Didier Dupont
email: didier.dupont@rennes.inra.fr

Giuseppina Mandalari

Daniel Mollé

Julien Jardin

Odile Rolet-Répécaud

Gabriel Duboz

Joëlle Léonil

Clare E N Mills

Alan R Mackie

E N Clare Mills

Food processing increase casein resistance to simulated infant digestion

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

Introduction

Milk allergy mainly affects children through their first contacts with non-human milk products. Fortunately, up to 85% of them outgrow their allergy in the first 5-10 years of life [START_REF] Asero | IgE-Mediated food allergy diagnosis: Current status and new perspectives[END_REF]. Most if not all the milk proteins are potential allergens. Whey proteins like β-lactoglobulin (β-lg), α-lactalbumin, bovine serum albumin and lactoferrin that account for 20% of the total milk proteins are mostly globular proteins and several IgE-binding epitopes have been identified on these proteins in the past [START_REF] Jarvinen | IgE and IgG binding epitopes on α-lactalbumin and β-lactoglobulin in cow's milk allergy[END_REF]. However caseins (CNs) which represent c.a. 80% of the milk proteins have also been shown to be major allergens [START_REF] Docena | Identification of casein as the major allergenic and antigenic protein in cow's milk[END_REF]. This is quite surprising if we consider that for eliciting an allergic response, they must be partly resistant to the enzymatic degradation that occurs during digestion. CNs have a very flexible structure and are therefore extremely sensitive to proteolysis. Indeed, purified bovine CNs were shown to be rapidly cleaved by digestive proteases when submitted to various in vitro digestion models [START_REF] Guo | Susceptibility of β-lactoglobulin and sodium caseinate to proteolysis by pepsin and trypsin[END_REF][START_REF] Macierzanka | Emulsification alters simulated gastrointestinal proteolysis of β-casein and β-lactoglobulin[END_REF][START_REF] Mandalari | In vitro digestibility of β-casein and β-lactoglobulin under simulated human gastric and duodenal conditions: A multi-laboratory evaluation[END_REF][START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF].

Several hypotheses have been raised to explain the resistance of CNs to digestion.

The presence of phosphorylated sequences that could also explain the crosssensitization found in several patients [START_REF] Bernard | Post-translational phosphorylation affects the IgE binding capacity of caseins[END_REF], the protection of CNs by fat, heatdenatured whey proteins or by the dairy matrix and the resistance of CNs to digestion due to the immaturity of the both infant immune and digestive systems can all potentially contribute to CNs allergenicity.

Recently, Roth-Walter et al. [START_REF] Roth-Walter | Pasteurization of milk proteins promotes allergic sensitization by enhancing uptake through Peyer's patches[END_REF] showed that triggering of an anaphylactic response towards milk proteins requires two phases (1) sensitisation by thermally-induced milk protein aggregates through Peyer's patches and (2) efficient transfer of milk protein across the epithelial barrier. Although, this was only demonstrated for whey protein aggregates, we need to keep in mind that heat-treatment of milk will also result in the formation of aggregates between casein micelles and whey proteins via the formation of disulfide bonds between κand/or αs 2 -casein and whey proteins [START_REF] Guyomarc'h | Formation of soluble and micellebound protein aggregates in heated milk[END_REF][START_REF] Patel | Effects of heat and high hydrostatic pressure treatments on disulfide bonding interchanges among the proteins in skim milk[END_REF].

Heat treatment of milk could therefore be partly responsible for sensitization via the formation of milk protein aggregates. Hence understanding how milk processing, and particularly heat treatment, affects the digestion of milk proteins is of great importance. Most of the studies conducted so far on CN in vitro digestions were made using adult gut models which is almost irrelevant since the pathology mainly affects children. Therefore, we recently proposed a new infant in vitro digestion model dedicated to study the resistance of milk and egg allergen to digestion [START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF].

The objective of the present work was to determine if processing could modify the resistance of CNs to digestion and to identify resistant regions capable of eliciting an allergic response in infant. To reach this goal, a set of dairy samples was manufactured from a raw whole milk used as a reference sample and submitted to simulated digestion using the infant gut model. Milk protein hydrolysis was investigated using SDS-PAGE, immunoassays and mass spectrometry. Results obtained showed that milk processing increased CNs resistance to digestion. Some resistant area were identified and compared to known IgE-epitopes.

Materials and methods

Chemicals

Unless otherwise stated, chemicals were from commercial origin (Sigma, St-Louis, MO, USA).

Milk and dairy products

A 50-litres batch of raw milk collected in local farms was kindly provided by the Coopérative Laitière de Tourmont (France). Determination of fat, total protein and lactose was achieved using a Milkoscan infrared spectrophotometer. From this milk, s 1 -, -and -CN were purified as described previously [START_REF] Senocq | Monoclonal antibodies against bovine β-casein: production and epitope characterization[END_REF]. A yogurt, three raw (whole, homogenised, skimmed), two pasteurised (whole, homogenised) and three sterilised (whole, homogenised, semi-skimmed) milks were manufactured at INRA's pilot plant in Poligny (France). Milk homogenisation was performed on a APV homogeniser apparatus (LAB 60 type, COMPAS sarl, Voisins le Bretonneux, France). Pasteurisation time and temperature were chosen in order to be as close as possible to the conditions used in industry. Therefore, milk was heated during 30 s at 82°C using a laboratory tubular exchanger (INRA homemade). Milk sterilisation was done on pasteurised milk (30 s/82°C) by autoclaving the flasks at 120°C during 10 min. For yogurt manufacture, 30 g of milk powder were mixed with 708 g of skimmed milk and 292 g of whole milk in order to reach objectives in term of fat (12.5 g/l), protein (44.4 g/l) and lactose (66.8 g/l). The mix was incubated at 20°C during 33 min under gentle stirring. Then, the mix was pasteurised at 92°C during 10 min in a water bath and FYS 11 starters (Danisco A/S, Copenhagen, Denmark) consisting of a mixture of Streptococcus thermophilus and Lactobacillus delbrueckii subsp.

Bulgaricus were added at 3%. The mix was aliquoted into eight 100 ml pots and coagulation occurred after 2.5 h incubation at 45°C. After coagulation, yogurts were cooled and stored at 4°C until utilization.

In vitro infant digestion model

Prior to digestion, phospholipid vesicles were prepared as described previously [START_REF] Mandalari | Physiological phosphatidylcholine protects bovine β-lactoglobulin from simulated gastrointestinal proteolysis[END_REF].

Proteolysis was performed essentially as previously described [START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF] using triplicate incubations at 37°C. The concentrations of digestive enzymes, bile salts, surfactants… were chosen according to the data available in the literature on the newborn consuming real foods (mainly infant formula) [START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF]. Prior to digestion, samples (milks and yogurt) were diluted to 1 mg CN/ml in 0.15M NaCl, pH 6.5 in order to reduce the quantities of enzymes and surfactants necessary for conducting simulated digestions but the enzyme / substrate ratio (i.e. the digestive proteases / dietary proteins) was set in order to remain physiologically relevant. Therefore, digestion of purified proteins and/or more complex food matrices can be studied with this model as long as the enzyme / substrate ratio remains constant. Then, diluted samples were mixed with PC vesicles and the pH was adjusted to 3.0 with 0.5 M HCl solution.

Porcine gastric mucosa pepsin (EC 3.4.23.1, Sigma, activity: 3,300 U/mg of protein calculated using haemoglobin as a substrate) was added to give 22.75 U of pepsin/mg of total CN (0.05 mM, final concentration). Aliquots (100 µL) were removed over the 60 min digestion time course. Pepsinolysis was stopped by raising the pH to 7.0 using 0.5 M ammonium bicarbonate (BDH, Pole, Dorset, UK). Then, pH of samples subsequently subjected to duodenal proteolysis was adjusted to 6.5 by addition of 0.1 M NaOH and components added to give final concentrations as follows: 1 mM sodium taurocholate, 1 mM sodium glycodeoxycholate, 26.1 mM Bis-Tris buffer pH 6.5, 0.04 U/mg of total CN bovine -chymotrypsin (activity 40 U/mg of protein using benzoyltyrosine ethyl ester, BTEE, as substrate), 3.45 U/mg of total CN porcine trypsin (activity 13,800 U/mg of protein using benzoylarginine ethyl ester, BAEE, as substrate). Aliquots (100 µL) were removed over the 30 min digestion time course, and proteolysis stopped by addition of a two-fold excess of soybean Bowmann-Birk trypsin-chymotrypsin inhibitor above that calculated to inhibit trypsin and chymotrypsin in the digestion mix.

Antibodies

Twenty eight mouse monoclonal antibodies specific for s 1 -, s 2 -, -and -CN were taken from INRA's collection [START_REF] Johansson | Epitope characterization of a supramolecular protein assembly with a collection of monoclonal antibodies: The case of casein micelle[END_REF] in order to cover as much of the sequence of the CNs as possible. The specificity of these antibodies is represented in Fig. 1.

Rabbit polyclonal antibodies specific for s 1 -, -and -CN were raised following the protocol previously described by Senocq et al [START_REF] Senocq | Antipeptide antibodies recognizing plasmin sensitive sites in bovine β-casein sequence[END_REF].

SDS-PAGE

Samples taken at different stages of the digestion were analysed by SDS-PAGE as described previously [START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF].

Inhibition ELISA

Inhibition ELISA using s 1 -, -and -CN specific polyclonal antibodies was applied to the samples collected throughout digestion of these 3 proteins in order to determine the residual immunoreactivity of each protein during the digestive process.

ELISA plates (NUNC, Maxisorp, Roskilde, Denmark) were coated with 0.5 µg/mL s 1 -, -and -CN in 0.1 M bicarbonate buffer, pH 9.6 (100 µL per well) and incubated for 1 h at 37°C. Wells were rinsed between incubation steps for 15 s with four changes of 250 µL phosphate-buffered saline, 0.05% Tween 20 (PBS-T, Sigma) using a Model 1575 Immunowash microplate washer (Bio-Rad, Hercules, CA, USA).

Blocking of the remaining binding sites was performed with 250 µL fish gelatin (Sigma) at 10 g/L in PBS-T for 1 h at 37°C. Serial dilutions of s 1 -, -and -CN in PBS-T were used as standards (concentrations ranging from 0 to 100 µg/mL).

Digested and undigested samples diluted in PBS-T (four dilutions from 1:1000 to 1:5000, 75 µL) were incubated in test tubes with 75 µL of rabbit polyclonal antibodies specific for s 1 -, -and -CN diluted at 1:3000, 1:7000 and 1:67000 respectively and incubated for 1 h at 37°C. One hundred microlitres of the mixture was then added to each ELISA plate well and further incubated for 1 h at 37°C. The reaction was revealed by incubating 100 µL of goat anti-rabbit Ig alkaline phosphatase conjugate (Sigma) diluted 1/3000 in PBS-T for 1 h at 37°C. Finally, 100 µL p-nitrophenyl phosphate (Sigma) at 1 g/L 1 M diethanolamine-HCl, 1 mM MgCl 2 , 0.1 mM zinc acetate were incubated in the wells. After 30 min at 37°C, the absorbance at 405 nm was read against a blank using a Benchmark Plus microplate spectrophotometer (Bio-Rad). Results were expressed as percentage of residual immunoreactivity in comparison with the one of the undigested sample.

