Vincent Breton

Ignacio Blanquer

Vicente Hernandez

Nicolas Jacq

Yannick Legre

Mark Olive

Tony Solomonides

Roadmap for a European Healthgrid

Keywords: Grids, medical research, healthcare, medical informatics, medical imaging

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The concept of grids for health was born in Europe in 2002 and has been carried forward through the HealthGrid initiative [1]. This European initiative has edited, in collaboration with CISCO, a white paper setting out for senior decision makers the concept, benefits and opportunities offered by applying newly emerging Grid technologies in a number of different applications in healthcare [START_REF] Breton | on behalf of the Healthgrid White Paper collaboration, The Healthgrid White Paper[END_REF]. Starting from the conclusions of the White Paper, the EU funded SHARE project aims [3] at identifying the important milestones to achieve the wide deployment and adoption of healthgrids in Europe. The project will devise a strategy to address the issues identified in the action plan for a European e-Health [START_REF]Action plan for a European e-Health Area[END_REF] and set up a roadmap for technological developments needed for successful take up of healthgrids in the next 10 years.

In a previous paper [START_REF] Breton | Proposing a roadmap for Healthgrids[END_REF], we presented an analysis of the adoption of grids for biomedical sciences and healthcare in Europe, identifying bottlenecks and proposing actions. These actions have been further assessed within the framework of the SHARE European. The present paper proposes a technical roadmap for the adoption of healthgrids for medical research in Europe.

The Goal

Our goal is to offer to healthcare professionals an environment created through the sharing of resources, in which heterogeneous and dispersed health data as well as applications can be accessed by all users as a tailored information providing system according to their authorisation and without loss of information. The environment should allow multiple usages depending on the user community and offer to any groups involved in medical and life sciences research means to share information and access resources on demand. Persistent infrastructures are needed for the users who are looking for fully functional services for medical research and aim at scientific production. For the users who are designing distributed applications and projects, toolkits are also needed to enable use of the Grid in a secure, interoperable and flexible manner. These toolkits should allow creating virtual organizations, administering it, manipulating biological and medical data, and interfacing with the infrastructure resources and services whenever needed. The environment should be open and evolving: for those who wish to integrate resources, tools and content, standard specifications should be documented.

The infrastructure itself should give access through standard interfaces to public and private, free and non free computing and storage resources; data stored in public and private databases. The data accessible on the environment include molecular data (ex. genomics, proteomics), cellular data (ex. pathways), tissue data (ex. cancer types), personal data (ex. EHR), population (ex. epidemiology); services for medical research such as analysis tools (workflow, data mining); services for on line patient care (telemedicine, visualization), For the majority of the healthcare professionals, the technical aspects should be completely hidden behind a friendly user interface. As stated above, our goal is to offer services to the healthcare professionals but legal issues may very significantly delay the deployment of healthgrids for healthcare. As a consequence, we are not sure how relevant a purely technical roadmap like the one presented in this paper is for healthcare. As a consequence, we expect the technology to get adopted primarily for medical research. Our roadmap will therefore refer specifically to medical research.

The starting point

The state of the art of healthgrids has been described in several documents [START_REF] Breton | on behalf of the Healthgrid White Paper collaboration, The Healthgrid White Paper[END_REF][START_REF] Breton | Proposing a roadmap for Healthgrids[END_REF][START_REF]Healthgrid technology baseline report[END_REF][START_REF] Breton | HealthGrid, a new approach to eHealth[END_REF]. We summarize here the main conclusions reached by their authors. Large scale grid infrastructures are available for scientific production in Europe; these infrastructures offer unprecedented opportunities for intensive computing. Several toolkits are now available which start to offer grid services in a secure, interoperable and flexible manner; these toolkits have still to be tested at a large scale on biomedical applications. Europe has witnessed in the last few years the emergence of e-science environments such as my Grid [START_REF] Stevens | MyGrid, personalised bioinformatics on the information grid[END_REF] or VL-e [START_REF] Rauwerda | The Promise of a virtual lab[END_REF]; these environments are science portals for distributed analysis which offer scientists the possibility to carry out their experiments in a familiar environment while using the most recent web service technology and developments. Successful deployment of CPU intensive biomedical applications has been convincingly achieved on grid infrastructures world wide and several projects in Europe and world wide are successfully deploying biomedical grid applications; most of the success stories involve groups which have been active in grid projects for more than 5 years, which shows the present difficulty to use these environments for scientific production. The successful deployments in Europe are almost strictly limited to CPU intensive applications while very few applications involving manipulation of distributed medical data have been demonstrated so far; those were done at a prototype level by recognized team of computer experts in relation to clinicians.

The documents also highlighted several issues which had to be addressed in order to enable the healthgrid vision.

First issues are grid technology: web services are identified as the most promising technology to enable the HealthGrid vision despite their present limitations. There are today many different grid middleware (Globus, gLite, SRB, Unicore, GRIA, etc.) but none of them fulfil the requirements for a healthgrid. The ones which have demonstrated their scalability (gLite, Unicore) have limited functionalities particularly in the area of data management. Some which offer powerful and demonstrated data management functionalities (SRB) do not provide job management services. Moreover, these middleware are not so far built on web services and therefore do not offer standard interfaces. More recent grid middleware based on web services have not yet demonstrated their robustness and scalability.

Second issues are grid deployment: the deployment of grid nodes in healthcare centres such as hospitals or medical research laboratories is still extremely limited. This is due mainly to the present limitations of the existing European grid middleware which do not offer the necessary functionalities for secure manipulation of medical data. Other present limitations include the absence of an easy to install middleware distribution and the lack of friendly user interfaces to the grid for non experts. Just like Windows allows users to use their PC without any knowledge of the operating system, there is a need for user friendly interfaces to the grid.

Third issues are standardization: the definition and adoption of international standards and interoperability mechanisms is required for storing medical information on the grid. This includes for instance recording and ensuring consent, anonymization and pseudonymization.

Fourth issues are communication: there is an evident lack of information on the grid technology in the biomedical community. Communication on grids has been mostly focussed on the particle physics and computer sciences academic communities. Raising the awareness of the biomedical community is a major challenge for the coming years.

