1. Introduction
Following the fundamental work of Taylor (1923) , flow patterns between two concentric cylinders, depicted in Fig. 1 , have been extensively studied. In practical applications involving Taylor–**Couette** **flows** (bio and chemical reactors, filtration, etc.), the inner cylinder usually rotates while the outer one stays at rest. Annular centrifugal contactors based on such geometry showed their great potential in the nuclear industry where they are particularly suitable for small-scale studies of solvent liquid–liquid extraction processes, as shown by Davis and Weber (1960) . This flow is known to exhibit multiplic- ity of stable regimes, ranging from laminar **Couette** flow to turbulence through a sequence of successive hydrodynamic instabilities ( Fig. 1 ), as the rotation rate of the inner cylinder is increased ( Andereck et al., 1986 ).

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2.1 What is the physical origin of the localisation?
There are several explanations for the origin of the localisation; many of them are reviewed and discussed by Schall and van Hecke (2010).
Coussot et al. (2002) used MRI methods to measure the local velocity in three-dimensional cylindrical **Couette** **flows** of other EVP materials such as carbopol gel, and more generally yield stress fluids such as bentonite suspensions and cement paste. Despite the apparent simplicity of shear **flows**, a common description of these experiments is still lacking (see Ovarlez et al. (2009) for a review): on the one hand, thixotropic materials exhibit an intrinsic critical shear rate, i.e. these materials cannot flow homogeneously at a shear rate smaller than a critical value, which is characteristic of the material; on the other hand, non-thixotropic materials may or may not exhibit a critical shear rate (that does not seem to be intrinsic to the material), as discussed below in section 2.3. We focus here on non-thixotropic materials and try to explain this peculiar behaviour.

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With this method, the effect of dilute (1–2%) and moder- ately concentrated (4–8%) suspensions is compared through mixing indicators in the Taylor Vortex Flow and Wavy Vor- tex Flow regimes. The conclusion drawn from the exper- imental results is a clear and systematic enhancement of mixing under two-phase flow conditions. Both the particle size and concentration promote mixing i) during the early stages following dye injection, ii) on the long time behav- iour towards homogeneous spatial distribution of tracer. Due to the finite size of particles interacting with the flow, yielding shear-induced hydrodynamic interactions and col- lisions, mixing is significantly enhanced in dilute suspen- sions compared to the single-phase flow. The effect is further enhanced by large particles. While particles are crossing the fluid streamlines, they drag fluid and participate to stretching and folding of tracer concentration spatial gradients. This mechanism scales with the square of the particle radius, because the dye in the boundary layer of the particle sur- face is transported along particle trajectories. These observa- tions regarding mixing in two-phase Taylor–**Couette** **flows** are consistent with the microscale mechanisms evidenced

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cylinders. The dispersive characteristics are analysed for Taylor Vortex Flow, Wavy Vortex **Flows** and fully Turbulent Taylor-**Couette** **flows**. Experiments based on flow visualization, PIV and PLIF measurements are compared to direct numerical simulations of Navier- Stokes equations coupled to Lagrangian tracking of fluid elements and bubbles. In vortical **flows**, bubble accumulation is driven by a competition between added-mass effect, lift and buoyancy forces. At low to moderate Reynolds numbers, the flow is strongly coherent and bubble accumulation patterns can be predicted theoretically (stability analysis of fixed points). When turbulence sets in, small scale structures enhance dispersion. This complex situation where large-scale coherent structures interact with fine scale turbulence leads to bubble mixing which have been analyzed by numerical simulations. Several distributions of bubbles are observed depending on the respective magnitude of turbulence and buoyancy force.

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Sophie Charton · Eric Climent
Received: date / Accepted: date
Abstract This paper reports on original experimental data of mixing in two-phase Taylor-**Couette** **flows**. Neu- trally buoyant particles with increasing volume concen- tration enhance significantly mixing of a passive tracer injected within the gap between two concentric cylin- ders. Mixing efficiency is measured by planar laser in- duced fluorescence coupled to particle image velocime- try to detect the Taylor vortices. In order to achieve re- liable experimental data, index matching of both phases is used together with a second PLIF channel. From this second PLIF measurements, dynamic masks of the par- ticle positions in the laser sheet are determined and used to calculate accurately the segregation index of the tracer concentration. Experimental techniques have been thoroughly validated through calibration and ro- bustness tests. Three particle sizes were considered, in two different flow regimes to emphasize their specific roles on the mixing dynamics.

