg under microgravity conditions (see Fig.

3

) exhibits some oscillations until de-pinning occurs. The detailed reasons for this strange behavior are under investigations, but the oscillations in volume may be related to higher mechanical coupling to the rocket vibrations due to its initial volume being larger than that of the other drops (see table 1). It might also have resulted from the release of gas bubbles inside the drop during evaporation, as can be observed from the side view images of the drop. The global evaporation rate of drop 5DPµg (microgravity) is thus excluded in the subsequent analysis.

The effect of gravity on the evaporation rate clearly appears in Figure 4: its value is roughly halved under microgravity conditions as compared to Earth conditions; this is in agreement with previous works [START_REF] Carle | Contribution of convective transport to evaporation of sessile droplets: Empirical model[END_REF][START_REF] Carle | Experimental evidence of the atmospheric convective transport contribution to sessile droplet evaporation[END_REF][START_REF] Almohammadi | Sessile drop evaporation under an electric field[END_REF]. Indeed, the average evaporation rate of the sessile drops of HFE-7100 under microgravity is 56% and 45% lower than that under Earth conditions without and with the electric field, respectively. Interestingly, the analytical diffusion-limited evaporation rate enables us to conclude that the average evaporation rate of HFE-7100 sessile drops under microgravity conditions in the absence of an electric field seems to be mainly controlled by diffusion. Furthermore, note that the average evaporation rate under Earth conditions with an electric field is 6% lower than the average rate without one, whereas the average evaporation rate under microgravity conditions with an electric field is 19% higher than the average rate without one. That is to say, the effect of an electric field on the evaporation rate of HFE-7100 is opposite under microgravity and Earth conditions, as it is for liquid water drops [START_REF] Almohammadi | Sessile drop evaporation under an electric field[END_REF].

Figure 5 shows side views of the sessile drops under the four considered conditions. For a fair comparison, compare Figure 5a with Figure 5c and Figure 5d with Figure 6 (drop 4DP1gEF), as these drops were of comparable volumes (see Table 1). Also note that no drop in microgravity without an electric field had a lower initial injected volume (see Fig. 3b).

Therefore, the minimum volume for drop 7DPµg under microgravity at t = 30 s can be used for comparison of the interface.

The interface shape of the sessile drops resulted from body and surface forces acting on them. As it clearly appears in Figure 5a and Figure 5b, the shape of a sessile drop under microgravity is exactly spherical in comparison to that in Figure 5a. In contrast, sessile drops exhibit clear cone formation under microgravity conditions with an electric field (see Fig. 5d, 5c and Fig. 6). Along with the influence on the interface (see Fig. 5), which is in agreement with other experiments [START_REF] Saccone | Effects of force fields on interface dynamics, in view of two-phase heat transfer enhancement and phase management for space applications[END_REF][START_REF] Brutin | Sessile drop in microgravity: Creation, contact angle and interface[END_REF][START_REF] Kabov | The effect of wetting hysteresis on drop spreading under gravity[END_REF][START_REF] Almohammadi | Sessile drop evaporation under an electric field[END_REF][START_REF] Abdella | Interfacial deformation of liquid drops by electric fields at zero gravity[END_REF], the de-pinning process is also associated with the gravitational and electrical forces individually or in combination. Based on these comparisons, we can see the correlation between the body and surface force conditions and the volumes (see Fig. 3) and contact angles (contact angles were measured by using the ImageJ R software plugin known as DropSnake, which is based on B-spline snakes (active contours)) during de-pinning irrespective of the shape of the sessile drop interface shape. The fact that de-pinning is anticipated in the presence of an electric field can be attributed to the fact that the radial electric force is directed inwards, causing striction of the interface [START_REF] Saccone | Effects of force fields on interface dynamics, in view of two-phase heat transfer enhancement and phase management for space applications[END_REF]. Accordingly, the order of de-pinning based on the volume and contact angle and body and surface force conditions was as follows: drop 6DPµgEF with an electric field (under microgravity conditions) at volume = 1.83 µL and contact angle θ = 18.7 • de-pinned at the highest volume and contact angle and did so earlier than drop 2DP1gEF with an electric field (under Earth conditions), which de-pinned at volume = 0.81 µL and contact angle θ = 14.6 • , and drop 4DP1gEF with electric field (under Earth conditions), which de-pinned at volume 0.76 µL and contact angle θ = 13.4 • . Drop 3DP1g without an electric field (under Earth conditions) at volume = 0.43 µL and contact angle θ = 8.6 • and drop 1DP1g without an electric field (under Earth conditions) at volume = 0.43 µL and contact angle θ = 7.6 • de-pinned with the smallest volumes and contact angles. According to the above correlations, it can be predicted that for sessile drops 5DPµg and 7DPµg (under microgravity without an electric field) the volume (and contact angle) should have been either equal to or higher than the volume (and contact angle) of drops 2DP1gEF and 2DP1gEF (under Earth conditions with an electric field) at de-pinning. The influence of the substrate grooves in the de-pinning dynamics could itself be a subject of investigation [START_REF] Grishaev | Study of edge effect to stop liquid spillage for 344 microgravity application[END_REF].

