H Trannois
email: harold.trannois@u-picardie.fr

J Fortin

C Drocourt
email: cyril.drocourt@u-picardie.fr

F Dubois
email: dubois@lmgc.univ-montp2.fr

GIDE : Graphic Interface for Discrete Element

In this paper we propose a graphic display tool for the results of calculations carried out using a discrete element code: Graphic Interface for Discrete Element Code (GIDE). This is a post-processing application written in C++ based on portable open source libraries, making GIDE compatible with different OS (Windows, Linux, Unix, MacOS etc.).

Introduction

Traditionally, when carrying out a numeric investigation of the mechanical behaviour of a deformable body undergoing divers stresses, use is made of the Finite Element Method (FEM) which is an important tool in the analysis of structures and more generally in engineering science [START_REF] Zienkiewicz | the finite element method[END_REF]. It is based on the mechanics of continuous media; however, the bodies are not continuous, but the assumption of continuity affords a simplification making it possible to solve the problems of classical mechanics. However, the assumption of continuity appears difficult to accept for systems composed of several rigid or deformable parts, interconnected by links. We then refer to multi-body systems. Currently, numerous applications involve the study of such systems. In the field of sport, we study the movements of athletes. In civil engineering, the modelling of granular materials by a multibody system enables understanding of the origin of mechanical behaviour, whether it is microscopic, macroscopic etc. In the field of the automobile and transport, we seek continually to improve the performance, comfort and safety of cars, lorries and trains. In granular mechanics, geomaterials or masonry, numerical simulations based on the individual behaviour of grains or blocks are qualified as DEM or Distinct Element Method [START_REF] Cambou | Micromécanique des matériaux granulaires[END_REF], in contrast to the FEM strategy used when a homogenised behaviour law has been chosen, assimilating the granulate or masonry to a continuous medium.

For the moment in granulate mechanics, because of limitations in memory size and calculation time, discrete numerical simulations are limited to samples of a few thousand, or even a few tens of thousands of grains. For comparison, 1cm 3 of sand with 0.1mm diameter grains contains about 10 6 grains. Thus, another substantial problem in modelling granular media is to be able to define the average magnitudes (average stresses, average strains etc.), taking into account the overall behaviour of a granular medium considered as a continuous medium, and representative of the physics at the scale of the grain (contact efforts, volume efforts, local rotations and speeds etc.). The final aim is to obtain homogenised behaviour laws. This is the object of the micro-mechanical approaches that can be enriched with the results obtained by discrete numerical simulations of granular media, some quantities such as intergranular forces being difficult to measure experimentally.

A modelling problem is generally characterised by defining a real physical system in which certain quantities are a priori unknown, and others assumed to be known. The first step in the modelling process consists in making a series of simplifying assumptions that make it possible to model the problem: idealisation of the geometry, boundary conditions, stresses. The second step consist in selecting the relations that govern the model (taking account of friction, shock law, thermal effect, wearing phenomena; or remote interactions), eliminating certain variables between these relations, making simplifying assumptions (rigidity of bodies), then choosing the methods of discretising the equations thus obtained. Creating such a discretised model uses numerous implicit or explicit knowledge of the user: choice of time step, choice of stiffness, choice of coefficients of restitution, etc. We can thus obtain a system of algebraic equations which approximately represents the behaviour of the physical system being investigated [START_REF] Duran | Introduction à la physique des milieux granulaires[END_REF].

Solving the system supplies the unknowns, here the speeds and local reactions. The development of a numerical tool enabling the display in 2D and 3D of mechanical systems in unilateral dynamics, of large size, i.e. containing more than 10 000 particles, naturally involves a series of conditions to be satisfied by the model:

• Portability : it must be easy to integrate and to use in divers calculation environments ; • Performance: it must have minimum cost while enabling faithful modelling of the mechanical part of the system; • Reliability: it must be accurate and robust so as to represent as accurately as possible the behaviour of the mechanical system whatever the conditions to which it is submitted and the time interval that is simulated.

The aim of this paper is to propose a graphic display tool for the results of calculations performed with the aid of a Discrete Element code [START_REF] Fortin | Numerical simulation of granular materials by an improved discrete element method[END_REF]: Graphic Interfaces for Discrete Elements (GIDE). This is a post-processing application based on portable open source libraries, making GIDE compatible with various OS (Windows, Linux, Unix, MacOS etc.). GIDE is a vector tool; this alternative approach to current display tools allows the discrete aspect of bodies to be conserved; two bodies interconnected by a third are not transformed into a 3D image, they remain 3 graphically representative elements that can be selected individually.

The choice of libraries has been a determining factor, whether with HDF [START_REF]Hierarchical Data Format[END_REF] for the handling of data files or OpenSceneGraph (OGS) [START_REF]D graphics toolkit[END_REF][10] for managing of 3D scenes in OpenGL.

GIDE is also a post-processing application enabling a body to be tracked in time or data to be extracted with the aid of the tool called "capteur" in reference to the tool that can be found when experiments are carried out.

Finally, GIDE is equipped with a filter allowing the importing/conversion of data files from various calculation codes (MULTICOR, LMGC90 [START_REF] Dubois | Lmgc90 une plateforme de développement dédiée à la modèlisation de problèmes d'intéraction. 6 ème colloque national en calcul des structures[END_REF]).

Technology choice

GIDE was designed to be as open as possible through recourse to recognised and free libraries. We have been particularly attentive to the documentation of the code and the development environment through collaborative work. The tools used are all from open sources. The result is an application entirely uncoupled from EE calculation code.

Data format

First version of the data format

In order not to be intrusive, the initial aim of GIDE was to be capable of functioning without any modification of upstream software, such as MULTICORPS or other simulators, only via the transformation of data from these applications. The transformation is based on an XML file describing the organisation of the data and a conversion tool (included in the GIDE). Thus, all results data files are transformed into the format HDF5 before use.

The format HDF5 was adopted as the native GIDE format; its advantages are:

• A data model allowing the representation of complex data,

• A portable file format, • A library that can be used on different platforms, with interfaces in C, C+ +. Fortran and Java, • Optimised performance for access time and size of data,

• Tools for the display and processing of data in the format HDF. Reading and writing are optimised in order to exploit various types of architecture: simple file on a standard file system, several files on a standard file system, several files on a parallel file system and other situations. The first version of the organisation of data in the HDF file offers the following architecture (Figure 1) :

Fig. 1. HDF representation

Starting from the root '/' w access the simulation of global data, then we access each scene and finally the block of data for each type of discrete element. In the HDF file there are sets of data called DATASET which are gathered into groups called GROUP. Each GROUP or DATASET has a name; it is the name of the DATASET that determines the representation of a discrete element. The tree representing the organisation of the data looks like Figure 1. Access to the first DATASET is via path/scene/pas01/. The data are not only organised in the form of a tree in the files, but also in the memory. OSG also organises the various elements of a 3D scene in the form of a tree and even more in the form of a graph. GIDE re-uses the mechanisms of OSG for this part; the discrete elements of GIDE are specialisations of the GROUP of OSG. While data are being read, the tree is built up of the scene for each step in the simulation, Figure 2.

Fig. 2. Representation in the memory

Second version of the data

The second phase in the development of the GIDE project was to integrate the recording of data directly in the format HDF5 into the simulator application such as MULTICORPS. The constraints were as follows:

• Ability to consult the data from the simulator without going through the GIDE graphic interface, • Enable the selection of only a part of the variables from the set of time steps, which the first version of the file format did not allow, • Not cause any increase in the current processing time of simulator applications, and possibly improve this.

The first point is resolved natively by the choice of HDF5 format, since the files saved in this format can be consulted with ancillary tools such as "hdfview", which even allow the exporting of a selection of data.

The second point made us think about an alternative in the representation of data in the HDF5 file, and the solution adopted is a matrix: With the possibilities afforded by the DATASET of the HDF files, there are then two solutions possible:

• Build the matrix initially with the desired size, from the start of the program, and update it at each time step (Figure 4), • Use a redimensionable matrix of initially empty size, and expand it at each time step, with the size of a vector of dimension [ 1: N ] (Figure 5). For a solution to the third point a series of tests on the various solutions had to be devised, and the best solution selected. The possible solutions are as follows:

• Write the data in text form, using standard recording primitive (current solution), • Write the data in raw binary form, using the standard recording primitives (reference solution), • Use the HDF5 format of the first version of the GIDE, • Use an initialised matrix at maximum size, • Use a matrix of empty size, increasing it by one vector at each time step.

Step Each of these solutions was tested with the recording of 4096 variables of the « long int » type (32 bits), over 4096 time units. These tests were carried out with the "time" command under Linux, and the result is given in the table of figure 6 The good results from solution B are due to the possibility in binary mode of recording a vector in a single operation, stating the memory address and the length of this vector. However, this solution does not satisfy the prerequisites overall.

Aside from this point, it clearly appears that solution "E" gives the best results, and this is therefore the solution selected and integrated in the GIDE. Moreover it offers the advantage of satisfying our three initial constraints.

3D motor

The choice of 3D library is crucial. Though this interface is considered as a navigation tool in an ocean of data, the first contact with the user is nevertheless visual and this is his first expectation: to display the results of the simulation. There was no question of making an nth 3D library, but rather of choosing the most appropriate one from existing ones. Four constraints were set:

First constraint: Targeting the material and operating system to the users.

Most work stations are individual laptops; these are not supercomputers or CAD stations and the post-processing of results is for the moment a poorly furnished second order activity. The use of the API 3D standards Direc3D or OpenGl is therefore strongly indicated! They allow currently available graphic cards to be driven and use to be made of their internal calculator thereby leaving the main processor free.

For the operating system, we decided on portability over the three commonly used OS: Linux, Windows, MacOS. This constraint eliminates the API OpenGl which is for the moment only supported by Windows. GIDE therefore uses the API OpenGl to drive the graphic cards.

Second constraint: Displaying a large number of elements.

This constraint turns out to be difficult to respect. It is in contradiction with the first. Without making use of a dedicated display machine a large number of bodies cannot be displayed. The threshold of 10 000 is often the limit beyond which manipulation of the display becomes jerky and even blocks for small configurations. It is in the organisation of the bodies that we hope to overcome this constraint. To do this we use graphs of scenes, techniques developed initially for an IRIS project [START_REF]IRIS performer: a high performance multiprocessing toolkit for real-time 3D graphics[END_REF] and then taken up by one of its authors in an OSG project involving more general use [10].

