3D Scanning Radar for the Remote Reading of Passive Electromagnetic Sensors

Keywords:

This communication reports for the first time the remote reading of passive electromagnetic sensors using a 24GHz Frequency Modulated Continuous Wave (FMCW) scanning radar. It shows that the detection, identification and reading of such sensors are possible by the analysis of 3D radar images.

I. INTRODUCTION

Passive (battery-less) ElectroMagnetic (EM) sensors are very good candidates for measuring physical quantities in harsh environment (e.g., high radiation or extreme temperature) and/or for applications requiring sensing devices with low-cost of fabrication, small size and long-term measurement stability. Such EM sensing devices convert the variation of a physical quantity (such as, e.g., pressure, temperature or gas concentration) into a known/specific variation of a given EM wave descriptor (see, e.g., [1][2]). The wireless measurement of a physical quantity from the analysis of the Radar Cross Section (RCS) variability of such passive sensors using a millimeter-wave (mm-wave) Frequency-Modulated Continuous-Wave (FMCW) radar was proposed for the first time in 2008 [3] while the proof-ofconcept was demonstrated in 2010 [4]. Compared with lower frequencies, mm-waves improve the sensor immunity to objects located at its vicinity by increasing the electrical length separation distance to them. Moreover they allow reducing the sensor and antenna sizes and designing high gain antennas for beamforming, multi-beam or beam-steering radar. Since 2010, this wireless and long-range (>30 meters) sensing technique based on RCS-variability measurement has been successfully applied to the remote estimation of many physical quantities (see a very recent overview in [5]). Up to now mm-wave FMCW radar readers were used for the remote reading of passive sensors occupying known positions and consequently, the beam of the radar transmitting antenna could be pointed in the direction of the sensor for maximizing the measured electromagnetic echo level. Moreover a slight deviation from the sensor direction of this narrow beam causes a significant reduction of the measured echo level. Consequently, it is necessary in practice to accurately adjust the beam direction. To overcome these limitations and timeconsuming adjustment we propose here to perform the automatized 3D radar scanning for detecting, identifying and remotely reading passive electromagnetic sensors.

For the first time a 3D imaging FMCW radar with mechanical beam scanning is proposed in this communication for localization, identification and interrogation of passive electromagnetic sensors.

II. BEAM SCANNING FOR DETECTING AND REMOTE READING PASSIVE ELECTROMAGNETIC SENSORS

The sensor reading system used here is a Frequency-Modulated Continuous-Wave (FMCW) radar (central frequency: 24 GHz; bandwidth or excursion frequency: Δf=2GHz) with a transmitting parabolic antenna which offers high gain (34.3 dBi) and narrow beamwidth (Δθ=1 degree in azimuth and Δφ=1 degree in elevation). The transmitted signal (power PT=30dBm) or chirp has a linear sawtooth variation of frequency with time (modulation period TR=18ms). For such modulation the frequency is tuned linearly as a function of time by using a Voltage-Controlled Oscillator (VCO). The signal backscattered by sensors is received by a 1x4 patch array antenna (beamwidth of the main lobe: 80 degrees in azimuth and 24 degrees in elevation). The received signal is mixed with the front-end transmitted signal and filtered for obtaining I-and Q-channels of the Difference Frequency Signal (DFS). A Fast Fourier Transform with a 2 blocks zeropadding of 512 samples is finally applied to the digitalized DFS (1024 samples per ramp) for deriving the beat frequency spectrum of I-and Q-channels.

Fig. 1 shows the whole system mounted on the experimental setup. The beam steering is performed mechanically. A computer is used to control the movement of the transmitting antenna (and consequently to monitor the antenna beam) and to perform the data acquisition, the signal processing and radar echoes visualization.

The sensor reading system is used to perform the automatized 3D radar scanning for detecting, identifying and remotely reading passive electromagnetic sensors. A sensor is here passive scatterer composed of an antenna (horn antenna; gain: 20dBi) connected to a length of TEM transmission line (a 50-Ω coaxial cable) which is in turn connected to a sensing device, that is, a resistance which depends on the physical quantity of interest. The length of the transmission line is used here for separating in the beat frequency spectrum the structural scattering mode and the sensing mode (or antenna scattering mode). Only the radar echo associated to the sensing mode depends on the physical quantity of interest. The structural scattering mode is used here for detecting the sensors.

As shown in Fig. 2 four sensors are located on a 2x2 array. The physical length DCABLE of transmission lines used by these sensors are: 20cm, 50cm, 75cm and 100cm. With a velocity factor VF of 0.83, these lengths result respectively in D.Henry, P.Pons and H.Aubert (CNRS-LAAS, Toulouse University, France) the following electrical lengths: 24cm, 60cm, 90cm and 120 cm. The distance D between the sensors and the radar is set to 3 meters while the distance DMEAS provided by the radar is giving by:

        + + = D V D D D F CABLE ANT MEAS 2 2 2 1 (1)
where DANT=30cm denotes the distance between the Vivaldi horn antenna and the parabolic reflector (see Fig. 1).

According to (1), the expected value of DMEAS is 3.75m.

f c d ∆ = 2 (2)
where c designates the celerity of light in the propagation channel (here an empty corridor of dimensions 1m60 x 2m50 x 20m). In our experimental setup the theoretical depth resolution is then 7.5cm. Angular resolutions in azimuth and elevation are given by the main lobe beamwidth of the transmitting antenna, that is, Δθ=1 degree and Δφ=1 degree.

As displayed in Fig. 3, a resolution cell of volume V at the distance D can then be defined by the following relationship: Scanning sectors of 12 degrees in azimuth (with 0.5 degree step) and of 13 degrees in elevation (with a 0.72 degree step) are considered. The resistance RL of the sensing device which loads the sensor transmission line is here 50Ω (match load) or 0Ω (short-circuit). These two values are viewed as the maximum and the minimum resistances taken by the sensing device as the physical quantity of interest varies. For one radar echo acquisition in a given direction in the chosen 3D scanning region, 160 points are recorded which leads to a total of 72000 data points for the entire region.

) 2 sin( 2 . . 2 2 3 1 ) ( 3 3 ϕ θ ∆ ∆               - -       + = d D d D D V
Fig. 4 shows the cut-plane of the resulting radar image for φ=12.9 degrees and for RL=0Ω (the measured echo level is between -40dB to -15dB). Radar echoes from sensors #3 and #4 can be detected. As expected, the structural scattering mode is found between 3.82m and 4.1m (Note that this mode can be used for identifying the sensor though a specific EM barcode). The sensing modes are apparent when RL=0Ω. The measured distances between echoes gives 90cm and 128cm, with a ±10 cm error due to the spreading of the echo spot associated to the structural scattering modes. These distances corresponds to the lengths of line used by the sensors #3 and #4, respectively, and could then be used for identifying the two sensors. If sensors are far enough from each other to be distinguished by the radar (a minimum resolution of 5cm at D=3.0m), the distance between these two sensors can be estimated. From elementary geometrical considerations, a

Parabolic reflector

Vivaldi horn antenna Patch array antenna Linear actuator (elevation) distance of 24 cm is derived from such measurement with a precision of ±10cm.

A 3D visualization of the radar image can be advantageously used for detecting the echoes due to the sensing modes. In Fig. 5 this visualization is shown for measured echo levels higher than -20dB. The sensing modes are apparent when RL=0Ω (see Fig. 5(b)) and disappear, as expected, when the sensor antenna is matched (RL=50Ω, see Fig. 5(a)). From the measurement of the number N of resolution cells for which the echo level P is higher than an arbitrary value PMIN, the echo rate τ of a scanning 3D region occupying by one sensor is defined by:

T MIN MIN N P P N P ) ( ) ( > = τ (5)
where NT denotes the total number of resolution cells in the scanning region. The echo rate takes its highest value when PMIN is close to the noise level. The contribution of the sensing mode to this echo rate may then be estimated from the socalled echo rate difference Δτ of a sensor defined by:

) ( ) ( ) ( 1 2 MIN MIN MIN P P P τ τ τ - = ∆ (6)
where τ1 and τ2 denote the echo rates when RL=0Ω and RL=50Ω, respectively. When the resistance RL of the sensing device changes due to the variation of the physical quantity of interest, the Δτ changes accordingly. The value of RL may then be directly derived from the determination Δτ (note that Δτ is minimal when the resistance RL matches the sensor antenna while it is maximal when RL=0Ω or when RL is infinite). For the sake of illustration Fig. 6 displays the measured echo rate difference of sensor #2 as a function of the chosen threshold echo level PMIN (θ sweeps from -3 to 0 degree and, φ from 0 to 6.48 degrees). Δτ reaches its highest value (13%) for PMIN=-40dB and the full-scale dynamic range is reached when the sensing device resistance is shortcircuited. When the resistance RL increases, the echo rate difference peak decreases. When RL=50Ω the echo rate difference is 0.

Fig. 1 .

 1 Fig. 1. Transmitting parabolic antenna of the FMCW Radar using a mechanical beam steering

  Fig. 3. Definition of a resolution cell of volume V at a distance D

Fig. 4 .

 4 Fig. 4. Radar image in the cut-plane φ=12.9 degrees of a given scanning sector (RL=0Ω).

Fig. 5 .

 5 Fig. 5. 3D radar image of passive electromagnetic sensors for a minimum echo level of -20 dB: (a) RL=50Ω and (b) RL=0Ω.


Hortense Lauwick

Yang Sun

Huriye Akdas-Kilig

Sylvie Dérien
email: sylvie.derien@univ-rennes1.fr

Mathieu Achard
email: mathieu.achard@univ-rennes1.fr

Access to 3-oxindoles from Allylic Alcohols and Indoles

The site and regioselective allylation reaction of 2substituted indoles was carried out by using a ruthenium(IV) precatalyst containing phosphine-sulfonate chelate. Mono-, di-and tri-allylated indoles could be selectively obtained according to reaction conditions and gave only water as byproduct. The preparation of 3-oxindole derivatives was then successfully performed owing to the oxidation under air of the corresponding allylated indoles. Diallylated pseudoindoxyls were proved to be good synthons to perform cyclization through Ring Closing Metathesis affording the corresponding tricyclic adducts. Photophysical properties of 3-oxindoles have been measured, some compounds showing a strong fluorescence in water.

