Flows and heterogeneities with a vane tool: Magnetic resonance imaging measurements
Guillaume Ovarlez, Fabien Mahaut, François Bertrand, Xavier Chateau
aa r X i v : . [ c ond - m a t . s o f t ] M a y Flows and heterogeneities with a vane tool: MRI measurements
Guillaume Ovarlez ∗ , Fabien Mahaut, Fran¸cois Bertrand, Xavier ChateauUniversit´e Paris Est - Laboratoire Navier (UMR 8205 ENPC-LCPC-CNRS)2, all´ee Kepler, 77420 Champs-sur-Marne, FranceJuly 3, 2018 Synopsis
We study the local flow properties of various materials in a vane-in-cup geometry. Weuse MRI techniques to measure velocities and particle concentrations in flowing Newtonianfluid, yield stress fluid, and in a concentrated suspension of noncolloidal particles in a yieldstress fluid. In the Newtonian fluid, we observe that the θ -averaged strain rate component d rθ decreases as the inverse squared radius in the gap, in agreement with a Couette analogy.This allows direct comparison (without end-effect corrections) of the resistances to shearin vane and Couette geometries. Here, the mean shear stress in the vane-in-cup geometryis slightly lower than in a Couette cell of same dimensions, and a little higher than whenthe vane is embedded in an infinite medium. We also observe that the flow enters deeplythe region between the blades, leading to significant extensional flow. In the yield stressfluid, in contrast with the usually accepted picture based on simulation results from theliterature, we find that the layer of material that is sheared near the blades at low velocityis not cylindrical. There is thus a significant extensional component of shear that should betaken into account in the analysis. Finally and surprisingly, in the suspension, we observethat a thin non-cylindrical slip layer made of the pure interstitial yield stress fluid appearsquickly at the interface between the sheared material and the material that moves as a rigidbody between the blades. This feature can be attributed to the non-symmetric trajectoriesof the noncolloidal particles around the edges of the blades. This new important observationis in sharp contradiction with the common belief that the vane tool prevents slippage, andmay preclude the use of the vane tool for studying the flows of pasty materials with largeparticles. I Introduction
Experimental investigations of the rheology of concentrated suspensions often involve a vane-in-cup geometry (see Barnes and Nguyen (2001) for a review). The vane tool offers two mainadvantages over other geometries. First, it allows the study of the properties of structuredmaterials with minimal disturbance of the material structure during the insertion of the tool[Dzuy and Boger (1983); Alderman et al. (1991)]. It is thus widely used to study the propertiesof gels and thixotropic materials [Alderman et al. (1991); Stokes and Telford (2004)] and for insitu study of materials as e.g. in the context of soil mechanics [Richards (1988)]. Second, it issupposed to avoid wall slip [Keentok (1982); Dzuy and Boger (1983); Saak et al. (2001)], whichis a critical feature in concentrated suspensions [Coussot (2005)]; the reason for this belief isthat the material sheared in the gap of the geometry is sheared by the (same) material thatis trapped between the blades. Consequently, it is widely used to study the behavior of pastymaterials containing large particles, such as fresh concrete [Koehler et al. (2006); Estell´e etal. (2008); Wallevik (2008); Jau and Yang (2010)] and foodstuff [Stokes and Telford (2004);Mart´ınez-Padilla and Rivera-Vargas (2006)]. ∗ corresponding author: [email protected] a priori simple linear problem (for Hookean or Newtonian materials) is complex to solvewith a vane tool. This linear problem was studied theoretically by Sherwood and Meeten (1991)and Atkinson and Sherwood (1992) in the general case of a N -bladed vane tool embedded in aninfinite linear medium. The analytical expression found for the torque vs. rotational velocity isin rather good agreement with macroscopic experimental data [Sherwood and Meeten (1991)].Note however two possible shortcomings of this theoretical approach for its use in practice: theblades are infinitely thin and there is no external cylinder.There is no such approach in the case of nonlinear media ( i.e. complex fluids). A practicalmethod used to study the flow properties of non-linear materials, known as the Couette analogy[Bousmina et al. (1999); A¨ıt-Kadi et al. (2002); Estell´e et al. (2008)], consists in calibrating thegeometry factors with Hookean or Newtonian materials. One defines the equivalent inner radius R i, e q of the vane-in-cup geometry as the radius of the inner cylinder of a Couette geometry thatwould have the same geometry factors for a linear material. For any material, all macroscopicdata are then analyzed as if the material was sheared in a Couette geometry of inner cylinderradius R i, e q . The nonlinearity (that affects the flow field) is sometimes accounted for as it is in astandard Couette geometry [Estell´e et al. (2008)]. This approach may finally provide constitutivelaw measurements within a good approximation [Baravian et al. (2002)].However, simulations and observations show that R i, e q is not a universal parameter of thevane tool independent of the properties of the studied material. While the streamlines go into thevirtual cylinder delimited by the blades in the case of Newtonian media [Baravian et al. (2002)],yielding an equivalent radius lower than the vane radius [Sherwood and Meeten (1991); Atkinsonand Sherwood (1992)], it was found from simulations [Barnes and Carnali (1990); Savarmand et al. (2007)] that the streamlines are nearly cylindrical everywhere for shear-thinning fluids iftheir index n is of order 0.5 or less, and thus that R i, e q = R i in these cases. Moreover, foryield stress fluids, simulations and photographs of the shearing zone around a four-bladed vanerotating in Bingham fluids [Keentok et al. (1985)], simulations of Herschel-Bulkley and Cassonfluids flows in a four-bladed vane-in-cup geometry [Yan and James (1997)], and simulations ofBingham fluids flows in a six-bladed vane-in-cup geometry [Savarmand et al. (2007)], all showthat at yield ( i.e. at low shear rates), the material contained in the virtual cylinder delimitedby the blades rotates as a rigid body, and that it flows uniformly in a thin cylindrical layernear the blades. This is now widely accepted [Barnes and Nguyen (2001)] and used to performa Couette analogy with R i, e q = R i ; the yield stress τ y is then simply extracted from torque T measurements at low velocity thanks to τ y = T / (2 πHR i ), where H is the vane tool height(neglecting end effects) [Nguyen and Boger (1992)].The flow field in a vane-in-cup geometry and its consequences on the geometry factors havethus led to many studies. However, only theoretical calculations, macroscopic measurementsand simulation data exist in the literature: there are no experimental local measurements of theflow properties of Newtonian and non-Newtonian materials induced by a vane tool except thequalitative visualization of streamlines made by Baravian et al. (2002) for Newtonian media,and the photographs of Keentok et al. (1985) for yield stress fluids. Moreover, while the mainadvantage of the vane tool is the postulated absence of wall slip, as far as we know, this widelyaccepted hypothesis has been neither investigated in depth nor criticized. In order to providesuch local data, we have performed velocity measurements during the flows of a Newtonianmedium and of a yield stress fluid in both a coaxial cylinder geometry and a vane-in-cup geome-try. We have also performed particle concentration measurements in a concentrated suspensionof noncolloidal particles in a yield stress fluid, which is a good model system for complex pastessuch as fresh concrete [Mahaut et al. (2008a,b)]. Our main results are that:2i) in the Newtonian fluid, the θ -averaged strain rate component d rθ decreases as the inversesquared radius in the gap, as in a Couette geometry, which allows direct determination(without end-effect corrections) of the value of R i, e q : it is here found to be lower than R i ,but slightly higher than for a vane in an infinite medium; the flow enters deeply the regionbetween the blades, leading to a significant extensional flow;(ii) in the yield stress fluid, in contrast with results from the literature, the layer of materialthat is sheared near the blades at low velocity does not have a cylindrical shape;(iii) in the suspension of noncolloidal particles in a yield stress fluid, the noncolloidal particlesare quickly expelled from a thin zone near the blades, leading to the development of a thinslip layer made of the pure interstitial yield stress fluid, in sharp contradiction with thecommon belief that the vane tool prevents slippage.In Sec. II, we present the materials employed and the experimental setup. We present theexperimental results in Sec. III: velocity profiles obtained with a Newtonian oil and with a yieldstress fluid are presented in Sec. IIIA and Sec. IIIB, while Sec. IIIC is devoted to the case ofsuspensions, with a focus on the slip layer created by a shear-induced migration phenomenonspecific to the vane tool.Throughout this paper, we use cylindrical coordinates ( r, θ, z ). All flows are supposed to be z invariants ( i.e. there are no flow instabilities). We define the θ -average ¯ f ( r ) of a function f ( r, θ ) as ¯ f ( r ) = (1 / π ) R π f ( r, θ ) d θ . II Materials and methods
A Materials
We study three materials: a Newtonian fluid, a yield stress fluid, and a concentrated suspensionof noncolloidal particles in this yield stress fluid.The Newtonian fluid is a silicon oil of 20 mPa.s viscosity.The yield stress fluid is a concentrated water in oil emulsion. The continuous phase is dodecaneoil in which Span 80 emulsifier is dispersed at a 7% concentration. A 100 g/l CaCl solution isthen dispersed in the oil phase at 6000 rpm during 1 hour with a Sliverson L4RT mixer. Thedroplets have a size of order 1 µ m from microscope observations. The droplet concentration is75%, and the emulsion density is ρ f = 1 .
