A Bertrandias
email: aude.bertrandias@centralesupelec.fr

) H Duval

) J Casalinho

M L Giorgi

Dripping to jetting transition for cross-flowing liquids

Received and accepted dates, other relevant dates

We experimentally study drops formed from a nozzle into an immiscible, crossflowing phase. Depending on the operating conditions, drops are generated either in dripping or jetting mode. We investigate the impact of the continuous and dispersed phase velocities, dispersed phase viscosity and interfacial tension on the drop generation mode and size. We find that a dripping to jetting transition (DJT) takes place at a critical inner Weber number, function of the outer capillary and Ohnesorge numbers. Two jetting regimes occur depending on the phase velocity ratio. When the continuous phase velocity is significantly greater (resp. lower) than the dispersed phase velocity, jet narrowing (resp. widening) occurs. In jet widening, the critical inner Weber number depends little on the outer capillary number whereas in jet narrowing, it sharply decreases as the outer capillary number increases. We propose a comprehensive model to describe the DJT based on the attached drop equation of motion. The model satisfactorily predicts the DJT and the effect of the outer capillary number on the critical inner Weber number. It also well accounts for the drop diameter in jet narrowing.

I.

 In jetting mode, the liquid thread breaks up by Plateau-Rayleigh instabilities. In certain operating conditions, drops much smaller than the nozzle diameter may be produced. The same trend is expected for membrane emulsification operated in jetting mode. Thus, it is of high interest to study the dripping to jetting transition (DJT) in this process.

 The simplest case is the dripping faucet, where a dispersed phase flows from a nozzle into a stagnant, immiscible outer phase. Smith and Moss 8 studied mercury jets into gases and found that above a critical velocity (named the jetting velocity), the liquid exits the nozzle as a jet. They proposed an empirical expression for the jetting velocity, which can be recovered from a simple balance between the jet momentum flux and the retaining capillary force. Scheele and Meister 9 investigated the DJT for fifteen liquid-liquid couples and established the jetting velocity from a force balance, which further includes the excess pressure force. The maximum error between their data and predictions is of 30.2%. Richards et al. 10 studied drop formation before and after the DJT by computational fluid dynamics (CFD) and obtained drop sizes that compare well with Scheele and Meister's data. 9 Clanet and Lasheras 11 studied the DJT for water flowing from a stainless steel nozzle into air and found that the DJT occurs at a critical inner Weber number We in function of Bond numbers (Bo, Bo # ). We in compares the inner momentum to the capillary force. It is built with the nozzle inner diameter and mean dispersed phase velocity in the

nozzle. Bo and Bo # compare buoyancy to the retaining capillary force. Bo (resp. Bo # ) is built with the nozzle inner (resp. outer) diameter. When the dispersed phase does not wet the nozzle, only Bo is relevant. Clanet and Lasheras 11 extended Taylor's model [START_REF] Taylor | The dynamics of thin sheets of fluids. III. Disintegration of fluid sheets[END_REF] for the recession of a free, liquid edge. Their model, adapted to drop recession and growth, predicts the jetting velocities. Their calculations are in good agreement with their own data and Scheele and Meister's data [START_REF] Scheele | Drop formation at low velocities in liquid-liquid systems: Part II. Prediction of jetting velocity[END_REF] (maximum error of 13.7% and 20%, resp.).

In coflowing liquids, Cramer et al. [START_REF] Cramer | Drop formation in a co-flowing ambient fluid[END_REF] examined the critical continuous phase velocity for the DJT. They found that it decreases for increasing dispersed phase flow rates or viscosity ratios 𝜁 = 𝜂 '( /𝜂 *( and for decreasing interfacial tensions. Utada et al. [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF] proposed a state diagram of the DJT in coflowing liquids in a Ca #-. -We 23 space. Ca #-. compares the viscous force (exerted by the outer fluid) to the capillary force. It is built with the nozzle inner diameter, the outer (continuous) phase viscosity and velocity. Two jetting regimes occur depending on the fluid velocity ratio. If the outer velocity is greater than the inner one, the inner liquid is stretched by the outer fluid and jet narrowing occurs. If the outer velocity is lower than the inner one, the outer fluid slows the inner fluid and jet widening occurs. [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF][START_REF] Castro-Hernández | Scaling the drop size in coflow experiments[END_REF] In jet widening, Castro-Hernández et al. [START_REF] Castro-Hernández | Scaling the drop size in coflow experiments[END_REF] showed that inertial or viscous forces drove the DJT depending on the inner Reynolds number Re 23 (built with the mean dispersed phase velocity and nozzle inner diameter). They proposed a unified scaling to predict drop size in both the widening and narrowing regimes (relative errors of 30%). Chen et al. [START_REF] Chen | Emulsion droplet formation in coflowing liquid streams[END_REF] studied both regimes by CFD. They noted that drop detachment in jet widening is due to high pressures in the neck whereas in jet narrowing, it is due to velocity differences between the front and rear ends of the neck (linking the drop to the thread).

Two spatiotemporal instabilities may occur in coflowing liquids: an absolute instability (A), with disturbances advected up-and downstream or a convective instability (C), with advection only downstream. [START_REF] Leib | Convective and absolute instability of a viscous liquid jet[END_REF][START_REF] Huerre | Local and Global Instabilities in Spatially Developing Flows[END_REF] Linear stability analysis was performed for confined coflowing liquids. Concentric cylindrical capillaries [START_REF] Guillot | Stability of a Jet in Confined Pressure-Driven Biphasic Flows at Low Reynolds Numbers[END_REF] and rectangular channels [START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF] were studied at low Reynolds numbers. Analysis was extended to when liquid inertia is not negligible. [START_REF] Herrada | Spatiotemporal instability of a confined capillary jet[END_REF] A/C regions were typically provided as a function of the inner and outer phase flow rates. The A (resp. C) region coincides with the dripping (resp. jetting) region identified experimentally. Stability analysis was also performed for unbounded coflowing liquids: jet widening was reported as absolutely unstable and jet narrowing as convectively unstable. The widening regime was actually assimilated to a dripping regime, with drops formed at the end of the fluid thread. [START_REF] Utada | Absolute Instability of a Liquid Jet in a Coflowing Stream[END_REF] Compared to coflowing liquids, the DJT in membrane emulsification (i.e., for cross-flowing liquids) was very little investigated: Meyer and Crocker [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] performed experiments whereas Pathak [START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] examined the DJT by CFD. For both, membrane emulsification was mimicked by forcing a dispersed phase through a single, circular pore of a plane wall sheared by a continuous phase flow. Meyer and Crocker [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] found that the state diagram of the DJT depends on We 23 and Ca #-. , but also on the inner Ohnesorge number Oh '( (ratio of the viscocapillary to inertial-capillary time scale). Both authors proposed a correlation for the DJT by replacing Bo and Bo # by Ca #-. in Clanet and Lasheras' DJT criterion [START_REF] Clanet | Transition from dripping to jetting[END_REF] and by adjusting coefficients. However, no comprehensive models were developed in this configuration. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] The aim of this work is to study drop generation in cross-flow. The setup consists in a nozzle, which forms dispersed phase drops into a continuous cross-flowing phase. Both dripping and the DJT are studied for various phase velocities, interfacial tensions and dispersed phase viscosities. Dripping data are used to identify the drag force experienced by a drop attached to the nozzle. Then, a physical model is proposed to predict the jetting velocities and drop diameters at the DJT. This enables to gain insight on drop formation in membrane emulsification and more generally on the DJT for cross-flowing liquids.

II. EXPERIMENTAL

The systems investigated are reported in Table I. For the reference system, the continuous and dispersed phases are distilled water and dodecane, respectively. For systems 1 and 2, a surfactant (Sodium dodecyl sulfate, SDS) is added to the continuous aqueous phase to study the effect of the interfacial tension. We note that the SDS concentration of system 1 (resp. 2) is lower (resp. greater) than the critical micellar concentration (CMC) of SDS in water, i.e. 2 g.L -1 . For both systems, a good estimate of the interfacial tension when drops form is given by the value of the dynamic interfacial tension at the intermediate plateau. [START_REF] Deshiikan | SDS, Brij58 and CTAB at the dodecane-water interface[END_REF][START_REF] Bertrandias | Drop generation from a vibrating nozzle in an immiscible liquid-liquid system[END_REF] This value is greater than the equilibrium value. For system 2, we consider that micelles have no specific impact on the process examined: surfactant essentially modifies the intermediate plateau value. Last, for systems 3 and 4, paraffin is added to dodecane to study the effect of the dispersed phase viscosity.

We note 𝛾 the interfacial tension, 𝜌 '( and 𝜌 *( the dispersed and continuous phase densities, respectively, and 𝜂 '( and 𝜂 *( the dispersed and continuous phase viscosities. 𝛾 was measured for all systems by the rising drop method, with a tensiometer (Tracker, I.T. Concept, Teclis). 𝜂 '( was measured at 25.1°C, with a Ubbelohde viscosimeter (AVS310, Schött-Gerade). The tabulated values a are measured experimentally; b correspond to those at the intermediate plateau.