Indirect ELISA

This method was used to detect the s 1 -, s 2 -, -and -CN area resistant to digestion using 28 monoclonal antibodies specific for these 4 CNs. Briefly, 100 µL of digested and undigested whole raw, pasteurised and sterilised milks and yogurt were diluted 1:2000 in 0.1 M bicarbonate buffer, pH 9.6 were coated onto a micro-titre plate (NUNC) and incubated 1 h at 37°C. The remaining binding sites were blocked by incubating 250 µL fish gelatin (Sigma) at 10 g/L in PBS-T for 1 h at 37°C. Hybridoma culture supernatants were diluted 1:2 in PBS-T and incubated for 1 h at 37°C. Bound mouse Ig was detected by incubating 100 µL of goat anti-mouse Ig alkaline phosphatase conjugate (Sigma) diluted 1:3000 in PBS-T for 1 h at 37°C. Following the last rinsing, 100 µL p-nitrophenyl phosphate (Sigma-Aldrich) at 1 g/L in 1 M diethanolamine-HCl, 1 mM MgCl 2 , 0.1 mM zinc acetate were incubated in the wells and plates were read as described above.

Immunoblotting

The whole raw, pasteurised and sterilised milks and yogurt digested samples collected at the end of the gastro-duodenal digestion process were electrophoresed as described above. Immediately after separation, proteins and peptides were transferred onto a 0.2 m pore size nitrocellulose membrane (Bio-Rad) as previously described [START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF]. The membrane was then incubated at room temperature for 1 h periods in PBS-T, with, successively, 1% gelatine, s 1 -, -and -CN specific polyclonal antibodies at 1:2000, 1:500, and 1:1000 respectively or a mixture of s 2 -CN monoclonal antibodies specific from the area 36-75 (diluted 1:2). Reaction was revealed using either goat anti-rabbit (for polyclonal antibodies) or goat anti-mouse (for monoclonal antibodies) immunoglobulin alkaline phosphatise conjugate at 1:500 and Fast 5-bromo-4-chloro-3indolyl phosphate/nitro blue tetrazolium (Sigma) as substrate.

Nano-Liquid Chromatography/Tandem Mass Spectrometry (LC/MS/MS)

The yogurt and raw, pasteurised and sterilised whole milks digested samples collected at the end of gastro-duodenal digestion were analysed by LC/MS/MS in order to identify the peptides remaining after digestion. Digested samples were subjected to nanoscale reverse phase liquid chromatography as previously described [START_REF] Dupont | Comparative resistance of food proteins to adult and infant in vitro digestion models[END_REF]. The online separated peptides were analysed by ESI Q-TOF-MS/MS using a QSTARXL global hybrid quadrupole/time-of-flight mass spectrometer (Applied Biosystems, Framingham, CA) operated in positive ion mode.

To identify peptides, all data (MS and MS/MS) were submitted to MASCOT (v.2.1).

The search was performed against a homemade database dealing with major milk proteins which represents a portion of the Swissprot database (http://www.expasy.org). No specific enzyme cleavage was used and the peptide mass tolerance was set to 0.3 Da for MS and 0.15 Da for MS/MS. Three variable modifications (phosphorylation on serine and threonine, oxydation of methionine and deamidation of asparagines and glutamine residues) were selected. For each peptide identified, a minimum MASCOT score corresponding to a p-value below 0.05 was considered as a prerequisite for peptide validation with a high degree of confidence.

Statistical analysis

The effect of the casein type and the processing conditions on the residual immunoreactivity remaining after gastro-duodenal digestion of dairy products was tested by variance analysis using the R software package [START_REF]R : a language and environment for statistical computing[END_REF] running on the UNIX® system.

Hydrophobicity profile

CN hydrophobicity profiles at pH 3.0 and 6.5 were established as previously described by Sweet and Eisenberg [START_REF] Sweet | Correlation of sequence hydrophobicities measures similarity in 3-dimensional protein-structure[END_REF].

Results

Analysis of digested samples by SDS-PAGE

After dilution of the samples in the digestion buffer, the decrease of the pH to 3.0 resulted in the flocculation of all samples. The aggregates which were visible rapidly disappeared when pepsin was added and the reaction medium was clear after 20-40 min gastric digestion.

Within each type of milk (raw, pasteurised and sterilised) all the different samples (whole, homogenised, skimmed and/or semi-skimmed) submitted to in vitro digestion showed the same pattern in SDS-PAGE (data not shown). Therefore, in order to improve the clarity only the whole milk samples are shown as representative of their respective types.

Gastric phase

Fig. 2 shows the electrophoretic patterns obtained after submitting whole raw (a), pasteurised (b), sterilised milks (c) and yogurt (d) to gastric in vitro digestion. In raw and pasteurised milks, the intact CNs bands disappeared after 20-40 min and bands corresponding to low Mw compounds (between 3 and 6 kDa) appeared concomitantly at the bottom of the gels in the samples. In contrast, patterns obtained with whole sterilised milks strongly differed with those obtained with both raw and pasteurised milks. Bands corresponding to intact CNs were hardly visible and smears appeared on the gels. Sterilisation may have caused extensive protein denaturation or protein-lipid interactions altering the separation of proteins in SDS-PAGE. Finally, pattern of yogurt submitted to gastric digestion was similar to those obtained for raw and pasteurised milks, i.e. with a disappearance of intact CNs after 20 min digestion and the concomitant appearance of low Mw compounds.

Strong differences were also observed for the band at 18kDa corresponding to -lg. This protein was indeed shown to be highly resistant to digestion in nonheated samples. However, heat treatments applied to milk for manufacture of pasteurised, sterilised milks and yogurts resulted in an increased digestibility of -lg.

Duodenal phase

Fig. 3 shows the pattern obtained in SDS-PAGE when whole raw (a), pasteurised (b), sterilised milk (c) and yogurt (d) were submitted to gastro-duodenal digestion. All the digested samples showed an absence of bands corresponding to intact CNs. The bands at low Mw already observed at the end of the gastric phase seemed to resist the duodenal phase of digestion. Finally, β-lg gave an intense band in the digested raw milks, whereas its intensity was less in pasteurised milks and yogurt, and the protein was not detectable in digested sterilised milks.

Determination of casein residual immunoreactivity after digestion by inhibition ELISA and western-blotting

Fig. 4 shows that some residual immunoreactivity was detectable for s 1 -, and -CN in all the samples after gastro-duodenal digestion. Indeed, even though intact CNs were hardly visible by SDS-PAGE in whole sterilised milk, ELISA shows that the proteins were detectable in this sample at similar levels than in the other samples. αs 1 -CN residual immunoreactivity was significantly higher in digested yogurt than in the other dairy samples (p<0.001) and higher in digested sterilised milk than in raw or pasteurised milk although this was not statistically significant (p=0.0728). βand -CN residual immunoreactivities were higher in digested yogurt than in the other dairy samples (p<0.001). Therefore, it looks like that heat treatment of milk increases CN residual immunoreactivity, although the difference in microstructure between the yogurt and raw milk may also play a role. We have previously demonstrated the relationship existing between the residual immunoreactivity of a protein and the extent of its proteolysis [START_REF] Dupont | A new approach for monitoring proteolysis phenomena using antibodies specifically directed against the enzyme cleavage site on its substrate[END_REF]. Since residual immunoreactivity can be correlated with the resistance of CNs to digestion, our data show that milk processing into yogurt (and sterilised milks to a lesser extent) increases CN resistance to in vitro digestion. It has however to be emphasized that these residual immunoreactivities of CNs were obtained by ELISA with polyclonal antibodies, i.e. probes that are able to detect intact proteins as well as fragments of a proteins. Therefore, a 50% residual immunoreactivity does not mean that 1 molecule out of 2 present in the sample are still intact; the SDS-PAGE shows that it is much less than that.

Western-blotting of the digested raw, pasteurised and sterilised milk and yogurt samples revealed strong differences between the patterns of the digested samples that were not observable by SDS-PAGE probably because revelation of the bands is much more sensitive with specific antibodies. It confirmed an impact of the processing conditions on the composition of the digested samples (Fig. 5). It also confirmed the extensive degradation of s 1 -CN. Indeed no bands were detected with the s 1 -CN specific antibody except a faint one at 32 kDa on the digested yogurt. It is however interesting to note that smears were observable for both the digested yogurt and sterilised milk and that high-molecular weight bands were visible around 60 kDa in all samples. A mixture of monoclonal antibodies specific for fragments 16-35 and 36-55 of s 2 -CN revealed a major band at 10 kDa that was much more intense in pasteurised milk and yogurt than in raw milk. A second band around 17 kDa was only present in the digested pasteurised milk and yogurt. Intense smears were observed when these two monoclonal antibodies were applied to the digested sterilised milk.

For -CN a band at 3 kDa was visible in all samples, whereas one at 4.3 kDa was mainly observed in the digested sterilised milk. In contrast, -CN showed several bands in all the samples but at different molecular weight. Digested raw and pasteurised milks showed bands at 25 and 16 kDa whereas digested sterilised milk mainly showed one intense band around 60 kDa. Digested yogurt showed bands at 60, 25, 16 and 9 kDa.

Identification of the casein area resistant to digestion using a collection of specific monoclonal antibodies.

A collection of 28 different monoclonal antibodies specific for s 1 -, s 2 -, -or -CN was used to identify by indirect ELISA the areas that were resistant to in vitro digestion. Fig. 6 shows the residual immunoreactivity observed with these 28 antibodies on digested raw, pasteurised and sterilised milks and yogurt.

Most of the s 1 -CN specific monoclonal antibodies gave low residual immunoreactivity in all the analysed samples confirming that this protein is extensively hydrolysed during digestion. Antibodies specific for s 1 -CN (f149-166) and s 1 -CN (f185-199) showed a slightly higher residual immunoreactivity.

Areas of resistance (f36-55 and f56-75) on s 2 -CN were also revealed by specific antibodies on all the digested samples studied and to a lesser extent s 2 -CN (f191-207) was also resistant in all samples.

For -CN, the most resistant area in all the samples studied was the one recognized by the -CN (f76-93) specific antibody. The other -CN specific antibodies gave residual immunoreactivities lower or close to 10%.

For -CN, residual immunoreactivity was found with the 4 antibodies specific for the C-terminal half of the molecule, for all samples the highest response was obtained with the -CN (f131-150) specific antibody. Some differences in the pattern of the different samples were observed. Indeed, digested yogurt reacted almost exclusively with -CN (f131-150) specific antibody whereas residual immunoreactivity was found with -CN (f98-115), -CN (f116-130) and -CN (f151-169) specific antibodies in digested raw, pasteurised and sterilised milks. It has to be emphasized however that for this CN, only one antibody is specific for the N-terminal moiety of the protein and therefore other resistant areas could be present in this half. These results confirmed that -CN C-terminal moiety is resistant to in vitro digestion.