The data from the discrete element calculation code are particularly well suited to to the construction of graphs of scenes since the latter are made up of vector elements: circles, rectangles, spheres, cylinders, polyhedra etc., and it is these same basic elements that are used in simulations of the MED type. One of the major strengths of this approach lies in the elimination of calculations for masked bodies and other elements of the scene, called culling [START_REF]Occlusion Cullingnext term in OpenSG PLUS, Dirk Staneker[END_REF], and it is this strength that is particularly interesting in our case. From experience, MED simulations have bodies organised in heaps, which means that most of the bodies need not be represented as they are masked by those on the surface. For example, if we take a cube of 50 bodies along aside, the simulation will count 50 3 = 125 000 bodies for only 3 visible faces on a maximum of 2 rows, i.e. 50x50x2 = 15 000 bodies, which is the limit of manipulability on a laptop.

Third constraint: Select the bodies.

The currently available offer of applications for displaying MEF is very rich, unlike that for MED which is very poor.

What is the difference between the two which justifies such a disparity?

There is first of all the recency of MED which has only been used for a few years, but also the unsuitability of methods for representing discrete finite elements. It is for example impossible to select several 3D bodies graphically since the notion of group of nodes forming a body is in contradiction with the spirit of meshes used in MEF.

Representation in the form of vector elements is also a good response to this constraint! This reinforces the choice of a data structure in the form of a graph as in AutoCAD, Adobe Illustrator, Acrobat 3D etc.

In conclusion to this section, we stress the fact that GIDE is closer to to a vector image application than bitmap streaming. This explains why we will not further refer to remote display solutions [START_REF]Université d'Orléans LIFO[END_REF] suitable for representing a very number of data, which only partially meet our requirements.

The user interface / Documentation

The choice of library to use for user interface management: menu, dialog box, window management, mouse etc. must respect the above constraints which can be summarised as: Portability and Compatibility.

The choice of OSG library and the portability of the various OS (cross-platform) limit the possibilities.

OSG interfaces well with the libraries or toolkits: GLUT, FLTK, QT, GTK and WxWidgets and also with FOX and MCF which are not themselves crossplatform. Among these 5 only those richest in functions are selected, QT, GTK and WxWidgets.

At this stage all 3 are suitable, but preference is for WxWidgets with its better integration with the host "windows manager", and the look and feel of the native system are retained To summarise, the GIDE architecture is represented by the following diagram (Figure 8):

Characteristic of the application

In the previous chapter, we saw that GIDE is a vector type application allowing the selection of simulation elements as unitary entities. This characteristic is present in design applications such as computer aided design or drawing.

To make things clear, take two flagship applications from Adobe: Illustrator and Photoshop. The former, vectorial, is used to create illustrations, the patter, raster, to retouch images. Vector scan is used for creation and raster scan for display.

The scientific data visualisation application Paraview produces 3D without notions of vectors. So why do we proceed as we do? Why process the image as a set of vectors when there is no question of modifying the simulation -we have not the right to change the data!

WxWidgets

OpenSceneGraph

OpenGl HDF

C++

The idea is actually quite different here; the aim is not to allow modification of the image but to allow the bodies to be tracked in time, by tracing them or, for example, by numbering them within a particular zone.

Tracer

In GIDE a tracer is a visual marker applicable to a body. It enables one or more bodies to be followed throughout the simulation. To apply a marker a zone of space is selected. All bodies within this zone will be marked (Figure 9). The user has several tools at his disposal, to delimit a zone of space: { the point, the line, the plane, the cubic volume }. Each of these tools is directly manipulable with the aid of the mouse. Marking is then done by intersection of the zone with the set of bodies; all the bodies having an intersection, even partial, with the defined zone will be marked.

The following example illustrates the marking of a layer in a silo, and its follow-up during the flowing (Figure 10, 11, 12 and 13).

Sensor

A sensor is defined as an active element. With it, data can be extracted and calculations performed on them. This part is currently limited to applying a calculation formula to each of the bodies encountered by the sensor and to trace the result. Developments are in hand to script the sensors. A sensor will be able to execute a script (python), and the latter will be able to access the body via an API and to produce outputs (Figure 14).

The representation

The display of mechanical phenomena is the most important part. GIDE includes the main possibilities of 3D software: rotations, displacements etc. We have, however, added the capability of following a body in time and tracing the associated curves: we can thus at any moment display the information on this body. The interface currently allows:

• 3D display -of isovalues -in wire frame /hidden face mode, • Rotation, zoom, dynamic translation with the aid of the mouse, • The ability to cut through the structure to examine a field in non-visible parts, • Recording of mpeg formats for films, jpeg for images and svg for exploited data.

Flexibility of representation

In GIDE a discrete element is an indivisible entity, it is a unitary element. It is thus possible to select it, to manipulate it, to track it, to decorate it etc. This approach is very different from representations by meshing. It more closely matches the granular world and enables the virtual experimental medium to be observed and dissected, and its characteristics extracted for comparison with theory.

The drawing of an element in GIDE is done via plugins, affording a representation library adapted to each phenomenon under study. By default the name DATASET is the name of the plugins giving, for example, the possibility of changing the representation during display. Some simulations are done in 2D or the body is considered mechanically as a cylinder when it is actually comprised of spherical elements; in this case, the user needs to change the representation according to the desired approach.

The change in representation is done according to the available plugins. The following figures illustrate the example of the spherical representation becoming cylindrical (Figure 15). Apart from displaying the position of a body, a researcher often studies other physical phenomena: temperature, speed, electric potential etc. He should therefore be able to display these data. The notion of decorators has been implemented in GIDE; it is based on Design Pattern. It allows the graph representing the scene to be modified by adding nodes. The modifications are cumulable with each other. The representation of forces can for example be activated with that of temperatures (Figure 16).

Conclusion

The language C++ was chosen for the development. With this language the OpenGL graphic card capacities can be fully exploited, which is necessary for the processing of 3D scenes rich in bodies. Moreover, as libraries such as OSG are also written in C++, we have been able to exploit them and use them to the full via the inheritance. The data organisation tree in the HDF file has enabled a cache to be implemented; this is managed with the aid of a thread whose parametrisable number has up to now enabled smooth animations. GIDE has a threadsafe cache and OSG is also threadsafe. The graphic interface has bee developed with the aid of the WxWidget toolkit portable on all the OS. This is a display tool for better exploitation of data from a Discrete Element code. It provides a representation that supports the researcher's discourse. Finally, respecting the design patterns during development should allow other developers a relatively rapid learning curve. The presentation of GIDE will be through various digital applications from the code ED MULTICOR developed in the Laboratoire des Technologies Innovantes and the code LMGC90 developed in the LMGC of Montpellier.

Fig. 3

 3 Fig. 3 Representation matrix

Fig. 4 .

 4 Fig. 4. Internal representation 1

Fig. 5 .

 5 Fig. 5. Internal representation 2

Fig. 7 .

 7 Fig. 7. Cube in 3D (40 000 bodies)

Fig. 8 .

 8 Fig. 8. GIDE Architecture

Fig. 9 .

 9 Fig. 9. Figure showing marking of an element

Fig. 10 .Fig. 11 .Fig. 12 .

 101112 Fig. 10. Emptying of a silo

Fig. 14 .

 14 Fig. 14. Curve

Fig. 15 .

 15 Fig. 15. Example of change of representation

Fig. 16 .

 16 Fig. 16. Speed vector

  .

	A	B	C	D	E
	real 0m4.596s	0m0.351s	0m0.974s 11m36.040s 0m0.557s
	user 0m4.023s	0m0.001s	0m0.485s	0m9.820s	0m0.227s
	sys 0m0.417s	0m0.292s	0m0.411s	7m1.452s	0m0.292s
		Fig. 6. Results of the tests

Acknowledgements: we thank Frédéric Dubois of the LMGC (UMR 5508) for the numerous discussions and proposals for the development of GIDE.


Keywords: Public health (I18), accounting (M40), specific statistics (C46) social statistics, public health, pandemic management, non-pharmaceutical interventions (NPI)

Infection, hospitalization and mortality statistics have played a pivotal role in forming social attitudes and support for policy decisions about the 2020-21 SARS-CoV-2 (COVID-19) pandemic. This article raises some questions on some of the most widely-used indicators, such as the case fatality rate, derived from these statistics, recommending replacing them with information based on regular stratified statistical sampling, coupled with diagnostic assessment. Some implications for public health policies and pandemic management are developed, opposing individualistic and holistic approaches.

Introduction

Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write (Samuel S. Wilks, 1951) Starting January 2020, public-health authorities, social media and the press reported daily the number of cases, hospitalisations and deaths related to the SARS-CoV-2 (COVID-19 thereafter) pandemic. These numbers have been the focus of public, medical and policy attention, and driven public sentiment towards epidemic and its evolution (Infantino 2021; Craven McGinty 2020).

Numbers carry an aura of objectivity, and being scientific. Infection, hospitalizations and deaths statistics appear as facts about the consequences and changing status of the pandemic. Monitoring the current statistics is supported by our urgent wishes to control the pandemic, and forecast its future. These numbers make headline news, ranking high in the critical medical and political decisions on pandemic management (Abbasi 2020; Asselineau et al. 2021). 1 The problem is that the reported numbers are poor proxies for the underlying phenomenon (Section 1), and they do not deliver reliable information on it (Section 2). Section 1 discusses how reported numbers on the COVID-19 pandemic do not capture its impact and evolution in a reliable and trustworthy manner. Their relationship with the underlying phenomenon is blurred and potentially misguiding. Section 2 addresses the role of those numbers as guidance for decision-making, discussing how properly-accounted numbers may provide relevant and useful information for pandemic management. An overview of the main argument is given in the concluding section.

-Where do the numbers come from?

Daily counting of cases, hospitalisations and deaths gathers the hazardous wish to report on the pandemic as it is, that is, to deliver the full and complete picture about its diffusion in real time. Unfortunately, we do not and cannot know how the virus has been spreading. And even if we knew it, we do not and cannot have sufficient social control over its diffusion to track and count all and every case. And even if an almighty authority had such control, we should yet design an appropriate measurement frame of reference to gather those numbers properly.