01 g cm − . The emulsion behavior, measured throughcoupled rheological and MRI techniques described in Ovarlez et al. (2008) (see Fig. 1), is wellfitted to a Herschel-Bulkley behavior τ = τ y + η HB ˙ γ n with yield stress τ y = 20 . η HB =6.8 Pa s . , and index n = 0 . ρ p = 1 .
05 g cm − ,diameter d = 250 µ m) suspended at a 40% volume fraction in the concentrated emulsion de-scribed above. The density matching between the particles and the yield stress fluid is sufficientto prevent shear-induced sedimentation of the particles in the yield stress fluid [Ovarlez et al. (2010)]; in all experiments, we check that the material remains homogeneous in the verticaldirection by means of MRI density measurements. B Rheometry
The rheometric experiments are mainly performed within a six-bladed vane-in-cup geometry(vane radius R i = 4 .
02 cm, outer cylinder radius R o = 6 cm, height H =11 cm). The shaftradius is 1.1 cm and the blade thickness is 6 mm. Other experiments are performed with awide-gap Couette geometry of slightly different inner cylinder radius R i = 4 .
15 cm, due to thepresence of sandpaper (the other dimensions were identical). The inner cylinder of the Couette3 .01 0.1 1 10 100020406080 S h ea r s t r e ss ( P a ) Shear rate (s -1 ) Figure 1: Constitutive law of the emulsion measured locally in the gap of a Couette cell throughMRI techniques [Ovarlez et al. (2008)]. The empty squares are local data; the solid line is aHerschel-Bulkley fit to the data τ = τ y + η HB ˙ γ n with τ y = 20 . η HB =6.8 Pa s . , and n = 0 . C MRI
Proton MRI [Callaghan (1991)] was chosen as a non-intrusive technique in order to get measure-ments of the local velocity and of the local bead concentration inside the sample. Experimentsare performed on a Bruker 24/80 DBX spectrometer equipped with a 0.5T vertical supercon-ductive magnet with 40 cm bore diameter and operating at 21 MHz (proton frequency). Weperform our experiments with a home made NMR-compliant rheometer, equipped with the ge-ometries described in the previous section. This device was already used in a number of previousrheo-nmr studies [Raynaud et al. (2002); Rodts et al. (2004); Ovarlez et al. (2006)], and is fullydescribed in Raynaud et al. (2002). The volume imaged is a (virtual) rectangular portion of4 cm in the axial (vertical) direction with a width (in the tangential direction) of 1 cm and alength of 7 cm (in the radial direction, starting from the central axis). Velocity and concentra-tion profiles V θ ( r ) and φ ( r ), averaged over the vertical and tangential directions in this volume,are obtained with a resolution of 270 µ m in the radial direction. This volume is situated at themagnet center (so as to minimize the effects of field heterogeneities) and sufficiently far fromthe bottom and the free surface of the rheometer so that flow perturbations due to edge effectsare negligible. We checked that the velocity and concentration profiles are homogeneous alongthe vertical direction in this volume, which justifies averaging data over this direction.Details on the sequence used to obtain velocity profiles can be found in [Raynaud et al. (2002); Rodts et al. (2004)]. While it is possible to get 2D or 3D maps of 2D or 3D velocityvectors [Rodts et al. (2004)], such measurements may actually take minutes and require complexsynchronization of the MRI sequences and of the geometry position. However, we will show inthe following that the azimuthal velocity alone provides a valuable information that can besufficient for most analyses; in particular it allows computation of the θ -averaged strain ratecomponent ¯ d rθ and thus the θ -averaged shear stress ¯ τ rθ (and the torque T ) in the case of theNewtonian oil. That is why we have chosen to limit ourselves to 1D profiles of 1D velocitymeasurements, namely the azimuthal velocity V θ ( r, t ) as a function of the radius r and time4 , for which a single measurement may take as little as 1 s; this has allowed us to perform asufficient number of experiments, with various materials, geometries, and rotational velocities.Depending on the time over which this measurement is averaged as compared to 2 π/ ( N Ω)where N = 6 is the number of blades and Ω is the rotational velocity, this measurement mayprovide either a time (or θ )-averaged azimuthal velocity ¯ V θ ( r ) = (1 / π ) R π V θ ( r, θ )d θ or aninstantaneous (transient) azimuthal velocity V θ ( r, t ). In this latter case, the θ dependence ofthe azimuthal velocity V θ ( r, θ ) at a given radius r can then be easily reconstructed by simplyreplacing the time t dependence by an angular θ dependence with θ = Ω t . It should also be notedthat due to incompressibility of the materials we study, the V r ( r, θ ) field can be reconstructedthanks to (1 /r ) ∂ r ( rV r ) + (1 /r ) ∂ θ ( V θ ) = 0 with V r ( R o , θ ) = 0; however, this derivation from theexperimentally measured values of V θ ( r, θ ) cannot be very accurate. Finally, from V r ( r, θ ) and V θ ( r, θ ), we are also able to evaluate the strain rate components d rr ( r, θ ) = − d θθ ( r, θ ) = ∂ r V r and d rθ ( r, θ ) = (1 / (cid:2) (1 /r ) ∂ θ ( V r ) + r∂ r ( V θ /r ) (cid:3) . Note that the derivative ∂ x f with respect tocoordinate x of experimental data f ( x i ) measured at regularly spaced positions x i was computedas: ∂ x f ( x i ) = [ f ( x i +1 ) − f ( x i − )] / [ x i +1 − x i − ].The NMR sequence used in this work to measure the local bead concentration is a modifiedversion of the sequence aiming at measuring velocity profiles along one diameter in Couettegeometry [Hanlon et al. (1998); Raynaud et al. (2002)], and is described in full detail in Ovarlez et al. (2006). The basic idea is that during measurements, only NMR signal originating fromthose hydrogen nuclei belonging to the liquid phase of the sample ( i.e. both the oil and waterphase of the emulsions) is recorded: the local NMR signal that is measured is thus proportionalto 1 − φ , where φ is the local particle volume fraction. A rather low absolute uncertainty of ± .
2% on the concentration measurements values was estimated in Ovarlez et al. (2006). Allvolume fraction profiles are measured at rest, after a given flow history. This is possible becausethe particles do not settle in the yield stress fluid at rest: the volume fraction profile inducedby shear is gelled by the interstitial yield stress fluid.
III Experimental results
In this section, we study successively the flow properties of the Newtonian oil, the yield stressfluid, and the suspension.