In summary, the highest residual immunoreactivities were observed with antibodies s 2 -CN (f36-55), s 2 -CN (f56-75), -CN (f76-93), -CN (f131-150) and -CN (f151-169), regardless of the sample. Given that the antibodies used in the study did not fully cover the whole CN sequences, other area not monitored here could also have resisted to digestion. From the percentage of residual immunoreactivity, it appears that both -CN and s 2 -CN are the most resistant CNs to in vitro digestion followed by -CN, s 1 -CN being extensively hydrolysed.

Identification of peptides in digested samples using LC-MS-MS

LC-MS-MS was applied to digested raw, pasteurised and sterilised milks and yogurt in order to identify the remaining peptides. Only the peptides identified unambiguously are presented here. were identified in the digested raw and pasteurised milks respectively whereas a higher number of peptides were identified in the more heavily processed dairy matrices (42 and 51 peptides identified in the digested sterilised milk and yogurt respectively). The data obtained with immunoassays (higher residual immunoreactivity found in high heated digested samples) suggest that the lower number of peptides identified in the low heated milks could be attributed to a higher degree of proteolysis leading to shorter fragments unidentifiable by LC-MS-MS.

Almost no peptides were found in the area 44-79 which is a highly phosphorylated area, with phosphoserine residues in 46, 48, 64, 66, 67, 68 and 75.

There are several reasons why part of a sequence can not be detected by LC-MS-MS:

-the area can be extensively hydrolysed leading to the formation of small peptides with Mw lower than 500 Da and therefore not validated during MASCOT database search -the area can be resistant to digestion leading to fragments larger than 2500 Da i.e. not detectable in our conditions -post-translational modifications can lower the abundance of peptides in the digested samples leading to concentrations too low to be detected or limit the ability of peptides to ionise or fragment.

In the present case, the absence of bands specific for large s 1-CN fragments in western-blotting (Fig. 5) as well as the weak residual immunoreactivity detected by ELISA with s 1 -CN specific monoclonal antibodies (Fig. 6) suggested that these phosphorylated fragments were extensively hydrolysed.

For s 2 -CN, no peptides were identified in the 1-88 (with the exception of peptide I 71 -K 80 that was observed in the digested sterilised milk) and 126-150 regions (Fig. 7b). Fragment 1-88 also corresponds to a phosphorylated area with phosphoserine residues in position 8, 9, 10, 16, 31, 56, 57, 58 and 61. Westernblotting showed an intense band around 10 kDa that was specifically detected by monoclonal antibodies specific from the s 2 -CN N-terminal moiety (Fig. 5), results that were confirmed by ELISA (Fig. 6). Taken together these results showed that the sequence s 2 (f1-88) can resist in vitro digestion and does not generate shorter peptides in sufficient amounts to be detected by mass spectrometry. Compared to the other CNs, -CN gave a lower number of identified peptides (Fig. 7d). Areas 1-16, 77-95, 106-137 and 147-160 gave no peptides. Part of this absence of detected peptides may be attributed to post-translational modifications. Indeed, -CN shows a site of phosphorylation in S 149 and glycosylation sites in positions 131, 133, 135, 136 (variant A) and 142. -CN is also characterized by a pyro-Glu residue at its N-terminal extremity that, when hydrolyzed, would lead to uncharged peptides, i.e. undetectable by the LC-MS-MS technique we used. However, western-blotting analysis (Figure 5 d) showed several bands corresponding to intact quite large fragments of -CN that partly resisted digestion and probably gave peptides in concentrations too low to be detected by mass spectrometry and/or too short to be validated during MASCOT database search. Furthermore, ELISA with specific monoclonal antibodies on digested samples showed a high residual immunoreactivity of the C-terminal part of -CN confirming the partial resistance of this area to in vitro digestion.

Caseins hydrophobicity profiles at pH 3.0 and their area resistant to digestion

Fig. 8 shows the charge and hydrophobicity profiles of s 1 -, s 2 -, -and -CN at pH 3.0 (a) and 6.5 (b), i.e. the pH used for the gastric and duodenal phases of in vitro digestion respectively. It shows that for s 1 -CN the most hydrophobic part at pH 3.0 is 149-199 and it was the one that showed a slightly higher resistance to digestion for s 1 -CN. For s 2 -CN, the N-terminal moiety is more hydrophobic than the C-terminal one and, again, it was this area that showed the highest residual immunoreactivity after in vitro digestion. For -CN, the main hydrophobic part is located in 55-92 area and the area showing the highest residual immunoreactivity after in vitro digestion was the 76-93. Finally, for -CN most of the hydrophobicity is located in the 120-169 zone which corresponds to the area that was found resistant to in vitro digestion. Therefore, it looks as if there may be a relationship between hydrophobicity at pH 3.0 and resistance to in vitro digestion. For the duodenal phase, when pH is raised to 6.5, the hydrophobic areas mentioned above become more hydrophilic and more accessible to digestive enzymes and typical trypsin and chymotrypsin cuts can be observed after peptide identification by LC-MS-MS. This is particularly the case for s 1 -CN (f149-199) for which 6 chymotrypsin cuts were observed (Fig. 7a). However, s 2 -CN (f1-88) showed only 2 trypsin cuts appearing during duodenal digestion (Fig. 7b), -CN (76-93) only one typical chymotrypsin cut (Fig. 7c) whereas -CN (f106-169) showed none (Fig. 7d). It is interesting to note that the 3 latter areas were the most resistant observed by ELISA (Fig. 6).

Discussion

This study shows that food processing, and particularly heat treatment of milk, increases CNs resistance to simulated infant digestion. Inhibition ELISA and westernblotting with polyclonal antibodies revealed a lower resistance to hydrolysis of the CNs in unheated samples than in samples submitted to high heat treatments such as sterilised milk and yogurt. Mass spectrometry analysis of the digested samples showed that a higher number of peptides were identifiable in the digested heavily processed dairy products (sterilised milks and yogurt) than in the original raw milk.

The lower number of peptides identified in the low-heated milks was therefore attributed to a higher degree of proteolysis leading to short undetectable peptides.

The hypothesis that heat-treatments applied to milk increase the resistance of CN to digestion has already raised by Chatterton et al. [START_REF] Chatterton | In vitro digestion of novel milk protein ingredients for use in infant formulas: Research on biological functions[END_REF] and was recently confirmed by Almaas et al. [START_REF] Almaas | In vitro digestion of bovine and caprine milk by human gastric and duodenal enzymes[END_REF] who observed that raw milk was digested significantly faster with human proteolytic enzymes than pasteurised and high-heated milk.

Our results emphasize the complementarity of LC-MS-MS and immunoassays for estimating the resistance of protein area to digestion. Indeed, only few peptides were detected by LC-MS-MS in the 2 large areas αs 2 -CN (f1-88) and -CN (f106-169) that were found to be highly immunoreactive when specific monoclonal probes were used.

Analysis of digested samples by western-blotting showed that among CNs, s 2 -and -CN were the ones giving most, if not all, the largest fragments at the end of digestion, which was quite surprising since they are present in milk at lower concentrations than β-and s 1 -CN (ratio 1:1:3:3 for , s 2 , s 1 , ). Therefore, the persistence of peptides from -and s 2 -CN must be a consequence of their structure rather than abundance. Both of these proteins have at least two cystein residues in their sequence compared to s 1 -and β-CN, which have none. Consequently when milk is submitted to severe heat treatments, -and s 2 -CN have been shown to generate heat-induced aggregates with whey proteins through the formation of disulfide bridges with the cystein residues of β-lactoglobulin and -lactalbumin [START_REF] Guyomarc'h | Formation of soluble and micellebound protein aggregates in heated milk[END_REF][START_REF] Patel | Effects of heat and high hydrostatic pressure treatments on disulfide bonding interchanges among the proteins in skim milk[END_REF][START_REF] Lowe | Heat-induced redistribution of disulfide bonds in milk proteins. 2. Disulfide bonding patterns between bovine beta-lactoglobulin and kappa-casein[END_REF]. Although the resistance of these protein aggregates to digestion has not yet been investigated, it is probable that aggregation with whey proteins will result in an increased protection of -and s 2 -CN. It has recently been demonstrated that whey protein aggregation could play a key role in milk allergic patient sensitization [START_REF] Roth-Walter | Pasteurization of milk proteins promotes allergic sensitization by enhancing uptake through Peyer's patches[END_REF]. However, the exact role of casein-whey protein aggregates on sensitization has not been investigated so far.

Use of a collection of highly specific monoclonal antibodies allowed the identification of 3 major areas of resistance, β-CN (f76-93), -CN (f106-169) and s 2 -CN (f36-75) that are hydrophobic at pH 3.0, i.e. in the conditions used for the gastric digestion. This high hydrophobicity is a factor which may contribute to resistance of these areas to pepsinolysis either through changes in conformation or aggregation.

At the beginning of the duodenal phase, when the pH is raised to 6.5, these areas will become more hydrophilic and more accessible for proteases. Depending on the number of potential trypsin and chymotrypsin cleavage sites, they will be hydrolyzed to different extents. It is interesting to note that identification of the peptides by LC-MS-MS revealed only one typical chymotrypsin cut for -CN (76-93) whereas none was observed for s 2 -CN (f36-75) and -CN (f106-169). This is probably why these 3 areas were the most resistant observed by ELISA. Another possible explanation is that the two resistant fragments -CN (f106-169) and s 2 -CN (f36-75) are known to carry several post-translational modifications. -CN (f106-169) has up to 6 glycosylation sites (in position 121, 131, 133, 136, 142 and 165) whereas s 2 -CN (f36-75) contains 4 phosphoserine residues (in position 56, 57, 58 and 61). Peptides carrying post-translational modifications have been previously shown to highly resist to in vivo gastro-duodenal digestion and their presence has been detected in intestinal fluids. Indeed, CN phosphopeptides have been identified in ileostomy fluids collected from human volunteers fed with milk [START_REF] Meisel | Detection of caseinophosphopeptides in the distal ileostomy fluid of human subjects[END_REF]. Moreover, -CN (f106-169) has been previously shown to be particularly resistant to digestion. Ledoux et al. [START_REF] Ledoux | Intraluminal immunoreactive caseinomacropeptide after milk protein ingestion in human[END_REF] found it in the jejunum of volunteers consuming 15 N labelled dairy diets. This peptide was also shown to resist digestion and be absorbed at the intestinal level and was subsequently detected in plasma [START_REF] Chabance | Casein peptide release and passage to the blood in humans during digestion of milk or yogurt[END_REF]. All these results tend to reinforce the relevance of the in vitro digestion model we used in the present study.

