Benjamin Faure
email: benjamin.n@gmail.com

Pierre Alphonse
email: alphonse@chimie.ups-tlse.fr

Co-Mn-oxide spinel catalysts for CO and propane oxidation at mild temperature

Keywords:

a b s t r a c t Co x Mn 3-x O 4 oxides (0 ≤ x ≤ 3) were prepared by controlled decomposition of mixed oxalates near 200 • C, followed by a calcination at 300 • C. These oxides are amorphous for x < 0.9. For higher cobalt fraction they have a cubic spinel structure and their crystallite size grows with the cobalt fraction. These materials have a large surface area; the highest values, exceeding 250 m 2 /g, were obtained for x ≈ 2. The spinel oxides exhibit an outstanding catalytic activity for propane oxidation at mild temperature (20-200 • C). They are also active for CO oxidation at ambient temperature. This high activity was correlated both with the surface area and the cobalt concentration. The most efficient material is Co 2,3 Mn 0,7 O 4 , which has a better activity than cobalt oxide catalysts reported in the literature.

Introduction

Catalytic oxidation is a very effective method for the abatement of low concentrations of Volatile Organic Compounds (VOCs). Currently, the most active catalysts are supported noble metals [START_REF] Liotta | Catalytic oxidation of volatile organic compounds on supported noble metals[END_REF][START_REF] Ousmane | Supported Au Catalysts For Low-Temperature Abatement of Propene and Toluene, As Model VOCs: Support Effect[END_REF][START_REF] Santos | Oxidation of CO, ethanol and toluene over TiO2 supported noble metal catalysts[END_REF]. However these catalysts are very expensive and their activity can be strongly inhibited by CO [START_REF] Patterson | The effect of carbon monoxide on the oxidation of four C6-C8 hydrocarbons over platinum, palladium and rhodium[END_REF], water or chloride [START_REF] Marecot | Propane and propene oxidation over platinum and palladium on alumina: effects of chloride and water[END_REF]. For low temperature applications, like VOCs removal in indoor air, precious metals can be replaced by transition metal oxides [START_REF] Busca | Transition metal mixed oxides as combustion catalysts: preparation, characterization and activity mechanisms[END_REF]. Especially spinel cobalt oxide (Co 3 O 4 ) was reported to be the best catalyst for the total oxidation of CO [START_REF] Royer | Catalytic oxidation of carbon monoxide over transition metal oxides[END_REF] and hydrocarbons [START_REF] Busca | Transition metal mixed oxides as combustion catalysts: preparation, characterization and activity mechanisms[END_REF][START_REF] Solsona | Complete oxidation of short chain alkanes using a nanocrystalline cobalt oxide catalyst[END_REF]. Spinel oxides, with the general formula AB 2 O 4 , contain cation sites of different coordination (tetrahedral and octahedral) with the oxide anions arranged in a cubic close-packed lattice. Partial substitution of cobalt by manganese gives mixed Co-Mn spinel oxides, which can be better catalysts than Co 3 O 4 for the oxidation of VOCs [START_REF] Zhu | Mesoporous MCo2O4 (M = Cu, Mn and Ni) spinels: structural replication, characterization and catalytic application in CO oxidation[END_REF][START_REF] Todorova | Complete n-hexane oxidation over supported Mn-Co catalysts[END_REF][START_REF] Puertolas | The different catalytic behaviour in the propane total oxidation of cobalt and manganese oxides prepared by a wet combustion procedure[END_REF].

Most often the catalysts reported in the literature are synthesized at high temperature (>500 • C). This requirement, unavoidable for automotive catalytic converters, becomes useless for VOCs abatement at mild temperature (<300 • C). Actually it is expected that metastable nanocrystalline oxides, with very large porosity and surface area, will be highly active catalysts. This kind of materials can be easily obtained by thermal decomposition of metal oxalates. Indeed a close control of the decomposition allows prepar-ing mixed oxides with very high surface area (300-500 m 2 /g) [START_REF] Drouet | Synthesis of mixed manganites with high surface area by thermal decomposition of oxalates[END_REF][START_REF] Iablokov | High catalytic activity in CO oxidation over MnOx nanocrystals[END_REF]. Moreover this easy and inexpensive method is also very convenient to obtain mixed oxides [START_REF] Robin | Etude des oxalates métalliques comme matières premières pour la preparation de solution solides d'oxydes[END_REF].

The goal of this study was the synthesis of large surface area Co x Mn 3-x O 4 oxides (0 ≤ x ≤ 3) by controlled decomposition of mixed oxalates and the evaluation of the catalytic performance of these metastable materials for the total oxidation of carbon monoxide and propane at mild temperature • C). Carbon monoxide is produced in large amount by transportation, industrial and domestic activities. It is extremely toxic and catalytic oxidation into CO 2 constitutes the best solution for CO removal from indoor air [START_REF] Royer | Catalytic oxidation of carbon monoxide over transition metal oxides[END_REF]. Thus low cost, precious metal free catalysts, working at room temperature are highly demanded. Propane is largely used as domestic and industrial fuel. Besides it is also the third most common motor vehicle fuel in the world behind gasoline and Diesel fuel. On the other hand it is generally admitted that alkanes are the least reactive among VOCs and a catalyst able to remove propane at mild temperatures is expected to be active for other VOCs as well.

Experimental

Synthesis of oxides 2.1.1. Preparation of mixed oxalates

Mixed oxalates Co x/3 Mn (3-x)/3 C 2 O 4 .2H 2 O were precipitated at room temperature by quick introduction of an aqueous solution of cobalt and manganese nitrates (200 mL; 0.2 M) into an aqueous solution of ammonium oxalate (200 mL; 0.22 M) under vigor-ous stirring. After 30 min, the precipitate was filtered, thoroughly washed with deionized water and dried in air at 70 • C.

Thermal decomposition of oxalates

The thermal decomposition of oxalates was carried out in a vertical tubular fixed-bed flow reactor under atmospheric pressure. The internal diameter of reactor was 1 cm. The flow rate of the inlet gas (4% O 2 in Ar) was 100 cm 3 /min. The outlet gas composition was followed using a mass spectrometer (HPR20-QIC from Hiden Analytical). The temperature of the reacting solid was recorded by a K thermocouple positioned inside the powdered sample. This setup allowed controlling both the temperature of the reacting material and the composition of the atmosphere. The temperature was increased at 2.5 • C/min until CO 2 emission was detected; from then the decomposition was done in isothermal conditions. For example, in the case of manganese oxalate this temperature was about 210 • C. When CO 2 emission was over, to ensure a total decomposition, even in the core of particles, the partial pressure of O 2 was augmented gradually to 20%; then the temperature was increased at 5 • C/min up to 300 • C and maintained to this value for 1 h.

Thermal analysis (TGA-DSC)

The thermal decomposition of oxalates was studied by thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC), using a constant heating rate (5 • C/min), on a TGA-DSC-1 Mettler-Toledo device in the temperature range 30-600 • C. The flowing gas was a mixture 20% O 2 in Ar. About 5 mg of oxalate powder were placed in a 40 mL aluminium pan and the reference was an empty aluminium pan.

Powder X-ray diffraction (PXRD)

The crystal structure was investigated via powder X-ray diffraction. Data was collected, at room temperature, with a Bruker AXS D4 -2 diffractometer, in the Bragg-Brentano geometry, using filtered Cu K␣ radiation and a graphite secondary-beam monochromator. Diffraction intensities were measured by scanning from 20 to 80 • (2) with a step size of 0.02 • (2).

A quantitative estimation of the lattice parameters and peak broadening was accomplished by profile fitting of the whole XRD patterns using the Fullprof software [START_REF] Rodríguez-Carvajal | Recent developments of the program FULLPROF[END_REF]. The peak profiles were modeled by pseudo-Voigt functions. The parameters refined were zero shift (2), background, cell parameters and peak shape. The size and strain contribution to the integral breadth of each reflection were calculated by the software. The instrumental broadening contribution was evaluated by using an ␣-alumina sample (NIST Standard Reference Material 1976b).

The structural changes versus temperature were followed by High Temperature X-ray Diffraction (HTXRD) with a Bruker AXS D8 diffractometer (using Ni-filtered Cu K␣ radiation) equipped with a high temperature chamber Anton Paar HTK1200N. Diffraction intensities were recorded in synthetic air flow (20% O 2 in N 2 ), at fixed temperature, every 10 • C, in the range 100-500 • C. The heating rate between each step was 10 • C/min. The time needed to record each pattern was about 15 min.

Specific surface area, pore size distribution

Specific surface area and pore size distribution were calculated from nitrogen adsorption-desorption isotherms collected at 77 K, using an adsorption analyzer (Micromeritics Tristar II 3020). The specific surface areas were computed from adsorption isotherms, using the Brunauer-Emmett-Teller (BET) method [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF]. The pore size distributions (PSD) were computed from desorption isotherms by the NLDFT method [START_REF] Seaton | A new analysis method for the determination of the pore size distribution of porous carbons from nitrogen adsorption measurements[END_REF] (with Quantachrome Autosorb-1 software using silica equilibrium transition kernel at 77K, based on a cylindrical pore model).

Pore volume (V pore ) was calculated from the adsorbed volume at a relative pressure of 0.995 (V sat ) by:

V pore = N 2 gas density N 2 liq. density V sat = 0.00155 V sat
Prior to analysis, to remove the species adsorbed on the surface, the oxalate samples (about 0.5 g) were degassed for 16 h at 70 • C whereas the oxide samples (about 0.1 g) were degassed for 16 h at 90 • C (final pressure <10 -3 Pa).

Electron microscopy

Transmission electron microscopy analyses were performed with a Jeol JEM-1400 operating at 80 kV. Samples were prepared by putting a drop of an ethanol suspension of particles on a carboncoated copper grid.

Scanning electron microscopy analyses were performed with a SEM FEG FEI Quanta-250 at 20 kV. The samples were prepared by putting a drop of an ethanol suspension of particles on an aluminum sample holder. Before analysis, the samples were covered with a thin layer (5 nm) of Pt by sputter coating.

Chemical analysis by X-ray fluorescence

The elemental composition was determined on powder samples by X-ray fluorescence with a Bruker S2 Ranger working with a maximum voltage of 50 kV and a current of 2 mA.