The infrared images in Figure 6 reveal some characteristic patterns associated with the thermo-capillary instabilities that occurred for drop 6DPµgEF in microgravity conditions at time t = 18.3 s, which corresponds to a volume of 2.01µL, calculated using the side view image (refer Fig. 6) in which the maximum sessile drop height is 0.41 mm. The thermocapillary instabilities first appeared near the periphery of the sessile drop during evaporation and before de-pinning, and they remained visible up to complete evaporation (see Fig. 6). In the ground experiment, however, there were instability patterns for drop 4DP1gEF stating at time t = 15 s and volume = 1.67 µL (maximum interface height of 0.24 mm); the patterns began to appear at time t = 12 s and volume = 1.10 µL. Similarly, instability patterns appeared in all the sessile drops in the ground reference experiment (see Fig. 3), for which volume and time are reported in Figure 3. The thermo-capillary instabilities appeared as soon as the maximum drop height was below a critical value of approximately between 0.2 to 0.3 mm for Earth conditions and around 0.4 mm for the microgravity conditions, which is associated with thermo-capillary instabilities referred to as Marangoni instabilities. Interestingly, the above critical thickness for HFE-7100 under Earth conditions fully agrees with Chauvet et al. [START_REF] Chauvet | Threshold of bénard-marangoni instability in drying liquid films[END_REF]. Therefore, as the injected volume of most of the microgravity drops exceeded that of the drops in the Earth reference experiment, longer evaporation times would have been required for the former to reach the critical height at which thermo-capillary instabilities are observed. As a result, flushing of the largest microgravity sessile drops was unfortunately performed before instability patterns could be observed.

In conclusion, it is worth mentioning that it was very difficult to carry out repeatable injection of prescribed sessile drop volume both under Earth and microgravity conditions. The exact reasons for the formation of oversized sessile drops under microgravity conditions are still under investigation. In future microgravity experiments, it would therefore be preferable to perform sessile drop volume injection with real-time feedback control. Under our experimental conditions, the results provide evidence for the effect of microgravity conditions on the sessile drop evaporation rate, indicating that the rate under microgravity conditions is nearly half that under Earth conditions for HFE-7100. Furthermore, the effect of an electric field on the evaporation rate is opposite under microgravity and Earth conditions. The experimental results also demonstrate the relationship between thermo-capillary instabilities and the measured critical height of the sessile drop interface. For temperature differences between substrate and ambient in the range of 2-3 • C with a base diameter of 4 mm, the measured critical height for the appearance of thermo-capillary instabilities is approximately between 0.2-0.3 mm for Earth conditions and around 0.4 mm for the microgravity conditions for HFE-7100. It is also noteworthy that meanwhile they strongly change the fluid flow structure in the sessile drop, these thermo-capillary instabilities do not significantly influence the evaporation rates. Through the application of different combinations of volumetric forces (an electric field and gravity), we also demonstrate the role of gravity on the shape of the sessile drop interface and its influence on the de-pinning of sessile drops. To concrete the above evidence, module will re-fly again (as a baseline, in 2022). One of the main objectives of the reflight is to better control the actual injected volumes so as to ensure a better data comparison among the different testing conditions.