Have these areas that are resistant to digestion been associated with food allergy phenomena? A closer look at the literature available on CNs IgE epitopes, shows that it is quite difficult to give a clear answer to that question. For β-CN, Vila et al. [START_REF] Vila | Role of conformational and linear epitopes in the achievement of tolerance in cow's milk allergy[END_REF] found in sera from 36 allergic children that most of the IgE epitopes were located at the C-terminal extremity of this protein and corresponded to area 151-160, 167-176, 175-184 and 193-202. In another study, Chatchatee et al. [START_REF] Chatchatee | Identification of IgE and IgG binding epitopes on beta-and kappa-casein in cow's milk allergic patients[END_REF] identified 6 major (1-16, 45-54, 57-66, 83-92, 107-120 and 135-144) and 3 minor (149-164, 167-178, 173-184) IgE epitopes in persistant cow milk allergy patients. Among those, epitope 83-92 was the most frequently recognized (found in 13 patients out of [START_REF] Senocq | Antipeptide antibodies recognizing plasmin sensitive sites in bovine β-casein sequence[END_REF] and lies within the area we found resistant to digestion in the present study.

In the same paper, 8 IgE epitopes were identified on -CN (9-26, 21-44, 47-68, 67-78, 95-116, 111-126, 137-148, 149-166), the later 3 corresponding to part of the caseinomacropeptide (f106-169) we identified as being resistant to digestion.

Several studies have been published on the identification of s 1 -CN IgE epitopes [START_REF] Vila | Role of conformational and linear epitopes in the achievement of tolerance in cow's milk allergy[END_REF][START_REF] Chatchatee | Identification of IgE and IgG binding epitopes on beta-and kappa-casein in cow's milk allergic patients[END_REF][START_REF] Spuergin | Allergenic epitopes of bovine s 1 -casein recognized by human IgE and IgG[END_REF][START_REF] Nakajima-Adachi | Determinant analysis of IgE and IgG4 antibodies and T cells specific for bovine s 1 -casein from the same patients allergic to cow's milk: existence of s 1 -casein-specific B cells and T cells characteristic in cow's-milk allergy[END_REF][START_REF] Jarvinen | B-cell epitopes as a screening instrument for persistent cow's milk allergy[END_REF][START_REF] Cocco | Mutational analysis of sequential IgE-binding epitopes in s 1 -casein, a major cow's milk allergen[END_REF][START_REF] Elsayed | Evaluation of the allergenicity and antigenicity of bovine-milk s 1 -casein using extensively purified synthetic peptides[END_REF]. The epitopes identified were numerous covering almost the whole sequence of the protein (Fig. 9) making the designation of IgE immuno-dominant epitopes difficult which is also the case for s 2 -CN.

Finally, it also has to be mentioned that the relevance of IgE binding to short linear peptides for allergenic activity of food allergens is quite controversial. In a recent study, Albrecht et al. [START_REF] Albrecht | Relevance of IgE binding to short peptides for the allergenic activity of food allergens[END_REF] demonstrated that peptides identified as major sequence epitopes on two food allergens (Ara h 2 from peanut and Pen a 1 from shrimp) show little contribution to the IgE binding of the allergens studied. It is probably also the case for most of the linear IgE epitopes already described.

However, since CNs are rheomorphic proteins lacking a defined conformational structure, they are likely to induce antibody responses against sequential structures.

This work was supported by the European Commission through the funding of Didier Dupont's Marie Curie Intra-European Fellowship (project n°39515). This research was partly funded by the Biotechnology and Biological Research Sciences

Fig. 7

 7 Fig.7shows a diagrammatic representation of the peptides originating from

  β-CN digestion led to a very significant number of peptides in all samples covering almost the whole sequence of this protein (Fig 7c). Only digestion of area 1-42 led to a low number of peptides in sterilised milk or yogurt. β-CN shows phosphorylated serine residues in position 15, 17, 18, 19 and 35. Since ELISA with monoclonal probes indicates a low residual immunoreactivity of fragment 1-19, it is probable that the absence of peptide identified by LC-MS-MS is rather due to an extensive hydrolysis of this fragment during digestion.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. SDS-PAGE analysis of whole raw (a), pasteurised (b), sterilised milks (c)

Figure 3 .

 3 Figure 3. SDS-PAGE analysis of whole raw (a), pasteurised (b), sterilised milk (c)

Figure 4 .

 4 Figure 4. Residual immunoreactivity of αs 1 -(), β-()and κ-casein (□) as

Figure 5 .

 5 Figure 5. Western-blotting of (from the left to the right) raw whole, pasteurised,

Figure 6 .

 6 Figure 6. Residual immunoreactivity of αs 1 -, αs 2 -, β-and κ -CN fragments determined in digested raw, pasteurised, sterilised milks and yogurt after gastro-duodenal

Figure 7 .

 7 Figure 7. Identification of peptides derived from αs 1 -(a), αs 2 -(b), β-(c) and κ-casein (d) by LC-MS-MS after gastroduodenal digestion of (from bottom to top) whole raw

Figure 8 .Figure 9 .

 89 Figure 8. αs 1 -, αs 2 -, κand β-casein hydrophobicity profile as calculated at pH 3.0 (a) and 6.5 (b) according to Sweet and Eisenberg [17] Figure 9. αs 1 -casein IgE epitopes as identified in different recent studies

Council (BBSRC, UK) and the EU through EuroPrevall FOOD-CT-514000. The authors wish to thank Dr. Stephane Pezennec (INRA) for his assistance in the statistical treatment of data.

The authors have declared no conflict of interest.


Francisco M Rocha

Yoël Forterre

Bloen Metzger

Henri Lhuissier

Drag of a shear-thickening suspension on a rotating cylinder

Keywords: Shear-thickening suspension, Couette Flow, Jamming, Drag

We investigate experimentally the two-dimensional flow of a shear-thickening suspension around a rotating cylinder to which a constant torque is applied. While for low torques both the drag and the flow are steady and close to those for a Newtonian fluid, above the onset torque for discontinuous shear thickening the average velocity of the cylinder saturates and large periodic oscillations of the cylinder velocity are observed. The oscillations result from a hydrodynamic instability of the flow: slow-acceleration phases are followed by high-deceleration phases, triggered by the propagation of a thickening front, and so on. The slow-acceleration phases set the oscillation period, which is limited by the cylinder inertia and inversely proportional to the applied torque. Combined analyses of the cylinder motion and the flow reveal that the front nucleates when the shear rate at the cylinder surface reaches the discontinuous shear-thickening threshold, typically. In addition, the characteristics (duration, stress) of the deceleration are set by the interplay between the thickening front propagation and the suspension and cylinder inertiae or the container size. Since for a slow acceleration the shear rate at the cylinder surface is essentially the cylinder angular velocity, this description of the unsteadiness elucidates the saturation of the average velocity. More generally, it illustrates how the hydrodynamics of a shear-thickening suspension with a strongly re-entrant rheology can lead to a marginally re-entrant, though steep, drag curve.

Introduction

Dense suspensions of microscopic particles are often steeply shear-thickening media. This peculiar rheology has been interpreted fruitfully as a frictional transition driven by the stress level between the particles [START_REF] Seto | Discontinuous shear thickening of frictional hard-sphere suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF][START_REF] Wyart | Discontinuous shear thickening without inertia in dense nonbrownian suspensions[END_REF]) and has received important numerical and experimental confirmations [START_REF] Guy | Towards a unified description of the rheology of hard-particle suspensions[END_REF][START_REF] Lin | Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions[END_REF][START_REF] Dong | Analog of discontinuous shear thickening flows under confining pressure[END_REF][START_REF] Clavaud | Revealing the frictional transition in shear-thickening suspensions[END_REF][START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF][START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF][START_REF] Clavaud | The darcytron: a pressure-imposed device to probe the frictional transition in shear-thickening suspensions[END_REF]. On the other hand, much less is known about simple hydrodynamics questions such as the unsteady structures of the flow, the spatial and temporal fluctuations of the stresses, the role of the suspension inertia or the mechanism which selects the effective drag in the regime when the suspension shearthickens. Progresses on these questions would be helpful, practically, for optimizing the transport and flowability of shear-thickening suspensions (LaFarge 2013; [START_REF] Abdesselam | Rheology of plastisol formulations for coating applications[END_REF][START_REF] Blanco | Conching chocolate is a prototypical transition from frictionally jammed solid to flowable suspension with maximal solid content[END_REF]. They might also be useful to appreciate and push the limits of rheological characterizations.

Important results have already been obtained for the flow around a solid displaced with a constant force. Eighty years ago, [START_REF] Freundlich | Dilatancy and its relation to thixotropy[END_REF] reported that the average velocity of a sphere displaced in a shear-thickening suspension inside a narrow tube becomes independent of the applied force for high loads, while the instantaneous velocity is not steady but oscillates around the average value. Later, the oscillations have been modeled based on a heuristic drag law [START_REF] Von Kann | Velocity oscillations and stop-go cycles: The trajectory of an object settling in a cornstarch suspension[END_REF] but the reality of this drag law and how it actually emerges from the unsteady flow past the sphere still need to be clarified. Recently, numerous studies on small-gap rheological configurations have reported similar saturations of the average velocity of the tool, as well as oscillations under a constant applied torque [START_REF] Boersma | Time-dependent behavior and wall slip in concentrated shear thickening dispersions[END_REF][START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Nagahiro | Experimental observation of shear thickening oscillation[END_REF][START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Rathee | Localized stress fluctuations drive shear thickening in dense suspensions[END_REF][START_REF] Chacko | Dynamic vorticity banding in discontinuously shear thickening suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Ovarlez | Density waves in shear-thickening suspensions[END_REF][START_REF] Sedes | Fluctuations at the onset of discontinuous shear thickening in a suspension[END_REF][START_REF] Gauthier | A new pressure sensor array for normal stress measurement in complex fluids[END_REF]. Such oscillations have been modeled, first, by [START_REF] Nakanishi | Shear thickening oscillation in a dilatant fluid[END_REF]; [START_REF] Nakanishi | Fluid dynamics of dilatant fluids[END_REF], who have introduced two key elements: a shear-thickening steady-state rheology with a negatively-sloped portion (∂σ/∂ γ < 0, with σ and γ the shear stress and shear rate, respectively) and a delayed relaxation to the equilibrium state (involving an intrinsic strain scale, γ 0 ). Combined with the inertia of the flow, the shear-weakening (∂σ/∂ γ < 0) is destabilizing, while the delay regularizes the instability at small wavelengths (short timescales). Later, [START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF] have added the inertia of the tool in the modeling. However, their model treats the suspension as uniform and spaceless, which does not permit to capture the heterogeneous structures of the flow observed in experiments [START_REF] Rathee | Localized stress fluctuations drive shear thickening in dense suspensions[END_REF][START_REF] Ovarlez | Density waves in shear-thickening suspensions[END_REF][START_REF] Gauthier | A new pressure sensor array for normal stress measurement in complex fluids[END_REF][START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF]) and requires the introduction of a second relaxation scale (intrinsic time scale) to obtain the periodic oscillations reported experimentally. Last, additional heuristic modifications of the rheology have also been proposed to capture the saturation of the mean velocity of the tool [START_REF] Baumgarten | A general constitutive model for dense, fine-particle suspensions validated in many geometries[END_REF], but these modifications are equivalent to changing the steady-state rheology and disregard the oscillatory nature of the mean velocity saturation.