Introduction

Pseudoindoxyl derivatives containing the indolin-3-one core structure and a quaternary stereogenic center at the 2-position is an attractive class of compounds with highly interesting photophysical or biological properties (Figure 1). [1,2] They are found in various biomolecules such as the spirocyclic Aristotelone, [1a] Brevianamides, [1b] Mytragynine pseudoindoxyl [1c,d] as well as in the Isatisine A, [1e] Duocarmycin A,C [1f,g] and Fluorocurine. [1h] Following the seminal observations of Perkin and Plant [3] that 10,11-dihydroxyhexahydrocarbazoles are unstable under basic conditions, reexamination by Witkop demonstrated the formation of the key 3-hydroxyindolenines which undergo a semi-pinacolic rearrangement to afford the corresponding 3-oxindoles. [4] Various oxidative approaches have been described in the literature to access 3-hydroxyindolenines involving the use of strong oxidants such as dimethyldioxirane, [5] meta chloro perbenzoic acid [6] and others. [7] Alternative pathways involve photo-oxygenation of the substituted indoles and the transient formation of hydroperoxides. [8] Alternative approaches to 3-oxindoles are also reported in the literature. [START_REF] Liu | Miscellaneous transformations leading to pseudo indoxyls: a)[END_REF] Due to the presence of a rigid conjugated electron-donor acceptor fragment Restricting Intramolecular Rotation (RIR), several pseudoindoxyls exhibit fluorescent properties. In 1983, during the isolation of various alkaloids of Aspidosperma oblongum, Potier and al. isolated pseudoindoxyl derivatives as "Produit jaune, amorphe, très fluorescent à l'uv". [1c] Recently, 3oxindoles have been used as fluorophores (Figure 1). [2] Among them, Lipid green® has been used for the fluorescent labelling of lipid droplets. [2b,c] Other reports highlight their facile postfunctionalization to tune their photophysical properties. [2a] Therefore the development of sustainable transformations to access 3-oxindole derivatives minimizing the steps and waste is highly desirable. Transition metal complexes-catalyzed allylic substitution become a method of choice for the formation of various Carbon-Carbon and Carbon-Heteroatom bonds. [START_REF]Comprehensive Organic Synthesis[END_REF] In these transformations, the use of dissymmetric allylic substrate raises the control of regioselectivity and several ruthenium precatalysts have shown excellent activities for the formation of branched products. [11] Interestingly, recent developments toward the introduction of an allylic or an alkyl fragment were developed via transition metalcatalyzed allylic substitution, borrowing hydrogen methodology or more recently hydroelementation starting from simple allylic or aliphatic alcohols [START_REF] Sundararaju | For recent reviews on allylic substitution with allylic alcohols : a)[END_REF][13][14][15][16][17] and alkynes or allenes [18] providing an useful atom economical toolbox to access to allylated or alkylated indoles. Recently, some of us reported the synthesis of an allylation ruthenium precatalyst by designing an allylic ruthenium(IV) complex featuring phosphine-sulfonate chelate which proven efficiency in regio-and chemo-selective allylation from various allylic alcohols. [16j,19] Herein we disclose a facile access to various substituted indole derivatives starting from allylic alcohols via a rutheniumcatalyzed regio-and site-selective allylation followed by a solidstate oxidation on the resulted allylated indoles to afford the corresponding indoxyls generating water as the sole byproduct of the overall reaction sequence.

Results and Discussion

The allylation of 2-methyl indole 1a in the presence of catalytic amount of our previously described Ru-1 {[Ru(Cp*)(H2CCHCH2)(dppbs)]PF6} was first evaluated (Table 1). [16j] During the stoichiometric treatment of 1a with allyl alcohol 2a, selective C(3)-allylation occurred in dichloromethane in less than three hours reaction time providing 3aa with almost complete conversion (entry 1). Increasing the amount of allyl alcohol 2a to 2 equivalents, resulted in the selective N-and C(3)-allylation to afford the product 4aa (entry 2). Higher amount of allyl alcohol along with the catalyst loading had a crucial influence on the selectivity and favored the formation of the diallylated compound 4aa (entry 2). 2-methylindoles are useful scaffolds for the generation of 2-vinylindolines and oquinodimethanes. [START_REF] Aj | [END_REF] However, alkylation of 2-methylindole usually require harsh reaction conditions or the use of strong base such as organolithium reagent or KHMDS to ensure conversion. [START_REF] Aj | [END_REF] Astonishingly, this catalytic system was not limited to the N-or C( 3) allylation and for the first time, the base free allylation of the methylene moiety was achieved leading to the tri-allylated product 5aa accompanied with traces of the tetraallylated side product (entry 3). It is important to note that this last transformation was very sensitive to the reaction conditions and lower selectivity and conversion were observed with lower catalyst loading and/or insufficiently deaerated solvent. Finally, almost complete formation of the tri-allylated indole 5aa was obtained in the presence of 7 mol% of Ru-1 (entries 4 and 5). Having established our best reaction conditions toward selective allylation, we next examined the scope of the transformation with 2-methylindole 1a and 2-phenyl indole 1b (Scheme 1). As expected, indoles 1a and 1b react smoothly with allyl alcohol 1a leading to the selective formation of mono-allylated products 3aa and 3ba in up to 90% isolated yields. In the case of unsymmetrical allylic alcohols, the allylation afforded the branched indoles with complete regioselectivity toward the Table 1. Ruthenium-catalyzed site-selective allylation of 2-methylindole 1a with allyl alcohol 2a.

Entry [a] Ru-1

(mol%) ratio 1a/2a T (°C), t (h)
conv. (%) [b] Ratio 3aa/4aa/5aa [c] 1 3 mol% 1:1 30°C, 3h 99%(3aa:90%) 100:0:0 2 5 mol% 1:2 30°C, 1h 99%(4aa:95%) 0:100:0 3 5 mol% 1:4 20°C, 16h 99% 0:68:32 4 7 mol% 1:4 20°C, 16h 99% 0:0:100 [d] 5 7mol% 1:5 20°C, 16h 99%(5aa:88%) 0:0:100 [d] 6 7 mol% 1:3.5 30 °C, 21h 99% 0:7:93

[a] Experimental conditions: all reactions were performed under an inert atmosphere of argon and carried out with allyl alcohol 2a (0.4-2 mmol), 2methylindole 1a (0.4 mmol), metallic precursor Ru-1 in a closed Schlenk tube in degassed dichloromethane CH2Cl2 (0.5 mL). [b] Conversions were determined by GC and numbers in parentheses correspond to the isolated yield of the major product after column chromatography. [c] ratio was determined by GC analysis on the crude reaction mixture. [d] traces of tetra-allylated product was detected.

branched compounds between 70 and 96% isolated yields. Interestingly, reverse prenylation was also observed starting from 2-Methyl-3-buten-2-ol 2f affording allylated indole 3af in up to 93% isolated yield. [21] The reaction of 1a with linear allylic alcohol 2g and 2h provided the access to the branched indoles 3ab and 3ad in up to 97:3 ratio. In the light of the selective formation of branched products, the diallylation reaction provided the access to the dibranched indoles 4 with isolated yields ranging from 86 and 99% without diastereoselectivity. Interestingly, as a reminiscence of the Potier statements, during the isolation by column chromatography on silica gel of the colourless indole 3aa, we observed its partial decomposition highlighted by the apparition of a yellow band. [1c] Analysis of the side product demonstrated the formation of the fluorescent pseudoindoxyl 6aa. The latter arose from the oxidation of the indole 3aa involving the transient formation of hydroperoxides followed by a semi pinacolic rearrangement. [8,22] Thus, having prepared our set of substituted indoles, we next examined their reactivity toward oxidation. To achieve this task, solvent less oxidation of the suspended indole 3ac on SiO2 was chosen to prevent the formation of undesired side products arising intermolecular side reactions (Table 2). [8] To our delight, 20% of conversion was observed after one hour under sun light (entry 1).

It is important to note that shorter reaction time, highlighted the transient formation of the putative hydroxyindolenine intermediates as a diastereoisomeric mixture which were detected by GC/MS and 1 H NMR. [8d,23] Longer reaction time improved the conversion and almost exclusive formation of the 3-oxindole 6ac was obtained after three-day reaction time (entry 4). Reaction time can be shortened and Scheme 1. Regio-and Site-selective allylation of 2-substituted indoles 1.

10.1002/chem.201800348

Accepted Manuscript

Chemistry -A European Journal when similar reaction was performed under UV irradiation (λ= 312 nm) product 6ac was obtained in less than five hours with up to 71% of isolated yield (entry 7). The reaction rate was greatly affected by the presence of the N-H moiety. Using similar protocol, the N-substituted indole 4aa provided less than 10% of the corresponding oxindoles after one-week reaction time. [9f] As expected, owing to their higher oxidation potential, 2phenylindoles 3b(a,b,e) did not undergo oxidation under these conditions. [8] The semi pinacolic rearrangement was also examined with branched indoles 3ab, 3ac and 3ae (Scheme 2). As a result, no or very low diastereoselectivities were observed with oxindoles 6ab and 6ac. [a] A: Compound 3ac was suspended on SiO2 and the reaction mixture was placed in air at room temperature without stirring. B: Compound 3ac was suspended on SiO2 and the reaction mixture was placed in air at room temperature in the dark without stirring. C: Compound 3ac was suspended on SiO2 and the reaction mixture was placed in air at room temperature under UV irradiation without stirring. D: Compound 3ac was suspended on SiO2 and the reaction mixture was placed in air at room temperature under UV irradiation under stirring. [b] Conversions were determined by 1 H NMR. Numbers in parentheses correspond to the isolated yield of the oxindole 6ac after column chromatography.

In contrast, substituted indole 3ae allowed up to 40% of diastereoisomeric excess. Interestingly, configuration of the major diastereoisomer was unambiguously confirmed by X-ray studies. It is noteworthy that intermolecular H-bonding interactions prevail in solid state and no noticeable π-π stacking interactions were detected. [24] Oxidation of indole 3aa occurred at faster rate and under a modified procedure, pseudoindoxyl 6aa was obtained in up to 84% yield. Taking together, the nature of the allylic substituent had a profound impact on the transformation rate and the overall reactivity follows the decreasing order 3aa>3ab=3ac>3af>3ae.

Scheme 2. oxidation of the C(3)-allylated indoles 3 over SiO2. X-ray structure of the major diastereoisomer of 6ae. CCDC 1533366 contains the supplementary crystallographic data for 6ae

Keeping in mind, the low reactivity of the disubstituted indoles 4 towards solid state oxidation allowed us to next investigate the possibility of introducing a different second allylic substituent on the nitrogen atom to further perform cyclization through Ring Closing Metathesis (RCM) on the resulting compounds (Scheme 3). As expected, the use of our allylation catalyst Ru-1 led to the expected formation of the corresponding diallylated pseudoindoxyls 7 in 55-98% isolated yields (Scheme 3). A rapid screening of the reaction conditions demonstrated that Hoveyda-Grubbs II precatalyst allowed the formation of the tricyclic adducts in up to 84% isolated yields. [9m,25] Owing to the rigid structure of the tricyclic adducts, diastereoisomers of 8ab,ac,ae were easily separated after isolation by column chromatography.

Scheme 3. Allylation/Metathesis sequence on oxindoles 6. 10.1002/chem.201800348

Accepted Manuscript

Chemistry -A European Journal According to the previous independent reports of Kitamura, [26] Pregosin [27] and us, [START_REF] Jiang | Synthesis and Applications in Catalysis of Metal Complexes with Chelating Phosphinosulfonate Ligands[END_REF] an overall plausible mechanism for the allylation of indoles 1 is depicted in Figure 2. It involves the activation of the allylic alcohol 2 by the protonated sulfonate fragment on the ruthenium(II) intermediate I followed by the oxidative addition and the release of a water molecule. Outer sphere approach of the nucleophilic indole 1 on the πruthenium(IV) intermediate II followed by proton transfer and ligand exchange on IV with allylic alcohol 2, release the allylated indole 3. Concerning the pseudoindoxyls 6, according to previous reports on 2,3-disubstituted indoles oxidation and related compounds, formation of a hydroperoxide A tends to be the key intermediate. [4,8b,c,28,29d] Homolysis of this latter followed by reduction via comproportionation with 3, water or the surface involving hydroxy radical might explain the formation of hydroxyindolenine B. [29] Finally, the semi pinacol rearrangement of B assisted by the acidic R3SiOH groups led to the formation of the expected pseudoindoxyl 6.

The photophysical properties of uncyclized and cyclized pseudoindoxyl compounds 7aa and 8aa are given in Table 3, and the absorption and emission spectra, measured in dichloromethane and more interestingly in water of 7aa and 8aa are presented in Figure 3. The UV-vis absorption spectra of 7aa and 8aa in dichloromethane display a broad band at λ = 407 and 410 nm, respectively, presenting a shoulder which can be ascribed to an intramolecular charge transfer (ICT) transition. Compounds 7aa and 8aa are also strongly fluorescent in solution at 298 K, with fluorescence quantum yield up to 59%. Solvatochromic and fluorosolvatochromic effects were observed by using two different solvents (Table 3). Interestingly, they positive fluorosolvatochromism: increasing the polarity of the solvent leads to a red shift (Stock shift ≈ 45 nm) of the emission band (for 7aa: em(CH2Cl2) = 455 nm; em (H2O) = 498 nm), consistent with a more pronounced charge-transfer character due to hydrogen bonding interactions with the oxindole and the molecules of water. [2b,23,28] Unlike the absorption spectra, we can notice a shoulder is present at longer wavelength on all the emission spectra in dichloromethane and water for both compounds meaning that the shape of the emission is not affected by the solvent. The effect of the allylic substituent had a minimal incidence of the absorption and emission properties on the resulting tricyclic adducts whereas the quantum yield was reduced by the introduction of the phenyl moiety presumably due to higher vibrational degrees of freedom.

