Serge Marlet

Pierre Ruelle

Akissa Bahri

Water reuse in Tunisia: stakes and prospects

Water reuse in Tunisia: Stakes and prospects. In the arid and semi-arid region, countries like Tunisia are facing increasingly serious water shortage problems. According to forecasts, increased domestic and industrial water consumption by the year 2020 may cause a decrease in the volume of fresh water available for Tunisian agriculture. It is therefore important to develop additional water resources as well as protect the existing ones. One way to cope with these problems is to reuse wastewater in agriculture. Therefore and before launching the water reuse policy, a research program was undertaken at the beginning of the 1980s. The objectives of the work were to characterize wastewater chemical and biological composition, to establish impacts of wastewater application on the water-soil-plant system, and to evaluate the long-term changes in soil properties connected to irrigation practices. The results showed the feasibility of water reuse provided that some precautions are taken. A national reuse policy has thus been elaborated and implemented, and water reuse was made an essential component of the Tunisian national water resources strategy. Reuse is up to now mainly practiced for crop irrigation and irrigation of recreational facilities, such as golf courses. Other reuse opportunities such as groundwater recharge are screened. In this paper, the water reuse achievements are presented. This is done by reviewing the last decades' research results and the overall reuse framework. Stakes and prospects are also presented.

Résumé

Réutilisation des eaux usées en Tunisie : enjeux et perspectives. Dans les régions arides et semi-arides, des pays tels que la Tunisie font face à des problèmes croissants de déficit en eau. Selon les prévisions, une augmentation de la consommation de l'eau à des fins domestiques et industrielles vers l'an 2020 risque d'entraîner une diminution du volume d'eau disponible pour le secteur agricole. Il est par conséquent important de développer de nouvelles ressources en eau ainsi que de protéger celles qui existent. La réutilisation agricole des eaux usées est un des moyens de faire face à ces problèmes. A cet effet et avant de passer au stade d'une exploitation à grande échelle, un programme de recherche a démarré depuis 1980 sur ce sujet. L'objectif de ce travail était de déterminer les conditions d'utilisation agricole des eaux usées traitées tenant compte de leur composition physico-chimique et biologique, des impacts de l'application d'eaux usées sur le système eau-sol-plante, et d'évaluer les effets à long-terme de leur application sur les propriétés du sol tenant compte des pratiques d'irrigation. Les résultats obtenus ont montré la faisabilité de la réutilisation des eaux usées moyennant des précautions d'emploi. Une politique nationale de réutilisation des eaux usées a ainsi été élaborée et mise en place. La réutilisation des eaux usées est devenue une composante essentielle de la stratégie nationale des ressources en eau. Les eaux usées sont essentiellement utilisées à des fins agricoles et pour l'irrigation d'espaces récréationnels tels que les terrains de golf. D'autres opportunités de réutilisation des eaux usées telles que la recharge de nappes sont à l'étude. Les principaux résultats de recherche obtenus durant les dernières décades, le cadre général de la réutilisation des eaux usées ainsi que les enjeux et les perspectives sont présentés dans ce papier.

Introduction

In the arid and semi-arid region, countries such as Tunisia are facing increasingly more serious water shortage problems. Problems of water scarcity will intensify because of population growth, rise in living standards, and accelerated urbanization which threaten the water supply in general and agriculture in particular and lead to both an increase in water consumption and pollution of water resources. Continuing increase in demand by the urban sector has led to increased utilization of fresh water for domestic purposes, on the one hand, and production of greater volumes of wastewater, on the other. Agriculture in competition with other sectors will face increasing problems of water quantity and quality considering increasingly limited conventional water resources and growing future requirements and a decrease in the volume of fresh water available for agriculture. Around the cities of the region, competition with other sectors often makes water the main factor that limits agricultural development. Policy makers have then been compelled to develop additional water resources as well as to preserve the existing ones. Reclaiming water is among various measures designed to encourage integrated and efficient management and water use was therefore made an important component of the national water resources strategy. To strengthen the development and utilization of reclaimed water, research work was carried out. The main research results related to the agricultural use of reclaimed water are presented in this paper as well as the overall water reuse framework.

Water resources and wastewater characteristics General

Tunisia extends from the Mediterranean coast in the North to the Sahara desert in the South and its total surface area is 164 150 km 2 . The Tunisian coastline extends over 1300 km. Average annual rainfall is around 594 mm in the North, 289 mm in the Centre, and 156 mm in the South. The annual precipitation in Tunisia is on average equal to 37 billion m 3 . The annual evaporation varies between 1300 mm in the north to about 2500 mm in the south. The water resources are about 4.7 billion m 3 of which 2.7 billion m 3 are from surface water and 2 billion m 3 from groundwater. The volume of water brought on-line during the terms of the Eighth Economic, Social, and Development Plan (1997) was about 3100 million m 3 (Mm 3 ), i.e., 66% of the potential water resources, from large dams, hillside-dams, open shallow wells, deep tubewells, and springs. The country's five-year development plans emphasize water reuse and water harvesting. Reclaimed water is now a part of Tunisia's overall water resources balance (Table I) (Bahri, 1998b). It is actually considered as an additional water resource and as a potential source of fertilizing elements (UNDP, 1987a). Water reuse has been made an integral part of overall environmental pollution control and water management strategy. It is also considered as a complementary treatment stage and consequently, as a way of protecting coastal areas, water resources, and sensitive receiving bodies.

Wastewater treatment and characteristics

Most residents of large urban centres have access to adequate sanitation systems and the wastewater treatment facilities generally follow conventional designs. The sanitation coverage in the sewered cities is about 78%; this rate, related to the whole urban population (5.8 million), is 61%. Concerning industry, compliance with the Tunisian standards (INNORPI, 1989a) to discharge wastewater into the sewerage system is required. So, preliminary treatment plants to fulfil the discharge requirements stated in the regulations must be implemented. Subsidies are given to equip industrial units with pre-treatment processes.

Of the 240 Mm 3 of wastewater discharged annually, 140 Mm 3 (58%) are treated in 61 treatment plants (WWTP) of which around 41 have a daily capacity less than 3500 m 3 and 10 above 10 000 m 3 , Choutrana being the largest with 120 000 m 3 /d. Five treatment plants are located in the Tunis area, producing about 62 Mm 3 /yr or 54% of the country's treated effluent.

Several of the plants are located along the shoreline to protect coastal resorts and prevent sea pollution. Municipal wastewater is mainly domestic (about 88%) and processed biologically up to a secondary treatment stage. The treatment processes vary from plant to plant depending on wastewater origin and on local conditions. Out of 61 treatment plants, 44 are based on activated sludge (medium or low rate), 3 on trickling filters, 14 on facultative or aerated ponds. Sanitation master plans have been designed for several towns. The annual volume of reclaimed water is expected to reach 290 Mm 3 in the year 2020. The expected amount of reclaimed water will then be approximately equal to 18% of the available groundwater resources and could be used to replace groundwater currently being used for irrigation in areas where excessive groundwater mining is causing salt-water intrusion in coastal aquifers.

There are, however, some constraints related to reclaimed water quantity and quality:

• Reclaimed water is consistent with the Tunisian standards regarding water quality required for agricultural reuse (INNORPI, 1989b). However, salinity and the microorganisms content are the two major constraints related to secondary effluent quality. Reclaimed water is often salt-affected (Table II) due to sea-or groundwater seepage into the sewerage network, to the plant location (near a salty lake), and to industrial activities. This salt load limits the range of crops to irrigate and the benefits related to water reuse, and may affect the soil chemical and chemical properties. • Tunis concentrates 54% of the country's effluent, which means that large storage infrastructures are needed. On the other hand, peri-urban irrigated areas are mainly devoted to the production of eaten raw vegetables, which is a major constraint to reuse development in the case of crop restrictions. Availability of agricultural land is another limiting factor, especially along seashores. Transfer of water for reuse may then be required even though expensive.

Water reuse implementing approaches

A gradual approach to expanding reuse since the mid 1960s has been adopted (UNDP et al., 1992). The strategy has consisted of 1) extending wastewater treatment to all urban areas; 2) conducting pilot-and demonstration-scale irrigation operations on agricultural and green areas; 3) establishing large scale irrigation schemes; and 4) implementing a policy calling for an increase in the percentage of treated effluent that is to be reused. Different development phases can then be distinguished.

First phase : Reclaimed water to safeguard citrus production. Some of the wastewater from Tunis has been used since the early 1960s to irrigate 600 ha of citrus fruit orchards located at La Soukra (8 km North East of Tunis). The reason for using the wastewater was to reduce the impact of salt water intrusion due to excessive pumping of groundwater. The reuse has enabled citrus fruit orchards to be saved. Effluents were thus used, mainly during spring and summer, either exclusively or as a complement to groundwater. Irrigation of vegetables was not allowed.

Second phase : Planned reclaimed water reuse. The water reuse policy was launched at the beginning of the 1980s. The main applications of water reuse are agricultural irrigation, and landscape irrigation. Some pilot projects have been launched or are under study for groundwater recharge, irrigation of forests and highways, and wetlands development. Water reuse was implemented after the construction of existing treatment plants. However, for new plants, treatment and reuse are combined already at the planning stage. About 35 Mm 3 of reclaimed water is annually allocated for reuse. A total flow of about 28 Mm 3 /yr. of treated effluent (approximately 20% of the treated effluent) is being reused. In some areas, irrigation with effluents is well established and most of the volume allocated is being used, while in new areas where irrigation is just beginning, the reclaimed water usage rate is slowly increasing. The area currently equipped is about 6500 ha, 80% of which are located around Tunis. Other projects are being implemented extending the area to 9000 ha. The use of secondary treated effluents in Tunisia is for a restrictive irrigation from which all vegetable crops, whether eaten raw or cooked, are forbidden. The main crops irrigated with treated wastewater are fruit trees (citrus, grapes, olives, peaches, pears, apples, grenades, etc.) (28.5%), fodder (alfalfa, sorghum, berseem, etc.) (45.3%), industrial crops (sugarbeet) (3.8%), and cereals (22.4%). 57% of the equipped area are sprinkler irrigated and 48% surface irrigated. Some farmers use localised irrigation systems. Cattle (milking cows, calves, sheep, and goats), not grazed on pastures irrigated with treated wastewater, is also fed with forage crops cultivated on the irrigated areas.