The road to grid infrastructures for medical research

In the previous section, we have listed a number of issues which need to be addressed in order to build the environment we have set up as our goal. In the following, we focus this study on medical research where adoption of grids is less depending on the evolution of EC legal texts. On the road to healthgrids, we have identified milestones which correspond to important steps forward in the services offered to the medical research community. We need to stress at this point that we are strongly convinced that a bottom-up approach has to be adopted on the road to healthgrids. Starting from the services made available on the existing grid infrastructures, a persistent distributed environment for medical research has to be built. This environment will progressively be enriched with new functionalities as technology progresses. In the United States, the Biomedical Informatics Research Network [1] is a very successful example of this bottom-up approach. In parallel to the building of this persistent environment, more volatile projects using grid toolkits or e-science environments to manage distributed data and knowledge for specific medical applications are very important to develop high level data integration services and spread the grid culture in the medical community. These services will later be made available on the grid infrastructures through standardized interfaces.

Milestones

Grid deployment is still very limited in European healthcare centres. Several factors contribute to this situation. One is the present human cost for deploying grid elements. Another important factor is the present situation in hospitals where none of the resources is accessible from the outside world. Our first deployment milestone called MD1 is the "successful permanent deployment of computing grid nodes inside European medical research centres". The goal of this first deployment is to create an environment where groups active in medical research can find resources for large scale simulation and modelling. This requires the development of friendly user interfaces. As a consequence of the present organization, a grid node in a hospital would have to be located outside the hospital firewall. Only anonymized or pseudonymized data could be transferred from inside the hospital firewall to the grid node. Technical solutions exist but they have not been deployed yet. These issues must be addressed and successful deployment of grid nodes inside European healthcare centres must be achieved. The services made available on these nodes are dependent on the technology. As documented in [START_REF]Healthgrid technology baseline report[END_REF], the presently available grid infrastructures are mostly offering services for large scale computing.

As documented in [START_REF]Healthgrid technology baseline report[END_REF], some data management services are emerging which are relevant to manipulate medical images. These services have to reach their full maturity for the second milestone called MD2, "Successful permanent deployment of data grid nodes inside European medical research centres". The challenge here is to store medical data on the grid and this will only be achieved once the grid middleware will allow doing it in a secure way.

Once medical data are securely stored on the grid, the next issue is to deploy the services to query these data, to build relationships between and to provide appropriate representation to the researchers and to healthcare professionals. Building relationships between these data requires agreeing on standards for representing and storing them, in order to develop knowledge management services to manipulate the data. Once the standards are defined and widely adopted, the next grid deployment milestone MD3 can be achieved, namely a "Successful permanent deployment of knowledge grid nodes inside European medical research centres". The difference here between the data grid and the knowledge grid is that data stored on the data grid are just exploited through simple queries while a knowledge grid offer services to manipulate concepts while ignoring the underlying data model and the grid storage architecture.

Reaching the milestones

For each of the milestones described above, we need the right technology. We are going to discuss now the technological issues as well as the standardization issues which have to be addressed on the road to healthgrids.

Grid technology issues

We need a grid operating system that allows handling distributed computing for milestone MD1, distributed data storage and query for milestone MD2, data integration and knowledge management for MD3. There are definitely progresses being made in this direction and several software stacks are now available which provide relevant services based on OGSA (GT4 [START_REF] Foster | Globus Toolkit Version 4: Software for Service-Oriented Systems[END_REF], OGSA-DAI [11], GRIA [12], etc). Beside the fact that these toolkits are not yet offering all the functionalities required by a grid environment for healthcare, a major issue is that most of them have not yet been tested on medical applications and/or at a significant scale. On the other hand, several academic consortia are developing grid middleware which are deployed on the large scale infrastructures in Europe. These consortia are aware of the requirements of the medical community but these middleware are not yet based on web services because their development effort started before the migration of grid standards to web services. These two trends need to converge to deliver a "tested, robust and scalable grid middleware based on web services that allow job and data management and which complies with EC countries laws on manipulation of personal medical data". Such a middleware is not for tomorrow and we rather foresee a progressive evolution of the grid operating systems similar to the one observed for personal computers.

Once a grid operating system is developed, a crucial issue is to make it easy to install in a healthcare centre. The availability of a public distribution of the middleware allowing a quick installation and configuration of the grid elements is required. This distribution requires significant resources because a middleware is very complex software with multiple modules and dependencies. Moreover, the technology is still evolving quickly as its standards are still under definition and only few implementations of these standards are now available. As a consequence, the distribution will have to be regularly updated to adapt to this rapid evolution path and be kept backward compatible so that sites using old versions are still able to belong to the same virtual organization as sites configured with newer versions of the distribution. In summary, a major challenge for the development of healthgrids is the "availability of a free, easy to install and configure, robust and documented distribution of the grid middleware, accompanied by a significant user support".

Standardization issues

As already mentioned above, standards are absolutely necessary to the deployment of services which integrate data in bioinformatics and medical informatics, data coming from different medical disciplines and data coming from different countries in Europe. The adoption of standards for the exchange of biological and medical information is still limited to a few specific fields. These standards are needed to build data models, to produce ontologies and to develop knowledge management services. Moreover, they need to be compatible with grid standards so as to allow their implementation on the healthgrids. The largest initiatives in the medical informatics field such as DICOM and HL7 are just starting to study the interface between their standards and web services technology. We suggest focussing the effort on two topics, medical imaging and Electronic Health Records, in a first stage. In view of the importance of these standardization efforts, we introduce standardization milestones MS1 corresponding to the "Production of a standard for the exchange of medical images on the grid based on DICOM", and MS2 corresponding to the "Production of a standard for the exchange of Electronic Healthcare Records on the grid" compatible with HL7.

Communication issues

The lack of information on grids is frequently identified as one of the key reasons why they have raised so far very little interest in the field of medical research. On the other hand, the services offered by the grids have been too limited to really make them a serious alternative to the existing computing models. A convincing communication relies on success stories demonstrating the impact of grids for medical research. So we identify the need for a demonstration environment offering a very easy access to the grid for non experts and providing some convincing services for medical research. The evolution of this demonstration testbed should follow the evolution of the healthgrid through the different deployment milestones. On this dissemination environment, dedicated efforts to promote the technology can be developed.