Compound

Solvent λabs/nm (εmax) [a] λem/nm Φem [b] Stokes shift/cm [a] Units = 10 3 Mol -1 cm -1 . [b] Fluorescence quantum yield (%) with ± 10% error versus quinine sulphate.

Conclusions

In conclusion, through the use of a well-defined π-allylic ruthenium(IV) precatalyst, we achieved the site and regioselective allylation of 2-substituted indoles. Sustainable solid state oxidation under air of the corresponding indoles 3 in the absence of photosensitizer gave the corresponding 3-oxindole 10.1002/chem.201800348

Accepted Manuscript

derivatives. This reaction sequence releasing only water as side product provides a promising alternative to the use of strong oxidant. Finally, solid state oxidation in the presence of photosensitizer would provide interesting results to reluctant substrates.

Figure 1 .

 1 Figure 1. Examples of biorelevant or synthetic pseudoindoxyls and applications.

Figure 2 .

 2 Figure 2. Proposed overall mechanisms.

Figure 3 .

 3 Figure 3. Absorption and Emission spectra of 7aa in CH2Cl2 (red), in H2O (blue) and of 8aa in CH2Cl2 (green), in H2O (black).

Table 2 .

 2 Solid state oxidation of C(3)-allylated indole 3ac.

	Entry	Conds. [a]	6ac/SiO2 (mmol/g SiO2)	t (h)	Conv. [b]
	1	A	0.6	1	20%
	2	A	0.3	1	22%
	3	A	0.3	27	66%
	4	A	0.3	72	91%(73%)
	5	B	0.3	27	38%
	6	C	0.3	4.5	84%
	7	D	0.3	4.5	98(71%)

Table 3 .

 3 Photophysical properties of 7aa, 8aa, 8ab, and 8ae in H2O and CH2Cl2.

Acknowledgements

H. L. thanks the MESRI-Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation for fellowship.

Keywords: ruthenium • allylation • oxidation • metathesis • fluorescence


G Bardan
email: bardan@imft.fr

E Knobloch

A Mojtabi

H Khallouf

Toulouse Archive Ouverte

Keywords:

We present a numerical and analytical study of doubly di usive convection driven by horizontal thermal and solutal gradients in square and rectangular enclosures with no-slip walls subjected to high-frequency vibration. The two vertical walls of the enclosure are maintained at di erent but uniform temperatures and concentrations while the horizontal walls are assumed to be impermeable and insulating. The resulting system is described by time-averaged Boussinesq equations. These equations possess a doubly di usive quasi-equilibrium solution provided the thermal and solutal buoyancy forces are equal and opposite. This solution is linearly stable up to a critical value of the stability parameter independently of the strength and orientation of the vibration. The solutions in the neighborhood of the bifurcation point are described analytically as a function of the strength and orientation of the vibration, and the larger amplitude states are computed numerically using a spectral collocation method. For vertical oscillation increasing the vibration amplitude decreases the subcriticality of the solutions and may even reverse it; the opposite occurs with horizontal vibration.

Introduction

Many convective phenomena involve both heat and mass transfer. Doubly di usive processes in enclosures, such as those arising in geology or in chemical applications, serve as a generic model of such systems. A review of doubly di usive convection in these systems is given by [START_REF] Viskanta | Double di usive natural convection[END_REF]. Recently, doubly di usive convection in enclosures has received increased attention since it is closely related to transfer processes in crystal growth [START_REF] Wilcox | Transport phenomena in crystal growth from solution[END_REF].

In the present work we restrict our attention to a uid layer subjected to horizontal temperature and concentration gradients with the property that the thermal and concentration buoyancy forces exactly balance. In this case there exists an exact equilibrium solution with zero velocity and linear temperature and concentration proÿles, and instability is caused by the di erence between the solutal and thermal di usivities, measured by the Lewis number Le. When Le = 1 the di usive solution is stable for all Rayleigh numbers. When Le = 1 there is a critical value of the Rayleigh number at which this solution undergoes a bifurcation and above which motion is observed. The square cavity case was previously investigated by [START_REF] Krishnan | A numerical study of the instability of double di usive convection in a square enclosure with a horizontal temperature and concentration gradients[END_REF] for Pr = 1 and Le = 3:161 whose numerical study indicated that the onset of convection occurs at a critical thermal Rayleigh number Ra Tconv = 3000 (Ra Tconv (Le -1) = 6481). Below this point the purely di usive solution is stable. [START_REF] Gobin | Double di usion in a vertical uid layer: onset of the convection regime[END_REF] have studied theoretically the case of an inÿnite vertical layer with free-slip boundary conditions and showed that the critical Rayleigh number obeys the relation Ra Tc (Le -1) = 6122. With no-slip boundary conditions the corresponding result is Ra Tc (Le-1)=17 172 [START_REF] Ghorhayeb | Double di usive convection in a vertical rectangular cavity[END_REF]. These results indicate that the lowest value of the convective threshold reported by [START_REF] Krishnan | A numerical study of the instability of double di usive convection in a square enclosure with a horizontal temperature and concentration gradients[END_REF] in fact corresponds to a turning point on a ÿnite amplitude subcritical branch [START_REF] Gobin | Double di usion in a vertical uid layer: onset of the convection regime[END_REF]. The full solution branch was constructed using Newton's method, combined with a continuation scheme, by [START_REF] Xin | Bifurcation analysis of double-di usive convection with opposing horizontal thermal and solutal gradients[END_REF]. The subcritical branch corresponds to a clockwise cell while the supercritical branch corresponds to a three-cell solution consisting of a large counterclockwise cell slanted across the cavity with smaller clockwise cells located in opposite corners. Mechanisms responsible for oscillatory doubly di usive convection have been discussed by [START_REF] Chang | Unsteady thermosolutal opposing convection of liquid-water mixture in a square cavity-II. Flow structure and uctuation analysis[END_REF] and by [START_REF] Alavyoon | On natural convection in vertical porous enclosures due to opposing uxes of heat and mass prescribed at the vertical walls[END_REF].

A possible application of interest arises in materials processing under reduced gravity conditions when convection due to buoyancy forces is strongly reduced. In zero gravity the desired basic state may be set up in a melt when the temperature and concentration gradients are parallel. However, any residual gravity and in particular uctuations in the e ective gravity due to orientation changes of the vehicle and on-board activities introduce signiÿcant perturbations. As a rule, the residual acceleration ÿelds on board of a spacecraft are non-stationary and the measured oscillation frequencies range from 10 -2 Hz to 10 or 100 Hz [START_REF] Belyaev | Mathematical modelling and measurements on the orbital station Mir[END_REF][START_REF] Rogers | Summary report of mission acceleration measurements for STS-66 launched November 3[END_REF][START_REF] Rogers | Summary report of mission acceleration measurements for STS-66 launched November 3[END_REF]. Crew activity or orbital manoeuvres give rise to time-dependent accelerations (g-jitter) with amplitudes as high as 10 -4 to 10 -3 g and uctuating direction. The literature on the e ects of g-jitter on materials processing is quite extensive; for a recent comprehensive review the reader is referred to [START_REF] Nelson | An examination of anticipated g-gitter on the Space Station and its e ects on materials processing[END_REF]. A recent review by [START_REF] Alexander | Low gravity experiment sensitivity to residual acceleration: a review[END_REF] summarizes the important concerns and discusses the e ect of residual accelerations on heat and mass transfer in low-gravity materials experiments. [START_REF] Alexander | Analysis of low gravity tolerance of Bridgman-Stockbarger crystal growth II. Transient and periodic accelerations[END_REF] also analyze low-gravity-tolerance (or Bridgman-Stockbarger) crystal growth. The e ect of gravity modulation on the onset of convection for unidirectional solidiÿcation was considered by [START_REF] Murray | The e ect of gravity modulation on solutal convection during directional solidiÿcation[END_REF] while [START_REF] Wheeler | Convective stability for the Rayleigh-BÃ enard and directional solidiÿcation problems: high-frequency gravity modulation[END_REF] analyzed the high-frequency limit of this problem. In this limit the method of averaging may be used to determine the e ect of large-amplitude gravity modulation on both the primary instability and its subsequent evolution. In contrast, low-frequency (¡ 1 Hz) sinusoidal modulation has to be treated via Floquet theory.

Full Navier-Stokes simulations of the BÃ enard problem for a one-component uid in a ÿnite box carried out by [START_REF] Birigen | Numerical simulation of 3-D BÃ enard convection with gravitational modulation[END_REF] agree with the stability calculations of [START_REF] Gresho | The e ects of gravity modulation on the stability of a heated uid layer[END_REF]. The e ect of vibration on the onset of convection in a horizontally unbounded two-component uid has also been studied. [START_REF] Gershuni | On free convection in vibrational ÿeld in weightlessness[END_REF] computed the ow in an enclosed di erentially heated cavity at Ra = 10 4 for ÿnite frequencies and various values of the vibrational Rayleigh number and showed that a strong resonance can take place that increases the heat ux by more than a factor of two. The resonance frequency depends on both the Rayleigh number and the amplitude of the vibration. Finally, [START_REF] Terrones | Convective stability of gravity-modulated doubly cross-di usive uid layers[END_REF] describe the e ect of time-dependent sinusoidal gravity perpendicular to the boundaries on the onset of convection in a horizontal doubly cross-di usive uid layer. The topology of the neutral curves is more complex than that encountered in constant-gravity multiply di usive layers, leading to new types of behavior not possible in the absence of modulation.

The averaging method, suitable for high-frequency (¿ 1 Hz) modulation, was ÿrst used for thermovibrational convection by [START_REF] Simonenko | On the e ect of high frequency vibrations on the origin of convection[END_REF] and this is the approach adopted in the present paper. Using the averaged equations [START_REF] Gershuni | On free convection in vibrational ÿeld in weightlessness[END_REF][START_REF] Gershuni | On convective instability of uid in vibrational ÿeld in weightlessness[END_REF], 1988) performed linear stability analyses for a uid layer with various boundary conditions. These authors introduced the vibrational analog of the Rayleigh number R v to represent the strength of the vibration source, and studied the Rayleigh-BÃ enard problem as a function of the vibration angle, and the interaction between natural and thermo-vibrational convection. More recent motivation for extending this work has been provided by the development of space experiments and the use of mechanical vibration in industrial processes requiring control of convective motions. In addition to the stability analysis, numerical simulations of closed cavities, either heated from below or from the side, have also been done [START_REF] Chernatynsky | Transient regimes of thermovibrational convection in a closed cavity[END_REF][START_REF] Khallouf | Numerical study of two-dimensional thermovibrational convection in rectangular cavities[END_REF]. For binary mixtures [START_REF] Braverman | On vibrational convective instability of plane uid layer in weightlessness[END_REF] considered the case when the thermal and solutal gradients and the vibration axis are parallel and perpendicular to the layer; a solution describing a long wave mode was obtained. More recently, [START_REF] Gershuni | On the vibrational convective instability of a horizontal binary-mixture layer with Soret e ect[END_REF] analyzed the linear stability of a binary mixture in a plane horizontal layer with the Soret e ect subject to static gravity and a longitudinal high-frequency vibration, focusing on long-wave modes. Situations involving both instability mechanisms, gravitational and vibrational, were studied numerically for representative values of the parameters. Three types of instability can be distinguished: monotonic cellular, monotonic long wave and oscillatory cellular. In the case of the normal Soret e ect only the monotonic instability takes place and thermodi usion plays a destabilizing role; in contrast, in the case of the anomalous Soret e ect the vibrational instability depends strongly on the Soret parameter. To the best of our knowledge, no papers have appeared on the corresponding problem in enclosures.