A water reuse program has been established and experimental research has been conducted on the subject. The institutional and legal framework of water reuse in agriculture has been set up to regulate the treatment and the distribution of irrigation with reclaimed water, to supervise the Water Code and other enactments application and to control the sanitary aspects. Water reuse in agriculture is regulated by the 1975 Water Law and by the 1989 decree (Decree No. 89-1047). In separate documents, reclaimed water quality standards for reuse (INNORPI, NT 106.03, 1989b), wastewater disposal standards in receiving waters (INNORPI, NT 106.002, 1989a), a list of crops that can be irrigated as well as the specifications determining the conditions of reclaimed water reuse have also been set up. The reclaimed water quality criteria for agricultural reuse were developed using the FAO guidelines [START_REF] Ayers | Water Quality for Agriculture[END_REF], the WHO guideline (1989) for restricted irrigation (< 1 helminth egg per litre), and other Tunisian standards related to irrigation or water supply.

Responsibility from wastewater collection to use in agriculture is shared among various ministries: the Ministry of Agriculture (the National Sewerage and Sanitation Agency, the General Directorate for Agricultural Engineering, the Regional Commissariats for Agricultural Development), the Ministry of Public Health, the Ministry of Tourism and Handicrafts. Users' associations are also involved in water reuse operations.

As municipal wastewater discharges at a more or less constant rate throughout the year and as its volume will increase with urban, tourism, and industry development, wastewater will be reused for agricultural purposes -the area irrigated with treated wastewater is planned to expand up to 20-30 000 ha, i.e. 7-10% of the overall irrigated area, with 14 500 ha located around the Great Tunis-and in other activity sectors. A new wastewater treatment plant is planned for the city of Tunis The Tunis-West project with a design capacity in the year 2016 of 105 000 m 3 /d (41 Mm 3 /yr.). The treatment plant will be operated as a BOT. Interseasonal storage (9 Mm 3 in 2001, 15 Mm 3 in 2016) in hillside dams is included for water resources protection purposes, and increase of the resource. In a first phase, an irrigation scheme covering 1 000 ha is planned. The total irrigated area should cover about 6000 ha. Farmers' willingness to reuse reclaimed water is taken into account. Water users' associations will manage the irrigation system. The water distribution system at the plot level will be optimised and irrigation saving methods will be encouraged.

Third phase : Development of reclaimed water reuse. Actually, there are several other water reuse opportunities when the water quality is in adequacy with the intended end use of the effluent [START_REF] Asano | Wastewater reclamation, recycling and reuse: past, present, and future[END_REF]. Diversification of the reuse options by developing non-agricultural uses such as municipal, industrial, and environmental uses is on the way.

Based on the on-going reuse projects, a study aimed at developing a strategy to promote water reuse was launched in 1997 (Bechtel and Scet, 1998). The study showed that the strategy should be oriented towards the substitution of conventional water by reclaimed water for the high-rated water activities or the creation of a new demand based on strategic projects. Promotion of reclaimed water reuse should be based on (1) a real water demand, (2) the definition of appropriate water quality standards for the different uses, (3) a relevant regulation, (4) clarified and identified responsibilities for the different interested parties, and

(5) an efficient control on all the uses. The legal and institutional framework should be strengthened. Reclaimed water reuse should be more integrated to water resources management. By upgrading the water quality and with more widespread information, reclaimed water reuse should gain wider acceptance in the future. Projects aimed at developing water reuse have been proposed such as the implementation of the water reuse strategies for the Great Tunis and other major cities, groundwater recharge of some coastal aquifers, industrial reuse of reclaimed water, etc. [START_REF] Serah | Etude de la stratégie nationale de valorisation des eaux usées traitées -Rapport de synthèse[END_REF].

Research regarding use of reclaimed water

A research program co-financed by UNDP was undertaken from 1981 to 1987 by the Rural Engineering Research Center (INRGREF actually). The aim was to determine the conditions to use reclaimed water and sewage sludge in agriculture taking into account their composition, soil types, different crops, and sanitary aspects (UNDP, 1987a). The methodology that was adopted is mainly studies of real field conditions. Research is still continuing. A brief review of the results to date is presented here.

Physical-chemical and biological quality of wastewater

Chemical elements, bacteria, and parasitic content of raw and reclaimed water discharged from different wastewater treatment plants and reused in agriculture have been monitored. It was found that the chemical composition of effluents depended on the treatment process employed, the plant location, the proportion of industrial water compared to domestic, the seepage of brackish/sea water into the sewerage network, and finally on the quality of the water supply. The composition was characterized by a moderate salinity and sodicity for most of the treatment plants (Table III). This means, consequently, soil salinization risks. Alkalinization risks may not be important because of the high concentration of Ca and the elevated electrical conductivity of the effluents. The fertilizing elements concentration was 30.0 mg N/L, 3.6 mg P/L, and 52.0 mg K/L, on average. Reclaimed water had a high variability of the organic parameters which, for N and P, may be a constraint for fertilization purposes, and a low trace elements content (Table IV), far below the expected toxicity thresholds (Bahri, 1998a). The concentration of almost all regulated elements in raw and reclaimed water were below the maximum concentration recommended for agricultural reuse by the Tunisian standards (INNORPI, 1989b) and had a high fertilizing content which may widely exceed the needs of plant growth for nitrogen and potassium. Over-application of nitrogen exceeding requirements for crop growth may present some risks for crops and/or groundwaters. Average faecal coliforms and streptococci removal (log units) was about 0-2 in biological wastewater treatment plants such as activated sludge, around 1-2 in aerated lagoons, and from 2 to 5 in stabilization ponds (Trad-Raïs, 1992, 1995). The different treatment processes did not result in a complete removal of pathogenic bacteria such as salmonella except effluents from stabilization ponds which were free of these pathogens. Consequently, stabilization ponds, compared to intensive biological treatment systems, may produce a higher bacterial effluent quality [START_REF] Trad-Raïs | Surveillance bactériologique et parasitologique des eaux usées brutes et traitées de la ville de Tunis[END_REF]) consistent with WHO guidelines (WHO, 1989). Reclaimed water had a certain parasitic load, which depends on the treatment process. Stabilization pond effluents were free of parasites.

Soil physical properties, chemical composition, and bacterial contamination

Application of reclaimed water on alluvial and sandy-clayey to sandy soils of experimental stations, has led to little modifications of their physical properties: bulk density of the soil surface layer (0-5 cm) was the same whatever the irrigation water quality (reclaimed water or groundwater) was. Irrigation with reclaimed water did not affect the soil trace element content but, on the contrary, it affected the chemical composition of La Soukra clayey-sandy soils [START_REF] Bahri | Utilization of treated wastewater and sewage sludge in agriculture in Tunisia[END_REF]. An increase in the electrical conductivity and in the sodium adsorption ratio and a change in the soil solution composition (bicarbonate-calcic → sodium-chloride) were noticed. This increase was, in this case, within the usual range found when using conventional water. Concerning the bacterial level of contamination, no significant effects of reclaimed water application were recorded on the soil over two years. Microorganisms die-off in the soil was shown to be sensitive to factors such as the climate (temperature, insolation, precipitation, and relative humidity), soil texture, and plant canopy (UNDP, 1987a).

Plant growth, yield, chemical composition, water consumption, and bacterial contamination

Tests were conducted during four years on two species, a fodder crop, sorghum (Sorghum vulgare) and a vegetable, pepper (Capsicum annuum), and were compared to a control irrigated with well water; the first was flood-irrigated, the second was furrow-irrigated. Irrigation with reclaimed water had a favourable effect on the growth of sorghum and led to a significant increase of N, P, and K contents in plant tissues (Rejeb, 1992) with, however, a lower efficiency in the use of these elements compared to mineral fertilizing [START_REF] Rejeb | Comparaison de l'utilisation de l'azote des eaux usées traitées et de l'ammonitrate par le sorgho fourrager (Sorghum sudanense)[END_REF]. No effect was noticed for pepper. Fertilizing element (N, P, and K) uptake differed with the crop species but remained low compared to the soil residual load entailing nitrate groundwater pollution risks. Concerning trace elements, only Fe and Zn concentrations in sorghum leaves increased significantly without reaching thresholds values; the same was observed for B (Rejeb, 1992). Similar studies were conducted on maize, alfalfa, and barley. The use of reclaimed water compared to irrigation with groundwater resulted in higher annual and perennial crop yields. Recycling of wastewater nutrients can then offer both environmental and economic benefits. In pot experiments, N, P, and K efficiency of reclaimed water was compared to mineral fertilizing (ammonium nitrate). Results showed that global effects of effluents were less important than that of mineral fertilizers; however, compared to a non-fertilized control, wastewater application led to a higher dry matter production. Therefore, irrigation with reclaimed water has to be considered as a complementary fertilization that has to be taken into account when the fertilizer amount to apply is to be evaluated. Water consumption of citrus trees did not depend on the water quality (UNDP, 1987a). Tests conducted during two years showed that citrus fruits produced on plots irrigated with reclaimed water and not in contact with the soil were free of faecal germs. Contamination level of citrus fruits picked up on the soil of plots irrigated with reclaimed water was significantly higher than that of fruits picked from the trees (UNDP, 1987a). As to annual crops, experiments were made first to evaluate the bacterial quality of fodder crops irrigated with effluents and to compare them to non-irrigated fodder crops (winter crops) or to fodder irrigated with groundwater (Trad-Raïs, 1991). Results showed that faecal contamination of forage crops sampled from control plots was not equal to zero. This contamination, due to natural factors, was higher for summer forage crops. Bacterial quality of forage crops irrigated with reclaimed water depended on the crop specie, the number of days since the last irrigation occurred and the climatic conditions. For both sorghum and alfalfa, 7 to 10 days were required between the last irrigation and the cutting to achieve natural decontamination. The natural die-off of faecal microorganisms on sorghum plants was quicker in summer than in autumn. As to pepper, tests did not show any particular fruit contamination.