A Newtonian fluids
In this section, we study the flows observed with a Newtonian fluid. We first present a basictheoretical analysis of the flows in a vane-in-cup geometry as compared to flows in a standardCouette geometry, which provides the basis for a Couette analogy. The θ -averaged azimuthalprofiles ¯ V θ ( r ) are then shown, and are compared to predictions of the Couette analogy. The fullvelocity field V θ ( r, θ ), V r ( r, θ ) is finally presented and analyzed. The stress balance equation projected along the azimuthal axis is:(1 /r ) ∂ r ( r τ rθ ) + ∂ θ ( τ θθ ) − ∂ θ p = 0 (1)where τ ij is the deviatoric stress tensor and p the pressure.The strain rate tensor component d rθ is given by: d rθ ( r, θ ) = 12 (cid:16) (1 /r ) ∂ θ ( V r ) + r∂ r ( V θ /r ) (cid:17) (2)We recall that the constitutive law of a Newtonian fluid of viscosity η is: τ ij = 2 ηd ij (3)5 In all the following analysis, we assume a no-slip boundary condition at the walls of the innertool and of the cup. Couette geometry
In a standard – coaxial cylinders – Couette geometry, due to cylindrical symmetry, Eq. 1 becomes ∂ r ( r τ rθ ) = 0 which means that the whole shear stress distribution τ rθ ( r ) in the gap is knownwhatever the constitutive law of the material is. If a torque T (Ω) is exerted on the inner cylinderdriven at a rotational velocity Ω, τ rθ ( r ) is given by: τ rθ ( r ) = − T (Ω)2 πHr (4)For a Newtonian fluid of viscosity η , it follows that the strain rate component d rθ ( r ) is given by: d rθ ( r ) = − T (Ω) η πHr (5)As Eq. 2 becomes d rθ ( r ) = (1 / r ∂ r ( V θ /r ) with cylindrical symmetry, due to the boundaryconditions V θ ( R i ) = Ω R i and V θ ( R o ) = 0, from R R i R o d rθ ( r ) /r d r = Ω, one gets alternatively d rθ ( r ) = − Ω R i R o / [ r ( R o − R i )]. This yields the following azimuthal velocity profile: V θ ( r ) = Ω R i r R o − r R o − R i (6)Finally, the viscosity η of a Newtonian fluid is obtained from the measured torque/rotationalvelocity relationship T (Ω) through η = T (Ω)Ω R o − R i πHR o R i (7)These equations will be used for the comparison with the flows observed in a vane-in-cupgeometry, in particular to determine the radius R i, e q of the equivalent Couette geometry. Vane-in-cup geometry
In a vane-in-cup geometry, there is a priori no cylindrical symmetry and all quantities a priori depend on θ . However, averaging Eq. 1 over θ yields ∂ r ( r ¯ τ rθ ) = 0. This means that whatis true in a Couette geometry, τ rθ ( r ) = τ rθ ( R i ) R i /r , is still true on average with a vane-in-cup geometry: ¯ τ rθ ( r ) = ¯ τ rθ ( R i ) R i /r independently of the material’s constitutive law. Notethat this derivation is true only between R i and R o ; this is not true for the material betweenthe blades as the unknown τ ij distribution in the blades contributes to the θ -average. Thelink between this stress distribution and the torque T (Ω) exerted on the vane tool may thenseem difficult to build. However, it can be equivalently computed on the outer cylinder as T = R π τ rθ ( R o , θ ) HR o d θ = 2 πHR o ¯ τ rθ ( R o ). This means that Eq. 4 is still valid for the θ -averaged shear stress in the vane-in-cup geometry, for R i < r < R o :¯ τ rθ ( r ) = − T (Ω)2 πHr (8)From the θ -averaged Eq. 3, this means that the θ -averaged strain rate distribution in aNewtonian fluid, for R i < r < R o , is:¯ d rθ ( r ) = − T (Ω) η πHr (9)6rom the θ -averaged Eq. 2 ¯ d rθ = (1 / r ∂ r ( ¯ V θ /r ), this means that the θ -averaged azimuthalvelocity profile of a Newtonian fluid of viscosity η in a vane-in-cup geometry for R i < r < R o ,with a boundary condition ¯ V θ ( R o ) = 0 is given by:¯ V θ ( r ) = T (Ω) η R o − r πHR o r (10)Finally, the only difference with a standard Couette flow, as regards these θ -averaged quan-tities, is that we do not know the value of ¯ V θ ( R i ); we only know that V θ ( R i , πk/n ) = Ω R i , for k integer, where N is the number of blades. This means that ¯ d rθ ( r ) and ¯ V θ ( r ) follow the samescaling with r and Ω as in the standard Couette geometry, but with a different prefactor.Nevertheless, these equations provide a new insight in the Couette analogy. The usual wayof performing the Couette analogy consists in defining the radius of the equivalent Couettegeometry R i, e q as the radius that allows measuring the viscosity η of a Newtonian fluid withthe standard Couette formula. From Eq. 7, η should then be correctly obtained from thetorque/rotational velocity relationship T (Ω) measured in a vane-in-cup geometry with: η = T (Ω)Ω R o − R i, e q πHR o R i, e q (11)From Eqs. 7 and 11, it means in particular that the torque T vane exerted by the vane tool isdecreased by a factor T vane T Couette = R i, e q R i − R i /R o − R i, e q /R o (12)as compared to the torque T Couette exerted by the inner cylinder of a Couette geometry of sameradius R i at a same rotational velocity.Here, from Eqs. 10 and 6, we see that from the local flow perspective, there is a Couetteanalogy in the sense that the θ -averaged azimuthal velocity (and shear) profiles will be exactlythe same as in a Couette geometry. This defines a radius R i, e q of the equivalent Couette geometry,such that ¯ V θ ( r ) and ¯ d rθ ( r ) for R i < r < R o are given by:¯ V θ ( r ) = Ω R i, e q r R o − r R o − R i, e q (13)¯ d rθ ( r ) = − Ω R i, e q R o r ( R o − R i, e q ) (14)Of course, these two definitions of R i, e q are equivalent: combining Eqs. 10 and 11 yields Eq. 13.This point of view provides an additional meaning to the Couette analogy, namely thesimilarity of the average flows, and offers a new experimental mean to determine R i, e q , which ismore accurate than calibration. In rheological measurements, the T (Ω) relationship has to becorrected for end effects [Sherwood and Meeten (1991); Mart´ınez-Padilla and Quemada (2007);Savarmand et al. (2007)] and the Couette analogy has to be calibrated on a reference materialof known viscosity. Here, the ¯ V θ ( r ) or ¯ d rθ ( r ) measurements provide the value of R i, e q directlywithout any correction, as only shear in the gap is involved in the analysis, and independent ofthe viscosity of the material. This will be illustrated in the following. Vane in a finite cup vs. vane in an infinite medium
The only theoretical prediction of the stress field associated with a vane tool is that of Atkinsonand Sherwood (1992) for an infinite N -bladed vane embedded in an infinite linear medium.7n this case, it is shown that the torque T vane (Ω , N, R i , R o = ∞ ) exerted on the vane is wellapproximated by T vane (Ω , N, R i , R o = ∞ ) T Couette (Ω , R i , R o = ∞ ) = 1 − N (15)where T Couette (Ω , R i , R o = ∞ ) is the torque exerted on a cylinder of same radius R i as the vanein an infinite medium ( i.e. with R o = ∞ ). Eq. 15 is in agreement with experimental results[Sherwood and Meeten (1991)].Sherwood and Meeten (1991) argue that, as the stress distribution varies as 1 /r in a Couettegeometry, ( R i /R o ) should be of the order of 1% or less in order to nullify the influence of theouter boundary; this is clearly the case in their experiments and in field experiments where thevane is embedded e.g. in a soil; this is clearly not the case in our experiments and in mostrheological experiments that make use of a vane-in-cup geometry. However, when the cup tovane radius ratio R o /R i is not large, no generic theoretical expression exists in the literature.Nevertheless, bounds of the value of the torque T vane (Ω , N, R i , R o ) can be derived usingclassical results of linear elasticity [Salen¸con (2001)]. Our starting points are the variationalapproaches to the solution of the Stokes equations describing the flow of an incompressiblelinear material induced by the rotation of an inner tool (of any shape) at a rotational velocityΩ within a cup. In this framework, it can be shown that [Salen¸con (2001)]: Z S v rσ ′ θn d S − η Z ω f τ ′ ij τ ′ ij d ω ≤ T vane (Ω , N, R i , R o ) ≤ η Ω Z ω f d ′ ij d ′ ij d ω (16)where S v denotes the inner tool-fluid interface and ω f the domain occupied by the fluid. Inthe first inequality, σ ′ ij is any stress tensor complying with the stress balance equations, σ ′ θn isthe azimuthal component of the surface forces applied by the tool on the fluid and τ ′ ij is thedeviatoric stress tensor associated to σ ′ ij . In the second inequality, d ′ ij is the strain rate tensorassociated with any velocity field V ′ complying with the incompressibility constraint and theboundary conditions prescribed on the tool-fluid and cup-fluid interfaces.Eq. 16 leads in particular to the expected inequalities: T vane (Ω , N, R i , R o = ∞ ) ≤ T vane (Ω , N, R i , R o ) ≤ T Couette (Ω , R i , R o ) (17)The lower bound is obtained by using the velocity field defined by V ′ = V Ω ,N,R i ,R o for r < R o and V ′ = 0 for r ≥ R o , where V Ω ,N,R i ,R o is the solution for the N -bladed vane of radius R i in a cup of radius R o . V ′ complies with the boundary conditions for the N -bladed vane ofradius R i in an infinite domain problem. Then, putting this test velocity field within the secondinequality (16) with R = ∞ and using T vane (Ω , N, R i , R o ) = 2 η Ω Z ω ( N,R i ,R o ) d ij d ij d ω (18)yields the lower bound of the inequality (17) for the quantity T vane (Ω , N, R i , R o ). In Eq. 18, d ij denotes the strain rate tensor associated with V Ω ,N,R i ,R o while ω ( N, R i , R o ) is the domainoccupied by the fluid.The upper bound is obtained using the test velocity field defined by V ′ = V ′ θ ( r ) e θ with V ′ θ defined by Eq. 6 for R i ≤ r ≤ R o and by V ′ θ = Ω r for r ≤ R i . It is easily checked that V ′ complies with the velocity boundary conditions for any vane-in-cup geometry with vane radius R i and cup radius R o . Putting this test velocity field into the second inequality (16) then yieldsthe upper bound of inequality (17).Finally, combining inequalities (17), Eq. 15 and Eq. 7 yields (cid:18) − N (cid:19) (cid:18) − R i R o (cid:19) ≤ T vane (Ω , N, R i , R o ) T Couette (Ω , R i , R o ) ≤ θ = ± π/N by σ ′ rθ = − τ (cid:18) rR i (cid:19) m ; σ ′ θθ = − ( m + 2) τ θ (cid:18) rR i (cid:19) m ; σ ′ rr = − τ θ m + 2 m + 1 (cid:18) rR i (cid:19) m (20)with m > − r ≤ R i and by σ ′ rθ = − τ (cid:18) rR i (cid:19) − ; σ θθ = 0 ; σ ′ rr = − τ θ m + 2 m + 1 (cid:18) rR i (cid:19) − (21)for R i ≤ r ≤ R o . This stress field complies with the balance equations within the fluid domain.Let us recall that a stress field does not need to be continuous to comply with the balanceequations (of course, this stress field is not the solution of the problem). Putting this stressfield into the first inequality (16) and using a numerical optimization tool to choose the optimalvalue of the parameter m yields a new lower bound for the N -bladed vane in cup problem,which depends on N and R i /R o . In some cases, this test stress field improves the lower boundof inequality (19): e.g., for the geometry we use in this study ( N = 6 , R o /R i = 1 . .