In this paper we present an extensive discussion of the e ect of high-frequency vibration on doubly di usive convection in square and rectangular enclosures. We describe the results of a numerical study of the in uence of the direction of the vibration axis on thermosolutal convection in cavities bounded by two vertical walls maintained at di erent but uniform temperatures and concentrations making equal and opposite contributions to the net buoyancy force. Since the main purpose of conducting experiments in space lies in eliminating or signiÿcantly decreasing natural convection, it is important to understand the in uence of any residual acceleration on the convection amplitude. In Section 2 we summarize the mathematical model and the time-averaged form of the Boussinesq equations. In Section 3 we show that the onset of convection is una ected by the presence of vibration. This is not so, however, for the nonlinear problem and we describe in Section 4 the e ects of vibration on the weakly nonlinear regime. In square cavities convection sets in via a transcritical bifurcation with the subcritical branch turning towards higher Rayleigh numbers at a saddle-node bifurcation. For larger aspect ratios the primary bifurcation may be a (subcritical) pitchfork. In either case the subcritical solutions acquire stability at a secondary saddle-node bifurcation; the location of this turning point is a ected by vibration leading to the possibility of control of the onset of ÿnite amplitude convection. These results are described in Section 5. The paper concludes with a brief discussion in Section 6.

Problem description and basic equations

The system considered is two-dimensional thermosolutal convection in a container. Fig. 1 shows the ow conÿguration and coordinate system. The ow domain is (x; z) ∈ = [0; L] × [0; H ]. All the physical properties are taken to be constant. The vertical walls at x = 0 and L are kept at constant and uniform temperatures  1 and  2 , respectively, and at constant and uniform concentrations c 1 and c 2 , with  1 ¿  2 and c 1 ¿ c 2 . The top (at z = H ) and bottom (at z = 0) horizontal walls are insulated. All the boundaries are assumed rigid and the horizontal ones impermeable. The uid cavity with its boundaries is subject to linear harmonic oscillations.

The binary uid in the cavity is considered to be Newtonian and to satisfy the Boussinesq approximation. The Dufour and Soret e ects are neglected. The thermophysical properties are constant except for the uid density in the buoyancy term which depends linearly on the local temperature and concentration. The equation of state has the form

(Â; c) = ref (1 -ÿ T (Â -Â ref ) -ÿ C (c -c ref ));
(1) The presence of vibration changes the gravitational acceleration according to

where ref = (Â ref ; c ref ), ÿ T = -1= ref (@ =@Â) c and ÿ C = -1= ref (@ =@c) Â are,
g → g -b! 2 sin(!t)k; (2)
where k = cos x + sin z is the unit vector along the vibration axis and = (x; k) is the vibration angle, b is the displacement amplitude and ! is the angular frequency. Using the velocity u, the pressure p, the temperature  and the concentration c as independent variables, the system is then described by the nondimensional primitive equations ∇:u = 0;

(3)

@u @t + (u:∇)u = -∇p + ∇ 2 u + (Gr T Â + Gr C c)z + (Gr TV Â + Gr CV c) ! sin( !t)k; (4) @Â @t + (u:∇)Â = 1 Pr ∇ 2 Â; (5) @c @t + (u:∇)c = 1 Sc ∇ 2 c: (6)
These equations have been nondimensionalized with respect to the viscous di usion time in the horizontal using L as the lengthscale. Velocities are expressed in terms of =L, where is the kinematic di usivity, while the temperature and concentration di erences from their reference values  2 and c 2 are nondimensionalized using  1 - 2 and c 1 -c 2 , respectively. The nondimensional boundary conditions are therefore u = 0 on @ ;

(7)

 = 1; c = 1 for x = 0; ∀z; (8)
 = 0; c = 0 for x = 1; ∀z; (9) @ @z = @c @z = 0 for z = 0 and A; ∀x: (10

)
The problem is speciÿed by seven nondimensional parameters: the thermal Grashof number Gr T = gÿ T (Â 1 -Â 2 )L 3 = 2 , the solutal Grashof number Gr C = gÿ C (c 1 -c 2 )L 3 = 2 , the modiÿed vibrational thermal Grashof number Gr TV = b!ÿ T (Â 1 -Â 2 )L= , the modiÿed vibrational solutal Grashof number Gr CV = b!ÿ C (c 1 -c 2 )L= , the dimensionless vibration frequency ! = !(L 2 = ), the Prandtl number Pr = =a (a is the thermal di usivity), the Schmidt number Sc = =D (D is the mass di usivity) and the aspect ratio A = H=L. The Lewis number is deÿned to be Le = Sc=Pr. The modiÿed vibrational Grashof numbers (thermal or solutal) are obtained by replacing g by b! 2 in the Grashof number and rescaling by the ratio between the viscous di usive time and the vibrational time !L 2 = . In the limit of high frequency and small amplitude the e ect of vibration is determined by the product b! appearing in the deÿnition of the modiÿed vibrational Grashof numbers Gr TV and Gr CV . In the following we use the Grashof numbers as parameters of the problem, instead of the Rayleigh numbers; these are related by the simple relation Ra T = Gr T Pr.

In the asymptotic case of high-frequency small amplitude vibration the method of averaging can be applied to obtain a closed system of equations for the slow evolution of the mean velocity U , mean temperature T , mean concentration C and mean pressure P. The method is analogous to that employed by [START_REF] Schlichting | Berechnung ebener periodischer Grenzschichtstr omungen[END_REF] to compute the boundary condition on inviscid ow in the bulk due to the presence of an oscillatory viscous boundary layer, and results in the following set of equations [START_REF] Landau | Mechanics[END_REF][START_REF] Gershuni | Thermal Vibrational Convection[END_REF] ∇:U = 0;

(11)

@U @t + (U :∇)U = -∇P + ∇ 2 U + Gr T (T + NC)z + 1 2 Gr 2 TV (W T + N W c ):∇(T k -W T + N (Ck -W c )); (12) @T @t + (U :∇)T = 1 Pr ∇ 2 T; (13)
@C @t + (U :∇)C = 1 Sc ∇ 2 C; (14)
∇:W T = 0; curl(W T ) = curl(T k); (15)
∇:W c = 0; curl(W c ) = curl(C k); (16)
where N ≡ Gr C =Gr T = Gr CV =Gr TV . These equations describe the evolution of the mean ÿelds on a timescale much longer than the vibration period =2 =!; the new term in Eq. ( 12) is the consequence of the quadratic nonlinearity in the Navier-Stokes equation. The quantities W T and W c represent the solenoidal parts of the vectors T k and Ck, respectively, and also vary slowly with time. In terms of these quantities the uctuating velocity ÿeld u is given by u = -Gr TV cos !tW T -Gr CV cos !tW c . Eqs. ( 15), ( 16) are to be solved subject to the boundary conditions ( 7) -( 10) together with

W T :n = W c :n = 0 on @ ; ( 17
)
where n is the unit outward normal.

The averaged equations are valid under the assumptions: (i) the frequency must be su ciently high (but not acoustic) that the vibration period is small compared to all the characteristic hydrodynamic times min(L 2 =a; L 2 =D; L 2 = ), (ii) the displacement amplitude must be small in the sense that bmin(L=ÿ T ( 1 - 2 ); L=ÿ C (c 1 -c 2 )). [START_REF] Simonenko | A justiÿcation of the averaging method for a problem of convection in a ÿeld rapidly oscillating forces and other parabolic equations[END_REF] proves the convergence of solutions of the averaged system to averaged solutions of the system (3) -(6). A numerical description of this convergence is given by [START_REF] Khallouf | Simulation numà erique de la convection thermo-vibrationnelle par une mà ethode spectrale[END_REF], while [START_REF] Gershuni | Thermal Vibrational Convection[END_REF] show that the new terms act as a vibrational body force directed opposite to the kinetic energy gradient.

Linear stability

In the following we write R v ≡ 1 2 Gr 2 TV and refer to it as the vibrational Grashof number. The thermal Grashof number Gr T will be written as Gr.

Mechanical equilibrium and its stability

We refer to states with U = 0 as a mechanical quasi-equilibrium. Such states have zero mean velocity but the uctuating velocity does not necessarily vanish. In order for a quasi-equilibrium to exist the following equations must be satisÿed:

Gr(T 0 + NC 0 )z + R v (W T0 + N W c0 ):∇(T 0 k -W T0 + N (C 0 k -W c0 )) = ∇P 0 ; (18)
T 0 = 0; ( 19
)
C 0 = 0; (20) ∇:W T0 = 0; curl W T0 = curl(T 0 k); (21)
∇:W c0 = 0; curlW c0 = curl(C 0 k) (22)
subject to the boundary conditions ( 7) -( 10) and ( 17). Eqs. ( 19) and ( 20) and the uniqueness of the Helmholtz decomposition lead to the solution T 0 = C 0 = 1 -x and hence to W T0 = W c0 . Eq. ( 18)

now becomes Gr(1 + N )T 0 z + R v (1 + N ) 2 W T0 :∇(T 0 k -W T0 ) = ∇P 0 : (23)
One can check that this equality holds if and only if N = -1. This assumption implies that the contributions to the buoyancy force from the thermal and solutal gradients exactly cancel. Two cases are of particular interest:

• Horizontal vibration. In this case k = x and W T0 = W c0 = 0. Consequently in this case we have a true mechanical equilibrium, i.e., the oscillatory components of the velocity also vanish.

• Vertical vibration. In this case k = z and W T0 = W c0 = W vert = 0. In this case the solution U = 0 is a quasi-equilibrium.
It is convenient to rewrite equations ( 11) -( 16) as evolution equations for two-dimensional perturbations about this equilibrium state. We denote the perturbations by (U ; P ; T ; C ) and introduce the following streamfunction representations: U x = -@ =@z; U z = @ =@x;

(24)

W Tx = -@ 1 =@z; W Tz = @ 1 =@x; (25)
W Cx = -@ 2 =@z; W Cz = @ 2 =@x: (26)
As a result a positive streamfunction corresponds to a clockwise cell. Eliminating p we obtain

@ @t   T C   =    2 Gr @ @x -Gr @ @x -@ @z Pr 0 -@ @z 0 Sc      T C   +   N 1 ( ; ) N 2 ( ; T ) N 2 ( ; C )   -R v   N 3 ( 1 -2 ; T -C ) -N 1 ( 1 -2 ; 1 -2 ) 0 0   (27) 1 = sin @T @x -cos @T @z ; (28) 2 = sin @C @x -cos @C @z ; ( 29
)
where for all pairs (f; g) of real functions

N 1 (f; f) = @f @z @ 3 f @x @z 2 + @ 3 f @x 3 - @f @x @ 3 f @z @x 2 + @ 3 f @z 3 ; (30) N 2 (f; g) = @f @z @g @x - @f @x @g @z ; (31) N 3 (f; g) = -sin @ 2 f @x 2 @g @z + @f @x @ 2 g @x @z - @ 2 f @x @z @g @x - @f @z @ 2 g @x 2 + cos @ 2 g @z 2 @f @x + @g @z @ 2 f @x @z - @ 2 g @x @z @f @z - @g @x @ 2 f @z 2 : (32)
At the boundaries of the cavity the streamfunctions vanish and thus @ @x x=0;1 = @ @z z=0; A = ( ) @ = 0: (33)

T (x = 1; z) = C (x = 1; z) = T (x = 0; z) = C (x = 0; z) = 0 ∀z ∈ [0; A]; (34)
@T @z (x; z = 0; A) = @C @z (x; z = 0; A) = 0 ∀x ∈ [0; 1]; (35) 1 = 2 = 0 along @ : (36)
Eqs. ( 27) -( 29) with boundary conditions ( 33) -( 36) are invariant under rotations by about the point ( 1 2 ; A=2). This rotation is described by the operator S deÿned by

S         T C 1 2         (x; z) =         -T -C 1 2         (1 -x; A -z) (37)
and is a generalized re ection since S 2 = 1. The resulting symmetry group Z 2 ≡ {I; S} plays an important role in the bifurcation analysis described below. In particular, it is known that in the presence of this symmetry group [START_REF] Crawford | Symmetry and symmetry-breaking bifurcation in uid dynamics[END_REF], the conduction state can only lose stability to states that are either symmetric or antisymmetric with respect to S. The former occurs when the marginally stable eigenfunction is invariant under S, the latter when it breaks invariance under S.