An original setup was designed (Fig. 1). It consists in a cell with a horizontal channel through which the continuous phase flows. A vertical nozzle of inner diameter 𝐷 ( = 0.32 mm emerges into the channel. The continuous phase is pumped in a closed cycle by a gear pump (MDG M15T3B, Iwaki Co.), perpendicular to the nozzle axis (cross-flow). The flow rate is adjusted (OVAL MIII LSF41, OVAL Corporation) with a 0.1 L.h -1 precision. The dispersed phase is forced through the nozzle with a syringe pump (PHD ULTRA, Harvard Apparatus) at a volumetric flow rate 𝑞 '( (accuracy ≤ 0.25%), giving a mean dispersed phase velocity in the nozzle 𝑣 '( . A cold light (KL 2500 LCD, Schott) illuminates the setup. A high-speed camera (v310, Phantom) allowing up to 3250 fps at full resolution (1280 × 800 px²) is mounted with a macro lens (Macro MP-E 65mm f/2.8, Canon) of magnification × 5. Images are captured through the cell windows and are analysed with ImageJ to obtain data such as drop size 𝐷 ' , with the scale (238 px/mm) set by the nozzle outer diameter (433 ± 2 µm). Average relative standard deviations in drop diameters are of 3.4%, 8.7% and 7.7% in dripping, jet widening and narrowing, respectively. In order to obtain the so-called unperturbed continuous phase velocity at the location of the bound drop center of mass (denoted 𝑣 *( ), the continuous phase flow is analyzed by particle image velocimetry (PIV), without dispersed phase injection (see supplementary material A, Fig. A1). Analysis is performed for the range of continuous phase volumetric flow rates 𝑞 *( tested in this work. The average continuous phase velocity is essentially uniform above the nozzle except in a thin adjacent layer (see supplementary material A, Fig. A2). We theoretically estimate the boundary-layer thickness at a distance 𝐷 ( 2 from the leading edge of the nozzle [START_REF] Morrison | An introduction to fluid mechanics[END_REF] for all 𝑞 *( and find a thickness of 0.06 mm for the highest 𝑞 *( to 0.2 mm for the lowest 𝑞 *( : the boundary-layer thickness is always small compared to the minimum drop diameter formed at the given 𝑞 *( (0.302 mm and 1.019 mm, resp.). The drop is thus mainly located above the shear layer that develops above the nozzle. This differs from Meyer and Crocker's 21 or Pathak's [START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] work, where the drop is entirely located in the shear flow set up by the continuous phase flowing parallel to the plane wall. In the following, the continuous phase velocity 𝑣 *( seen by a growing drop corresponds to the velocity measured in the uniform flow above the shear layer.

Drop formation by dripping is studied as a function of the continuous phase velocity 𝑣 *( seen by the growing drop. The parameters tested are reported in Table II (dripping trials). We consider that the Reynolds number characteristic of the continuous phase flow in the channel inside the cell is close to the Reynolds number in the cylindrical tubes on the sides of the cell, in series with this channel (see Fig. 1). We find Re = 4𝜌 *( 𝑞 *( /𝜋𝐷 D 𝜂 *( from 330 to 3300, with 𝐷 D the tube diameter. According to Morrison [START_REF] Morrison | An introduction to fluid mechanics[END_REF] , the continuous phase flow entering the cell is laminar (Re < 2100) or transitional (2100 < Re < 3300). The inner Reynolds number Re 23 = 4𝜌 '( 𝑞 '( /𝜋𝐷 ( 𝜂 '( ranges from 2 to 38, so the dispersed phase flow is laminar in the nozzle.

Then, the DJT is studied. The 𝑣 *( values tested are given in Table II (DJT trials) and the dispersed phase velocity 𝑣 '( is increased slowly until the DJT. The onset of jetting is defined as in the literature [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] : it occurs when 𝐿 G /𝐷 ' > 1, with 𝐿 G the thread length prior to drop break off (from the nozzle surface to the drop base) and 𝐷 ' the detached drop diameter. We obtain jetting velocities with a precision of 2.9 to 7.7% in jet widening and 2.6 to 11.1% in jet narrowing (due to the chosen increment in dispersed phase flux). We note that these velocities correspond to a transition from dripping to jetting. For the transition from jetting to dripping, the value may vary due to hysteresis phenomena. [START_REF] Clanet | Transition from dripping to jetting[END_REF] Re 23 ranges from 7 to 130, so the dispersed phase flow is laminar in the nozzle. In the tubes on the sides of the cell, Re ranges from 330 to 6600, so for the highest 𝑞 *( , i.e. 𝑣 *( , the continuous phase flow is turbulent in the channel. [START_REF] Morrison | An introduction to fluid mechanics[END_REF] After a series of trials (either PIV or dripping and DJT trials), the cell is filled with 3 vol% Mucasol (Merz) for 24h and is rinsed with distilled water. The nozzle surface is then hydrophilic: the organic dispersed phase does not wet the nozzle surface, so its outer diameter does not impact drop generation.

III. DROP GENERATION BY DRIPPING WITH CROSS-FLOW

Drop diameters 𝐷 ' were measured as a function of the continuous phase velocity 𝑣 *( (Fig. 2), at constant 𝑣 '( = 0.031 m.s -1 . As 𝑣 *( increases, 𝐷 ' decreases due to the increasing shear exerted by the continuous phase flow. Drop diameters scale as 𝑣 *( JK.MN to 𝑣 *( JK.OK , depending on the system. A lower interfacial tension 𝛾 leads to smaller drops (systems 1 and 2, Fig. 2). The capillary force decreases with 𝛾, so a drop detaches earlier, also found by Xu et al. [START_REF] Xu | Experimental and theoretical approaches on droplet formation from a micrometer screen hole[END_REF] No significant impact on 𝐷 ' is seen with the dispersed phase viscosity 𝜂 '( (possibly a slight decrease in 𝐷 ' with increasing 𝜂 '( ) (Fig. 2). In membrane emulsification, Timgren et al. [START_REF] Timgren | Effects of cross-flow velocity, capillary pressure and oil viscosity on oil-in-water drop formation from a capillary[END_REF] numerically found a decrease in drop size with a thousand-fold increase in 𝜂 '( , expected as the drag coefficient then notably increases. [START_REF] Clift | Bubbles, Drops and Particles[END_REF] In our case, the difference may not be large enough to significantly impact drop size.

The influence of 𝑣 '( was also tested on the reference system (not shown). We find that 𝑣 '( does not impact drop diameter, as found for a stationary outer phase [START_REF] Bertrandias | Good vibrations -Transition in drop generation from an immersed capillary tube[END_REF] or in cross-flow [START_REF] Xu | Experimental and theoretical approaches on droplet formation from a micrometer screen hole[END_REF] . For these trials, drops form in dripping mode. Ca #-. = 𝜂 *( 𝑣 *( /𝛾 ranges from 3.1×10 -3 to 9.7×10 -3 , We 23 = 𝜌 '( 𝐷 ( 𝑣 '( N /𝛾 from 1.1×10 -3 to 2.0×10 -1 and Oh '( = 𝜂 '( / 𝜌 '( 𝐷 ( 𝛾 = 1.2×10 -2 . This is in fact in the dripping region of the state diagram reported in the literature. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] FIG. 2: Drop diameter 𝐷 ' function of the continuous phase velocity 𝑣 *( , 𝑣 '( = 0.031 m.s -1 . See Table I for symbols.

In order to account for the drop diameter scaling observed in dripping mode, we consider that drag and the retaining capillary force are the main forces involved. Buoyancy is neglected since the buoyancy to drag force ratio is on average less than 5%. As Peng and Williams 2 , we assume that the drop subjected to the forces discussed above stays spherical and rotates about the nozzle edge until detachment. The fraction of the drop that could remain attached to the nozzle tip after break off is neglected. The torque balance (TB) about the nozzle edge reads:

𝜋𝐷 ' N 4 𝐶 Q 𝜌 *( 2 𝑣 *( N 𝐷 ' 2 = 𝜋𝐷 ( 𝛾 𝐷 ( 2 (1)
with 𝐶 Q the drag coefficient. In Eq. ( 1), we neglect the velocity of the drop center of mass 𝑣 ' . Indeed, if we assume that in dripping mode, the main contribution to the drop center of mass motion is due to drop growth, a rough estimate of 𝑣 ' is given by 𝑣 ' 𝑡 ≈ 0.5 d𝐷 ' d𝑡 , where 𝐷 ' increases in time according to 𝐷 ' 𝑡 = 6𝑞𝑡/𝜋 Z [ . For all systems and phase velocities tested, we find 𝑣 ' ≪ 𝑣 *( (38 to 589 times). We further rearrange Eq. ( 1) as:

𝐷 ' 𝐷 ( = 𝑘 Oh *( Ca #-. N/[ (2)
with 𝑘 = 8/𝐶 Q Z [ and Oh *( = 𝜂 *( / 𝜌 *( 𝐷 ( 𝛾 the outer Ohnesorge number. We note that Eq. ( 2) may also be written as 𝐷 ' 𝐷 ( = 𝑘 We out JZ [ , with We out the outer Weber number, built with the continuous phase velocity. If we assume that 𝐶 Q is approximately constant (corresponding to the Newton regime), from Eq. ( 2), we find that the drop diameter scales as 𝑣 *( JN [ . This is in good agreement with our experimental data (𝑣 *( JK.MN to 𝑣 *( JK.OK ). However, this scaling differs from the one found by Meyer and Crocker [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] , i.e. 𝐷 ' 𝐷 ( ~ Ca out JZ N . In their trials, drops are entirely located in the shear layer that develops along the plane wall and their diameters seem better described by a force balance.

For a solid sphere in an infinite fluid, the Newton regime holds for particle Reynolds numbers Re b = 𝜌 *( 𝑣 *( 𝐷 ' /𝜂 *( from 10 3 to 10 5 . [START_REF] Morrison | An introduction to fluid mechanics[END_REF] In this range, 𝐶 Q ≈ 0.4. In our case, Re b = 92 to 360, so 𝐶 Q may vary. We estimate 𝐶 Q from Eq. ( 1) and find that 𝐶 Q decreases with increasing Re b (data points, Fig. 3). Our 𝐶 Q values are higher than reported for a viscous sphere in an infinite fluid by Feng and Michaelides 30 (dashed line, Fig. 3). Indeed, in our trials, the drop is not isolated but bound to a nozzle. We propose to model the variations of 𝐶 Q as a function of Re b by the law of Eq. ( 3), in the form obtained by Abraham [START_REF] Abraham | Functional Dependence of Drag Coefficient of a Sphere on Reynolds Number[END_REF] for a solid sphere:

𝐶 Q = 𝐶 K 1 + 𝛿 K Re b Z N N ( 3
)
We perform a weighted least squares minimization to identify the free coefficients of Eq. ( 3) from the data of the reference system, systems 1 and 2. These systems are characterized by a viscosity ratio 𝜁 = 1.5, with Re b = 92 to 360. We find 𝐶 K = 0.14 and 𝛿 K = 17.7. The data are well described by the adjusted law (solid line, Fig. 3). In the Stokes limit ( Re b ≪ 1), Eq. ( 3) reads 𝐶 Q ≈ 𝐶 K 𝛿 K N /Re b ≈ 45/Re b close to the drag coefficient of a solid sphere at a wall in creeping flow, i.e. 𝐶 Q ≈ 1.7× 24/Re b ≈ 41/Re b . [START_REF] O'neill | A sphere in contact with a plane wall in a slow linear shear flow[END_REF] We only retain that the order is satisfactory as our drop deviates from a solid sphere and the coefficients were adjusted far from the Stokes regime.