Security issues

In this document, we are not dissociating the security issues from the other technological issues because they have to be addressed from the lowest middleware layers. Deployment of a data grid for medical research discussed above will only be possible when the middleware will be able to provide all the necessary guarantees in terms of management of personal data. Here is what we perceived as the specific technical challenges related to the handling of medical data on the grid: manipulation of personal data on the grid must strictly obey regulations; these regulations change from country to country in Europe. Services for anonymization and pseudonymization of medical data must be provided. Medical data belong to the patient; a mechanism must be set up to allow any given individual to access to all his/her data or the grid. In the perspective of the usage of grids for healthcare, authentication of health professionals on the grid can not be handled by requesting all of them to get a grid certificate; a mechanism must be set up so that professional cards can be used to provide authentication on the grid.

Summary

We have identified 5 milestones on the road: MD1, called "Computing grid", corresponding to the successful permanent deployment of computing grid nodes inside European medical research centres; MD2, called "Data grid", corresponding to the successful permanent deployment of data grid nodes inside European medical research centres; MD3, called "Research K-Grid", corresponding to the successful permanent deployment of knowledge grid nodes inside European medical research centres; MS1, called "Grid DICOM", corresponding to the production of a standard for the exchange of medical images on the grid; MS2, called "Grid EHR", corresponding to the production of a standard for the exchange of Electronic Healthcare Records on the grid.

The milestones MD1, MD2, MD3 correspond to the deployment of infrastructures while MS1 and MS2 are related to the availability of standards. The figure 1 illustrates how the different milestones follow each other on the road and how the progress on the road will depend on the availability of the grid operating system as well as its distribution. For each of the three deployment milestones, computing grid, data grid and knowledge grid, the figure illustrates how two environments are needed, one for scientific production and one for demonstration.

We estimate to about 10 years the time necessary to reach the goal of deploying an healthgrid after successfully achieving the different milestones discussed previously. The technology available today allows reaching the first milestone MD1 today. We estimate about 2 years are needed to achieve MD2 while at least 5 years will be needed for MS1, MS2 and MD3. Finally, our experience shows that about three years will be needed from the day a first knowledge grid is deployed to the day it is robust enough to become an environment for scientific production.

Risk analysis

There are three keys to the successful progress on the technical road to healthgrids: the evolution of the technology, including development and distribution of the needed grid middleware; the deployment of stable infrastructures with guaranteed level of services; the deployment of medical applications on these infrastructures. As a consequence, we identify three key technical risks to achieve the vision: the absence of concentration of the critical mass of expertise to develop the grid middleware and its distribution; the absence of agreed standards to share medical images and Electronic Health Records on the grid; the non-adoption of the healthgrid infrastructures by the research community so that they are not used by medical applications. To these technical risks, we add a fourth major risk which is the absence of evolution of the legal texts to allow sharing of medical data: this could completely prevent the deployment of an healthgrid.

Short term roadmap (2-3 years)

Successful achievement of the 10-year target depends on immediate actions. We recommend developing R&D activities along three lines: 1. Develop healthgrid infrastructures 2. Deploy biomedical grid applications on the existing infrastructures; 3. Deploy biomedical grid applications using OGSA compliant grid toolkits and escience environments. Two out of the three lines (2 and 3) are now actively pursued around the world as biomedical applications are deployed on almost all grid infrastructures and many projects are now under development using web services toolkits or e-science environments. The three lines are needed because there is no dedicated grid environment for medical research and there are no OGSA compliant grid toolkits and e-science environments available on the existing infrastructures. However a convergence of these research axes should be achieved in about 2 to 3 years when the middleware deployed on the healthgrid and on the grid infrastructures will offer web service interfaces to their grid services so that the grid toolkits and e-science environments will be available to all healthgrid users. In parallel to these R&D activities, we recommend to pursue actively the definition of standards for the sharing of medical images and electronic health records on the grid. Rather than developing new standards, grid experts should as much as possible get involved in the already existing medical informatics standardization bodies.

Development of healthgrid infrastructures

Having a dedicated infrastructure for medical research is a key to the adoption of grids for medical research. It also solves the problem of handling priorities on a multidisciplinary grid where the services made available have been chosen through a consensus process and specific requirements for the medical community like security issues have low priority for other research communities. This dedicated infrastructure should deploy the most up to date services relevant to biomedical research as soon as they have demonstrated their scalability and robustness. The infrastructure will initially mostly focus on computing services, but should aim at deploying very rapidly secure data management services and later knowledge management tools. In order to offer immediate service to the community, the healthgrid should be built using technologies which are interoperable with the existing infrastructures while keeping the perspective to offer as soon as possible web services interfaces to the grid services.

Deployment of biomedical grid applications on the existing infrastructures

One can wonder why it is interesting to keep deploying biomedical applications on the existing multidisciplinary infrastructures once an healthgrid is running. The first reason is that one should not reinvent the wheel and take advantage of the services already offered by infrastructures like EGEE [START_REF] Gagliardi | Building an infrastructure for scientific Grid computing: status and goals of the EGEE project[END_REF] and DEISA [14]. The second reason is that the amount of resources and storage available on the healthgrid will hardly ever compare to what EGEE and DEISA provide. The healthgrid should provide the interface and the high level services needed for medical research but whenever there is a need for heavy CPU or storage resources, mechanisms should be available to use EGEE or DEISA to address these needs. As a consequence, through the collaboration between the developers of both healthgrid and infrastructure projects, tools should be designed to achieve this submission in the most transparent way. This means also that deployment of biomedical grid applications will increase on infrastructures like DEISA and EGEE once the healthgrid is deployed. These infrastructures should offer the largest possible palette of services relevant to medical research and exposed through web services interfaces. As a consequence, a continuous effort must be maintained to develop the synergy between the projects and to improve the environment for the deployment of biomedical grid applications on these infrastructures.