Catalytic tests

The activities of catalysts were tested for CO and C 3 H 8 total oxidation. These tests were performed, at ambient pressure, in a tubular fixed bed flow glass reactor (internal diameter = 6 mm). The catalyst mass was always close to 0.05 g. The catalyst powder was packed in the tube giving a 2-3 mm bed length. The volumetric flow rate was 1.63 mL s -1 giving a contact time of 0.03 s. The size of catalyst particles was about 10 m. The reactor operates at differential conditions only for propane oxidation, at mild temperatures (conversion <10%). The reactants were dosed by mass flow controllers (Brooks 5850). The catalyst temperature was controlled by a K-type thermocouple positioned inside the catalyst bed. For CO oxidation the temperature range was 30-200 • C and the inlet gas composition was 0.8% CO + 20% O 2 in Ar. For C 3 H 8 oxidation the temperature range was 30-300 • C and the inlet gas composition was 0.4% C 3 H 8 + 20% O 2 in Ar. The catalyst temperature was increased at a heating rate of 200 • C/h. The gas phase composition during the tests was monitored by mass spectrometry (HPR20-QIC from Hiden Analytical). Before the CO oxidation test, the catalysts were first pretreated with 20% O 2 in Ar for 60 min at 200 • C. The C 3 H 8 oxidation test was done after CO test without any pretreatment.

Results and discussion

Characterization of oxalate precursors

XRD

The XRD pattern (Fig. 1) of manganese oxalate corresponds to the monoclinic structure with the space group C2/c (PDF # 00-025-0544) whereas cobalt oxalate has the orthorhombic structure with the space group Cccm (PDF # 00-025-0250). The structure of mixed Co-Mn oxalates depends on their manganese content. For oxalates containing more Mn than Co, pattern matching with the monoclinic Fig. 1. Examples of XRD patterns of Co x/3 Mn (3-x)/3 C2O4.2H2O after the profile fitting with the FULLPROF software [START_REF] Rodríguez-Carvajal | Recent developments of the program FULLPROF[END_REF]. The upper patterns were indexed for the monoclinic structure (PDF # 00-025-0544) whereas the lower patterns were indexed for the orthorhombic structure (PDF # 00-025-0250).

Table 1

Structure parameters of Co x/3 Mn (3-x) /3C2O4.2H2O determined by profile fitting of XRD patterns with FULLPROF software [START_REF] Rodríguez-Carvajal | Recent developments of the program FULLPROF[END_REF]. D is the crystallite size. structure gives a better agreement whereas, when Co equals or exceeds Mn, a best fit is obtained with orthorhombic structure. The lattice parameters and the crystallite size are reported in Table 1.

x Space group a (nm) b (nm) c (nm) ˇ ( • ) D (nm) 0 C2/c 1.
Both a and b lattice parameters decrease when the proportion of Co increases indicating that the smaller Co 2+ ions (r = 89 pm HS) substitute for the larger Mn 2+ ions (r = 97 pm HS). The crystallite size is constant, at about 50 nm, for the monoclinic structure and drops near 20 nm when the structure becomes orthorhombic. However the crystallite size of cobalt oxalate is larger, at about 30 nm, than that of Co-rich mixed oxalates.

Scanning electron microscopy

The SEM images for several compositions are shown in Fig. 2. The particle morphology changes according to the chemical com- position. The particles of manganese rich oxalates (upper images) are very irregular in size and shape whereas the particles of cobalt rich oxalates are more uniform. These particles are at least ten times larger than the crystallite size determined from XRD. For 2 ≤ x < 3 the particles aggregate in ball-shaped units (lower left image). We did not observe such aggregates for cobalt oxalate, which gives rod like particles (lower right image).

Specific surface area and pore volume

The BET surface area (S BET ) and pore volume (V pore ) of oxalates are reported in Table 2. The surface areas of all the compounds containing cobalt are similar, in the range 4-6 m 2 /g. They are at least 5 times lower than the surface area of manganese oxalate which has also the largest pore volume. For this oxalate we suspected that the

Decomposition of oxalate precursors

Thermal analysis of oxalate decomposition

The TGA-DSC analysis curves for several compositions are plotted in Fig. 3. The decomposition occurs in two main separated stages: below 200 • C the endothermic dehydration giving the anhydrous oxalate, followed by the exothermic decomposition near 300 • C. For an oxalate containing 2H 2 O per mole, the mass loss due to dehydration must decrease slightly from 20.1% for x = 0 to 19.7% for x = 3. We obtain the expected value for manganese oxalate whereas the mass loss is only 19% for cobalt oxalate, indicating a number of water molecules slightly less than 2. Besides, for cobalt oxalate, the endothermic dehydration peak has a small shoulder on the right side equivalent to about 10% of the whole area. This peak is always present whatever the mass of sample analyzed or the heating rate. It could be the indication that we have a mixture between the orthorhombic form and a small amount of the monoclinic polymorph with the space group C2/c (PDF # 00-025-0251) not detected on the XRD patterns.

Thermal decomposition of transition metal oxalates hydrates has been studied for many years [START_REF] Dollimore | The thermal decomposition of oxalates: part I, Var. Surf. Area Temp[END_REF][START_REF] Broadbent | The thermal decomposition of oxalates. part VII. The effect of prior dehydration conditions upon the subsequent decomposition of cobalt oxalate[END_REF][START_REF] Dollimore | The thermal decomposition of oxalates. Part X. Nitrogen adsorption data on solid residues from the isothermal heat treatment of manganese(II) oxalate dihydrate[END_REF][START_REF] Brown | Thermal decomposition of manganese(II) oxalate in vacuum and in oxygen[END_REF][START_REF] Nagase | Thermal dehydration and decomposition reactions of bivalent metal oxalates in the solid state[END_REF][START_REF] Mu | Thermal decomposition of carbonates, carboxylates, oxalates, acetates formates and hydroxides[END_REF][START_REF] Dollimore | The thermal decomposition of oxalates. A review[END_REF][START_REF] Gao | The thermal decomposition of oxalates. Part 26. A kinetic study of the thermal decomposition of manganese(II) oxalate dihydrate[END_REF][START_REF] Coetzee | Kinetics of the thermal dehydrations and decompositions of some mixed metal oxalates[END_REF][START_REF] Nohman | Thermal and chemical events in the decomposition course of manganese compounds[END_REF][START_REF] Nikumbh | Thermal and electrical properties of manganese (II) oxalate dihydrate and cadmium (II) oxalate monohydrate[END_REF][START_REF] Maciejewski | Monitoring of the gas phase composition: a prerequisite for unravelling the mechanism of decomposition of solids. Thermal decomposition of cobalt oxalate dihydrate[END_REF][START_REF] L'vov | Kinetics and mechanism of thermal decomposition of nickel, manganese, silver, mercury and lead oxalates[END_REF][START_REF] Malecka | Kinetics of thermal decomposition of manganese(II) oxalate[END_REF][START_REF] Mohamed | A comparative study of the thermal reactivities of some transition metal oxalates in selected atmospheres[END_REF]. It was observed that the decomposition product for these oxalates depends on the reducibility of the metallic cation involved [START_REF] Nagase | Thermal dehydration and decomposition reactions of bivalent metal oxalates in the solid state[END_REF][START_REF] Dollimore | The thermal decomposition of oxalates. A review[END_REF][START_REF] Malecka | Kinetics of thermal decomposition of manganese(II) oxalate[END_REF]. With cations presenting a low reducibility, like Mn 2+ , the decomposition leads to the metal oxide following the reaction:

MnC 2 O 4 → MnO + CO + CO 2 H = 151 kJ/mol (1)
Whereas for more reducible cations like Co 2+ the decomposition produces the metal:

CoC 2 O 4 → Co + 2CO 2 H = 94 kJ/mol (2)
However in air, Co is oxidized in Co 3 O 4 :

3Co + 2O 2 → Co 3 O 4 H = -891 kJ/mol (3)
Similarly MnO will be also oxidized in air. According to the reaction conditions the product was reported as Mn 3 O 4 [START_REF] Donkova | Mechanism of decomposition of manganese(II) Oxalate dihydrate and manganese(II) oxalate trihydrate, Thermochim[END_REF], Mn 2 O 3 [START_REF] Brown | Thermal decomposition of manganese(II) oxalate in vacuum and in oxygen[END_REF][START_REF] Brown | Thermochemistry of decomposition of manganese(II) oxalate dihydrate[END_REF], MnO 2 [START_REF] Gao | The thermal decomposition of oxalates. Part 26. A kinetic study of the thermal decomposition of manganese(II) oxalate dihydrate[END_REF] or amorphous MnO x [START_REF] Iablokov | High catalytic activity in CO oxidation over MnOx nanocrystals[END_REF].

Moreover the manganese or cobalt oxides formed are known to be good catalysts for the CO oxidation [START_REF] Royer | Catalytic oxidation of carbon monoxide over transition metal oxides[END_REF]:

CO + 1/2O 2 → CO 2 H = -283 kJ/mol (4)
Therefore, since these oxidation reactions are very exothermic, the overall decomposition reaction is largely exothermic too.

The effect of cobalt fraction on some characteristics of mixed oxalate dehydration and decomposition is illustrated in Fig. 4. Four features are followed, the onset temperature of dehydration (T i ), the width of the decomposition peak (in • C), the enthalpy of dehydration and the enthalpy of decomposition. The upper-left plot shows that the T i of mixed oxalates is close to the T i of cobalt oxalate (130 • C) whereas the dehydration of Mn oxalate begins at least 30 • C below. The lower-left plot shows that the enthalpy of dehydration is rather similar for all the samples analyzed, at about 520 J/g (95 kJ/mol oxalate); only the enthalpy observed for manganese oxalate is slightly lower. This value is very close to the enthalpy reported by Maciejewski et al. [START_REF] Maciejewski | Monitoring of the gas phase composition: a prerequisite for unravelling the mechanism of decomposition of solids. Thermal decomposition of cobalt oxalate dihydrate[END_REF].

We observed that the onset temperature of decomposition augments linearly with the cobalt content of oxalates, going from 230 • C for manganese oxalate to 255 • C for cobalt oxalate. The upper-right plot shows that the width of the decomposition peak, which is linked to the rate of the reaction, markedly decreases when the proportion of cobalt increases. Nevertheless the enthalpy of decomposition, estimated by integration of the exothermic peak, does not rise much with the cobalt fraction as shown by the lowerright plot. This is confirmed by the red line in this plot, which corresponds to the values calculated from thermodynamic data ( f H • 298 ) [START_REF] The | Tables of chemical thermodynamic properties[END_REF][START_REF] Kubaschewski | Materials Thermochemistry[END_REF] assuming the formation of an ideal solid solution having the spinel structure (cf. § 3.2.2). Therefore we think that the increase of the decomposition rate with the proportion of cobalt could be explained by the very high reactivity of metal cobalt in oxygen containing atmospheres. Thus to avoid a temperature overshoot, leading to a fast growth of crystallites, the decomposition of cobalt-rich oxalates shall be performed in isothermal condition and low oxygen partial pressure.