Chemical and bacterial groundwater composition

Besides reclaimed water reuse for agricultural purposes, seasonal recharge of the shallow and sandy aquifer of Nabeul has been performed since 1985. Activated sludge effluents that were not used for irrigation during winter season were infiltrated and stored into the aquifer, thus increasing the volume farmers can pump during summer season to irrigate citrus orchards. Artificial groundwater recharge is operated at experimental scale in infiltrationpercolation basins and in the riverbed in the Cap Bon area. Groundwater recharge efficiency was proven not only by the increase of the water level in the wells but also by the improvement of the production of the surrounding wells. This experiment allowed an underground storage and an additional treatment step as wastewater slowly infiltrated through the unsaturated zone (7 months after the end of the recharge, nitrogen concentrations were about few milligrams per litre) through the unsaturated zone (UNDP, 1987b). However, no clear conclusion could be drawn about the effect of reclaimed water on the bacterial and chemical composition of shallow groundwater since the initial contamination level of most of the wells was relatively high and subject to seasonal variations. According to the state of the art of soil-aquifer treatment [START_REF] Bouwer | Role of groundwater recharge in treatment and storage of water reuse[END_REF][START_REF] Brissaud | Infiltration percolation as a tertiary treatment[END_REF], improved operation of this facility would lead to a groundwater quality meeting unrestricted irrigation requirement [START_REF] Rekaya | Recharge de la nappe de l'Oued Souhil par des effluents secondaires[END_REF]. Performing coastal aquifer recharge, where the hydrogeological context is favourable, would make water reuse well accepted by farmers. This subject is still under study and other sites are screened for further studies including the comparison of different treatment processes preceding recharge.

Recreational reuse

Since the beginning of the 1970s and with the development of tourism, a policy was set up for golf course irrigation with reclaimed water. Golf course irrigation means a high rate of water reuse and a water demand that lasts all the year long through varying climatic conditions. Actually, the eight existing golf courses are irrigated with secondary-treated effluent. Some are irrigated with reclaimed water blended with conventional water (surface or ground-water). Irrigation water is in compliance with the WHO guidelines (1989) for water reuse on recreational areas with free access to the public (2.3 log. units /100 mL) during winter and part of spring [START_REF] Bahri | Reuse of reclaimed wastewater for golf course irrigation in Tunisia[END_REF]. Polishing secondary effluents through lagooning or seasonal storage would lower health hazards and contribute to increase this demand of reclaimed water [START_REF] Bahri | Agronomic and health aspects of storage ponds located on a golf course irrigated with reclaimed wastewater in Tunisia[END_REF]. Supplying reclaimed water to golf courses, green belts, and hotel gardens would result in an optimization of both investment and operation costs.

Long-term effects

Investigations in La Soukra's perimeter, which has been irrigated for more than twenty years with reclaimed water have been conducted. The results did not show notable effects on soils, crops, or groundwater. A study conducted in this scheme to evaluate health impacts of reclaimed water reuse could not set up a clear cause-effect relationship between the observed diseases and the reuse practice.

Research needs

Additional scientific work is needed to reduce persistent uncertainty about the potential impacts on human health and the environment from exposure to reclaimed water. A number of knowledge gaps are identified in the following:

• Improvement of existing treatment processes and appropriate selection/combination of treatment methods. • Development of cost-effective and innovative wastewater treatment technologies, especially energy-saving and reliable processes (biotechnologies for degradation of refractory organics, etc.). • Disinfection treatment processes.

• Storage systems: planning, operation, improvement of reclaimed water quality.

• Achieving best use of nutrients without adverse impacts such as over-fertilization problems and groundwater pollution. • Fate of microorganisms and contaminants (refractory trace organics, pharmaceutically active chemicals, etc.) in the water-soil-plant system and evaluation of the soil's absorptive capacity to assimilate, and detoxify pollutants in agricultural and groundwater recharge applications. • Improvement of irrigation systems (filtration, distribution (localized, etc.), etc.).

• Long-term effects of reclaimed water reuse on the soil-plant-aquifer system.

• Risk assessment studies on water-soil-plant-animal-human exposure pathways.

Decentralized management of wastewater treatment and reuse for small communities. Assessment of the reclaimed water market and screening other reuse opportunities: groundwater recharge, municipal, industrial uses, etc. Evaluation of the socio-economic feasibility of reclaimed water reuse.

Conclusion

During the last decades, Tunisia has gained experience in the field of irrigation with emphasis on the use of marginal water for agricultural crops. To try on a rational basis to make a better use of the water resources on account of their rather limited quantity, research should continue along some of the lines already carried out with a broadening of it with emphasis on socio-economic aspects. This also requires a strengthening of the human potential of researchers, material means, and integrated multidisciplinary efforts.

Table I . Accessible (A) and available (B) water in Tunisia (Mm 3 /yr) for different time- horizons (Ministry of Agriculture, 1998)

 I

		1996		2010		2020	2030
		A	B	A	B	A	B	A	B
	Large dams	1 340 871 1 800 1 170 1 750 1 138 1 750 1 138
	Hillside-dams and lakes	65	59	100	50	70	35	50	45
	Tubewells and springs	997	997 1250 1 150 1 250 1 000 1 250 1 000
	Open wells	720	720	720	720	720	620	720	550
	Reclaimed water	120	120	200	200	290	290	340	340
	Desalinated water	7	7	10	10	24	24	49	49
	Total	3 249 2 774 4 080 3 300 4 104 3 107 4 159 3 122

Table II . Reclaimed water annual flow and salinity level

 II

	EC (µS/cm)	Annual flow (m 3 /yr)	%
	Slightly saline: 700 -3 000	14 006 664	12
	Medium saline: 3000 -6 000	92 740 287	80
	Highly saline: 6000 -14 000	9 231 807	8
	Total	115 978 758	100

Table III . Average element concentration for effluents (in mg/L)

 III

	Parameter	Mean	CV (%)	Range	NT*
	pH	7.6	1.7	7.5 -7.9	6.5 -8.5
	SS	42.5	112.8	15 -191	30 (a)
	COD	173.6	87.9	61 -640	90 (a)
	BOD 5	35.3	52.1	18 -70	30 (a)
	EC (dS/m)	4.1	41.0	2.4 -8.9	7.0
	TDS (g/L)	2.6	41.4	1.5 -5.6	-
	SAR	8.5	44.9	5.1 -17.6	-
	N	42	36.7	27 -85	-
	P	3.6	45.3	1.6 -6.5	-
	K	52	51.2	18 -120	-

* Tunisian standards

(INNORPI, 1989b)

.

Table IV . Element concentration for wastewater (mg/L) and sewage sludge (mg/kg)

 IV

	Parameter	Influent	Effluent	Sewage sludge
	pH	7.1 -7.9	7.5 -7.9	6.1 -8.4
	EC (a)	3.0 -10.4	2.4 -8.9	2.7 -11.8
	B	0.7 -1.7	0.3 -1.4	12 -111
	Cd	0.004 -0.010	0.004 -0.008	1 -39
	Co	0.012 -0.041	0.012 -0.031	11 -34
	Cr	0.010 -0.109	0.009 -0.023	33 -490
	Cu	0.040 -0.160	0.011 -0.025	44 -431
	Ni	0.027 -0.051	0.021 -0.049	12 -293
	Pb	0.050 -0.347	0.035 -0.066	57 -1 580
	Zn	0.074 -0.482	0.023 -0.063	170 -1 500
	(a): EC is measured as dS/m at 25°C in water or 1:5 extract in sludge.

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TRAD-RAÏS, M. 1991. Contamination bactérienne des fourrages irrigués avec les eaux usées traitées, Ann. INRAT, 64, 12, 16 p. TRAD-RAÏS, M. 1992. Efficacité bactériologique des principaux procédés de traitement des eaux usées urbaines, Archs. Inst. Pasteur Tunis,


Feng Qi

A LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTION INVOLVING THE q-GAMMA FUNCTION

Keywords: 2010 Mathematics Subject Classification. Primary 44A10, Secondary 05A20, 26A48, 26D07, 33B15 logarithmically completely monotonic function, q-gamma function, inequality, q-multinomial coefficient, q-multivariate beta function

In the paper, the author proves the logarithmically complete monotonicity of a function involving the q-gamma function and derives some inequalities for the q-multinomial coefficient and the q-multivariate beta function. These conclusions generalize corresponding ones for the gamma function, the multinomial coefficient, and the multivariate beta function respectively.

then we call f a completely monotonic function; if f : I → (0, ∞) is positive and (-1) n [ln f (x)] (n) ≥ 0, n ∈ N, then we call f a logarithmically completely monotonic function; if f : (0, ∞) → [0, ∞) can be written in the form

f (x) = a x + b + ∞ 0 1 s + x d µ(s),
where a, b are non-negative constants and µ is a measure on (0, ∞) such that ∞ 0 1 1+s d µ(s) < ∞, then we call f a Stieltjes transform. For more details on definitions and properties of these kinds of functions, please see [15, Chapter XIII], [START_REF] Schilling | Bernstein Functions-Theory and Applications[END_REF]Chapter 1], and [START_REF] Widder | The Laplace Transform[END_REF]Chapter IV]. Among these three kinds of functions, there are the following relations:

(1) a function f is completely monotonic on (0, ∞) if and only if it is a Laplace transform, that is, there is a positive measure µ on [0, ∞) such that

f (x) = ∞ 0 e -xt d µ(t);
(2) the set of all logarithmically completely monotonic functions is a strict subset of all completely monotonic functions; (3) the set of all Stieltjes transforms is a strict subset of all logarithmically completely monotonic functions on (0, ∞).

For details on these relations, please refer to [START_REF] Berg | Integral representation of some functions related to the gamma function[END_REF][START_REF] Guo | A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function[END_REF][START_REF] Qi | A complete monotonicity property of the gamma function[END_REF][START_REF] Qi | Some completely monotonic functions involving the gamma and polygamma functions[END_REF][START_REF] Schilling | Bernstein Functions-Theory and Applications[END_REF] and closely related references therein. For information on new developments of these kinds of functions, please refer to [START_REF] Guo | A completely monotonic function involving the tri-gamma function and with degree one[END_REF][START_REF] Guo | On the degree of the weighted geometric mean as a complete Bernstein function[END_REF][START_REF] Qi | Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions[END_REF][START_REF] Qi | Integral representations and properties of some functions involving the logarithmic function[END_REF][START_REF] Qi | Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions[END_REF][START_REF] Qi | Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean[END_REF][START_REF] Schilling | Bernstein Functions-Theory and Applications[END_REF] and closely related references therein.

