Nathalie Morel
email: nathalie.morel@sciences-po.fr

The political economy of domestic work in France and Sweden in a European perspective

While there is a growing body of literature dealing with the development of migrant domestic work in Western countries, so far there has been very little attention paid to the development of formal domestic services as an economic activity actively structured through public policy. Yet the development of the domestic services sector in Europe is part of a specific political and economic strategy, which has been actively promoted by national governments and national lobby groups, but also by the European Commission since the 1990s, so that it seems warranted to speak of a new 'political economy of domestic work'.

The aim of this paper is to analyse the rationale behind this public intervention in favour of the development of domestic services, to highlight the economic, political and social issues it raises and to see how the policies implemented interact with existing welfare / care systems, employment regimes, and prevailing gender and social norms. This paper does so through an analysis of the policy discourse at the EU level, and a comparison of the policies implemented and their consequences in France and Sweden.

The choice of these two countries is guided by the fact that while they represent very contrasted social models, France and Sweden are the two countries that have gone furthest in terms of the support provided to domestic services, and they have done so through the introduction of a same policy instrument, namely a 50 % tax reduction on domestic services. We suggest that the uncovering of similar trends in the logic and modes of public intervention and in the social, economic and political consequences of this public intervention in two strongly contrasted national models could be revealing of more global trends in Europe, linked to more profound transformations of welfare states, of labour markets, and of societies more generally.

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While domestic work as a form of employment had progressively disappeared in most European countries during the course of the 20 th century, it has been on the increase everywhere in Europe over the past decade or so. This development of domestic employment has been highlighted in a number of sociological works, which usually emphasize the role of global socio-economic transformations to explain the expansion in both the demand and supply of domestic work. On the supply side, the rise in inequalities, the growth in unemployment, the development of a reserve of unskilled labour, transnational economic inequalities and migrations, especially the strong rise in female immigration, have been underlined (Andersson, 2000 ;Hochschild, 2001 ;Lutz, 2008 ;Gallotti, 2009 ;Widding Isaksen, 2010 ;Österle et al. 2011 ;Williams and Gavanas, 2008 ;Williams, 2011). On the demand side, the most salient factors put forward have to do with the sharp increase in female labour force participation, the transformations in family structures and increased geographical distances between family members which have reduced mutual aid possibilities between generations, the rise of new social needs linked to population ageing, the absence or insufficient provision of child-and elderly-care services, the transformation of socio-cultural norms, not least the increased emphasis that is placed on personal and professional selfrealization and the concomitant depreciation of housework and care, changing attitudes towards the domestic division of labour and the wish for more free time for leisure and family (cf. Cancedda, 2001 ;Yeandle, 2002;Lutz, 2008 ;Gallotti, 2009 ;D'Souza, 2010 ;Österle et al. 2011).

These studies also very much focus on the informal nature of the employment relationship, as work in this area is often undeclared. Current research on the development of domestic work has also been very much focused on migrant domestic workers and on what many authors, following Hochschild or Parreñas, refer to as the "global care chain" (Hochschild, 2001;Parreñas, 2001;Ehrenreich and Hochschild, 2003). These analyses developed first in the US and in the other Anglo-speaking countries which have in common the fact that social care services are largely lacking, which has created an important market for private domestic services, not least through the use of migrants, sometimes illegal but sometimes also 'imported' legally (various countries have set up schemes to facilitate the granting of temporary work-permits for people coming to work as domestic workers). The use of migrant domestic workers for care work has also been highlighted in the countries of Southern Europe (Bettio et al., 2006;Scrinzi, 2008), where the phenomenon has recently become so widespread that Bettio et al. speak of a transformation of Southern European welfare states from a familialist care regime to a 'migrant-in-the-family' care regime. Recent research has shown that the use of migrant domestic workers is on the rise in other countries also (cf. Lutz, 2008;van Hooren, 2012), including in the Nordic welfare states (Gavanas, 2010;Widding Isaksen, 2010;Williams and Gavanas, 2008).

Here it is the ethnic dimensions of domestic employment and care provision that are at the core of the international literature on domestic work. While some authors are interested in the transnational processes that shape the global political economy of care (Andersson, 2000;Yeates, 2005;Mahon and Robinson, 2011;Williams, 2011), much work adopts a more socioethnographic approach with the focus being placed on the living and working conditions of these migrants, on their relations (of subordination) with their employers, but also on the transnational family situations that are created by these migrations.

There is thus now a wealth of literature dealing with domestic work, especially that which is performed -legally or not -by migrant workers, but one aspect that has not yet received much attention is the fact that many European countries have, over the past decade or so, developed specific policies to actively promote the development of domestic services, leading to an important growth in domestic employment, with formal employment figures in the personal and household services currently estimated at around 7.5 million in the EU 1 . Indeed, many European countries have set up schemes to subsidize the demand for domestic services, especially for childcare and elderly-care, through the introduction of cashfor-care schemes, vouchers or different socio-fiscal measures such as social contribution exemptions and/or tax reductions, indicating a new approach to care provision. Some countries have gone further in that they also subsidize non-care related domestic services such as cleaning, ironing, gardening, etc. The development of domestic employment is thus not only driven by a natural growth in supply and demand, there are in fact policies that structure this supply and/or demand. It thus seems warranted to speak of a 'political economy of domestic work', the delegation of domestic work and the development of domestic services being encouraged and structured through specific political and economic strategies, which have been actively promoted by national governments and national lobby groups, but also by the European Commission since the 1990s.

1 "Commission staff working document on exploiting the employment potential of the personal and household services" SWD(2012)95 final, p.4. This figure can only be a rough estimate as the personal and household services encompass activities that are classified in different NACE sectors.

The aim of this paper is to analyse the rationale behind this public intervention in favour of the development of domestic employment, to highlight the economic, political and social issues it raises and to see how the policies implemented interact with existing welfare / care systems, employment regimes, and prevailing gender and social norms. It is the contention of this paper that such an analysis can shed light in important ways on some of the broader transformations under way, in terms of welfare state transformation and its redistributive outcomes, labour market segmentation and the shaping of social stratification and symbolic social hierarchies.

To address these issues, the paper starts with an analysis of the discourse that has underpinned the development of this sector at the EU level. Part two discusses some of the issues and contradictions that this EU policy orientation raises. Part three then narrows the focus on two case studies, France and Sweden, to illustrate how some of the issues and contradictions identified in part two play out. The choice of these two countries is justified by the fact that while they represent very contrasted social models, bearers of different values and societal projects, and underpinned by specific institutions -notably in the field of social policy, they have nonetheless introduced a similar policy (namely a 50 % tax reduction on the costs incurred for domestic services, up to a ceiling which is roughly the same in both countries but considerably higher than in other European countries) in a field where these two countries have traditionally differed in substantial ways. Here the aim is to analyse how this policy is mediated by existing welfare, gender and labour market regimes and to what extent we observe similar consequences in terms of the governance and redistributive effects of social policy, of labour market transformation, and of the structuring of different social cleavages. I suggest that the uncovering of similar trends in the logic and modes of public intervention and in the social, economic and political consequences of this public intervention in two strongly contrasted national social models could be revealing of more global trends in Europe, linked to more profound transformations of welfare states, of labour markets, and of societies more generally. The domestic services sector has increasingly been promoted in Europe over the past ten years or so by various national and supranational actors such as the European Commission, national governments and national lobby groups of domestic services providers, which have increasingly organised themselves in European networks also.

While today national and transnational lobby groups are important actors behind the continued promotion of domestic services, the European Commission has, already since the early 1990s, been quite instrumental in promoting the development of this sector across Member states, according to a rationale that dovetails several preoccupations and strategies, and consequently a number of possible contradictions too. A brief overview of the European Commission's discourse and recommendations thus seems warranted in order to understand the particular policies that have increasingly been implemented in a large number of countries, and the issues they raise.

Promoting low-skilled jobs through labour cost reductions

This interest in the development of domestic services on the part of the EC can first be traced back to the 1993 White paper "Growth, competitiveness, employment" (COM 93( 700)). The central concern addressed in this White Paper is the issue of unemployment, (already) then at a very high level in Europe, especially amongst the low skilled. To remedy this situation, specific actions are recommended, the most important of which having to do with flexibilizing the labour market and addressing the disincentives to employing low skilled workers by reducing the costs for employers (see in particular chapter 9). The high level of statutory charges on labour in Europe compared to the US or Japan is analysed as an important factor in the slowdown in economic growth and especially in the inability to create employment, not least in the labour-intensive service sector. The high cost of labour is also viewed as promoting a parallel economy. This White Paper thus recommends that member states seek to reduce the statutory charges (taxes and social contributions of employers and employees) imposed on labour, these recommendations being especially directed towards Denmark, the Netherlands, Belgium, France and Germany, while the level of statutory charges in the UK is considered satisfactory, being well below the EU average.

Responding to social needs in a more cost effective way

At the same time, certain areas are identified as sources of new jobs. Here, 'local services', comprising care for children, the elderly and the disabled, domiciliary health care and various household services (e.g. meal preparations and housework), are highlighted as an important source of job creation. The White Paper notes that they are new social needs that need to be satisfied which are due to changes in lifestyles, the transformation of family structures, the increase in the number of working women and the new aspirations of the elderly and of very old people. It is then argued that the market cannot properly respond to these needs as the development of both supply and demand for these services comes up against barriers which, on the demand side, relate to the high cost of labour and on the supply side to the reticence to take up jobs which are perceived as degrading and low-skilled. As a result, the White Paper concludes, "the development of the services in question is either left to the undeclared employment market, or is publicly funded, which is expensive " (p.19). Calls are therefore made to develop new initiatives to stimulate both demand and supply, which could be based, on the demand side, on "incentives such as income tax deductibility, or the local issuing of 'vouchers' along the lines of luncheon vouchers, issued instead of providing the social services normally provided by employers and local authorities", and on the supply side, on "traditional subsidies for the setting-up of undertakings which could be increased in cases where a 'social employer' undertakes to employ formerly unemployed people" (p.19, my emphasis). Thus, while new social needs are identified, they are mainly addressed as a source of employment growth rather than as a social concern.