HT-XRD of oxalate decomposition

The thermal decomposition of the oxalates was also followed by HT-XRD. The samples were heated by step of 10 • C at 5 • C/min. Taking into account the time required to record each pattern, the heating rate on the average was about 0.5 • C/min. Thus, compared with TGA-DSC for which the heating rate was 5 • C/min, a peak shift toward lower temperature can be expected.

The relevant XRD patterns for several compositions are plotted in Fig. 5. The upper-left chart corresponds to manganese oxalate.

The first pattern, recorded at 150 • C, corresponds to the anhydrous oxalate which has an orthorhombic structure with the space group Pmna (PDF # 00-032-0646). This anhydrous oxalate decomposes from 230 • C to give an amorphous phase which crystallizes only at 400 • C in cubic Mn 2 O 3 (bixbyite) with the space group Ia3 (PDF # 00-041-1442). This transformation gives a small exothermal peak together with a slight mass loss at 450 • C on the TGA-DSC plot (see Fig. 3, x = 0).

The upper-right plot corresponds to the mixed oxalate Co 0.2 Mn 0.8 C 2 O 4 .2H 2 O (xCo = 0.6). The anhydrous oxalate pattern is observed from 140 • C. Like for manganese oxalate, this anhydrous oxalate decomposes from 220 • C to give an amorphous phase which crystallizes at 450 • C in a tetragonally deformed spinel (space group I41/amd) like Mn 3 O 4 (hausmannite, PDF # 00-024-0734). Mn 3 O 4 is a "normal" spinel because all Mn 3+ cations are located in the octahedral sites, giving the cation distribution Mn 2+ [Mn 3+ ] 2 O 4 [START_REF] Xiao | Determination of cations distribution in Mn3O4 by anomalous X-ray powder diffraction[END_REF]. Bordeneuve et al. [START_REF] Bordeneuve | Structural variations and cation distributions in Mn3-xCoxO4 (0 < x <3) dense ceramics using neutron diffraction data[END_REF], from neutron diffraction data recorded on Co x Mn 3-x O 4 ceramics, showed that, for x < 1, the substitution occurs only in the tetrahedral site where Co 2+ replaces Mn 2+ .

The deformation of the cubic spinel structure is correlated with a distortion of the coordination octahedron around Mn 3+ , usually interpreted as a consequence of the cooperative Jahn-Teller effect [START_REF] Jarosch | Crystal structure refinement and reflectance measurements of hausmannite, Mn3O4[END_REF].

The lattice parameters of hausmannite are a = 0.576 nm and c = 0.944 nm [START_REF] Jarosch | Crystal structure refinement and reflectance measurements of hausmannite, Mn3O4[END_REF]. Profile matching (using FullProf software [START_REF] Rodríguez-Carvajal | Recent developments of the program FULLPROF[END_REF]) gives a = 0.575 nm and c = 0.938 nm for the pattern recorded at 450 • C and a = 0.576 nm and c = 0.938 nm for the pattern recorded at 500 • C. The lower c value in our samples could originate from partial oxidation of Mn 3+ into non-distorting Mn 4+ in the octahedral sites because, in hausmannite structure, the coordination octahedron around Mn 3+ are elongated approximately parallel to [001]. Profile matching also provides an estimation of the crystallite size D; we found D=12 nm for 450 • C and D = 14 nm for 500 • C.

It is worth noticing that the crystallization observed at 450 • C is associated with a small mass loss on TG curve but, unlike manganese oxalate, gives no detectable thermal event (Fig 3, x = 0.6). This could be an indication that, in this case, the crystallization involves only little change in the structural arrangement of atoms, because the amorphous state has a proto-spinel structure as it was previously assumed for nickel manganite spinels prepared by low temperature decomposition of mixed oxalates [START_REF] Laberty | A wide angle X-ray scattering (WAXS) study of nonstoichiometric nickel manganite spinels NiMn2ᮀ 3ı/4 O 4 + ı[END_REF].

The lower-left plot corresponds to the mixed oxalate Co 0.3 Mn 0.7 C 2 O 4 .2H 2 O (x = 0.9). The dehydration occurs above 130 • C giving a compound with a XRD pattern similar to the monoclinic cobalt oxalate with the space group P21/n (PDF # 00-037-0719). This compound decomposes at 240 • C generating a poorly crystalline phase corresponding to the cubic spinel (space group Fd-3 m). Then, from 350 • C, this phase is progressively converted in the tetragonal spinel. Profile matching gives a = 0.574 nm, c = 0.930 nm and D = 16 nm for the pattern recorded at 500 • C.

The critical concentration of distorting Mn 3+ required in the octahedral sites to trigger the Jahn-Teller effect is about 55% [START_REF] Jarosch | Crystal structure refinement and reflectance measurements of hausmannite, Mn3O4[END_REF]. Thus the transient formation of the cubic spinel could be explained by the partial oxidation, after the oxalate decomposition, of more than 45% of Mn 3+ into non-distorting Mn 4+ . From 350 • C Mn 4+ cations are reduced in Mn 3+ inducing the progressive conversion in the tetragonal spinel.

In the case of cobalt oxalate (lower-right plot) the dehydration occurs above 140 • C, giving the monoclinic cobalt oxalate (PDF # 00-037-0719), which starts to decompose at 230 • C to yield the cubic spinel Co 3 O 4 (space group Fd-3m, PDF # 00-042-1467). Profile matching, for the pattern recorded at 300 • C, gives a = 0.809 nm and D = 12 nm.

Characterization of oxides used as catalysts 3.3.1. XRD

Whatever the decomposition temperature of oxalates, the oxides used as catalysts were heated in air at 300 • C for 1 h. The crystal structure, lattice parameter and crystallite size of these materials are reported in Table 3. As shown in the previous section, the oxides for which x < 0.9 are amorphous. The XRD pattern of Co 0.9 Mn 2.1 O 4 shows very broad lines corresponding to a cubic spinel compound but is is too poorly crystallized to evaluate its cell parameter and crystallite size. For x > 0.9 all the products have a cubic spinel structure. The cell parameters of the mixed oxides are slightly larger than the cell parameter of Co 3 O 4 . Neutron diffraction data on ceramics showed that, for 1 < x < 2, the tetrahedral sites are fully occupied by Co 2+ and the substitution now occurs in octahedral sites where Mn 3+ is replaced both by Co 2+ and Co 3+ bearing in mind that each Co 2+ also implies the oxidation of one Mn 3+ in Mn 4+ to preserve the global electro neutrality [START_REF] Bordeneuve | Structural variations and cation distributions in Mn3-xCoxO4 (0 < x <3) dense ceramics using neutron diffraction data[END_REF]. Because Co 2+ and Mn 3+ ions have a larger diameter (respectively, 79 and 72 pm) than LS Co 3+ ions (55 pm) it is expected that the cell parameter of mixed oxides be larger than that of Co 3 O 4 . The biggest cell parameter is observed for x = 2 probably because this oxide contains the largest amount of Co 2+ . Furthermore the increase of cobalt content is associated with a significant growth of the crystallite.

Electron microscopy

At medium magnification, the SEM images of oxalates before and after decomposition are similar as shown in the case of Co 2.3 Mn 0.7 O 4 in Fig. 6 (left and center images). Thus, despite a mass loss close to 60%, when the reaction conditions allows con-Table 3 Microstructural and textural properties of mixed oxides (CoxMn3-xO4) obtained after heating in air at 300 • C. Structural parameters was determined by profile fitting of XRD patterns with FULLPROF software [START_REF] Rodríguez-Carvajal | Recent developments of the program FULLPROF[END_REF]; D is the crystallite size. BET surface area (SBET) and pore volume (Vpore) were calculated from N2 adsorption isotherms at 77 K. trolling the decomposition rate, the external shape of the oxalate particles is kept. It is possible on the central micrography to make out many cracks indicating that the oxide particles are highly fractured but the porosity is more easily evidenced on the TEM image which clearly shows the void between crystallites (right image).

The crystallite size is in good agreement with the size obtained from XRD.

Specific surface area, pore volume and pore size distribution (PSD)

The BET specific surface areas (S BET ) and the pore volume (V pore ) of the materials heated at 300 • C, calculated from the N 2 adsorption isotherms, are reported in Table 3. In the case of mixed oxides, we observe that S BET is strongly dependent of the decomposition conditions so that, for several preparations done in similar conditions, S BET can differ by about 20%. Monometallic and amorphous oxides have the lowest surface and pore volume. The highest S BET is obtained for x ≈ 2; above this value the increase of cobalt content is associated with a diminution of the surface area and pore volume.

The high surface area and porosity of oxides strongly contrast with the low textural properties measured on oxalates (see Table 2). This large interface is created because there is almost no shrinkage of the oxalate particles induced by the big mass loss occurring during decomposition process.

Some representative isotherms and their associated PSD, calculated using NLDFT method, are plotted in Fig. 7. The porosity of these material spans a wide range, from micropores to macropores; however their PSD varies considerably according to the cobalt fraction. The approximate correlation between the crystallite size and the position of the main peak of PSD lets suppose that the mesopores correspond to the inter-crystallite spaces. The pores in the macropore range are probably due to the inter-granular porosity corresponding for example to the void between the particles.

Catalytic activity

The catalytic activity of the monometallic and mixed oxides, for CO and C 3 H 8 total oxidation, is presented in Fig. 8. These measurements were not done at steady state but with a dynamic ramp rate of temperature (200 • C/h) and on-line monitoring using mass spectrometry. For CO oxidation, to reach the maximum activity, the catalysts were heated in a dry gas flow (20% O 2 in Ar). For propane oxidation this pre-treatment was not required to obtain optimum activity. We detected only CO 2 and H 2 O as products of oxidation and the carbon balance was close to 100% within 2%. All the catalysts were tested over the temperature range (20 tests). Some representative CO (left) and C 3 H 8 (right) conversion curves versus temperature (light-off curves) are shown on the upper plots of Fig. 8. We noticed that the same oxalate decomposed in what 7. N2 adsorption-desorption isotherms and their associated pore size distribution (PSD) calculated using NLDFT method [START_REF] Seaton | A new analysis method for the determination of the pore size distribution of porous carbons from nitrogen adsorption measurements[END_REF]. seems to be the same conditions could produce two catalysts having significant differences in activity. This is illustrated in the lower charts where the conversion observed at 60 • C for CO (left) and 200 • C for C 3 H 8 (right) are plotted versus the amount of cobalt. Despite the scattering of the results, the effect of the substitution of manganese by cobalt seems rather similar for both reactions. Especially we observe that, for x < 0.9, the substitution does not change the activity whereas for x = 0.9 the activity is clearly better (about 3 times). Besides Co 3 O 4 has a lower activity than Co 2.3 Mn 0.7 O 4 which seems to be the best composition.