The classical Euler's gamma function Γ(z) can be defined [START_REF] Qi | Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities[END_REF] by

Γ(z) = ∞ 0 t z-1 e -t d t, (z) > 0 or by Γ(z) = lim n→∞ n!n z n k=0 (z + k) , z ∈ C \ {0, -1, -2, . . . }.
The logarithmic derivative of Γ(z), denoted by ψ(z) = Γ (z) Γ(z) , and the derivatives ψ (i) (x) for i ∈ N are respectively called the digamma (or psi) funciton and polygamma functions. See [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]Chapter 6] and [START_REF]NIST Handbook of Mathematical Functions[END_REF]Chapter 5].

The q-analogue Γ q (x) of the gamma function Γ(x) for q > 0 and x > 0, called the q-gamma function, can be defined [3, pp. 493-496] and [6, Section 1.10] by

Γ q (x) =            (1 -q) 1-x ∞ i=0 1 -q i+1 1 -q i+x , 0 < q < 1; (q -1) 1-x q ( x 2 ) ∞ i=0 1 -q -(i+1)
1 -q -(i+x) , q > 1; Γ(x), q = 1.

(1.1)

They satisfy

lim q→1 + Γ q (z) = lim q→1 -Γ q (z) = Γ(z) and Γ q (x) = q ( x-1 2 ) Γ 1/q (x).
The q-analogue ψ q (x) of the digamma function ψ(x) for q > 0 and x > 0, called the q-digamma function, is defined by

ψ q (x) =    Γ q (x) Γ q (x) ; q = 1, ψ(x), q = 1.
The q-analogues ψ

(k)
q (x) of the polygamma functions ψ (k) (x) for k ∈ N are called the q-polygamma functions. From (1.1), we obtain that, (1) when 0 < q < 1 and x ∈ (0, ∞),

ψ q (x) = -ln(1 -q) + (ln q) ∞ k=0 q k+x 1 -q k+x = -ln(1 -q) + (ln q) ∞ k=1 q kx 1 -q k ,
(2) when q > 1 and x ∈ (0, ∞),

ψ q (x) = -ln(q -1) + (ln q) x - 1 2 - ∞ i=1 q -ix 1 -q -i .
The formula (1.11) in the paper [START_REF] Ismail | Inequalities and monotonicity properties for gamma and q-gamma functions[END_REF] and its corrected version [START_REF] Ismail | Inequalities and monotonicity properties for gamma and q-gamma functions[END_REF] reads that

ψ q (x) = -ln(1 -q) - ∞ 0 e -xt 1 -e -t d γ q (t) (1.2)
for 0 < q < 1 and x > 0, where

γ q (t) =      -ln q ∞ k=1 δ(t + k ln q), 0 < q < 1 t, q = 1 (1.3)
and δ(t) represents the Dirac delta function, that is, d γ q (t) is a discrete measure with positive masses | ln q| at the positive points k| ln q| for k ∈ N. Accordingly, we obtain

∞ 0 e -xt d γ q (t) = - q x ln q 1 -q x and ∞ 0 te -xt d γ q (t) = q x (ln q) 2
(1 -q x ) 2 for 0 < q < 1 and x > 0. Differentiating with respect to x on both sides of (1.2) yields

ψ (k) q (x) = (-1) k+1 ∞ 0 t k e -xt 1 -e -t d γ q (t), 0 < q < 1, k ∈ N. (1.4)
In [10, p. 1245 , it was presented that, when 0 < q < 1, the identity

ψ (k-1) q (x) -ψ (k-1) q (x + 1) = (ln q) d k-1 d x k-1 q x 1 -q x
is valid for x ∈ (0, ∞) and k ∈ N. One can also find these knowledge in [START_REF] Qi | A completely monotonic function related to the q-trigamma function[END_REF][START_REF] Qi | Bounds for the ratio of two gamma functions[END_REF][START_REF] Qi | Certain logarithmically N -alternating monotonic functions involving gamma and qgamma functions[END_REF][START_REF] Qi | Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions[END_REF][START_REF] Qi | Comments on two completely monotonic functions involving the q-trigamma function[END_REF] and closely related references.

Motivations

Let a = (a 1 , a 2 , . . . , a m ) with

a i > 0 for 1 ≤ i ≤ m, let p = (p 1 , p 2 , . . . , p m ) with m i=1 p i = 1 and p i ∈ (0, 1) for 1 ≤ i ≤ m, let m i=1 a i a 1 , a 2 , . . . , a m = Γ 1 + m i=1 a i Γ(1 + a 1 )Γ(1 + a 2 ) • • • Γ(1 + a m )
denote the multinomial coefficient, and let

B(a 1 , a 2 , . . . , a m ) = Γ(a 1 )Γ(a 2 ) • • • Γ(a m ) Γ(a 1 + a 2 + • • • + a m )
be the multivariate beta function. In [START_REF] Ouimet | Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex[END_REF] and [START_REF] Qi | Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions[END_REF], the authors considered the function

Q(x) = Q a,p;m (x) = Γ 1 + x m i=1 a i m i=1 Γ(1 + xa i ) m i=1 p xai i = x m i=1 a i xa 1 , xa 2 , . . . , xa m m i=1 p xai i = m i=1 a i m i=1 a i m i=1 p xai i x m-1 B(xa 1 , xa 2 , . . . , xa m ) , x ∈ (0, ∞), m ∈ N (2.1)
and, among other things, proved the following theorem.

(x) = Q a,p;m (x) defined in (2.1
) is logarithmically completely monotonic on (0, ∞).

For more information on the backgrounds, history, motivations, origins, and applications of the function Q(x) = Q a,p;m (x), please refer to [START_REF] Alzer | Complete monotonicity of a function related to the binomial probability[END_REF][START_REF] Leblanc | On a uniformly integrable family of polynomials defined on the unit interval[END_REF][START_REF] Ouimet | Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex[END_REF][START_REF] Qi | Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions[END_REF] and closely related references therein.

In this paper, we consider the q-analogue of the function Q(x) = Q a,p;m (x) and investigate its logarithmically complete monotonicity.

A logarithmically completely monotonic function

Let a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m and let p = (p 1 , p 2 , . . . , p m ) with

m i=1 p i = 1 and p i ∈ (0, 1) for 1 ≤ i ≤ m. Define Q q (x) = Q q;a,p;m (x) = Γ q 1 + x m i=1 a i m i=1 Γ q (1 + xa i ) m i=1 p xai i (3.1)
for q ∈ (0, 1), x ∈ (0, ∞), and m ∈ N. It is clear that

lim q→1 -Q q (x) = lim q→1 -Q q;a,p;m (x) = Q(x) = Q a,p;m (x).
Theorem 3.1. The function Q q (x) = Q q;a,p;m (x) defined in (3.1) is logarithmically completely monotonic on (0, ∞).

Proof. From (1.3) for 0 < q < 1 and the well-known property δ(at) = 1 |a| δ(t), it follows that

γ q t τ = -ln q ∞ k=1 δ t + kτ ln q τ = -τ ln q ∞ k=1 δ(t + kτ ln q)
for τ > 0. This implies that

d γ q t τ = τ d γ q (t). (3.2) Direct calculation gives ln Q q (x) = ln Γ q 1 + x m i=1 a i - m i=1 ln Γ q (1 + a i x) + x m i=1 a i ln p i , [ln Q q (x)] = m i=1 a i ψ q 1 + x m i=1 a i - m i=1 a i ψ q (1 + a i x) + m i=1 a i ln p i , and
[ln Q q (x)] = m i=1 a i 2 ψ q 1 + x m i=1 a i - m i=1 a 2 i ψ q (1 + a i x).
Making use of (1.4) for k = 1 and the equality (3.2), it follows that

ψ q (τ x + 1) = ∞ 0 t 1 -e -t e -(τ x+1)t d γ q (t) = ∞ 0 t e t -1 e -τ xt d γ q (t) = ∞ 0 h v τ e -vx d γ q v τ = τ ∞ 0 h v τ e -vx d γ q (v),
where τ > 0 and h(t) = t e t -1 . Hence, we have

[ln Q q (x)] = ∞ 0 m i=1 a i 3 h v m i=1 a i - m i=1 a 3 i h v a i e -xv d γ q (v). (3.3)
By calculus, we have

d d t t 3 h 1 t = d d t t 2 e 1/t -1 = e 1/t (2t + 1) -2t (e 1/t -1) 2 , d 2 d t 2 t 3 h 1 t = 2t 2 -e 1/t 4t 2 + 2t -1 + e 2/t 2t 2 + 2t + 1 (e 1/t -1) 3 t 2 → 0, t → 0 + , d 3 d t 3 t 3 h 1 t = e 1
/t 4e 1/t + e 2/t + 1 (e 1/t -1) 4 t 4 > 0 on (0, ∞). As a result, the function t 3 h 1 t is strictly convex on (0, ∞). Recall from [14, p. 650] that

(1) a function φ : [0, ∞) → R is said to be star-shaped if φ(αx) ≤ αφ(x) for all α ∈ [0, 1] and x ≥ 0; (2) a real function φ defined on a set S ⊂ R n is said to be super-additive if

x, y ∈ S implies x + y ∈ S and φ(x + y) ≥ φ(x) + φ(y); (3) if φ is a real function defined on [0, ∞), φ(0) ≤ 0, and φ is convex, then φ is star-shaped, but convexity is not a property of all star-shaped functions; (4) if φ : [0, ∞) → R is star-shaped, then φ is super-additive.

Consequently, since lim t→0 + t 2 e 1/t -1 = 0, the function t 3 h 1 t is star-shaped, and then super-additive, on (0, ∞). As a result, it follows inductively that m i=1

a i v 3 h 1 m i=1 (a i /v) ≥ m i=1 a i v 3 h 1 a i /v
which can be rearranged as

m i=1 a i 3 h v m i=1 a i ≥ m i=1 a 3 i h v a i .
Combining this with (3.3) yields that the second derivative [ln Q q (x)] is completely monotonic on (0, ∞).