57 while the lower bound given by inequality (19) is 0 .
45. Nevertheless, such animprovement is not obtained for all parameter sets ( N , R i /R o ). It is thus necessary to computethe two lower bounds for each value of ( N , R i /R o ) in order to obtain the more accurate lowerestimate of the torque. Although application of variational approaches to the derivation ofestimates of the applied torque of a vane-in-cup problem is not classical, it is believed thatsuch a strategy is able to provide useful results when no theoretical prediction of the solution isavailable for particular geometries. Lower bounds of T vane /T Couette computed using the approachpresented above are displayed in Tab. 1 and are compared below to our results and to data inthe literature. θ -averaged profiles We first study the θ -averaged azimuthal velocity profiles ¯ V θ ( r ) observed during the flows ofa Newtonian oil (Fig. 2). As shown above, these profiles can be used to check the validityof the Couette analogy and to determine the Couette equivalent radius R i, e q . The azimuthaldependence of the velocity profiles between two adjacent blades of the vane tool will then beconsidered.In Fig. 2a we observe that the velocity profiles in the gap of a Couette geometry are, as ex-pected, in perfect agreement with the theory for a Newtonian flow (Eq. 6). This first observationcan be seen as a validation of the measurement technique.In the vane-in-cup geometry (Fig. 2b), we first note that the θ -averaged dimensionless az-imuthal velocity profiles ¯ V θ ( r, Ω) / Ω R i measured for several rotational velocities Ω are super-posed, as expected from the linear behavior of the material. We also remark that the materialbetween the blades rotates as a rigid body only up to r ≃ . r, θ ) plane will be determined inSec. IIIA3 (Fig. 4b). We finally observe that the theoretical velocity profile for a Newtonianfluid in a Couette geometry of radius equal to that of the vane lies above the data, as expectedfrom the literature. This is also consistent with the observation that the shear flow enters theregion between the vane blades.In order to test the Couette analogy, we have chosen to plot the θ -averaged strain rate ¯ d rθ vs. the radius r in Fig. 3. This allows us to distinguish more clearly the difference betweenthe experimental and theoretical flow properties than would the velocity profiles, because thevelocity profile always tends to the same limit ( V ( R o ) = 0) at the outer cylinder whereas the9 .2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00.00.20.40.60.81.0 D i m e n s i on l e ss v e l o c it y NewtonianRotational velocity 20rpm 10rpm 5rpm 2rpm
Radius (cm)(a) (b) D i m e n s i on l e ss a v e r a g e v e l o c it y Radius (cm)
Figure 2: a) Dimensionless velocity profile V θ ( r, Ω) / Ω R i of a Newtonian oil in a Couette geometry( R i = 4 .
15 cm), at various rotational velocities Ω ranging from 2 to 20 rpm; the solid lineis the theoretical profile for a Newtonian fluid. b) Dimensionless θ -averaged velocity profile¯ V θ ( r, Ω) / Ω R i of a Newtonian oil in a six-bladed vane-in-cup geometry ( R i = 4 .
02 cm) for Ωranging from 1 to 9 rpm; the vertical dashed line shows the radius of the vane; the dotted lineis the theoretical profile for a rigid body rotation (for r < R i ); the solid lines are the theoreticalprofiles for a Newtonian fluid in Couette geometries of radii, from right to left: (i) R i = 4 .
02 cm,(ii) R i, e q = 3 .
90 cm, and (iii) R i,th = 3 .
82 cm corresponding to the Atkinson and Sherwood(1992) theory in an infinite medium.
Couette prediction R i = 4.02cm R i,eq = 3.90cm R i,th = 3.82cm S t r a i n r a t e ( s - ) Radius (cm)
Figure 3: θ -averaged strain rate ¯ d rθ vs. radius r for a Newtonian oil sheared at 1 rpm in asix-bladed vane-in-cup geometry. The vertical dashed line shows the radius of the vane. Thesolid lines are the theoretical strain rate profiles for a Newtonian fluid in Couette geometries ofradii: (i) R i = 4 .
02 cm (light grey), (ii) R i, e q = 3 .
90 cm (black), and (iii) R i,th = 3 .
82 cm (darkgrey) corresponding to the Atkinson and Sherwood (1992) theory in a infinite medium.10train rate profile does not. Note that velocity measurements could not be performed close tothe blades, which explains why strain rate data are missing from 4 to 4.2 cm.In Fig. 3, we first note that ¯ d rθ is zero up to ≃ d rθ then increases when r tends towards R i as the material ismore and more sheared between the blades. In the gap of the geometry, ¯ d rθ decreases when r increases. As expected, the theoretical strain rate profile for a Newtonian material in a Couettegeometry of radius equal to that of the vane falls well above the data at any radius r (thiswas less obvious on the velocity profiles). We then observe that the data are well fitted to thetheoretical strain rate profile (Eq. 14) for a Newtonian material flowing in an equivalent Couettegeometry of inner cylinder radius R i, e q = 3 .
90 cm ( R i, e q = 3 . ± .
005 cm was obtained from afit of the velocity profile to Eq. 13). This confirms that the θ -averaged strain rate ¯ d rθ decreasesas the inverse squared radius in the gap, in agreement with the Couette analogy.From Eq. 12, we find T vane /T Couette = 0 .
90 (let us recall that we do not need to consider endeffects here because we determine the shear rate within the gap, and hence only the contributionto the torque from the material sheared in the gap). This value can now be compared to datafrom the literature. For a six-bladed vane tool in an infinite medium, the Atkinson and Sherwood(1992) theory would imply a theoretical T vane /T Couette = 0 .
83, which is 8% lower than what wemeasure, and corresponds to a theoretical “equivalent radius” R i,th = 3 .