What is interesting to note here is that while the main concern of this White Paper is with fostering employment -and the promotion of local services appears as a way of neatly tying in employment objectives with social concerns -the arguments put forward go a bit beyond the simple development of a new sector. Also outlined here is the idea that these services should come to replace the expensive, already existing, publicly funded social services, with no explanation as to how that would foster employment.

In the 1994 White Paper "European social policy -A way forward for the Union" (COM 94(333)), 'local services' are only mentioned in passing as a necessary support to women's employment and as part of a broad employment strategy to provide work for the low-skilled: "It is important to ensure that, as well as supporting high productivity jobs, the Union maximises its ability to generate and sustain jobs at other levels, particularly in the unskilled, semi-skilled and personal and local services fields" (p.12).

In its Communication on "The European Employment Strategy: recent progress and prospects for the future" (COM 95(465)), the Commission once again highlights the employment potential of local services, which lend themselves to "activities for reintegrating the long-term unemployed, young people with problems and unemployed women" and "are geared towards providing the best means of meeting the new needs of society through new occupations". The schemes that have been set up in a number of countries (France, Belgium, Denmark, Germany) are highlighted and their development encouraged. These schemes include the development of service vouchers and "the adjustment of legal, financial or fiscal provisions to enable households to become employers or consumers of domestic services" (p.30). Reducing indirect labour costs, especially for the low skilled, is also very high on the agenda in this document. The same arguments are re-iterated in the Commission draft for the 1997 Joint Employment Report, which emphasises that household related services need to be exploited much more decisively, following the example of France and Belgium.

Promoting female employment

The 1997 European Employment Strategy and the 2000 Lisbon Agenda mark the beginning of a strong activation turn, with a clear focus on the re-commodification of all groups of society, and especially on promoting women's employment. This activation turn also marks a shift in the EU's gender-mainstreaming approach. Indeed, while until the 1990s Commission documents showed a concern with promoting gender equality through measures that aimed at enabling women to work but also at redistributing paid and unpaid work more equally between men and women, the 'feminist' potential of EU discourse and policies start to give way to economic concerns. This translates into policies that make women's labour force participation a necessity rather than an opportunity and which focus on supporting women's caring responsibilities rather than on redistributing them between women and men, as well as on encouraging flexible forms of employment (Stratigaki, 2004;Lewis, 2006;Radulova, 2009).

In this context, and even more so starting with the Lisbon strategy in 2000, and reinforced with the Barcelona summit of 2002, what are now termed alternatively 'domestic services' or 'personal services', and eventually 'personal and household services' or 'household and social services', really come into focus, with a new rationale being put forward linked to this growing focus that is placed on promoting female labour force participation (with a Lisbon target of a 60% female employment rate within a 10-year horizon). The issue becomes both one of providing women with the necessary services to support their gainful employment, and of creating new sources of jobs for women.

The need to develop childcare services to meet the Barcelona targets (a childcare coverage of 33% for children under 3, and of 90% for the 3 to school age children) has provided important impetus for the development of different forms of childcare. The Barcelona targets only dealt with the quantitative aspects of childcare provision, however, and as different observers have highlighted, the question of the quality of the care provided has not really been addressed. As a result, cheaper, private forms of care have been promoted across most countries, notably through socio-fiscal measures such as tax deductions and/or credits and social contribution exemptions, thus leading to a rapid growth in the use of private child-minders rather than to the development of public childcare services (cf. Morel, 2007).

The level of education and qualification of these private child-minders varies between countries, ranging from no vocational training at all to limited vocational training, but education levels and vocational training remain in all cases much lower than in the public childcare services (cf. Platenga & Remery, 2009). Promoting domestic forms of childcare has thus proven to be an important source of job creation for the low skilled, as well as for older women entering or re-entering the labour market after a long break.

Supporting the productive potential of the high-skilled

The Lisbon strategy's aim to turn Europe into 'the most dynamic and competitive knowledge-based economy in the world' has also led to a focus on the more highly skilled and productive workforce. In this respect, enabling the more highly-skilled to invest more time into paid work has been a recurring theme, not least in discourses related to the reconciliation of work and family life where the fear is expressed that the human capital of high-skilled women might be wasted or under-used if they cannot free themselves enough from caring and household tasks. In this context, the promotion of personal and household services becomes increasingly presented as a gender-equality measure that will relieve women from domestic tasks and enable them to invest more time into paid work, thus giving them the same opportunities as men on the labour market.

Responding to unmet social needs

Finally, in the past decade, while fostering the development of domestic services has continued to be considered as an important strategy for creating jobs, especially for the lowskilled and the long-term unemployed, at the discursive level such services have increasingly been presented as a means to respond to unmet social needs, especially with respect to childcare and elderly-care, but also with respect to the needs of increasingly time-pressed dual-earner families or single parents who struggle to combine paid work with family life and household tasks such as cleaning, cooking, etc. Promoting greater freedom of choice in terms of the services available to individuals, not least for the elderly, has also been highlighted as a reason for encouraging the development of domestic services. Last but not least, the development of personal and household services has been put forward as a desirable solution to meet the increasing demand for long-term care under increased budget constraints. This keen interest in promoting personal and household services is re-affirmed in the Europe 2020 strategy which was defined in 2010, and its clearest expression can be found in the 2012 "Commission staff working document on exploiting the employment potential of the personal and household services" (SWD(2012)95 final). In a time of high unemployment across Europe, the focus is once again on the employment potential of domestic services (the term 'domestic services' is used interchangeably with that of 'personal and household services' in this document). After defining personal and household services as "a broad range of activities that contribute to the well being at home of families and individuals", including "child care, long term care for the elderly and for persons with disabilities, cleaning, remedial classes, home repairs, gardening, ICT support, etc.", the document highlights the key characteristics of personal and household services that make them interesting from the point of view of employment policies (p.4-5, my emphasis):

• Low import content (activities produced locally)

• High employment content, implying a potentially important effect on job creation in case of public support;

• Varying levels of technical skills requirements • Low productivity in some of the tasks involved, but a potential for indirect productivity increases if clients of PHS are able to focus more on their own, higher-productivity work;

• Given the predominance of undeclared workers in these sectors, public intervention is not likely to trigger a shift of employment from other sectors;

• A growing need for these services due to population ageing and to the need to increase female participation in the labour market. PHS activities help to improve work-life balance as well as earn back effects via an increase of working hours or a return on the job market.

This document also takes up the issue of the high prevalence of undeclared work in domestic services, and thus the necessity to set up public support to the personal and household services sector in order to fight this. Policies implemented in France, Germany and Austria are then used as examples, with France in particular being used as a success story having supposedly created, between 2005 and 2009, "500,000 new jobs subject to social contributions in private households, mainly for groups on the fringes of the labour market.

Informal employment could thus be reduced by 70%". What is interesting to note is that while this figure corresponds to what the French government had been hoping to achieve, real job creation over that period was in fact well below this figure, with only 390 000 jobs having been created since 2005 (furthermore, this figure drops drastically when recalculated in full time equivalent)2 (Agence Nationale des Services à la Personne 3 ), just as the 70% reduction in informal employment is a misinterpretation of the fact that Marbot (2008) had calculated that 70% of employment growth in this sector can been attributed to a 'whitening' of informal jobs 4 . While the 'true' figure is readily available in government documentation, it is the 'wishful' figure that is used in this Commission document 5 .

In the German example, it is the creation of "Minijobs" and the reduction in employers' social contribution rates for the employment of domestic workers that are cited as examples of good practice. All in all, as in previous documents, the policy measures put forward include simplified procedures for hiring household workers, subsidized vouchers, social contribution exemptions or reductions to reduce the cost of labour, and tax deductions, all of which are expected to foster the creation of new jobs in the form of self-employment or through the development of SMEs.

In many ways, this document neatly sums up all the underlying arguments that have developed over time at EU level for justifying the promotion of domestic services: the job creation potential for the low-skilled of these local services which cannot be displaced abroad; the fact that these jobs hold a promise of social inclusion for "groups on the fringes of the labour market"; their contribution to supporting female employment by facilitating the worklife balance and by creating employment for women; the 'productivity boost' argument that holds that the high-skilled can devote more time to productive tasks if they can delegate their unproductive household tasks; and the cost effectiveness argument for responding to new social needs, and in particular to child-and elderly-care needs.

All in all, the development of domestic services is presented as some kind of panacea that responds to both employment and social concerns. In all these documents, only two elements are mentioned as deserving special attention. One has to do with the need to give due attention to the working conditions of household workers in terms of their access to social protection rights, and the other is a concern with the quality of care, especially with respect to childcare, with the need to improve vocational skills in this field mentioned in a few documents. While these are indeed important elements, this policy orientation towards promoting domestic services seems to me revealing of broader economic and societal choices which raise a number of more crucial issues not addressed in any of the Commission documents and which carry important social ramifications.