Other important results have been obtained about impulsive dynamics under imposed displacement. Experiments of indentation into a bulk or floated layer of suspension [START_REF] Waitukaitis | Impact-activated solidification of dense suspensions via dynamic jamming fronts[END_REF][START_REF] Waitukaitis | Dynamic jamming fronts[END_REF][START_REF] Peters | Quasi-2d dynamic jamming in cornstarch suspensions: visualization and force measurements[END_REF][START_REF] Peters | Direct observation of dynamic shear jamming in dense suspensions[END_REF] have revealed that, for impulsive motion, the transport of shear inside the suspension is not diffusive anymore, but involves steep propagative thickening fronts, coined jamming or shear fronts. The latter have been rationalized as kinematic shocks stemming from the shear-weakening rheology and a relaxation strain scale (γ 0 ) [START_REF] Han | Shear fronts in shear-thickening suspensions[END_REF]). However, it remains to understand how these shocks nucleate and decay, the history of stress they impose to the flow, how their propagation interacts with the motion of the solid, and how all these phenomena actually set the drag on the solid.

In this paper, we address those hydrodynamic questions through experiments on the flow of a floated layer of shear-thickening suspension (cornstarch) around a rotating cylinder to which a constant torque is applied. This configuration has several advantages. The flow in the suspension layer, floated above a non-miscible low-viscosity oil [START_REF] Peters | Direct observation of dynamic shear jamming in dense suspensions[END_REF], is essentially two-dimensional, which gives access to local and instantaneous measurements of the suspension flow and facilitates the evaluation of the stresses. Importantly, it is also a model configuration for the dynamics of a solid moved inside a fluid, with a simple but non-uniform base-state flow, from which the fundamental aspects of the solid motion and its interaction with the flow can be addressed. The manuscript is organized as follows. §2 presents the experimental set-up. §3.1 introduces the overall behavior with the different flow regimes. §3.2 and §3.3 analyse the steady regime observed for low torques and each phase of the oscillatory regime observed for high torques, respectively. §3.4 comments on how the dynamics of the two regimes set the curve of the effective drag of the suspension on the cylinder. Last, a discussion is given in §4.

Experimental set-up

The experimental set-up is sketched in figure 1a. It consists of a cylinder of radius R = 14.05 mm immersed in a layer of an aqueous suspension of corn-starch particles (Agrana Starch Maisita) contained in a cylindrical vessel with radius R out = 95.0 mm. The cylinder is driven in rotation by a rheometer head (Anton Paar MCR 501), which applies a controlled torque Γ to the cylinder and measures the current angular velocity Ω(t) (the actual torque delivered by the rheometer remains constant within less than 1%). The suspension layer, of thickness h, is floated on a Perfluorotributylamine bath (3M, Fluorinert), which is a water-non-miscible, high density (1.86 g/cm 3 ) and low viscosity (4 mPa s) liquid. This ensures a close to stress-free boundary condition at the bottom of the suspension layer and a quasi two-dimensional experimental configuration [START_REF] Peters | Direct observation of dynamic shear jamming in dense suspensions[END_REF][START_REF] Han | Shear fronts in shear-thickening suspensions[END_REF]. To avoid slippage, the vertical surfaces of both the cylinder and the outer wall are roughened with sandpaper (3M, P100).

The suspension layer is imaged from below, through the transparent bath, using an inclined mirror and a high-speed camera (Phantom v711) operating at 500 fps. The flow is monitored by seeding the suspension/bath interface with black glass tracers (diameter of 40-70 µm). The velocity field, u = u r e r + u θ e θ , is measured with conventional Particle Image Velocimetry (PIV) methods. A typical image of the layer and of the measured flow field are shown in figure 1b.

The suspension rheology is measured with a small-gap cylindrical Couette rheometer (figure 2). The suspension is pre-sheared with a constant shear rate of 0.1 s -1 for 30 s, after which an increasing logarithmic sweep in torque is imposed and shear rates are obtained by averaging over sufficient durations to be independent of the sweep. The resultant data set is fitted with Wyart-Cates flow rule, η(ϕ) = η S (ϕ J (σ)ϕ) -2 and ϕ J (σ) = ϕ 0 (1e -σ ⋆ /σ ) + ϕ 1 e -σ ⋆ /σ , where η S is a numerical factor, ϕ J denotes the suspension jamming volume fraction, ϕ 0 and ϕ 1 are the jamming volume fractions of the frictionless and the frictional viscosity branches, respectively, and σ ⋆ is the onset shearstress for frictional contact activation [START_REF] Wyart | Discontinuous shear thickening without inertia in dense nonbrownian suspensions[END_REF]. We use the same fitting procedure as described in Darbois [START_REF] Texier | Surface-wave instability without inertia in shear-thickening suspensions[END_REF], which yields η S ≃ 0.60 mPa s, ϕ 0 ≃ 0.480, ϕ 1 ≃ 0.407 and σ ⋆ ≃ 7.0 Pa, i.e., ϕ DST = ϕ 0 -2e -1/2 (ϕ 0ϕ 1 ) ≃ 0.390 for the onset volume fraction of discontinuous shear-thickening (figure 2). All experiments are performed at a fixed volume fraction ϕ = 0.44 > ϕ 1 , for which the fitted suspension rheology is not only re-entrant (∂ σ γ < 0) above the critical stress σ c ≃ 3.0 Pa (defined by ∂ σ γ| σc = 0) and critical shear rate γc ≃ 5.4 s -1 , but is actually expected to jam above the shear stress σ J = σ * / ln ϕ0-ϕ1 ϕ0-ϕ ≃ 11.6 Pa. The volume fraction is computed from the dry mass of starch assuming a fixed starch density of 1.55 g/cm 3 . The suspension is thoroughly mixed between each run to homogenize the volume fraction and avoid bias due to sedimentation.

It should be noted that while measurements at the microscopic scale support the frictional transition scenario for cornstarch suspensions [START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF], their rheology is more complex than the simple Wyart-Cates model, raising the possibility of a more complex scenario [START_REF] Gauthier | Shear-thickening in presence of adhesive contact forces: the singularity of cornstarch[END_REF]. In particular, cornstarch rheology exhibit yield-stress and aging, due to adhesion between the particles [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions[END_REF][START_REF] Galvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. A generalization of the Wyart-Cates model including adhesion has been proposed [START_REF] Guy | Towards a unified description of the rheology of hard-particle suspensions[END_REF], but its application to cornstarch is still debated. In the present study, the main ingredients of the Wyart-Cates framework, i.e., a quasi-Newtonian low-stress branch followed by a jamming transition at high stress, will prove sufficient to describe observations.

Results

Overall behavior of the cylinder dynamics

To obtain the typical dynamics of the cylinder for a broad range of applied torques, Γ , we consider, first, a protocol where Γ is increased sufficiently slowly such that the cylinder motion and the flow both adjust to the current forcing at all times. Figure 3a shows the angular velocity of the cylinder, Ω, for torques ranging from ≈ 0.2Γ c to

Steady ow

Unsteady ow Figure 3. The two regimes of cylinder motion (ϕ = 0.44 > ϕDST). (a) Applied torque versus angular velocity of the cylinder for a slowly increasing ramp of applied torque (h = 15 mm). (b) Time evolution of the cylinder speed for constant torques applied at t = 0 s (Γ/Γc = 0.36, 0.72, 1.07, 1.43 and 1.79, highlighted by dashed lines in (a)). See also Sup. Data movie 1.

over 90Γ c , where Γ c ≡ 2πhR 2 σ c ≃ 56 µN m is the critical torque for which the average stress at the cylinder surface (r = R) is equal to the onset stress of discontinuous shearthickening σ c ≃ 3.0 Pa, with h = 15 mm. Below Γ c , the velocity increases monotonically with increasing torque in a quasi-Newtonian fashion. By contrast, above Γ c the velocity is found to saturate, on average, at a value close to 4 rad/s, even though the torque is further increased by almost two orders of magnitude. Concomitantly, large-amplitude oscillations in Ω are observed. The ramp is not extended to higher torques because large out-of-plane deformations of the suspension layer are observed there.

The temporal dynamics of the cylinder in the low and large torque regimes is illustrated in figure 3b for a constant torque applied at t = 0 s, which is the case we will consider in the following of the paper. Below Γ c (blue disks) the cylinder velocity increases monotonically from 0 to its equilibrium steady value. Conversely, sufficiently above Γ c (Γ ≳ 1.4Γ c , pink squares, see also Sup. Data movie 1) the motion is found to be unsteady at all times. After a short initial transient of a few cycles, the motion settles down to periodic oscillations, with short and steep decreases in velocity separated by much slower and longer increases.

In the following, we consider successively the steady regime obtained for low torques and the oscillations observed for high torques.

Low-torque steady regime

The low-torque steady regime allows to verify and precise the suspension rheology from local, in situ measurements. For each constant-torque experiment in the low-torque regime we measure the steady flow field of the suspension (figure 4a) and extract the radial profile of the azimuthal velocity component, u θ (r) (figure 4b). The velocity is found to vanish beyond a no-flow radius R y (= 20.6, 34.9 and 47.2 mm, respectively, for Γ/Γ c = 0.35, 0.70 and 1.10), which suggests that the suspension has a small yield stress, σ y = Γ/(2πhR 2 y ) ≃ 0.38 ± 0.11 Pa, with a value in good agreement with those already reported for aqueous cornstarch suspensions at similar volume fractions [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions[END_REF]. Nevertheless, in the flowing region near the cylinder, the velocity profiles are found to match closely the steady Newtonian profile

u θ (r) ΩR =      r R 1/r 2 -1/R y 2 1/R 2 -1/R y 2 , r ⩽ R y , 0 , r ⩾ R y , (3.1)
which is represented by solid lines in figure 4b. This indicates that, in spite of the nonuniform stress field (σ(r) ∝ 1/r 2 ), no significant effect of shear-induced migration is observed in the low-torque regime, in agreement with previous observations [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions[END_REF]. Equivalently, the close to Newtonian behavior of the suspension (above σ y ) is illustrated in figure 4c, which presents the local stress, σ(r) = Γ/(2πhr 2 ), assuming steadiness and rotational invariance, as a function of the shear rate, γ(r) = -r∂ r (u θ /r), measured locally from the flow. The measurements obtained for different applied torques and different radii collapse on the same quasi-Newtonian flow curve, which matches (above σ y ) the frictionless part of the Wyart-Cates flow rule obtained from the global rheology measurements of figure 2. The match is fairly good up to the largest stress, σ ≃ 3 Pa, that can be reached in the steady regime. Importantly, this latter value also matches the discontinuous shear-thickening onset (σ c ≃ 3.0 Pa) obtained from the global rheograms (figure 2). This indicates that unsteadiness is triggered by the discontinuous shear-thickening of the suspension at the cylinder surface and that the onset condition is approximately σ(R) ≃ σ c , or equivalently, Γ ≃ Γ c . The instability mechanism, per se, will be clarified in the following sections.

High-torque unsteady regime

We turn, now, to the periodic oscillations observed for a constant applied torque exceeding Γ c , starting with the slow-acceleration phases of the oscillations.