82 cm when the vaneis embedded in a cup of radius R o = 6 cm. Figs. 2b and 3 show that the flow characteristicspredicted with this value of the equivalent radius can be distinguished from our experimentaldata and fall slightly below the data (a discrepancy could be expected as ( R i /R o ) is not smallin our experiment). Study N R o /R i r s /R i ǫ/R i T vane T Couette
LowerboundAtkinson and Sherwood (1992) (th.) 4 ∞ et al. (1998) (num.) 4 2 – – 0.65 0.56Mart´ınez-Padilla and Quemada (2007) (exp.) 4 1.825 0.6 0.045 0.67 0.52Zhu et al. (2010) (num.) 4 1.14 0.14 0.1 0.73 0.26Barnes and Carnali (1990) (num.) 4 1.12 0.35 0.12 0.61 0.24Atkinson and Sherwood (1992) (th.) 6 ∞ et al. (2002) (exp.) 6 2.55 – – 0.7 0.7Zhang et al. (1998) (num.) 6 2 – – 0.74 0.63Present study (exp.) 6 1.49 0.27 0.15 0.90 0.57Table 1: Ratio T vane /T Couette between the torque measured when straining a linear medium(viscous or elastic) in a vane-in-cup geometry and that measured in a coaxial cylinder geometryof similar dimensions, obtained in various theoretical, numerical and experimental studies ofthe literature; only data corrected for (or free from) end effects are shown. The number N ofblades, the cup to vane radius ratio R o /R i , the shaft radius to vane radius ratio r s /R i , and theblade thickness to vane radius ratio ǫ/R i , are displayed when provided in the manuscripts. Thetheoretical lower bound computed using variational approaches in Sec. IIIA1 is also provided.We have gathered experimental and numerical data from the literature where the cup to vaneradius ratio R o /R i is not large in Tab. 1; only data corrected for (or free from) end effects areshown. First, it should be noted that all torque data obey the theoretical inequalities computedusing variational approaches in Sec. IIIA1. However, no clear trends emerge from the comparisonof the data. The relative impact of the various geometrical parameters that may affect the flowfield, namely the cup to vane radius ratio R o /R i , the shaft radius to vane radius ratio r s /R i ,and the blade thickness to vane radius ratio ǫ/R i , cannot be determined at this stage. Forexample, in very similar geometries, Zhu et al. (2010) find a torque ratio T vane /T Couette close to11hat of Atkinson and Sherwood (1992) whereas Barnes and Carnali (1990) find a much lowertorque ratio. The only noticeable difference between these two studies (apart from the numericalmethod) is that r s /R i is higher in Barnes and Carnali (1990), but our data, with a rather largevalue of r s /R i , show different features. We actually note that our study is the only one to reporta torque ratio higher than in an infinite medium; all other data report torque ratios up to 19%lower than expected in an infinite medium. In the general case of a finite vane-in-cup geometry,it thus seems that numerical investigations are still needed, and that, at this stage, a calibrationhas to be performed to get the geometry factors. We also expect that the bounds obtained usingvariational approaches in Sec. IIIA1 can be improved. θ dependent profiles To better characterize the flow field, we now study the dependence of the velocity profiles onthe angular position θ . We have performed experiments in which we measure one azimuthalvelocity profile per second while the vane tool is rotated at 1 rpm, yielding 10 profiles betweentwo adjacent blades. (a) Angle 3(cid:176) 9(cid:176) 15(cid:176) 21(cid:176) 27(cid:176) V e l o c it y ( m / s ) Radius (cm) (b) A ng l e ( (cid:176) ) R a d i u s ( c m ) rigid shear Figure 4: a) Azimuthal velocity profile V θ ( r, θ ) of a Newtonian oil sheared at 1 rpm in a six-bladed vane-in-cup geometry, for various angular positions, θ , between one blade ( θ = 0˚) andmidway between adjacent blades ( θ = 30˚). The vertical dashed line shows the radius of thevane. The dotted line is the profile for a rigid body rotation (for r < R i ) and the theoreticalprofile for a Newtonian fluid in a Couette geometry of radius R i (for r > R i ). b) Two-dimensionalplot of the limit between rigid motion and shear (empty circles) for a Newtonian material in thesix-bladed vane-in-cup geometry; the grey rectangles correspond to the blades.In Fig. 4a, we plot the velocity profiles V θ ( r, θ ) measured at different angles θ . We firstobserve that the velocity profile which starts near from a blade tip (corresponding to θ = 0˚bydefinition) is very different from the velocity profile in a Couette geometry of same radius: itstarts with a much steeper slope, which means that the blades tip neighborhoods are regionsof important shear as already observed by Barnes and Carnali (1990). We then observe that,as expected from the θ -averaged velocity profiles, the shear flow enters more and more deeplythe region between the blades as θ tends towards 30˚(corresponding to midway between twoadjacent blades); at this angular position, the rigid rotation stops at R l ≃ .
05 cm. From all thevelocity profiles, we finally extract a 2D map of the limit R l ( θ ) between rigid rotation and shear,which is depicted in Fig. 4b. This provides an idea of the deviation from cylindrical symmetry,and will be compared in the following to the case of yield stress fluids. Note that eddies are likelyto be present in the “rigid” region [Moffatt (1964); Atkinson and Sherwood (1992)]; however,12e did not observe any signature of their existence: they can thus be considered as second-orderphenomena. R a d i a l v e l o c it y ( m / s ) Radius (cm)
Angle ((cid:176)) 0 6 12 18 24 30 36 42 48 54 60
Figure 5: Radial velocity profile V r ( r, θ ) of a Newtonian oil sheared at 1 rpm in a six-bladed vane-in-cup geometry, for various angular positions θ between two adjacent blades (from θ = 0˚to θ = 60˚).As explained in Sec. II, from the V θ ( r, θ ) measurement and from the material incompress-ibility, we are able to reconstruct the radial velocity profile V r ( r, θ ) (see Fig. 5). This also allowsus to compute the strain rate components d rθ ( r, θ ) and d rr ( r, θ ) = − d θθ ( r, θ ), which are plottedin Fig. 6. Of course, due to the limited number of profiles between two adjacent blades, thismethod provides only a rough estimate of these quantities. In addition to their interest for fu-ture comparison with models and simulations, these data allow us to evaluate the contributionof the extensional flow to dissipation; here, in a Newtonian medium, the local power density isgiven by: p d ( r, θ ) = 2 η ( d rθ + 2 d rr ). (a) S t r a i n r a t e | d r | ( s - ) Radius (cm)
Angle: 3(cid:176) 9(cid:176) 15(cid:176) 21(cid:176) 27(cid:176) average (b) S t r a i n r a t e d rr ( s - ) Radius (cm)
Angle ((cid:176)) 6 12 30 36 42 48 60
Figure 6: Strain rate profiles d rθ ( r, θ ) (left) and d rr ( r, θ ) (right) vs. radius r for various angularpositions θ between two adjacent blades (from θ = 0˚to θ = 60˚).In Fig. 5, we first observe that V r ( r, θ ) ≃ θ = 0˚and θ = 30˚; there is thus noextensional flow in these regions of space, as seen in Fig. 6. This is actually expected from thefore-aft symmetry of the flow around these angular positions. V r and its spatial variations (i.e. d rr ) are maximal at θ ≃ d rθ is maximal near the blades: at r ≃ R i it is more than 4 times larger at θ = 0˚than at θ = 30˚. We then find that d rθ (and thusthe shear stress τ rθ ) decreases more rapidly from the blades (at θ = 0˚) than the 1 /r scaling of13he Couette geometry, whereas it does not vary much with r midway between adjacent blades(it even seems to slightly increase with r as already observed in simulations by Savarmand et al. (2007)). It is also worth nothing that at r ≃ R e , in contrast with what is observed at r ≃ R i ,the shear stress value is of order two times lower at θ = 0˚than at θ = 30˚.From the whole set of d rθ and d rr measurements (Fig. 6), we finally find that in regions where d rr is maximal, the contribution of the extensional flow to dissipation is of order 25%. Over thewhole gap, we then evaluate its average contribution to dissipation to be rather important, oforder 5 to 10%. This significant value may be a reason why the torque that has to be exertedto enforce flow is higher than that predicted by Atkinson and Sherwood (1992) in an infinitemedium. The confinement effect induced by a close boundary at a radius R e likely increases thecontribution of the extensional flow to dissipation as compared to the case of an infinite medium(although other effects may exist, as appears from the comparison of the data of Tab. 1). B Yield stress fluid
In this section, we study the flows induced by the vane tool with a yield stress fluid (a concen-trated emulsion). We focus on the behavior near the yielding transition, i.e. on low rotationalvelocities Ω. D i m e n s i on l e ss a v e r a g e v e l o c it y Radius (cm)
Figure 7: Dimensionless θ -averaged velocity profile ¯ V θ ( r, Ω) / Ω R i of a yield stress fluid (concen-trated emulsion) in a six-bladed vane-in-cup geometry for Ω ranging from 0.1 to 9 rpm; thevertical dashed line shows the radius of the vane; the dotted line is the theoretical profile for arigid body rotation (for r < R i ).In Fig. 7, we plot the θ -averaged azimuthal velocity profiles ¯ V θ ( r ) measured at several Ωvalues ranging from 0.1 to 9 rpm, corresponding to macroscopic shear rates varying between0.02 and 2 s − . We first observe that flow is localized: the material is sheared only up to aradius R c < R o . R c is found to increase as Ω increases. This is a classical feature of flows ofyield stress fluids in heterogeneous stress fields. It has been observed in Couette geometries[Coussot (2005); Ovarlez et al. (2008)], where it is attributed to the 1 /r decrease of the shearstress τ rθ , which passes below τ y at some R c (Ω) < R o at low Ω. In this case, when Ω tends to 0, R c tends to R i and the torque T at the inner cylinder tends to τ y ∗ πR i H . In the vane-in-cupgeometry, the same argument holds qualitatively thanks to Eq. 8. It implies that the flow hasto stop inside the gap at low Ω. However, in contrast with the case of the Couette geometry,as the whole stress field a priori depends on θ , this θ -averaged equation does not provide the14osition of the limit between the sheared and the unsheared material (which will determined atthe end of this section).We then observe that, although this effect is less pronounced than with a Newtonian material,the shear flow still enters the region between the blades, even at the lowest studied Ω. Closeexamination of the profiles shows that the material trapped between the blades rotates as a rigidbody only up to R l ≃ .
65 cm at Ω = 9 rpm, R l ≃ .
75 cm at Ω = 1 rpm, and R l ≃ .
85 cm atΩ = 0 . R l ≃ .