The issues this policy orientation raises.

First of all, such a policy orientation can be questioned in economic terms as the promotion of domestic services comes at a cost for public finances which is far from negligible and one may question the kind of economic model that such a policy orientation promotes, targeted as it is towards the development of low-skilled work through a policy that aims at reducing labour costs, rather than at up-skilling or at promoting activities with a higher added value (as the aim of turning Europe into the most competitive knowledge-based economy would seem to entail). This policy orientation also reflects societal choices in that the development of domestic work, as it is structured through these policies, can only prosper in a context of important socio-economic inequalities, and these policies implicitly promote a societal structure based on a distinction between 'productive' and 'unproductive' workers, where the more highly-skilled productive workers and/or wealthier individuals are encouraged to delegate the more 'basic' domestic and caring tasks to the low-skilled. By encouraging those who are better off to provide employment for the less skilled, social transfers are mediated through households rather than through collective means. These policies effectively transform social relations and symbolic hierarchies by turning a growing number of individuals into private employers.

While such a policy orientation reinforces or institutionalizes the increasing dualism in the labour market between the high-skilled and the low-skilled, it also increases or even generates inequalities in the access to care or to the possibility to reconcile work and family life. Indeed, even when subsidized, the cost of these services remains well beyond the cost of collective social services and thus inaccessible to those on lower incomes. Furthermore, the use of tax rebates to subsidize these services means that households paying no or low taxes cannot benefit from any subsidies, and as several studies have shown, the use of tax rebates strongly favours high income earners. This very skewed redistributive profile of this form of public expenditure should be cause for particular concern given that it is clear from all the Commission documents analysed, and from the schemes implemented in the various European countries, that while the term 'personal and household services' can cover a very wide and rather heterogeneous range of activities requiring different skills and performed under very different working arrangements (e.g. remedial tuition vs. cleaning), care related work (care for children, the elderly or people with disabilities) is considered as the core activity, and indeed corresponds to the types of services most used. This policy orientation thus leads to a transformation in the logic of public intervention, where the state finances care services through fiscal measures to subsidize the demand for private services rather than through the supply of public services. This policy orientation thus seems to lead to a 'fiscalization' of the welfare state, or at least of care policies. The redistributive consequences of this shift, the change in the mode of governance of social policy that it entails and how it affects the political legitimacy of social policy, as well as the effects on the quality of the care thus provided certainly deserve closer analysis. Such an approach to providing care also raises the issue of geographical inequalities as the availability of services is entirely dependent on a locally generated and market-driven offer. As we already know from the experience of childcare provision in different countries where the state subsidizes the demand rather than the provision of childcare services, market services tend to concentrate in the urban areas (and especially the wealthier areas). This policy orientation also seems to raise a number of contradictions. One of these has to do with the fact that these services are intended to fulfil care needs, and are thus directed towards vulnerable groups (children, the elderly, the disabled). As such the issue of the quality of the services provided and the need for specific skills is raised. Yet these services are expected to create jobs for the 'low-skilled', the 'long-term unemployed', 'marginalised individuals', 'groups on the fringes'. How the quality of care can be reconciled with this lack of skill is not addressed, other than to say that attention should be paid to improving vocational training, especially for childcare. But again, a contradiction arises between the emphasis that is placed on reducing labour costs / keeping the price of these services as low as possible and improving skills.

Another contradiction has to do with the Commission's stated aim to improve gender equality. Indeed, desegregation of the labour market has been a major stated objective of European employment policies (cf. European Commission, 1997), yet women represent the overwhelming majority of domestic services workers, and in fact women outside the labour market were particularly targeted as a group that could fill these new jobs. This gender segregation of the labour market affects gender equality in other ways too. Indeed, while the policy discourse is that these services help women to better reconcile work and family life and foster gender equality by allowing women to devote more time to paid employment, thus allowing them to invest in their professional life on a par with men, it is clear that these schemes affect women very differently depending on their socio-economic status. A clear division appears between the wealthier households / women who can avail themselves of these services, and those who cannot and/or provide these services. In many ways, one could argue that far from fostering greater gender equality, these policies in fact contribute to accentuating gender inequalities in that they allow wealthier households to sidestep the issue by delegating their domestic tasks to others, while the fundamental issue of the unequal division of household tasks between men and women at a societal level remains untouched.

Finally, although the issue of the working conditions is sometimes mentioned, especially with regard to access to social protection rights, there is no or little reflexion on the very nature of the work that is promoted and its consequences. One defining characteristic of domestic services is of course that they take place in a domestic sphere, which constitutes a very specific workplace. First of all, monitoring working conditions tends to be very difficult, as labour inspection is not normally allowed in private homes due to privacy laws. Secondly, workers are on their own which makes it difficult to share experiences, let alone unionize, and they are likely to be more subject to the arbitrariness of the employer. This is likely to be particularly the case when domestic workers are employed directly by a private employer rather than through a firm. Yet as the 2012 Commission document on "exploiting the potential of the personal and household services" indicates, it is precisely in the form of selfemployment or through the development of SMEs that the policies promoted are expected to create new jobs.

The self-employed are also less likely to be able to benefit from vocational training, minimum wage (i.e. a secured wage independently of the number of hours worked), income replacement in case of illness or accident, etc. Finally, the fact that these jobs are almost exclusively part-time jobs (because of the very nature of the work which requires working for different employers and thus involves much time spent in transport which is not paid) is not addressed. Yet the fact that most workers only work part-time (and often short part-time) means that they usually have limited access to social coverage (limited or no access to sickness benefits for instance if they have not cumulated enough hours of work over a given period).

These issues warrant closer analysis as most countries in Continental and Northern

Europe have now implemented some kind of scheme to promote the demand for domestic services (subsidized vouchers, social contribution exemptions for employers and/or tax reductions or credits), at least in the field of child and elderly-care, and some countries have gone even further in that they also subsidize 'comfort' services such as cleaning, ironing, cooking, gardening, home repairs and other activities taking place in a domestic setting. The extent of public support provided varies between countries, France and Sweden being the two countries that subsidize these services the most, and this for both care-related and more comfort related services.

The comparison of these two countries should prove fruitful on several accounts. Firstly, while both countries have introduced a very similar policy, the reasons put forward for the introduction of this measure have been framed partly differently, and the kind of political, public and academic debates that such a measure has unleashed are of a very different nature. Indeed, while there is a wide political consensus around this scheme in France, and very little public or academic debate, this scheme has been much more politicized and controversial in Sweden, engaging the different political parties, trade unions, feminists and the media, with the focus being in particular on the gender and social cleavages that such a policy structures.

Contrasting the political justifications for this scheme and the debates -or absence thereofthat this policy measure has unleashed is therefore interesting in that it sheds light on the contrasted normative assumptions, values and norms for public action that guide these two countries and the extent to which these are being reformulated.

Secondly, while these two countries represent very contrasted social models, bearers of different values and societal projects, and underpinned by specific institutions -notably in the field of social policy, they have nonetheless introduced a similar policy in a field where these two countries have traditionally differed in substantial ways. This allows for an analysis of how this policy is mediated by these different socio-political contexts and the extent to which we observe similar consequences in terms of the governance and redistributive effects of social policy, of labour market transformation, and of the structuring of different social cleavages.

Developing domestic services in France and Sweden.

The measures implemented and the debates surrounding those measures

As can be seen from the Commission documents analysed above, France has been a pioneer when it comes to subsidizing the demand for domestic services through socio-fiscal measures. Starting in 1987, France introduced employer social contribution exemptions for elderly people above the age of 70 or disabled people who directly employed a home-help worker. The AGED was introduced the same year for the employment of a private childminder and offered both social contribution exemptions and a generous tax deduction.

Another benefit, the AFEAMA, was set up in 1990 to cover the cost of social contributions when parents employ a registered child-minder to care for children in her home. These benefits were increased in 1994. The AFEAMA and the AGED were intended to develop cheaper forms of care than day-care institutions and to increase employment, partly by making it easier to hire someone and partly by bringing women working on the black market into the formal labour market (Morel, 2007). This promotion of domestic services in the field of childcare marked a considerable break away from the policy logic that prevailed in the 1970s and 1980s and which was based on the development of public childcare services (Jenson and Sineau, 2001;Morgan, 2002). Thus, while these schemes had a 'social' objective in that they aimed at responding to care needs, they did so in a way that aimed at fostering another, more central objective for the government which was to create employment for the low-skilled. From 1991 onwards, these employment objectives become paramount. A new law is voted which allows any individual to deduct 50% of the total amount (wage + social contributions) spent on domestic services from their income tax. While the ceiling for this tax deduction was initially relatively low (3811 / year) it has been progressively increased up to 13 720 in 1995, then decreased in 1998 by the then socialist government, and re-increased since 2004 to 12 000 . This ceiling can be raised depending on the composition of the household (number of children, presence of a disabled person…).

Since 1993, individuals can use service cheques (called "Chèque emploi service universel -CESU" since 2006) to remunerate household workers. These cheques have a fixed value and simplify the administrative procedures for the employer (the worker paid with these cheques is automatically declared without the employer having to do anything, a special agency takes care of calculating the social contributions, no need to deliver any payslips), and since 1996 a new system allows for a third party (e.g. companies) to contribute to the financing of these service cheques for employees wishing to employ a household worker (along the same principle as luncheon vouchers).