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1. Introduction
Following the fundamental work of Taylor (1923) , flow patterns between two concentric cylinders, depicted in Fig. 1 , have been extensively studied. In practical applications involving Taylor–**Couette** **flows** (bio and chemical reactors, filtration, etc.), the inner cylinder usually rotates while the outer one stays at rest. Annular centrifugal contactors based on such geometry showed their great potential in the nuclear industry where they are particularly suitable for small-scale studies of solvent liquid–liquid extraction processes, as shown by Davis and Weber (1960) . This flow is known to exhibit multiplic- ity of stable regimes, ranging from laminar **Couette** flow to turbulence through a sequence of successive hydrodynamic instabilities ( Fig. 1 ), as the rotation rate of the inner cylinder is increased ( Andereck et al., 1986 ).

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Indeed, there are simple stationary solutions to the Navier-Stokes equations for fluid motion between two infinitely long, rigid walls (moving/fixed) maintained at different temperatures (cold upper wall or vice versa) with no-slip boundary condition. If the temperature difference between the walls is small enough then conduction would be the only means of heat transfer and one expects a linear temperature variation between both walls to prevail. The base flow, under the assumption that the buoyancy force is the only temperature effect in the momentum equation, could be plane Poiseuille or plane **Couette** flow depending on whether the walls are stationary or moving relative to each other. Hereafter, the former is referred to as Rayleigh-B´enard-Poiseuille flow (RBP ) and the latter is referred to as Rayleigh-B´enard-**Couette** flow (RBC). From an experimental as well as a theoretical point of view, the Rayleigh-B´enard convection problem is the simplest and most easily accessible case, in which the onset of instabilities can be readily studied. Plane **Couette** and plane Poiseuille flow represent prototype shear **flows** in which the onset of instabilities depends strongly on the initial conditions and hence, on the background disturbance field. Thus, the linear stability analysis of Rayleigh-B´enard- Poiseuille and Rayleigh-B´enard-**Couette** **flows** is expected to be of fundamental interest in hydrodynamic stability.

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Most of these studies reveal either a continuous [1,5–7] or discontinuous [2–4] transition between the flowing and non-flowing regions. These differences have not been un- derstood yet and continue to excite debate [2,9]. A scalar visco-elasto-plastic model including viscous drag [10,11] successfully reproduced the exponential decay of velocity that was observed in the bidimensional plane [9] and cylin- drical [1] **Couette** **flows** of a foam between two glass plates. In both cases, the localisation was interpreted as the com- petition between the internal viscosity of the foam and the external friction from the glass plates. Recently, the data presented in [1] were re-analysed in [12] and, in addi- tion to the velocity field, two tensorial informations were extracted both in stationary and transient regimes: the statistical elastic strain tensor and the plastic rearrange-

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The mean velocity proﬁles obtained from PIV measurements and DNS simulations are compared in Fig. 6 at three different locations that are relevant for the mixing study: the center of a vortex, and two consecutive vortex boundaries where the ﬂuid is ejected toward either the inner cylinder (inﬂow zone) or the outer one (outﬂow zone). An excellent agreement can be observed in every cases representative of the different Taylor–**Couette** ﬂow regimes we investigated. Moreover, beyond the mean ﬂow values, a good agreement is also achieved regarding velocity ﬂuctuations ( Fig. 6 , bottom), where deviation is only observed in the near-wall regions presumably due to measurement issues in the curved areas. The maximum of spurious velocities we measured was of the order of 1% of the inner cylinder speed regarding axial com- ponent; and 2% for the radial component.