No such a thing as a case exists, as such. Case detection depends on the healthcare testing for infected people, and the time and conditions of such testing. Even if reliable tests were available promptly and uniformly over territories, not all the population would be tested every day for contagion; therefore, the number of reported cases depends on the ways tests have been performed, to whom and when. It follows that indicators such as the case fatality rate -based upon the ratio between the number of deaths and the number of cases over a certain time window -are misleading and cannot help understanding the ongoing evolution of the underlying epidemic (Brown 2020).

The number of deceased might appear more solid at the first glance. Nothing seems surer than death and taxes, as the adage goes. But even this number depends on the ways the decease causes are defined and reported. The fact that someone died for medical causes directly related to the COVID-19 disease depends on the healthcare standards which are applicable and the ways these standards have been applied. What if someone died in a car accident while having tested positive for contagion, should this decease be counted? This would imply a broad definition of deceased with the COVID-19 (association). And what if someone died of pneumonia while having tested positive, should this decease be counted? This would imply a narrower definition of deceased due to COVID-19 (causation) including directly related medical complications. Public health authorities have generally chosen to count deceases when a death occurred in a person with a laboratory-confirmed positive COVID-19 test and either: died within 28/60 days of the first specimen date, or COVID-19 was mentioned anywhere on the death certificate (even if not as primary cause), applying indeed a broad definition (Heneghan and Oke 2020; Craig 2021). 2 No international standard was developed and enforced.

Similar problems occur with the number of either hospital admissions, or people in hospital beds due to or with COVID-19. Their number may include those who tested positive in hospitals, including those already there for other reasons (likely nosocomial infection), and those who had COVID-19 prior to admission and were admitted for other reasons than COVID-19. This counting depends on hospital infection management procedures, and it is further complicated by the origin of admitted people (which could come from nursing or residential care homes), their ongoing treatments and conditions, the overall length of stay and the number of people discharged from hospital with a diagnosis of COVID-19 (Mahon, Jefferson and Heneghan 2020).

Since the number of cases, hospitalisations and deaths depends on both the testing process and the overarching measurement standards, we should be careful when comparing those numbers through time, as well as across different contexts such as countries. Not to forget the fact that measurement standards have been repeatedly amended over time since the pandemic began.

A look at available statistics shows that the relative number of reported cases over the population in various countries differs widely, as does the relative number of reported deaths. 3 For instance, mid-September 2020 (mid-June 2021), around 2% (10%) of US population tested positive, relative to less than 0.5% (7%) in Italy, implying a four-time (+43%) larger contagion in the former country. Does this 2 In fact, when monitoring for SARS-CoV-2 vaccine breakthrough infections, the US CDC COVID-19 Vaccine Breakthrough Case Investigations Team applied a different standard with three mutually exclusive categories (CDC 2021): asymptomatic cases; patients known to be hospitalized but asymptomatic or hospitalized for a reason unrelated to COVID-19; and hospitalised patients likely due to COVID-19. This latter standard allows sorting out asymptomatic people, while disentangling hospitalisations and deceases unrelated to COVID-19. However, a dualistic distinction between vaccinated and unvaccinated patients -vaccinated patients being counted as such since a certain time lapse after the final dose -does not allow disentangling breakthrough infections and other collateral effects which would occur throughout the individual vaccination process. 3 Data retrieved from The Johns Hopkins Coronavirus Resource Center (CRC), https://coronavirus.jhu.edu/ difference depend on distinctive contagion diffusion patterns, or just the number of tests and the ways those tests were performed and reported? Moreover, the case-based lethality rate (that is, the ratio of the number of deceased over the number of reported cases, cumulated through the whole pandemic time window, the latter being dependent on data availability) over the population was around 3% (1.8%) in US and 12% (2.99%) in Italy. 4 Was the virus more lethal in the latter, or was this difference resulting from a different way to count for it?

In this context, reports on excess deaths -that is, deaths in excess of previous years or historical patterns -mingle up a combination of deaths that were not properly attributed to COVID-19, plus collateral damage from containment policies (such as undetected medical conditions and postponed medical procedures including screenings). 5 For instance, a statistical survey by the Italy's Institute of Statistics (ISTAT 2020a) on mortality over the first trimester 2020 shows that, on total excess deaths (25.354 units), only 54% were COVID-19 diagnosed (13.710 units).

In a nutshell, we have been counting the ongoing pandemic impact by testing disparate people who carry COVID-19 virus in their upper respiratory tract; but this condition does not tell us much about their underlying medical status, neither the reasons for their hospitalisation or death. In fact, mass testing with low viral incidence may certainly lead to an overwhelming number of false positives even though single tests are reliable and properly performed. 6 Coronaviruses are known to spread rapidly and mutate frequently over time. It is likely that their circulation becomes endemic, notwithstanding pre-existing immune defences, emergent natural immunity from prior infection and vaccination campaigns. For instance, influenza vaccination has been deployed since the 1940s, somewhat protecting vulnerable people without being able to eradicate viral presence and circulation (Montag 2021). In these circumstances, our current way of counting affords the hazard to keep alerting for the virus prevalence even though its potential danger has diminished and become under control by natural immunities, vaccination and therapies. Imagine if we test for cold all people being admitted to hospitals: We would certainly obtain correctly identified, or true positives) and specificity (proportion of negatives that are correctly identified, or true negatives) are around 80% and 95% respectively. Accordingly, a baseline of 10 000 random tests on a population with 1% viral incidence (implying 100 real cases; an illustrative parameter in line with the results of the phase III studies of COVID-19 vaccines, see Montag 2021) would yield 80 true positives, 20 false negatives, and 495 false positives (5% of the 9 900 negative test results, true negatives being 9 405). Therefore, out of the 575 positive test results, only 80 (13.9%, the latter being the 'positive predictive value' of such testing) are true positives while 495 (86.1%) are false positives. With low viral incidence (1%), even a test with 99% sensitivity and 99% specificity will generate only a 50% positive predictive value (all computations were confirmed by the Bayesian Clinical Diagnostic Model at https://kennis-research.shinyapps.io/Bayes-App/). See also Fenton et al. (2020).

seasonal waves of persons hospitalised with cold, but this latter circumstance would not add any relevant information about their personal medical condition for public health purposes.

Led by weak and misguiding data-gathering, public health and policy decision-makers enacted draconian measures such as repeated and prolonged lockdowns, disrupting social life. But, at the outset of the pandemic (or at latest about four-six months after it), we had sufficient ex ante knowledge to choose an alternative data-gathering strategy that would have served the societies better. The next section will outline this alternative strategy by addressing the use of numbers for pandemic management and policy.

-Not all numbers provide information

If numbers of cases, hospitalisations and deaths are poor proxies for knowing about the ongoing pandemic impact, should we abandon our wish to gather reliable information about its diffusion? How could we manage our response without such information? And since guidance is needed, how can we know about it? Attention paid to the number of cases, hospitalisations and deaths was based on the naïve belief that we can gather the full picture of the ongoing phenomenon in real time, and even forecast it through time and space. As a matter of fact, some scholars have been pursuing a sophisticated version of this very same line of reasoning. They apply models which demand to track individuals and their social interaction, based upon a key parameter called "effective reproduction number" (R t ), a pivotal measure of how fast the virus has been spreading. This parameter points to the average number of people who become infected by an infectious person at some point of time. When R t is above 1.0, the virus will continue spreading over the population; when R t is below 1.0, the virus will progressively stop its diffusion. This 'individualistic' approach aims to track and control individual behaviour. This focus is consistent with social distancing and social interaction breakdown as policy responses, as well as mass vaccination of the entire population, in view to eradicate virus transmission and circulation.

The problem is that tracking individuals in view to command their social interaction would demand quite an impossible effort, not to mention the violation of liberties and breach of fundamental rights which would be required to perform such effort. In fact, even a less ambitious estimation of that parameter R t would be so dependent on timing and circumstances that it would not provide but poor guidance for decision-making. To be sure, these models are recent scientific advances fostered by computational capabilities of today. But their overarching reasoning seems inappropriate to provide policy guidance, especially in exceptional circumstances. 7 At a time when computational capabilities were far more limited, social scientists developed alternative approaches to understand and gather information on social phenomena. Among others, Carl Friedrich Gauss did not become one of the founding fathers of social statistics by tracking individuals throughout a certainly untraceable population. He sampled across the former rigorously, in view to infer reliable information on the latter.

Today, publicly available numbers are poorly defined and have been inconsistently collected. Our overwhelming obsession with case counting, hospitalisation alerts and the death toll is simply meaningless, from a Gaussian perspective. Even research studies have used datasets sampled from patients admitted to hospital, people tested for active infection, or people who volunteered to participate. But Gauss would highlight the struggle of interpreting observational evidence from those non-representative samples. According to Griffith et al. (2020), 'the problem of collider bias (sometimes referred to as selection bias, sampling bias, ascertainment bias, Berkson's paradox) has major implications for many published studies of COVID-19 and is seldom addressed'.

Instead, what counts would be a reliable estimation of infection diffusion over the whole population, and the correlated risk exposure by certain classes of vulnerable people. This modest approach does not require forecasting the pandemic diffusion in real time, but understanding its featuring patterns as a social phenomenon. At least two statistical indicators would be relevant: the risk for the entire population to develop serious illness and encounter severe outcome, and the lethality rate by relevant classes. By disentangling vulnerable classes, this information would also identify nonvulnerable ones, which do neither get infected (protected by pre-existing immune defences or lack of exposure), nor develop symptoms or serious illness and severe outcome. These indicators may be provided through regular stratified statistical sampling, coupled with diagnostic assessment.