Until 1996, only household services bought by private employers directly, or through the intermediary of associations who took care of the recruitment and administrative tasks but where the private individual still remained the direct employer, could benefit from these socio-fiscal exemptions, but since 1996 these fiscal advantages have also been extended to for-profit companies who function as direct service providers (i.e. they, rather than private households, are the employers). Finally, since 1999, the VAT for both associations and for-profit companies involved in household services is only 5,5% (raised to 7% since 2012) instead of 19,6% for other goods and services.

In 2002, a new benefit for the dependent elderly was created, the Personalised Autonomy Benefit (APA), which is a cash benefit calculated on the basis of dependency level and household income and which can be used to directly remunerate a home-help worker or a service provider. Here too, the aim was to increase the solvency of households in order to support the demand for domestic services.

Indeed, the stated aim of these different measures has been to actively promote the development of household services, this sector having been identified since the late 1980s as an important source of employment that needs to be tapped into. While these schemes were initially introduced by a Left-wing government, they have been expanded under successive Right-wing governments who have raised the ceiling of the amount that can be deducted from taxes and so as to encompass a larger range of services 6 . Especially since the 2005 Borloo Plan which aimed at creating 500 000 new jobs in household services over a three-year period 7 , these schemes have become a cornerstone in the government's employment policy. A new plan to expand this sector further has been launched in 2009. Since 2008, the scheme also functions as a tax credit in order to enable those on low income to benefit from this scheme, but it can only be claimed by people who are actively employed.

There has been little debate in France regarding these schemes, most critiques emanating from economists who have called into question the job creation figures or from some commentators on the Left who have questioned the public cost (in terms of income foregone) in a time of budgetary constraints as well as the redistributive profile of these measures, but these debates have not engaged the public and there is a fairly broad agreement on the Left and the Right in favour of the promotion of household services. While these policies were initially presented as a way to promote employment growth by lowering the cost of low-skilled workers and to fight informal employment, and while job creation remains the central objective, the policy discourse around these schemes has evolved over time to centre more on the social function of these services and the increasing need for 25 activities are identified, which include child-minding, home-help, cleaning, private tuition, IT support, gardening…

The objective of 500 000 jobs was based on an extrapolation of very dubious data. Indeed, as the Court of Auditors has pointed out, this figure is far from reliable, the ANSP counting all workers who fit the following definition: "any remunerated person having worked at least once during the year, whatever the length of the service provision, with one or more employer". This figure came out to 75000 for the year 2004. It was then decided to double this figure for the following three years (arriving at 450 000 jobs) and to even that out to 500 000…(Cour des Comptes, 2010). See also footnote 2. them (especially with respect to elderly-care), on the contribution of these services to facilitating the work-life balance (although there has been no discourse around the promotion of gender equality), and on their contribution to promoting 'free choice' in child-and elderlycare (cf. Morel, 2007).

Sweden also introduced a 50% tax reduction on household services in 2007 for the purchase of services up to a ceiling of 100 000 SEK [12 090 ] per year and per person (i.e. a single person can deduct maximum 50 000 SEK but in a two-earner household each can deduct that amount, bringing the total deduction to 100 000 SEK, i.e. on a par with the French ceiling).

Unlike the French example, in Sweden the introduction of this measure followed a protracted and very ideologically charged debate (the so-called 'maid-debate') that began in the early 1990s between the proponents of this measure (Right-wing parties) and the parties on the political Left, a debate that has also engaged feminists, trade unions, the media and the public at large. The various proposals put forward in favour of this measure since the 1990s as well as the 2006 Government Bill have underlined that such a measure would contribute to reducing informal employment (and thus improve working conditions for household workers) and to creating jobs for the low skilled and peopled marginalized from the labour market (immigrants, long-term unemployed, youth). It would also lead to productivity gains for the whole economy if unpaid household work became paid, not least as delegating household work can allow more productive households to specialize in what is most advantageous to them and make the economy more efficient (Prop. 2006/7:94, p.32). Such a measure is also presented as a way of promoting greater gender equality on the labour market, notably by helping women to better reconcile work and family life and by allowing them to invest more time on the paid labour market, and of promoting greater freedom of choice with respect to the services made available, not least for the elderly.

The gender equality argument raised considerable debate, opponents from the political Left, women's organizations and trade unions pointing out that not only did such a scheme not address the fundamental issue of the unequal distribution of household tasks between men and women and in fact cemented these gender inequalities, it also created considerable class inequalities between women, some women being encouraged to make a career on the backs of other women who could not afford household services themselves. Interestingly, in the different government propositions one does find some mention of the fact that not all women would benefit in the same way from this measure, but the argument is made that it is important to free the productive potential of the more highly skilled.

Finally, the decreasing coverage that has taken place in elderly-care services is pointed out as affecting women's investment in paid employment. The tax deduction is thus supposed to support women's employment by facilitating the elderly's access to home help services.

Officially, these private domestic services are not intended to replace public social care services, but rather to complement them and offer a greater range of more individualised services, and to help cover the needs of those who fall below the threshold for accessing public care services, not least due to a shrinking of resources. It is nonetheless worth noting that the rules that apply for claiming a tax reduction indicate that "the work must be carried out in, or in direct connection to, the employer's home or the home of his/ her ascendants" (Prop. 2006/07:94, own translation). Not only is the scheme clearly intended to be used to provide services to the elderly, it is also assumed that children will/can pay for these services, which marks a clear shift from previous normative assumptions and institutional arrangements regarding the role of the family in elderly-care provision in Sweden.

Impact of these measures

Regarding the impact of these measures, several aspects should be taken into account.

One relates to the impact in terms of job creation, both with respect to numbers but also to the quality of these jobs, as well as to the public cost of supporting these jobs. The profile of household workers, in particular in terms of gender and skill level, but also in terms of ethnicity, are also of interest for an understanding of the kind of labour market segmentation / segregation that these policies potentially promote. Skill levels are also an important issue in the way they relate to the quality of care-related service provision to vulnerable groups.

Another aspect relates to the redistributive profile of these schemes: who are the beneficiaries of these schemes, i.e. who benefits from these public tax expenditures? Finally, the type of services used, and by whom, is of interest in so far as it can provide an indicator of the extent to which these domestic services serve to fulfil care needs, i.e. the extent to which social care systems are being redesigned through fiscal means.

As pointed out earlier, it is very difficult to obtain any reliable or consistent data in the French case regarding the number of jobs created by these schemes, both because this sector is politically loaded and the government as well as the National Association of Household Services (ANSP) have a vested interest in showing that these measures are indeed creating employment, but also because of the specificity of this sector where irregular periods of employment and part-time work is the norm and problems of double counting are high (household workers often work for several employers, both private household and service enterprises or associations providing services) 8 . According to official figures, the household services sector currently employs nearly 2 million people, but according to the French Economic Observatory (OFCE) this figure is closer to 1,2 million. Likewise, where the ANSP and the government estimate that 319 000 new jobs were created over a three year period (2005)(2006)(2007), the OFCE estimates this figure to be 157 800 only, which falls further to 52 000 when recalculated in full-time equivalent (Jany-Catrice, 2009).

Nonetheless, all estimates do indicate that there has indeed been some job creation.

However, this job creation comes at a fairly high public cost. In 2009, public expenditure on domestic services reached 6,6 billion euros in the form of social contribution exemptions (2,16 billion) and tax reductions (4,44 billion), which corresponds to a 40% increase since 2006 (Cour des comptes, 2010). While different economists have shown the high public cost associated to these job creations, it has been argued that some of this public cost is off-set by the fact that these new workers contribute in the form of social contributions and taxes and that less has to be spent on unemployment and social assistance. While it is difficult to factor in all these parameters, it should nonetheless be noted that it is rather dubious whether any of these workers earn enough to even pay taxes, and social contributions are not paid by all employers since there are exemptions for the elderly and for childcare, which form the core of the activity. Also, as Flipo et al. (2007), Marbot (2008) and others have shown, much of this job 'creation' -up to 70% -actually corresponds to a whitening of informal labour, meaning that net job creation is limited, and also shows that the tax deduction has mainly had a windfall effect for households who were already purchasing domestic services on the informal market.

This windfall effect should be questioned both from the perspective of the efficacy of this policy instrument in terms of creating jobs, and from the perspective of the redistributive effects of this policy. For who are the people buying domestic services and benefiting from these tax reductions? If one takes a closer look at the profile of domestic services users, it appears that age is a major distinguishing criteria. In 2005, 54.2% of households who used domestic services were over 70, even though this age group only represents 20.1% of the For a critical overview of the different data used and their limitation in the French case, see Jany-Catrice, 2009. population. Furthermore, 19.3% of the 70 to 80 years old, and up to 40.3% of the over 80 use domestic services (Marbot, 2011;see also Carbonnier, 2009). In fact, nearly half the public support to the domestic services sector in the form of social contribution exemptions and tax deductions (3 billion euros) is related to services for the dependent elderly (Cour des comptes, 2010). Socio-professional status is another discriminating factor, with a strong overrepresentation of people in executive professions (Marbot, 2011). The use of domestic services is also very much concentrated in the larger urban areas. Not all users can benefit from the tax deduction or credit, however, and as various studies have shown, it is the wealthiest households who are the main beneficiaries. In 2005, 32% of households who declared a household worker belonged to the wealthiest 10% of households, whereas only 3.5% belonged to the poorest 10%. Furthermore, the wealthiest 10% stood for 53.6% of total expenditure. If one adds to that that only those who pay a positive amount in income tax qualify for the tax deduction, the distributional impact of the tax cut is strongly in favour of more affluent: 73.1% of the tax reduction went to the wealthiest 10% (Marbot, 2011). Furthermore inequalities have been increasing since 2001 both with respect to those having recourse to household services and in terms of those benefiting from tax advantages (Carbonnier, 2009). While the introduction of a tax credit in 2008 has enabled more households to benefit from this scheme, the effect is nonetheless quite limited (Cour des comptes, 2010), and does not benefit the elderly since one has to be actively employed to benefit from it. Thus the main users (the elderly) of domestic services are penalized by this system (although they do benefit from social contribution exemptions instead).