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The mean velocity proﬁles obtained from PIV measurements and DNS simulations are compared in Fig. 6 at three different locations that are relevant for the mixing study: the center of a vortex, and two consecutive vortex boundaries where the ﬂuid is ejected toward either the inner cylinder (inﬂow zone) or the outer one (outﬂow zone). An excellent agreement can be observed in every cases representative of the different Taylor–**Couette** ﬂow regimes we investigated. Moreover, beyond the mean ﬂow values, a good agreement is also achieved regarding velocity ﬂuctuations ( Fig. 6 , bottom), where deviation is only observed in the near-wall regions presumably due to measurement issues in the curved areas. The maximum of spurious velocities we measured was of the order of 1% of the inner cylinder speed regarding axial com- ponent; and 2% for the radial component.

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Although the exact solutions for the Hartmann flow and the MHD **Couette** flow have been achieved for more than seventy years, the solutions for a heat transfer in flow between concentric rotating cylinders, also known as Taylor **Couette** **flows**, under external magnetic field have been restricted to high Hartmann numbers.

We can always rewrite Eq. (44) and express the spatial coordinate, *
z , as a function of the
spin velocity, * , (i.e., plot z by varying * * instead of plot * by varying z ) so as to avoid * encountering complex valued solutions or transition of the real valued solution from one root to another as shown in Rosensweig [22] for ferrofluid **Couette** **flows** subjected to uniform magnetic fields. However, even by this method, we will still encounter the problem of multi-valued solutions and of finding the most physically likely solution that satisfies the stable micro-particle rotation requirement for the present electrorotation **flows**. Moreover, as will be shortly shown in the following, since the linear velocity profile, u * , and the 2D volume flow rate, Q , solutions depend on integrations of the spin velocity profile, *

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IPGP, CNRS UMR 7154
(Dated: Received 18 December 2012; revised manuscript received 22 February 2013; published 18 March 2013) We investigate dynamo action in three-dimensional numerical simulations of turbulent spherical **Couette** **flows**. Close to the onset of dynamo action, the magnetic field exhibits an intermittent behavior, characterized by a series of short bursts of the magnetic energy separated by low-energy phases. We show that this behavior corresponds to the so-called on-off intermittency. This behavior is here reported for dynamo action with realistic boundary conditions. We investigate the role of magnetic boundary conditions in this phenomenon.

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We can always rewrite Eq. (44) and express the spatial coordinate, *
z , as a function of the
spin velocity, * , (i.e., plot z by varying * * instead of plot * by varying z ) so as to avoid * encountering complex valued solutions or transition of the real valued solution from one root to another as shown in Rosensweig [22] for ferrofluid **Couette** **flows** subjected to uniform magnetic fields. However, even by this method, we will still encounter the problem of multi-valued solutions and of finding the most physically likely solution that satisfies the stable micro-particle rotation requirement for the present electrorotation **flows**. Moreover, as will be shortly shown in the following, since the linear velocity profile, u * , and the 2D volume flow rate, Q , solutions depend on integrations of the spin velocity profile, *

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1 Synthèse et Conclusions
Cette étude avait pour but de caractériser l’influence d’une phase disper- sée sur le mélange en écoulement de Taylor-**Couette**. En effet, le mélange axial contrecarre le mécanisme de séparation dans la colonne **Couette** et doit donc être, sinon minimisé, du moins quantifié dans les colonnes d’ex- traction. fAfin de distinguer les différents mécanismes physiques à l’origine du mélange dans un tel système, nous avons choisi de travailler avec des billes sphériques calibrées de Polyméthacrylate de méthyle (PMMA), de diamètre 800 µm à 1,5 mm, en nombre suffisant pour atteindre des réten- tions jusqu’à 32% pour les propriétés hydrodynamiques, et 8% pour l’étude du mélange en colonne **Couette**. Ces particules sont mises en suspension dans une solution aqueuse de Thiocyanate de Potassium (KSCN) et de Di- méthylsulfoxyde (DMSO). Nous nous sommes particulièrement intéressés aux deux premières instabilités de l’écoulement de Taylor-**Couette**, à savoir les régimes de Taylor Vortex Flow (TVF) et Wavy Vortex Flow (WVF). Dans ces deux régimes, des études précédentes réalisées en monophasique ont mis en évidence un lien étroit entre les propriétés hydrodynamiques et l’efficacité du mélange. Par exemple, lorsque le nombre de Reynolds aug- mente, les mélanges intra-vortex (Desmet et al. (1996)) et inter-vortex (Ako- nur and Lueptow (2003)) sont accélérés. Lorsque le nombre de Reynolds dé- passe une valeur critique, un changement de régime (du TVF vers le WVF) apparaît. En régime ondulatoire (WVF), des études numériques par Rud- man (1998) et expérimentales par Nemri et al. (2014) s’accordent à préciser l’importance de la longueur d’onde axiale, représentant la taille d’une paire de vortex adjacents, sur le mélange. Enfin, le mélange intra-vortex et le mé- lange inter-vortex sont d’autant plus rapides que le nombre d’onde azimutal