By seeking to track and control individual behaviour, individualistic approaches advocated extreme containment measures such as lockdowns and forced vaccination for all. COVID-19 crisis management has been featured by these extreme containment measures, certainly driven by quite a catastrophic misapplication of the precautionary principle (Bhattacharya 2021). 8 Notwithstanding national variants in policy-making, a scare and persuade (and mandate) strategy has been enacted and maintained up to date. 9

Facing ex ante uncertainty about the infection fatality rate, immunities prior and after infection, and the correlates of disease severity, public-health authorities and experts seemed assuming the worst 8 In fact, since summer 2021, no precaution was apparently taken when virtually all citizens were pushed to get injected with genetically engineered experimental vaccines, whereas, in 1990, EU political leaders restrained use of genetically modified organisms (GMO) for human food and animal feed (EU Council 1990). Some public health authorities further deployed campaigns of serial vaccination shots (boosters) certainly aimed to maintain high antibody levels in the population, affording the hazard that antibody-dependent enhancement (ADE) and negative collateral effects may become more likely by repeated mass vaccination for all. Moreover, they pushed vaccination for classes of non-vulnerable people, including when class-specific riskbenefit assessment was uncertain or even contrary to individual well-being. In this context, it is worth remembering that, according to Article 7 of the United Nations' International Covenant on Civil and Political Rights (UN 1976), "no one shall be subjected without his free consent to medical or scientific experimentation", a principle which cannot be derogated even "in time of public emergency which threatens the life of the nation and the existence of which is officially proclaimed" (Article 4). 9 Just one illustrative example from the minutes and papers by the UK Scientific Advisory Group for Emergencies (SAGE) meeting held on 22 March 2020 (SAGE 2020), recommending that "the perceived level of personal threat needs to be increased among those who are complacent, using hard-hitting emotional messaging", while foreshadowing measures such as "use media to increase sense of personal threat", "enacting legislation to compel required behaviours" and "use of social disapproval for failure to comply". See also Kelly (2020) and Rothwell and Desai (2020) on misinformation distorting COVID-19 policies and behaviors. case scenario and acted as if over three out of a hundred infected people will die (Ioannidis 2021)10 ; immunity prior or after infection does not occur; and virtually all the population is potentially exposed and at serious risk of hospitalization and death after infection. However, holistic approaches were able to timely understand the real threat for the population as a whole, along with the factual risks at stake for identifiable vulnerable people. But their claims for a statistical approach to data-gathering remained isolated and unsuccessful (Ioannidis 2020;[START_REF] Lipsitch | Defining the Epidemiology of Covid-19 -Studies Needed[END_REF][START_REF] Kulldorff | COVID-19 Counter Measures Should be Age Specific[END_REF]Finley 2020, interviewing professor Ioannidis;[START_REF] Fenton | COVID-19 infection and death rates: the need to incorporate causal explanations for the data and avoid bias in testing[END_REF][START_REF] Lenzer | The COVID Science Wars. Shutting down scientific debate is hurting the public health[END_REF]Tierney 2021a). 11 At least since spring 2020, information driven from statisticallysound studies was able to reliably identify both the outstanding risk for all, and the relevant classes of vulnerable people, rejecting the worst suppositions behind the forecasting models and their rationale for excess precaution. For instance, a statistical survey by ISTAT (2020b) run between May and July 2020 showed that people who have encountered COVID-19 and showed that estimated seroprevalence was six times higher than the reported cases.12 Since April 2020, the UK's Office for National Statistics (ONS 2021) started running a representative sample testing survey series based on blood test results for antibodies against COVID-19 taken from a randomly selected sub-sample of individuals aged 16 years and over within the community population (referring here to private residential households, and excluding those in hospitals, care homes and/or other institutional settings). Accordingly, development of serious illness and severe outcome was limited to a tiny share of population, the vast majority of cases being mild or asymptomatic (Petersen and Phillips 2020; Jenkins 2020). Similar studies on mortality showed that increased and excess mortalities have been concentrated on the oldest and fragile people, often already beyond median and mean age at the time of death (longer than their cohort's life expectancy when born), affected by chronic diseases and other comorbidities, and living in healthcare facilities with limited life expectancy. 13 Statistical surveys of incidence and caused severity may provide reliable information on infection incidence, ongoing immunisation by prior immune defence, infection or vaccination, and the likelihood that infection leads to severe forms involving hospitalisation or death. Actual severity may vary depending on strain virulence; healthcare infection management; population structural conditions (especially age and comorbidities as far as COVID-19 is concerned); and exposure. Since an epidemic and its caused severity may spread in non-linear ways, survey sampling may be disaggregated by areas and repeated regularly over time. In a similar way, statistical inference may help generating reliable information on ongoing hospital infection treatment processes and associated outcomes. Possibly, local networks of healthcare institutions may be actively involved in the data-gathering process, contrary to received procedures which appear to rely only on idiosyncratic mass testing, thus neglecting sufficient clinical and laboratory evidence (including postmortem examination) and bypassing the medical profession. This 'holistic' approach based upon statistical surveys and inference can generate the relevant and reliable knowledge which is useful for research advancement, policy-making and pandemic management. Public-health authorities may run (and might have run) statistical surveys since the beginning of the pandemic, enabling the scientific community, policy-makers and the citizenry to properly understand its diffusion patterns and emerging risks. Through representative statistical surveys, we can know reliably the factual impact of the ongoing epidemic diffusion over the whole population, and which classes are factually at serious risk. This holistic approach advocates focused protection of vulnerable people, including through voluntary vaccination campaigns, while trusting pre-existing immune defences and emergent natural immunity for non-vulnerable ones.

Concluding remarks

'Muor giovane colui ch'al cielo è caro' [Menandro] (Giacomo Leopardi, 1831) Reviewing the handling of the H1N1 pandemic (the so-called Swine Flu) in 2010, the Parliamentary Assembly of the Council of Europe (CE 2010, par. 1) was alarmed about the way in which the H1N1 influenza pandemic has been handled, not only by the World Health Organization (WHO), but also by the competent health authorities both at the level of the European Union and at national level. It is particularly troubled by some of the consequences of decisions taken and advice given leading to the distortion of priorities of public health services across Europe, the waste of large sums of public money and unjustified fears about health risks faced by the European public at large. Since early 2020, a scare-and-persuade (and mandate) strategy has been in place to manage the COVID-19 pandemic. 14 It resulted in disrupting social interaction, breaching social trust, violating fundamental rights, and spending unprecedented amount of resources, mostly unrelated to providing actual healthcare. 15 Relying on mass testing and tracing, this strategy has struggled to control and forecast pandemic diffusion in real time. The entire population is sought to be 14 See also Tierney (2021b) on unlearned AIDS lessons for COVID-19 pandemic management, and Ferguson (2021) providing a comparative benchmark with the flu pandemic (so-called 'Asian flu') management of 1957-58. 15 According to OECD (2021, p. 5), "OECD governments borrowed USD 18 trillion from the markets in 2020, equal to almost 29% of GDP". In addition, "central banks in 28 OECD countries purchased government bonds in 2020 with more than half of the net purchases occurring during the period between March and May. Total net purchases by major central banks reached USD 4.5 trillion in 2020, more than half of the new securities (i.e. excluding securities issued to roll over existing debt) issued by OECD sovereigns in the year" (ibidem, p. 15). Central banks further provided material financial assistance to private financial and economic actors, including through those sovereign debt management policies. In this context, Sunder (2021) discussed the impact of central bank liquidity provision on US financial markets. vaccinated in hope of fully eradicating virus transmission, through discriminatory measures against the unvaccinated in many cases.

Daily numbers of cases, hospitalisations and deaths have been an integral part of that strategy. But numbers are not facts. They result from measurement processes which should be carefully considered (and designed) when interpreting those numbers. Not all numbers provide reliable guidance for decision-making, but some of them do it better than others.

Those daily numbers of cases, hospitalisations and deaths are meaningless when we wish knowing what has been going on with the pandemic diffusion. On the contrary, statistical techniques are available to reliably understand its emerging patterns and implied risks. Strategies based upon such a 'holistic' approach would advocate stratified sample testing and focused protection of certain classes of vulnerable people, including their selective vaccination, while relying on mutual trust and the collective emergence of natural immunity for the others (Gupta 2021). Table I provides a dualistic comparison between individualistic and holistic approaches to pandemic management. 16 On the one hand, individualistic approaches aim to track and control individual behaviour (and then advocate for draconian measures such as lockdowns and forced vaccination for all); on the other hand, holistic approaches aim to understand the whole of pandemic as a social phenomenon, then advocating for focused protection of vulnerable people based upon a holistic understanding. Pandemic management and policy require seeking for a balance between individual rights and other people's protection within the same risk class. The vexata quaestio is whether draconian measures such as repeated and prolonged lockdowns and forced vaccination for all do apply excess precaution through measures which are neither strictly required by the exigencies of the situation (necessity) nor proportional to factual risks (proportionality), resulting in violating the former in the name of the latter. Especially once evidence of the real threats at stake for all and each one exists, pandemic may be managed in alternative and less disruptive ways.17 And this has been the case since spring 2020 for the COVID-19 pandemic. 18 Neither individuals nor societies can beat the death, eventually. But we can keep living while doing our best effort to cope with old and new diseases, applying sound public health management and policies which respect fundamental rights.

Paris, 12 July 2021

Table I . Dualistic overview on alternative approaches to pandemic management

 I

		Individualistic Approach	Holistic Approach
	Approach Design	Individual behaviour and tracking	System behaviour and sampling
	Accounting framework	Individual tracing and counting	Regular statistical sampling
	Recommended policies	Preventative lockdowns and mass	Focused protection and targeted
		vaccination for all	vaccination of vulnerable people
	Methodological and	Methodological individualism,	Methodological holism,
	epistemological stances	forecasting purpose	understanding purpose
	Overall political philosophy	Draconian precautionary measures	Informed management and
		by authoritarian public health	containment measures respectful
			of individual liberty and
			responsibility

To provide just one illustrative example of this number fetishism, the French government pro-vaccination campaign TV spot titled 'the debate' ('Le débat', disseminated since

August 2021) deploys the slogan: 'you can argue about everything but numbers' ('On peut débattre de tout sauf des chiffres'). URL: https://youtu.be/iLi5kkTGfRo

Data retrieved from The Johns Hopkins Coronavirus Resource Center (CRC), https://coronavirus.jhu.edu/

Mortality in a population is a non-linear phenomenon over time and space. Descriptive statistics may be insufficient to grasp its dynamic. A valid analysis of excess mortality should consider the evolving age structure of population[START_REF] Toubiana | International Covenant on Civil and Political Rights (ICCPR), adopted and opened for signature[END_REF]) and the evolving piling-up of vulnerable people[START_REF] Herby | Sweden's "Dry Tinder" Accounts for Many Covid-19-deaths[END_REF], among other factors.