METHODS

Fluid property measurements

In all cases, the liquid used was 99.9% pure HFE-7100 (3M TM Novec TM 7100 Engineered Fluid, a hydrofluoroether also known as methoxy-nonafluorobutane (C4F9OCH3)). It was chosen for its volatility, semi-transparency in the infrared wavelengths, perfect wetting, non-toxicity, and being non-flammable. For more on the properties of HFE-7100, please refer to [26].

Hardware description

The ARLES experiment module was designed and manufactured by the SSC under the guidance of the ESA, and Science team proposition based on the required scientific objectives. The overall design of the experiment module is subdivided into two parts (see Fig. 1a): the Main Evaporation Cell (MEC), which is for single drop experiment systems, and the Multi-drop Cell, which is for multi-drop experiment systems to be executed in parallel. For safety reasons, a neutral gas nitrogen ( N 2 ) atmosphere was used.

Main Evaporation Cell : The chamber volume of the Main Evaporation Cell (MEC) is 4 litre. It is sized to maintain a low vapor concentration throughout the whole experiment even if the N 2 evacuation fails during the flight. The cell thickness was chosen to withstand the expected pressure differences during the filling and emptying of the gas ( N 2 ). Figure 1 and 2 shows the main cell used to perform sessile drop evaporation of a pure fluid on a heated substrate with and without an electric field. The substrate was a thin single-crystal silicon wafer coated with a 50-nm-thick platinum layer, whose surface roughness was less than 1 micron RMS, deposited by atomic layer deposition (ALD). The substrate had 50 × 50 µm grooves with 4 ± 0.1 mm in diameter to force the pinning of a sessile drop with a diameter of 4 mm (see Fig. 1d). The central hole for the fluid injection was 0.7 mm in diameter. The substrates were manufactured at MICAS TU Leuwen. An infrared camera was mounted on the lid of the main evaporation cell, where a ZnSe window served as the passage for infrared wavelengths.

The interferometry camera observed the single-drop evaporation process through the side observation windows of the MEC.

Multi-drop Cell : The multi-drop experiment system is for the analysis of different fluids with nanoparticle suspensions, and the related pattern formation on the substrates after the evaporation process, and its consequent functionalisation. As such, it is not part of our analysis.

Heatflux, temperature and pressure measurements: Two T-type thermocouples monitored the substrate temperature. One thermocouple was placed close to the center hole and the other one close to the edge of the substrate. These Heater: The heaters were custom made and manufactured by NEL Technologies Ltd. They are capton patch heaters with an etched resistive pattern. For the MEC, the heater was designed to provide 5 W of uniform heating power at 24 V.

The heater was driven from 24 V PWM.

Electrode: The positive high voltage potential was connected to a conical shape electrode, which was located above the substrate, concentric with the substrate grooves and the drop injection inlet hole, at a distance of 6 mm from the substrate (see Fig. 2). On the other hand the substrate is connected to negative voltage potential.