Deployment of biomedical grid applications using OGSA compliant grid toolkits and e-science environments

The services offered for medical research evolve with the grid operating system technology and we expect therefore to witness the evolution of the applications deployed with the progress of the middleware. On the other hand, most of the present European biomedical grid projects require already a high level of data integration which is beyond the capacities of the existing grid infrastructures. Nowadays, we observe a gap between the needs of these projects and the services made available on the infrastructures. How to address this gap is an important issue. The roadmap exposed above presents the bottom-up approach where infrastructures will progressively offer new services with the evolution of the technology. Toolkits such as OGSA-DAI or GRIA are very relevant to address the specific needs of medical grid projects and their use should be strongly encouraged. As well, high level environments such as my Grid or VL-e allow now biologists and healthcare professionals to start manipulating concepts they are familiar with while accessing potentially distributed data. The next step is to achieve the deployment of these environments on grid infrastructures and evolve them towards improved usability by end users.

Development of standards

The R&D activities described above are going to require the manipulation of medical data on the grid. As a consequence, standards for the exchange of medical data will be more and more required. At all costs, grid experts should avoid developing new standards but rather should as much as possible get involved in the already existing medical informatics standardization bodies. There, they can disseminate the concept of grids and work on extending the existing standards to distributed environments. The Open Grid Forum is a potentially very interesting place where grid experts involved in the different medical informatics standardization bodies could meet and investigate the interface between these standards and the existing grid standards. However, this would require a deep reorganization of the present Life Sciences Research group at OGF which is not working properly. The Healthgrid initiative provides the right framework to coordinate the development of the different standards in collaboration with the OGF and the different medical informatics standardization bodies.

Conclusion

In this document, we are proposing a 10-year roadmap to achieve the goal to offer to healthcare professionals an environment created through the sharing of resources, in which heterogeneous and dispersed health data as well as applications can be accessed by all users as a tailored information providing system according to their authorisation and without loss of information. Starting from the state of the art on healthgrids described in several documents [START_REF] Breton | on behalf of the Healthgrid White Paper collaboration, The Healthgrid White Paper[END_REF][START_REF] Breton | Proposing a roadmap for Healthgrids[END_REF][START_REF]Healthgrid technology baseline report[END_REF][START_REF] Breton | HealthGrid, a new approach to eHealth[END_REF], we have described a way toward the environment. We have identified 5 milestones on the road:

• MD1, called "Computing grid", corresponding to the successful permanent deployment of computing grid nodes inside European medical research centres, • MD2, called "Data grid", corresponding to the successful permanent deployment of data grid nodes inside European medical research centres, • MD3, called "Research K-Grid", corresponding to the successful permanent deployment of knowledge grid nodes inside European medical research centres, • MS1, called "Grid DICOM", corresponding to the production of a standard for the exchange of medical images on the grid, • MS2, called "Grid EHR", corresponding to the production of a standard for the exchange of Electronic Healthcare Records on the grid.

Achieving these different milestones require the availability of a grid operating system providing all the needed functionalities as well as an easy-to-install distribution of this middleware.

We have identified major technical risks which can prevent the vision to happen:

• the absence of concentration of the critical mass of expertise to develop the grid middleware and its distribution, • the absence of agreed standards to share medical images and Electronic Health Records on the grid, • the non-adoption of the healthgrid infrastructures by the research community so that they are not used by medical applications, To these technical risks, we have added a fourth major risk which is the absence of evolution of the legal texts to allow sharing of medical data which can prevent the deployment of an healthgrid.

Finally, we have proposed in the next 2 to 3 years to develop R&D activities along three lines:

• Develop healthgrid infrastructures,

• Deploy biomedical grid applications on the existing infrastructures,

• Deploy biomedical grid applications using OGSA compliant grid toolkits and e-science environments. In parallel to these R&D activities, we recommend to pursue actively the definition of standards for the sharing of medical images and electronic health records on the grid in the already existing medical informatics standardization bodies. We consider that the Healthgrid initiative provides the right framework to coordinate the development of the different standards in collaboration with the OGF and the different medical informatics standardization bodies.

Figure 1 .

 1 Figure 1. Illustration of the key challenges and the milestones on the road to healthgrids

Correspondence : breton@clermont.in2p3.fr

Acknowledgements

The SHARE project is co-funded by the European Commission under contract n°FP6-2005-IST-027694.


Francesco Viola

François Gallaire

Benjamin Dollet
email: benjamin.dollet@univ-grenoble-alpes.fr

Sloshing in a Hele-Shaw cell: experiments and theory

The response of the free liquid surface in a Hele-Shaw cell subjected to a horizontal oscillation is investigated. We study the low-oscillation-amplitude regime and we show, by varying the fluid viscosity, ν, and the forcing frequency, ω, that the ratio between the Stokes viscous length, 2ν/ω, and the cell thickness greatly affects the amplitude and phase lag of the gravity waves. In particular, the sloshing system undergoes an underdamped/overdamped transition for sufficiently large viscosities. A consistent theoretical model, based on a modification of Darcy's law to include unsteadiness, is then introduced to rationalize the experimental observations. Contrary to traditional sloshing wave theory, the viscous flow dissipation comes at leading order in the analysis, rather than as a higher order asymptotic correction to the inviscid sloshing dynamics. The analytical expression for the resonance curves agrees well with experimental results without tunable parameter.

Introduction

The motion of the free liquid surface in a closed basin is encountered in many industrial applications and can have a detrimental effect on the stability of liquid carriers and aerospace vehicles [START_REF] Ibrahim | Recent advances in liquid sloshing dynamics[END_REF][START_REF] Ibrahim | Liquid Sloshing Dynamics: Theory and Applications[END_REF][START_REF] Veldman | The numerical simulation of liquid sloshing on board spacecraft[END_REF][START_REF] Faltinsen | Sloshing[END_REF]. In the case of a harmonic forcing imposed to the container, the wave amplitude increases significantly at resonance. If the amplitude of sloshing motion becomes large enough, geyser formation, wave overturning [START_REF] Royon-Lebeaud | Liquid sloshing and wave breaking in cylindrical and square-base containers[END_REF]) and spilling [START_REF] Mayer | Walking with coffee: Why does it spill?[END_REF] can occur, thus potentially affecting safety and operational efficiency. With the purpose of reducing sloshing motion at resonance, tanks have been designed in such a way to change the system's natural frequencies with respect to the ones of possible excitations, or to increase viscous dissipation. Dissipation devices such as rigid lids and elastic membranes at the free surface or baffles placed in the container, are widely used. Another option consists in designing fuel tanks filled with a porous medium or Hele-Shaw cells as proposed for tanks of aerospace vehicles [START_REF] Dweyer | Hele-shaw and porous medium flow for space fuel systems[END_REF]. This last solution consists in a narrow container where the separation between two plates is small compared to the other two dimensions. In this way viscous friction is enhanced and viscous forces dominate over inertial ones. Interestingly, due to the strong confinement, the transversal velocity is negligible with respect to the in-plane components and the resulting motion is nearly two-dimensional.