Slow-acceleration phases

The slow-acceleration phases are visco-inertial transients. They can be described simply by considering the inertia of the cylinder, the constant applied torque and the frictionless Newtonian-effective drag of the suspension. Angular momentum theorem applied to the cylinder implies

I Ω = Γ -2πhR 2 σ(R, t) , (3.2)
with I = 104 ± 1 µN m s 2 the angular moment of inertia of the cylinder plus rotor of the rheometer, which is characterized using an in-built function of the rheometer (when the cylinder is in air). During the acceleration the shear stress is below σ c everywhere in the suspension, momentum has time to diffuse across the flow (ρR 2 /η 0 ∼ 10 -1 s ≪ Ω/ Ω ∼ 1 s, with ρ ≃ 1240 kg/m 3 the suspension density and η 0 ≃ 0.38 Pa s the nonfrictional viscosity of the suspension), and the inertia of the flowing part of the liquid (∼ ρhR 4 ∼ 10 -7 N m s 2 ) is much smaller than the cylinder inertia (I ∼ 10 -4 N m s 2 ). This means that the shear stress at the cylinder surface is almost equal to the steady, frictionless, Newtonian-effective value, σ(R, t) ≃ η 0 γ(R, t) ≃ 2η 0 Ω(t), with γ(R, t) ≃ 2Ω(t) the steady shear rate for a large gap (see (3.1) in the limit R/R y ≪ 1). Integrating (3.2) with σ(R, t) = 2η 0 Ω(t) gives

Ω(t) = Γ 4πη 0 hR 2 (1 -e -Γc Γ t-t ′ τacc ) , with τ acc = Iσ c 2η 0 Γ , (3.3)
which is found to agree quantitatively with the measured acceleration phases, as shown in figure 5a.

Transitions from slow-accelerations to fast-decelerations

During the acceleration phase, the rate of acceleration of the cylinder decreases, which means that the stress, (Γ -I Ω)/2πhR 2 , actually transmitted to the suspension increases, until a value of approximately σ c is reached. Because of the slowness of the acceleration the condition is reached for

Ω(t) = Ω c ≈ γc /2 , (3.4)
after the typical acceleration time (Γ/Γ c ) ln( 1 1-Γc/Γ ) τ acc set by (3.3). Equation (3.4) is valid for a suspension that would be strictly Newtonian up to σ c . In practice there is a continuous shear-thickening for stresses just below σ c , which augments the actual value of Ω c relative to γc /2. For the volume fraction we consider (ϕ = 0.44), the critical steady cylinder velocity based on the Wyart-Cates rheology (given by

Ω c = ∞ R γdr r = 1 2 σc 0 dσ η ) is Ω c ≃ 1.35 γc /2
. This value is found to be within a factor 2 of the critical velocity, Ω c ≈ 2 γc /2, observed experimentally, with no trend on the suspension layer thickness (see inset of figure 7c).

Besides this non-trivial relation between the stress level at the cylinder, σ(R), and the steady velocity, Ω, it must be noticed that the condition for unsteadiness is presumably more complex than σ(R) ⩾ σ c . As will be discussed in §4, the criteria is actually expected to depend slightly on the Reynolds number of the flow, as well as on the mode of destabilization. This latter point brings us to the fast deceleration phases.

Fast-deceleration phases

The deceleration between two acceleration cycles lasts only a fraction of a second, which is shorter than the temporal resolution of the rheometer. To resolve the dynamics we use high-speed imaging of the cylinder and a PIV procedure, which yield both timeresolved measurements of the cylinder velocity Ω(t) and the average stress at the cylinder surface, σ(R, t) = (Γ -I Ω)/2πhR 2 , from the measured deceleration rate, Ω. A typical evolution of Ω(t), Ω(t) and σ(R, t) is shown in figure 5. The cylinder decelerates from Ω c ≈ 7.5 to about 2 rad s -1 within approximately 0.1 s. This indicates that the drag of the suspension increases steeply. The average stress, σ| r=R , peaks at almost 10 2 σ c , which is about 30 times as large as the constant stress, Γ/2πhR 2 , applied by the rheometer and is much larger than the shear stress, σ J ≈ 10 Pa, at which the suspension jams, according to the fitted Wyart-Cates rheology.

To understand the peak of drag experienced by the cylinder, the flow of the suspension has to be considered. Figure 6a presents the evolution of the suspension velocity field measured during the deceleration phase (see also Sup. Data movie 2). It shows that the beginning of the cylinder deceleration ( Ω = 0, defining t = 0) coincides with the emergence of a velocity front, which propagates from the cylinder to the outer wall with a velocity of order 1 m/s. The front is found to nucleate over a small portion of the cylinder surface. After a short transient that will be discussed below, it propagates within an angular sector of approximately 180 • around the nucleation direction (the latter being random with respect to both the cylinder and the laboratory). As shown from the velocity profiles along the front propagation direction presented in figure 6b, this velocity front is also a 'thickening' front separating an inner high-stress and low-shear (thickened) region from the outer, unperturbed, low-stress region. The corresponding thickening front position, R f , the cylinder angular velocity, Ω, and the deceleration, -Ω, are plotted versus time in figures 6c-e, respectively (the front position is defined as the radius, in the front propagation direction, at which u θ = RΩ/2). These analyses confirm that the deceleration phase is triggered by the inception of the thickening front and show that the front propagates at a close to constant speed, V f , which is transverse to the suspension velocity and much larger than the latter. This front is of the same nature as those, coined jamming or shear fronts, reported previously in imposed-velocity configurations [START_REF] Peters | Quasi-2d dynamic jamming in cornstarch suspensions: visualization and force measurements[END_REF][START_REF] Peters | Direct observation of dynamic shear jamming in dense suspensions[END_REF]. For high levels of stress these fronts have been rationalized as kinematic shocks with a typical shock width, ∼ 10η 0 /ρV f , limited by viscosity, and a typical propagation velocity, V f ∼ ∆u/γ 0 , proportional to the increment of transverse velocity of the suspension across the shock, ∆u, and with the inverse of the relaxation strain scale, γ 0 [START_REF] Han | Shear fronts in shear-thickening suspensions[END_REF]. The fronts observed in the present study are found to match these scalings. The typical shock width, ≈ 10 mm, is of order 10η 0 /ρV f ∼ 5 mm, and the relative propagation velocity, V f /RΩ c ≈ 7, is consistent with reported values [START_REF] Peters | Quasi-2d dynamic jamming in cornstarch suspensions: visualization and force measurements[END_REF][START_REF] Han | Shear fronts in shear-thickening suspensions[END_REF].

t out t max t rel -Ωmax
The important point, now, is to elucidate how the interaction between the front propagation and the cylinder motion sets the oscillation and the effective drag of the suspension. To do so we model the velocity field and the stresses based on the features of the front observed experimentally. We assume that the velocity of the suspension remains essentially azimuthal (u θ ≫ u r ), that the front is sharp, such that the shear stress immediately upstream of the thickening front vanishes (σ| r>R f ≪ ρV f u θ ), that azimuthal stress gradients can be neglected and that the angular momentum imparted by the rheometer during the deceleration is small (Γ × 0.1 s ≪ IΩ c ). In this case, angular momentum conservation of the cylinder and the suspension gives

IΩ(t) + S(t)Ω(t) = IΩ c , (3.5)
where IΩ c is the initial angular momentum of the cylinder (before deceleration) and IΩ(t) and S(t)Ω(t) are the current momenta of the cylinder and of the suspension in the thickened region, respectively. Equation (3.5) simply expresses that the cylinder slows down as it shares momentum with the thickened region. Experimental observations (figure 6a) indicate that the front propagates typically over a half space, i.e. within an angle span ∆θ ≈ π. Inside the thickened region (r < R f (t)) the azimuthal velocity is roughly uniform and close to RΩ(t) (figure 6b). Based on these observations the angular momentum of the suspension reads

S(t)Ω(t) = ∆θρhR 3 R 3 f (t) -R 3 Ω(t) , (3.6)
and, from (3.5), the cylinder velocity is expected to follow

Ω(t) = Ω c 1 + α R 3 f (t)/R 3 -1 , where α = ∆θρhR 4 3I , (3.7)
represents the characteristic angular moment of inertia of the suspension (at scale R) relative to that of the cylinder. Considering R f (t) = R + V f t, as a consequence of the approximately constant front speed, V f (figure 6c), gives the following time evolution

Ω(t) = Ω c 1 + (α 1/3 + t/τ dec ) 3 -α , with τ dec = R V f α -1/3 . (3.8)
In the present case where the cylinder inertia is large (α -1 ≳ 10 2 in the experiments) the maximal deceleration rate is

-Ωmax ≃ 2 4/3 3 Ω c τ dec ∼ ∆θρhR 4 I 1/3 V f Ω c R , (3.9)
which is reached at

R f,max ≃ (2α) -1/3 R , and
t max ≃ (2 -1/3 -α 1/3 ) τ dec ∼ τ dec , (3.10)
when the moment of inertia of the thickened region (∼ ∆θρhRR f 3 ) compares with I. Equation (3.9) indicates that the maximal deceleration is expected to scale with the suspension layer thickness as -Ωmax ∝ h 1/3 or, equivalently, that the maximal stress, σ max ∝ -Ωmax /h, follows h -2/3 . We test this prediction experimentally by varying h systematically from 6 to 35 mm, which is the thickness range accessible with our set-up, while keeping a fixed value of Γ/Γ c = 3.5. For each thickness, three deceleration events are analyzed with high-speed PIV of both the cylinder and the suspension. The latter allows to verify that the front characteristics remain close to the heuristic features we adopted, i.e. ∆θ ≈ π and a front velocity, V f ≈ 1.1 m/s, independent of h (insets of

(R f (t)/R = 1 + L 2 + (V f t) 2 -L with L 2 = R 2 + (h/2) 2 , see text).
figure 7c). We also verify that the cylinder velocity at the onset of deceleration (Ω c ) is essentially independent of h.

Figure 7a shows that the maximal deceleration rate obtained experimentally is well captured by (3.9), both in magnitude and trend. It also indicates that the peak stress during the deceleration, σ max = -I Ωmax /(∆θhR 2 ), can reach as much as 300 σ c for the thinnest layer. As for the instant of the maximal deceleration, the magnitude of t max is reasonably well captured by (3.10) (see figure 7b), which consistently predicts that, for all thicknesses investigated, the maximal deceleration of the cylinder occurs before the time t out at which the front reaches the outer wall (t max /t out < 1). However, the model is found to underestimate the exact value, especially for thick layers. We will comment this point in the next paragraph.

Finally, figure 7c compares (3.8) with the evolution of the cylinder velocity observed during a whole deceleration event for a thickness h = 35 mm. The time origin is taken at the onset of deceleration ( Ω = 0). The match in the maximal deceleration rate is recovered but the deceleration rate at the beginning of the deceleration phase is found to be overestimated by (3.8), which indicates that the initial growth of the thickened region is not as fast as assumed by the model. This suggests that the thickened region nucleates at a single locus on the cylinder wall and has to extend across the layer and azimuthally before propagating radially only. This hypothesis is consistent with the initial growth of the angular span of the front, ∆θ, observed at the layer surface (see figure 6a). It is also in line with the slight underestimation of t max by the model. Indeed, as shown in figure 7b-c, a better match is found for both the early stage of deceleration and the arrival time, t out , by considering heuristically that the average front radius follows

R f (t)/R = 1 + L 2 + (V f t) 2 -L, with L 2 = R 2 + (h/2) 2 ,
to account for the azimuthal component, ∼ R, and the vertical component, ∼ h/2, an initially localized front has to propagate through. Importantly, the maximal deceleration is almost unchanged by the lag, provided h remains small relative to R f,max (see figure 7a).