The boost of activity for x = 0.9 is associated with a strong increase of S BET which rises from 100 to 230 m 2 /g. In an attempt to dissociate the effect of specific area from the influence of cobalt concentration, we calculated, from the conversion rate and S BET , the intrinsic activity, A i , defined as the number of reactant moles converted per second and per m 2 of catalyst. This intrinsic activity, plotted against the cobalt fraction, is shown in Fig. 9. In the case of CO oxidation A i does not appear to be directly dependent of cobalt fraction below x = 1.5. Then a strong improvement of activ- ity is observed for higher cobalt content. For propane oxidation, a similar correlation between A i and cobalt fraction is observed for x > 1.5. Below x = 1.5, the correlation is less clear. Several works have demonstrated that the octahedral sites are almost exclusively exposed at the surface of the spinel oxides [START_REF] Shelef | Ion scattering spectra from spinel surfaces[END_REF][START_REF] Jacobs | The surface of catalytically active spinels[END_REF][START_REF] Omata | Active site of substituted cobalt spine1 oxide for selective oxidation of CO/H2. Part II[END_REF]. Moreover it was also shown that the catalytic activity of cobalt oxides was due to Co 3+ ions in octahedral sites [START_REF] Omata | Active site of substituted cobalt spine1 oxide for selective oxidation of CO/H2. Part II[END_REF][START_REF] Xie | Low-temperature oxidation of CO catalysed by Co3O4 nanorods[END_REF]. Hence, for low cobalt content, it is expected that the catalytic activity does not change much because the substitution occurs only in the inactive tetrahedral sites where Co 2+ replaces Mn 2+ . When the tetrahedral sites are fully occupied by Co 2+ (x > 1) the substitution occurs in octahedral sites creating active Co 3+ ions [START_REF] Bordeneuve | Structural variations and cation distributions in Mn3-xCoxO4 (0 < x <3) dense ceramics using neutron diffraction data[END_REF].

The apparent activation energy for propane oxidation was determined from Arrhenius plots in the conversion range 0-10%. Except for manganese oxide, it was found almost constant at 60 ± 10 kJ/mol whatever the cobalt concentration. For MnO x the activation energy is slightly higher at 75 ± 10 kJ/mol.

In the case of the CO oxidation the conversion is too high to allow the calculation of activation energy.

For the best catalyst (Co 2.3 Mn 0.7 O 4 ) we followed the effect of the oxalate decomposition temperature (in the range 220-300 • C) on the catalytic activity for propane oxidation. Although the variations in activity were of the same order of magnitude as the differences between replicates it seems that the optimum decomposition temperature is 280 • C.

The long-term stability of the conversion was tested with the best catalyst (Co 2.3 Mn 0.7 O 4 ). The left plot of Fig. 10 shows that the CO conversion decreases by about 8% during the first half of the test. Then it seems to remain stable until the end of the test. Unexpectedly we observed variations of temperature associated with variations of conversion. We could not determine if it was the temperature change that induced conversion change or the converse. As regards propane oxidation, the right plot of Fig. 10 indicates that the conversion was stable for more than 14 h at 160 • C.

To compare the catalytic activity of our materials with the data reported in the literature we calculated the specific activity defined as the number of reactant moles converted per second and per gram of catalyst.

As it was demonstrated that CO conversion was strongly dependent upon the amount of water in the inlet gas [START_REF] Xie | Low-temperature oxidation of CO catalysed by Co3O4 nanorods[END_REF][START_REF] Yu Yao | The oxidation of hydrocarbons and CO over metal oxides: III. Co3O4[END_REF][START_REF] Grillo | Low temperature oxidation of carbon monoxide: the influence of water and oxygen on the reactivity of a Co3O4 powder surface[END_REF] we limited our comparison to propane oxidation. This comparison revealed (Table 4) that Co 2.3 Mn 0.7 O 4 activity was more than 50% higher than that of Co 3 O 4 catalysts found in the most recent publications and was similar to catalysts for which precious metal have been added to enhance the activity.

Conclusion

Cobalt-manganese mixed oxalate dihydrates crystallize in the monoclinic structure when the cobalt fraction is lower than 0.5 and in orthorhombic structure otherwise. The controlled decomposition of these oxalates near 200 • C, followed by a calcination at 300 • C, strongly restrains the shrinkage of particles and the crystallite sintering, producing mixed oxides Co x Mn 3-x O 4 with a very large surface area. For x < 0.9 these materials are amorphous. For x ≥ 0.9 they have a cubic spinel structure and their crystallite size increases with the cobalt fraction.

These spinel oxides exhibit an outstanding catalytic activity for propane oxidation. They are also active for CO oxidation even at ambient temperature. This high activity is correlated both with the surface area and the cobalt concentration. For manganese oxide the apparent activation energy for propane oxidation is 75 ± 10 kJ/mol whereas it is 60 ± 10 kJ/mol and nearly independent of cobalt fraction for the other catalysts. The most efficient material is Co 2,3 Mn 0,7 O 4 , which has an activity more than 50% higher than the best Co 3 O 4 catalysts reported in the literature.

Fig. 2 .

 2 Fig. 2. SEM images of some oxalates.

Fig. 3 .

 3 Fig. 3. TGA-DSC curves for some oxalates.

Fig. 4 .

 4 Fig.[START_REF] Patterson | The effect of carbon monoxide on the oxidation of four C6-C8 hydrocarbons over platinum, palladium and rhodium[END_REF]. Effect of cobalt fraction on the onset temperature of dehydration T i (upper-left), the width of the decomposition peak (upper-right), the enthalpy of dehydration (lower-left) and the enthalpy of decomposition (lower-right).

Fig. 5 .

 5 Fig. 5. XRD patterns recorded at increasing temperature during the decomposition of oxalates in air.

Fig. 6 .

 6 Fig. 6. SEM images of Co0.77Mn0.23C2O4.2H2O oxalate (left image) and Co2.3Mn0.7O4 oxide (middle image). TEM micrograph of Co2.3Mn0.7O4 oxide (right image).

Fig. 8 .

 8 Fig. 8. Examples of the CO (upper-left) and C3H8 (upper-right) conversion curves versus temperature (light-off curves). Effect of cobalt fraction on the conversion observed at 60 • C for CO (lower-left) and 200 • C for C3H8 (lower-right).

Fig. 9 .

 9 Fig. 9. Intrinsic activity plotted against the cobalt fraction for CO oxidation at 60 • C (left) and propane oxidation at 200 • C (right).

Figure 10 .

 10 Figure 10. CO (left) and propane conversion with time on stream for Co2.3Mn0.7O4 catalyst.

Table 2

 2 BET surface area (SBET) and pore volume (Vpore) of Co x/3 Mn (3-x)/3 C2O4.2H2O determined from N2 adsorption isotherms at 77 K.

	x	SBET (m 2 /g)	CBET	Vpore (cm 3 /g)
	0	26 ± 2	130	0.095 ± 0.008
	0.6	2.4 ± 0.2	120	0.010 ± 0.001
	0.9	4.0 ± 0.3	130	0.030 ± 0.002
	1.6	5.0 ± 0.4	150	0.062 ± 0.005
	2.0	4.4 ± 0.3	110	0.044 ± 0.003
	2.3	6.0 ± 0.5	160	0.074 ± 0.005
	3	4.0 ± 0.3	170	0.024 ± 0.002

  The values of SBET and Vpore are the average of several measurements.

	x	Structure	a (nm)	D (nm)	SBET (m 2 /g)	CBET	Vpore (cm 3 /g)
	0	Amorphous			90 ± 5	110	0.16 ± 0.01
	0.6	Amorphous			100 ± 10	140	0.21 ± 0.02
	0.9	Cubic spinel			230 ± 20	160	0.28 ± 0.03
	1.6	Cubic spinel	8.11	6	270 ± 30	50	0.33 ± 0.04
	2.0	Cubic spinel	8.12	7	260 ± 30	40	0.48 ± 0.05
	2.3	Cubic spinel	8.10	9	220 ± 20	35	0.29 ± 0.03
	3	Cubic spinel	8.09	16	60 ± 5	50	0.27 ± 0.01

Table 4

 4 Comparison of catalyst activity for propane oxidation at 200 • C.

	Ref.	Catalyst	Catalyst mass (g)	Flow rate (cm 3 /min)	Inlet C3H8 concentration (%)	%C3H8 conv. at 175 • C	Activity at 175 • C(mol s -1 g -1 )
	[49]	Co3O4	0.25	50	0.80	21	0.25
	[50]	4%Au/Co3O4	0.25	50	0.80	32	0.38
	[51]	Co3O4	0.05	98	0.37	5	0.27
	this work	Co2,3Mn0,7O4	0.05	98	0.37	8	0.43

Acknowledgments

This work was financially supported by the DGE and the Regional Council of Midi-Pyrénées in the framework of the SOFTAIR project.


Benjamin Lalanne

Nicolas Abi Chebel

Jiří Vejražka

Sébastien Tanguy

Olivier Masbernat

Frédéric Risso

Non-linear shape oscillations of rising drops and bubbles: Experiments and simulations

This paper focuses on shape-oscillations of a gas bubble or a liquid drop rising in another liquid. The bubble/drop is initially attached to a capillary and is released by a sudden motion of that capillary, resulting in the rise of the bubble/drop along with the oscillations of its shape. Such experimental conditions make difficult the interpretation of the oscillation dynamics with regard to the standard linear theory of oscillation because (i) amplitude of deformation is large enough to induce nonlinearities, (ii) the rising motion may be coupled with the oscillation dynamics, and (iii) clean conditions without residual surfactants may not be achieved. These differences with the theory are addressed by comparing experimental observation with numerical simulation. Simulations are carried out using Level-Set and Ghost-Fluid methods with clean interfaces. The effect of the rising motion is investigated by performing simulations under different gravity conditions. Using a decomposition of the bubble/drop shape into a series of spherical harmonics, experimental and numerical time evolutions of their amplitudes are compared. Due to large oscillation amplitude, non-linear couplings between the modes are evidenced from both experimental and numerical signals; modes of lower frequency influence modes of higher frequency, whereas the reverse is not observed. Nevertheless, the dominant frequency and overall damping rate of the first five modes are in good agreement with the linear theory. Effect of the rising motion on the oscillations is globally negligible, provided the mean shape of the oscillation remains close to a sphere. In the drop case, despite the residual interface contamination evidenced by a reduction in the terminal velocity, the oscillation dynamics is shown to be unaltered compared to that of a clean drop.