More generally, Hele-Shaw cells have been adopted over the years to study twophase flows, providing useful insight into viscously driven instability phenomena. A paradigmatic example is the viscous fingering formation, also called Saffman-Taylor instability, that occurs when a less viscous fluid is injected into a Hele-Shaw cell, filled with a different immiscible and more viscous fluid. The interface between the fluids develops an instability leading to the formation of finger-like patterns [START_REF] Saffman | The penetration of a fluid into a porous medium or heleshaw cell containing a more viscous liquid[END_REF][START_REF] Bensimon | Viscous flows in two dimensions[END_REF].

The mathematical description of the flow in a Hele-Shaw cell is based on the parabolic flow profile approximation in the narrow direction. The in-plane mean velocity u is thus proportional to the pressure gradient, with the proportionality constant scaling inversely with the viscosity, µ, and with the cell gap square,

ζ 2 u = - ζ 2 12µ ∇p. (1.1)
This is nothing more than the Darcy equation that governs flows in porous media, which are widely studied in geophysics, for instance in the context of oil recovery. Darcy's law, however, is not valid in cases where inertia, through the advection of momentum, plays a relevant role, such as waves [START_REF] Kalogirou | Variational finite element methods for waves in a hele-shaw tank[END_REF] and Kelvin-Helmholtz instability in a Hele-Shaw cell where the instability threshold was found by [START_REF] Gondret | Shear instability of two-fluid parallel flow in a Hele-Shaw cell[END_REF] to be governed by inertial effects. It is not mathematically straightforward to consistently reintroduce convective terms in the depth averaged Hele-Shaw equations (see [START_REF] Ruyer-Quil | Inertial corrections to the darcy law in a hele-shaw cell[END_REF]; [START_REF] Plouraboué | Kelvin-helmholtz instability in a hele-shaw cell[END_REF]).

In this work, we consider the case of inertial effects in a Hele-Shaw cell flow limited to the unsteady term in the Navier-Stokes equations. This scenario corresponds to a pulsatile flow where the fluid's motion reduces to a two-dimensional oscillating Poiseuille flow (similarly to the Womersley flow in arteries [START_REF] Womersley | Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known[END_REF])) and it is well suited to investigate sloshing waves in narrow containers, which is the main aim of this work. Our approach is to perform experiments in a narrow cell of fixed gap exposed to a lateral harmonic forcing of angular frequency, ω. The boundary layer to cell's gap ratio, δ /ζ, is varied by filling the cell with liquids of different viscosities, ν, up to the point the thickness of the oscillating boundary layers, δ = 2ν/ω, becomes comparable to the cell gap, ζ, and, as a result, the dissipation is intense in the whole fluid volume, in contrast to the case of large containers. The amplitude and phase of the free surface shape are monitored. The experimental results are then rationalized through a generalized Darcy's law for oscillating flows, and the analytical model is then compared to the experimental data.

Experimental methods

The experimental set-up used to generate the sloshing waves in the narrow container and to measure the free-surface displacement is shown in figure 1(a). A Plexiglas cell of height H = 15 cm, length = 10 cm and width ζ = 0.3 cm is fixed to a single axis linear motion actuator (Aerotech PRO 165). The container is partially filled from the top with a column of liquid of height h 0 = 10 cm, that is of the same order of the container's length, h 0 ≈ , in order to limit viscous dissipation at the bottom of the container. Indeed, the effect of a finite depth h 0 / enters the eigenfrequency of the fundamental mode, which is the most sensitive to finite-depth effects, through a factor tanh (πh 0 / ) which equals the infinite-depth limit within less than 1% for h 0 = [START_REF] Ibrahim | Liquid Sloshing Dynamics: Theory and Applications[END_REF]. We use different solutions of 95% ethanol plus 5% isopropanol that is mixed with glycerol (0%, 5%, 25%, 40%, 45%, 55% and 62% in volume of glycerol). The properties of the different solutions are measured with an Anton Paar viscosimeter (SVM 3000) and summarized in Table 1.

Sloshing is generated by imposing a lateral harmonic motion of angular frequency ω = 2πf , and amplitude a. The actuation is set and controlled by the Aerotech Soloist CP software and spanning in sequence the frequency range from f = 0.1 Hz to f = 4.5 Hz with step 0.1 Hz. The oscillation amplitude imposed to the cell is set equal to a = 6 mm, which was seen to generate sloshing waves in the linear regime (see Sec. 4 for further details). To ensure that the steady regime is reached, the free-surface dynamics is recorded only after a certain number of cycles, typically 15, through a Blaser camera coupled with a Nikon 18-108/2.5 lens. The acquisition frequency is adapted to the excitation frequency: 5 frames per second (fps) for f 0.4 Hz, 25 fps for 0.5 Hz < f < 0.9 Hz, and 50 fps for f > 1 Hz. Lighting is provided by a LED panel (25 cm × 15 cm) placed behind the cell. Due to the dependence of viscosity on temperature and high volatility of ethanol, experiments are performed within short time scales, typically less than 30 minutes, thus avoiding in this way fluid evaporation, and the room temperature is recorded for each experiment. The liquid level is measured before and after the experiment to ensure fluid evaporation is negligible.

In order to extract the location of the air/liquid interface, one needs first to subtract the horizontal motion of the cell, since the camera is not attached to the vibrated tank.