Complete monotonicity of [ln Q q (x)] implies that the first derivative [ln Q q (x)] is strictly increasing on (0, ∞). Hence, from (1.2), it follows that

[ln Q q (x)] ≤ lim x→∞ m i=1 a i ψ q 1 + x m i=1 a i - m i=1 a i ψ q (1 + a i x) + m i=1 a i ln p i = m i=1 a i lim x→∞ ψ q 1 + x m i=1 a i - m i=1 a i lim x→∞ ψ q (1 + a i x) + m i=1 a i ln p i = - m i=1 a i ln(1 -q) + m i=1 a i ln(1 -q) + m i=1 a i ln p i = m i=1 a i ln p i < 0.
By definition, the function Q q (x) is logarithmically completely monotonic on (0, ∞). The proof of Theorem 3.1 is complete.

Inequalities for the q-multinomial coefficient

Let a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. Then the q-multinomial coefficient for q ∈ (0, 1) is

m i=1 a i a 1 , a 2 , . . . , a m q = Γ q 1 + m i=1 a i Γ q (1 + a 1 )Γ q (1 + a 2 ) . . . Γ q (1 + a m ) . (4.1)
With the aid of logarithmically complete monotonicity of Q q (x) = Q q;a,p;m (x), we now present some inequalities for the q-multinomial coefficient.

Theorem 4.1. Let q ∈ (0, 1), , m ∈ N, and a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. If x j > 0 for 1 ≤ j ≤ and λ j ∈ (0, 1) with j=1 λ j = 1, then F. QI j=1 λ j x j m i=1 a i a 1 j=1 λ j x j , a 2 j=1 λ j x j , . . . , a m j=1 λ j x j q ≤ j=1 x j m i=1 a i x j a 1 , x j a 2 , . . . , x j a m λj q (4.2) and the equality in (4.2) holds if and only if

x 1 = x 2 = • • • = x .
Proof. The logarithmically complete monotonicity in Theorem 3.1 indicates that the function Q q (x) is logarithmically convex on (0, ∞). Hence, one acquires

Q q j=1 λ j x j ≤ j=1 Q λj q (x j ).
Using the expression

Q q (x) = x m i=1 a i xa 1 , xa 2 , . . . , xa m q m i=1 p xai i leads to j=1 λ j x j m i=1 a i a 1 j=1 λ j x j , a 2 j=1 λ j x j , . . . , a m j=1 λ j x j q m i=1 p ai j=1 λj xj i ≤ j=1 x j m i=1 a i x j a 1 , x j a 2 , . . . , x j a m q m i=1 p aixj i λj
which can be rearranged as (4.2). The proof of Theorem 4.1 is complete. Theorem 4.2. Let q ∈ (0, 1), , m ∈ N, and a = (a 1 , a 2 , . . . , a m ) with a i > 0 for

1 ≤ i ≤ m. If x j > 0 for 1 ≤ j ≤ , then j=1 x j m i=1 a i x j a 1 , x j a 2 , . . . , x j a m q < j=1 x j m i=1 a i a 1 j=1 x j , a 2 j=1 x j , . . . , a m j=1 x j q . (4.3)
Proof. In [2, Lemma 3], it was proved that, if g : [0, ∞) → (0, 1] is differentiable and g g is strictly increasing on (0, ∞), then the inequality g(x)g(y) < g(x + y) is valid for x, y ∈ (0, ∞). As a result, we can inductively deduce

j=1 g(x j ) < g j=1 x j .
Applying this inequality to the logarithmic convexity of the function Q q (x) reveals j=1 x j m i=1 a i x j a 1 , x j a 2 , . . . , x j a m q m i=1 p aixj i < j=1 x j m i=1 a i a 1 j=1 x j , a 2 j=1 x j , . . . , a m j=1 x j q m i=1 p ai j=1 xj i which can be rewritten as (4.3). The proof of Theorem 4.2 is complete. Theorem 4.3. Let q ∈ (0, 1), m ∈ N, and a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. If 0 < a ≤ c and x > 0, then (a + x) m i=1 a i (a + x)a 1 , (a + x)a 2 , . . . , (a + x)a m q c m i=1 a i ca 1 , ca 2 , . . . , ca m q ≤ a m i=1 a i aa 1 , aa 2 , . . . , aa m q (c + x) m i=1 a i (c + x)a 1 , (c + x)a 2 , . . . , (c + x)a m q (4.4)

and the equality in (4.4) holds if and only if a = c.

Proof. For 0 < a < c, set

V (x) = ln Q q (a + x) + ln Q q (c) -ln Q q (a) -ln Q q (c + x). Because V (x) = Q q (a + x) Q q (a + x) - Q q (c + x) Q q (c + x
) and the logarithmically complete monotonicity of Q q (x) hints that the function

Q q (x)
Qq(x) is strictly increasing on (0, ∞), we see that V (x) < 0 and V (x) < V (0) = 0. Consequently, it follows that ln Q q (a + x) This can be simplified as (4.4). The proof of Theorem 4.3 is complete.

+ ln Q q (c) ≤ ln Q q (a) + ln Q q (c + x) which is equivalent to ln (a + x)

Inequalities for the q-multivariate beta function

Let a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. Then the q-multivariate beta function for q ∈ (0, 1) is

B q (a 1 , a 2 , . . . , a m ) = Γ q (a 1 )Γ q (a 2 ) • • • Γ q (a m ) Γ q (a 1 + a 2 + • • • + a m ) .
In [3, p. 544], it was proved that

Γ q (x + 1) = 1 -q x 1 -q Γ q (x)
with Γ q (1) = 1 for q ∈ (0, 1). Applying this recurrence relation to (4.1) yields

m i=1 a i a 1 , a 2 , . . . , a m q = (1 -q) m-1 1 -q m i=1 ai m i=1 (1 -q ai ) 1 B q (a 1 , a 2 , . . . , a m )
.

(5.1) Substituting (5.1) into those inequalities in Theorems 4.1 to 4.3, we can conclude several inequalities for the q-multivariate beta function B q (a 1 , a 2 , . . . , a m ). Equivalently speaking, Theorems (4.1) to (4.3) can be respectively reformulated as follows.

Let q ∈ (0, 1), , m ∈ N, and a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. If x j > 0 for 1 ≤ j ≤ and λ j ∈ (0, 1) with j=1 λ j = 1, then

F. QI m i=1 1 -q ai j=1 λj xj 1 -q j=1 λj xj m i=1 ai B q a 1 j=1 λ j x j , a 2 j=1 λ j x j , . . . , a m j=1 λ j x j ≥ j=1 m i=1 (1 -q xj ai ) 1 -q xj m i=1 ai B q (a 1 x j , a 2 x j , . . . , a m x j ) λj (5.2)
and the equality in (5.2) holds if and only if

x 1 = x 2 = • • • = x .
The inequality (5.2) implies that the function m i=1 (1 -q aix ) 1 -q x m i=1 ai B q (a 1 x, a 2 x, . . . , a m x) for m ∈ N is logarithmically concave with respect to x ∈ (0, ∞). Generally, we claim that the function 1 -q x m i=1 ai m i=1 (1 -q aix ) 1 B q (a 1 x, a 2 x, . . . , a m x) for m ∈ N is logarithmically completely monotonic with respect to x ∈ (0, ∞).

Let q ∈ (0, 1), , m ∈ N, and a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. If x j > 0 for 1 ≤ j ≤ , then 1 (1 -q) ( -1)(m-1) j=1 m i=1 (1 -q aixj ) 1 -q xj m i=1 ai B q (a 1 x j , a 2 x j , . . . , a m x j ) > m i=1 1 -q ai j=1 xj 1 -q j=1 xj m i=1 ai B q a 1 j=1

x j , a 2 j=1

x j , . . . , a m j=1

x j .

Let q ∈ (0, 1), m ∈ N, and a = (a 1 , a 2 , . . . , a m ) with a i > 0 for 1 ≤ i ≤ m. If 0 < a ≤ c and x > 0, then 1 -q (a+x) m i=1 ai 1 -q (c+x) m i=1 ai m i=1 (1 -q (c+x)ai ) m i=1 (1 -q (a+x)ai ) B q ((c + x)a 1 , (c + x)a 2 , . . . , (c + x)a m ) B q ((a + x)a 1 , (a + x)a 2 , . . . , (a + x)a m ) ≤ 1 -q c m i=1 ai 1 -q a m i=1 ai m i=1 (1 -q aai ) m i=1 (1 -q cai ) B q (ca 1 , ca 2 , . . . , ca m ) B q (aa 1 , aa 2 , . . . , aa m ) (5.3) and the equality in (5.3) holds if and only if a = c.

Theorem 2 . 1 (

 21 [START_REF] Ouimet | Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex[END_REF] Theorem 2.1] and[START_REF] Qi | Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions[END_REF] Thereom 2.2]). The function Q

m

  i=1 a i (a + x)a 1 , (a + x)a 2 , . . . , (a + x)a m x)a 1 , (c + x)a 2 , . . . , (c + x)a m

Acknowledgements. The author is thankful to Dr. Frédéric Ouimet (Université de Montréal, Canada) and Dr. Li Yin (Binzhou University, China) for their careful corrections to and valuable comments on the original version of this paper.