I. INTRODUCTION

Interfacial dynamics plays a major role in dispersed two-phase flows since it governs the shape and motion of drops or bubbles. Shape oscillations involve eigenmodes characterized by their frequency and damping rate. These oscillation modes are excited by fluctuations of the surrounding flow. Depending on the time scales of these fluctuations relative to the characteristic time scales of interface dynamics, the shape may undergo large oscillations, which can either cause the breakup or just relax towards the equilibrium shape. [START_REF] Kang | Bubble dynamics in time-periodic straining flows[END_REF][START_REF] Risso | Oscillations and breakup of a bubble immersed in a turbulent field[END_REF][START_REF] Galinat | Dynamics of drop breakup in inhomogeneous turbulence at various volume fractions[END_REF] In this way, the interfacial dynamics also critically affects the size distribution of bubbles or drops in multiphase flows.

The knowledge of bubble or drop eigenmodes is therefore a crucial issue for multiphase flows. They can be derived from the linearized Navier-Stokes equations when considering small oscillation amplitudes. The case of a drop oscillating around a spherical equilibrium shape in the absence of gravity is well known since the fundamental works of Miller and Scriven 4 and Prosperetti. [START_REF] Prosperetti | Normal mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid[END_REF][START_REF] Prosperetti | Free oscillations of drops and bubbles: The initial-value problem[END_REF] However, in many practical situations, drop shape eigenmodes are modified by various physical mechanisms such as attachment to a support, action of surfactants, gravity, or drop motion. In the absence of gravity and of imposed external flow, the cases of a drop constrained to a support [START_REF] Strani | Free vibrations of a drop in partial contact with a solid support[END_REF][START_REF] Strani | Viscous oscillations of a supported drop in an immiscible fluid[END_REF][START_REF] Bostwick | Capillary oscillations of a constrained liquid drop[END_REF][START_REF] Prosperetti | Linear oscillations of constrained drops, bubbles, and plane liquid surfaces[END_REF] and of a free drop in the presence of surfactant [START_REF] Lu | Shape oscillations of drops in the presence of surfactants[END_REF] have been solved. Dealing theoretically with buoyancy and drop motion is difficult owing to the coupling between the shape oscillations and the drop motion. Recently, the linear oscillations of a bubble or a drop rising in a liquid otherwise at rest have been investigated using direct numerical simulations. [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] For both bubbles and drops, the mode frequency is found to be a slightly decreasing function of the Weber number based on the rising velocity. The damping rate for the drops and bubbles shows totally different trends. The damping rate of a drop is an increasing function of the Weber number, whereas the damping rate of a bubble is almost unaffected by the rising motion when it has reached its terminal shape and velocity. For millimeter-sized drops and bubbles rising at constant velocity within a liquid, the shape eigenmodes are weakly affected by the rising motion provided the Weber number based on terminal velocity is less than one.

We may question the relevance of these conclusions from numerical simulations for practical situations. Is it possible to produce experimental conditions in which they are valid? An experiment, in which a drop or a bubble undergoes axially symmetric damped oscillations, with conditions similar to those in computations, has been described in Ref. 13. The drop is initially attached to the tip of a capillary. A sudden translation of the capillary brings about the drop detachment by causing a significant deformation of the interface. The drop then rises under the action of buoyancy and its shape oscillates. Two complications generally take place in experiments. The first arises from the fact that several modes are present simultaneously. Their amplitudes are difficult to control and usually exceed the linear limit, which is around 10% of the radius. [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] Second, the presence of surface-active contaminants can hardly be avoided, especially in the case of drops in another immiscible liquid.

The purpose of the present work is thus to investigate the moderate non-linear shape oscillations of a rising bubble (Sec. III) and a rising drop (Sec. IV) that appear in such an experimental device, by means of a detailed comparison of experimental observations with numerical simulations. In the numerical simulations, the same initial shape is prescribed as that measured in the experiments at the instant of detachment. Computations are also carried out for zero-gravity and high-gravity conditions. In the experiments, great care is taken to keep the fluids pure. For the bubble case, water remains pure enough so that the rising velocity is that of a clean bubble. For the drop case, some surface-active contaminants remain present. Owing to this variety of conditions, the comparison allows us to examine and disentangle the influence of both gravity and slight interface contamination on the oscillation dynamics.

II. PHYSICAL AND NUMERICAL EXPERIMENTS

A. Experimental procedure

The experimental setup is the same as that used by Abi Chebel et al. [START_REF] Chebel | Shape oscillations of an oil drop rising in water: Effect of surface contamination[END_REF] At a bottom of a vessel (11 × 11 cm in width and 26 cm height) filled with water, a bubble of air or a drop of heptane is injected through a glass capillary (250 µm in diameter). When the bubble/drop has reached the desired size, the capillary is rapidly moved vertically by the action of an electromagnetic coil, leading to its detachment. The motion of the capillary also induces relatively large, but axially symmetric initial deformation of the bubble/drop, due to which its shape oscillates during the free rise. The translation and the shape oscillations of the bubble/drop are recorded by a high-speed camera with a resolution of 5.9 µm/pixel for the drop and 6.8 µm/pixel for the bubble; the framing rate is 18 000 images per second. The position of the interface is detected in the recorded images by searching for gray level threshold, which is determined using Otsu's method. [START_REF] Otsu | Threshold selection method from gray-level histograms[END_REF] The interface identification includes also image interpolation, which increases the effective resolution of the detection of boundary location up to about one-fifth of a pixel size.

In the experiments, care is taken to ensure the purity of the system. Water is initially distilled and purified using a Watrex Ultrapur 40 system. Then, it is additionally cleaned within the experimental setup by circulating through a filter with activated-carbon cartridge in order to remove the residual surface-active contaminants. As we will show in Sections III and IV, this treatment is enough to ensure that the bubble interface is free of contaminants. However, for the heptane drop, measurements of terminal velocity indicate that the interface is still contaminated. The main source of contamination is the plastic syringe (Braun), in which the heptane (Lachner, p. a. grade) is held prior to its injection.

B. Numerical procedure

Axisymmetric simulations of the oscillating and rising air bubble or heptane drop are carried out with the aim to mimic experimental conditions. The numerical methods used are that proposed by Sussman et al. [START_REF] Sussman | A level-set approach for computing solutions to incompressible two-phase flows[END_REF][START_REF] Sussman | A sharp interface method for incompressible two-phase flows[END_REF] and Liu et al.; 17 they are implemented in our in-house code, the DIVA code (see the works of Tanguy and Berlemont [START_REF] Tanguy | Application of a level-set method for simulation of droplet collisions[END_REF] and Lalanne et al. [START_REF] Lalanne | On the computation of viscous terms for incompressible two-phase flows with level set/ghost fluid method[END_REF] for details and validations). In the framework of an Eulerian one-fluid approach, the DIVA code simulates two-phase flows by solving the incompressible Navier-Stokes equations using a projection method, and the interface is captured by means of the Level-Set method. In this method, the interface is defined as the zero-level curve of a distance function and its evolution in time is obtained by solving an advection equation. In complement, the Ghost-Fluid method is implemented to compute the viscosity and pressure jumps across the interface in order to satisfy the balance of the normal stresses.

The ability of the code to calculate rising velocity of bubbles and drops and free-gravity shape-oscillations has been proved in Ref. 12, where the validation was based on mesh convergence studies, on comparison of the computed terminal velocity with the theory of Moore [START_REF] Moore | The boundary layer on a spherical gas bubble[END_REF] and on comparison of the computed shape oscillations with the linear theory. [START_REF] Miller | The oscillations of a fluid droplet immersed in another fluid[END_REF][START_REF] Prosperetti | Normal mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid[END_REF] To ensure high accuracy, a numerical resolution of 32 uniform cells over the equivalent radius of the bubble or drop is employed in this study. The radius of the cylindrical calculation domain is eight times the radius of the inclusion in order to avoid confinement effects. The flow in the capillary through which the drop/bubble is injected, is not explicitly modeled since we study only the relaxation of the inclusion during its ascending motion. The numerical simulations are initialized in the following ways:

• The initial shape of the inclusion is matched to that observed in experiments just before its detachment. At this instant, the deformation reaches a maximum, and thus, the calculation recovers all the surface energy related to the deformation. • Initial velocities of both fluids are set to zero in the simulation. This assumption is equivalent to neglecting the velocities of the fluids (not measured) and of the inclusion generated by the motion of the capillary prior to the detachment.

C. Physical parameters

The physical parameters for the drop and the bubble cases are given in Table I, including values of density and viscosity of the internal (ρ i , µ i ) and the external (ρ o , µ o ) phases, equivalent radius R of the undisturbed bubble/drop, and the interfacial tension σ. The problem is characterized by the initial shape of the interface and by a set of four independent dimensionless numbers (Table II) consisting of the density ratio ρ o /ρ i , the viscosity ratio µ o /µ i , and the two Reynolds numbers. The first is the

= ρ o V ∞ 2R/µ o
based on the terminal velocity V ∞ . The choice of these two Reynolds numbers is particularly relevant to assess the role of the rising motion on the oscillations since their ratio compares the characteristic velocities involved in the two kinds of motion. In order to measure the importance of gravity and inertia effects upon the bubble/drop deformation, the Bond number Bo = (ρ o -ρ i )g (2R) 2 σ and the Weber number

W e ∞ = ρ o V 2 ∞ 2R σ
are also introduced. The value of the Bond number is low for both experiments with the bubble and the drop (Bo = 0.08).

Note that in this study, the Reynolds number of oscillation is high enough to consider that shape-oscillations are dominated by inertia, the role of viscous effect being limited to cause a slow damping of the oscillations. Moreover, since Re OSC and Re ∞ are of the same order of magnitude, a coupling between the translational and oscillating motions is possible. In addition to the experimental Bond number (Bo = 0.08), a case with no gravity (Bo = 0) and a case with augmented gravity (Bo = 0.40) will also be numerically simulated to better analyze this point.

D. Mode extraction

Figs. 1(a) and 1(b) show several computed shapes of the bubble and of the drop, respectively, superposed on the corresponding image taken in the experiment just before the detachment. The uppermost contour corresponds to the computed terminal shape once the steady motion is reached.