Finally, as Carbonnier (2009) has suggested after carrying out an ex post analysis of the impact of the raise in the tax reduction ceiling that was implemented in 2002, the fiscal cost of this raise has probably been greater than what the budgetary cost of a direct financing of these new jobs would have been. The change in ceiling thus appears more as an aid to (wealthy) households than as a measure to substantially increase employment in personal services.

While the (anti-)redistributive effects of this policy already raise a number of serious issues, one should also question the types and quality of jobs that are thus being created and supported at a high public cost. A few figures give some idea of the quality of these jobs: around 80% of household workers are directly employed by households, and therefore cannot benefit from vocational training, medical check-ups, or the protection of Labour inspection or the help of trade-unions in the case of work litigation (CERC, 2008). Contracts are short-term and employment is very much part-time, household employees working 12 hours / week on average (except for child minders who tend to work full-time).

Commercial service providers, where working conditions may be a little better in terms of access to training, etc., account for only 15% of total employment in this sector (Jany-Catrice, 2009).

In terms of work protection, workers are covered by different collective agreements depending on the nature of their work (e.g. cleaning or caring for an elderly) and of their employer (private employer or commercial enterprise) and several collective agreements can co-exist within the same commercial enterprise. Given that household workers usually work for different employers (e.g. both through a commercial enterprise or an association and directly employed by a household) and carry out different types of tasks at different times of day, they are constantly going in and out of different collective agreements which makes for a great lack of legibility and a difficulty in knowing one's rights. Finally, due to the very atomized work situation of household employees, these cannot really unionize, nor have trade unions shown much interest in this sector.

Although the need to better structure and to improve the quality of the household services sector has been repeatedly highlighted, the priority that has been given to subsidizing the demand for services has clearly occurred at the expense of the structuring of a quality, professional offer.

Regarding the profile of household workers, skill levels are low: today, only 30% of workers in household services hold some kind of qualification (Cour des comptes, 2010, p.523) which raises issues in terms of the quality of care provided to the main users of domestic services, namely the elderly. Finally, one should mention the gender profile of these jobs: 98% are women. It would also be worth looking at the ethnic profile of these workers, but this is made difficult in the French case as one is not allowed to collect any data related to origins. As a proxy, one can mention Eurostat data which indicate that 21,1% of foreign women work in domestic services in France, against only 3,8% of French women. Furthermore, one can mention the agreement that has been signed in 2008 between the Ministry of Immigration and the Ministry of the Economy, which states that newly arrived immigrants should be directed towards the National Agency for Household Services (ANSP) in order to be trained as household workers. This agreement concerns 10,000 immigrants per year 9 . Here, clearly, being an immigrant is associated to a specific aptitude to perform domestic work or, conversely, is equated to a lack of skills which would allow them to pursue other lines of work.

If one now turns to Sweden, a similar picture appears. While the introduction of the tax reduction scheme is much more recent than in France, its popularity has increased rapidly, and public expenditure on this scheme now exceeds what had been forecasted when the scheme was proposed. In 2010, tax expenditure for this scheme amounted to 1,340 million SEK (about 162,6 million euros) which represented an 80% increase from the previous year (Sköld and Heggeman, 2011).

In terms of job creation, the number of people employed in the formal domestic services sector has gone up from 29 100 in 2003/4 to 41 000 in 2009/10 (Almega, 2011), i.e. nearly 12 000 new jobs over a 6-year period.

The redistributive profile of the scheme is similar to that found in France. Today, about 4,5% of households benefit from tax deductions on domestic services, and this benefits mainly the wealthier households. Of the total tax reduction in 2009, 64% went to households in the upper income quartile, while only 7% went to households in the lowest income quartile (Sköld and Heggeman, 2011).

The profile of users is also similar: dual-earner couples with children and the elderly are the main users. People over 65 in fact represent over one third of claimants (Meagher and Szebehely, 2010). The use of domestic services is also very much concentrated to the largest urban areas.

Finally, the profile of household workers is also similar to France with respect to skill levels (only 12% of workers have more than a secondary education, 36% have no secondary education at all), but less so with respect to gender: only 69% of workers are women. Finally, 43% of household employees are born abroad (Almega, 2011).

Working conditions are potentially better than in France, however, as the market for domestic services has been structured in such a way that most household workers are actually employed by service enterprises rather than by private households. Workers are thus likely to be better covered by collective agreements and to have access to various benefits such as 9 http://www.immigration.gouv.fr/spip.php?page=dossiers_det_int&numrubrique=324&numarticle=1347 training, etc. In fact, 24% of household workers have received some training from their employer (Almega, 2011). Household workers also benefit from trade-union representation.

Conclusion

Here we have only sketched out some of the main characteristics of the domestic services field, as it has been structured through state intervention in two contrasted social models. While these different elements warrant closer analysis, a couple interesting trends already stand out.

One important trend has to do with the economic priorities that transpire through the focus that is placed on creating low-skilled, bad quality employment at a high public cost, for the benefit of the privileged few. A similar tendency to reinforce or even foster socioeconomic inequalities between the low-skilled and the high-skilled, between immigrants and non-immigrants, in quite an intentional and well-assumed fashion also appears.

important trend has to do with the care content of these domestic services. Both in France and in Sweden, these policies seem to lead to a privatization and to some form of re-domestication of care in the sense that care is increasingly provided through private domestic services rather than public social services. What is interesting, however, is that this doesn't correspond to a simple roll-back of the welfare state, with the responsibility for care provision being placed back onto the family, but rather to a transformation in the logic of public intervention, where the state still finances care services, but it does so by subsidizing the demand for private domestic services rather than the supply of public services, and it does so through tax expenditures rather than government spending. Clearly the redistributive consequences of this shift towards 'fiscal welfare' warrant closer analysis, as does the impact of this shift on the quality of care provided.

While these trends seem to reflect broader transformations at the European level, the consequences and intensity of these changes are likely to vary in different welfare systems. In our two case studies, the social inequalities and symbolic hierarchies that this policy fosters seem to represent a much more frontal attack on the Swedish egalitarian model than it does in France where social stratification has long been important. Likewise, contrary to France where care services for children and the dependent elderly were largely insufficient, Sweden has for a long time offered universal social care services for both childcare and elderly-care, thus limiting the demand for private domestic services. As some commentators have noted, the introduction of this tax deduction for private domestic services can be interpreted as a

Here estimates vary greatly. The Government and the National Agency for Personal Services (ANSP) give the figure of

000, refusing to adjust that to full time equivalent. Here the figure corresponds to the number of people having worked at least once during the year in these services, independently of duration of work. Since most workers in this field (over 70%) work part-time, and mostly short part-time (around 12 hours a week), the number of jobs created in full time equivalent would be much lower than the official figure. According to the National Statistics Office (INSEE), the number of full-time equivalent jobs created between 2006 and 2008 is only 108 000 (cf. Cour des Comptes, 2010). Other sources give a figure of 102 000 FTE between 2005 and 2009. 3 http://www.servicesalapersonne.gouv.fr/chiffres-cles-(2064).cml 4 Furthermore, Marbot's calculation concerns the period 1996-2006, i.e. a much longer period. 5 To be fair, this Commission document reproduces the wrong information published in another document from a project coordinated by the Institut für Wirtschaft, Arbeit und Kultur and supported by DG Employment, Social Affairs and Inclusion "Creating Formal Employment Relationships in the Domestic Services Sector: Successful Strategies", 2011.

"privatisation by stealth" of elderly-care service, with the elderly being encouraged to buy private services instead of the services they used to receive from the public sector. This could well lead to an incremental dismantling of public care provision which would seriously reshape both the underlying principles, aims and outcomes of Swedish care policy.

• Anderson, B. (2000), Doing the dirty work ? The global politics of domestic labour. ZED Books, NY.


Bijan Mohammadi
email: bijan.mohammadi@univ-montp2.fr

Jukka Tuomela
email: jukka.tuomela@joensuu.fi

Involutive formulation for electroneutral microfluids

Keywords: electrophoresis, stacking, overdetermined PDE, involution 1991 MSC: 35K55, 58J10, 65M60, 76W05

In this paper we study a microfluidic flow model where the movement of several charged species is coupled with electric field and the motion of ambient fluid. The main numerical difficulty in this model is the net charge neutrality assuption which makes the system essentially overdetermined. Hence we propose to use the involutive and the associated augmented form of the system in numerical computations. The numerical experiments show that our approach gives significantly better results than the standard approach. Our methodology is also applicable to other overdetermined systems.

Introduction

Over the past 15 years, integrated electrokinetic microsystems have been developed with a variety of functionalities including sample pretreatment, mixing, and separation. Electric fields are used either to generate bulk fluid motion (electroosmosis) or to separate charged species (electrophoresis). In this article we study a widely used mathematical model for electrophoresis [START_REF] Brahadwaj | Design and optimization of on-chip capillary electrophoresis[END_REF]2,[START_REF] Hak | Handbook series for Mechanical Engineering[END_REF][START_REF] Lin | Instability of electrokinetic microchannel flows with conductivity gradients[END_REF][START_REF] Probstein | Physicochemical Hydrodynamics[END_REF][START_REF] Squires | Instability of electrokinetic microchannel flows with conductivity gradients[END_REF].