According to[START_REF] Watson | Interpreting a COVID-19 test result[END_REF], COVID-19 test sensitivity (proportion of positives that are

On the role plaid by the "effective reproduction number" (Rt) in COVID-19 management and policy, and its late introduction in epidemiologic studies, see also[START_REF] Miller | Afterword: Quantifying, Mediating and Intervening: The R Number and the Politics of Health in the Twenty-First Century[END_REF].

A preprint was made available online at medRxiv since 14 July 2020. DOI: https://www.medrxiv.org/content/10.1101/2020.05.13.20101253v3

[START_REF] Chikina | Modeling strict age-targeted mitigation strategies for COVID-19[END_REF] developed a simple model of age-targeted mitigation strategies, published on 24 July 2020. See also[START_REF] Chin | Effect estimates of COVID-19 non-pharmaceutical interventions are non-robust and highly model-dependent[END_REF].

ISTAT (2020b) estimated 1 501 537 infected people (against 243 846 persons who tested positive over the same time window), that it, an incidence of 2.5% over total population (against 0.40%).

Altogether, this information might have fostered an understanding of the syndemic nature of the COVID-19 threat, requiring a "more nuanced approach" in view to protect the health of our communities[START_REF] Horton | Offline: COVID-19 is not a pandemic[END_REF].

Methodological individualism and methodological holism refer here to methodological and epistemic stances; see also[START_REF] Gupta | We may already have herd immunity -an interview with Professor Sunetra Gupta[END_REF].

In this context, it is worth remembering that, in the middle of the COVID-19 pandemic, the Council of Europe's resolution No. 2361 (CE 2021) asks governments, 'with respect to ensuring a high vaccine uptake', to 'ensure that citizens are informed that the vaccination is not mandatory and that no one is under political, social or other pressure to be vaccinated if they do not wish to do so' (7.3.1); and to 'ensure that no one is discriminated against for not having been vaccinated, due to possible health risks or not wanting to be vaccinated' (7.3.2). According to Article 6 of UNESCO (2005)'s Universal Declaration on Bioethics and Human Rights: 'any preventive, diagnostic and therapeutic medical intervention is only to be carried out with the prior, free and informed consent of the person concerned, based on adequate information. The consent should, where appropriate, be express and may be withdrawn by the person concerned at any time and for any reason without disadvantage or prejudice.' And the Article 5 of the Convention on Human Rights and Biomedicine (CETS 1997) states that '(i) an intervention in the health field may only be carried out after the person concerned has given free and informed consent to it. (ii) This person shall beforehand be given appropriate information as to the purpose and nature of the intervention as well as on its consequences and risks. (iii) The person concerned may freely withdraw consent at any time.'

According to[START_REF] Packalen | Lockdown Harms and the Silence of Economists, Collateral Global[END_REF], 'politicians, civil servants, and even scientists now constantly hide behind the "follow the science" mantra rather than admit that science merely helps us make more informed decisions. We no longer dare acknowledge that -because our choices always involve tradeoffs -the virtue of pursuing one course of action over another always rests not just on the knowledge we get from science but also on our values. We have seemingly forgotten that scientists merely produce knowledge about the physical world, not moral imperatives about actions that involve trade-offs. The latter requires understanding our values.'


Franco Nicolas

Pinan Basualdo

Ronald Terrazas

Benoit Scheid

Aude Bolopion

Michaël Gauthier

Pierre Lambert

Franco N Piñan Basualdo

R Terrazas Mallea

Effect of insoluble surfactants on a thermocapillary flow

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Whenever a gradient of temperature exists along a fluid interface, an interfacial flow will arise due to the resulting surface stress [START_REF] Scriven | The Marangoni effects[END_REF] . This effect, known as thermocapillary effect, has a strong impact in industrial processes, like welding 2 and solidification [START_REF] Preisser | Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface[END_REF] . More recently, several applications have been developed in the field of microfluidics: a thermocapillary pump [START_REF] Amador | Thermocapillary-driven fluid flow within microchannels[END_REF] , droplets generation [START_REF] Zhao | Uniform breaking of liquid-jets by modulated laser heating[END_REF] and migration [START_REF] Ryazantsev | Thermo-and soluto-capillarity: Passive and active drops[END_REF][START_REF] Hu | Instability of thermocapillary-buoyancy convection in droplet migration[END_REF] , bubbles migration [START_REF] Rahman | Cooperative micromanipulation using the independent actuation of fifty microrobots in parallel[END_REF][START_REF] Lu | Numerical study of thermocapillary migration of a bubble in a channel with an obstruction[END_REF] , and particles manipulation [START_REF] Dietrich | Microscale Marangoni surfers[END_REF] . These applications require understanding and controlling the thermocapillary flow. Experimentally, it can be difficult to obtain a perfectly clean fluid interface. Adsorbed impurities act like surfactants that rigidify the interface [START_REF] Lucassen-Reynders | Gibbs elasticity, surface dilational modulus and diffusional relaxation in nonionic surfactant monolayers[END_REF][START_REF] Champougny | Surfactant-induced rigidity of interfaces: a unified approach to free and dip-coated films[END_REF] , effect known as surface elasticity, and can affect the thermocapillary flow. The presence of such an adsorbed layer of surfactants has been observed to cause symmetry breaking instabilities [START_REF] Mizev | On the shear-driven surfactant layer instability[END_REF][START_REF] Koleski | Azimuthal instability of the radial thermocapillary flow around a hot bead trapped at the water-air interface[END_REF] under some conditions. The stable regime has also been studied [START_REF] Shmyrov | On the extent of surface stagnation produced jointly by insoluble surfactant and thermocapillary flow[END_REF] in a thin water layer, under 2D configuration. Therefore, in this manuscript we close the gap and study the effect of an adsorbed layer of surfactants on the symmetric thermocapillary flow in a cylindrical configuration and a thick water layer.

In particular, we analyse the case where an infrared (IR) laser is used to generate a localized temperature increase at an air-water interface, inducing a thermocapillary flow. Such a configuration has been previously used to manipulate particles at the air-water interface [START_REF] Terrazas | Laser-induced thermocapillary convective flows: A new approach for noncontact actuation at microscale at the fluid/gas interface[END_REF][START_REF] Piñan Basualdo | A microrobotic platform actuated by thermocapillary flows for manipulation at the air-water interface[END_REF] . Water was selected as a fluid due to its high surface tension and low viscosity, giving place to high fluid velocities a) Electronic mail: Franco.Pinan.Basualdo@ulb.be (≈ 30 mm/s). However, since water is a polar fluid, impurities can easily be trapped at the interface, which makes almost impossible to have a completely pure air-water interface. In Section II, we first introduced the theoretical frame and our particular problem. Then in Section III, we present the developed numerical model and compare its results and with three experimental measurements: Particle tracking velocimetry, convection cell radius and thermography measurements. Finally, in Section IV we present the developed analytical model and compare its predictions with the simulation's and experiments' results. We find that no matter how small the amount of impurities is, the induced surface elasticity always has an effect on the flow, similarly to how a low fluid viscosity can completely change the flow profile compared to the inviscid case. The developed models help us to better understand the flow generation and give us insights towards the miniaturization of the thermocapillary manipulation platform.

II. PHYSICAL CONSIDERATIONS

In this section, we first introduce the theoretical frame of the work. Then, we present the particular problem we focus on, and we introduce the non-dimensional parameters governing the system.

A. Theoretical frame

Surface tension (σ ) can be defined as the required energy to increase the area of an interface per unit area (σ = ∂ E/∂ A). It is known that surface tension depends on the temperature and on the concentration of some species [START_REF] Nguyen | Influence of hydrophilicity on the thermal-driven surfactant flow at the air/water surface[END_REF] . In particular, some big molecules, known as surfactants, that have a hydrophobic tail and a hydrophilic head, adsorb at air-water or oil-water interfaces. They adsorb with their heads in the water phase and their tails in the other one, and reduce the surface tension [START_REF] Eastoe | Dynamic surface tension and adsorption mechanisms of surfactants at the air-water interface[END_REF] . When a gradient of surface tension exists along the interface, whether it is due to a temperature or concentration gradient, a surface stress arises (τ σ = ∇ S σ ). This stress is balanced by the viscous shear stress of each phase, resulting in

µ 1 ∂ U 1 ∂ n -µ 2 ∂ U 2 ∂ n = ∇ S σ (1)
where U i and µ i are the velocity vector and the dynamic viscosity of the fluid i respectively, and n is the interface normal vector pointing from phase 1 to phase 2. The operator ∇ S is the surface gradient operator (∇ S σ = ∇σ -n (n • ∇σ )). In case of a liquid-gas interface, the gas' phase viscous stress can be neglected in (1). If we also assume that there is only one species of surfactant, we can write

µ ∂ U ∂ n = -(γ T ∇ S T + γ Γ ∇ S Γ) ( 2
)
where T is the temperature, Γ is the surface surfactant concentration, γ T = -∂ σ /∂ T and γ Γ = -∂ σ /∂ Γ are positive values hereby assumed constant. This last approximation is valid for low surfactant concentrations (for higher concentrations ∂ σ /∂ Γ = -E S /Γ where E S if the surface elasticity [START_REF] Champougny | Surfactant-induced rigidity of interfaces: a unified approach to free and dip-coated films[END_REF] ). It can be seen that, through (2), the momentum equation gets coupled with the energy and the surfactant transport equations at the interface. The surfactant transport equation can be complex for soluble surfactants, due to adsorption dynamics, or deformable interfaces [START_REF] Stone | A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface[END_REF] , but in the case of an insoluble surfactant and a non-deformable interface, this equation is reduced to

∂ Γ ∂t + ∇ S • (UΓ) = D S ∇ S 2 Γ, (3)
where D S is a surface diffusion coefficient.