For both the experimental and computational results, the interface contour is extracted and decomposed on the basis of the axisymmetric spherical harmonics, given by the Legendre polynomials P l as

r(θ,t) = R       a 0 (t) + ∞ l=2 a l (t) P l (cos θ)       , (1)
where r and θ are the spherical coordinates defined in Fig. 1(a). The decomposition of the interface is written in the frame of reference in which the amplitude of the first harmonic vanishes, a 1 = 0, which describes the vertical translation of the inclusion. Note that the normalized amplitude a 0 of

a l (0) l = 0 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 l = 8 l = 9 l = 10 l = 11 l = 12
Bubble 0.9904 0.1810 -0.0718 0.0694 -0.0552 0.0458 -0.0451 0.0389 -0.0358 0.0321 -0.0359 0.0263 Drop 0.9912 0.1858 -0.0644 0.0555 -0.0351 0.0273 -0.0232 0.0183 -0.0225 0.0211 harmonic l = 0 is slightly lower than unity since this harmonic ensures volume conservation by compensating for the increase of volume by higher harmonics. In the following, harmonic 0 will be disregarded as its variations are negligible compared to those of the other harmonics. By decomposing the shape observed in experiments short time before detachment, the initial level-set function for the numerical simulation is obtained. To ensure good accuracy, terms with order up to l = 12 (bubble) and l = 10 (drop) are used. Coefficients describing the initial shape at t = 0 are given in Table III, and the corresponding shape is also shown in Figs. 1(a) and 1(b) (lowermost contour).

From the time evolution of the amplitude of the different harmonics a l (t), we will extract the frequencies ω l and damping rates β l of the corresponding oscillation modes. These values will be compared against the values given by the linear theory ω th l and β th l , which are computed from the expressions of Ref. 5 and given in Table IV.

III. RESULTS FOR THE BUBBLE

Fig. 2(a) plots time evolution of the Reynolds number Re ASC = ρ o V (t)2R/µ o based on the instantaneous rise (ascension) velocity V (t) of the center of mass of the bubble, and Fig. 2(b) shows the evolution of the aspect ratio χ(t), which is defined as the ratio of horizontal to vertical maximum bubble dimension.

Even though the initial velocity of the bubble is different in the experiment and simulation (the initial velocity in the experiment is about 5% of the terminal velocity because of the impetus received by the bubble from the moving capillary), its evolution is comparable from the beginning of the first oscillation period until reaching the same terminal velocity. Since the simulation assumes a clean interface boundary condition, this demonstrates that the experimental device is clean enough for the drag force to match the case of a non-contaminated bubble. To reinforce this conclusion, let us consider a balance of forces to reconstruct the evolution of the rising velocity. The balance between the added-mass, drag, and buoyancy forces acting on the bubble reads

4 3 πR 3 d dt [(ρ i + ρ o C M (t)) V (t)] = -ρ o C D (t) πR 2 2 V (t) 2 + (ρ o -ρ i )g 4 3 πR 3 . (2)
At each instant, the added-mass coefficient C M is taken as that of the ellipsoid with the same aspect ratio χ, given by

C M = β 0 /(2 -β 0 ), with β 0 = (ζ 2 + 1)ζ cot -1 ζ -ζ 2 and ζ = ( χ 2 -1) -1/2 .
For the drag coefficient C D , we use the following:

• for Re ASC < 50, the expression for a spherical bubble given by Mei et al. [START_REF] Mei | A note on the history force on a spherical bubble at finite Reynolds number[END_REF] is • for Re ASC ≥ 50, the expression for a spheroidal bubble given by Moore [START_REF] Moore | The velocity of rise of distorted gas bubbles in a liquid of small viscosity[END_REF] is

C bubble D = 16 Re ASC        1 + 8 Re ASC + 1 2 1 + 3.315 Re 1/2 ASC -1       (3)
C D = 48 Re ASC G( χ)       1 + H( χ) Re 1/2 ASC       , (4)
where G( χ) and H( χ) are two functions of the aspect ratio given in Moore's paper.

Using the known evolution of the aspect ratio, either from the experiment or from the numerical simulation, the velocity evolution is obtained by the integration of Eq. ( 2) and the results are also shown in Fig. 2(a). We observe that this model matches well both experimental and numerical results. The possible influence of surface-active contaminants on the bubble motion is thus excluded. Note that the shape oscillations affect periodically the rising velocity of the bubble due to periodic evolution of its added-mass: velocity oscillates at mode 2 frequency as previously demonstrated in Refs. 23 and 12. In Fig. 2, we also observe that the bubble velocity in the simulation without gravity is not exactly zero. This is due to the presence of odd modes in the deformation that breaks the symmetry between the lower and the upper parts of the bubble and consequently generate small displacements of its center. Nevertheless, this motion is negligible compared to the velocity variations observed in the case with gravity.

Concerning the shape, Fig. 2(b) shows that the oscillations cause significant variations of the aspect ratio of the bubble, ranging between 0.8 and 1.5. The bubble oscillates around a mean shape, which remains nearly spherical and flattens only slightly, reaching a final aspect ratio of χ ∞ = 1.04 once the terminal velocity is attained and oscillations decay. Hydrodynamic stresses resulting from the rising motion, which deform the bubble, are of low magnitude (W e ∞ = 0.3); consequently, we observe comparable fluctuation amplitudes of χ in the simulations at Bo = 0.08 and at Bo = 0.

The shape oscillations are studied in detail by decomposing the interface in the basis of the spherical harmonics (Eq. ( 1)). The evolution of amplitudes for orders l = 2-7 is shown in Fig. 3. Amplitudes are normalized by their initial value and time by the theoretical period of oscillation of each mode, T th l = 2π/ω th l . Note that ω th l increases with the order of the mode: for instance, ω th 7 ≈ 6 ω th 2 . In Fig. 3, the physical time corresponding to the same dimensionless time is hence decreasing with the mode order.

The agreement between the experimental and numerical results on the evolution of various harmonics is excellent for lower harmonics. Discrepancies appear for harmonics of higher order (l = 7 or superior), highlighting the limitations of the two approaches used here. In the experiments, the main limitation is the temporal resolution. By contrast, in the numerical calculation, this issue can be solved by reducing the time step as necessary. The numerical tool is rather limited by its spatial resolution, which becomes problematic when the oscillation amplitude is low. We have estimated [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] that the resolution is sufficient if the oscillation amplitude exceeds 15% of the grid spacing, that is, 0.5% of the undisturbed radius of the particle. We observe in Fig. 3 that the oscillations are well reproduced whatever the tool is considered and a disagreement appears only at mode 7. Comparisons of Fig. 3 suggest that both measured and calculated modes are reliable, including low-amplitude oscillations of harmonics 3-6, which clearly show complex modulations. Note that such results would be questionable if obtained with only one of the approaches. The good agreement of both instantaneous shape and velocity between the experiment and the simulation justifies a deeper investigation of oscillating modes and non-linear effects.

Two other representations are used to characterize the oscillations. Fig. 4 shows power spectra of the a l signals for spherical harmonics l = 2-7, normalized by their variances. The theoretical natural frequencies ω th l of modes l = 2-7 are highlighted. Fig. 5 plots the time evolution of the local maxima of |a l (t)| for l = 2-5 in the both cases without and with gravity (Bo = 0 and Bo = 0.08, respectively). For clarity purposes, only numerical data are displayed in this figure but experimental signals exhibit similar evolutions of local maxima. On this graph, straight lines represent the theoretical damping rates β th l in the case without gravity. In Fig. 4, the spectrum of each spherical harmonic l exhibits a dominant frequency very close to the theoretical eigenfrequency ω th l , for harmonics 2-5. For each of these harmonics, secondary peaks of smaller amplitude occur at lower-order harmonic frequencies. (For harmonic 6 and 7, dominant frequency is no longer the corresponding theoretical frequency but that of the harmonic of adjacent lower order). Figure 5 shows that even though the local extrema of the oscillation amplitudes of each harmonics are not distributed along a straight line, they oscillate around a line corresponding to a constant damping rate β l , which is only about 10% higher than β th l for harmonics l = 2-4. Thus, the linear theory, which (i) does not consider the effect of the rising motion and (ii) assumes oscillations of low amplitudes, still correctly predicts the main trends of the bubble oscillations at moderate amplitudes a 2 (0) ≈ 0.2 in terms of both dominant frequency and damping rates. We also observe in Fig. 5 that the decay of oscillations is comparable for the cases at Bo = 0.08 and Bo = 0, except for harmonic 3 which will be discussed later. This proves that the low discrepancy between β l and β th l mainly comes from the non-linearity of the oscillations and not from the rising motion. Hence, gravity can be disregarded in this range of parameters, in accordance with the theoretical study of Meiron [START_REF] Meiron | On the stability of gas bubbles rising in an inviscid fluid[END_REF] and our previous numerical study. [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] In particular, we can consider that the bubble shape oscillates around a spherical shape. Eigenvectors of oscillation modes are therefore the spherical harmonics defined by the P l polynomials and the eigenfrequencies are given by ω th l . Consequently, the presence of a secondary frequency ω th n , with n l in the a l signal is undoubtedly the signature of non-linear interactions between the modes. The non-linear coupling between the modes is exhibited by the modulation of amplitude of harmonics 3-7 (Figs. 3 and5). At Bo = 0.08, Fig. 5 shows that the extrema of amplitudes of harmonics 3-5 oscillate around an average decay close to β th l at a frequency that is almost the same for all harmonics and close to that of mode 2. This suggests periodic energy exchanges between the modes. Remarkably, no perturbation is seen on the amplitude of a 2 .

As apparent from the spectra (Fig. 4), a 2 is the only harmonic that does not contain any other frequency than its natural one. This is linked to a more general observation that is valid for all modes: only secondary frequencies corresponding to modes of lower orders are present in the signal of a given mode. For example, signal of harmonic 5 is dominated by ω 5 but it also contains three lower frequencies, ω 4 , ω 3 , and ω 2 . In return, it does not contain ω 6 and ω 7 (Fig. 4). Moreover, most of the energy is distributed on ω 5 and ω 4 , peaks at ω 2 and ω 3 being small. Interactions between the modes of very different orders are weak, probably because the fast damping of a high-order mode does not allow it to disturb a low-order mode.

Let us consider now a peculiar characteristic of mode 3, which is the only one that shows a difference between the cases with and without gravity. At Bo = 0, a 3 is not affected by the presence of the other modes: it contains a single peak at its natural frequency (Figs. 3 and4) and its decay is monotonous (Fig. 5). On the contrary, at Bo = 0.08, a 3 shows significant secondary oscillations at ω 2 , which also modulates its decay. The weak effect of gravity is therefore both sufficient and necessary to generate a notable coupling between modes 2 and 3. A possible interpretation of this phenomenon is that the coupling between the odd and even modes requires that the fore-aft symmetry is broken. Since mode 3 is not affected by odd modes of higher order, the slight average deformation generated by the rising motion at Bo = 0.08 is the only way to allow mode 3 to be affected by mode 2. For mode 5, the presence of mode 3 is sufficient to allow the coupling with mode 4. The same argument holds for odd modes of higher orders.