Image acquisition and analysis

To perform fluid flow visualization of the drops, high-resolution infrared images were captured by a commercial off-the-shelf (COTS) bolometric (non-cooled) Xenics, Gobi 640 camera. The images are 640 × 480 pixels (H × V) with a noise-equivalent temperature difference (NETD) of 50 mK at 30 • C and an infrared wavelength region of 8 -12 µm. The images were recorded via an infrared optical path consisting of an AR-coated ZnSe window (Ø75 × 6 mm). The depth-of-field (DoF) of the infrared camera with an image pixel density of 17 µm is 0.7 mm at 9 cycles mm -1 . To visualize and track the evolution of the interface of the sessile drops, we used images from the side view camera of the interferometer. The interferometer images have a fieldof-view of 15 × 15 mm with an image pixel density of 10.78 µm pixels -1 (11.2 µm pixels -1 for microgravity conditions). The interferometry fringes were removed from the raw images with the help of the ImageJ software. The cleaned images without fringes were used for the analysis (see Fig. 5e). The image acquisition rate for all the images was 25 Hz.

Experimental procedure

The SubOrbital Express rocket (MASER 14) launch took place successfully on Monday, June 24, 2019, from the Esrange Space Center in northern Sweden. The atmospheric replacement was executed 60 s after the launch by feeding in the N 2 while the experimental cells were connected to an exhaust port in the outer structure. At the start of the microgravity phase (100 km level) t = 70.4 s, the experiment liquid was injected to create the first drop of HFE-7100 upon the heated substrate.

After a delay corresponding to the estimated drop evaporation time, the atmosphere in the chamber was flushed. After the flushing sequence, another drop was injected, and the evaporation cycle with diagnostics was repeated. The outside pressure was 0 bar during the microgravity period. At the bottom of the ARLES experiment module is an N 2 pressure vessel for flushing the single-drop cell after each consecutive drop. Flushing was performed to prevent the evaporated liquid from condensing in the experiment cell. The ground test experiment were executed in the same way as during the flight. The only difference was the membrane vacuum pump, which was connected to the exhaust of the module. All rights reserved. Disclosure to third parties of this document or any part thereof, or the use of the information contained herein for other purposes than here intended, is not permitted except with the prior and written permission of SSC.

EXPERIMENT MODULE AND GSE

3.1 System description

Overall design

The overall design of the experiment module reflects the split between the two experiment sets, to be executed in parallel. The first set being the single drop experiment systems and the second the multi-drop experiment systems.

Main evaporation cell

The single-drop system will inject a series of Novec 7100 droplets onto a heated substrate. The droplets will evaporate during coverage of diagnostic instruments, sensors and stimuli MASER-1429217191-159 ver 3.41

ENG-18 Design Report ARLES

All rights reserved. Disclosure to third parties of this document or any part thereof, or the use of the information contained herein for other purposes than here intended, is not permitted except with the prior and written permission of SSC.

Internal

Page 28 In the bottom of the cell a liquid loop & GN2 loop is implemented to thermalize the experiment cell and GN2. The liquid loop is connected to a heat exchanger in order to heat or cool the experiment cell prior to launch. The inlet and outlet of the liquid loop is connected to the module umbilical blocks.

Three windows are implemented, two for side observation and interferometry and one for the top infrared observation. The top observational window is a ZnSn glass window.

In the bottom of the cell the readout electronics for the heatflux sensor and the T-type thermocouple is housed.

MASER-1429217191-159 ver 3.41

ENG-18 Design Report ARLES

All rights reserved. Disclosure to third parties of this document or any part thereof, or the use of the information contained herein for other purposes than here intended, is not permitted except with the prior and written permission of SSC.

Internal

Page 24

Thermal model of main evaporation cell

The simulation on the updated Modell of the latest design is pending.

Thermal distribution calculation of heater block

To have near uniform temperature distribution over the substrate the aluminium heaterblock needs to be larger than the substrate. The temperature profile of a cylinder with a uniform heating is described by: The bar l denotes the de-pinning stage of the sessile drops. For the better interpretation, please also refer Table 1 The frames illustrate the injection, instability pattern, and de-pinning stages, respectively, for drops 6DPµgEF and 4DP1gEF under microgravity (top) and Earth's gravitational conditions (bottom) (see Table 1 and refer to the supplementary materials for complete movies).