Analytical results

In this section we summarize a number of results that can be obtained analytically. These results are based on a severe truncation of a Galerkin expansion but nonetheless describe correctly the qualitative behavior of the solutions as a function of the system parameters. Precise and complete bifurcation diagrams for speciÿc parameter choices are presented in Section 5.

The linearized time-dependent equations are obtained by neglecting the nonlinear terms. The resulting linear problem is independent of the vibrational Grashof number (i.e., T 0 =C 0 and W T0 =W c0 ), and the computation of the vibrational streamfunctions decouples. Solving the corresponding equations (27) gives T and C . The vibrational part of the velocity represented by 1 and 2 is then obtained from Eqs. ( 28) and ( 29). Because of the physical boundary conditions, the value of the thermal Grashof number Gr is positive. Ghorayeb and Mojtabi (1997) showed that in the static case the onset of convection occurs at a critical Grashof number Gr (0) given by Gr (0) |Sc -Pr| = 17 172 (for A = 1) and 7650 (for A = 2:6). This result continues to hold in the presence of vibration. In fact the linear stability results for the static case (without vibration) obtained by Ghorayeb and Mojtabi (1997) and [START_REF] Bardan | Nonlinear doubly di usive convection in vertical enclosures[END_REF] are recovered for arbitrary values of A. These authors show that the primary bifurcations are always stationary.

The linear stability problem is solved by means of a Galerkin method using the following expansions:

(x; z) = n i=0 m j=0 a ij sin( x) sin(i x) sin( z=A) sin(j z=A); (38) T (x; z) = n i=0 m j=0 b ij sin(i x) cos(j z=A); (39)
C (x; z) = n i=0 m j=0 c ij sin(i x) cos(j z=A); (40)
Due to the Z 2 symmetry the associated eigenmodes are either symmetric with respect to S, i.e., S( ; T ; C ; 1 ; 2 )=( ; T ; C ; 1 ; 2 ), or antisymmetric, i.e., S( ; T ; C ; 1 ; 2 )=-( ; T ; C ; 1 ; 2 ). The symmetric eigenmodes contain an odd number of cells whereas the antisymmetric ones contain an even number of cells. Eqs. ( 38) -( 42) indicate that the contribution to the symmetric modes comes from terms with i + j even while the contribution to the antisymmetric modes comes from terms with i + j odd. We used the software MAPLE to ÿnd the values of Gr c ; a ij ; b ij ; c ij ; d ij and e ij at which a bifurcation to a symmetric mode (A = 1) or an antisymmetric one (A = 2:6) occurs. For the symbolic computations described below we use n = m = 4 for , n = m = 3 for T and C and n = m = 2 for 1 and 2 .

In the next section we study ÿrst the codimension one cases arising when Gr is increased for A = 1 and 2:6. We use center manifold reduction [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF] to describe the weakly nonlinear problem with particular attention devoted to its dependence on the vibrational Grashof number R v and the vibration angle . We only consider the case Sc ¿ Pr corresponding to most binary mixtures, i.e., Le ¿ 1.

Nonlinear analysis

Weakly nonlinear analysis

In this section we use a multiple scale analysis to reduce Eqs. ( 27) -( 29) to a simpler form, called a normal form, valid near onset of the primary instability, i.e., for |Gr -Gr (0) |=Gr (0) 1. We expand the perturbations in powers of a small parameter ¿ 0,

= (1) + 2 (2) + • • • ; (43)
with similar expressions for T ; C ; 1 ; 2 , and write

@ @t = @ @t (1) + 2 @ @t (2) + • • • ; (44) Gr = Gr (0) + Gr (1) + 2 Gr (2) + • • • : (45)
At order we recover the linear eigenvalue problem. Since C (1) = Le T (1) , the critical eigenmode can be written in the form

        
(1)

T (1) C (1) (1) 1 (1) 2          = K(t (1) )         f 1 (x; z) Pr f 2 (x; z) Sc f 2 (x; z) Pr f 3 (x; z) Sc f 3 (x; z)         ; (46)
where the amplitude K depends on the slow time t (1) and the functions f i depend only on the spatial variables. These functions and the corresponding critical Grashof number Gr (0) are computed as described in Section 3. At order 2 we obtain

   2 Gr (0) @ @x -Gr (0) @ @x -@ @z Pr 0 -@ @z 0 Sc       (2) T (2) C (2)    = -    0 Gr (1) @ @x -Gr (1) @ @x 0 0 0 0 0 0       (1)
T

(1) C (1)    -    N 1 ( (1) ; (1) ) N 2 ( (1) ; T (1) ) N 2 ( (1) ; C (1) )    + @ @t (1)    (1) T (1) C (1)    -R v    N 1 ( (1) 1 -(1) 2 ; (1) 1 -(1) 2 ) -N 3 ( (1) 1 -(1) 2 ; T (1) -C (1) ) 0 0    : (47)
The solvability condition for these equations now yields the evolution equation. To obtain this equation we need the solution to the adjoint eigenvalue problem

    2 @ @z @ @z -Gr * (0) @ @x Pr 0 Gr * (0) @ @x 0 Sc        * T * C *    =    0 0 0    (48)
with identical boundary conditions. Thus Gr * (0) = Gr (0) and

   * T * C *    =    (Pr -Sc)f * 1 (x; z) Pr f * 2 (x; z) -Sc f * 2 (x; z)    : (49)
The solvability condition is therefore

(e + (Pr + Sc)f) @K @t (1) = Gr (1) (Pr -Sc)aK + (b + R v (Sc -Pr) 2 c + (Pr + Sc)d)K 2 (50)
with Gr (1) indicating the distance from onset. The coe cients are given by the expressions

a = @f 2 @x f * 1 d x d z; (51) b = N 1 (f 1 ; f 1 )f * 1 d x d z; (52) c = (N 1 (f 3 ; f 3 ) -N 3 (f 3 ; f 2 ))f * 1 d x d z; (53) d = N 2 (f 1 ; f 2 )f * 2 d x d z; (54)
e = f 1 f * 1 d x d z; (55) f = f 2 f * 2 d x d z; (56)
where the asterisk denotes the adjoint eigenfunctions. In the following two subsections we summarize the results obtained analytically from the truncated Galerkin equations.

Case

A = 1: The symmetric case S(f 1 (x; z)) = f 1 (x; z) .
For aspect ratio A = 1 the critical eigenmode is invariant under the symmetry S. The bifurcation is therefore expected to be transcritical. Our symbolic computations conÿrm this expectation, giving a = -0:142, b = -159:3, c = c( ), d = 184 960, e = 41:5 and f = 126:7. Consequently, the purely di usive solution (K =0) is stable if and only if Gr (1) ¡ 0, i.e., for Grashof numbers smaller than the critical thermal Grashof number, while the convective solution (K = 0) is stable for Gr (1) ¿ 0 and unstable for Gr (1) ¡ 0. Fig. 2 shows the coe cient c as a function of the angle of vibration . Note that this function is -periodic because vibration at angle or + is the same. The convective solution is given by

(x; z) ≈ (1) = f 1 (x; z) a b + R v (Sc -Pr) 2 c + (Pr + Sc)d (Gr -Gr (0) )(Sc -Pr): (57)
In the following we refer to the branch that bifurcates supercritically (subcritically) from the quasiequilibrium state as the supercritical (subcritical) branch, regardless of its subsequent appearance. The supercritical branch is initially stable, while the subcritical one is initially unstable. Subsequent saddle-node and=or Hopf bifurcations may change these stability properties.

For Pr = 1, Sc = 11 and R v = 0 we ÿnd that the convective branch has a very small slope in the ( ( 1 2 ; 1 2 ); Gr) plane, of the order of 10 -6 . This result can be conÿrmed by direct numerical simulation. Since the numerical solution of the linear problem shows that f 1 ¿ 0 at x = z = 1 2 whenever Le ¿ 1, while the solution of the adjoint problem implies that a ¡ 0, d ¿ 0 and b ¡ 0 with Pr+Sc ¿-b=d (≈ 10 -3 ), it follows that on the stable supercritical branch ( 1 2 ; 1 2 ) ¡ 0 and hence that the corresponding ow is counterclockwise. This is because when Le ¿ 1 heat di uses faster than concentration and consequently the direction of motion is determined by the concentration gradients. At the left of the cavity these carry heavier uid from the top to the bottom producing a counterclockwise cell. As the strength of the vibration increases this picture gradually changes because of the dependence of c. If =0 or 1:4 (mod ) the coe cient c vanishes and the resulting bifurcation diagram is independent of the vibration strength.

If 0 ¡ ¡ 1:4 (mod ) c ¿ 0 and the slope decreases (in magnitude) as R v increases. Finally, if 1:4 ¡ ¡ (mod ) c ¡ 0 and the magnitude of the slope instead increases with R v and the slope becomes vertical at

R v = - b + (Pr + Sc)d (Sc -Pr) 2 c ¿ 0: (58)
At this point the perturbation expansion breaks down and a di erent scaling must be employed. This scaling brings in a cubic term in the amplitude K, indicating that at degeneracy the bifurcation is a pitchfork. The unfolding of this degeneracy is correctly described by including a small quadratic term in K as in Eq. ( 50). Golubitsky and Schae er (1985) discuss additional phenomena described by the universal unfolding of the pitchfork. In the present problem these require relaxing the requirement N = -1.

Although for A = 1 the mode truncation used predicts that Gr (0) (Sc -Pr) = 19 040 (compared with the exact value Gr (0) (Sc -Pr) = 17 172) the analytical calculation captures correctly the qualitative dependence of the solutions on the system parameters. Numerically accurate bifurcation diagrams for vertical vibration and Le = 11; 51 can be found in Section 5.2.2.

4.3.

Case A = 2:6: The antisymmetric case S(f 1 (x; z)) = -f 1 (x; z) .

When A = 2:6 the critical eigenmode is antisymmetric S( (x; z; t (1) )) ≡ (1 -x; A -z; t (1) ) = -(x; z; t (1) ) ( 59)

and we now expect a nondegenerate pitchfork bifurcation. Our symbolic calculations conÿrm that b = c = d = 0 for all values of m and n. Thus Gr (1) = 0 (K = 0). Consequently, K does not depend on t (1) .