Zaiyi Shen

Alois Würger

Juho S Lintuvuori

Hydrodynamic interaction of a self-propelling particle with a wall

Keywords: XX, XX, XX No code given

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

Introduction

Artificial microswimmers have become an important tool to study the structures and dynamics of motile microorganisms in a laboratory [START_REF] Zöttl | Emergent behavior in active colloids[END_REF][START_REF] Bechinger | Active particles in complex and crowded environments[END_REF]. A typical example of artificial swimmer is provided by an active Janus particle: a spherical micrometer sized colloidal particles rendered motile by local gradients ( e.g. chemical or thermal) [START_REF] Bechinger | Active particles in complex and crowded environments[END_REF]. These local gradients give a rise to a phoretic slip velocity tangential to particle surface, resulting a squirming motion of the particle [START_REF] Bechinger | Active particles in complex and crowded environments[END_REF]. A popular theoretical model for describing the squirming motion was introduced by Lighthill [START_REF] Lighthill | On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers[END_REF] (so called squirmer model). In this model, a continuous slip velocity is assigned to the particle surface, leading to a time-independent squirming motion [START_REF] Magar | Nutrient uptake by a self-propelled steady squirmer[END_REF]. The squirmer model has been instrumental in theoretical and simulation studies of the hydrodynamics of spherical selfpropelling particles [START_REF] Llopis | Hydrodynamic interactions in squirmer motion: Swimming with a neighbour and close to a wall[END_REF][START_REF] Pagonabarraga | The structure and rheology of sheared model swimmer suspensions[END_REF][START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF][START_REF] Zöttl | Hydrodynamics determines collective motion and phase behavior of active colloids[END_REF][START_REF] Li | Hydrodynamic interaction of microswimmers near a wall[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF][START_REF] Lintuvuori | Hydrodynamics defines the stable swimming direction of spherical squirmers in a nematic liquid crystal[END_REF]. The hydrodynamic nature of the motion is given by the squirming parameter β, where β < 0 corresponds to pushers and β > 0 to pullers, respectively.

Both experimental and simulation studies have shown that the swimmers have an affinity to accumulate near surfaces [7-9, 12, 13,10]. Experiments have shown that the synthetic swimmers can be trapped by a colloidal crystal [START_REF] Brown | Swimming in a crystal[END_REF] and solid geometries can be used to guide the particles [START_REF] Das | Boundaries can steer active janus spheres[END_REF][START_REF] Simmchen | Topographical pathways guide chemical microswimmers[END_REF]. Specifically simulations [START_REF] Pagonabarraga | The structure and rheology of sheared model swimmer suspensions[END_REF][START_REF] Li | Hydrodynamic interaction of microswimmers near a wall[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF]

and detailed

Send offprint requests to: juho.lintuvuori@u-bordeaux.fr theoretical calculations [START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF] have shown that the trapping of the squirmers by a no-slip wall is strongly dependent of the value of β: typically |β| 4 is required [START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF]. Also specific surface interactions can influence the observed dynamics. For example, including a short range repulsion near the wall, can lead to both decaying or periodic cyclic motion near the surface [START_REF] Pagonabarraga | The structure and rheology of sheared model swimmer suspensions[END_REF][START_REF] Li | Hydrodynamic interaction of microswimmers near a wall[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF].

In the experimental realisation of artificial microswimmers based on two faced Janus particles, the two hemispheres are expected to have different interaction with the medium. An example of this is provided by differing catalytic rates (e.g. reaction with hydrogen peroxide H 2 O 2 in the case of chemical swimmers [START_REF] Brown | Ionic effects in selfpropelled pt janus swimmers[END_REF][START_REF] Brown | Swimming in a crystal[END_REF][START_REF] Ebbens | Synthetic running and tumbling: an autonomous navigation strategy for catalytic nanoswimmers[END_REF][START_REF] Sabass | Efficiency of surfacedriven motion: Nanoswimmers beat microswimmers[END_REF][START_REF] Ebbens | Size dependence of the propulsion velocity for catalytic janus-sphere swimmers[END_REF][START_REF] Ebbens | Electrokinetic effects in catalytic platinum-insulator janus swimmers[END_REF][START_REF] Wang | Enhanced active motion of janus colloids at the water surface[END_REF][START_REF] Das | Boundaries can steer active janus spheres[END_REF]) or for example different heat conduction properties leading to a temperature gradient across the particle surface for thermophoretic colloids [START_REF] Volpe | Microswimmers in patterned environments[END_REF][START_REF] Buttinoni | Active brownian motion tunable by light[END_REF]. Thus it is natural to assume that the slip velocity would have a discontinuity across the equator of the particle. This is in stark contrast with the squirmer model, where the tangential slip velocity varies continuously over the the particle surface [START_REF] Magar | Nutrient uptake by a self-propelled steady squirmer[END_REF]. While the far-field topology of the flow field is expected to only depend on the nature of the swimmer (e.g. pusher versus puller), the near-field effects could change dramatically, when the two different mobilities on the opposite hemispheres are taken into account. The near-field effects have been shown to be important in determining the behaviour of the swimmer near a no-slip surface [START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF]. It should be noted that recent theoretical and simulation work suggest 1. Schematic of the microswimmer near a wall, defining the shortest distance d between the particle surface and the wall as well as the angle ϕ between the particle direction and the wall, used in the text. vs(θ) is the slip velocity at the particle surface at an angle θ from the swimmer direction.

d θ v s (θ) φ X Z Fig.
that phoretic interactions can dominate particle-particle interactions [START_REF] Liebchen | Phoretic interactions generically induce dynamic clusters and wave patterns in active colloids[END_REF][START_REF] Scagliarini | Non-equilibrium phases in suspensions of self-propelled colloidal particles controlled by phoretic mobility and hydrodynamics[END_REF], and can lead to the experimentally observed clustering of synthetic swimmers [START_REF] Theurkauff | Dynamic clustering in active colloidal suspensions with chemical signaling[END_REF][START_REF] Palacci | Living crystals of light-activated colloidal surfers[END_REF][START_REF] Buttinoni | Dynamical clustering and phase separation in suspensions of selfpropelled colloidal particles[END_REF]. These interactions could also have an effect on the surface accumulation of synthetic swimmers. For example, the interplay between a chemical distribution and a hydrodynamic flow has been shown to affect the swimmer states observed for a single spherical swimmer approaching a noslip wall [START_REF] Uspal | Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering[END_REF][START_REF] Mozaffari | Self-diffusiophoretic colloidal propulsion near a solid boundary[END_REF].

In this work we develop a hydrodynamic model for a Janus swimmer in the framework of lattice Boltzmann (LB) simulations. We do not explicitly deal with phoretic interactions, but instead build a model for a Janus swimmer by considering a discontinuous slip velocity over the particle surface to take into account the varying mobilities in the opposing hemispheres. The model is based on LB implementation of squirmers [START_REF] Llopis | Hydrodynamic interactions in squirmer motion: Swimming with a neighbour and close to a wall[END_REF][START_REF] Pagonabarraga | The structure and rheology of sheared model swimmer suspensions[END_REF], but instead of continuous varying slip velocity, we assign two different mobilities on the two hemispheres u 1 and u 2 . To compare to the traditional squirmer model, we map the two mobilities to the squirming parameter β. Via extensive simulations we compare the hydrodynamic interactions between the active colloid and no-slip wall, for the two models: our active Janus model and classical squirmer model. Our results demonstrate that qualitatively the Janus swimmers behave in an agreement with the squirmers. However we show that for certain values of β the Janus swimmers can exhibit multiple different modes of the cyclic motion and stronger affinity to accumulate at the surface, this is especially true to low values of |β|.

Computational model

Chemically or thermally driven Janus particles have become a widely studied class of artificial micro-swimmers. In most cases they consist of colloidal spheres with an active cap which typically covers half of their surface. The particles' activity results in a finite slip velocity u which, because of different physical or chemical properties, takes different value on the uncoated and coated hemispheres. When defining the polar angle θ with the respect to the particle's axis (Fig. 1), one has v s (θ) = u 1 sin θ for cos θ ≥ 0 u 2 sin θ for cos θ < 0

where prefactors u 1 and u 2 describe the slip velocity on the upper and lower hemispheres, and arise from different material properties of the two hemispheres; the factor sin θ is due to the spherical geometry. More intricate dependencies may arise from the particle's material properties [START_REF] Bickel | Flow pattern in the vicinity of self-propelling hot janus particles[END_REF] or from its activity [START_REF] Bickel | Polarization of active janus particles[END_REF].

Expanding the step function in Eq. ( 1) in terms of Legendre polynomials P n (cos θ), one has

v s (θ) = sin θ ∞ n=0 p n P n (cos θ). (2)
The even coefficients vanish except for the first one, p 0 = 1 2 (u 1 + u 2 ), whereas the odd ones are finite,

p 1 = 1 2 (u 1 - u 2 ), p 3 = -1 8 (u 1 -u 2 ),.
.. The first two terms of this series correspond to the standard squirmer model,

v s (θ) = 3 2 u 0 sin θ(1 + β cos θ), (3)
where u 0 ≡ 2 3 p 0 is the unperturbed bulk swimming speed and β ≡ p 1 /p 0 is the squirming parameter, which can be used to distinguish between β > 0 pullers and β < 0 pushers. The parameters between the Janus particle and squirmer models are related through

u 0 = u 1 + u 2 3 , β = 3 2 u 1 -u 2 u 1 + u 2 , (4)
implying u 1,2 = u 0 ( 3 2 ± β). We use lattice Boltzmann method (LBM) to simulate the squirmers. In the LBM a no-slip boundary condition at solid-fluid interface can be realised by using a bounce back on links method [START_REF] Ladd | Numerical simulations of particulate suspensions via a discretized boltzmann equation. part I. theoretical foundation[END_REF][START_REF] Ladd | Numerical simulations of particulate suspensions via a discretized boltzmann equation. part II. numerical results[END_REF]. When considering a mobile particle, the bounce back on links needs to be modified to take into account the rotational and translational motion of the particle surface. In order to simulate the squirming motion, the boundary condition at the particle surface is modified to include the surface slip flow [START_REF] Llopis | Hydrodynamic interactions in squirmer motion: Swimming with a neighbour and close to a wall[END_REF][START_REF] Pagonabarraga | The structure and rheology of sheared model swimmer suspensions[END_REF]. To model the active Janus particle we use the slip velocity given by eq. (1) while for squirmer standard slip flow given by eq. ( 3) is used.

To stop the particles to penetrate the wall, we employ a short range repulsive potential

V (d) = σ d ν (5)
which is cut-and-shifted by to ensure that the potential and resulting force go smoothly to zero at the interaction range d c = 1.2 in simulation units (SU) corresponding to repulsion range of ≈ 0.15R. The = 0.06 and σ = 1.0 are constant in the reduced units of energy and length, respectively. The ν = 12 controls the steepness of the repulsion.