Mathematically this leads to a complicated system of nonlinear PDEs. The system is naturally composed of 2 blocks. The first is the Stokes system which governs the motion of the ambient fluid where the charged species of interest are. The second block is a nonlinear convection diffusion system for charged species and elliptic equation for electric potential. The real challenge comes from the physically reasonable charge neutrality assumption: charge distributions for different species add up to zero. This constraint makes the system essentially overdetermined, and leads immediately to problems with standard numerical methods which are designed to deal only with square systems (as many equations as unknowns).

Our method, already used in [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF][START_REF] Mohammadi | Involutive methods for Navier-Stokes equations[END_REF], uses the involutive and its associated augmented form of the system in numerical computations. The augmented form can be constructed once the involutive system and its compatibility operator are known. Typically this leads us back to the class of square systems where again standard methods are applicable. All relevant constraints of the problem are now explicitly included in the augmented system, hence the numerical errors related to constraints can effectively be controlled. The computations thus become more stable because one does not need to worry about possible instabilies due to nonrespect of constraints. Also the results will be more reliable, and they are more likely to reproduce essential qualitative properties of the solution than the standard methods. An example of this is presented below. Note that our approach is not restricted to microfluidic systems, but on the contrary can be applied in a wide variety of situations.

The content of the article is as follows. In section 2 we briefly recall what is meant by involutive, completed and augmented systems. In section 3 we present the physical model and then in section 4 derive the corresponding completed systems, both in stationary and time dependent case. Then in section 5 the relevant augmented systems are introduced and in section 6 we present our numerical results. Our method gives clearly better results than the standard approach. Finally in section 7 we conclude with some perspectives for future work.

2 Involutive, completed and augmented systems

Involutive systems

The important concept of involutivity is unfortunately quite difficult to define precisely. However, for our purposes it is sufficient to explain the idea in concrete terms and indicate how to work with this notion constructively. For more details we refer to [3,[START_REF] Krupchyk | Overdetermined elliptic systems[END_REF][START_REF] Krupchyk | Shapiro-Lopatinskij condition for elliptic boundary value problems[END_REF][START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF][START_REF] Mohammadi | Involutive methods for Navier-Stokes equations[END_REF][START_REF] Pommaret | Systems of Partial Differential Equations and Lie Pseudogroups[END_REF][START_REF] Seiler | Involution -the formal theory of differential equations and its applications in computer algebra and numerical analysis[END_REF][START_REF] Spencer | Overdetermined systems of linear partial differential equations[END_REF][START_REF] Tarkhanov | Complexes of differential operators[END_REF].

Let us consider the system ∇ × y + y = 0. Taking the divergence we see that if y is a solution, then it must also satisfy ∇•y = 0. This new equation is called a differential consequence or integrability condition of the initial system. Hence we have two systems:

S : ∇ × y + y = 0 S ′ :    ∇ × y + y = 0 ∇ • y = 0 (1)
We say that S ′ is the involutive form of S because no more new first order differential consequences can be found. So informally we may say that a system is involutive, if it contains all its differential consequences (up to given order).

There are many tricky issues involved when one actually tries to compute the involutive form, but the important point is that these constructions can be in fact carried out. Hence the approach we are proposing here is potentially useful for solving quite general systems of PDE.

It turns out that for the purposes of numerical computation it is sometimes convenient to use not the "full" involutive form of the system, but to add just some of the integrability conditions to the original system. Hence we will use the term completed system to indicate that we may not use the full involutive system.

But since completed systems usually have more equations than unknowns while the numerical methods for PDEs are designed for square systems, it is not obvious how to generalize for example the finite element method to this more general case. We outline below one possible approach.

Augmented systems

Let us consider our problem in a general form

A 0 y = f (2)
and let us suppose that A 0 is already in completed form. Now since A 0 is in general overdetermined, there are typically no solutions for arbitrary f ; hence there are some compatibility conditions for f . These conditions are given by a compatibility operator A 1 such that A 1 A 0 = 0 and (2) has a solution only if

A 1 f = 0.
Let us now introduce some function spaces V i such that A i : V i → V i+1 . Let us suppose that A 0 is injective, A 1 is surjective, and that image A 0 = ker A 1 ; in other words the following complex is exact:

0 / / V 0 A 0 / / V 1 A 1 / / V 2 / / 0
This suggests that we can decompose V 1 as follows:

image A 0 ) ⊕ image A T 1 ) ≃ V 1
where A T 1 is the formal transpose of A 1 . Of course to be able to write equality instead of ≃ we should specify carefully the relevant vector spaces. However, proceeding formally, this decomposition suggests that it is indeed possible to find some functional framework such that the combined operator (A 0 , A T 1 ) would be bijective or at least Fredholm.

So instead of trying to solve the original system (2) in some least square sense, we introduce an auxiliary variable ỹ and solve

A 0 y + A T 1 ỹ = f (3)
We call this system the augmented system. This formulation is reasonable because the augmented system is square, hence standard software is readily available. Also all the relevant information about the original system is contained in the completed operator A 0 which means that the results will be reliable. The drawback is that we have introduced an extra variable z which increases the computational cost. However, we can use ỹ in error control as explained in [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF].

3 Governing equations of ionic microfluids 3.1 Physical background.

Electrokinetic systems have been developed to perform a variety of functions including chemical separations, pre-concentration, and mixing [2,[START_REF] Squires | Instability of electrokinetic microchannel flows with conductivity gradients[END_REF]. Examples of separation assays include on-chip capillary zone electrophoresis and isoelectric focusing. Preconcentration methods include field amplified sample stacking and isotachophoretic preconcentration. These applications involve the convective-diffusion-electromigration of multiple ionic and neutral species.

Below we will discuss a fairly general convective diffusion system of equations applicable to electrokinetic microfluidics. We assume uniform and constant electrical permittivity and low Reynolds number flows. We also assume species are dilute enough to apply the Nernst-Planck equations and the dilute approximation for Fick's law [START_REF] Probstein | Physicochemical Hydrodynamics[END_REF].

Our formulation is applicable in thin electrical double layers, and accounts for net charge accumulation in the bulk resulting from a coupling between electric fields and conductivity gradients. The principal computational difficulty in this model is due to the net electroneutrality assumption. This assumption is valid on the length scales relevant to the phenomena under study [START_REF] Hak | Handbook series for Mechanical Engineering[END_REF][START_REF] Lin | Instability of electrokinetic microchannel flows with conductivity gradients[END_REF][START_REF] Probstein | Physicochemical Hydrodynamics[END_REF].

Initial model.

Let us consider the case where we have m different charged species in some ambient fluid. Let ρ e = F i z i C i = i z i c i be the net charge density where z i ∈ Z is the valence number of species i, C i is the molar concentration and F is the Faraday's constant. The charge induces an electric field which is supposed to come from the potential φ. Hence we get our first equation

-ε∆φ -ρ e = 0
where ε is the permittivity of the ambient fluid. The motion of ambient fluid is governed by Navier-Stokes equations, but because in typical applications the Reynolds number is very low we can as well use the Stokes system. We will further suppose that the ambient fluid is incompressible. Hence the Stokes system in the presence of electric field can be written as

ρu t -µ∆u + ∇p -ρ e ∇φ = 0 ∇ • u = 0
Here ρ is the density of the fluid and µ is the dynamic viscosity. Now the movement of different species are governed by equations c i t + ∇ • J i = 0 where the current density J i is given by

J i = -ν i c i ∇φ -d i ∇c i + c i u
where ν i is the mobility times the Faraday's constant and d i the diffusivity of species i. We have now introduced all necessary variables and parameters. Taking into account that ∇ • u = 0 one has:

∇ • J i = -ν i ∇c i , ∇φ -ν i c i ∆φ -d i ∆c i + ∇c i , u
This gives the system

ρu t -µ∆u + ∇p -ρ e ∇φ = 0 ∇ • u = 0 c i t -d i ∆c i -ν i c i ∆φ -ν i ∇c i , ∇φ + ∇c i , u = 0 , i = 1, . . . , m -ε∆φ -ρ e = 0 ρ e - i z i c i = 0
In the following we will only consider the net neutrality assumption case where ρ e = 0. Assuming this, and taking the divergence of the first equation and simplifying we obtain

ρu t -µ∆u + ∇p = 0 -∆p = 0 ∇ • u = 0 c i t -d i ∆c i -ν i c i ∆φ -ν i ∇c i , ∇φ + ∇c i , u = 0 , i = 1, . . . , m -∆φ = 0 - i z i c i = 0 (4)
Note that the system is naturally composed of 2 blocks: the Stokes system for variables (u, p) and a diffusion like system for variables (c, φ).

The main difficulty in solving (4) with standard methods is that the electroneutrality constraint (the last equation) is not respected. Of course one of the species could be deduced from the constraint:

c m = - 1 z m m-1 i=1 z i c i (5)
However, this is a real limitation because in this case the physical characteristics of one species are not taken into account during integration. Hence it is definitely interesting to include the electroneutrality constraint explicitly in the numerical model.