As for the late stage of the deceleration, the model deviates from the measurements close to the time, t out , at which the thickening front reaches the outer wall. This is consistent because the leading edge of the front is not stress-free anymore. However, the deviation is not towards a stronger deceleration of the cylinder, as could be expected for a suspension fully jammed from the cylinder to the wall, but towards an increasingly milder deceleration until the cylinder eventually re-accelerates about t rel ≈ 120 ms after the onset of deceleration, i.e, ≈ 60 ms after t out (see figure 6e and figure 7c). This brings us to the question of the release mechanism terminating the fast-deceleration phases.

Transitions from fast-decelerations to slow-accelerations

The deceleration of the cylinder results from the extension of the shear-thickened region. On the other hand, shear-thickening is conditioned to the shear stress sustained by the suspension. If the latter typically decreases below the critical value σ c at any radius inside the thickened region, the front propagation should stop. This should in turn lead to a large drop in shear stress all over the suspension. Since at this time the deceleration has already reduced the cylinder velocity and the shear rate in the suspension significantly relative to the critical condition for the nucleation of a new thickening front, the cylinder is expected to enter a new acceleration phase (where the suspension is essentially unstressed because most of the applied torque is absorbed by the accelerating cylinder). Therefore, understanding the end of the deceleration phases requires understanding the stress field in the suspension.

To do so it is insightful to consider, first, the case of a sufficiently large vessel, for which no interaction with the outer wall is expected. Conserving the same assumptions as in §3.3.3 and applying the angular momentum theorem to the cylinder and thickened suspension up to a fixed radius r, the shear stress in the expanding thickened region

(r < R f ) follows σ(r, t) = - (I + S <r ) Ω ∆θhr 2 = - 1 + α(r 3 /R 3 -1) r 2 /R 2 I Ω ∆θhR 2 , (3.11)
with S <r Ω(t) the angular moment of the thickened suspension at radii below r. The stress is minimal at the same radius, r min ≃ (2/α) 1/3 R, regardless of time, and it evolves as (using 3.8)

σ min (t) ≃ 3ρRΩ c V f 2 2/3 t τ dec -4
.

(3.12)

This means that the release condition, when the shear stress becomes typically lower than σ c , is expected at a time and a front position given by

t rel,∞ ≃ τ dec 3 2 2/3 V f RΩ c Re 1/4 , R f,rel ≃ R 3 2 2/3 V f RΩ c Re 1/4 α -1/3 , (3.13)
respectively, where

Re = ρR 2 Ω 2 c σ c ∼ ρR 2 γc η 0 , (3.14)
is the typical Reynolds number at the onset of the deceleration phase. Since σ c ∼ η 0 γc and both Ω c and V f scale like γc , the Reynolds number is essentially the square of the ratio between the cylinder size R and the characteristic viscous length η 0 /ρ γc of the suspension. In the experiments Re ≈ 5 implies R f,rel ≳ 12 R, which means that the release stress, ≈ σ c , is not expected to be reached before the front arrives at the outer wall (R out ≃ 6.8 R), even for the thickest suspension layer. This is consistent with the observation that, in all the experiments, the re-acceleration of the cylinder (t = t rel ) is posterior to the arrival the front at the outer wall (t = t out ).

In this case the interaction of the front with the wall has to be considered. High stresses should develop as the thickened region reaches the wall, and one could expect that the cylinder would re-accelerate only after a reflected shear front propagating inward (i.e. a stop front with even higher stresses and a low velocity for r > R f ) has reached the cylinder or decreased the stress below σ c . Such an inward propagating wave of low velocity is indeed observed systematically in experiments (see figure 8a). However, modeling the influence of a stop front on the cylinder deceleration under the same assumptions as in §3.3.3 does not match, quantitatively, the observations for t > t out . Indeed, as discussed in appendix A, if the dissipation law of the front is assumed to be unchanged by the reflection on the wall, the cylinder deceleration is expected to be slightly larger for a reflected front than for an ever outward-propagating front (sufficiently large vessel), whereas the observed deceleration is actually milder. This suggests that departure from axial symmetry may have a more significant effect on the stress field in the suspension after the interaction of the front with the wall than before, or that the front interaction with the wall leads to either an elastic response of the suspension [START_REF] Larsen | Elasticity of dilatant particle suspensions during flow[END_REF] or to out-of-plane motion, facilitating the release of stress. Nevertheless, the observation of an inward-propagating stop front suggests that the typical re-acceleration time (relative to the onset of deceleration) should follow

t rel ≲ 2 R out -R V f . (3.15)
Equation (3.15) is found to be consistent with the measurements presented in figure 8b. The observed re-acceleration time is found to be t rel ≈ 2t out for all layer thicknesses without significant dependence on the latter. Similarly, the cylinder release velocity is found to be close to Ω rel ≈ Ω c /4 without significant dependence on h. While this is compatible with the idea that the front reflection at the wall leads to the stress release, which allows the cylinder re-acceleration, further investigations would be required to be conclusive.

Effective drag on the cylinder

The quantitative picture of the dynamics obtained in the previous sections allows to model the effective drag (time-averaged drag) of the suspension on the cylinder, for both steady and unsteady regimes, that is, for any value of the applied torque.

Below Γ c , the motion is steady and a quasi-Newtonian drag, Γ ≃ 4πη 0 hR 2 Ω, is expected. Above Γ c , the cylinder velocity oscillates between a fraction of Ω c and Ω c , it essentially saturates at a value of order Ωc ≈ γc/2 because of the periodic instability of the suspension flow. Solid line: average velocity, τ -1 τ 0 Ω(t)dt, expected from (3.3) with Ω rel = Ωc/4 (i.e., t ′ = I 4πη 0 hR 2 ln(1-Γc 4Γ )). Symbols: measurements obtained for an increasing ramp of applied torque (short-time averaging of the data of figure 3a).

whose value Ω c ≈ γc /2 is independent of the applied torque. This means that the average velocity ⟨Ω⟩ = τ 0 Ω(t)dt/τ is expected to saturate at a value of order γc /2, almost independently of the applied torque, or, equivalently, that the effective dragvelocity curve of the cylinder should be infinitely steep at Ω ≈ γc /2. More precisely, in the limit when the duration of the deceleration phase can be neglected, the average of the velocity given by (3.3) is expected to decrease only minimally from ⟨Ω⟩ = Ω c at Γ c to ⟨Ω⟩ = (Ω c + Ω rel )/2 ≈ 0.6Ω c for larger torques (for Γ/Γ c ≳ IV f 4πη0hR 2 (Rout-R) ≈ 20, the deceleration phase has to be considered, but this only marginally changes the expectation to ⟨Ω⟩ ≈ 0.5Ω c ).

As shown in figure 9, this simple model is found to capture the main characteristics of the effective drag-velocity curve obtained experimentally by ramping up the torque from Γ/Γ c ≈ 0.2 to over 90 (the data are obtained by a short-time averaging of those presented in figure 3a). Though the drag-velocity curve obtained by averaging (3.3) is slightly re-entrant for applied torques above Γ c , it is found to be in agreement with the experimental average velocity within a factor two over the almost three orders of magnitude variation in applied torque.

Discussion and conclusions

The experiments and analyses presented above provide a simple picture of the dynamics of a shear-thickening suspension driven by a rotating cylinder. For low applied torques the flow is steady. The drag is close to Newtonian and set by the low-stress branch of the rheology. Above the onset torque for discontinuous shear-thickening, Γ c , the average velocity of the cylinder saturates, while large periodic oscillations are observed. Long phases of slow acceleration are followed by short phases of high deceleration, which are triggered by the propagation of a thickening front, and so on. The front is found to nucleate when the shear rate at the cylinder surface reaches, typically, the discontinuous shear-thickening threshold ( γ ⩾ γc ), i.e., when the rotation rate, Ω, compares with the critical shear rate, ∼ γc /2 (3.4). The oscillation period is controlled by the slow acceleration phases,

∼ τ acc (Γ/Γ c ) ln( 1 1-Γc/Γ ) ∼ I η0R 2 h ln( 1 1-Γc/Γ ) (3.
3), the duration of which is set by the applied torque and cylinder inertia. The peak stress, σ max (3.9), and intrinsic timescale of the deceleration, τ dec (3.8), also involve the inertia of the suspension. However, in the present case of an insufficiently large gap, the duration of the deceleration phases, ∼ t out (3.15), is set by the propagation time of the front across the gap. Several important conclusions can be drawn from these analyses.

The first one concerns the oscillations and their timescales. As mentioned in the introduction, similar velocity oscillations have been reported in small-gap Couette flows [START_REF] Boersma | Time-dependent behavior and wall slip in concentrated shear thickening dispersions[END_REF][START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Nagahiro | Experimental observation of shear thickening oscillation[END_REF][START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Rathee | Localized stress fluctuations drive shear thickening in dense suspensions[END_REF][START_REF] Chacko | Dynamic vorticity banding in discontinuously shear thickening suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Ovarlez | Density waves in shear-thickening suspensions[END_REF][START_REF] Sedes | Fluctuations at the onset of discontinuous shear thickening in a suspension[END_REF][START_REF] Gauthier | A new pressure sensor array for normal stress measurement in complex fluids[END_REF]. In this case, the oscillations have been explained as limit cycles involving the tool inertia and a uniformly sheared suspension with a relaxing shear-thickening rheology [START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF]. This formalism predicts small amplitude oscillations, close to the instability onset torque, and oscillations with a strict stop, for higher torques. It also requires a second rheological relaxation scale (timescale) to reproduce the periodic oscillations reported experimentally. However, in the large-gap experiments presented above no small amplitude oscillation is observed. The amplitude of the oscillations is found to be of order one, including just above the onset torque for unsteadiness (≈ Γ c ), and the tool is never close to stop, even for much larger torques. Moreover, none of the oscillation timescales is reflecting the intrinsic rheological timescale, γ 0 / γc , only. All timescales of the oscillation, including those for the deceleration phases (τ dec ∼ γ0 γc ( ρhR 4 I ) -1/3 and t rel,∞ ∼ τ dec ( Re γ0 ) 1/4 ∼ τ dec ( γc γ0 ρR 2 η0 ) 1/4 or t out ∼ γ0 γc Rout R depending, respectively, on whether the vessel is large enough or not), also involve the suspension inertia, the cylinder inertia or the gap size, and parameters of the steady rheology. As for small-gap configurations, it is not possible to conclude on whether there is also a single thickened region for each oscillation. However, it is worth noticing that, in the small-gap limit, the propagation time of the front across the gap is typically the intrinsic rheological timescale, γ 0 / γc (indeed for a gap width w, it is t out = w/V f ∼ γ 0 w/RΩ c ∼ γ 0 / γc ). Therefore, both times are of order a fraction of a second for a dense aqueous suspension of cornstarch and compare with the typical oscillation periods reported for small-gap configurations.