For system 3 (𝜁 = 2.0) and 4 (𝜁 = 3.6), we cannot carry out the above method since our trials do not cover a wide range of Re b for these viscosity ratios. We assume that 𝐶 Q are not significantly different than from Eq. ( 3) fitted on systems with 𝜁 = 1.5, consistently with the analysis of Fig. 2. In Section IV, Eq. ( 3) will be used to estimate the drag force experienced by a drop near the DJT.

IV. DRIPPING TO JETTING TRANSITION (DJT)

A. Experimental results For a set system and continuous phase velocity 𝑣 *( , the DJT is reached for a critical dispersed phase velocity (corresponding to the jetting velocity for the given 𝑣 *( ). We remind that the onset of jetting is defined when the length of the liquid thread connecting the drop to the nozzle reaches the drop diameter (see Section II). Typical snapshots of dripping and jetting are reported in Fig. 4. We note that in Fig. 4(a) and (b), the first image on the left represents "strict" dripping (as studied in Section III), with a drop rotating about the nozzle edge.

As in coflowing liquids, [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF][START_REF] Castro-Hernández | Scaling the drop size in coflow experiments[END_REF][START_REF] Chen | Emulsion droplet formation in coflowing liquid streams[END_REF] when the DJT is reached, two jetting regimes may be observed. When 𝑣 *( ≲ 𝑣 '( , the liquid thread is on average thicker than the nozzle inner diameter (Fig. 4(a), last image): this is the widening regime. On the contrary, when 𝑣 *( ≳ 𝑣 '( , the liquid thread gets thinner from the nozzle to the drop (Fig. 4(b), last image): this is the narrowing regime. Curiously, Meyer and Crocker 21 did not distinguish these regimes in their paper.

In jet widening, we also see that the thread undergoes surface oscillations (see supplementary material B, Fig. B1). These oscillations are essentially stationary in space. According to Utada et al. [START_REF] Utada | Absolute Instability of a Liquid Jet in a Coflowing Stream[END_REF] who studied jet widening in coflowing liquids, this behavior is characteristic of an absolute instability. To quantify this, as Utada et al. [START_REF] Utada | Absolute Instability of a Liquid Jet in a Coflowing Stream[END_REF] , we examine the variations of the neck diameter (between the thread and growing drop) as a function of time (see supplementary material B, Fig. B2). The neck diameter oscillates about its mean with an increasing amplitude until pinch off and subsequent drop detachment. The oscillation frequency is around 350 Hz for the reference system, consistent with the inertial-capillary time scale of this system.

As for coflowing or cross-flowing liquids [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF][START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] , we represent the DJT in the Ca #-. -We 23 space. We remind that We 23 is the inner Weber number ( We 23 = 𝜌 '( 𝐷 ( 𝑣 '( N /𝛾 ) and Ca #-. the outer capillary number (Ca #-. = 𝜂 *( 𝑣 *( /𝛾). Figure 5(a) presents the variations of We 23 built with the jetting velocity as a function of Ca #-. , for the five systems of Table II. The curves exhibit a plateau for lower Ca #-. values and a sudden decrease for higher Ca #-. . They are similar in shape to the curves established by Meyer and Crocker 21 and Pathak. [START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] We associate the plateau with jet widening and the sudden decrease with jet narrowing. For all systems, at the plateau, We 23 is in the order of 1, in the order of magnitude reported for jet widening in coflow. [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF] Our plateau values are also similar to those of Meyer and Crocker 21 and Pathak. [START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] The transition between widening and narrowing occurs at a critical value of Ca #-. , such that 𝑣 *( ≈ 𝑣 '( . This value is denoted Ca * for the reference system in Fig. 5(a). The same trends, i.e. a plateau followed by a marked decrease, are obtained for the drop diameters formed at the DJT as a function of Ca #-. (Fig. 5(b)). The variations in drop diameters at the DJT as a function of Ca #-. were not reported before. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] In jet widening, we expect that the drop diameter is controlled by a balance between the jet momentum and the retaining capillary force. In jet narrowing, the drop diameter should be controlled by a balance between the drag force experienced by the drop and the capillary force. As the continuous phase velocity increases, the thread gets thinner leading to a lower retaining capillary force, thus smaller drops.

As the interfacial tension 𝛾 decreases between the reference system, system 1 and 2, the plateau value for We 23 decreases and the transition from jet widening to narrowing is shifted towards higher Ca #-. . This result, to which we shall return, was not reported before. The influence of 𝛾 on the DJT was not examined in detail in cross-flow. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] In our trials, the dispersed phase viscosity 𝜂 '( does not affect the jetting velocity since the data for the reference system, system 3 and 4 collapse onto a unique curve (Fig. 5(a)). This is in good agreement with Meyer and Crocker's results. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] They found that the jetting velocity does not vary significantly with the inner Ohnesorge number Oh '( = 𝜂 '( / 𝜌 '( 𝐷 ( 𝛾 while Oh '( is below 3×10 -2 . Indeed, while Oh '( ≪ 1, the pinch-off time scale is in the order of the inertial-capillary time scale, thus it does not depend on 𝜂 '( . This corresponds to our experimental range: Oh '( increases from 1.2×10 -2 to 2.9×10 -2 as 𝜂 '( increases from 1.34 to 3.24 mPa.s. Drop diameters are also unaffected by 𝜂 '( (Fig. 5(b)). According to Utada et al. [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF] and Meyer and Crocker [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] , our DJT curves should overlap in the Ca #-. -We 23 space since Oh '( ≪ 1. However, as stated before, our DJT curves are shifted towards higher Ca #-. as the interfacial tension decreases. We attribute this to the hydrodynamic regime of the drag force experienced by a growing drop. Indeed, the particle Reynolds number Re b ranges from 114 to 424 in our DJT trials. These values are much higher than those reported by Utada et al. [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF] and Meyer and Crocker. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] Thus, the drag force in our trials rather corresponds to the Newtonian regime than to the Stokes regime. Since the ratio of the drag to capillary force is in the order of (Ca #-. /Oh *( )² = We out under Newton regime assumption (as opposed to Ca #-. under Stokes approximation), we plot We 23 as a function of Ca #-. /Oh *( (see supplementary material C). We remind that Oh *( is the outer Ohnesorge number (Oh *( = 𝜂 *( / 𝜌 *( 𝐷 ( 𝛾). We find that reference system and system 1 and 2 data then actually collapse in the narrowing regime.

B. Model

The DJT criteria proposed by Pathak [START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] and Meyer and Crocker [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF] are not adapted to our case. Indeed, in our trials, the growing drops are mainly located above the shear layer adjacent to the nozzle tip (see Section II) and the particle Reynolds number is far from the Stokes regime (see Section IV.A). Figure D1 (supplementary material D) shows the discrepancy between our DJT data for the reference system and previous authors' DJT criteria. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] We remind that these criteria are semiempirical since these authors simply replaced Bo and Bo # , the Bond numbers built with the nozzle inner and outer diameters, by Ca #-. in Clanet and Lasheras' criterion [START_REF] Clanet | Transition from dripping to jetting[END_REF] and adjusted the coefficients to fit their data. In this section, we propose a comprehensive model to account for the jetting velocity and drop diameters at the DJT.

Firstly, the footage shows that thread dynamics occur essentially along the 𝑥-axis (Fig. 4): the drop forms at the end of a thread that tends to align with the continuous phase flow. Thus, we approximate our configuration by a coflow configuration (Fig. 6). A force balance is now more relevant than a torque balance since the drop can no longer be considered as a sphere rotating about the nozzle edge.

We revisit Clanet and Lasheras' approach [START_REF] Clanet | Transition from dripping to jetting[END_REF] developed for a liquid injected downwards into a stagnant gas under gravity and amend the drop equation of motion to consider coflowing liquids. We thus add a drag force induced by the continuous flowing phase since 𝜂 *( is orders of magnitude higher than the dynamic viscosity of air and we account for the added mass effect since 𝜌 *( ~𝜌'( . We neglect buoyancy since for the largest drops (obtained in the widening regime), the buoyancy to capillary force ratio is lower than 3%.

In Clanet and Lasheras' scenario [START_REF] Clanet | Transition from dripping to jetting[END_REF] , drops are generated as follows: a first drop detaches (at 𝑥 = 0 in Fig. 6), leaving a thread behind that recedes at a velocity d𝑥/d𝑡, due to surface tension effects. During recession, a mass (drop) 𝑀 forms (sphere in Fig. 6). It recedes until it reaches a distance 𝑥 nop (closer to the nozzle, Fig. 6). The mass then progresses the other way, once momentum and drag overcome surface tension effects. It is assumed that pinch off begins at 𝑥 nop . From that point, the drop no longer grows. The drop travels a distance 𝑙 ' until detachment. If 𝑥 nop > 𝑙 ' , dripping occurs. Oppositely, if 𝑥 nop < 𝑙 ' , the detachment point advances each time, leading to jetting. The left-hand side of Eq. ( 5) corresponds to the drop effective inertia and includes 𝑀 o 𝑡 the added mass due to the surrounding continuous phase. We suppose that the added mass is equal to half of the displaced volume of continuous phase, as for a solid sphere in an infinite medium. It is thus given by 𝑀 o 𝑡 = 0.5𝑀 𝑡 𝜌 *( /𝜌 '( . The first term on the right-hand side of Eq. ( 5) is the retaining capillary force, with 𝐷 Ds 𝑥 the thread diameter at the location 𝑥 . The second term is the jet momentum flux entering the drop. The last term is the drag force, in which the frontal area corresponds to that of a sphere with the given mass. The drop mass diameter is given by:

𝐷 n (𝑡) = 6 𝜋 𝑀 𝑡 𝜌 '( Z [ (6)
In the drag force, the drag coefficient 𝐶 Q (Re b ) is estimated from Eq. ( 3), with the coefficients adjusted before and Re b is built with 𝐷 n (𝑡) (Eq. ( 6)) and the relative velocity 𝑣 *( + d𝑥 d𝑡 . At the DJT, we find Re b = 114 to 424 (estimated with the experimental 𝐷 ' and 𝑣 *( ). This is in the same range as where the coefficients of Eq. (3) were adjusted. However, Eq. ( 3) was established for drops detaching in dripping mode: we here neglect the deviations in 𝐶 Q that may arise at the DJT when the thread linking the drop to the nozzle is larger. We assume that 𝐶 Q essentially depends on the drop mass diameter and the particle Reynolds number and little on the drop shape details. 𝑥 nop is determined from the numerical integration of system (4-6) (see Appendix). As stated above, the DJT occurs at 𝑥 nop = 𝑙 ' . 𝑙 ' is the detachment distance by pinch off, such that:

d𝑥 𝑣 Ds (𝑥) u v K = 𝜏 G ( 7
)
𝜏 G is the necking time or pinch-off time (in the order of the inertial-capillary time), given by:

𝜏 G = 𝑘′ 𝐷 x [ 𝑥 nop 𝜌 '( 8𝛾 Z N (8)
We estimate 𝑘′ ≈ 8.26 ± 0.43 by Clanet and Lasheras' method [START_REF] Clanet | Transition from dripping to jetting[END_REF] : we measure the neck diameter variations in dripping mode as a function of time during pinch off, we fit the variations to an exponential form to find 𝜏 G and we plot 1/𝜏 G as a function of 𝜏 * = 8𝛾/𝐷 ( [ 𝜌 '( Z N

. This was done for the reference system, systems 1 and 2 (see supplementary material E). These systems differ only by their interfacial tension and are characterized by a viscosity ratio 𝜁 = 1.5. We find 𝜏 G 2.6 times higher than Clanet and Lasheras 11 for a liquid in air, which depicts a lower instability growth rate. This is in agreement with results reported in the literature: Rayleigh [START_REF] Rayleigh | On the instability of jets[END_REF] found a growth rate (made dimensionless by 𝜏 * ) of 0.34 for an inviscid liquid in air. On the other hand, Funada and Joseph 34 found a growth rate 1.3 to 4.8 times lower for water in benzene (𝜁 = 1.5) in the range of our Re 23 , from a fully viscous analysis.

Returning to the model, the jetting velocity at a set 𝑣 *( is obtained as follows (see Appendix for details): 𝑣 '( is scanned and for each 𝑣 '( , the system (4-6) is integrated as a function of time, until 𝑥 = 𝑥 nop . If 𝑥 nop coincides with 𝑙 ' (Eq. ( 7)), the corresponding 𝑣 '( value is the jetting velocity at the given 𝑣 *( . The resulting drop diameter is deduced from Eq. ( 6): 𝐷 ' ≈ 𝐷 n (𝑡 nop ), where 𝑡 nop is the time taken for the mass to travel from 0 to 𝑥 nop . This method may be repeated for each 𝑣 *( tested to reproduce our experimental results.

C. Comparison with experiments

To implement the above model, we must describe the mean dispersed phase velocity along the thread. Two limit cases will be considered: (1) the thread diameter and mean velocity are negligibly affected by the continuous phase flow in both jetting regimes and (2) the mean thread velocity quickly reaches the continuous phase velocity in the narrowing regime.

C.1. Uniform nozzle-sized thread

First, we consider the simplest scenario where the thread diameter is not affected by the continuous phase in either jetting regime. In this case, we assume 𝑣 Ds 𝑥 = 𝑣 '( and 𝐷 Ds 𝑥 = 𝐷 ( from 𝑥 = 0 to 𝑥 nop . Then, the thread cross-section reads 𝑆 Ds (𝑥) = 𝜋𝐷 ( N 4. We call this the uniform nozzle-sized thread limit. We solve system (4-6) under this assumption by the method described in the Appendix. The results are reported in Fig. 7 (solid lines).

The variations of the critical inner Weber number We 23 function of the outer capillary number Ca #-. are satisfactorily reproduced (Fig. 7(a), solid lines). The same conclusion may be drawn for the variations of the dimensionless drop diameter as a function of Ca #-. (Fig. 7(b), solid lines). Furthermore, the effect of the interfacial tension on the transition is well accounted for.

The model predicts a plateau in jet widening (for small Ca #-. ), but in our experiments, the plateau is more pronounced and extends for higher Ca #-. . Furthermore, in jet widening, We 23 and especially 𝐷 ' are overpredicted. The difference in plateau values for We 23 between the reference system and system 2 is also not reproduced (Fig. 7(a)). We checked that the differences in plateau values cannot be attributed to the relative effect of buoyancy. We attribute them to the thread surface oscillations (see supplementary material B), which affect the retaining capillary force and pinch-off time. These oscillations cannot be accounted for in the framework of the present model.

Lastly, we note that We 23 and 𝐷 ' are overestimated at high Ca #-. for the reference system (and systems 3 and 4, not shown in Fig. 7). This may be explained by the narrowing of the thread which becomes significant at high 𝑣 *( but is neglected in the present case. In the next section (IV.C.2), we attempt to take this effect into account.

C.2. Maximal narrowing

We measure the thread diameter for the reference system in the narrowing regime: it decreases by less than 10% for 𝑣 *( ≤ 0.73 m.s -1 (Ca out = 1.3×10 -2 ) but decreases by more than 50% for the highest 𝑣 *( . In the latter case, narrowing effects can no longer be neglected and our hypothesis 𝐷 Ds 𝑥 = 𝐷 ( of Section IV.C.1 is no longer valid.

We suppose that the momentum diffusion across the thread is rapid and fully efficient in the narrowing regime. In this case, we may consider that if 𝑣 *( > 𝑣 '( , the thread is characterized by 𝑣 Ds 𝑥 = 𝑣 *( , 𝐷 Ds 𝑥 = 𝐷 ( (𝑣 '( 𝑣 *( ) Z N and 𝑆 Ds 𝑥 = 𝜋𝐷 Ds N (𝑥) 4 from 𝑥 = 0 to 𝑥 nop . This gives the maximal narrowing limit. According to Castro-Hernandez et al. [START_REF] Castro-Hernández | Scaling the drop size in coflow experiments[END_REF] , maximal narrowing is observed when Re 23 < 1. In our trials, Re 23 = 7 to 130, so their assumption is not strictly applicable. However, it will give us an overestimate of the narrowing effect on the DJT and drop diameter. If 𝑣 *( < 𝑣 '( (jet widening), we still neglect the effect of the continuous phase on the thread size and assume that 𝐷 Ds 𝑥 = 𝐷 ( and 𝑣 Ds 𝑥 = 𝑣 '( , as in Section IV.C.1.

As before, we solve system (4-6) under these new assumptions (see Appendix) and we report the results in Fig. 7 (dashed lines). For the reference system (and systems 3 and 4, not shown), our data for We 23 and 𝐷 ' in jet narrowing lie in between the maximal narrowing limit (dashed lines, Fig. 7(a) and (b)) and the uniform nozzle-sized thread limit (solid lines, Fig. 7(a) and (b)).

We note that the data for system 2 are surprisingly well described by the uniform nozzle-sized thread limit. This is probably due to the compensation of different errors related to the uniform nozzle-sized approximation, the use of the drag coefficient estimated in the dripping regime and the assumption that forces act only along the 𝑥-axis.

V. CONCLUSIONS While the dripping to jetting transition (DJT) is well documented for liquid-air [START_REF] Smith | Experiments with Mercury Jets[END_REF][START_REF] Clanet | Transition from dripping to jetting[END_REF] or liquidliquid coflow [START_REF] Utada | Dripping to Jetting Transitions in Coflowing Liquid Streams[END_REF][START_REF] Castro-Hernández | Scaling the drop size in coflow experiments[END_REF][START_REF] Cramer | Drop formation in a co-flowing ambient fluid[END_REF][START_REF] Chen | Emulsion droplet formation in coflowing liquid streams[END_REF] , liquid-liquid cross-flow [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] has received little attention and no comprehensive model was proposed to describe the DJT in this configuration. Also, drop diameters specifically at the DJT were either not measured or not reported. [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] In the present work, we studied liquid-liquid cross-flow for different phase velocities, interfacial tensions and dispersed phase viscosities. Contrary to previous work [START_REF] Meyer | Universal Dripping and Jetting in a Transverse Shear Flow[END_REF][START_REF] Pathak | Numerical simulation of membrane emulsification: Effect of flow properties in the transition from dripping to jetting[END_REF] , the growing drops are mainly located above the shear layer that develops above the nozzle. Furthermore, since the inner Ohnesorge number (characteristic of the dispersed phase flow) is much lower than 1, drop pinch-off is controlled by the inertial-capillary time scale.

In strict dripping, we found that the drop diameter is well described by a simple torque balance about the nozzle edge, taking into account the drag force experienced by the drop and the retaining capillary force. This result was used to estimate the drag coefficient for an attached drop as a function of the particle Reynolds number.

The DJT occurs at a critical inner Weber number function of the outer capillary and Ohnesorge numbers. Two jetting regimes occur (widening and narrowing) depending on the phase velocity ratio. In jet widening (when the dispersed phase velocity is greater than the continuous phase one), the critical inner Weber number depends little on the outer capillary number whereas in the narrowing regime, it sharply decreases as the outer capillary number increases. Furthermore, when the outer Ohnesorge number increases, the transition between widening and narrowing is shifted to higher values of the outer capillary number. The hydrodynamic regime of the drag force experienced by a growing drop is actually inertial and not viscous.