B. Particular problem

In our case [START_REF] Terrazas | Laser-induced thermocapillary convective flows: A new approach for noncontact actuation at microscale at the fluid/gas interface[END_REF][START_REF] Piñan Basualdo | A microrobotic platform actuated by thermocapillary flows for manipulation at the air-water interface[END_REF] , we study the geometry shown in Fig. 1. A cylindrical volume of water whose top surface, in contact with air and where a layer of insoluble surfactants is adsorbed, is heated from above by an infrared laser beam pointed at the centre of the water-air interface. Qualitatively, once the laser is turned on, the temperature at the centre increases, creating a positive radial surface tension gradient, giving place to a positive radial flow. Because of mass conservation, a negative radial flow arises deep below the surface, generating a toroidal convection cell as shown by the arrows in Fig. 2a (Multimedia view). The radial flow at the free surface generates a depletion of surfactants around the laser spot, increasing the surface tension and counteracting the effect of the temperature increase.

Besides the insolubility of the surfactants, we make further assumptions to simplify the modelling: the interface deformations, the heat transfer to the air phase, and the buoyancy forces are neglected.

Indeed, surface deformations are considerable only for thin fluid layers [START_REF] Sen | Steady thermocapillary flows in two-dimensional slots[END_REF][START_REF] Messmer | Confined thermo-capillary flows in a double freesurface film with small Marangoni numbers[END_REF] . The vertical deformation of the interface in a similar configuration [START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF] has been predicted to be ∆H/R L = γ T ∆T /σ 0 . If we consider ∆T ≈ 2 K (the measured temperature increase for a laser power P L = 40 mW), we obtain ∆H/R L ≈ 0.004, thus validating our assumption.

On the other hand, the heat transfer towards the air phase can be compared to the heat conduction in the water phase through the Biot number Bi = hR L /κ, where h is the heat transfer coefficient to the air phase. Typical values for free convection of air [START_REF] Kosky | Chapter 14 -mechanical engineering[END_REF] are h < 25 W/m 2 K, thus obtaining Bi < 0.03 and validating our assumption.

Finally, the strength of buoyancy forces can be compared to thermocapillary forces through the dynamic Bond number Bo d = ρβ gR 2 L /γ T = 0.005, where β = 200 × 10 -6 K -1 is the thermal expansion coefficient of water, and g = 9.8 m/s 2 is the gravitational acceleration.

In order to explain the absorption of the energy provided by the laser, we consider the light attenuation coefficient of water α = 3280 m -1 defined for the used wavelength (λ = 1455 nm). Assuming a collimated Gaussian beam with a waist radius R L , the absorbed volumetric power qL (energy per unit volume and unit time) is [START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF] qL

= 2P L α πR L 2 exp -2 r R L 2 -α z (4)
where P L is the laser power, r the radial coordinate, and z the vertical coordinate, with origin at the interface. Finally, (3) and (4) together with the mass, momentum (Navier-Stokes), and energy conservation complete the set of equations. Zero velocity, homogeneous temperature (T = T 0 ) and homogeneous surfactant concentration (Γ = Γ 0 ) are used as initial conditions. Regarding boundary conditions, we considered no slip, thermal insulation and zero surfactant flux boundary conditions on the lateral recipient walls, no slip and constant temperature on the bottom wall, and zero vertical velocity, (2), and thermal insulation at the air-water interface. For the detailed set of equations in cylindrical coordinates under the assumption of a flat interface, see the Supplementary Material. These equations can be made non-dimensional by using R L as the characteristic length, t = tµ/ρR 2 L as the non-dimensional time, Ũ = UρR L /µ as the non-dimensional velocity, P = PρR 2 L /µ 2 as the non-dimensional pressure, Θ = (T -T 0 )κR L /P L as the non-dimensional temperature, and Γ = Γ/Γ 0 as the non-dimensional surfactant concentration. The resulting set of equations is

∇ • Ũ = 0 (5) ∂ Ũ ∂ t + Ũ • ∇ Ũ = -∇ P + ∇2 Ũ (6) ∂ Θ ∂ t + Ũ • ∇Θ = 1 Pr ∇2 Θ + qL (7)
where qL = 2ψ exp -2r 2ψ z /π is the dimensionless absorbed volumetric power, ψ = αR L describes the aspect ratio of the heated volume, and Pr = µc p /κ is the Prandtl number. At the inter-

C S = ργ Γ Γ 0 R L /µ 2 20 to 80 face, (3) and (2) result ∂
Γ ∂ t + ∇S • ( ŨC) = 1 Sc ∇2 S Γ (8)
∂ Ũ ∂ ñ = -C T ∇S Θ +C S ∇S Γ ( 9
)
where Sc = µ/ρD S is the surface Schmidt number, C T = ργ T P L /µ 2 κ is the thermocapillary number, and C S = ργ Γ Γ 0 R L /µ 2 is the solutocapillary number. For the numerical values of these nondimensional parameters, see Table II. For the detailed set of non-dimensional equations in cylindrical coordinates under the assumption of a flat interface, see the Supplementary Material.

III. RESULTS

In this section, we first present our numerical model and its results, observing three different regimes. Secondly, we compare the simulation results with three different experimental measurements (with an uncontrolled initial surfactant surface pressure): particle tracking velocimetry (PTV), convection cell radius and thermography measurements. Finally, a controlled quantity of surfactants was added to the interface and the thermography measurements was repeated, validating our numerical model without any fitting parameter.

A. Simulation results

Finite elements, axisymmetric, time dependent simulations were performed using Comsol multiphysics with the parameters shown in Table II, Hr = 10, and Rr = 40. For more details on the geometry discretization, see Supplementary Material. The results have been observed to be independent on the recipient height Hr and its radius Rr provided they are larger than a minimum threshold value ( Hr > 7.5, Rr > 30 respectively) for the standard case (C T = 1 × 10 4 and C S = 30).

Depending on the values of C S and C T , three regimes were observed: the convective, the transition and the conductive regime. If C S is small enough and C T large enough, we observe the convective regime. In this regime, a central area with no surfactant and a positive radial flow is obtained as shown in Fig. 2 (Multimedia view). Beyond a given radius, the surface tension gradients induced by temperature and concentration cancel each other and, according to (9), the shear stress at the surface vanishes. This radius coincides with R cell , the radius at which the surface radial velocity vanishes, (see the nondimensional Rcell = R cell /R L in Fig. 2b). In addition, in this regime the maximum fluid velocity and temperature increase have been found to be independent of C S . This is the regime we observe experimentally without added surfactant.

The transition regime is obtained for intermediate values of C S and C T . In this regime, Rcell approaches unity and the maximum fluid velocity and temperature were found to depend on C S (the first one decreases and the second one increases with C S ).

Finally, for large enough values of C S and small enough values of C T , we obtain the conductive regime. In this regime, the effect of the temperature is not strong enough to create a complete depletion of surfactants at the laser spot. The stable state is a zero fluid velocity in the whole domain, since the surfactants completely counteract the temperature gradient. Therefore, the temperature profile is governed by heat conduction alone. We can find the limiting value for this regime as the point where the maximum temperature surface pressure is equal to the surfactant surface pressure:

C Slim = C T Θ c
where Θ c is the value of Θ at the centre of the interface for a conduction only case. By performing a conduction only simulation to estimate Θ c , we find C Slim ≈ 0.2C T .

B. Experimental results

The experimental validation of the above described numerical results was performed through three experimental observations (with an unknown surface contamination): particle tracking velocimetry (PTV), convection cell radius and thermography measurements. All this measurements were performed in the convective regime. The PTV measurement allowed us observe the effect of contamination, but it could not be performed exactly at the interface. The convection cell radius measurement could be performed at the interface, but it does not allow us to observe the radial distribution. Finally, the thermography measurement allowed us to obtain the temperature distribution at the interface. Then, a controlled quantity of surfactant was added to the interface and the thermography measurements was repeated to validate our numerical model without any fitting parameter. However, with the added surfactant, the system was in the transient regime.

PTV measurements

To measure the flow velocity close to the air/water interface, the particle tracking velocimetry (PTV) method [START_REF] Dracos | Particle tracking velocimetry (ptv)[END_REF][START_REF] Ferrari | Particle tracking velocimetry and accelerometry (ptva) measurements applied to quasi-twodimensional multi-scale flows[END_REF] was used. A sketch of the setup is shown in Fig. 3a. The optical system is composed of a green laser (MGL-III-532 200 mW) which is coupled to a fiber (Thorlabs P5-460B-PCAPC-1). Following the design methodology presented in Ref. 27, a set of convex and concave lenses was used to generate a thin collimated laser sheet, which was then used to illuminate the air-water interface. Polyamide seeding particles with a diameter of 50 µm (Dantec Dynamics PSP-50) were submerged in the water bulk and stirred before each measurement. Because the seeding particles were only slightly denser than water (ρ = 1030 kg/m 3 ) the convective flow was strong enough to move them along. The laser sheet was placed so that it would be as close as possible to the interface but ensuring that enough particles could be detected by the camera to reconstruct the flow velocity around the laser spot using the software presented in Ref. 28. As the exact experimental immersion depth of the laser sheet is unknown and due to its strong influence on the fluid velocity, it was decided to turn it into a fitting parameter. The immersion depth mainly affects the maximum value for the radial velocity while C S mainly affects the point at which the velocity becomes zero. First an immersion depth of 80 µm (comparable with the size of the particles) was fitted using the maximum velocity value. Then, C S was fitted between 30 and 75 to rationalize the measurements. The process is shown in Fig. 3b. The variation between each experimental measurement can be explained by a different initial surfactant concentration in each water sample.

Our model considering an initial surface contamination succeeds in predicting a limited range of the laser effect, in contrast with the model that did not consider the surface contamination. Moreover, our model can explain the difference between measurements as the result of different contamination levels among different water samples.

Convection cell radius at the surface

To eliminate the immersion depth dependence, we decided to look directly at the interface. Some aluminium flakes (typical size 71 µm) were deposited on the water surface, where they float thanks to the surface tension [START_REF] Vella | Floating versus sinking[END_REF] . They are used as tracers to observe the surface flow since their inertia is low (Stokes number St < 0.3). Then, the laser was pointed towards a cluster of these floating flakes for at least 10 s, where the generated thermocapillary flow repelled the flakes from the laser. The mean radius R cell of the area without flakes was estimated using the area entrapped by the dashed line shown in Fig. 4a (Multimedia view) where the central circle represents the laser spot. For the simulations results, since the surface velocity never becomes exactly zero, the non-dimensional radius Rcell was defined as the point at which the non-dimensional radial velocity at the surface ũ becomes lower than 0.3. The obtained Rcell value is not very sensitive to the threshold value in the range 0.3 to 1, but quickly increases for lower values (for more details, see Supplementary Material). The comparison between experimental measurements and simulations can be seen in Fig. 4b where the measurements agree with the results obtained with a fitting parameter C S = 25 (which is comparable to the lower value used to fit the PTV measurement C S = 30). In this case, we did not observe a considerable variation among different water samples.