Second, and as detailed in §3.4, the picture discussed above illustrates how a steep, but marginally re-entrant, effective drag curve can emerge from the strongly re-entrant rheology of the suspension. Because of the large inertia of the cylinder (α ≪ 1) the condition for the nucleation of a thickening front ( γ ⩾ γc ) sets a maximal cylinder velocity, Ω c ≈ γc /2, which is, essentially, independent of the applied torque. This implies that the average velocity, ⟨Ω⟩, saturates at a value ∼ γc . This saturation is a direct consequence of the flow instability, which means that the drag is intrinsically inertial above Γ c . Ironically, it turns out that a very similar effective drag curve would be expected if the flow was stable for shear stresses above σ c . Indeed, the steady shear stress follows σ/σ c = (Γ/Γ c )(R 2 /r 2 ), which means that, for a fully re-entrant flow curve as in figure 9a, the suspension would be jammed for radii below r J ≈ R(Γ/Γ c ) 1/2 and frictionless beyond. Therefore, the steady rotation velocity of the 'cylinder plus jammed suspension' would also be ≈ Γ/4πη 0 hr 2 J = Ω c , independently of the applied torque (above Γ c ). Nonetheless, this match should not be given too much significance. It omits the order one fluctuations of the motion as well as the inertial nature of the drag and it is merely coincidental. Indeed, the two scenarii (saturation by an instability at the velocity for which the shear stress at the solid surface is critical, or steady saturation when the drag of the 'solid plus jammed suspension' balances the applied force) do not predict the same average velocity for other configurations such as the drag of a translating cylinder or a translating sphere, for instance.

Third, and as briefly evoked in §3.3.2, the criteria for the onset of unsteadiness, Ω ≳ Ω c (or, equivalently, σ| r=R ≳ σ c ), is not expected to be strict nor independent of any other consideration. Indeed, for a Wyart-Cates rheology with a single relaxation scale, γ 0 , the dynamic for a slowly accelerating cylinder is expected to depend not only on the current velocity relative to the critical (Ω/Ω c ) but also on the critical Reynolds number, Re/γ 0 ∼ ρR 2 Ω c /(γ 0 η 0 ), which is the ratio between the typical viscous time, ρR 2 /η 0 , and the intrinsic rheological time, γ 0 / γc . In the present case where Re/γ 0 ≈ 30, the onset of instability is actually observed for Ω ≈ Ω c , while the destabilization consists in the nucleation and fast expansion of a thickened region resulting in a peak torque that is noncorrelated with the applied torque. It remains to be studied whether the onset velocity can be significantly shifted by varying the Reynolds number (as for zero-dimensional [START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF] or long-wavelength linear models (Darbois [START_REF] Texier | Shear-thickening suspensions down inclines: from kapitza to oobleck waves[END_REF], as well as whether the mode of destabilization is significantly altered for low Reynolds numbers or not.

Last, the role of the vessel and the release mechanism deserve a comment. In the present experiments, the thickening front interacts with the outer wall of the vessel before the cylinder re-accelerates, which means that some momentum is transferred to the vessel. This is certainly also the case for small-gap configurations, but the analyses considering the deceleration in a large vessel presented in §3.3.3 (R out ⩾ R f,rel ) suggests that this is not mandatory and that periodic oscillations could also be observed in a large or open system. As for the condition for the fading of the thickening front, it is certainly more complex than a decrease of the stress below σ c somewhere within the thickened region. It might actually be that the stress remains below the critical stress, σ J , at which the steady-state rheology is jammed for a significant deformation, ∼ γ 0 . Indeed, for a strainrelaxing Wyart-Cates rheology, a purely propagative front is possible only for a stress jump above σ J , while the thinning of the suspension requires a typical deformation γ 0 . These subtleties should not change, significantly, the expectation for the onset of the re-acceleration (t rel ), but they call for further studies to understand better the details of the release mechanism with or without interaction of the front with a wall.

All these considerations emphasize the importance of considering not only the rheology of the suspension but also its hydrodynamics and the interaction with the often large inertia of the moving solid to understand the unsteady flow of shear-thickening suspensions driven by a moving solid above the thickening threshold. They might prove useful for understanding other drags or flows, such as those past a sphere or a cylinder. Supplementary data. Supplementary movies are available at XXXXXXXXXX. Funding. This work was financially supported by the Agence Nationale de la Recherche (grants ANR-18-CE30-0024 and ANR-21-CE30-0015-01).

Declaration of interests. The authors report no conflict of interest.

Figure 1 .

 1 Figure 1. Experimental set-up. (a) Sketch of the setup. (b) Typical view of the cylinder (red), tracer-seeded suspension layer (black and white background) and measured flow field (arrows).

Figure 2 .

 2 Figure 2. Rheology of the suspension. (a) Shear stress versus shear rate for different volume fractions, ϕ. (b) Frictionless and frictional viscosity branches obtained from low and large shear-rates measurements, respectively. (a-b) Symbols stand for measurements with the small-gap cylindrical Couette device. Lines show the fitting Wyart-Cates law (see text).

Figure 4 .

 4 Figure 4. Low-torque steady regime: local rheology & onset of unsteadiness (h = 15 mm). (a) Steady velocity field around the cylinder for a constant applied torque Γ = 0.30Γc. (b) Radial profiles of the azimuthal velocity for Γ/Γc = 0.35, 0.70 and 1.10. Solid lines: Newtonian profiles given by (3.1). (c) Corresponding local shear stress versus local shear rate. Solid line: Wyart-Cates flow rule fitting the global rheometry in figure 2 (ϕ = 0.44). Red dashed line: onset shear stress of discontinuous shear-thickening, σc.

Figure 5 .

 5 Figure 5. High-torque unsteady regime. (a) Long-time evolution of the cylinder velocity (rheometer measurements, Γ = 3.5Γc, h = 35 mm). Solid-line: equation (3.3) for the acceleration phases. (b-c) Focus on a deceleration phase resolved with high-speed imaging. t = 0 stands for the onset of deceleration ( Ω = 0). (b) Cylinder velocity normalized by the velocity, Ωc, at the deceleration onset. (c) Deceleration (left-hand vertical axes) and shear-stress at the cylinder surface normalized by the discontinuous shear-thickening stress, σc (right-hand).

Figure 6 .

 6 Figure 6. Spatio-temporal analysis of a deceleration phase. (Γ = 3.5Γc, h = 35 mm). (a) Velocity fields at t = 8, 28, 48 and 68 ms after the cylinder has started decelerating (t = 0, see also Sup. Data movie 2). Colour encodes the relative norm of the velocity, u 2 θ + u 2 r /RΩc. (b) Radial profiles of the azimuthal velocity component in the direction where the thickening front nucleates and propagates (white arrow in a). (c-e) Time evolution of the thickening front position (c), of the cylinder velocity normalized by the deceleration onset velocity (d), and of the cylinder deceleration rate (e). Red dashed line: arrival time, tout, of the front at the outer (vessel) wall. Green-dashed line: release time, t rel , at which the next acceleration phase starts.

Figure 7 .

 7 Figure 7. Early part of the deceleration phase: maximal deceleration. (a) Normalized maximal deceleration of the cylinder or, equivalently, maximal shear stress in the suspension versus thickness of the suspension layer. (b) Maximal deceleration time normalized by the front arrival time at the outer wall, tout, versus h. (c) Normalized velocity of the cylinder during a deceleration phase (h = 35 mm). Insets from top to bottom: cylinder velocity at the onset of deceleration, front velocity and front angular width, ∆θ, versus h.(a-b-c) The pink solid lines show (3.7-3.10) with tout = (Rout -R)/V f for an homogeneous nucleation of the thickening front (R f (t)/R = 1 + V f t). The black dashed lines show the same equations for a localized nucleation (R f (t)/R = 1 + L 2 + (V f t) 2 -L with L 2 = R 2 + (h/2) 2 , see text).

Figure 8 .

 8 Figure 8. Late part of the deceleration phase: release. (a) Velocity profiles in the suspension (azimuthal velocity component, same direction as in figure 6) between the front arrival time, tout (R f = Rout), and the release time, t rel (onset of re-acceleration, Ω = 0). (b) Release velocity. The mean observed value is Ω rel ≃ 0.24 Ωc. (c) Release time relative to the front arrival time (time origin at the onset of the deceleration phase). Symbols: experiments. The solid line indicates the upper bound t rel ⩽ 2tout of (3.15), see text. The mean observed value is t rel ≃ 2.0 tout.

Figure 9 .

 9 Figure 9. Effective drag of the suspension. (a) The local rheological flow rule is strongly re-entrant for σ > σc. (b) The average cylinder velocity, ⟨Ω⟩, is marginally re-entrant for Γ > Γc:it essentially saturates at a value of order Ωc ≈ γc/2 because of the periodic instability of the suspension flow. Solid line: average velocity, τ -1 τ 0 Ω(t)dt, expected from (3.3) with Ω rel = Ωc/4 (i.e., t ′ =

† Email address for correspondence: henri.lhuissier@univ-amu.fr

Data availability statement. Data available upon request. Appendix A. Stress and deceleration for a reflected front

This appendix derives the deceleration of the cylinder and the shear stress in the suspension, which would be expected after the thickening front reaches the outer wall (t > t out ) if the front was reflected by the wall into an inward-propagating thickening front (stop front) preserving the axial symmetry of the flow.

For an inward propagating front, the outer side of the front is not stress-free as during the initial (outward propagating) front. Under the same assumptions as in §3.3.3, the angular momentum balance of the cylinder and flow motion becomes

with Γ out the resistive torque applied by the outer wall.

The thickening fronts are propagative structures, which are intrinsically dissipative. Assuming that the dissipation relation of the reflected front is the same as for the initial front (i.e., half the kinetic energy flux across the front), implies that the torque follows Γ out = -2 ṠΩ(t), which is positive since Ṡ = -ρ∆θhR 2 V f < 0. This yields

which matches, in t = t out , the velocity Ω = I I+S Ω c obtained for the initial front (3.7). Consistently, for a reflected front with the same velocity as the initial one, the deceleration is also continuous at t = t out . After t out , the deceleration predicted by (A 2) is slightly larger, though close in practice, than the one expected in the absence of outer wall (3.7).

Similarly, the minimal stress in the suspension is not much different from the one expected in the absence of outer wall (3.12). Indeed, the relation (3.11) between the stress on the inner side of the front (r < R f ) and the cylinder deceleration, i.e., σ(r, t) = -(I + S <r ) Ω

remains valid. Accordingly the stress is minimal at the same radius r min ≃ (2/α) 1/3 R, provided r min ⩽ R f , or at the inner side of the front (r = R f ), otherwise. Therefore the minimal stress is expected to follow

σ min (t) ≃ I(I + S) 5)