= + (

  [START_REF] Fuchs | Evaporation and Droplet Growth in Gaseous Media[END_REF] and Fig.5bfor a side view). The parameters for the analytical calculation are the base radius L = 2 mm, contact angle θ = 45.6 • , D ef f = 5.4 10 -6 m 2 s -1 , P sat = 27268 Pa, M l = 0.25 Kg mol -1 , P amb = 105100 Pa and T amb = 25.36 • C. The calculated theoretical value of the diffusion-limited evaporation rate for this drop (7DPµg) under microgravity conditions at time t = 30 s is 0.095 µL s -1 . The experimental value for the time evolution of the sessile drop volume is calculated from post-processing the side view of the drop shape (see Fig.5e). The experimental values under Earth and microgravity conditions without electric field are 0.198 µL s -1 and 0.087 µL s -1 , respectively. This technique is more accurate in the constant contact area evaporation mode with an uncertainty maximum up to ±0.05 µL for the volume and of ±0.015 µL for the evaporation rate.Drop shapes result from body forces equilibrium during the evaporation process. Figure5shows the comparison of the sessile drop under gravity only (see Fig.5a), microgravity only (see Fig5b), both gravity and electric field (see Fig.5c) and finally microgravity and electric field (see Fig.5d). The combination of body forces results in changes of interface curvature, contact angle, and thus in the de-pining stage. Figure5eis only intended to show the comparison between raw images from experiments (top) and clean ones (bottom) after post processing. The cleaned images have been later used to calculate the time evolution of drop volumes reported in Fig.3.

1 K

 1 sensors were incorporated in the heat flux sensor by the CAPTEC manufacturer. The heat flux sensor with integrated thermocouples determined the heat flux to the drops and substrate temperature with a sensitivity of 2 µV W -1 m 2 ). Along with the substrate temperature, we used a set of PT-100 sensors to monitor the cell wall temperature and the ambient temperature inside the MEC (see Fig.1for the position). The temperature measurement rate was 30.4 Hz, with an uncertainty of ±2.from the true temperature in the worst case. The passband of the filter was 4.56 Hz. A dedicated µ-TC interface board performed the readout of the heat flux sensor and the thermocouples. The pressure sensor measured pressure in the range of 0-1.6 bar with an accuracy of ±0.2 % inside the MEC throughout the experiment.

Figure 3 . 1 - 1

 311 Figure 3.1-1 ARLES experiment module

Figure 3 . 4 - 2 :

 342 Figure 3.4-2: Cross-section of main evaporation cell

Figure 3 . 3 - 2

 332 Figure 3.3-2 Thermal model of main evaporation cell

Figure 1 :

 1 Figure 1: Overview of the ARLES experimental setup. (a) Experiment module on-board the MASER 14 rocket, divided into two parts: the main evaporating and multi evaporating cells. (b-c) Main evaporating cell (MEC) with a detailed schematic. (d) Platinum layered surface crystal silicon wafer substrate (top view) with grooves. Images (a), (b), and (c) are credited to the European Space Agency (ESA) and Swedish Space Corporation (SSC).

Figure 2 :

 2 Figure 2: Schematic of the Main evaporating cell (MEC) of the ARLES experiment (left). Axisymmetric electric field around sessile drop (right). Readers are advised to refer to web version of this figure for better display.

9 7DFigure 3 :

 93 Figure 3: Volume of sessile drops on the heated substrate versus time. (a) Earth's gravity; (b) Microgravity conditions. The bar l denotes the de-pinning stage of the sessile drops. For the better interpretation, please also refer Table 1 along with figure. Reader are advise to referred to web version of the figure.