Except for the coupling between modes 2 and 3, gravity does not affect the oscillations at Bo = 0.08. An additional simulation at Bo = 0.40 has been carried out in order to show the effect of a significant gravity. In this case, the larger rising velocity of the bubble (Re ∞ = 358, W e ∞ = 2.4) results in a larger and faster deformation of its shape with a terminal aspect ratio χ ∞ = 1.5. The spectra of the oscillation amplitude are shown in Fig. 4. New secondary peaks are present and, in particular, the frequency of harmonic 2 clearly appears in the oscillations of each spherical harmonic until l = 7. However, this is not the signature of enhanced non-linear couplings. As shown from numerical simulations conducted in the linear regime, [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] the eigenmodes of oscillations around a shape that is deformed by the rising motion no longer correspond to the spherical harmonics. The decomposition into spherical harmonics is therefore not able to separate the different eigenmodes. In the present case, at Bo = 0.40, the secondary peaks in a l spectra result both from the fact that spherical harmonics do not describe the eigenmodes of oscillations of an oblate bubble and from non-linear couplings between the modes. During the transient rising stage, while the bubble deforms due to inertia, the oscillating frequency ω 2 is found to decrease slightly in time compared to the value without gravity, reaching 0.9 ω th 2 when the bubble rate of deformation no longer evolves, i.e., when its rising velocity is close to the terminal velocity. This value is consistent with the theoretical prediction of Meiron [START_REF] Meiron | On the stability of gas bubbles rising in an inviscid fluid[END_REF] and the numerical result of Lalanne et al. [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] for a bubble at χ ∞ = 1.5. At this time, the damping rate of amplitude of spherical harmonic 2 is measured to be constant, with a value close to β th 2 . To conclude, for the experimental bubble at Bo = 0.08, the analysis of the rising velocity and shape-oscillations reveals that its interface can be considered as clean in experiments since both experiments and numerical simulations give the same results and compare well with the model incorporating the drag force that corresponds to the contaminant-free case. Effect of buoyancy on the oscillations is globally negligible (except on the coupling of oscillations of harmonics 3 and 2), because the terminal shape of the bubble is nearly spherical, so that the eigenmodes of oscillation correspond to the spherical harmonics. The dominant frequency of each mode is given by the linear theory without gravity, whereas its global damping rate is only 10% larger than the linear prediction. Non-linear couplings between the different modes of oscillation have been analyzed from the spectra of each spherical harmonic and the temporal decay of its amplitude. Modes of lower frequency influence modes of higher frequency, whereas the reverse is not observed.

IV. RESULTS FOR THE DROP

We consider now the case of the liquid drop. Figs. 6(a) and 6(b) show the time evolutions of Re ASC and χ in the experiment and in the simulations for Bo = 0.08 and Bo = 0. The origin of time has been defined as the instant of detachment of the drop from the capillary. The experimental evolution of Re ASC for negative times illustrates what happened before the drop release: the capillary moves up and down and the detachment of the drop takes place during the deceleration stage. Contrary to the previous experiment with the bubble, the initial velocity of the drop at the instant of detachment is large and even exceeds the terminal rising velocity. The initial condition in the simulation (with zero velocity) is hence significantly different from the experimental condition in this case. Nevertheless, different initial conditions for velocity should not cause differences between the results once the terminal velocity, which is governed by the equilibrium between buoyancy and drag forces, is attained. Yet, we observe in Fig. 6(a) that the experimental terminal rising velocity is only 60% of the terminal velocity in the simulation. This discrepancy suggests that surface-active contaminants are adsorbed at the drop interface and they partially immobilize it by being convected to the rear of the drop; it results in an increase of the drag that becomes comparable to that exerted at the surface of a solid particle exactly like if the interface was fully immobilized. [START_REF] Cuenot | The effect of sligthly soluble surfactants on the flow around a spherical bubble[END_REF] However, this residual contamination induces a negligible variation of surface tension between heptane and water, since we find the value for a clean interface. To confirm the presence of surfactant in the experiment, we reconstruct the time evolution of the drop velocity using the force balance between buoyancy, drag, and added-mass forces during the unsteady rising motion of the drop. The force balance is the same as for bubbles, given by Eq. ( 2). The added-mass coefficient C M is calculated from the aspect ratio χ(t) in the same way as for the bubble, and the value of χ(t) is taken either from the experiment or from the Direct Numerical Simulation (DNS) results. The expressions of the drag coefficient C D are chosen as follows:

• for the experimental case, we use the drag coefficient of a solid sphere proposed by Schiller and Naumann, [START_REF] Schiller | A drag coefficient correlation[END_REF] to assess the assumption of a contaminated interface, • for the numerical case, we use the correlation of Rivkind and Ryskin 27 valid for a spherical drop with a clean interface,

C D = 24 Re ASC (1 + 0.15 Re 0.687 ASC ), (5)
C D = C bubble D + µ i /µ o C solid sphere D 1 + µ i /µ o , (6)
where C bubble D is the drag coefficient of a spherical bubble [START_REF] Moore | The boundary layer on a spherical gas bubble[END_REF] (Eq. ( 4)), and C solid sphere D

is the drag coefficient of a solid sphere [START_REF] Schiller | A drag coefficient correlation[END_REF] (Eq. ( 5)).

Evolutions of the rising velocity obtained by solving the analytical model of force balance (2) for both experimental and the numerical cases (using their corresponding initial velocity: that measured at the instant of detachment for the experimental case, zero for the numerical case) are plotted in Fig. 6(a). It can be shown that this model reproduces well both the experimental results with contaminatedinterface assumption (black and dashed red curves) and the numerical results with clean-interface assumption (dark and dashed light blue curves); thus, it contains the main physics of the problem. The rising velocity oscillates slightly at frequency ω th 2 due to varying added-mass C M , but these oscillations are of lower amplitude than in the bubble case as a consequence of a higher density ρ i of the inner fluid. The small discrepancy that exists between measured or simulated velocity and that from the model comes from neglecting the effect of drop deformation on the drag coefficient and also from neglecting the history force in the force balance (which is more important than in the case of bubbles). However, the agreement of the model with our results allows us to conclude that the presence of surfactants explains the slower rising velocity in the experiments by immobilizing the interface, leading to a drag force similar to that of a solid sphere.

Fig. 6(b) shows that the oscillations cause large variations of χ, ranging between 0.8 and 1.4. Surprisingly, in spite of interface contamination, the evolution of χ is the same in the experiment and in the simulation, indicating that the contaminants do not affect significantly the drop-shape dynamics in the experiment. It also confirms that they are present in residual concentration only, which is consistent with the fact that the measured interfacial tension is close to that of a clean interface. The drop shape has an aspect ratio of χ ∞ = 1.02 after reaching its steady rise, consistent with the low values of Bond and Weber numbers, Bo = 0.08 and W e ∞ = 0.23. The buoyancy forces are hence not sufficiently intense compared to the surface tension forces to deform significantly the drop and consequently, the time evolution of χ in the simulation without gravity is the same as in the case at Bo = 0.08.

To analyse more precisely the shape oscillations of the drop, let us consider in Fig. 7 the decomposition of its shape into spherical harmonics (Eq. ( 1)), at each instant of measurement or calculation done at Bo = 0.08. Results of the simulation without gravity are also included in the figure and time has been normalized by the measured frequency of the different harmonics (which is, and discussed later). Graphs of Fig. 7 show an excellent agreement between the experimental and numerical results, including the high order spherical harmonics with low amplitudes (cf. Table III). This confirms the ability of both experimental and numerical tools to handle the non-linear oscillations of the drop. A strong conclusion can be drawn: the contamination by residual surfactants, responsible of the low value of the experimental drop terminal velocity, does not affect the shape-oscillations dynamics in present experiments, for none of the studied spherical harmonics. As it has been previously demonstrated by Abi Chebel et al., [START_REF] Chebel | Shape oscillations of an oil drop rising in water: Effect of surface contamination[END_REF] non-ultrapure experimental conditions affect the shape-oscillation dynamics through an increase of the damping rate of oscillations, whereas in cleaner conditions (like those in present experiments), shape oscillation dynamics recover those predicted by theories for contaminant-free interface, even though the terminal rising velocity is affected by residual contamination. The fact that (i) we observe an increase of the dissipation of the energy associated to the rising motion due to surface contamination and (ii) we do not observe an increase of the dissipation of the energy associated to the oscillating motion is probably related to the location where dissipation of energy takes places. For rising drops, energy of the translating motion is mainly dissipated in a boundary layer that develops along the interface and in the drop wake. For oscillating drops, the linear theory of oscillation (Ref. 5) shows that dissipation is mainly due to the existence of an oscillating boundary layer close to the interface (see Fig. 8(a) of Ref. 12 for an illustration). Our observations lead us to think that the boundary layer associated with rising motion and that associated with interface oscillations have a distinct dynamics since they are differently affected by contaminants. Further investigations of these boundary layers are required to validate this interpretation.

In accordance with the observed evolution of the aspect ratio, simulations at Bo = 0.08 and Bo = 0 provide a very similar behavior for all a l signals, confirming that the effect of gravity and of rising motion on shape-oscillations are negligible at Bo = 0.08. This is consistent with the fact that comparable oscillations at Bo = 0.08 are obtained in the experiment and in the simulation, even though the rising velocity is very different in the two cases. Indeed, the Weber number is too low to induce an effect of the rising motion on the oscillation dynamics (as already shown in Ref. 12 in the case of linear oscillations). Therefore, the mean shape around which the drop oscillates is almost a sphere with a negligible deformation at steady state (a 2 = -0.012). Consequently, the eigenmodes of oscillation still coincide with the spherical harmonic basis, as it is the case at Bo = 0, and the oscillation dynamics is not influenced by gravity.

The same representations as for the bubble case are used to characterize the oscillations of the different harmonics. Frequencies are analyzed from the power spectra of the a l signals for l = 2-7 (Fig. 8), and the time evolutions of the local extrema of |a l (t)| (Fig. 9) allow us to characterize the damping rates of the oscillation amplitudes for harmonics l = 2-5. We compare again measured ω l and β l with the linear theory of oscillation and we investigate the non-linear effects and the influence of gravity on the oscillation dynamics. Concerning the dominant frequency in each power spectrum, both the numerical simulation (at Bo = 0.08 and Bo = 0) and the experiment are in agreement with ω th l for all harmonics within the expected accuracy, ranging between 0.5% and 5% depending on the value of l.