We proposed to model the DJT in liquid-liquid cross-flow by revisiting an approach originally developed by Clanet and Lasheras for a liquid injected into a stagnant gas under gravity. The model describes the recession dynamics and the growth of the drop until pinch off. In the present case, the driving force for drop detachment is not buoyancy but the drag force exerted by the continuous phase flow. We distinguished two limit cases for the thread profile that exits the nozzle and enters the drop: a uniform nozzle-sized thread limit and a maximal narrowing limit. Jetting velocities and drop diameters measured at the DJT in jet narrowing are well accounted for and lie in between model predictions in the two limit cases. Furthermore, the effect of the outer Ohnesorge number on the DJT is well reproduced. In jet widening, the agreement is less satisfactory. Discrepancies are attributed to thread surface oscillations which appear in jet widening. However, we may conclude that the main features of the DJT in cross-flow are captured which highlights the insight and the robustness of Clanet and Lasheras' original model.

SUPPLEMENTARY MATERIAL

See supplementary material for insight on: A, continuous phase flow analysis using Particle Image Velocimetry (PIV); B, oscillations in jet widening; C, collapsed data in jet narrowing; D, comparison with Pathak's and Meyer and Crocker's DJT criteria; E, estimation of the necking time.

FIG. 1 :

 1 FIG. 1: Cell cross-section by CAD (to scale). Cell (left): (a) continuous phase flow axis; (b) optical axis; (c) dispersed phase flow axis; (d) cell binding point; (e) frame for zoom. Zoom in the cell (right): 1, dispersed phase inlet (microfluidic

FIG. 3 :

 3 FIG. 3: Drag coefficient 𝐶 Q function of the particle Reynolds number Re b . Experimental results from Eq. (1), see Table I

  for symbols. Feng and Michaelides predictions for 𝜁 = 1.5 (dashed line); our fit from Eq. (3) with 𝐶 K = 0.14 and 𝛿 K = 17.7 (solid line).

FIG. 4 :

 4 FIG. 4: Reference system snapshots (Oh *( = 7.0×10 -3 ): (a) jet widening, Ca #-. = 5.1×10 -3 ; (b) jet narrowing, Ca #-. = 1.3×10 -2 . Continuous phase flow from right to left.

FIG. 5 :

 5 FIG. 5: Characteristic values at the DJT, see Table I for symbols: (a) We 23 function of Ca #-. . Widening to narrowing limit Ca * for the reference system (dashed line); (b) Drop to pore diameter ratio 𝐷 ' /𝐷 ( function of Ca #-. .

  FIG. 6: Model configuration with the different variables involved.The drop equation of motion during recession reads:dd𝑡 𝑀(𝑡) + 𝑀 o (𝑡) d𝑥 d𝑡 = 𝜋𝐷 Ds 𝑥 𝛾 -𝜌 '( 𝑞 '( d𝑥 d𝑡 + 𝑣 Ds 𝑥 -𝜋 4 𝐷 n N 𝑡 𝐶 Q 𝜌 *( 2 𝑣 *( + d𝑥 d𝑡 N

FIG. 7 :

 7 FIG. 7: Characteristic values at the DJT, see Table I for symbols. (a) We 23 function of Ca #-. . (b) Drop to pore diameter ratio 𝐷 ' /𝐷 ( function of Ca #-.. Simulation results for the reference system (grey); system 2 (black). Uniform nozzle-sized thread limit of Section IV.C.1 (solid line); maximal narrowing limit of Section IV.C.2 (dashed line).

TABLE I :

 I Investigated system composition and physicochemical properties. Reproduced from A. Bertrandias, H. Duval, J. Casalinho and M. L. Giorgi, Phys. Fluids, 28, 102103 (2016), with the permission of AIP Publishing.

	System and symbol	Dispersed phase	𝜂 '( (mPa.s)	𝜌 '( (kg.m -3 )	Continuous phase	𝜂 *( (mPa.s)	𝜌 *( (kg.m -3 )	𝛾 (mN.m -1 )
	Reference (◊)	Dodecane	1.34	750	Distilled water	0.89	997	50.7 ± 3.5 a
	1 ( )	Dodecane	1.34	750	Distilled water and SDS (0.1 wt%)	0.89	997 ± 1.4 a	22.3 ± 0.5 a,b
	2 (♦)	Dodecane	1.34	750	Distilled water and SDS (2 wt%)	0.89	1001 ± 1.2 a	6.6 ± 0.2 a,b
	3 (⊲)	Dodecane (75 wt%) and paraffin (25 wt%)	1.79 ± 0.23 a	772 ± 1.2 a	Distilled water	0.89	997	53.5 ± 2.4 a
	4 (⊳)	Dodecane (50 wt%) and paraffin (50 wt%)	3.24 ± 0.42 a	790 ± 1.1 a	Distilled water	0.89	997	50.0 ± 1.2 a

TABLE II :

 II Operating conditions investigated.

	System	𝑣 *( (m.s -1 )	Dripping trials	𝑣 '( (m.s -1 )	DJT trials 𝑣 *( (m.s -1 )
	Reference	0.18 -0.55	0.016, 0.031, 0.063, 0.14, 0.21	0.17 -1.03
	1	0.11 -0.50		0.031	0.17 -0.59
	2	0.10 -0.40		0.031	0.17 -0.40
	3	0.11 -0.55		0.031	0.23 -1.03
	4	0.10 -0.60		0.031	0.23 -0.94

ACKNOWLEDGEMENTS

We thank T. Martin and J. Trubuil for their contribution in designing the setup and for their work on the setup manufacturing.

APPENDIX: NUMERICAL METHOD TO OBTAIN THE JETTING VELOCITY

For a set 𝑣 *( , the jetting velocity is obtained by an iterative procedure. 𝑣 '( is varied with a 0.001 m.s -1 increment. For each 𝑣 '( value, the recession dynamics of a growing drop is computed by integrating system (4-6) using the fourth order Runge-Kutta method. The initial conditions are 𝐷 ' 0 = 0 and 𝑥 0 = 0. According to Eq. ( 5), 𝑥 0 is then given by Eq. (A1) for the uniform nozzle-sized thread limit and by Eq. (A2) for the maximal narrowing limit. with 𝜌 y = 𝜌 '( + 𝜌 *( 2. Integration is performed until the drop stops and changes direction. That point is denoted 𝑥 nop . The above iterative procedure is stopped as soon as 𝑥 nop = 𝑙 ' , the latter calculated from Eq. (7). At this point, the 𝑣 '( value corresponds to the jetting velocity.


Imad Ez-Zazi •
email: imad.ezzazi@gmail.com

Mounir Arioua
email: m.arioua@ieee.org

Ahmed El Oualkadi
email: ahmed.eloualkadi@ieee.org

Pascal Lorenz
email: lorenz@ieee.org

A El Oualkadi

Keywords: WSN, ECC, LDPC, RS, Multi-hop, MTE I. Ez-zazi

In this paper, we have proposed a hybrid adaptive coding and decoding scheme for multi-hop wireless sensor networks (WSNs). Energy consumption and transmission reliability are used as performance metrics for multi-hop communications in WSNs. The presented scheme takes into account distance, channel conditions and correction codes performance to decide coding and decoding procedure, and considers Reed Solomon (RS) code and Low Density Parity Check (LDPC) code to provide error protection on the transmitted data. The proposed approach aims to reduce the decoding power consumption and to prolong the lifetime of the network as well as improve the reliability of the transmission. Simulation results show that our performed scheme enhances both energy efficiency and communication reliability of multi-hop sensor networks.

A hybrid adaptive coding and decoding scheme for multi-hop wireless sensor networks

1 Introduction
Wireless sensor networks (WSNs) are becoming a popular research area in recent years. They are considered as one of the most promising technologies due to their unique characteristics, low cost, easy deployment and flexibility. WSNs are employed in many applications such as medical monitoring, military, security, environment control and industry [START_REF] Ahmad Fahmy | WSNs Applications. Wireless Sensor Networks[END_REF]. They are composed of numerous smaller, cheaper and intelligent sensor nodes which are capable of sensing, processing and communicating the gathered information to the base station [START_REF] Ammari | Centralized and clustered k-coverage protocols for wireless sensor networks[END_REF]. These sensor nodes are typically battery powered that can not be renewable for a long time [START_REF] Yadav | A review on energy efficient protocols in wireless sensor networks[END_REF]. Therefore, reducing energy consumption in order to prolong network lifetime is the most crucial design consideration in WSNs [START_REF] Kim | Low power routing and channel allocation of wireless video sensor networks using wireless link utilization[END_REF].

Communication reliability is another prime requirement in WSNs [START_REF] Lee | High reliable in-network data verification in wireless sensor networks[END_REF][START_REF] Mahmood | Reliability in wireless sensor networks: A survey and challenges ahead[END_REF]. The increase of transmission energy results in successful and reliable transmission. However, the rise of transmission energy profoundly affects the sensor nodes energy and the network lifetime. It is worth noting that designing optimal WSN design implies providing reliable communication with the minimum required energy consumption. Error control coding (ECC) is an efficient method used to improve link reliability and lower the required transmission energy [START_REF] Howard | Error control coding in low-power wireless sensor networks: When is ECC energy-efficient?[END_REF][START_REF] Elshabrawy | Network throughput analysis of IEEE 802.15.4 enabled wireless sensor networks with FEC coding under external interference[END_REF]. In low power application such as wireless sensor network, applying low power ECC techniques is primordial requirement [START_REF] Vuran | Error control in wireless sensor networks: a cross layer analysis[END_REF][START_REF] Alrajeh | Error Correcting Codes in Wireless Sensor Networks: An Energy Perspective[END_REF][START_REF] Salija | Optimum energy efficient error control techniques in wireless systems: a survey[END_REF]. Advanced coding schemes with complex decoding algorithms provide higher coding gain but require higher energy consumption. Thus, a trade-off between coding gain and decoder complexity should be considered in order to optimize the error correction schemes in WSN [START_REF] Abughalieh | Turbo codes for multihop wireless sensor networks with decode-and-forward mechanism[END_REF].