Thermography of the surface

An experimental temperature measurement was obtained imaging the interface from above with an infrared camera Gobi-384. This eliminates both the immersion depth effect and the aluminium flakes distortions of the flow. On the other hand, thermography measurements at the fluid interface are not easy [START_REF] Rubio | Without bounds: A scientific canvas of nonlinearity and complex dynamics[END_REF] , since the expected temperature increase are quite small (2 K and 8 K for 40 mW and 300 mW laser power respectively). Such a technique has been successfully used to study the evaporation of sessile water droplets [START_REF] Girard | Infrared thermography investigation of an evaporating sessile water droplet on heated substrates[END_REF] . A thermal image is taken before turning on the laser and another one 30 s after the laser has been turned on. The temperature map shown in Fig. 5a was obtained as the difference between both images. The temperature reading depends on both the camera sensor and ambient temperature and these can change during the measurement. Therefore, sometimes it was necessary to add a constant offset (< 0.5 K) to the temperature map to make the temperature increase vanish at large distances from the laser spot. Then, the point with the maximum temperature is defined as the laser centre, and the temperature and distance from that centre is computed for each pixel. The results are compared to the simulations as shown in Fig. 5b, fitting C S to 60. The same experiment was repeated for C T varying from 1 × 10 4 to 4 × 10 4 and the fitting parameter C S resulted always between 25 and 60. It is worth mentioning that while the convection cell radius strongly depends on C S , in the conductive regime the maximum temperature does not.

Controlled surfactants

Finally, we added a controlled quantity of a known insoluble surfactants to the interface to compare simulations and experiments without any fitting parameter. Oleic acid (OA) was chosen as a surfactant since it has been used in previous works [START_REF] Shmyrov | On the extent of surface stagnation produced jointly by insoluble surfactant and thermocapillary flow[END_REF] . First, a Langmuir trough (MicroTrough XS, Kibron Inc.) was used to measure the change of surface tension as a function of the surface concentration of OA. The corresponding C S value as a function of the surface concentration is shown in Fig. 6. Two different regimes are clearly observed. For concentrations below a threshold value (Γ e = 1.37 molecule/nm 2 ) the variation of surface pressure with the OA concentration is very low. Meanwhile, for concentrations larger than Γ e , the surface pressure quickly increases until reaching a saturation. face. Smaller concentrations produce results that are hard to distinguish from those produced with no added surfactants, and at larger concentrations (above Γ e ), the surface pressure becomes large enough to almost stop the flow. These molecules, besides surface elasticity, could also induce surface shear and dilatational viscosity [START_REF] Earnshaw | Surface viscosity of water[END_REF][START_REF] Choudhury | Enhanced stability of free viscous films due to surface viscosity[END_REF] . In our configuration, like in many other experimental setups, the effect of surface elasticity and surface dilatational viscosities cannot be distinguished. Then, the previously described procedure was followed to perform the thermography measurement. The results agree with the simulation using the corresponding C S value. When the effect of surfactant is much weaker than the thermocapillary flow, there is a good agreement with simulations as shown in Fig. 7a. On the other hand, when the surfactant concentration becomes higher, there is an agreement with the simulations in the maximum temperature increase as shown in Fig. 7b, but the plateau observed in the simulation is not observed in our experimental measurement. This can be due to second order effects not accounted for in our model, like the induced surface viscosity, or the fact that the added surface surfactant concentration is close to Γ e where the relationship between surface concentration and surface tension stops being linear, as shown in Fig. 6. Moreover, we cannot guarantee that the initial surfactant concentration was homogeneous over the entire workspace.

We would like to mention that with the added OA, we observe some instabilities at higher laser powers (C T > 4 × 10 4 ). These instabilities can be explained by the competition between the thermocapillary and solutocapillary effects [START_REF] Mizev | On the shear-driven surfactant layer instability[END_REF][START_REF] Koleski | Azimuthal instability of the radial thermocapillary flow around a hot bead trapped at the water-air interface[END_REF] . We will not discuss these instabilities here since they are out of the scope of this article.

IV. DISCUSSION

In this section we present an analytical model we have developed to explain our simulations and experimental results. Similar configurations have been analytically analysed [START_REF] Bickel | Effect of surface-active contaminants on radial thermocapillary flows[END_REF] under the assumption of negligible momentum and thermal convection could be neglected (Re << 1 and Pe T << 1), decoupling the temperature and momentum equations. In such a case, both the fluid velocity and the temperature increase turn out to be proportional to the laser power. However, for the case of water and a convection dominated scenario like ours (local Péclet number Pe = Pr ũ ≈ 100), no analytical model has been reported. Therefore, we developed a simple model using the concept of boundary layers as it was done in Ref. 35 for the solutocapillary effect. First we analyse the clean interface case (without surfactant). We assume that away from the laser beam (r > 1) there are both a viscous and a thermal boundary layers whose thicknesses are [START_REF] Schlichting | Boundary-layer theory[END_REF] respectively

δµ ∝ r ũS (10a) δT = δµ Pr 1/2 ∝ r Pr ũS ( 10b
)
where ũS is the non-dimensional radial fluid velocity and the sub-index S means at the interface. If we apply (7) to the interface for r > 1 ( qL ≈ 0) under a steady state condition and neglecting the conduction term in the radial direction, we obtain ũS

∂ Θ/∂ r = Pr -1 ∂ 2 Θ/∂ z2 . If we approximate at the interface ∂ 2 Θ/∂ z2 ∝ -Θ S / δ 2 T , the result is ∂ Θ ∂ r ∝ - Θ S r . (11)
From a surface stress balance point of view, we can approximate at the interface ∂ ũ/∂ z ∝ ũS / δµ and use (11) in (9) obtaining

Θ S ∝ r1/2 ũ3/2 S C T . (12)
Then, since most of the energy brought by the laser will be evacuated through the thermal boundary layer by convection, i.e. P L ∝ r δ av ρ c p (T S -T 0 ) u S , where δ av = Pr -1/4 δ µ is the geometric mean between δ µ and δ T . In non-dimensional form, this balance results

1 ∝ r Pr 3/4 δµ Θ S ũS . (13)
Finally, combining ( 12) and ( 13) for r = 1, and assuming ũmax ≈ ũS (r = 1) and

Θ max ≈ Θ S (r = 1) we obtain ũmax = AC 1/2 T Pr -3/8 (14)
Θ max = BC -1/4 T Pr -9/16 . (15)
We can fit the simulations results with A = 0.43 and B = 0.54 as shown in Figs. 8a and 8b respectively. Experimental temperature results also agree with both simulations and model results. As a comparison, in the case of a conduction dominated scenario [START_REF] Bickel | Effect of surface-active contaminants on radial thermocapillary flows[END_REF] , it was found that ũmax ∝ C T . If now we consider the presence of surfactants, we assume that in the convective regime (1 << Rcell ), ( 14) and ( 15) are still valid. Under this assumption, we find R cell as the point at which the thermocapillary and solutocapillary effect cancel each other (C T Θ S ≈ C S ). Therefore, we propose Rcell = f (C T Θ max /C S ), if we assume a power law, we obtain

Rcell = E C 3/4 T C -1 S Pr -9/16 ξ . (16)
The simulation results can be fairly fitted with E = 1.3 and ξ = 2/3 as shown in Fig. 8c. The discrepancy between the model and simulations for Rcell > 20 can be explained by the fact that we simulated a finite volume of water ( Rr ≈ 40). We simulated all the combinations of Pr =1, 3, 7 and 14, C T =(5, 10, 20 and 40)×10 [START_REF] Preisser | Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface[END_REF] ; and C S =10, 20, 40 and 80. When Rcell ≈ 1 (approaching the transition regime), the velocity ũmax becomes smaller than the value predicted by the model and Θ max becomes larger than the value predicted by the model.

V. CONCLUSION

In this manuscript, we analysed the effect of an adsorbed layer of surfactants on a laser-induced thermocapillary flow. Qualitatively, an infrared laser is pointed towards a water volume from above, locally heating the air-water interface. The temperature gradient triggers an interfacial thermocapillary flow away from the laser spot. Due to the mass conservation, a flow towards the laser spot arises deep below the surface, generating a toroidal-like convection cell around the laser spot. The interfacial flow generates a depletion of surfactants around the laser spot, increasing the surface tension and counteracting the effect of the temperature. Analysing the problem, we found 4 governing non-dimensional parameters (besides geometric ones): Pr = µc p /κ is the Prandtl number, Sc = µ/ρD S is the surface Schmidt number, C T = ργ T P L /µ 2 κ is the thermocapillary number (nondimensional laser power), and C S = ργ Γ Γ 0 R L /µ 2 is the solutocapilary number (non-dimensional surfactant concentration).

A numerical thermo-soluto capillary model was developed and applied to the problem described above. This model suggests that a (no matter how small) surfactant concentration affects the thermocapillary flow, reducing the size of the convection cell. Its results were compared with three different experimental observations (PTV, convection cell radius and thermography). The PTV and convection cell radius measurements are more sensitive to the surface contamination, but they necessarily modify the flow by introducing particles in it. On the other hand, the thermography measurement allowed us to obtain the temperature distribution at the interface, but this measurement is less sensitive to the surface contamination than the convection cell radius. Firstly, we performed these measurements on different water samples with an uncontrolled surface contamination, obtaining good agreement with the same range for C S between 25 and 75. Then, a known quantity of oleic acid was added as surfactant and the thermography measurements was repeated. The result was in good agreement with the simulations using the corresponding value for C S = 220 and C S = 440, thus validating the numerical model without fitting parameters.

Also, a simplified analytical model was developed for sufficiently low values of C S . We observed a good agreement between this model and the numerical one in the explored range of parameters (Pr, C T and C S ). We are confident that this simple model can be extended to other configurations as long as thermal convection dominates over conduction.