Concerning the global damping rates, experiment and simulation are again in good agreement with a discrepancy of 3% for harmonics l = 2-4 and show that a 2 and a 3 decay at a rate higher by 8% than β th l . Since the same value of β 2 is found in the simulations at Bo = 0.08, at Bo = 0, and in the experiment, we can attribute this slight increase of the damping rate of mode 2 to non-linear effects that are not considered in the theory. Nevertheless, an increase of 8% is moderate compared to the influence of gravity or surfactants which have been shown to increase the damping rate β more radically (by a factor of two or three) at larger Bond numbers [START_REF] Lalanne | Effect of rising motion on the damped shape oscillations of drops and bubbles[END_REF] or in less clean conditions. [START_REF] Chebel | Shape oscillations of an oil drop rising in water: Effect of surface contamination[END_REF] The linear theory hence provides correctly the main trends of frequency and global damping rates of the oscillations, which remain close to ω th l and β th l . Non-linear effects in the oscillations are however noticeable. In Fig. 8, the presence of several frequencies is visible in harmonics 4-7. This reveals the couplings between the modes, which is identical in the experiment and in the simulation at Bo = 0.08. As in the case of the bubble, only secondary frequencies of modes of lower orders are visible in the spectrum of a given mode. In particular, non-linear interactions are not detected in the dynamics of mode 2. In Fig. 9, we observe that non-linear mode coupling results in modulations of the decay rate of a given mode. These modulations have the same dominant frequency for l = 3 and l = 4 and this frequency is close to ω th 2 , as in the bubble case. Amplitude variations at Bo = 0.08 and Bo = 0 are similar, except for mode 3. This suggests again that the coupling between the odd and even modes requires the breaking of the fore-aft symmetry.

An additional simulation has been carried out for augmented-gravity condition (Bo = 0.40). This simulation has been stopped before the drop has reached its terminal velocity, at time t * = 8T th 2 , when the rising Reynolds number of the drop is Re ASC (t * ) = 278 and its mean aspect ratio is χ(t * ) = 1.15. The global damping of the oscillations β 2 is observed to increase as the drop accelerates, reaching a value of 1.3 β th 2 at t * when the instantaneous Weber number is W e(t * ) = 1.4, in agreement with Fig. 7(b) of Ref. 12. Even if the deformation of the mean shape remains moderate, it is sufficient to change the eigenmodes of oscillations that no longer correspond to the spherical harmonics. Consequently, the spectra of a l signals (especially those for l = 3-5) exhibit several low frequencies that are not present at Bo = 0.08: the peak at ω th 2 in the signal of harmonic 3 becomes significant and ω th 2 and ω th 3 appear in harmonics 4 and 5. To summarize, in the case of the drop at Bo = 0.08, the comparison between the simulation and experiment proves that, despite all the precautions taken to clean the experimental device, the interface is contaminated. This strongly affects the rising motion but does not have any significant effect on the shape-oscillation dynamics of the drop. The analysis of these shape-oscillations through a modal decomposition shows the existence of interactions between the modes due to their initial amplitudes reaching 0.2R. Nevertheless, the dominant frequency and the global damping rate of each mode of oscillation are still accurately predicted by the linear theory of oscillation. The gravity effects can be disregarded for the shape-oscillations of that drop (except for a non-linear effect of second order visible on the oscillation amplitude of mode 3).

V. CONCLUSION

This study has compared experimental observations and numerical simulations of a rising bubble or a drop, the shape of which oscillates. Despite some differences in the initial conditions between the simulation and the experiment (non-initialized-fluid velocities and truncated series describing the initial shape), a good agreement between the results validates both approaches. An interest in performing such a comparison is to separate the influence of various factors, which are all involved in the experiment, but are easily discriminated only in a simulation. This methodology has been applied here to interpret the experimental results (i) to conclude on the degree of contamination of the fluids in the experimental device and on the effect of this contamination on oscillation dynamics, and (ii) to assess the influence of gravity, in order (iii) to finally analyze the existence of non-linear couplings in the shape oscillations of a bubble and a drop.

Concerning contamination, the comparison has shown that the shape oscillations observed in experiments do not differ from pure conditions for both the bubble and the drop, though the drop rising velocity corresponds to the case of a drop with immobilized interface suggesting a residual contamination. To our knowledge, experiments in which the rising velocity of a drop corresponds to that of a drop with clean interface have not been reported. However, the present results indicate that it is easier to obtain experimentally a drop that oscillates as a clean drop, rather than a drop that reaches the rise velocity of an uncontaminated drop.

Concerning effects of gravity on millimeter-sized bubbles and drops, the simulations allow us to conclude that buoyancy does not affect significantly the interface dynamics provided the mean oscillating shape remains close to a sphere (as it is the case here), even though the rising and oscillating velocities are of the same order. Thus, it is still relevant to decompose the interface shape in spherical harmonics, which correspond to the eigenmodes of the oscillations.

Concerning the oscillation nonlinearities, couplings between the modes are easily observable from the temporal evolution and the spectrum of amplitudes of the different spherical harmonics. Exchanges of energy induce modulations at mode 2 frequency in the temporal decay of the oscillation amplitude associated with each mode. Nevertheless, for initial amplitudes of magnitude of 0.2R like those of our experiments, the dominant frequency and the global decay rate associated with each mode of oscillation are still accurately given by the linear theory, with a maximum discrepancy of 10% for the damping rate.

Thus, using such a complementarity between the experimental and numerical results in this paper, it has been demonstrated that it is easy to produce shape oscillations for rising drops or bubbles that are not influenced by buoyancy. This result might avoid the need for costly experiments in micro-gravity conditions, for example. Bubble or drop oscillations might also be considered for an investigation of the interface dynamics in more complex situations, e.g., in the presence of surfactants at non-negligible concentrations.

FIG. 1 .

 1 FIG. 1. (a)Air bubble in water. Experimental picture of the capillary with the bubble just before detachment, superposed with the initial shape for the simulation (yellow) and several other shapes from both the experiment (continuous lines) and the simulation (black symbols) at t = 3T th 2 /2, 4T th 2 , 13T th 2 /2, and 69T th 2 /4 (for terminal shape), T th 2 = 2π/ω th 2 being the theoretical period of oscillation of mode 2 in the absence of gravity. (b) Heptane drop in water. Experimental picture of the capillary with the drop just before detachment, superposed with the initial shape for the simulation (yellow) and several other shapes from both the experiment (continuous lines) and the simulation (black symbols) at t = 3T th 2 /2, 4T th 2 , 13T th 2 /2, and 31T th 2 /4 (for terminal shape).

FIG. 3 .

 3 FIG. 3. Bubble case. Time evolution of spherical harmonic amplitudes a l for l = 2-7. Black: Experiments at Bo = 0.08. Blue: DNS at Bo = 0.08. Dashed red: DNS at Bo = 0.

FIG. 4 .

 4 FIG.4. Bubble case. Power spectra of amplitudes a l of spherical harmonics for l = 2-7, normalized by the variance of the signal. Black: Experiments at Bo = 0.08; blue (thick): simulations at Bo = 0.08; red (thin): simulations at Bo = 0; and green (thin): simulations at Bo = 0.40. Note that for case at Bo = 0.40, the bubble oscillates around an oblate spheroid which flattens slowly with time; this low frequency evolution has been filtered from the oscillations at high frequency when calculating the spectra.

FIG. 5 .

 5 FIG.5. Bubble case. Symbols: Time evolution of extrema of |a l | for l = 2-5, extracted from the numerical signals of the simulation without gravity (thin lines with crosses) and with gravity (Bo = 0.08, thick lines with squares). Lines without symbols show theoretical damping β th l from the linear theory without gravity.

FIG. 6 .

 6 FIG. 6. Drop case. (a) Instantaneous Reynolds number of rising Re ASC . (b) Time evolution of the drop aspect ratio χ.

FIG. 7 .

 7 FIG. 7. Drop case. Time evolution of spherical harmonic amplitudes a l for l = 2-7. Black: Experiments at Bo = 0.08. Blue: DNS at Bo = 0.08. Dashed red: DNS at Bo = 0.

FIG. 8 .

 8 FIG.8. Drop case. Power spectra of amplitudes a l of spherical harmonics for l = 2-7, normalized by the variance of the signal. Black: Experiments at Bo = 0.08; blue (thick): simulations at Bo = 0.08; red (thin): simulations at Bo = 0; and green (thin): simulations at Bo = 0.40. Note that for case at Bo = 0.40, the drop oscillates around an oblate spheroid which flattens slowly with time; this low frequency evolution has been filtered from the oscillations at high frequency when calculating the spectra.

FIG. 9 .

 9 FIG. 9. Drop case. Symbols: Time evolution of the absolute values of the extrema of |a l | for l = 2-5, extracted from the numerical signals of the simulation without gravity (fine lines with crosses) at Bo = 0 and with gravity (thick lines with squares) at Bo = 0.08. Lines: Theoretical damping β th l from the linear theory.

TABLE I .

 I Physical parameters of the experiment and the simulation.

	Parameters	R (mm)	ρ o (kg/m 3 )	ρ i (kg/m 3 )	µ o (Pa s)	µ i (Pa s)	σ (N/m)
	Bubble Drop	0.388 0.579	998.3 998.21	1.226 684	1.02 × 10 -3 1.0 × 10 -3	1.78 × 10 -5 4.0 × 10 -4	0.072 0.049

TABLE II .

 II Dimensionless parameters for the two investigated configurations. The terminal velocity used to calculate Re ∞ et W e ∞ is that given by the numerical simulation to ensure that it is not affected by any effect of surface contamination. Re OSC = √ ρ o σR/µ o , based on the characteristic velocity induced by the oscillations σ/(ρ o R), and is the reciprocal of the Ohnesorge number. The second is the rising Reynolds number Re ∞

	oscillation Reynolds number
	Parameters	ρ o /ρ i	µ o /µ i	Re OSC	Re ∞	Bo	W e ∞
	Bubble	814.3	57.3	164	134	0.08	0.33
	Drop	1.5	2.5	168	115	0.08	0.23

TABLE III .

 III Initial non-dimensional amplitudes of the spherical harmonics obtained by fitting the experimental contours by Eq. (1) with l ≤ 12.

TABLE IV .

 IV Theoretical frequencies ω th l and damping rates β th l for modes l = 2-7, computed from expressions of Ref. 5. Bubble case. (a) Instantaneous Reynolds number of rising Re ASC . (b) Time evolution of the bubble aspect ratio χ.

	FIG. 2.
	Mode		l = 2	l = 3	l = 4	l = 5	l = 6	l = 7
	Bubble	ω th l (rad s -1 ) β th l (s -1 )	3831 125	6995 219	10 492 337	14 332 480	18 498 646	22 971 836
	Drop	ω th l (rad s -1 ) β th l (s -1 )	1177 55	2212 103	3 363 163	4 631 233	6 008 314	7 486 404

ACKNOWLEDGMENTS

The experiments have been carried out with support of Ministry of Education, Youth and Sports of the Czech Republic (Project No. LD 13018).