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The earliest account of the struggle an experimentalist may face trying to set up genuine pCf was published by Reichardt ( 1959 ). Probably owing to these practical diffi- culties, experimental studies have remained rather scarce throughout. Aydin and Leutheusser ( 1991 ) give a short summary of the early work that focussed, just like the later investigations by Bech et al. ( 1995 ) and Malerud et al. ( 1995 ), mostly on fully turbulent states with Re of Oð1000Þ. Tillmark and Alfredsson ( 1992 ) pioneered the detailed exploration of the transitional regime in pCf by triggering turbulent spots, an approach that was subse- quently followed by Daviaud et al. ( 1992 ) and Hegseth ( 1996 ). A different kind of perturbation was introduced by Dauchot and Daviaud ( 1995 ) who used a wire spanned in the neutral velocity plane along the spanwise direction to slightly deform the velocity profile. Their analysis was later on extended experimentally (Bottin et al. 1997 ; Bot- tin and Chate´ 1998 ; Antar et al. 2003 ) and numerically (Barkley and Tuckerman 1999 ). Bottin and Chate´ ( 1998 ) combined the analysis of turbulent spots with an investi- gation of the reverse transition by so-called quench experiments in order to perform a statistical analysis of turbulence lifetimes. Some of the latest experimental results are due to Prigent et al. ( 2003 ) who observed oblique turbulent stripes, closely related to the ‘‘spiral’’ or ‘‘barbers’s pole’’ turbulence in Taylor **Couette** flow when reducing Re starting from a turbulent state. This pattern is the subject of a numerical analysis by Barkley and Tuck- erman ( 2007 ), while simulations by Schmiegel and Eck- hardt ( 2000 ) suggest that turbulence might persist down to Re & 280 if the Reynolds number is reduced carefully. All the experiments reported in the literature so far utilize flow visualizations or point measurements by Laser Doppler velocimetry or hot wire anemometry. To bridge the gap to modern spatially and temporally resolved techniques, we designed a flow facility that is suitable for three-dimen- sional particle tracking velocimetry (3D-PTV). Specifi- cally, this means using a considerably larger gap width (3 cm instead of few millimeters in most other recent realizations) to obtain proper optical access making it somewhat more costly to reach large aspect ratios. Our newly build setup enables us to present for the first time truly 3D velocity fields along with quantities such as the turbulent kinetic energy and the Reynolds shear stress tensor. We will present these results together with a comparison to direct numerical simulations.

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By perturbing the flow, a critical Reynolds number has been determined, above which an artificially triggered turbulent spot can persist.. The study of the spatiotemporal evolution of th[r]

In trying to identify steps in the transition process, a first distinction can be made from the nature of the primary bifurcation. Supercritical transitions are characterized by continuous growth of the distance to the basic state in the direction of some known unstable mode. Rayleigh-Bénard convection is a typical example. A scenario then develops, for which classical tools of (weakly) nonlinear analysis are available. Another, strongly different, situation oc- curs when the primary bifurcation is subcritical. In that case, our understanding is much more limited and entirely relies on our capability to determine branching solutions that become stable at finite distance of the basic state. In the realm of hydrodynamics, the plane **Couette** flow (PCF) corresponds to this situation since it is known to be linearly stable for all Reynolds numbers [3], whereas transition to turbulence is observed in experiments [4]. The PCF there- fore seems to be a good prototype for studying “globally subcritical” transitions to turbulence [5], and has been the subject of numerous studies accordingly.