05 cm with a Newtonian fluid in the same geometry. V e l o c it y ( m / s ) Radius (cm)
Angle 3(cid:176) 9(cid:176) 15(cid:176) 21(cid:176) 27(cid:176) 33(cid:176) 39(cid:176) 45(cid:176) 51(cid:176) 57(cid:176)
Figure 8: Azimuthal velocity profile V θ ( r, θ ) of a yield stress fluid (concentrated emulsion)sheared at 0.1 rpm in a six-bladed vane-in-cup geometry, for various angular positions θ betweentwo adjacent blades (from θ = 0˚to θ = 60˚). The vertical dashed line shows the radius of thevane.As in Sec. IIIA3, to better characterize the flow field, we have performed experiments inwhich we have measured 10 azimuthal profiles between two adjacent blades at 0.1 rpm. InFig. 8, as for a Newtonian fluid, we observe that there is a strong θ -dependence of the velocityprofiles. The velocity profile that starts near from a blade tip (at θ = 0˚) has a much steeperslope than the profile measured midway between adjacent blades (at θ = 30˚); again, this showsthat the blade tip neighborhoods are regions of high shear. Meanwhile the flow stops at a radius R c which is larger at θ = 30˚(4.5 cm) than at θ = 0˚(4.3 cm). Note also that there maybe slight fore-aft asymmetry, as sometimes observed with yield stress fluids flows [Dollet andGraner (2007); Putz et al. (2008)], but we did not study this point further. From these velocityprofiles, we have reconstructed a 2D map of the flow field (Fig. 9), indicating both the boundarybetween the region of rigid body rotation (between the blades) and the sheared region, and theboundary between the sheared region and the outer region of fluid at rest ( i.e. the positionwhere the yield criterion is satisfied).Flow is found to occur in a layer of complex shape which is far from being cylindrical even atthis very low velocity. These observations are in contradiction with the usually accepted picturefor yield stress fluid flows at low rates [Barnes and Nguyen (2001)], namely that the materialcontained in the virtual cylinder delimited by the blades rotates as a rigid body, and that itflows uniformly in a thin cylindrical layer near the blades. Our results contrast in particularwith previous numerical works which showed that the yield surface is cylindrical at low ratesfor Bingham fluids, Casson fluids, and Herschel-Bulkley materials with n = 0 . et al. (1985); Yan and James (1997); Savarmand et al. (2007)]. With apparently similar conditionsto those in some of the Yan and James (1997) simulations, we find an important departurefrom cylindrical symmetry. This means that further investigation on the exact conditions underwhich this symmetry can be recovered is still needed. Possible difference between our work and15hat of Yan and James (1997) is that the blade thickness is zero in this last study.It is particularly striking and counterintuitive that R c is largest at the angular position( θ = 30˚) where shear at R i is smallest (similar observation was made by Potanin (2010)).As in Sec. IIIA3, this points out the importance of the extensional flow in this geometry, with θ -dependent normal stress differences which have to be taken into account in the yield criterion,and which thus impact the yield surface. It thus seems that the link between the yield stress τ y and the torque T measured at yield with a vane-in-cup geometry is still an open question,although the classical formula probably provides a sufficiently accurate determination of τ y inpractice. (a) flowrest A ng l e ( (cid:176) ) R a d i u s ( c m ) rigid (b) flowrest A ng l e ( (cid:176) ) R a d i u s ( c m ) rigid Figure 9: Two-dimensional plot of the limit between rigid motion and shear (circles) and betweenshear and rest (triangles) for a yield stress fluid (concentrated emulsion) sheared in the six-bladedvane-in-cup geometry at 0.1 rpm (left) and 1 rpm (right). The grey rectangles correspond tothe blades.The same 2D map as above is plotted for Ω = 1 rpm in Fig. 9; the same phenomena areobserved, with enhanced departure from cylindrical symmetry, consistent with the observationthat R l decreases when Ω increases. This result was also unexpected, as simulations find uniformflows for shear-thinning material of index n ≤ . etal. (2007)]; we would have expected the same phenomenology in a Herschel-Bulkley material ofindex n = 0 . R l to tend to R i when increasing Ω). This observation also shows thata Couette analogy can hardly be defined for studying the flow properties of such materials ina vane-in-cup geometry because the equivalent Couette geometry radius R i, e q would probablydepend also on Ω (as recently shown by Zhu et al. (2010)).Let us finally note that this departure from cylindrical symmetry has important impact onthe migration of particles in a yield stress fluid (see below). C Concentrated suspension
In this section, we investigate the behavior of a concentrated suspension of noncolloidal particlesin a yield stress fluid (at a 40% volume fraction).A detailed study of their velocity profiles would a priori present here limited interest: suchmaterials present the same nonlinear macroscopic behavior as the interstitial yield stress fluid,and their rheological properties (yield stress, consistency) depend moderately on the particlevolume fraction [Mahaut et al. (2008a); Chateau et al. (2008)].On the other hand, noncolloidal particles in suspensions are known to be prone to shear-induced migration, which leads to volume fraction heterogeneities. This phenomenon is well16ocumented in the case of suspensions in Newtonian fluids [Leighton and Acrivos (1987b);Abbott et al. (1991); Phillips et al. (1992); Corbett et al. (1995); Shapley et al. (2004); Ovarlez et al. (2006)] but is still badly known in yield stress fluids (some studies exist however inviscoelastic fluids [Tehrani (1996); Huang and Joseph (2000); Lormand and Phillips (2004)]).In the model of Leighton and Acrivos (1987b) and Phillips et al. (1992), migration is relatedto shear-induced diffusion of the particles [Leighton and Acrivos (1987a); Acrivos (1995)]. In awide gap Couette geometry, the shear stress heterogeneity is important (Eq. 4); the shear rategradients then generate a particle flux towards the low shear zones, which is counterbalancedby a particle flux due to viscosity gradients. A steady state, which results from competitionbetween both fluxes, may then be reached, and is characterized by an excess of particles in thelow shear zones of the flow geometry (near the outer cylinder in a wide-gap Couette geometry[Phillips et al. (1992); Corbett et al. (1995); Ovarlez et al. (2006)]). Note that there are othermodels [Nott and Brady (1994); Mills and Snabre (1995); Morris and Boulay (1999); Lhuillier(2009)] in which particle fluxes counterbalance the gradients in the particle normal stresses, andwhich can be used directly for non-Newtonian media.As the development of migration depends on the spatial variations of shear, one may wonderhow the azimuthal heterogeneities of shear introduced by the vane tool affect migration; a relatedquestion is that of the relevance of the Couette analogy for this phenomenon. In the following,we thus focus on the particle volume fraction distribution evolution when the material is sheared.
Behavior at high shear rate
We first study the behavior at high shear rate, in the absence of shear localization. We shear thesuspension in both the standard Couette geometry and the vane-in-cup geometry at a rotationalvelocity Ω = 100 rpm. In this first set of experiments, we only study the steady-state ofmigration. At Ω = 100 rpm, this steady-state is reached in less than 30 min (which correspondsto a macroscopic strain of order 50000, consistently with strainscale evaluations from data of theliterature [Ovarlez et al. (2006)]). In Fig. 10 we plot the steady state volume fraction profilesobserved after shearing the suspension at Ω = 100 rpm during 1h. (a) V o l u m e fr ac ti on ( % ) Radius (cm) V o l u m e fr ac ti on ( % ) Radius (cm) (b) V e l o c it y ( m s - ) Radius (cm)
Figure 10: a) Steady-state volume fraction vs. radius at Ω = 100 rpm in both the Couettegeometry (empty circles) and the vane-in-cup geometry (squares). In the vane-in-cup geometry,the volume fraction profile is determined in a 1 cm thick slice situated exactly between twoadjacent blades (see Fig. 11b). The inset is a zoom; the line is a fit of the data measured inthe Couette geometry to the Phillips et al. (1992) model with K c /K µ = 0 .
42. b) θ -averagedazimuthal velocity profile ¯ V θ ( r ); the dotted line is the theoretical rigid motion induced by therotation of the vane tool; the vertical dashed line shows the radius of the vane.17s expected, we first observe that the material is strongly heterogeneous in the Couettegeometry: the volume fraction varies between 37% near the inner cylinder and 43% near theouter cylinder (note that the NMR technique we use do not allow quantitative measurementsnear the walls). This heterogeneity is quantitatively similar to that observed in Couette flows ofNewtonian suspensions at a same 40% particle volume fraction [Corbett et al. (1995)]; the profilesare actually well fitted to the Phillips et al. (1992) model (see Eq. 16 of Ovarlez et al. (2006))with a dimensionless diffusion constant K c /K µ = 0 .
42 which is close to that found by Corbett et al. (1995) ( K c /K µ = 0 . R l = 3 . R l corresponds to the transitionbetween rigid motion and shear between the blades. The volume fraction then increases basicallylinearly with the radius up to a 40.5% volume fraction at a radius r = 3 .
85 cm which is close tothe vane radius. The volume fraction then increases only slightly (between 40.5% and 42.5%)in the gap of the geometry: the heterogeneity is here much less important than in a standardCouette geometry.To get further insight into the new strong depletion phenomenon we have evidenced, we haveperformed 2D magnetic resonance images of the material. Such images provide a qualitativeview of the spatial variations of the particle volume fraction as only the liquid phase is imaged.Images are coded in grey scales; a brighter zone contains less particles. In Fig. 11b, we firstsee an image of the homogeneous material. Before any shear, as expected, the light intensityis homogeneous in the sample (intensity variations correspond to noise). After a 1h shear atΩ = 100 rpm, we observe very bright and thin curves on the image: they correspond to zoneswhere the volume fraction suddenly drops down to a value close to zero. These curves are notcircles. More precisely, between two adjacent blades, a depleted zone goes from the edge of oneblade (at θ = 0˚, r = 4 .