Figure 4 :

 4 Figure4: Evaporation rates over the constant contact angle mode. Earth conditions: average evaporation rates of sessile drops 1DP1g and 3DP1g (without an electric field) and 2DP1gEF and 4DP1gEF (with an electric field); Microgravity conditions: evaporation rates of sessile drops 7DPµg (without an electric field) and 8DPµgEF (with and electric field). The parameters of sessile drop 7DPµg were used for the calculation of the analytical diffusion-limited evaporation rate without an electric field[START_REF] Picknett | The evaporation of sessile or pendant drops in still air[END_REF]. Error bars are calculated estimating the minimum and maximum evaporation rate experimentally measured.

2 ' 3 'FFigure 5 :

 235 Figure 5: Comparison of the sessile drop interface under the effect of gravitational and electrical field forces. (a) Drop 1DP1g on the ground at t = 2.3 s. (b) Drop 7DPµg under microgravity at t = 30 s. (c) Drop 2DP1gEF on the ground with an electric field at t= 2.4 s. (d) Drop 6DPµgEF under microgravity with an electric field at t = 6.8 s. (e) Image from a side view camera with interferometry lines (top) and after cleaning (bottom). The cleaned images are used to measure volume over the time (See Fig. 3)

Figure 6 :

 6 Figure6: Time series of infrared images (top view) during the evaporation of liquid HFE-7100 sessile drops on a heated substrate under microgravity and Earth conditions with electric field (EF). The frames illustrate the injection, instability pattern, and de-pinning stages, respectively, for drops 6DPµgEF and 4DP1gEF under microgravity (top) and Earth's gravitational conditions (bottom) (see Table1and refer to the supplementary materials for complete movies).

Table 1 :

 1 Summary of experiments with internal notation

	θ
	(Deg.)

Table 3 .3-1: Substrate heater block

 3

  along with figure. Reader are advise to referred to web version of the figure.

					Diffusion limited without EF
	Evaporation rate (μL s -1	0.198	0.186	0.087	0.109	0.095

) Earth's gravity(1g) Microgravity(μg) Analytical Without electric field (EF) With electric field (EF)

ACKNOWLEDGEMENTS

The present work was carried out in the framework of the European Space Agency research project AO-1999-110: EVAPO-RATION. We thank all of the ARLES Science Team Members for their contribution in making possible the Sounding Rocket Experiment in the framework of the ESA MASER-14 Campaign : Science Team Coordinator: Dr. C. S. Iorio, Université Libre de Bruxelles, University of Pisa, University of Alberta, University of Darmstadt, Université de Liège, Chinese Academy of Science, University of Edinburgh, University of Loughborough, Trinity Collegue Dublin, Siberian Branch Russian Academy of Science. The authors would also like to thank Dr. D. Mangini, Dr. B. Toth, and Dr. A. Verga for their interest in and support of the activities linked to ARLES and for the fruitful discussions. The research team of Prof. A. Ferrari of the University of Cambridge is also gratefully acknowledged for the enormous support in preparing the fluids with nanoparticle

suspensions tested both on the ground and in microgravity conditions. The authors are grateful to A.I. Garivalis from University of Pisa for his contribution in EF calculations and analysis. The authors would also like to thank the Swedish Space Corporation, with particular gratitude to the project manager, Mr. M. Lundin.

We would also like to acknowledge the financial support of the French National Space Agency (Centre national d'études spatiales: CNES) research grant for the DROPS experiment on the FLUIDES space mission.

DATA AVAILABILITY

The data collected during this study is available from the corresponding authors upon reasonable request.

COMPETING INTERESTS

The authors declare that there are no competing interests.

AUTHOR CONTRIBUTIONS

S.K. performed processing of the raw data. S.K., M.M., and D.B. contributed to the analysis of the results and writing the manuscript. P. D. M. contributed on the application of electric field on sessile drop. All authors approved the final manuscript.

ADDITIONAL INFORMATION supplementary information:

A supplementary movie for Figure 6 is available online.

Correspondence:

Correspondence and requests for materials can be addressed to S.K. (email: sanjeev.kumar@univ-amu.fr) and D.B. (email: david.brutin@univ-amu.fr).

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