To track its vertical central axis y, we threshold and binarise each image (Fig. 1(b,c)). The external edges of the cell then appear as black vertical lines (Fig. 1(c), which are automatically detected in a home-made Matlab program. The y axis is set as the middle line between these two edges. The interface appears as a bold black line, because of meniscus effects. We simply describe the interface by a curve of equation y = h(x, t), h being taken as the position of the top of the bold line. We checked that, apart from an offset irrelevant to the dynamics and practically constant along all experiments, defining h by the middle line or the bottom of the bold line would not change anything. The relevant quantity describing the interface is actually the deviation of h from the inteface position at rest h 0 , δh(x, t) = h(x, t) -h 0 . From this, we deduce the time-averaged amplitude of the interface motion ∆h(x) and phase difference ϕ(x) between the interface oscillation and the imposed displacement ã(t):

∆h(x) = 2 δh(x, t) 2 t , ϕ(x) = arccos[ ã(t)δh(x, t) t / ã(t) 2 t δh(x, t) 2 t ]. (2.1)
Here, time averaging • t is performed over an entire number of cycles. This ensures that if the horizontal displacement is ã(t) = a cos ωt, then the interface displacement is δh(x, t) = ∆h(x) cos[ωt + ϕ(x)] in the case of a pure harmonic motion in time.

Theory

Let us consider the response to forcing of a liquid contained in a rectangular container where the cell gap is much smaller than its width ζ/ 1. The forcing is applied along the direction x (see Figure 1(a)), and consists of an oscillatory motion of amplitude a and angular frequency ω. Hence, the acceleration of the cell in the lab frame is -aω 2 cos ωt . Primed variables, as time t here, design dimensional variables, in contrast to their dimensionless counterparts to be introduced later. In the non-inertial frame of the cell, the fluid obeys the continuity equation and the Navier-Stokes equation with a pseudo-force acting in the x-momentum:

∇ • v = 0, ∂v ∂t + (v • ∇ )v = - 1 ρ ∇ p + ν∆ v -ge y -aω 2 cos ωt e x , (3.1)
with v = u e x + v e y + w e z the velocity field. In all the following, we neglect the nonlinear advection term on behalf of the small amplitude of the oscillations. The equations of motion are then linear, and we can write v (x ; t ) = R[v (x )e iωt ] and p (x ; t ) = R[p (x )e iωt ] with i the imaginary unit and R taking the real part. We henceforth drop the hat referring to v (x ) and p (x ) simply as v = (u , v , w ) and p . Equations are made dimensionless by using along the directions x and y, and ζ along direction z. The forcing amplitude and frequency provide a scale aω for the in-plane velocity components, whereas continuity equation imposes the transverse component to scale as aω, due to the strong confinement in the z-direction: ≡ ζ/ 1. With these choices, dimensionless spatial scales, velocity components and pressure write:

x = x , y = y , z = z ζ , u = u aω , v = v aω , w = w aω , p = p ρ aω 2 . (3.2)
Furthermore, we separate the static and dynamic pressure contributions by setting p = x + g aω 2 y + q.

(3.3) Projecting (3.1) along x, y and z, we thus get the following dimensionless equations at leading order in :

∂u ∂x + ∂v ∂y + ∂w ∂z = 0, iu = - ∂q ∂x + 1 2 δ 2 ∂ 2 u ∂z 2 , iv = - ∂q ∂y + 1 2 δ 2 ∂ 2 v ∂z 2 , 0 = ∂q ∂z , (
u = i ∂q ∂x 1 - cosh(1 + i)z/δ cosh(1 + i)/2δ , v = i ∂q ∂y 1 - cosh(1 + i)z/δ cosh(1 + i)/2δ . (3.5)
Therefore, the averaged velocity along z satisfies a Darcy-like equation

u = 1/2 -1/2 u dz = iα∇q, where α = 1 - 2δ 1 + i tanh 1 + i 2δ (3.6)
is the effective permeability taking into account the effect of the depth-averaged pulsatile flow profile. Similarly to Darcy's law (1.1), equation (3.6) prevents the no-slip condition for the in-plane velocity to be imposed (see Appendix A for a boundary layer expansion valid near the lateral walls). Figure 2(b) shows the ratio α/ω (to which the dimensional solution is proportional) versus the forcing pulsation ω. For sufficiently small frequency, depending on the fluid viscosity, α/ω becomes constant, thus meaning that the flow profile does not depend on ω and is equal to a quasi-steady Poiseuille flow.

To obtain a governing equation for the z-averaged velocity u , we also average the continuity equation. Using the impermeability condition for the spanwise velocity, w = 0 at z = ±1/2, and (3.5), we find that the pressure field obeys Laplace equation

∂ 2 q ∂x 2 + ∂ 2 q ∂y 2 = 0.
(3.7)

Looking for solutions by separation of variables with the requirement that they must vanish as y → ∞ (we restrict our analysis to the deep water regime, consistently with experiments), we find that q must be of the type

q = ∞ n=0
A n e -kny sin k n x, (3.8) and the no-penetration condition u = 0 at x = ±1/2 results, through equation (3.6), in a Neumann condition on q and sets the possible wavenumbers to k n = (2n + 1)π, with n ∈ N.

Let us define the gain η as the ratio of the displacement of the interface ∆h to the forcing amplitude a, η = ∆h/a, where ∆h is defined by (2.1). We assume that the slope of the interface remains small, thus, the kinematic boundary condition can be then applied at y = 0 and simplifies to ∂η/∂t = v. Using equation (3.6), the z-averaged kinematic condition thus writes η = α ∂q ∂y (x, y = 0), that, combined with (3.8), yields an equation for the averaged interface position

η = -α ∞ n=0 k n A n sin k n x.
(3.9)

In the case of much larger than the capillary length c = σ/ρg with σ the surface tension (typically c = 2 mm in our experiments), surface tension effect is negligible, and the dynamic boundary condition reduces to the continuity of normal stress at the interface. Since the atmospheric pressure is an offset irrelevant to the dynamics, it simply writes p (x, y = aη/ ) = 0. From equation (3.3), we thus get 0 = x + q(x, y = 0) + g ω 2 η(x) and using equations (3.8), (3.9) and the Fourier series of x in sin k n x, we can solve for the coefficients A n . The interface response to the forcing thus reads

η = 4 π αω 2 ∞ n=0 (-1) n 2n + 1 sin[(2n + 1)πx] ω2 -(2n + 1)α , (3.10)
where ω = ω/ω 0 . Note that when viscosity is small (α ≈ 1), we recover the inviscid resonance frequencies, ω 2 n = (2n+1)πg/ . In the limit of small frequencies, equation (3.10) reduces to η = -π ω2 x: the interface assumes a quasi-steady tilted shape prescribed by the acceleration of the cell in the lab frame when ω2

1.