02 cm) to the edge of another blade (at θ = 60˚, r = 4 .
02 cm), anddescribes a concave r ( θ ) curve whose minimum is r = 3 . θ = 30˚. Note that as thevolume fraction profile is averaged over a slice which is 1 cm thick in the azimuthal direction(see Fig. 11b), the fact that we measure a minimum of 5% at r = 3 . r ≃ .
02 cm just beforethe blade is necessarily found at a radius slightly higher than 4.02 cm after the blade as there18a)(b) (c)Figure 11: 2D magnetic resonance image of a suspension of particles in a yield stress fluid ina vane-in-cup geometry: (a) after a 1h shear at Ω = 100 rpm (corresponding to a macroscopicstrain of order 75000), and (b) before any shear. The crosses in Fig. 11a correspond to the limitbetween rigid motion and shear for the Newtonian oil of Fig. 4b. The white rectangle in Fig. 11bshows the slice in which the volume fraction profiles of Figs. 10a and 12 are measured. (c) is azoom of image (a) near the edges of a blade. The images are taken in the horizontal plane ofthe geometry, at middle height of the vane tool, and correspond to vertical averages over 2 cm.The vane tool rotates counterclockwise. 19re no particles at r = 4 .
02 cm. This feature is reminiscent of the fore-aft asymmetry that isobserved in the bulk of noncolloidal suspensions [Parsi and Gadala-Maria (1987)] and that leadsto their non-Newtonian properties [Brady and Morris (1997)]. It thus seems that, in addition tothe shear-induced migration mechanism intrinsic to suspensions, the vane tool induces a specificmigration mechanism which has its origin in the direct interactions between the particles andthe blades; this effects leads to the full depletion that is observed at the transition between thesheared and the unsheared material. Such direct effect of a flow geometry on migration has alsobeen observed in microchannel flows of colloidal suspensions [Wyss et al. (2006)], and also ledto full particle depletion. See also Jossic and Magnin (2004). The kinetics of the phenomenonwill be briefly discussed below.The rest of the volume fraction profile results from a complex interplay between shear-inducedmigration and the fore-aft asymmetry around the blades; this leads to the rapid increase of thevolume fraction between 3.1 cm and 4.02 cm. After 4.02 cm the flow lines do not meet the bladeedges, and the phenomenon evidenced above should have basically no effect on the heterogeneitythat develops in the gap of the geometry. On the other hand, the mean volume fraction shouldbe slightly higher due to mass conservation; it is indeed observed to be equal to around 42%.Nevertheless, as the mean radial shear rate heterogeneity is basically similar to that observed ina standard Couette geometry (see previous sections), we would a priori expect the heterogeneityto be somehow similar. However, we observe that the volume fraction profile is only slightlyheterogeneous: there is less than 5% variation of the volume fraction in the gap, to be comparedto the 15% variation observed in the Couette geometry. Clearly, this means that the Couetteanalogy is irrelevant as regards this phenomenon, and that the details of shear matter. Here,the extensional flow that adds to shear may be at the origin of this diminution of migration. Amore detailed analysis is out of the scope of this paper.
Behavior at low shear rate
Let us now study the behavior at low shear rate. Low shear rates are typically imposed with theaim of measuring the yield stress of such materials. Starting from a homogeneous suspensionat rest, we apply a rotational velocity Ω = 1 rpm (without any preshear), and we measure theevolution of the particle volume fraction in time. The corresponding volume fraction profiles aredepicted in Fig. 12.In Fig. 12, we observe that, although shear is much less important than in the previousexperiments, particle depletion also appears between the blades. Comparison of the velocityprofile and the volume fraction profile shows that depletion also appears between the bladesat the transition zone between the sheared and the unsheared materials. This phenomenonappears with a very fast kinetics: the lower volume fraction value in the measurement zone is36% after only a 5 minute shear (corresponding to a macroscopic strain of order 50). Afterwards,it continues evolving slowly: the minimum observed volume fraction is of order 33% after a 1hshear and of order 32% after a 14h shear (corresponding to a 10000 macroscopic strain). Notethat the radial position of the minimum value of the volume fraction slightly decreases in time; itlikely corresponds to progressive erosion of the material between the blades (we did not measurethe velocity profiles to check this hypothesis).We also note that migration is negligible in the rest of the sheared material as expectedfrom the theory of migration briefly described above (a larger strain would be needed to observesignificant migration). Nevertheless, we note some particle accumulation (with a volume fractionvalue of 43%) at R c =4.7 cm after a very long time. This corresponds to the yield surface as flowis localized at low velocity (see velocity profile Fig. 12). Migration profiles usually result from anequilibrium between various sources of fluxes. On the other hand, the unsheared material doesnot produce any particle flux while it receives particles from the sheared region. This particleaccumulation is thus the signature that the migration phenomenon is indeed active, althoughnot observable on the profile measured in the sheared zone. It is probable that this accumulation20 V e l o c it y ( m s - ) Radius (cm) V o l u m e fr ac ti on ( % ) Radius (cm)
Shearing time 5 min 1 h 14 h
Figure 12: Volume fraction vs. radius at Ω = 1 rpm measured in the vane-in-cup geometry afterdifferent times of shear: 5min, 1h, 14h. The material is homogeneous at the beginning of shear.The inset presents the θ -averaged azimuthal velocity profile ¯ V θ ( r ) measured in the first stagesof shear; the dotted line is the theoretical rigid motion induced by the rotation of the vane tool;the vertical dashed line shows the radius of the vane.process would stop only (after a very long time) when there are no more particles in the shearedregion.As above, 2D magnetic resonance images of the material provide an insight in the phe-nomenon. In Fig. 13, we observe again that particle depletion is enhanced at the rear of theblades; this confirms that this phenomenon is likely due to direct interactions between the bladesand the particles, leading to the asymmetry of the particles trajectory around the blades. This a priori occurs with any particle whose trajectory is close to the blades, explaining why particledepletion appears so rapidly. There is probably no way to avoid it. Note that the images arehere much brighter very close to the blades than midway between two adjacent blades; thiswould mean that the particle volume fraction is probably close to 0 near the blades, althoughwe observe volume fraction of order 32% between two blades.Finally, let us note that the bright line provides a good idea of the boundary between thesheared material and the material that moves as a rigid body. We see as in Sec. IIIB that thisis far from being cylindrical even at this low velocity. Consequences: slip with a vane tool
We finally present some consequences of this phenomenon. From the above observations, ourconclusion is that depletion sets up quickly and is probably unavoidable. Then two situationshave to be distinguished. If linear viscoelastic properties of a suspension of large particles aremeasured at rest on the homogeneous material in its solid regime, without any preshear, thenthese measurements pose no other problem than that of the relevant Couette analogy to be used(see Sec. IIIA). If a yield stress measurement is performed at low imposed rotational velocity,starting from the homogeneous material at rest, then this measurement is likely valid as long asonly the peak value or the plateau value at low strain (of order 1) is recorded. On the other hand,any subsequent analysis of the material behavior will a priori be misleading: irreversible changeshave occurred and the material cannot be studied anymore. More generally, any measurementperformed after a preshear will be incorrect and any flow curve measurement will lead to wrongevaluation of the material properties. In these last cases, the consequence of the new particledepletion phenomenon we have evidenced is a kind of wall slip near the blades, whereas there21a) (b)Figure 13: (a) 2D magnetic resonance image of a suspension of particles in a yield stress fluidin the horizontal plane of a vane-in-cup geometry after a 14h shear at Ω = 1 rpm. (b) is a zoomof image (a) between two adjacent blades. The image is taken in the horizontal plane of thegeometry, at middle height of the vane tool, and corresponds to a vertical average over 2 cm.The vane tool rotates counterclockwise.are no walls. Here the “slip layer” is made of the (pure) interstitial yield stress fluid in a zoneclose to the blades, as would be observed near a smooth inner cylinder. This contrasts with thecommon belief that the vane tool prevents slippage.In order to illustrate this feature, we present some results of Mahaut et al. (2008a): Mahaut et al. performed classical upward/downward shear rate sweeps with a six-bladed vane-in-cupgeometry ( R i = 1 .