V W (d) = V (d) -V (d c ) -(d -d c ) ∂V (d) ∂d | d=dc (6)
We carried out our simulations in a rectangular simulation box with the size of 96 × 96 × 96, with a no-slip wall at z = 0.5 and z = 96.5 and periodic boundary conditions along x and y (for a schematic see Fig 1) . Unless otherwise mentioned, we carried out the simulations using a particle with radius R = 8SU. We used a kinematic viscosity η = 1/6SU and fixed the unperturbed bulk swimming speed u 0 = 10 -3 SU, but considered a large range of the squirming parameter β ∈ [-16 , 16]. We control our system in a Stokes regime by the small Reynolds number (Re) giving the ratio of inertial and viscous forces. Using the parameters above, we find Re = u 0 R/η ≈ 0.05.

Results

Flow field around a single particle in the bulk

To model the active Janus particle, we consider a surface slip velocity with two mobilities on the opposite sides of the particle as given by the equation (1), while the squirmer boundary conditions are given by the equation (3). In Fig. 2 we compare both the surface slip velocity The key difference between the two types of the swimmers is the discontinuity of v s (θ) for a Janus swimmer (green line in Fig. 2(A)) while for a squirmer v s (θ) varies continuously (red line in Fig. 2(A)). This is clearly visible for the graphs for a strong β = -5 pusher and β = +5 puller (Fig. 2(A)). These correspond to the ratio of the mobilities u 1 /u 2 = -13/7 and u 1 /u 2 = 7/13 for β = +5 and β = -5, respectively. The β = ±1.5 corresponds to the special case where u 1 = 0 (u 2 = 0) for β = -1.5 pusher (β = +1.5 puller) as shown in Fig. 2(A). For β = 0 neutral swimmer the requirement is u 1 = u 2 , thus the two models have an identical surface slip velocity v s (θ) (middle panel in Fig. 2(A)).

Despite the large differences between the surface flow of these two models (as shown in Fig. 2(A)) the topology of the bulk flow away from the particle surface is remarkably similar between the two models. This is clearly apparent from the visualisation of the streamlines for the Janus (squirmer) swimmer as shown in top (bottom) of Fig. 2(B). These correspond to the surface slip-velocities as presented in Fig. 2(A) for a pusher (β = -5, β = -1.5), neutral (β = 0) and a puller (β = +1.5, β = +5). Thus in far-field these two models should have almost identical behaviour. However, for the dynamics near a no-slip wall, the near-field lubrication effects can become very impor- tant [START_REF] Zöttl | Hydrodynamics determines collective motion and phase behavior of active colloids[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF]. Thus the differences on the surface slip velocity between these two models, could lead to a different dynamics near a boundary. In what follows the hydrodynamic interaction between these two models and a no-slip wall is analysed via lattice Boltzmann simulations.

Different swimming modes near the surface

To study the dynamics of a squirmer and an active Janus particle near a no-slip surface we place the particle initially in the X-Z plane with ϕ = -45 • and d = 3R/8. To quantify the dynamic state of the swimmers, during the course of the simulation, we measure the shortest distance between the particle surface and the wall d, and the angle ϕ between the swimmer director and the wall (see Fig. 1 for a schematic of the set-up). We consider both standard squirmer modeled by a surface slip velocity given by eq. 3 and Janus swimmer which has two independent surface flow mobilities, leading to a discontinuous surface flow (eq. 1). We map out the approximate phase diagrams for standard squirmer and a Janus swimmer near a surface. In both cases seven distinct swimming modes are observed for both the standard squirmer (A-G; Fig. 3) and the Janus swimmer (I-VII; Fig. 5). We start by considering the standard squirmer model (eq. 3). We explore the β values from -16 to 16. In Fig. 3, the time evolution of the distance between the wall and the particle surface d, the angle between the particle direction and wall ϕ and the absolute velocity of the particle u are plotted. We observe seven distinct swimming modes, when β is varied: After an initial re-orientation, strong pushers β < -9.3 swim near the surface while leaning to the external soft repulsion (d < 0.15R; shaded region in the top row in Fig. 3; mode A). The particle moves in a steady state with a constant d, ϕ and u. The swimmer direction remains oriented away from the wall ∼ ϕ ∈ [23 • , 30 • ], with ϕ increasing with β. For -9.3 < β < -4.2, the particle undergoes a periodic motion, where it escapes slightly from the wall and returns to the repulsion range due to the reorientation by hydrodynamic torques (mode B). This is in agreement with previous observations of β = -5 pusher near a no-slip wall with an additional repulsion at the surface [START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF]. The periodic oscillation in d, ϕ and u show an increase in the period and amplitude with increasing β (2nd column in Fig. 3) However the average value of ϕ and u decrease with increasing β, while averaged d increases. For -4.2 < β < 3.0 we observe no trapping by the surface (mode C in the Fig. 3). After an initial interaction and re-orientation, the particle escapes from the wall. This agrees well with the more precise theoretical calculations of a spherical squirmer near no-slip boundary [START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF], where no stable fixed points in d, ϕ space were found for β ≈ [-3, +4] range.

In the moderately strong puller regime 3.0 < β < 6.2, the particle is again trapped by the wall, with a decaying cyclic motion (mode D in Fig. 3). In a steady state the particle swims parallel to the surface with a constant d and ϕ, while pointing towards the wall, ϕ < 0 (D in Fig. 3). This observation is in agreement with previous simulations [START_REF] Li | Hydrodynamic interaction of microswimmers near a wall[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF] and theoretical calculations [START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF]. In this mode, the d and ϕ are observed to decrease when β increases. For β = +5, we observe a steady state d ≈ 0.22R (light red line in top row Fig. 3D) which is in excellent agreement with theoretical calculations by Ishimoto and Gaffney [START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF], as well as previous simulations [START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF]. The excellent agreement with our simulations and theoretical calculations [START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF] which do not suffer from resolution issues, implies that our resolution is enough to describe the hydrodynamics correctly. To further evaluate the convergence, we vary the particle size as R = 6, 8, 12 and 16 and redo the simulations for the case of F in Fig. 3 with the swimming parameter β = 8. The initial conditions are set to ϕ = -30 • and d = 5R/32. In Fig. 4, we show d(t) for different resolutions of the simulation. The results show a good convergence.

When the β is increased (6.2 < β < 7.6; mode E in Fig. 3), the particle undergoes a sustained cyclic motion. Compared to the mode D, both the amplitude and period are decreased. Further increasing the squirming parameter (7.6 < β < 8.8), a re-entrant behaviour of the decaying cyclic motion is observed (mode F in Fig. 3). It should be noted, that in all the cases where the cyclic motion is observed (D, E and F), in the steady state the swimmers do not interact with the external repulsion at the wall, i.e. d(t) > 0.15R (shaded region in the top row of Fig. 3). Thus all the particle-wall interactions are purely hydrodynamic. This is opposite for very strong pullers (mode G) β > 8.8. Here the particle swims along the boundary, while continuously leaning on the external repulsions d < 0.15R (G in Fig. 3. This is similar to what was observed to strong pushers, but for pullers d decreases with β, as oppose to increasing d with β as seen for very strong pushers (mode A in Fig. 3.) Next we turn to the active Janus particle, where the two hemispheres have different mobilities leading to discontinuous slip velocity at the particle surface (eq. 1 and Fig. 2(A)). In Fig. 5, we explore the β values from -9 to 9 for the Janus swimmer. Similarly to the standard squirmers, we observe seven different swimming modes (I-VII): (I) β < -4.7, corresponding to the mode A for a strong pusher squirmer. The particle moves along the wall, while leaning on the soft external repulsion. Decreasing the magnitude of the β the particle undergoes a cyclic swimming motion, escaping slightly from the wall and then returning to the repulsion regime due to the re-orientation by the hydrodynamic torques (mode II for -4.7 < β < -3.7). This is similar with the mode B observed with squirmers.

For Janus pushers with -3.7 < β < -2.8 we observe a steady swimming near the boundary (mode III in Fig. 5). Here, in a steady state the particle points away from the wall ϕ > 0 in agreement with the squirmer pushers. However for the Janus particle we observe that in the steady state the particle swims outside the repulsion range i.e. d(t) > 0.15R (Fig. 5; mode III), suggesting that there exists a stable fixed point in the (d, ϕ) space. This is markedly different from squirmers, where the steady state dynamics for pushers only observed to undergo a cyclic motion or lean to the external repulsion. For -2.8 < β < -2.4, we observe a re-entrant behaviour of the periodic dynamics similar to mode II, but with larger amplitude and longer period (mode IV).

Similarly to squirmers, we observe no trapping of the Janus swimmers by the wall for small values of |β|, as shown in Fig. 5(V). Notably the unstable region is reduced for the Janus swimmers -2.4 < β < 1.9, when compared to standard squirmers -4.2 < β < 3.0 (modes C and V in Fig. 3 and Fig. 5 for Squirmer and Janus swimmers, respectively).

For Janus pullers (β > 0) we observe two stable swimming modes: (VI) 1.9 < β < 4.9, the particle is trapped by the no-slip surface with a well defined steady state d beyond the external repulsion and pointing towards the wall, ϕ < 0 in agreement with squirmers. Finally strong pullers (VII) β > 4.9, are strongly trapped by the surface and in steady state they lean on the repulsion, while retaining the orientation towards the wall, similarly to strong squirmer pullers.

For rod-like swimmers, the initial conditions in d, φ space has been shown to play a role on the final observed steady state [START_REF] Shum | Hydrodynamic analysis of flagellated bacteria swimming near one and between two no-slip plane boundaries[END_REF][START_REF] Pimponi | Hydrodynamics of flagellated microswimmers near free-slip interfaces[END_REF]. In order to test our results against different initial conditions, we carried out further simulations with d 0 = 0.375R, 0.75R, 1.125R, 1.5R and ϕ 0 = -15 • , -30 • , -45 • , -60 • , -75 • in the strong trapping regions (G in Fig. 3 with β = 12 and I in Fig. 5 with β = -6, for a standard squirmer and for a Janus swimmer, respectively). The results are shown in Fig. 6, we see that the early time evolution is affected by different initial angle and distance while the steady state swimming modes remains the same.