In order to obtain the ÿrst nonzero Gr (i) for i = {1; 2; 3; : : : ; n}, we solve the order 2 problem for (2) , T (2) and C (2) . The solution takes the form

         (2) T (2) C (2) (2) 1 (2) 2          = K 2 (t (2) )        
g 1 (x; z; Pr; Sc; ; R v ) g 2 (x; z; Pr; Sc; ; R v ) g 3 (x; z; Pr; Sc; ; R v ) g 4 (x; z; Pr; Sc; ; R v ) g 5 (x; z; Pr; Sc;

; R v )         : (60)
At order 3 , we obtain

    2 Gr (0) @ @x -Gr (0) @ @x -@ @z Pr 0 -@ @z 0 Sc        (3) T (3) C (3)    = -     0 Gr (2) @ @x -Gr (2) @ @x 0 0 0 0 0 0       
(2)

T (2) C (2)    -    N 1 ( (1) ; (2) ) N 2 ( (1) ; T (2) ) N 2 ( (1) ; C (2) )    -    N 1 ( (2) ; (1) ) N 2 ( (2) ; T (1) ) N 2 ( (2) ; C (1) )    -R v    N 1 ( (1) 1 -(1) 2 ; (2) 1 -(2) 2 ) -N 3 ( (1) 1 -(1) 2 ; T (2) -C (2) ) 0 0    -R v    N 1 ( (2) 1 -(2) 2 ; (1) 1 -(1) 2 ) -N 3 ( (2) 1 -(2) 2 ; T (1) -C (1) ) 0 0    + @ @t (2)    (1) T (1) C (1)    : (61)
The solvability condition now yields (e p + (Pr + Sc)f p ) @K @t (2) = a p Gr (2) (Sc -Pr)K -g p (Pr; Sc; ; R v )K 3 ;

where a p ≈ 7:41, e p ≈ 2380, f p ≈ 8845 and g p ≡ g p (Pr; Sc; ; R v ).

In the static case (R v = 0) the pitchfork bifurcation is subcritical (g p ¡ 0) for any value of Pr and Sc, and the emerging solutions are unstable. However, as and R v vary the pitchfork can be transformed into a supercritical one and the convective solutions stabilized at small amplitude. This solution is given by 2 ≈ 2 (1) 2 = a p f 2 1 g p (Pr; Sc; ; R v ) (Gr -Gr ( 0) )(Sc -Pr):

(63)

Fig. 3 shows a contour plot of g p as a function of both and R v when Pr = 1 and (a) Sc = 11, (b) Sc = 51, again obtained from the truncated Galerkin expansion. This function describes correctly the functional dependence of the solution on the system parameters.

Numerical results

Numerical method

The system of equations ( 11) -( 16) was solved numerically using a spectral method [START_REF] Canuto | Spectral Methods in Fluid Dynamics[END_REF]. The method used is based on the projection di usion algorithm developed for solving the 2D-3D unsteady incompressible Navier-Stokes equations [START_REF] Khallouf | Simulation numà erique de la convection thermo-vibrationnelle par une mà ethode spectrale[END_REF]. Temporal integration consists of a semi-implicit second-order ÿnite di erence approximation. The linear (viscous) terms are treated implicitly using a second-order backward Euler scheme, while a second-order explicit Adams-Bashforth scheme is employed for the nonlinear (advective) parts. When applied to an advectiondi usion equation such as

@f @t + U :∇f = f (64) the method leads to 3 2 f n+1 -2f n + 1 2 f n-1 t = f n+1 -2(U :∇f) n -(U :∇f) n-1 : (65)
This equation can be written in the form of the Helmholtz equation

( -h)f n+1 = s; (66)
where h = 3=2 t is the Helmholtz constant and s is a scalar quantity containing all the terms known at time t n = n t (n is the time level and t is the time step). The temporal integration therefore transforms the system into a Helmholtz problem arising from the advection-di usion equations ( 13) and ( 14) coupled to the Poisson problems ( 15) and ( 16) with appropriate boundary conditions. The latter are solved using the Uzawa (or Poisson-like) formulation of [START_REF] Azaiez | Spectral methods applied to porous media[END_REF]. The Navier-Stokes equations ( 11) -( 12) are transformed into a generalized Stokes problem and solved by the projection-di usion method of [START_REF] Khallouf | Simulation numà erique de la convection thermo-vibrationnelle par une mà ethode spectrale[END_REF]. All the subproblems obtained are either Helmholtz or Poisson-like operators. A high-accuracy spectral method, namely one utilizing Legendre collocation points, is used in the spatial discretization of the Helmholtz and Poisson-like operators. Successive diagonalization is implemented to invert these operators. We mention that the Stokes and Darcy-Euler solvers are direct and guarantee an accurate spectral solution with divergence-free solenoidal ÿelds over the entire domain, including the boundaries. For the calculations discussed in this paper a grid with 15 × 15 mesh points in the (x; z) domain su ces to describe accurately the ow for Le = 11. We selected a 25 × 25 grid for our calculations at Le = 11 and 35 × 35 at Le = 51.

Numerical results

For the numerical study, we chose Sc = 11 and Pr = 1 in order to investigate a case in which there is a substantial di erence between the thermal and solutal disturbances. Some results for Le = 51 are also described. We consider two particular geometries, a square cavity (A = 1) and a rectangular cavity with A = 2:6. We begin with zero-gravity results, i.e., with thermo-vibrational convection, followed by a discussion of the interaction between vibrational and static gravity driven convection.

Zero gravity results for A = 1

For the square cavity the theoretical critical Grashof number is given by Gr (0) (Sc -Pr) = 17 172, i.e., Gr (0) = 1717. The corresponding instability is a steady one (see Sections 2 and 3). When Gr = 0 the system is controlled by the vibration angle and the vibration amplitude R v . Computations were done for 0 ¡ R v ¡ 50 000 for between 2 to -2 .

The ÿrst set of computations was performed with initial data for the di usive solution and conÿrmed the stability of the equilibrium or quasi-equilibrium solutions predicted by linear theory. For Le ¿ 1 larger amplitude solutions evolved either to this state or to thermo-vibrational convection consisting of a clockwise one-cell ow, depending on and R v . The results are summarized in Fig. 4 for selected values of R v (0 ¡ R v ¡ 50 000). The ÿgure shows that the range of angles for which a convective solution exists diminishes with decreasing R v and vanishes at R vc = 2610, obtained for =16

• . Note that, at high values of R v , the ow amplitude passes through a maximum for negative angles. Additional numerical simulations showed that for ¿ c = 73

• convective motion is suppressed regardless of the value of R v . For comparison, for a horizontally vibrating cavity the critical vibrational Rayleigh number is R v = 2860 whereas the conductive solution is unconditionally stable when vibrated vertically. With two-dimensional steady ow, the streamlines correspond to lines of constant streamfunction and a single contour map gives a complete portrayal of the ow ÿeld. Fig. 5 shows that the ow consists of a clockwise rotating single cell. In all three ÿelds (velocity, temperature and concentration) the resulting thermo-vibrational ow resembles natural convection.

Interaction between vibration and gravity driven convection for A = 1

In view of the weakly nonlinear results of Section 4 we now turn to three representative cases ( = 0; = 4 and = 2 ) illustrating the di erent types of interaction between vibration and static gravity driven convection. We are particularly interested in the location of the turning point on the subcritical convective branch.

Horizontal vibration ( =0): When the vibration is along the static gradient (Fig. 6) the supercritical regime branches from the pure di usive state at Gr(Sc -Pr) = 17 161 (for comparison the value obtained by linear stability analysis is 17 172). The numerical calculation conÿrms that the onset of the supercritical branch is independent of R v and that, near the bifurcation point, the slope of the supercritical branch is also independent of R v . On the supercritical branch the streamfunction is negative in the center of the cavity and the ow therefore consists (Fig. 7) of a dominant counterclockwise rotating cell with two identical recirculations in the upper right and lower left corners. The calculation also conÿrms the relation T = Le C between the thermal and concentration perturbations in the neighborhood of the bifurcation point. Note that the streamfunction has point symmetry while the thermal and solutal perturbations are antisymmetric (Fig. 7). Such a solution is invariant under the symmetry S as predicted by the linear stability analysis. The subcritical regime extends down to much lower convective thresholds (Gr = 676 for R v = 0) as reported by Ghorayeb and Mojtabi (1997) and terminates in a saddle-node bifurcation on the ÿnite amplitude branch. As R v increases, the value of Gr at this turning point decreases continuously and in zero gravity motion can exist for all values of R v ¿ 2860, in agreement with Fig. 4. The ow structure is qualitatively as shown in Fig. 5,i.e., it consists of a single clockwise roll. Fig. 6 shows that near Gr ≈ 1230 the convection amplitude is almost independent of the vibration strength. It is worth mentioning that the supercritical solution is obtained easily only if initial data with (nearly) the right symmetry properties are prescribed; otherwise the solution evolves towards the subcritical solution which appears to have a much larger domain of attraction.

Direct simulations of the system (3) -( 6) have also been carried out. The solution of the Navier-Stokes equation with a time-dependent force per unit volume requires a substantially longer computation time in comparison with the mean-ow equations and the same resolution requirements. The results of the latter calculations are recovered for all f = 2 ! ¿ 200 Hz (f1 Hz for water at 20

• C and L = 1 cm). The value f * ≈ 200 corresponds to the limit of the validity of the mean ÿeld treatment for our problem. Of course, as shown in Fig. 8 all of the variables now depend on time but their mean values over the vibration period are independent of the frequency whenever f ¿ f * . In this example, the value of the mean Nusselt number Nu is 1.1775 for f =100; 1:1758 for f =300 and 1.1756 for f = 500, i.e., Nu varies by less than 0:5% over this range of frequencies. For slower oscillations resonance phenomena may occur but these are beyond the scope of the present work.

Vibration at = 4 : For vibration at 45

• the supercritical branch is present for Gr ¿ 1715 and its slope decreases (in magnitude) with increasing R v (Fig. 9), as predicted by the weakly nonlinear analysis. In the limiting case R v → ∞, the supercritical branch becomes a purely di usive one. The subcritical branch behaves in much the same way as in the previous case. On this branch, the streamfunction is positive in the center of the cavity and the ow regime resembles that in Fig. 5, but with the vibration-independent solution moved to Gr ≈ 676. Note that this value corresponds to the turning point on the subcritical branch in the absence of vibration. In zero gravity (Gr = 0) thermo-vibrational convection can be produced for R v ¿ 5400.

Vertical vibration ( = 2 ): For vertical vibration the numerical results also conÿrm that the value of the critical Grashof number (Gr = 1715 when Le = 11) is independent of the vibrational Rayleigh number (see Fig. 10). The motion on the supercritical branch is a three-cell ow with the dominant cell being counterclockwise. The isolines of the temperature and concentration are almost parallel and vertical, and are consistent with the relation T = Le C found in weakly nonlinear theory. As R v increases the slope of the transcritical branch becomes steeper and steeper and for R v ¿ 13 000 it changes sign. In this regime there is a stable small amplitude supercritical branch with ¿ 0 which soon loses stability at a saddle-node bifurcation at which the branch turns around towards smaller Gr joining the original stable small amplitude ¿ 0 subcritical branch at a second saddle-node bifurcation. This behavior can be established by analyzing the degenerate transcritical bifurcation in which the coe cient of the quadratic term vanishes, as described in Section 4.2. This second ¿ 0 branch in turn loses stability at a further saddle-node bifurcation at the right where it turns to the left and joins the large amplitude subcritical branch (see Fig. 10); the saddle-node on the large amplitude subcritical branch moves towards higher Grashof numbers with increasing vibrational Rayleigh number. For Gr = 1600 and R v = 50 000, the values of the Nusselt number Nu and the corresponding dimensionless number describing the solute ux (the Sherwood number Sh) on the lower ¿ 0 branch are, respectively, 1:004 and 1:397; note that these do not satisfy the expected relation Nu -1 = Le 2 (Sh -1), see [START_REF] Bardan | Nonlinear doubly di usive convection in vertical enclosures[END_REF]. The corresponding ow structure is shown in Fig. 11. A ÿnite amplitude ¡ 0 branch is also present and plotted for R v = 20 000 in the enlargement. In contrast, the ow on the large amplitude branch consists of a clockwise single cell ow. The critical Grashof number at which a ÿnite amplitude solution ÿrst appears depends on R v and decreases gradually with increasing R v , becoming zero when R v exceeds 50 000. In this case a ÿnite amplitude vibration-driven state coexists with a stable conduction state in zero gravity.