Results

We now present the resonance curves for the different solutions. The time-averaged surface elevation, as defined by (2.1), over the forcing amplitude, ∆h(x = x g , f )/a, is a measure of the energy response of the sloshing wave and it is shown in Figure 3(a). Here the position x g = 45 mm is chosen as such to be off-centred enough that the displacement of the interface, hence the signal-to-noise ratio, is maximised, but not too close to the side walls, not to be influenced by them (see Figure 6 and Appendix A for a discussion of wall effects). As expected, at any given frequency, the gain is a decreasing function of viscosity. At increasing viscosity, we observe the disappearance of the characteristic resonance peak, which is reminiscent of an underdamped to overdamped transition, for viscosity roughly equal to 15 mm 2 •s -1 (solution 6, see Tab. 1). Note that this transition happens when the fluid bulk at the inviscid resonant frequency is almost completely filled by the oscillating boundary layers. This transition, 2δ ≈ ζ, takes place indeed for a viscosity in between that of solutions 6 and 7, as shown in Figure 2(a).

It is important to remark that, as mentioned in section 2, the energy gains reported in Figure 3(a) correspond to the linear resonance curves. As a consequence, the maximum wave amplification is here only limited by the viscous dissipation rather than nonlinear saturation that is expected at larger excitation amplitude. The hypothesis of linear regime has been verified by varying the forcing amplitude as a = 1.5, 3.0 and 6.0 mm. The energy gains were found to be unchanged, as shown in Figure 4 for base fluid. The other solutions with glycerol, which are more viscous and display oscillations of lesser amplitude, remain in the linear sloshing regime too.

This observation therefore allows us to compare the experimental results with our theoretical analysis (section 3) that is limited to linear sloshing. Assuming that we can identify the experimental measurements of the gain to the z-averaged free-surface elevation η , the corresponding resonance curves are reported in Figure 3(a) (dashed lines). For each solution, the value of the parameter α appearing in equation (3.10), is evaluated as defined in equation (3.6) by using the measured viscosity reported in Table 1. A good agreement is observed between the experiments and the theory without using any adjustable parameter. In particular, the resonance frequency is well reproduced and the disappearance of the resonance peak precisely captured. The theory, however, overestimates the wave response in the low viscosity case (namely for solution 1 and 2) close to the resonance peak, calling for a small additional dissipation to be considered, as discussed in section 5. In order to get a more complete description about resonant waves in a Hele-Shaw cell, the phase lag between the excitation and the response, as defined in equation (2.1), is reported in Figure 3(b). The phase lag tends to increase from -π to 0 at increasing frequency, except for the lowest viscosities at frequencies above 3.5 Hz, because of the influence of higher order modes. For these lowest viscosities which display underdamped resonance, the rate of increase of the phase is maximal at the resonance peak frequency, similarly to a classical forced harmonic oscillator. These measurements are in good agreement with the argument of η given in equation (3.10), further validating our theoretical approach.

In addition to local quantities, we present the shape of the whole oscillating liquid/air interface. Figure 5 shows the experimental and theoretical surface elevation, for a selection of representative frequencies, below (2.0, 2.6 Hz) and above resonance (3.2, 3.8 and 4.4 Hz), and for a set of viscosities in both underdamped and overdamped regime. Below resonance (circles in (a-d)), the shape of the interface is roughly a tilted line, as prescribed by the acceleration of the cell in the lab, in agreement with the discussion at the end of section 3. At moderate viscosities and close to resonance (squares in (a-c)) η matches a sinusoidal free-edge mode, whereas a tilted line mode is still observed for high viscosity (d). At increasing frequency, the shape becomes more complex, and the gain is no longer a monotonic function of |x| for the largest frequencies and the lowest viscosities. In particular, for forcing frequency f = 4.4 Hz higher modes contribute to the dynamics at low and moderate viscosities (see x-symbols in (a,b)). Although some discrepancies are present, especially at low viscosity (a), our model reproduces the main features of the free-surface shape observed in the experiments.

Conclusions and further discussions

In this paper, the behaviour of surface gravity waves in a Hele-Shaw cell subjected to a horizontal oscillation is investigated. The resonance curves are measured experimentally in the low amplitude regime and the oscillating boundary layer to cell's gap ratio is varied by filling the cell with liquids of different viscosities. With increasing viscosity and, as a consequence, thickness of the Stokes layers the viscous forces dominate over the inertial forces and the surface elevation of the sloshing wave decreases. In addition, the fluidic system undergoes to a transition from underdamped to overdamped regime when the two Stokes layers over the large faces of the cell fill the fluid bulk (2δ /ζ ≈ 1). Similarly to a damped harmonic oscillator, the phase lag increases from -π to 0 with frequency and it is equal to -π/2 at resonance. At moderate viscosities and close to resonance, i.e. higher wave amplitude, the surface wave is well described by the usual free-end edge modes. In contrast, when the exciting frequency is small the free-surface becomes roughly flat according to a quasi-steady non-inertial force acting on the system, which basically has the effect of tilting the gravity acceleration. This behaviour is further observed for higher viscosity at higher frequencies.

These results have been rationalized by taking into account a modification of Darcy's law where an unsteady term is introduced to deal with flow oscillation. The resulting governing equations can be rigorously deduced from the Navier-Stokes equations in the limit of small amplitudes and it is one of the main contributions of this work. In contrast to the traditional approach in sloshing theory where the viscous dissipation is found as a first order correction of the inviscid dynamics, in our formulation the viscous dissipation of the oscillating boundary layers comes at leading order. Hence, the hypothesis about the smallness of the fluid viscosity usually needed to determine the dissipation [START_REF] Faltinsen | Sloshing[END_REF] is not required here. An analytic expression for the instantaneous velocity and averaged free-surface elevation is found and used to compare against experimental reuslts. A good agreement is observed between the experiments and the theory, thereby validating our perturbative approach to predict the main characteristics of confined sloshing. In particular, the resonance frequency is well reproduced and the transition from underdamped to overdamped response correctly captured.