The reliability level provided by the link layer depends on the requirements of channel conditions for target applications [START_REF] Padilla | On the Influence of the Propagation Channel in the Performance of Energy-Efficient Geographic Routing Algorithms for Wireless Sensor Networks (WSN)[END_REF]. Furthermore, channel conditions and inter-node distances are different as the WSN deployment environment varies. In practice, operating over short distances in multi-hop wireless sensor network enables significant energy consumption in decoding than transmission energy [START_REF] Chouhan | Framework for energy consumption based design space exploration for wireless sensor nodes[END_REF]. Moreover, the effect of path loss, multipath and fading phenomenon is unimportant by the reason of shortness of internode distances. When inter-node distances are long, the radio energy usage is significantly larger than decoding energy consumption and the effects of the path loss, multipath and fading phenomenon are dominant which affect the reliability of transmission as well as the energy consumption. Therefore, the substantial compromise between the decoding process and channel condition constraints leads to the proposition of an optimized hybrid scheme as using the same error control scheme in the entire network is not always an adequate choice. In this paper, we propose an implementation of a hybrid adaptive coding and decoding scheme in multi-hop WSN with the advanced LDPC codes at long distance transmission and soft RS codes in short distance transmission for effective trade-off between performance and energy consumption. Different adaptive control schemes have been proposed for wireless sensor networks [START_REF] Kleinschmidt | An analytical model for energy efficiency of error control schemes in sensor networks[END_REF][START_REF] Kleinschmidt | An energy efficiency model for adaptive and custom error control schemes in Bluetooth sensor networks[END_REF]. Compared with existing optimized schemes, the proposed method is effectively suitable for short and large multi-hop networks when both free space and multipath propagation are presented. Our goal in this work is to reduce transmit and decoding energy usage as well as improve transmission reliability in multi-hop WSNs.

The rest of the paper is organized as follows. The error correction coding approach is discussed in section 2. In section 3, the radio propagation model and energy model for wireless sensor network are discussed. In section 4, the overall methodology and the proposed scheme are described. Simulations and results are presented in section 5 and finally, section 6 concludes the paper.

Error correction coding in WSN

Critical wireless sensor applications such as environment monitoring require the delivery of high priority events in a reliable and faster manner without any loss on the path from sensor nodes to the base station. However, the unstable nature of channels is the critical and unpredictable factor which disturbs the signal and causes data corruption [START_REF] Nithya | Energy Efficient Coded Communication for IEEE[END_REF]. All these constraints emphasize the need for reliability and robustness of the data transport and minimization of errors probability in spite of flawed channels [START_REF] Kiss | Improving transmission reliability in wireless sensor networks using network coding[END_REF]. Therefore, error correction coding is a prime solution for effective communications in WSN.

In error correction coding approach, errors are both detected and corrected at the receiver end. To enable the receiver to correct data, redundant information is combined with the original information at the transmitter. The capability of correcting errors induced by flawed channels in the received sequence is measured by providing better bit error rate (BER) performance for the same signal-to-noise ratio (SNR) compared to un-coded scheme or by providing the same BER at lower SNR than un-coded scheme [START_REF] Howard | Error control coding in low-power wireless sensor networks: When is ECC energy-efficient?[END_REF]. The coding gain is the difference in required SNR to achieve a certain BER for a given coding scheme compared to un-coded scheme. Efficient coding schemes characterized by higher coding gain and less transmission energy.

Several error correction codes have been investigated for WSNs, including BCH codes, Reed Solomon (RS) codes, convolutional codes, Turbo codes and Low Density Parity Check (LDPC) codes [START_REF] Pellenz | Error control coding in wireless sensor networks[END_REF][START_REF] Biroli | An LDPC Decoder Architecture for Wireless Sensor Network Applications[END_REF][START_REF] Li | A Low-Complexity Turbo Decoder Architecture for Energy-Efficient Wireless Sensor Networks[END_REF][START_REF] Soude | Towards an optimal Reed Solomon codes selection for sensor networks[END_REF]. RS codes have been proven suitable and energy efficient for wireless sensor network applications with simple and reasonable coding and decoding complexity [START_REF] Ez-Zazi | Performance analysis of efficient coding schemes for wireless sensor networks[END_REF]. Results presented in [START_REF] Chouhan | Integrated energy analysis of error correcting codes and modulation for energy efficient wireless sensor nodes[END_REF] show that using RS codes with short codeword length and BPSK modulation scheme in WSN saves about 50% of energy. LDPC codes have been proved to own better performance and several advantages over other block and convolutional codes [START_REF] Richardson | Efficient encoding of low-density parity-check codes[END_REF]. They are the most performant coding schemes and competitors of Turbo codes in terms of performance and complexity. Experimental results in [START_REF] Duc | On efficient design of LDPC decoders for wireless sensor networks[END_REF][START_REF] Choi | Implementation of Energy Efficient LDPC Code for Wireless Sensor Node[END_REF] exhibit that LDPC codes are good candidates for WSN applications. However, LDPC codes require complex decoders with large hardware and energy consumption [START_REF] Biroli | An LDPC Decoder Architecture for Wireless Sensor Network Applications[END_REF]. For this reason, authors of [START_REF] Duc | On efficient design of LDPC decoders for wireless sensor networks[END_REF] proposed the use of short-length LDPC codes in WSN in order to mitigate the LDPC decoder effect and provide acceptable BER. An adaptive iterative decoding (AID) approach exploiting a threshold on the number of iterations of decoding algorithm for a certain BER is proposed in [START_REF] Javaid | AID: An Energy Efficient Decoding Scheme for LDPC Codes in Wireless Body Area Sensor Networks[END_REF]. This approach with iterative LDPC decoding lowered the network energy consumption by 20-25%. The extra power consumption of LDPC decoders limits the use of these codes in multi-hop WSN which lead authors of [START_REF] Schmidt | Error correction in single-hop wireless sensor networks -a case study[END_REF][START_REF] Zhong | Does proper coding make single hop wireless sensor networks reality: the power consumption perspective[END_REF] to suggest the use of efficient single-hop sensor network instead of multi-hop approach with powerful coding schemes. This approach considers the decoding only at the base station in or-der to save energy and prolong the network lifetime. Moreover, several works have been interested in the optimization and use of strong coding techniques in multi-hop network design. For instance, a low power consumption decodeand-forward multi-hop approach for WSN is proposed in [START_REF] Abughalieh | Turbo codes for multihop wireless sensor networks with decode-and-forward mechanism[END_REF], where a Viterbi decoding of convolutional codes is implemented in intermediate nodes while the iterative decoding algorithm of Turbo codes is applied at the base station.

In this paper, we investigate the use of advanced coding schemes in multihop WSN where each node encodes and decodes data at the transmission and reception respectively. Both the strong LDPC codes and soft RS codes are adaptively considered based on the channel conditions and inter-node distance.

2.1 LDPC coding scheme LDPC codes are linear error correcting block codes described by a sparse (M × N ) parity-check matrix H [START_REF] Gallager | Low-density parity-check codes[END_REF]. The matrix H can be represented as a bipartite graph known as Tanner Graph [START_REF] Tanner | A recursive approach to low complexity codes[END_REF], with M check nodes and N variables nodes. Ones in the parity-check matrix H are presented by edges in Tanner graph and most of decoding algorithms require the exchange of information along the edges of the graph. Belief Propagation (BP) algorithm is the most algorithm used to decode LDPC codes due to its performance [START_REF] Mackay | Good error-correcting codes based on very sparse matrices[END_REF]. The BP algorithm emphasizes that variable nodes update the received information from the channel according to the parity check equations evaluated at the check nodes. This process is repeated several times until either the maximum number of iterations is reached. The decoding complexity associated with BP algorithm is evaluated based on the number of real multiplications and additions processes in each iteration [START_REF] Choi | Performance Analysis on Wireless Sensor Network using LDPC Codes over Node-to-node Interference[END_REF][START_REF] Sankarasubramaniam | Energy efficiency based packet size optimization in wireless sensor networks[END_REF].

E LDP C = n k=1 (3N.i + 6t.N.j -10N )E mult + (3N.j + N )E add ( 1
)
Where t is the correction capacity, N is the code word length, i and j are weight of row and column of parity check matrix for LDPC code, n is the number of optimal iterations, E add and E mult denote the energy consumption per m-bit in the addition and multiplication, respectively, of field elements in Galoi Field (GF (2 m )). These energy values are computed in [START_REF] Sankarasubramaniam | Energy efficiency based packet size optimization in wireless sensor networks[END_REF] for 0.18 µm, 2.5 V CMOS based implementation.

RS coding scheme

RS codes are efficient error correction schemes specified by their relevant performance for reasonable decoding complexity. RS code is composed of N symbols with a number of message symbols k and redundant symbols (N -k). It is characterized by a minimum distance of d = N -k+1 and an error correction capacity of t = (N -k)/2. A relevant decoding technique for binary RS codes is based on the Berlekamp-Massey (BM) and Chien's search (CS) which present a linear dependence on block length N . The computation energy of RS codes based on BM and CS algorithms is expressed in [START_REF] Sankarasubramaniam | Energy efficiency based packet size optimization in wireless sensor networks[END_REF] as:

E RS = (2t.N + 2t 2 )(E mult + E add ) (2)
The complexity of the decoding increases with increasing the codeword length N which require important amount of resources [START_REF] Duc | On efficient design of LDPC decoders for wireless sensor networks[END_REF]. Moreover, the amount of sensed and exchanged data by sensor nodes in WSN applications is typically limited. In this context, we considered short block length LDPC and RS codes for error correction in WSN.

Radio propagation & Energy model

Radio propagation model

Many radio propagation models known for wireless communications predict the signal-strength loss with distance-path loss. The free space and multipath propagation models are widely used for wireless sensor networks [START_REF] Giacomin | Radio Channel Model of Wireless Sensor Networks Operating in 2.4 GHz ISM Band[END_REF]. In the free space model, communications between the transmitter and the receiver are clear line-of-sight in which reflecting surfaces and effect of the earth surface are solely absent. The multipath propagation is as result of different signal paths between the transmitter and the receiver. These paths are the major detriment to signal propagation by giving rise to interference, distortion of the signal, loss of data and multipath fading.