We have examined one additional (and more realistic) value of the Lewis number. For Le = 51, corresponding to oxygen in water, the critical Grashof number is 343. The primary bifurcation remains transcritical and the e ect of vibration is qualitatively similar to that at Le = 11. However, the bifurcation diagram for vertical vibration shown in Fig. 12 reveals the presence of a new feature. The transcritical branch for R v = 2000 consists of a three-cell supercritical branch with a counterclockwise central cell; there is also a subcritical solution branch consisting of a small but ÿnite amplitude three-cell ow with a clockwise central cell. Once again we presume that this branch is connected to the transcritical bifurcation via an unstable subcritical branch. The branch itself loses stability with increasing Gr at a saddle-node bifurcation providing a connection to the large amplitude single clockwise cell states via another unstable segment of the subcritical branch. It is along Fig. 10. Bifurcation diagram in the ( 1 2 ; 1 2 ) -Gr plane for vertical vibration and di erent values of Rv when Le = 11, Pr = 1. The saddle-node bifurcation moves from Gr = 676 (for Rv = 0) to Gr = 1620 (for Rv = 20 000). The inset is an enlargement of the supercritical branch for di erent values of Rv. The slope of this transcritical branch increases (in modulus) as Rv increases and the branch becomes subcritical for Rv ¿ 13 000. At Rv = 20 000 there are as many as four stable branches for Gr ¿ Gr (0) , three at small amplitude and a fourth large amplitude branch, separated by unstable branches. For Rv = 50 000 stable convection is possible even when Gr = 0. Resolution is 25 × 25. this unstable segment that the form of the solution gradually changes from the three cell state to the single cell state as the central clockwise cell grows and the counterclockwise corner cells shrink. Such a transition must occur since the single cell state cannot connect to the three cell states present on either side of the transcritical bifurcation. As a result there is now an interval of Grashof numbers with three stable coexisting solutions. It is also noteworthy that for R v ¿ 2600 the transcritical branch again changes direction (see Fig. 12), generating an additional wiggle in the ¿ 0 branch, i.e., there are now three stable segments on the ¿ 0 branch instead of two. At the same time the ¡ 0 branch develops a subcritical segment. As a result the small amplitude ¡ 0 solutions become unstable although they once again acquire stability at a secondary saddle-node bifurcation. Altogether there are now four stable solution segments (cf. Fig. 12 for R v = 5000). Once again these results follow from the unfolding of the degenerate transcritical bifurcation. Our numerical simulations indicate that the value of R v at which this change takes place is inversely proportional to Le when the Prandtl number remains ÿxed. Fig. 13 shows the development of solutal boundary layers on the large amplitude ¿ 0 branch with increasing Le, as well as the constant concentration core produced by the expulsion of concentration gradients from the convective cell that occurs at these amplitudes.

Interaction between gravity and vibration driven convection when A = 2:6

In this section we brie y summarize the corresponding results for A = 2:6. In this case the primary instability leads to an antisymmetric eigenmode and the bifurcation is therefore a pitchfork. The resulting bifurcation diagram for horizontal vibration ( = 0) is shown in Fig. 14. As already mentioned the pitchfork for R v = 0 is subcritical. The initially unstable convection branch acquires stability at a saddle-node bifurcation at which the branch turns towards increasing Gr. The ÿgure shows that the location of this saddle-node bifurcation moves towards smaller Gr as R v increases and that for su ciently large R v stable ÿnite amplitude convection coexists with the (stable) conduction state even in zero gravity (Gr = 0). This is so for the case illustrated in Fig. 15; this ÿgure shows the streamlines, isotherms and isoconcentration lines for R v = 4000 and several values of Gr on the subcritical branch. For small values of Gr the ow consists of three cells of which the central one is counterclockwise while the two stronger outer cells are clockwise. By Gr = 400 the central cell has shrunk to nothing and for larger values of Gr the ow consists of a single large clockwise cell ÿlling the entire cavity. Note that these solutions are all symmetric with respect to the symmetry operation S. Such solutions cannot be connected to the original pitchfork bifurcation. Consequently, a symmetry-breaking bifurcation must occur along the solution branch as this is followed to smaller and smaller amplitudes, and it is these states that develop into the antisymmetric states born in the primary pitchfork bifurcation. In the present case these states are unstable.

For vertical vibration ( = 2 ) the situation is markedly di erent. In this case, as shown in Fig. 16, increasing R v decreases the subcriticality of the pitchfork; for su ciently large R v this bifurcation becomes supercritical and stable solutions can be found arbitrarily close to onset of the primary instability. Fig. 17 shows the evolution of the ow with decreasing Gr when R v = 5000. At large Gr the ow is again symmetric and consists of two strong cells both of which are clockwise. As Gr decreases these two cells merge into a single clockwise cell, leaving weaker counterclockwise cells in the top right and bottom left corners, but the overall symmetry with respect to S remains. However, as Gr decreases further, this symmetry is lost; we show the resulting solution at Gr = 795. By Gr = 770 the solution is very weak but almost entirely antisymmetric; this solution connects to the pitchfork bifurcation which for these parameter values occurs at Gr = 765.

Conclusion

We have described a theoretical and numerical study of doubly di usive convection in square and rectangular cavities in the special case permitting the existence of an equilibrium or quasi-equilibrium state. We explored the e ects of rapid vibration on this system as a function of both strength and direction. A close agreement between the theoretical and numerical results was found. In zero gravity convective motion appears in a square cavity with Le = 11 for a suitably chosen vibration axis whenever the vibration strength exceeds R v = 2610. The range of orientations for which instability occurs increases with R v , with an upper limit set by = 73

• ; for larger angles the conduction solution (i.e., the equilibrium state) is unconditionally stable. The interaction between static gravity and vibration-driven convection is similar to that found for the static case. When A = 1 the primary bifurcation is transcritical and when = 0 and = 4 leads to a stable supercritical branch and a stable ÿnite amplitude subcritical branch. In contrast for vertical vibration there may be several subcritical branches connected by unstable solutions, provided the vibrational Rayleigh number is su ciently high. On the small amplitude supercritical branch both horizontal and vertical vibration increases the strength of convective motion, while the opposite is the case when = 4 . In all cases su ciently high R v leads to bistability between the conduction state and vibration-driven convection in zero gravity. When A = 2:6 horizontal vibration is very e ective in driving thermo-vibrational convection; in contrast vertical vibration suppresses thermo-vibrational convection e ectively, and substantial buoyancy forces are required to sustain uid motion. This observation suggests a possible protocol that could be used to control convective motion induced by stray gravitational ÿelds or DC accelerations.

The reason di erent vibration directions have such di erent e ect can be understood by considering the direction of the e ective "vibration force" due to the oscillation of the cavity. This force is described by the new term on the right side of the Navier-Stokes equation ( 12) and represents the tendency of the uid to ow (on average) along gradients of density inhomogeneities (cf. [START_REF] Gershuni | Thermal Vibrational Convection[END_REF]. This ow is in addition to the doubly di usive ows present in the absence of vibration. When N = -1 this force is proportional to Gr(∇T -∇C); since for typical parameter values the concentration gradients in the stationary ow are stronger than those in the temperature the direction of the vibrational force is determined primarily by the former. For example, the isoconcentration contours shown in Fig. 13 for vertical vibration indicate that the overall vibration force in the left half of the cavity is towards to the lower left and conversely in the right half of the cavity. Thus the vibration-induced ow is counterclockwise and hence opposite to the doubly di usive ow. As a consequence, increasing R v reduces the overall circulation and the saddle-node bifurcation on the subcritical branch therefore moves towards higher values of Gr. This conclusion agrees with the results shown in Fig. 12. A similar conclusion holds for the ow shown in Fig. 17 at Gr =1680 and explains why the pitchfork bifurcation for A=2:6 becomes supercritical for su ciently large values of R v . The vibration-induced counterclockwise circulation weakens with decreasing Gr, however, and close to the primary pitchfork bifurcation the linear relation T = Le C indicates that the direction of the vibration-induced ow is determined instead by the temperature ÿeld. However, the vibration-induced ow remains in the same direction as the doubly di usive ow. Similarly, an examination of the contours of constant concentration for horizontal vibration (Fig. 15) shows that at Gr = 100 the vibration-induced circulation will be clockwise at the top and bottom of the cavity and counterclockwise in the center. It follows that in this case the vibration-induced ow is in the same direction as the overall ow and hence that increasing R v can only strengthen the observed circulation. As a result the saddle-node bifurcation on the subcritical branch moves towards lower values of the Grashof number as R v increases, as found in Fig. 14. In this case thermo-vibrational convection can be driven even under zero gravity conditions.

In certain cases we have seen that the direction of the transcritical bifurcation changes with increasing R v , producing a wiggle in the subcritical branch and increasing the multistability of the system. It appears likely that each time the transcritical bifurcation changes direction an additional wiggle is introduced on the subcritical branch, resulting ultimately in a steady-state branch with a large number of left-right oscillations as in the related problem studied by [START_REF] Tsitverblit | Bifurcation phenomena in conÿned thermosolutal convection with lateral heating: Commencement of the double-di usive region[END_REF].

Fig. 1 .

 1 Fig. 1. Sketch of the cavity conÿguration.

  respectively, the density at the standard temperature  ref =  2 and the standard concentration c ref = c 2 , and the thermal and solutal expansion coe cients. The quantity ÿ T is positive and, if c denotes the concentration of the heavier component, ÿ C is negative. As a result the thermal contribution to the buoyancy force favors a clockwise circulation while the solutal contribution promotes counterclockwise circulation.

  sin(i x) sin(j z=A); sin(i x) sin(j z=A):

Fig. 2 .

 2 Fig. 2. The function c( ) obtained using MAPLE with n = m = 4 for and n = m = 3 for T and C.

Fig. 3 .

 3 Fig. 3. Contour plot of gp(Pr; Sc; ; Rv) obtained using MAPLE with n = m = 4 for and n = m = 3 for T and C for Pr = 1 and (a) Sc = 11, (b) Sc = 51. For 1:4 ¡ ¡ 3:1, gp is initially negative but becomes positive as Rv increases.

Fig. 4 .

 4 Fig. 4. Graph of ( 1 2 ; 1 2 ) versus (degrees) for ÿnite amplitude solutions in zero gravity (Gr = 0) and di erent values of Rv when Le = 11, Pr = 1. The solid points denote the calculated values. Resolution is 25 × 25.

Fig. 5 .

 5 Fig. 5. Streamlines, isotherms and isoconcentration lines in zero gravity for Rv = 5000 and horizontal vibration (Le = 11, Pr = 1, A = 1). Resolution is 25 × 25.

Fig. 6 .

 6 Fig. 6. Bifurcation diagram in the ( 1 2 ; 1 2 ) -Gr plane for horizontal vibration and di erent values of Rv when Le = 11, Pr =1. The saddle-node bifurcation moves from Gr =676 (for Rv =0) to Gr =0 (for Rv =2860). The inset is an enlargement of the supercritical branch for di erent values of Rv. The vibration strength (Rv) does not a ect the transcritical branch in the neighborhood of the primary bifurcation. Resolution is 25 × 25.

Fig. 7 .

 7 Fig. 7. Streamlines, and lines of constant temperature and concentration perturbations for Gr = 2000 and Rv = 10 000 and horizontal vibration (Le = 11, Pr = 1, A = 1). Resolution is 25 × 25.

Fig. 8 .

 8 Fig. 8. The Nusselt number Nu as a function of the dimensionless time t for f = 100 (solid line), f = 300 (dotted line) and f = 500 (long-dashed line), computed from the primitive equations for Pr = 1, Le = 11, A = 1, Gr = 2000 and R v = 2000. Resolution is 33 × 33.

Fig. 9 .