Comparison between Janus swimmers and squirmers

To allow more detailed comparison we plot in Fig. 7, the steady state gap-size d, angle ϕ and velocity u as the function of β for both the standard squirmer and the active Janus particle when they move close to the wall. In general, the squirmer and Janus swimmer show similar trends for all the observables d, ϕ and u over the β range considered (Fig 7). The distance d increases when decreasing |β|. The largest oscillations (error bars in Fig. 7(A)) are observed near the boundaries between the trapping at the wall and no trapping (dashed lines and dotted lines in Fig. 7(A)). The biggest difference between the standard squirmer and active Janus particle, is the shift of the wall trapping for smaller absolute values of β, as can be seen both from the state diagrams (bottom panels in Fig. 3 and Fig. 5 for squirmers and active Janus particles, respectively). The Squirmers show no trapping at the wall The observed behaviour of the steady state angles ϕ, further strengthens the observation of the similar behaviour between the two types of swimmers (Fig. 7(B)). Both, squirmer and Janus swimmers, show the expected behaviour pushers swimming pointing on average away from the wall while pullers swim in the steady state pointing towards the wall, in agreement with previous simulations and theory [START_REF] Pagonabarraga | The structure and rheology of sheared model swimmer suspensions[END_REF][START_REF] Li | Hydrodynamic interaction of microswimmers near a wall[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF][START_REF] Ishimoto | Squirmer dynamics near a boundary[END_REF]. As β increases beyond the mode A (Fig. 3; squirmer) and I (Fig. 5; Janus) regimes, the ϕ decreases until the swimmer escapes from the wall (β ≈ -4.2 and β ≈ -2.4 for squirmer and Janus swimmers, respectively). When the particle escapes the surface, the value of ϕ corresponds to the reflection angle (regions between two dotted lines (squirmer) or two dashed lines (Janus) in Fig. 7(A)). The escape angle increases with β for a pushers and decreases with β for a pullers. The maximum reflection angle is observed with a neutral (β = 0) as seen from Fig. 7

(B).

A puller trapped at the surface show a monotonic decrease of ϕ when β is increased.

Finally we turn to the swimming speed u of the particles near the surface (third row in Fig. 3 and Fig. 5 as well as Fig. 7(C)). In all the cases where the swimmers are trapped near the surface, an increase of the swimming velocity u as compared to the free bulk swimming speed u 0 is observed. This can be up to u/u 0 ∼ 2.5 for high |β| values, as seen from Fig. 7(C). The trends between standard squirmer and Janus swimmers are very similar (circles and squares in Fig. 7, respectively): a linear increase of u/u 0 is observed when the absolute value of the squirming parameter |β| is increased. When the particle escapes the wall, u/u 0 ≈ 1 is recovered as required.

The swimming speed of a particle in the bulk is given by the surface average of the slip velocity [START_REF] Anderson | Colloid transport by interfacial forces[END_REF]. A more complex situation occurs for a particle close to a solid surface, since the area near the point of minimum distance plays an important role. Then the speed can be estimated by considering separately the hydrodynamic drag on the lubrication area and on the remainder of the particle surface. The latter contribution is well approximated by

F b = 6πηR(u || -u) (7)
where u is the particle velocity corresponding to the farfield, u the unknown actual speed, and F b the Stokes drag resulting from their difference. The lubrication area contributes an additional force

F I = λ6πηR(v s -u), (8)
which is proportional to the difference between swimming speed and the slip velocity at minimum distance, v s (θ contact ).

The prefactor λ = λ 0 + λ 1 ln(R/d) consists of a small constant and a logarithmic correction which arises from singular perturbation schemes in lubrication problems [START_REF] Happel | Low Reynolds Number Hydrodynamics[END_REF].

Since there is no net external force, F b + F I = 0, one obtains the particle velocity

u = u || + λv s 1 + λ . (9)
The orientational angle ϕ can in principle determined from a similar relation for the torques T b + T I = 0. The contribution from the far-field hydrodynamics u || can be estimated in the presence of a no-slip surface as u || ∼ -β sin(2ϕ) [START_REF] Spagnolie | Hydrodynamics of selfpropulsion near a boundary: predictions and accuracy of far-field approximations[END_REF][START_REF] Schaar | Detention times of microswimmers close to surfaces: Influence of hydrodynamic interactions and noise[END_REF][START_REF] Lintuvuori | Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls[END_REF]. In the steady state ϕ < 0 (ϕ > 0) for β > 0 puller (β < 0 pusher) (Fig. 7(C)), thus one arrives to a linear relation u || ∼ |β|. The surface slip velocity at contact u s includes the contribution both from the steady state swimming angle ϕ (Fig. 7(B)) and the surface flows (equations ( 1) and (3) for the surface slip velocity of the Janus particle and squirmer, respectively). Using the angle data from Fig. 7(B) and identifying that the angle between the swimmers direction and point on the particle surface closest to the wall is given by θ contact (β) = ϕ(β) + π/2 one can calculate the v s (β) = u(θ contact ) using eq. ( 1) and (3). In Fig. 8 v s is shown for the β values where a trapping of the swimmers at the surface was observed. Comparing the v s and the observed swimming speed u (Fig. 8 and Fig. 7(C)) one can see that v s >> u, implying that λ << 1 in the equation [START_REF] Li | Hydrodynamic interaction of microswimmers near a wall[END_REF]. Interestingly both v s and u show a linear scaling v s , u ∼ |β|, for both the Janus swimmers and squirmers (Fig. 8 and Fig. 7(C)).

Conclusions

In this work we have studied the hydrodynamics of an active spherical particle near a no-slip surface. We have developed a model for an active Janus particle. This corresponds to experimental situation where artificial microswimmers can be realised by chemically or thermally driven Janus particles. Typically they consists of colloidal spheres with active cap covering half of their surface. The particle motion results from a finite slip velocity, which takes different values on the two hemispheres due to the different material properties. This renders the slip velocity discontinuous at the particle's equator. We provide a straight forward mapping between the active Janus particle to standard squirmer model, and study the hydrodynamics of these two models via lattice Boltzmann (LB) simulations. Our results show that the far-field hydrodynamics between these two models are very similar, despite that the surface slip velocities (and thus near-field) differ. In order to study the near-field effects, we carried simulations probing the hydrodynamic interactions of the active particle near a no-slip surface. Generally the two models behave similarly over large range of the squirming parameter β. In both cases we could identify 6 distinct swimming modes of the particles trapped by the boundary. Some key differences arise as well. Notably, at low |β| we observe that the affinity of the particle to be trapped by the surface is increased for the Janus swimmers, when compared to the standard squirmer model. Finally we find that when the particles are trapped by the wall, their swimming speed is increased as compared to bulk, as has been seen for example in experiments of colloidal swimmers trapped at a fluid interface [START_REF] Wang | Enhanced active motion of janus colloids at the water surface[END_REF]. Interestingly our simulation data implies a linear relation between the swimming speed and the magnitude of the squirming parameter u ∼ |β|, for both active Janus particles and squirmers trapped at the boundary.

Authors contributions

All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

Fig. 2 .

 2 Fig. 2. (A) The surface velocity of an active Janus particle (green line) and a squirmer swimmer (red line) for five values of β corresponding to a pusher (β = -5, β = -1.5), neutral (β = 0) and puller (β = 1.5, β = 5). (B) The corresponding simulated flow fields in the bulk (lab frame): Active Janus particle (top row) and a standard squirmer (bottom row). The particle swimming direction is vertically up the page.

  v s (θ) and the bulk flow in lab frame, between active Janus particles and standard squirmer model, for five values of β corresponding to the pusher (β = -5, β = -1.5), neutral (β = 0) and puller (β = +1.5, β = +5) swimmers.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Squirmer swimming modes (A-G): Seven different swimming modes (A-G) observed for the standard squirmer for β ∈ [-16, +16] near a no-slip surface. The shortest distance between the particle surface and the wall d(t) (top row), the angle ϕ(t) between the swimmer orientation and the wall (2nd row) as well as the magnitude of the swimming velocity u(t)/u0 (3rd row), are used to define an approximate state diagram (bottom), giving the approximate boundaries between the swimming modes: (A) strong attraction, (B) periodic swimming, (C) no trapping, (D) decaying cyclic swimming, (E) periodic swimming, (F) decaying periodic swimming and (G) strong attraction. See text for more details. (The shading in the plots of d(t) (top row) corresponds to the range of the external soft repulsion.)

Fig. 5 .

 5 Fig.5. Active Janus particle swimming modes (I-VII): Seven different swimming modes (I-VII) observed for the active Janus particles for β ∈ [-9, +9] near a no-slip surface. The shortest distance between the particle surface and the wall d(t) (top row), the angle ϕ(t) between the swimmer orientation and the wall (2nd row) as well as the magnitude of the swimming velocity u(t)/u0 (3rd row), are used to define an approximate state diagram (bottom), giving the approximate boundaries between the swimming modes: (I) strong attraction, (II) periodic swimming, (III) stable swimming near the surface, (IV) periodic swimming, (V) no trapping, (VI) stable swimming near the surface and (VI) strong attraction. See text for more details. (The shading in the plots of d(t) (top row) corresponds to the range of the external soft repulsion.)

Fig. 6 .

 6 Fig. 6. Temporal evolution of d(t) for different initial conditions. (A) Tests of mode G in Fig. 3, β = 12. (B) Tests of mode I in Fig. 5, β = -6.

Fig. 7 .

 7 Fig. 7. Time averaged simulation results observed for active particles (radius R) for a steady state swimming near a no-slip surface. (A) shortest distance d/R between the particle surface and the wall, (B) the angle ϕ between the swimmer director and the wall and (C) normalised swimming speed u/u0 as a function of β, for standard squirmers (open circles) and Janus swimmers (open squares). The error bars give the amplitude of the oscillations. (The vertical dotted (dashed) lines gives the boundaries of the observed trapping at surface for squirmer (Janus) swimmers.)

Fig. 8 .

 8 Fig. 8. The surface velocity vs = vs(θcontact) on the particle surface at the point closest to the wall as a function of the squirming parameter β. vs(θcontact) is calculated using eqs. (1) and (3) with θcontact(β) = ϕ(β) + π/2, where ϕ(β) is from Fig. 7(C).

6 Acknowledgements Z.S. and J.S.L. acknowledges support by IdEx (Initiative d'Excellence) Bordeaux and computational resources from Avakas cluster. A.W. acknowledges support by the French National Research Agency through Contract No. ANR-13-IS04-0003.