To complete the model (4) we should now impose the relevant boundary conditions. However, now an essential difficulty appears concerning the potential φ. On one hand from physical point of view it is clear that values of φ on the boundary depend on other variables of the problem. However, in order to get physically reasonable boundary conditions one should model the quite complicated interactions of variables in a thin boundary layer. On the other hand in the intended application the precise behavior of the solution in the boundary layer is not very important and hence one would like to use some simple boundary conditions and model the interactions of potential with other variables in another way. Hence we drop the equation ∆φ = 0 in (4) and proceed with

ρu t -µ∆u + ∇p = 0 -∆p = 0 ∇ • u = 0 c i t -d i ∆c i -ν i c i ∆φ -ν i ∇c i , ∇φ + ∇c i , u = 0 , i = 1, . . . , m - i z i c i = 0 (6)
4 Completion of the initial model.

It is convenient to introduce the following quantities:

β = i z i ν i c i d i η = i z i c i d i ψ = i z i ν i c i (7)
These are all different kind of weighted averages of concentration fields and they appear naturally in the analysis of the system (6).

Stationary case

Let us consider first the stationary case of ( 6):

-µ∆u + ∇p = 0 -∆p = 0 -∇ • u = 0 -d i ∆c i -ν i ∇ • c i ∇φ + ∇c i , u = 0 , i = 1, . . . , m - i z i c i = 0 (8)
There seems to be too few equations: there is apparently no "natural" 2nd order equation for φ. However, we can get an independent equation for φ using the "constraint" i z i c i = 0. Differentiating this we obviously have

i z i ∆c i = 0
Multiplying the equations for c i in (8) by z i /d i and adding them to the above equation we obtain

- i z i ν i d i ∇ • c i ∇φ + i z i d i ∇c i , u = 0
Our current system can thus be written as

-µ∆u + ∇p = 0 -∆p = 0 -∇ • u = 0 -d i ∆c i -ν i c i ∆φ + ∇c i , u -ν i ∇φ = 0 , i = 1, . . . , m -∇ • β∇φ + ∇η, u = 0 - i z i c i = 0 (9)
Because the system is quasilinear, the principal symbol of its linearized version does not depend on the nonlinear terms of the original system. In fact the symbol is

σL = |ξ| 2    L 1 0 0 L 2   
where

L 1 =               µ 0 0 0 0 µ 0 0 0 0 µ 0 0 0 0 1 0 0 0 0               and L 2 =                  d 1 0 . . . 0 ν 1 c1 0 d 2 . . . 0 ν 2 c2 . . . . . . . . . . . . . . . 0 0 . . . d m ν m cm 0 0 . . . 0 β 0 0 0 0 0                 
Hence the linearised system is elliptic, if β = 0 because σL is injective for all ξ = 0 in this case. Here ci and β indicate the reference solution around which the system is linearized.

Time dependent case

Let us consider the system (6). To obtain an equation for φ we now multiply the equations for c i by z i , then operate with ∂ t -a∆ to the last equation and add. This gives

ρu t -µ∆u + ∇p = 0 -∆p = 0 -∇ • u = 0 c i t -d i ∆c i -ν i ∇ • c i ∇φ + ∇c i , u = 0 , i = 1, . . . , m -∇ • ψ∇φ - i z i (d i -a)∆c i = 0 - i z i c i = 0 (10)
Now arguing as above in the stationary case we expect that the system is wellposed if ψ = 0, because the (linearised version of the) next to last equation is elliptic in this case.

Note that the system as a whole is not parabolic according to standard definitions [START_REF] Eidelman | Parabolic equations[END_REF]. Instead we could say that it is of elliptic-parabolic type in the following sense:

-putting p = φ = 0 the linearised system is parabolic for (u, c) -putting u = 0, c = 0 the linearised system is elliptic for (p, φ)

Hence we could call (p, φ) elliptic variables and (u, c) parabolic variables. It seems that there is no "general" theory for these kind of systems, although in addition to Stokes system there are apparently quite many models of this type, for example the chemotaxis model [START_REF] Perthame | Transport equations in biology[END_REF].

5 From completed to augmented systems.

Stationary case

Our next task is to construct the augmented system corresponding to [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF], and in order to do that we need to find the compatibility operator. Let us denote y = (u 1 , u 2 , u 3 , p, c 1 , . . . , c m , φ) and write the system (9) as A 0 y = 0. Now the Stokes system is a subsystem of ( 9), hence one compatibility condition comes from that. The second one is simply given by our construction of the equation for φ. So if we define

A 1 =    ∇• 1 -µ∆ 0 . . . 0 0 0 0 0 0 z 1 /d 1 . . . z m /d m -1 -∆   
then it is easy to check that A 1 A 0 = 0. Now introducing ỹ = (ỹ 1 , ỹ2 ) the augmented system A 0 y + A T 1 ỹ = 0 can be written as

-µ∆u + ∇p -∇ỹ 1 = 0 -∆p + ỹ1 = 0 -∇ • u -µ∆ỹ 1 = 0 -d i ∆c i -ν i ∇ • c i ∇φ + ∇c i , u + z i d i ỹ2 = 0 , i = 1, . . . , m -∇ • β∇φ + ∇η, u -ỹ2 = 0 - i z i c i -∆ỹ 2 = 0 (11)
Again linearized version of this is elliptic, if β = 0. Hence the system should be well-posed under the same hypothesis as the system [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF]. Note also that the system ( 11) is nicely decoupled in two blocks so that the auxiliary variables do not interact. Moreover the structure of the block for c i and φ is such that the principal part also has a block structure so that all variables can be solved cyclically.

Time dependent case

Denoting the system (10) by K 0 y = 0 we get the following compatibility operator

K 1 =    ∇• 1 ρ∂ t -µ∆ 0 . . . 0 0 0 0 0 0 z 1 . . . z m -1 ∂ t -a∆   
Hence the augmented system K 0 y + K T 1 ỹ = 0 can be written as

ρu t -µ∆u + ∇p -∇ỹ 1 = 0 -∆p + ỹ1 = 0 ρỹ 1 t -∇ • u -µ∆ỹ 1 = 0 c i t -d i ∆c i -ν i ∇ • c i ∇φ + ∇c i , u + z i ỹ2 = 0 , i = 1, . . . , m -∇ • ψ∇φ - i z i (d i -a)∆c i -ỹ2 = 0 ỹ2 t -a∆ỹ 2 - i z i c i = 0 (12)
Note that this is also of elliptic-parabolic type and the auxiliary unknowns ỹ are parabolic. Again we expect that the system is well-posed if ψ = 0. The parameter a can be freely chosen, but looking at the last equation we see that it should be positive. Note that dropping the time dependence in [START_REF] Perthame | Transport equations in biology[END_REF] gives an alternative formulation of the stationary problem which is a bit different from [START_REF] Oddy | Multiple-species model for electrokinetic instability[END_REF].

The auxiliary variables ỹ1 and ỹ2 satisfy K 1 K T 1 ỹ = 0 and writing this out gives

ρ 2 ỹ1 tt -2ρµ∆ỹ 1 t + µ 2 ∆ 2 ỹ1 -∆ỹ 1 + ỹ1 = 0 ỹ2 tt -2a∆ỹ 2 t + a 2 ∆ 2 ỹ2 + 1 + |z| 2 ỹ2 = 0
These are 2-parabolic equations [START_REF] Eidelman | Parabolic equations[END_REF] (the latter for a > 0).

Implementation and examples

Physical setting

The derivation of completed and augmented systems presented above is valid in any dimension. Physically the problem is obviously strictly speaking 3 dimensional. In a typical application one could consider a horizontal channel of 100µm wide, 100µm deep and several centimeters long. However, the boundary layers along channel walls are only a few nanometers thick which implies that a 2 dimensional model is relevant and even a 1 dimensional model can be used [START_REF] Brahadwaj | Design and optimization of on-chip capillary electrophoresis[END_REF]. Indeed when one is mainly interested in stacking (see below) already 1 dimensional model gives sufficiently accurate results. However, if one wants to simulate how the propagating front of some concentration is distorted for example in curved channels then one must use a 2 dimensional model.

We did some computations both with 2 and 1 dimensional models. The channel width was chosen as 100µm and the length as 1mm. The computational cost of 2 dimensional case was about 100 times bigger than in 1 dimensional case. Such a dramatic difference is due to the fact that the resolution of the flow of the ambient fluid becomes trivial in 1 dimensional case. Hence we did only few test runs in 2 dimensional case and most of the computations were done with 1 dimensional model.

Because we have already analysed the augmented Stokes system in our previous papers [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF][START_REF] Mohammadi | Involutive methods for Navier-Stokes equations[END_REF] we will only discuss below how our formulation improves the quality of the solutions of concentrations of species c i . This is also justified by the fact that in this model the main interest and the main computational difficulty is related to concentrations while the motion of the ambient fluid is relatively unimportant and in any case considerably easier to compute.

Stacking

The aim of stacking is to amplify the local concentrations of species to make them observable by existing devices. We consider the case of 3 ionic species. The third species is the sample species of interest whose molar concentration is low and one would like to increase it through stacking. Typically the sample species is three or four orders of magnitude smaller than other species and consequently one needs to stack the sample species by the factor of about 1000 to make it observable. Representative experimental conditions are described in [START_REF] Brahadwaj | Design and optimization of on-chip capillary electrophoresis[END_REF][START_REF] Lin | Instability of electrokinetic microchannel flows with conductivity gradients[END_REF][START_REF] Oddy | Multiple-species model for electrokinetic instability[END_REF].