For both free space and multipath fading, the received power decreases as the distance between the transmitter and receiver increases [START_REF] Rappaport | Wireless communications: Principles and practice[END_REF]. For this work, both the free space model and the multipath fading model were used, depending on the distance between the transmitter and receiver. If the inter-node distance (distance between transmitter and receiver) is less than a crossover distance (d crossover ), the Friss free space model is used (d 2 attenuation). Otherwise (d > d crossover ), the two-ray ground propagation model is adopted (d 4 attenuation). The crossover distance for Friss and two-ray-ground attenuation models is defined in [START_REF] Heinzelman | Application-Specific Protocol Architectures for Wireless networks[END_REF] as:

d crossover = 4π √ Lh r h t λ (3)
Where L ≥ 1 is the system loss factor not related to propagation, h r and h t are the heights of the receiving and transmitting antennas above the ground respectively, and λ is the transmitted wavelength corresponding to the transmitting frequency.

The transmit power is attenuated according to the Friss free space equation for distance bellow d crossover as follow:

P r (d) = P t G t G r λ 2 (4πd) 2 L ( 4
)
Where P r (d) is the receive power for a transmitter-receiver separation of inter-node distance d, P t is the transmit power, G t and G r are the gains of the transmitting and receiving antennas respectively and d is the distance between the transmitter and receiver.

This attenuation is only considered when the transmitter and the receiver have direct line of sight communication, which occur when the inter-node distance is short (d < d crossover ). For all distances greater than d crossover , the transmit power is attenuated according to the two-ray ground propagation equation as follows:

P r (d) = P t G t G r h 2 t h 2 r d 4 L (5)
In this work, we adopted the experiments described in [START_REF] Heinzelman | Application-Specific Protocol Architectures for Wireless networks[END_REF] by using an omnidirectional antenna with the following parameters: G t = G r = 1, h t = h r = 1 m and no system loss (L=1). For low power WSN, we adopt the IEEE 802.15.4 standard with an operating frequency of 2.4 GHz and λ equals to 0.124 m [START_REF] Li | An Energy-Efficient Error Correction Scheme for IEEE 802.15.4 Wireless Sensor Networks[END_REF]. Substituting these parameter values into Equation 3, the obtained crossover distance is the following:

d crossover = 101 m 3.

Radio energy model

The first order radio energy model is adopted for our studies [START_REF] Heinzelman | Application-Specific Protocol Architectures for Wireless networks[END_REF][START_REF] Heinzelman | An application specific protocol architecture for wireless microsensor networks[END_REF][START_REF] Lu | Low-complexity and energy efficient image compression scheme for wireless sensor networks[END_REF]. This model considers the energy consumed at the transmitter and the receiver to send and receive an L-bits message over a distance d (Figure 1). In sensor nodes, the transmitter dissipates energy to run the radio electronics E T x-elec and the amplifier energy E T x-amp and the receiver dissipates energy to run the radio electronics. Thus, the energy expended to transmit an L-bits message over a distance d is described as:

E T x (L, d) = E T x-elec (L) + E T x-amp (L, d) (6)
Considering free space and multipath propagation, the equation 6 is formulated as:

E T x (L, d) = E elec .L + f riss-amp .L.d 2 if d < d crossover E elec .L + two-ray-amp .L.d 4 if d ≥ d crossover (7)
When receiving data, the energy expended is expressed as:

E Rx (L) = E elec .L (8)
Where E elec corresponds to the energy per bit required in transmit and receive electronics to process the information.

The parameters f riss-amp and two-ray-amp depend on the required received sensitivity and the receiver noise figure (RNF), as the transmit power Fig. 1 First order radio energy model with decoder bloc has to be adapted so that the power at the receiver is above a certain threshold P r-thresh . The minimum transmit power is determined based on this receive power threshold. If the radio bit rate is R, the transmit power is equal to the transmit energy per bit E T x-amp (1, d) multiplied by the bit rate R.

P t = E T x-amp (1, d).R (9)
Substituting in the value of E T x-amp (1, d) gives:

P t = f riss-amp .R.d 2 if d < d crossover T wo-ray-amp .R.d 4 if d ≥ d crossover (10)
Using Equations 4 and 5, the received power is expressed as:

P r = f riss-amp .R.Gt.Gr.λ 2 (4π) 2 if d < d crossover T wo-ray-amp .R.G t .G r .h 2 t .h 2 r if d ≥ d crossover (11)
The parameters F riss-amp and T wo-ray-amp are determined by setting equation 11 equals to P r-thresh :

F riss-amp = P r-thresh .(4π) 2 R.G t G r λ 2 (12)
T wo-ray-amp =

P r-thresh R.G t G r h 2 t h 2 r ( 13
)
The required transmit power P t as a function of the receiver threshold and the distance between the transmitter and receiver is determined by substituting equations 12 and 13 in equation 10 :

P t = (4π) 2 GtGrλ 2 .P r-thresh .d 2 if d < d crossover 1 GtGrh 2 t h 2 r .P r-thresh .d 4 if d ≥ d crossover (14)
The receiver threshold P r-thresh is determined by estimating the noise at the receiver. Considering the thermal noise floor equals to 99 dBm, the receiver noise figure of 17 dB and the required signal-to-noise ratio (SNR) of at least 30 dB to receive the signal with no errors [START_REF] Heinzelman | Application-Specific Protocol Architectures for Wireless networks[END_REF], the minimum receive power P r-thresh for successful reception is determined as:

P r-thresh ≥ 30 + (-82) = -52 dBm (15)
Accordingly, the received power must be at least -52 dBm or 6.3 nW for successful reception of the packet [START_REF] Heinzelman | Application-Specific Protocol Architectures for Wireless networks[END_REF]. Considering the experiment parameters used in this work (G t = G r = 1, h t = h r = 1 m, λ = 0.124), and as the IEEE 802.15.4 standard target a throughput (R = 250 kbps), the equations 12 and 13 are expressed as:

F riss-amp = 2.59 * 10 -10 J/bit/m 2 T wo-ray-amp = 2.52 * 10 -14 J/bit/m 4 (16)

Energy model for multi-hop WSN

The energy model for multi-hop wireless sensor networks using error correction codes takes into account decoding energy for all nodes except source node. Energy consumed in encoding is considered for all nodes except the last node. However, encoding in block codes is typically a simple process with insignificant computational energy [START_REF] Vuran | Error control in wireless sensor networks: a cross layer analysis[END_REF]. Thus, the encoding energy is not considered in the energy model.

The total energy required to transmit, receive and decode information is given as:

E T ot = m 1 E T x + m-1 1 E Rx + m-1 1 E Rx-Dec (17)
Where E T ot is the total energy consumed in the multi-hop network, E T x is the energy consumed in transmission by all nodes, E Rx is the energy consumed in receiving information by all nodes except source node, E Rx-Dec is the energy consumed in decoding process by all nodes except source node, m is the number of hopes and L is the packet size in bits.

In Minimum Transmission Energy (MTE) algorithm, nodes route data destined for the base station through intermediate nodes. These nodes act as a routers of other nodes data in addition to sensing the environment. Each data message must go through m transmits a distance r and m-1 receives. Thus, the total radio energy is expressed as:

E M T E = m.E T x (L, d = r) + (m -1)E Rx (L) (18)
For free space propagation:

E M T E = m(E elec .L + f riss-amp .L.r 2 ) + (m -1)E elec .L = L((2m -1)E elec + f riss-amp .m.r 2 ) (19)
For multipath propagation:

E M T E = m(E elec .L + two-ray-amp .L.r 4 ) + (m -1)E elec .L = L((2m -1)E elec + two-ray-amp .m.r 4 ) (20)
4 Proposed scheme

The main goal of the proposed scheme is to prolong energy lifetime as well as improve the reliability of communications in multi-hop wireless sensor networks. This is achieved by protecting data transmission from noise, avoiding extra transmissions, interferences, multipath fading and packet loss. Selection of the adequate error control coding usually depends on the distance and the transmission environment [START_REF] Howard | Error control coding in low-power wireless sensor networks: When is ECC energy-efficient?[END_REF]. Noisy environment with fading, shadowing and multipath typically requires more efficient control coding with strong decoding scheme to retrieve all data correctly. LDPC codes are the most efficient codes in the area used for reliable communication with BER performance closed to the Shannon limit [START_REF] Duc | On efficient design of LDPC decoders for wireless sensor networks[END_REF][START_REF] Roberts | An Improved Low Complex Hybrid Weighted Bit-Flipping Algorithm for LDPC Codes[END_REF]. However, these codes employ a complex iterative decoding in the receiver which raises the energy consumption. The suggested scheme is a hybrid adaptive algorithm aims to optimize the decoding usage in multi-hop network. In fact, it benefits from the efficiency and potency of LDPC code in multipath environment. While, in reliable free space environment it employs an efficient coding scheme as RS code with simple decoding algorithm. Through this combination, we aspire to minimize the energy computation consumed by decoders and ensure substantial reliability and flexibility. We employed MTE routing algorithm to minimize the inter-nodes propagation distance in order to frequently provide free space routes than multipath propagation. Each node in the considered network sends their information with MTE routing by using the nearest neighbor in the optimum path toward the base station. A simple network of multi-hop communication is shown in Figure 2, in which n nodes are separated a distance r apart from each other.

Inter-node distances depend on different applications and vary from few to hundreds of meters. For instance, in a WSN deployed for glacial environment monitoring [START_REF] Martinez | Glacial environment monitoring using sensor networks[END_REF], nodes were kept 20-25 m apart, whereas for volcano monitoring [START_REF] Werner-Allen | Deploying a wireless sensor network on an active volcano[END_REF], nodes were deployed 200-400 m apart. Nodes typically start with an important energy and keep near each other few meters apart and most of communications are successfully performed. After a certain time, nodes begin to die due to loss of energy and inter-node distances become longer which cause detriment and loss of communications.