 9 Fig. 9. Bifurcation diagram in the ( 1 2 ; 1 2 ) -Gr plane for vibration at = 4 and di erent values of Rv when Le = 11, Pr = 1. The inset is an enlargement of the transcritical branch of solutions for di erent values of Rv. The slope of this supercritical branch decreases (in modulus) as Rv increases. Resolution is 25 × 25.

Fig. 11 .

 11 Fig. 11. Lines of constant streamfunction, temperature and concentration perturbations for Gr = 1600, Rv = 50 000 and vertical vibration (Le = 11, Pr = 1, A = 1). Resolution is 25 × 25.

Fig. 12 .

 12 Fig. 12. Bifurcation diagram in the ( 1 2 ; 1 2 ) -Gr plane for vertical vibration and di erent values of Rv when Le = 51, Pr = 1, for comparison with Fig. 10. Resolution is 35 × 35.

Fig. 13 .

 13 Fig. 13. Streamlines, isotherms and isoconcentrations for Rv = 5000, Gr = 850 and vertical vibration when Le = 51, Pr = 1. Resolution is 35 × 35.

Fig. 14 .

 14 Fig. 14. Bifurcation diagram in the Nu-Gr plane for horizontal vibration and di erent values of Rv when Le = 11, Pr = 1, A = 2:6, and Rv = 0 (solid line), Rv = 2000 (dotted line) and Rv = 4000 (long-dashed line). Resolution is 33 × 33.

Fig. 15 .

 15 Fig. 15. Streamlines, isotherms and isoconcentrations for horizontal vibration and Rv = 4000 when Le = 11, Pr = 1 and A = 2:6. Resolution is 33 × 33.

Fig. 16 .

 16 Fig. 16. Bifurcation diagram in the Nu-Gr plane for vertical vibration and di erent values of Rv when Le = 11, Pr = 1, A = 2:6, and Rv = 0 (solid line), Rv = 1000 (dotted line), Rv = 2000 (dashed line) and Rv = 5000 (long-dashed line). Resolution is 33 × 33.

Fig. 17 .

 17 Fig. 17. Streamlines, isotherms and isoconcentrations for Rv =5000 and vertical vibration when Le=11, Pr =1, A=2:6 and Gr = 1680. For Gr = 770, Gr = 795 and Gr = 910 the contours of constant streamfunction, temperature and concentration perturbations are shown instead. Resolution is 33 × 33.

Acknowledgements

This work would not have been possible without the support of C.N.E.S. (Centre National d'Etudes Spatiales) and Università e Paul Sabatier.


N Habbachi
email: habbachinizar@yahoo.fr

H Boussetta
email: hatem_boussetta@yahoo.fr

M A Kallala
email: adel.kallala@gmail.com

A Boukabache

P Pons
email: ppons@laas.fr

Keywords: MEMS, capacitor, tuning, fluid

This letter presents the fabrication and the modeling of a continuously tuned microfluidic capacitor. On one hand its electrodes are realized by classical electrodeposition techniques and on the other hand, the lamination of SU-8 films allows superposing some microfluidic channels. According to the experimental results, the capacitor value increases continuously following the deionized (DI) water penetration in microchannels. The capacitance variations are comprised between Cmin = 0.52 pF and Cmax = 18.5 pF, allowing a wide tuning range that reaches 3460% at 500 MHz. The quality factor decreases from Qmax = 69 when the capacitor is empty to Qmin = 5.3 when it is fully filled with DI water. We have also investigated the theoretical aspects of our device by modeling the electric field and the current distributions inside the channels: when they are entirely filled with the DI water, the electric field is cancelled.

I. INTRODUCTION

Microfluidic MEMS are widely used to control the microwave devices in different configurations and many applications [START_REF] Entesari | Fluidics in Microwave Components[END_REF] requiring reconfigurability. The most used configurations are based on two complementary approaches: contact-based shortening [START_REF] Kondoh | High-reliability, high-performance RF micromachined switch using liquid metal[END_REF] and contactless RF-shortening [START_REF] King | Microfluidically switched frequency-reconfigurables slot antennas[END_REF]. The common advantages of these microfluidic-based structures can be summarized as follows: low insertion loss, high power handling, high frequency response, and ease of fabrication. Nevertheless, the liquid movement in microchannels needs an actuation pump and a reservoir tank that leads to increase the size and the complexity of the structure. In this letter we are interested by the contact-based RFshortening. Our main idea is to change some RF characteristics of a microwave capacitor by using the deionized (DI) water as a dielectric. In the second section we present the layout and the fabrication of our structure with detailed dimensions. The third section summarizes the performances of the device for different positions of the DI water between electrodes. The distribution of the electric field and the current density in the structure allows understanding some particular phenomena. Hence, the presence of a fluid with high dielectric constant could cancel the prevailing electric field between the capacitor electrodes. Knowing that our approach allows to control the different positions of the DI water in the channels, it is easy to predict a capacitor's range of values or to modify them on demand.

II. CAPACITOR DESIGN AND FABRICATION

The proposed structure is based on dual circular electrodes as shown on Fig. 1(a). This geometry has been used in microfluidic structures to control magnetic beads by modulating the magnetic field [START_REF] Fulcrand | Development of a flexible microfluidic system integrating magnetic micro-actuators for trapping biological species[END_REF] and to vary the inductance value of a microwave coil [START_REF] Gmati | Fabrication and evaluation of an on-chip liquid micro-variable inductor[END_REF].

III. MEASUREMENT RESULTS AND DISCUSSION:

The device is electrically characterized by using groundsignal-ground (GSG) probes and a vector network analyzer Agilent 8510.

A. Quality factor variation: The quality factor of the passive components represents an important parameter for RF circuits as VCO, LNA, and All these devices require a high quality factor in order to decrease their phase noise, insertion loss, and power consumption [START_REF] Chen | A low-power low-phase-noise LC VCO with MEMS Cu inductors[END_REF]. In our case, we study the influence of a particular fluid on the quality factor of our device. It will be measured for six particular positions of DI water present in the channel for frequency varying from 100 MHz to 10 GHz. The results reported in Fig. 3(a) show that the value of the quality factor decreases by following the DI water penetration in the micro channels. To highlight this influence, we have reported in Fig. 3 of the DI water positions in the channel: it can be observed that it decreases continuously from Qmax = 69 when the capacitor is empty to Qmin = 5.3 when the channel is full. This result can be explained by the modification of the electric field in the channels: it is due to the presence of the DI water acting as a partial dielectric layer. To search possible improvements of our approach, we have used a finite element method tool (HFSS software) to model the electric field distribution in the channels for each of the six positions of DI water: the principal results are summarized in Fig. 4. The first color map of the Fig. 4(a) shows the electric field distribution when the capacitor is empty. We can observe that the maximum electric energy is prevailing inside the micro channels namely between the electrodes. Knowing that the air permittivity is lower than the SU-8 one, the micro-channels represent the containment medium for the electric field. When we introduce the DI water between the capacitor electrodes, the electric field disperses into the SU-8 medium as shown in Fig. 4(b). Hence, we can observe in Fig. 4(c-f) that for the other fluid positions, the repartition of the electric field in micro-channels changes significantly. For the situation shown in Fig. 4(g) the capacitor device is fully filled with DI water, and we can observe that the electrical energy is transferred from the micro-channels medium to the whole structure in equiponderant way. These results can explain the drastic diminution of the quality factor of the device when the DI water occupies the different positions in the micro-channels.

B. Capacitance variations:

For each of the six positions of the DI water, we have carried out the capacitance measurements for frequency varying from 0.1 GHz to 4 GHz. The results reported on Fig 5 show similar behavior with noticeable changes in resonant frequencies from 690 MHz (POS 6) to 1.2 GHz (POS 1). It can be observed also that the capacitance value increases continuously with the presence of the DI water in the channels.

To evaluate the span of the capacitance variations, we have reported on Fig. 5(b) its values at 500 MHz for the six positions of DI water: Cmin = 0.52 pF when the device is empty; Cmax = 18.5 pF when it is fully filled. We obtain hence a tuning range of Tr = 3460%, which is very high compared to Tr = 800% of the conventional switched capacitor [START_REF] Baek | Widely Tunable Variable Capacitor with Switching and Latching Mechanisms[END_REF]. In the literature, continuous MEMS varactors with TR of 2000-3000% have been reported [START_REF] Nguyen | Angular Vertical Comb-Driven Tunable Capacitor With High-Tuning Capabilities[END_REF][START_REF] Lee | Piezoelectrically Actuated Tunable Capacitor[END_REF][START_REF] Pu | RF MEMS Zipping Varactor With High Quality Factor and Very Large Tuning Range[END_REF] and the ability to reach such tuning levels is possible with microfluidic approach.

(a) To go further in the analysis, we have investigated the electric current density variations occurring at the electrodes surface for 500 MHz. As we can see on Fig. 6(a-d), the electric current density increases continuously in the capacitor electrodes in response to the DI water displacement and reaches its maximal value for the POS4 (Fig. 6(e)). Furthermore, one can observe in Fig. 6(e-g) that the electric current density continues to increase on the electrode surface used as electrical ground according to the water positions. These observations could explain the increase of the capacitance value in response of the DI water displacement.

(b) (c) (d) (e) (f) (g)

C. Insertion loss variations:

In order to evaluate the influence of the DI water displacement on the insertion loss (IL) response, we reported in Fig. 7 the measurement results of the transmission gain at the frequency range from 100 MHz to 3 GHz for the six DI water positions. We can conclude from the reported results in Fig. 7(a) that the microwave capacitor acts as a tunable bandpass filter following the DI water displacement. The frequency band is larger when the capacitor is empty and decreases with DI water penetration in microchannels. In addition, we have collected the insertion loss value at 500 MHz for each DI water position as illustrated in Fig. 7(b). It is obvious that the insertion loss decreases from IL = -18.63 dB at (POS0) to IL = -0.44 dB when the capacitor is fully filled (POS6). These results reveal the importance of DI water that improves the transmission gain of the microwave capacitor.

IV. CONCLUSION

This letter investigates the effect of a dielectric fluid as DI water on the tunable capacitor performances. We have designed and fabricated a dual spiral capacitor by using a microfluidic based structure. The capacitance value is minimal when the structure is empty (Cmin = 0.52 pF) and increases continuously with DI water penetration in microchannels to reach Cmax = 18.5 pF. We obtain a high tuning range Tr = 3460% whereas the quality factor decreases from Qmax = 69 to Qmin = 5.3. The associated modeling allows to explain the cancellation of the distributed electric field in the micro channels and the increase of the current density in the electrodes in response to the DI water displacements. In the future, this theoretical investigation would be useful to explain the experimental results and to predict the response of microwave devices using dielectric fluids.

Fig. 1 :Fig. 2 :

 12 Fig. 1: (a) Tridimensional view of the microfluidic capacitor and its cross section; the six fluidic positions in the channels: (b) POS1 (c) POS 2 (d) POS3 (e) POS4 (f) POS5 (g) POS6.

Fig. 3 :

 3 Fig. 3: Measurement of the quality factor of our device (a) in response of frequency (b) in response of DI water positions at 500 MHz.

Fig. 4 :

 4 Fig. 4: Electric field distribution at 500 MHz for the different positions of water (a) POS0 (b) POS1 (c) POS2 (d) POS3 (e) POS4 (f) POS5 (g) POS6.

Fig. 5 .

 5 Fig. 5. Measurement results of the capacitance value (a) in response of frequency (b) in response of DI water positions at 500 MHz.

Fig. 6 :

 6 Fig. 6: Electric current distribution for the different DI water positions: (a) POS0 (b) POS1 (c) POS2 (d) POS3 (e) POS4 (f) POS5 (g) POS6.

Fig. 7 .

 7 Fig. 7. Measurement results of : (a) S21 gain in response of frequency ; (b) insertion loss in response of DI water positions at 500 MHz.