To model a single interface of conductivities, we choose a simple error function profile for the initial concentration. We consider the following initial conditions for the species that reproduces well the experimental conditions:

c 1 init (x) = 1 2 b 1 γ + 1 -(γ -1)erf(αx) c 3 init (x) = b 3 1 + erf(αx) z 2 c 2 init (x) = -z 1 c 1 init (x) -z 3 c 3 init (x) (13)
where b 1 = 100, b 3 = 0.1, α > 0 and γ > 0. Note that α controls the sharpness in the initial plug and γ indicates the initial concentration ratio between high and low conductivity regions in the sense that lim

x→-∞ c 1 init (x)/b 1 = γ, lim x→∞ c 1 init (x)/b 1 = 1
In fact a more important property of γ, both experimentally and numerically, is that it also indicates the final stacking ratio of c 3 . This can be seen as follows.

Let us consider the 1 dimensional model. Neglecting diffusion and the motion of the ambient fluid, the species conservation equation for different species becomes c i t + ∂ ∂x z i ν i c i φ x = 0 A constant current density j is applied in the axial direction defined by the difference of electric potential at the ends of the channel:

j = V left -V right L σ total
where L is the length of the channel and

σ total = L 0 σ(x, t)dx, σ(x, t) = i z 2 i ν i c i
Let us further suppose that σ total is independent of time. The electric field is then locally defined as:

E = φ x = j σ
Now in our setting of 3 species the concentration of sample species c 3 increases as it migrates from a region of high conductivity to a region of lower drift velocity. Hence the stacked sample keeps increasing with time and the concentration progressively approaches a maximum steady value. Now in the steady state the net flux of the species c 3 at the left and right edges balance:

(z 3 ν 3 c 3 E)| left = (z 3 ν 3 c 3 E)| right
Then considering the initial conditions [START_REF] Pironneau | Finite element methods for fluids[END_REF] we have

σ left = 3 i=1 z 2 i ν i c i left ≈ z 2 1 ν 1 c 1 left + z 2 2 ν 2 c 2 left ≈ z 2 1 ν 1 -z 1 z 2 ν 2 )b 1 γ
As j is constant this implies

c 3 left c 3 right = E right E left = σ left σ right ≈ γ
When diffusion and motion of the ambient fluid are present the above arguments can still be used to provide an upper bound for the stacking capacity. Indeed, both diffusion and the convective motion tend to distort the front where stacking happens and hence one expects a lower final stacking ratio in this case.

Boundary conditions

The boundary condition for the Stokes system is classical except along solid walls where, to avoid treating the nanometric double-layer region, one assumes the following slip boundary condition [START_REF] Hak | Handbook series for Mechanical Engineering[END_REF][START_REF] Probstein | Physicochemical Hydrodynamics[END_REF]:

u = -ζ ∇φ
Here ζ is a positive constant which depends on the material with which channels are built and also the fluid permittivity and dynamic viscosity. The flow is therefore parallel to the wall and to the electric field. For the electric potential we impose a Dirichlet boundary condition at the inlet and the outlet where a difference of potential is applied and homogeneous Neumann boundary condition is assumed along the wall. Together with the boundary condition for u along the wall, the former condition enforces the non-penetration boundary condition for the velocity.

The boundary condition for the concentration is Dirichlet at the inlet boundary, homogeneous Neumann along the walls and at the outlet. Initial and boundary conditions for the auxiliary variables ỹ are identically zero as these are compatible with the requirement that the solution of the initial and completed systems should match.

Test cases

We have used the following parameters in our computations, given in SI units.

Recall that mobility in our case is the usual mobility multiplied by Faraday's constant.

mobility

ν 1 = 5 • 10 -8 ν 2 = 3 • 10 -7 ν 3 = 3 • 10 -8 diffusivity d 1 = 2 • 10 -10 d 2 = 3 • 10 -10 d 3 = 2 • 10 -10 valence number z 1 = +1 z 2 = -1 z 3 = -2
We chose the value α = 4•10 4 in initial conditions ( 13) and a = (d 1 +d 2 +d 3 )/3 in [START_REF] Perthame | Transport equations in biology[END_REF]. We tried also some other values for a, but the results were essentially the same as long as a was of the same order of magnitude as diffusivity. This requirement is quite natural, considering that a is the diffusivity for the auxiliary variable ỹ2 .

As we showed above choosing the value of γ in [START_REF] Pironneau | Finite element methods for fluids[END_REF] approximately determines the ultimate stacking ratio. Otherwise choosing different values gave qualitatively similar results. The main difference is that reaching a higher stacking ratio takes longer time and hence a longer channel is needed. For this reason we will show below results for just one value of γ. We chose a moderate value which allowed a relatively short channel and hence avoided excessive computational cost. In fact all computations were done with a standard laptop.

We will show that our formulation [START_REF] Perthame | Transport equations in biology[END_REF] gives better results than the standard approach which we may identify as the model [START_REF] Krupchyk | Overdetermined elliptic systems[END_REF]. The effort to implement our method was relatively small because the augmented systems are so designed that the usual building blocks of numerical codes can quite easily be adapted to our context. Indeed our method was implemented by modifying the code which was used to obtain the results described in [START_REF] Brahadwaj | Design and optimization of on-chip capillary electrophoresis[END_REF].

Let us make some general remarks of the code which apply as well to the initial code as to our improved code. The system has a block diagonal structure which makes it possible to solve different variables cyclically. We use an explicit time marching fixed point algorithm which produces fully implicit solution of the system [START_REF] Saiac | Pratique de la simulation numérique[END_REF]. The advection-diffusion equations of species are solved using a mixed finite volume Galerkin implementation providing suitable conservation properties [START_REF] Pironneau | Finite element methods for fluids[END_REF]. Finally the equation for φ is solved using a classical central difference scheme.

Let us then discuss the results obtained. Figure 1 shows instantaneous isocontours of the electric field, velocity field, pressure, total charge density, as well as charge density of sample species for γ = 50. One can see that the propagating front for sample species where the stacking occurs is a little curved. Also it is clear that the errors in the total charge density are concentrated in this same region. To analyse the electroneutrality constraint and stacking in more detail we could now take slices of the solution in the middle of the channel, and see how they evolve in time. However, since the curving of the front does not play critical role here we may as well use 1 dimensional model to do this. In our test case doing 2 dimensional computations and then taking 1 dimensional slices produced essentially similar results as direct 1 dimensional simulations.

Figure 2 shows the time histories for c 3 , ỹ2 and ρ e when solving system ( 6) and ( 12) for γ = 50. To make stacking (and also the evolution of other quantities) easier to visualise we have plotted the solutions in different time instants so that the convective movement of the ambient fluid is eliminated. One can clearly see that our method preserves better the electroneutrality constraint.

In addition, and perhaps even more importantly, our method respects the upper bound for stacking ratio given by γ while in the standard method there is quite significant overshoot.

We may interprete the results as follows. The solution is most sensitive to errors in the region where rapid changes occur, i.e. in the front where the sample species is stacked. Now the biggest errors are rather naturally in the same region, and because our method reduces significanly the electroneutrality error in this region, the concentrations are also much better resolved.

Finally recall that the linearised version of ( 12) is well-posed only if ψ = 0.

Figure 2 shows that this is indeed the case in our examples. Note that there are natural physically relevant situations where this hypothesis is not satisfied. For example this happens when the background species have valences of opposite sign and whose mobilities are approximately equal. We hope to consider this case in a future work.

Concluding remarks

We have shown above how our methodology, which was applied in [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF][START_REF] Mohammadi | Involutive methods for Navier-Stokes equations[END_REF] to Stokes problem, can also be used to improve existing solvers of microfluid systems. The results show that our approach give clearly better results than the standard approach. It is also remarkable that implementing our method did not require a very big effort. Indeed starting from an existing code it was relatively straightforward to modify it for the present purposes. Moreover the computational cost of solving the augmented system was about the same as the initial system.

In the present paper we did not discuss using the information contained in auxiliary variables so it seems the computation of these quantities did not serve for any useful purpose. However, since the auxiliary variables should be identically zero in exact solution these could be used as error indicators in adaptive numerical schemes. In [START_REF] Mohammadi | Simplifying numerical solution of constrained PDE systems through involutive completion[END_REF] we indicated how this could be implemented, but we have not explored systematically this aspect.

There are also other physical models with algebraic constraints like the electroneutrality assumption. As an example we may cite the combustion problems where the conservation of mass leads to an algebraic constraint for mass fractions [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF]. However, it is important to point out that our approach is not restricted to microfluidic systems or flow problems in general, but can be useful in the analysis of rather arbitrary overdetermined systems.

Fig. 1 .

 1 Fig. 1. From top-left to bottom-right: instantaneous iso-contours for φ x , φ y , u 1 , u 2 , p and ρ e . Bottom figure shows iso-contours of c 3 .

Fig. 2 .

 2 Fig. 2. Evolution of c 3 (top-left) with the initial (6) and augmented (12) systems along the centerline of the channel for an applied electric field of 100kV /m and γ = 50. Top-right picture shows the evolution of ỹ2 . Middle left (resp. right): ρ e with (6) (resp. with (12)). Bottom picture shows that ψ remains nonzero. [2] G. Bruin, Recent developments in electrokinetically driven analysis of microfabricated devices, Electrophoresis 21 (3) (2000) 3931-3951. [3] P. Dudnikov, S. Samborski, Linear overdetermined systems of partial differential equations. Initial and initial-boundary value problems, in: M. Shubin (ed.), Partial Differential Equations VIII, Encyclopaedia of Mathematical Sciences 65,

Acknowledgements The authors would like to thank Professor Juan Santiago from Stanford Microfluidic Laboratory for his interest and comments on this manuscript.