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agrégats
Les résultats obtenus dans ce chapitre sont issus d’expériences de floculation en réacteur de Taylor-**Couette** en hydrodynamique séquencée. Les conditions de l’écoulement demeurent inchangées. Les mêmes taux de cisaillement seront étudiés et le séquençage sera identique à celui développé dans le chapitre précédent. Nous étudierons ainsi l’influence des conditions hydrodynamiques durant les six étapes conduites alternativement à faible et fort taux de cisaillement sur les propriétés des agrégats de latex. Nous nous intéressons à présent à l’influence du choix du coagulant sur les propriétés de taille et de forme des agrégats. Trois coagulants ont été choisis: le chlorure de sodium, le sulfate d’aluminium et le polychlorure de diallyl diméthyl ammonium (PolyDADMAC). Dans les conditions de l’expérience, le mécanisme à l’œuvre lors de l’ajout de chlorure de sodium ou de sulfate d’aluminium comme nous l’avons vu au chapitre II est principalement la neutralisation de charges. Le polymère quant à lui induit un mécanisme de floculation par pontage. L’étude de ces trois coagulants, mettant en avant deux mécanismes fréquemments observés d’agrégation, va permettre de discuter de l’impact de la physico-chimie sur la taille et la structure de flocs de latex placés sous contraintes hydrodynamiques. Nous nous intéresserons dans un premier temps à l’impact des trois coagulants préalablement cités sur la taille des agrégats puis, dans un second temps, aux conséquences sur leur forme.

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For the mixture of 65% glycerol, and for the Re and Reg ranges of the present study, the flow is a composite patterns flow, composed of wavy basic patterns. For the mixture of 65% glycerol, despite the fact that the gravity effect (Vb /V i) is comparable to Murai et al. (2008) [ 8 ], the difference in the geometry ( η larger than in Murai et al. [ 8 , 14 ]) makes the spiral pattern unstable. Indeed, in the geom- etry of Murai et al. [ 8 , 14 ], the entrapment of the bubbles preferentially near the inner cylinder and the increase of the axial wavelength in the bubbly flow make it easier the establishment of a steady spiral; this is not the case for the geometry of the present study which is characterized by preferential entrapment of the bubbles at midgap by the vortices and decrease of the axial wavelength induced by the bubbles. For the mixture of 65% glycerol,when increasing Reg (i.e., when increasing the air volumetric fraction α, the composite flow is less and less structured with more and more defects occurrences and transitions from different flow regimes are identified: For α < αIDC = 0.005%, the flow is a SCP flow [Fig. 6(b) ]. For α > αIDC = 0.005%, the SCP flow with punctual defects transits to the regime of the intermittency defect chaos (IDC). For the IDC regime, the defects appear randomly in space and time. They are no more localized in space and time, thus leading to defects spots that alternate in time and space with basic patterns [Fig. 6(c) ]. Defects spots are identified with space and time sequences where neither the toroidal, nor the spiral patterns prevail. A further increase in α above the critical value of αDDC = 0.01%, leads to the developed defect chaos regime: DDC [Fig. 6(d) ] for which the number of defects is high and for which the basic patterns do not persist beyond a time period of the azimuthal wave. In this case, there is a continuous switching between toroidal and spiral patterns. Interesting enough is the fact that Murai et al. (2008) [ 8 ] in the range of (Re, Qg) of their study did not observe unstructured composite **flows** (IDC or DDC). As mentioned before, the main difference between the two studies lies in the difference in the gap’s ge- ometry. Indeed, the geometry of the present study does not enable the development of a stable spiral pattern, which promotes the continuing switching between spiral and toroidal patterns (transition to DDC regime). The increase in the effective volume fraction (by the increase of Qg) destabilizes the toroidal pattern in favor of the spiral pattern; the increase in the axial pitch of the helicoidal bubble path (by the decrease of Vi) destabilizes the spiral pattern in favor of the toroidal pattern. As a consequence, an increase in the air volumetric fraction α, which is the ratio between Re g and Re is expected to destabilize both the spiral and toroidal patterns, thus leading to more and more defects occurrence.

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