The realized scheme takes into account inter-node distances and channel conditions to decide adequate error correction code. RS and LDPC codes are jointly applied to provide error protection on the transmitted data. RS code is operational when inter-node communication occurred over a distance d lower than the crossover distance. On the other hand, LDPC code is integrated and used when inter-node distance values exceed d crossover . Therefore, RS coding/decoding scheme is exclusively adopted in direct propagation and LDPC coding/decoding scheme when the multipath propagation model is selected. Figure 3 presents the mechanism of the implemented coding and decoding scheme. Using equations ( 1), ( 2), ( 19) and ( 20) of radio and decoder energy models, we obtain the total energy of the system for both RS and LDPC schemes as follow:

E T ot = E T ot-RS + E T ot-LDP C (21)
Where:

E T ot-RS = L((2m -1)E elec + f riss-amp .m.r 2 ) +(2t.N + 2t 2 )(E mult + E add ) (22)
And

E T ot-LDP C = L((2m -1)E elec + two-ray-amp .m.r 4 ) + n k=1 (3N.i + 6t.N.j -10N )E mult + (3N.j + N )E add (23)
Nodes adaptively decide the adequate error correction code to use according to distance and channel conditions. At the sender, nodes calculate and choose the optimum distance to send their data using MTE algorithm. At the receiver, nodes have to estimate the distance from the sender in order to decide which decoding algorithm has to be used. Received signal strength indicator (RSSI) is a measurement of the power presented in the received radio signal [START_REF] Benkic | Using RSSI value for distance estimation in wireless sensor networks based on ZigBee[END_REF][START_REF] Miroslav Botta | Adaptive Distance Estimation Based on RSSI in 802.15.4[END_REF]. When data packet is received, the value of RSSI feedback is used to estimate the distance from the transmitter as well as the path-loss phenomenon. Based on RSSI indicator, nodes decide which decoding scheme will be employed. The choice of the adequate coding and decoding method takes into account two crucial parameters: reliability of transmission and energy efficiency.

Simulation and results

MATLAB simulator tool is used to analyze the impact of the hybrid adaptive coding and decoding scheme on the energy of the entire network. For all simulations referred in our work, we followed the energy consumption models and parameters of reference [START_REF] Heinzelman | Application-Specific Protocol Architectures for Wireless networks[END_REF][START_REF] Heinzelman | An application specific protocol architecture for wireless microsensor networks[END_REF] and we used the values for the energy consumption related parameters as shown in Table 1. A packet size of L = 2000 bits is used, 50 sensors are randomly deployed in a network dimensions of 50m x 50m and 200m x 200m. The base station is placed at (x =25 m, y=-100 m) and (x =100 m, y=-100 m), respectively. All nodes begin with a starting energy level of 1 J . This value is commonly used in the literature since it provides small enough energy to quickly see the effect of the varying algorithms. In our simulations, MTE algorithm does not employ data aggregation strategy due to the fact that at each round every node is assumed to transmit its message in a TDMA scheme where there is only one message passing from source node to base station through the network at a time.

Concept of transmit power is used with respect of distance, less distance requires less transmit power while large distance requires greater transmit power. The total system energy versus transmission round in 50 m 2 and 200 m 2 network is shown in Figures 4 and5, respectively. Obtained simulations display the effect of distance and decoder complexity on total energy system and nodes lifetime. Increasing the inter-node distances, the total energy of the network is hastily consumed and most of nodes die quickly. In 50 m 2 sensor network, the simulation shows that the energy of the network with the un-coded scheme is completely consumed in 1514 rounds. The energy of the network implementing LDPC coding scheme is fully depleted in 1136 rounds which correspond to lifetime decrease of 25% when compared to the un-coded scheme. The energy of the network with RS coding scheme is completely dissipated in 1357 rounds which correspond to lifetime decrease of 10%. As expected, system with RS decoding is more energetically efficient than LDPC decoding scheme. Using LDPC decoding in the entire network intensely consumes and drains nodes energy. By using the proposed coding scheme, the energy of sensor network is wholly consumed in 1315 rounds which correspond to lifetime increase of 15% compared to LDPC coded scheme. In 200 m 2 sensor network with uncoded scheme, the total energy is completely consumed in 272 rounds. For LDPC coded scheme the energy is totally depleted in 213 rounds which correspond to lifetime decrease of 27 % when compared to uncoded scheme. For the RS coded scheme, the energy is dissipated in 244 rounds which corresponds to lifetime decrease of 10 % when compared to uncoded scheme. Apparently, by adopting the proposed coding scheme, the energy of sensor network is wholly consumed in 237 rounds which correspond to lifetime increase of 11% compared to LDPC coded scheme. Figures 6 and7 show the number of nodes alive in the case of unocoded scheme, RS coded scheme, LDPC coded scheme and proposed hybrid adaptive scheme in 50 m 2 and 200 m 2 network dimensions. Table 2 summarizes the round when 10% , 50% and 90% of nodes are dead in the networks. In 50

Round

Total System Energy (J) Uncoded scheme Hybrid adaptive scheme RS coding scheme LDPC coding scheme Fig. 5 Total system energy over 200 m 2 sensor network m 2 sensor network, when the WSN system transmits LDPC coded data, the percentages of 10%, 50% and 90% of dead nodes occur at 98 rounds, 282 rounds and 580 rounds, respectively. Using the proposed hybrid adaptive scheme, the percentages of 10%, 50% and 90% of dead nodes occur at 108 rounds, 395 rounds and 748 rounds, respectively. This corresponds to lifetime increase of 10%, 40% and 28%, respectively when compared to LDPC coded scheme. the proposed coding scheme outperformed LDPC coding scheme by optimizing the nodes lifetime of 26 % approximately. In 200 m 2 sensor network with LDPC coding scheme, the percentages of 10%, 50% and 90% of dead nodes occur at 68 rounds, 136 rounds and 181 rounds, respectively. In this case, the proposed coding scheme improves the nodes lifetime of 16%, 14%, 13%, respectively when compared to LDPC coded scheme. Accordingly, the proposed hybrid adaptive coding scheme outclasses the LDPC coding scheme in both 50 m 2 and 200 m 2 .

Adopting the proposed hybrid adaptive scheme efficiently saves the energy batteries of sensor nodes and extends the network lifetime by saving the energy of nodes consumed in the decoding process. In addition, the proposed coding framework improves the communication reliability by adaptively using the RS scheme in free space propagation and the LDPC scheme in multipath propagation. The aggregated data transmitted from nodes are decoded using the BP decoding of LDPC code in the base station which result in effective decoding and retrieving of the sensed data. In this paper, we have proposed a hybrid adaptive coding and decoding scheme based on inter-node distances and channel conditions to improve network lifetime and multi-hop transmission reliability. Extensive simulations have been carried out to demonstrate the advantages of the proposed method based on RS and LDPC codes. The results have shown that the new adaptive approach improves both energy consumption and communication reliability in the multihop wireless sensor networks.

This hybrid adaptive scheme can be adopted for different types of networks. The evolution and enhancement of the presented scheme should be done in the future to optimize energy consumption and transmission reliability in a clustered wireless sensor networks.

Fig. 2

 2 Fig. 2 Muti-hop network of n nodes

Fig. 3

 3 Fig. 3 Proposed error correction scheme for sender and receiver node

Fig. 4

 4 Fig.4Total system energy over 50 m 2 sensor network

Fig. 6

 6 Fig. 6 Number of nodes alive over 50 m 2 sensor network

Fig. 7

 7 Fig. 7 Number of nodes alive over 200 m 2 sensor network

  Imad EZ-ZAZI received engineer degree in Telecommunications and Networks from National School of Applied Sciences of Tetuan in 2014, Abdelmalek Essadi University. He is currently Ph.D student at the Department of Information and Communication Technologies at National School of Applied Sciences of Tangier, Abdelmalek Essaadi University, Morocco. His research interests include wireless sensor networks, channel and network coding in wireless communications, routing algorithms, modulation and signal processing. Mounir ARIOUA received his Ph.D degree in Telecommunications and Computer Science from Cadi Ayyad University of Marrakech in 2012. In 2012, he joined the National School of Applied Sciences of Marrakech as research assistant and assistant professor. Currently, he is professor at National School of Applied Sciences in the department of Engineering Technologies, University of Abdelmalek Essaadi. His research interests include wireless sensor networks, Internet of things, wireless networking and communication, wireless communications and mobile computing, real-time processing and embedded systems. Ahmed EL OUALKADI received Ph.D degree from Poitiers University, France, in 2004. From 2000 to 2003, he was a research assistant at the National Higher Engineering School of Poitiers. In 2004, he was an assistant professor at University Institute of Technology, Angoulme. In 2005, he joined the Catholic University of Louvain, Belgium. Currently, he is an associate professor at national school of applied sciences of Tangier, Abdelmalek Essaadi University. His main research interest is RFIC design for wireless communication, wireless sensor networks, embedded system applications and information technology. Pascal LORENZ received a Ph.D degree from the University of Nancy, France. Between 1990 and 1995 he was a research engineer at WorldFIP Europe and at Alcatel-Alsthom. He is a professor at the University of Haute-Alsace and responsible of the Network and Telecommunication Research Group. His research interests include wireless sensor networks, QoS, wireless networks and high-speed networks. He is the author/co-author of 3 books, 3 patents and 200 international publications in refereed journals and conferences.

Table 1

 1 Simulation parametersParameter ValueTx Amplifier, free space propagation, f s 2.59*10 -10 J/bit/m 2 Tx Amplifier, multipath propagation, mp 2.52*10 -14 J/bit/m 4

	Tx and Rx energy electronics E elec	50 nJ/bit
	Initial node energy (Joules)	1 J
	Number of nodes	50
	Frequency	2.45 GHz
	Bit rate	250 kbps
	Antennas gain	1 dB
	high of antennas	1 m
	dcrossover	101 m
	Packet size	2000 bits
	Addition energy E add	3.3*10 -5 m (mW/MHz)
	Multiplication energy E mult	3.7*10 -5 m 3 (mW/MHz)

Table 2

 2 Overall algorithm die-off statistics of LDPC and proposed schemes in 50 m 2 and 200 m 2 networks

	Coding scheme	LDPC scheme	Proposed scheme	% increase
	Network dimension	50 m 2	200 m 2	50 m 2	200 m 2	50 m 2	200 m 2
	10% Nodes Dead	98	68	108	79	10	16
	50% Nodes Dead	282	136	395	156	40	14
	90% Nodes Dead	580	181	748	205	28

Number of Nodes Alive Uncoded scheme Hybrid adaptive scheme RS coding sheme LDPC coding scheme