The theory, however, overestimates the wave response in the low viscosity case (namely for solution 1 and 2) close to the resonance peak. This is likely due to the fact that dissipative mechanisms other than the viscous dissipation at the vertical walls are not accounted in the model. So far we have not considered the additional dissipation at the free surface. In particular, the contamination of the free surface by impurities significantly affects both the damping rate and the eigenfrequency, which are highly sensitive to small local variations of the surface tension due to the presence of surfactants [START_REF] Davies | On the damping of capillary waves by surface films[END_REF]. [START_REF] Miles | Surface-wave damping in closed basins[END_REF] showed theoretically that the presence of a thin surface film, which was modeled as a viscoelastic media, can potentially double the damping rate. Additional dissipation is also expected in the tiny flow region in between the wetting film at the wall and the moving meniscus. In wetting conditions, as in our experiments, this dissipation is associated with a friction force scaling as the capillary number at the power 2/3. This result is identical to what is found when pulling a plate out of a bath and in Tanner's problem [START_REF] Cantat | Liquid meniscus friction on a wet plate: Bubbles, lamellae, and foams[END_REF], or when a foam layer is present over the sloshing liquid [START_REF] Sauret | Damping of liquid sloshing by foams[END_REF][START_REF] Viola | Foam on troubled water: Capillary induced finite-time arrest of sloshing waves[END_REF]. We believe that these dissipative phenomenon at the free liquid surface and at the dynamic meniscus, related to wetting effects [START_REF] Keulegan | Energy dissipation in standing waves in rectangular basins[END_REF][START_REF] Cocciaro | Capillarity effects on surface gravity waves in a cylindrical container: wetting boundary conditions[END_REF], are present in our experiments, but are overshadowed by the viscous dissipation in the oscillating boundary layers that becomes dominant when the confinement or the viscosity are large enough, as confirmed by the very good matching between theory and experiments for ν 3.7 mm 2 /s. and the matched velocity solution v + v satisfies the no-slip condition at x = /2, see Figure 6. The outer and matched solutions are indistinguishable at x g = 45 mm.

Figure 1

 1 Figure 1. a) Sketch of the experimental setup. b) Raw image of the air/liquid interface, taken from the experiment without added glycerol and forcing frequency of 3.5 Hz. (c) Binarised image obtained after thresholding.

Figure a )

 a Figure a) Stokes layers thickness at the walls, 2δ , over the cell width, ζ, as a function of the forcing frequency. Each line corresponds to a different solution used in the experiment, whereas the vertical line depicts the inviscid resonance frequency. b) Modulus (full line) and phase (dashed line) of α/ω as a function of ω for the less (blue lines) and most (red lines) viscous solutions used in the experiments. The grey area shows the frequency range covered by the experimental data.

Figure 3

 3 Figure 3. a) Resonance curves and b) phase lag for different solution viscosities (see Tab. 1) according to experiments (symbols) and theoretical results (lines). The gain and phase are defined as in equation (2.1) for the experimental data and as modulus and phase of the average surface elevation (3.10) for the theory.

Figure 4 .

 4 Figure 4. Experimental resonance curves for the base fluid for forcing amplitudes a = 1.5 mm ( ), 3.0 mm ( ) and 6.0 mm ( ).

Figure 5 .

 5 Figure 5. Shape of the oscillating interface at different frequencies according to experiments (symbols) and theory (lines), for different fluid viscosities, namely a) solution 1, b) solution 4, c) solution 6 and d) solution 8, (see Tab. 1 for the exact fluid properties). Since the data and prediction are symmetric with respect to x = 0, only the half cell x > 0 is presented.

Figure 6 .

 6 Figure6. Example of the side wall correction by asymptotic matching technique for solution 1 and forcing frequency equal to 2.6 Hz. The no-slip boundary condition at x = l/2 is satisfied by the matched solution v + v (dot-dashed line), which is given by adding the inner velocity correction (dashed line) to the vertical outer velocity v (solid line).

Table 1 .

 1 Physical properties of the solutions used at the temperature of the experiments.

	Solution	Composition	kinematic viscosity [mm 2 /s]	density [g/cm 3 ]	temperature [°C]
	1	base fluid (95% ethanol+5% isopropanol)	1.7934	0.8047	21.0
	2	base fluid + 5% glycerol	2.2783	0.8305	20.0
	3	base fluid + 20% glycerol	3.7069	0.8807	20.5
	4	base fluid + 25% glycerol	6.4688	0.9294	20.0
	5	base fluid + 30% glycerol	11.853	0.9786	20.0
	6	base fluid + 40% glycerol	15.319	0.9997	20.5
	7	base fluid + 45% glycerol	21.388	1.0225	20.0
	8	base fluid + 55% glycerol	40.594	1.0697	20.5
	9	base fluid + 62% glycerol	63.428	1.0987	20.5

Appendix A. Side wall correction: a matched asymptotic expansion

The leading-order solution derived in section 3 does not satisfy the no-slip boundary condition at the lateral walls, thus suggesting that the scaling x = x / is not suited to describe the flow close to x = ± /2. Let here set x = (x ∓ /2)/ζ and let look for an inner correction to the vertical velocity, v, such that the matched solution v + v tends to the outer solution, v, far from the wall and satisfies the no-slip condition at x = 0:

At leading order, the inner velocity v obeys the Helmholtz equation

which admits solutions of the kind

that satisfy the no-slip condition at z = ±1/2. Substituting the Fourier series (A 3) in the governing equation (A 2) we get

where exponentially growing solutions have been discarded due to the first condition (A 1). In order to determine the unknown coefficients B m (0), equation (A 3) is inserted in the no-slip condition (A 1). By projecting on the basis sin [mπ(z + 1/2)] and using equation (3.5) we obtain that