Such a system could be used to determine the level of contamination of an air-water interface. The convection cell radius measurement is very sensitive to the initial surface contamination level and could be used to characterize slightly contaminated interfaces where a simple surface tension measurement would not have enough resolution. On the other hand, the study of the flow gives us insights towards the miniaturization of the manipulation platform [START_REF] Piñan Basualdo | A microrobotic platform actuated by thermocapillary flows for manipulation at the air-water interface[END_REF] giving the parameters defining the range of the thermocapillary flow. For example, we see that by increasing slightly the surface concentration of surfactants, we could reduce the range of the thermocapillary flow without reducing its effect close to the laser spot.

SUPPLEMENTARY MATERIAL

See supplementary material for the expanded governing equations in cylindrical coordinates, details on the geometry discretization, a discussion on the chosen threshold value to define the simulations cell radius, and an analysis of the effect of each governing non-dimensional parameter in the surface velocity and temperature. FIG. 8: Comparison between the simplified model ( 14), ( 15) and ( 16), the simulation results and experimental measurements. For the simulations results, we simulated all the combinations of Pr =1, 3, 7 and 14; C T =(5, 10, 20 and 40)×10 3 ; and C S =10, 20, 40 and 80.

SUPPLEMENTARY MATERIAL A. Equations in cylindrical coordinates

The mass, radial and vertical momentum, and energy equations in cylindrical coordinates result:

1 r ∂ (r u) ∂ r + ∂ v ∂ z = 0 (S1) ρ ∂ u ∂t + u ∂ u ∂ r + v ∂ u ∂ z = - ∂ P ∂ r + µ 1 r ∂ ∂ r r ∂ u ∂ r - u r 2 + ∂ 2 u ∂ z 2 (S2) ρ ∂ v ∂t + u ∂ v ∂ r + v ∂ v ∂ z = - ∂ P ∂ z + µ 1 r ∂ ∂ r r ∂ v ∂ r + ∂ 2 v ∂ z 2 (S3) ρc p ∂ T ∂t + u ∂ T ∂ r + v ∂ T ∂ z = κ 1 r ∂ ∂ r r ∂ T ∂ r + ∂ 2 T ∂ z 2 + qL (S4)
where the cylindrical symmetry was assumed, u and v are the radial and vertical components of the velocity. The surface surfactant transport (3) and the surface stress balance (2) in cylindrical coordinates and assuming a flat interface result

∂ Γ ∂t + 1 r ∂ (r u Γ) ∂ r = D S 1 r ∂ ∂ r r ∂ Γ ∂ r (S5) µ ∂ u ∂ z = -γ T ∂ T ∂ r + γ Γ ∂ Γ ∂ r (S6)
The non-dimensional version of (S1), (S2), (S3) and (S4) result

1 r ∂ (r ũ) ∂ r + ∂ ṽ ∂ z = 0 (S7) ∂ ũ ∂ t + ũ ∂ ũ ∂ r + ṽ ∂ ũ ∂ z = - ∂ P ∂ r + 1 r ∂ ∂ r r ∂ ũ ∂ r - ũ r2 + ∂ 2 ũ ∂ z2 (S8) ∂ ṽ ∂ t + ũ ∂ ṽ ∂ r + ṽ ∂ ṽ ∂ z = - ∂ P ∂ z + 1 r ∂ ∂ r r ∂ ṽ ∂ r + ∂ 2 ṽ ∂ z2 (S9) ∂ Θ ∂ t + ũ ∂ Θ ∂ r + ṽ ∂ Θ ∂ z = 1 Pr 1 r ∂ ∂ r r ∂ Θ ∂ r + ∂ 2 Θ ∂ z2 + qL (S10)
where qL = 2ψ exp -2r 2 + ψ z /π is the dimensionless absorbed volumetric power, ψ = αR L describes the aspect ratio of the heated volume, Pr = µc p /κ is the Prandtl number. At the interface, the non-dimensional version of (S5), (S6) result

∂ Γ ∂ t + 1 r ∂ (r ũ Γ) ∂ r = 1 Sc 1 r ∂ ∂ r r ∂ Γ ∂ r (S11) ∂ ũ ∂ z = -C T ∂ Θ ∂ r +C S ∂ Γ ∂ r (S12)
where Sc = µ/ρD S is the surface Schmidt number, C T = ργ T P L /µ 2 κ is the thermocapillary number, and C S = ργ Γ Γ 0 R L /µ 2 is the solutocapillary number.

B. Simulations: geometry discretization and cell radius definition

Finite elements, axisymmetric, time dependent simulations were performed using Comsol multiphysics with the parameters shown in Table II, Hr = 10, and Rr = 40. We used a quadrilateral mesh to discretize the geometry, with finer elements close to the interface and close to the laser spot (symmetry axis). The minimum non-dimensional element size was 0.09 and the maximum is 0.44. The convergence of the simulations was verified for the standard case and the worst-case as shown in Fig. S1. For the worst case scenario, the simulation seems not to be completely converged, but the variation with respect to the converged value is less than 0.5 %. We used a P2+P1 discretization for the momentum, and a quadratic Lagrange for the energy and surfactant transport equations.

On the other hand, we need to define a threshold velocity value to define the size of the cell, since the surface velocity never reaches zero due, presumably, to numerical errors. The results are almost not sensitive to the threshold value in the range 0.3 to 1.0 as shown in Fig. S2. Therefore, we decided to take the lowest value in this range (0.3) as the threshold value for all simulation results.

C. Surface velocity and temperature dependence on the governing non-dimensional numbers

The effect of Pr and C T on the surface variables is shown in Figs. S3 and S4 respectively. It can be seen how decreasing the Prandtl number (Pr) has a similar effect than increasing the thermocapillary number (C T ).

The effect of C S on the surface variables is shown in Fig. S5. For C S < 40, the convection cell radius strongly depends on C S , but the maximum velocity remains almost constant. For C S > 40, the maximum fluid velocity starts to diminish as Rcell approaches unity. On the other hand, we observe how the temperature distribution remains almost constant for C S < 40. For C S > 40, the maximum temperature starts to increase due to the reduced convection. The effect of Sc and ψ on the surface variables is shown in Figs. S6 and S7 respectively. It can be seen that both the surface velocity and temperature distribution do not strongly depend on the Sc and ψ values. The fact that Sc does not have an strong impact on the system can be explained by the fact that the surfactant transport is dominated by convection and not diffusion.

FIG. 1 :

 1 FIG.1: Schematics of the system. An infrared laser (Keopsys Fiber Raman Laser CRFL-01-1455-OM1-B130-FA, wavelength λ = 1455 nm, output power of up to 1 W) is collimated and used to heat the air-water interface of a cylindrical volume of water (radius R r = 25 mm and height H r = 7.5 mm) from above. The laser waist diameter is 2R L = 1.26 mm (at the interface) and it is assumed to be constant (collimated beam).

FIG. 2 :

 2 FIG. 2: Simulation results in the convective regime for C T = 1 × 10 4 , and C S = 30. The results correspond to t = 25.

  FIG.3: PTV measurement setup and comparison between measurements and simulations.

FIG. 4 :

 4 FIG. 4: Convection cell superficial radius measurement and comparison with simulations

  FIG. 5: Thermography results and comparison with simulations for C T = 2 × 10 4

FIG. 7 :

 7 FIG. 6: Characterization of oleic acid (OA) as surfactant. Corresponding C S value as a function of the surface concentration Γ. Isotherm measurements performed with a Langmuir trough with a compression rate of 2.04 Å 2 /(min molecule). Two regimes are observed with a transition concentration Γ e = 1.37 molecule/nm 2 .

  (a) Maximum surface fluid velocity ũS . (b) Maximum surface temperature Θ S . (c) Convection cell radius Rcell .

FIG. S1 :

 S1 FIG. S1: Maximum non-dimensional velocity ũ (normalized by the converged value) as a function of the non-dimensional minimum element size for t = 25, C S = 40, Sc = 10, ψ = 2. Two curves are shown, the standard case ( C T = 1 × 10 4 and Pr = 7), and the worst-case scenario (C T = 4 × 10 4 and Pr = 14).

  FIG. S3: Surface simulation results for t = 25, C T = 1 × 10 4 , C S = 40, Sc = 10, ψ = 2, and different values of Pr.

( a )

 a FIG. S4: Surface simulation results for t = 25, C S = 40, Pr = 7, Sc = 10, ψ = 2, and different values of C T .

( a )

 a FIG. S7: Surface simulation results for t = 25, C T = 1 × 10 4 , C S = 40, Pr = 7, Sc = 10 and different values of ψ.

TABLE I :

 I Parameters' values used in this work.

	Density	ρ 997 kg/m 3
	Dynamic viscosity	µ 1.002 mPa s
	Thermal conductivity	κ 0.607 W/m K
	Specific heat capacity	c p 4181 J/kg K
	Surface tension (T = 20 °C) σ 0 72.9 mN/m
	Thermocapillary coeff.	γ T 155 µN/m K
	Laser wavelength	λ 1455 nm
	Light attenuation coeff.	α 3280 m -1
	Laser waist radius	R L 630 µm
	Recipient radius	R r 25 mm
	Recipient height	H r 7.5 mm
	Laser power	P L 20 mW to 200 mW
	Surfactant surface pressure γ Γ Γ 0 0 µN/m to 150 µN/m Surfactant diffusion coeff. D S 1 × 10 -7 m 2 /s

TABLE II :

 II Governing non-dimensional parameters and the values used in this work.

	Heated volume aspect ratio	ψ = αR L	2
	Prandtl number	Pr = µc p /κ	7
	Schmidt number	Sc = µ/ρD S	10
	Thermocapillary number	C T = ργ T P L /µ 2 κ 5 × 10 3 to 5 × 10 4
	Solutocapillary number

ACKNOWLEDGMENTS

This work was funded by BELSPO (IAP 7/38 MicroMAST), FNRS grant (PDR T.0129.18), and the EUR EIPHI program (Contract No. ANR-17-EURE-0002). The authors thank the GRASP laboratory in Université de Liège (ULiège) for lending us the Langmuir trough. B. S. thanks the FRS-FNRS for financial support.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.