25 cm, R o = 1 . H =4.5 cm, blade thickness=0.8 mm) in a pure con-centrated emulsion, and in the same emulsion filled with 20% of 140 µ m PS beads. In theseexperiments, constant macroscopic shear rates steps increasing from 0.01 to 10 s − and thendecreasing from 10 to 0.01 s − were applied during 30s, and the stationary shear stress wasmeasured for each shear rate value. The results are shown in Fig. 14.While the same curve is observed for the upward/downward shear rate sweeps in the case ofthe pure emulsion (as expected for a simple non-thixotropic yield stress fluid), the shear stressduring the upward shear rate sweep differs from the shear stress during the downward shear ratesweep in the case of the suspension. Moreover, any measurement performed on the suspensionafter this experiment gives a static yield stress equal to the dynamic yield stress observed duringthe downward sweep. This means that there has been some irreversible change. This irreversiblechange is actually the particle depletion near the blades we have observed in this paper. As theflow of the suspension is localized near the inner tool at low shear rate, it means that afterthe first upward sweep that has induced the particle depletion, during the downward shear ratesweep only the pure emulsion created by migration near the blades remains in the sheared layerat sufficiently low rotational velocity. This explains why the same apparent value of the yieldstress is found in the suspension during the downward sweep as in the pure emulsion with thisexperiment. On the other hand, the yield stress at the beginning of the very first upward sweepis that of the suspension as migration has not occurred yet.The conclusion is that the vane tool is probably not suitable to the study of flows of suspen-sions of large particles. 22 .01 0.1 1 100255075100125150175 S h ea r s t r e ss ( P a ) Shear rate (s -1 ) Pure emulsion
Emulsion + 20% of beads
Upward shear rate sweep Downward shear rate sweep
Figure 14: Shear stress vs. shear rate in a vane-in-cup geometry for upward/downward shearrate sweeps in a pure concentrated emulsion (open squares) and for the same emulsion filledwith 20% of 140 µ m PS beads (filled/open circles). Figure from Mahaut et al. (2008a). IV Conclusion
As a conclusion, let us summarize our main findings: • In the case of Newtonian fluid flows, our measurements support the Couette equivalenceapproach: the θ -averaged strain rate component d rθ decreases as the inverse squared radiusin the gap. Interestingly, the velocity profiles allow determining the Couette equivalentradius without end-effect correction and independently of the viscosity of the material.The torque exerted by the vane in our display is found to be higher (by 8%) than thetheoretical prediction of Atkinson and Sherwood (1992) for a vane embedded in an infinitemedium, and is thus much closer to the torque exerted by a Couette geometry of sameradius as the vane than expected from the literature. A key observation may be that thereis a significant flow between the blades which adds an important extensional component toshear, thus increasing dissipation. From a short review of the literature, it clearly appearsthat numerical investigations are still needed in the case of finite geometries. Variationalapproaches are also promising, although they do not yet provide tight bounds. • In the case of yield stress fluid flows, we find that the thin layer of material which flowsaround the vane tool at low velocity is not cylindrical, in contrast with what is usuallysupposed in the literature from simulation results. Consequently, a non negligible exten-sional component of shear has probably to be taken into account in the analysis. At thisstage, there are too few experimental and simulation data to understand the origin of thisdiscrepancy. It thus seems that progress still has to be made, in particular through simula-tions, which allow a wide range of parameters to be studied. This may help understandinghow the torque is linked to the yield stress of a material at low velocity, depending inparticular on the geometry. • An important and surprising result is the observation of particle depletion near the bladeswhen the yield stress fluid contains noncolloidal particles. This phenomenon is thus likelyto occur when studying polydisperse pastes like coal slurries, mortars and fresh concrete.It has to be noted that the phenomenon is very rapid, irreversible, and thus probablyunavoidable when studying flows of suspensions of large particles. It results in the creationof a pure interstitial yield stress fluid layer and thus in a kind of wall slip near the blades. Itcontrasts with the classical assumption that is made in the field of concentrated suspensionrheology where the vane tool is mainly used to avoid this phenomenon.23onsequently, we would say that, in the case of pasty materials with large particles, if accuratemeasurements are needed, the vane tool may finally be suitable only for the study of the solid(elastic) properties of materials and for the static yield stress measurements; as the yield stressmeasurement may induce irreversible particle depletion near the blades, any new measurementthen requires a new sample preparation. Furthermore, the vane can be used as a very accuratetool without any hypothesis nor any calibration to measure the relative increase of the elasticmodulus of materials as a function of their composition [Alderman et al. (1991); Mahaut et al. (2008a)]. In order to study accurately the flows of pasty materials with large particles, our resultssuggest that a coaxial cylinders geometry with properly roughened surfaces is preferable whenpossible. If the use of a vane tool cannot be avoided, one should keep in mind our observationsin order to carefully interpret any result.
References
Abbott, J. R., N. Tetlow, A. L. Graham, S. A. Altobelli, E. Fukushima, L. A. Mondy, andT. S. Stephens, “Experimental observations of particle migration in concentrated suspensions:Couette flow,” J. Rheol. , 773-795 (1991).Acrivos, A., “BINGHAM AWARD LECTURE–1994 Shear-induced particle diffusion in concen-trated suspensions of noncolloidal particles,” J. Rheol. , 813-826 (1995).A¨ıt-Kadi, A., P. Marchal, L. Choplin, A.-S. Chrissemant, and M.Bousmina, “Quantitative Anal-ysis of Mixer-Type Rheometers using the Couette Analogy,” The Canadian Journal of Chem-ical Engineering , 1166-1174 (2002).Alderman, N. J., G. H. Meeten, and J. D. Sherwood, “Vane rheometry of bentonite gels,” J.Non-Newtonian Fluid Mech. , 291-310 (1991).Atkinson, C., and J. D. Sherwood, “The Torque on a Rotating N-Bladed Vane in a Newto-nian Fluid or Linear Elastic Medium,” Proceedings of the Royal Society of London Seriesa-Mathematical Physical and Engineering Sciences , 183-196 (1992).Baravian, C., A. Lalante, and A. Parker, “Vane rheometry with a large, finite gap,” AppliedRheology , 81-87 (2002).Barnes, H. A., and J. O. Carnali, “The Vane-in-Cup as a Novel Rheometer Geometry for ShearThinning and Thixotropic Materials,” J. Rheol. , 841-866 (1990).Barnes, H. A., and Q. D. Nguyen, “Rotating vane rheometry - a review,” J. Non-NewtonianFluid Mech. , 1-14 (2001).Brady, J. F., and J. F. Morris, “Microstructure of strongly sheared suspensions and its impacton rheology and diffusion,” J. Fluid Mech. , 103-139 (1997).Bousmina, M., A. A¨ıt-Kadi, and J. B. Faisant, “Determination of shear rate and viscosity frombatch mixer data,” J. Rheol. , 415-433 (1999).Callaghan, P. T., Principles of Nuclear Magnetic Resonance Spectroscopy (Clarendon, Oxford,1991).Chateau, X., G. Ovarlez, and K. Luu Trung, “Homogenization approach to the behavior ofsuspensions of noncolloidal particles in yield stress fluids,” J. Rheol. , 489-506 (2008).Corbett, A. M., R. J. Phillips, R. J. Kauten, and K. L. McCarthy, “Magnetic resonance imagingof concentration and velocity profiles of pure fluids and solid suspensions in rotating geome-tries,” J. Rheol. , 907-924 (1995). 24oussot, P., Rheometry of Pastes, Suspensions and Granular Materials (John Wiley & Sons,New York, 2005).Dollet, B., and F. Graner, “Two-dimensional flow of foam around a circular obstacle: localmeasurements of elasticity, plasticity and flow,” J. Fluid Mech. , 181-211 (2007).Dzuy, N. Q., and D. V. Boger, “Yield stress measurement for concentrated suspensions,” J.Rheol. , 321 (1983).Estell´e, P., C. Lanos, A. Perrot, and S. Amziane, “Processing the vane shear flow data fromCouette analogy,” Applied Rheology , 34037-34481 (2008).Hanlon, A. D., S. J. Gibbs, L. D. Hall, D. E. Haycock, W. J. Frith, S. Ablett, and C. Marriott, “Aconcentric cylinder Couette flow system for use in magnetic resonance imaging experiments,”Measurement Science and Technology , 631-637 (1998).Huang, P. Y., and D. D. Joseph, “Effects of shear thinning on migration of neutrally buoyantparticles in pressure driven flow of Newtonian and viscoelastic fluids,” J. Non-Newtonian FluidMech. , 159-185 (2000).Jau, W.-C., and C.-T. Yang, “Development of a modified concrete rheometer to measure therheological behavior of fresh concrete,” Cement and Concrete Composites (2010), In Pressdoi:10.1016/j.cemconcomp.2010.01.001Jossic, L. and A. Magnin, “Structuring under flow of suspensions in a gel,” AIChE Journal ,2691-2696 (2004).Keentok, M., “The measurement of the yield stress of liquids,” Rheologica Acta , 325-332(1982).Keentok, M., J. F. Milthorpe, and E. O’Donovan, “On the shearing zone around rotating vanesin plastic liquids: theory and experiment,” J. Non-Newtonian Fluid Mech. , 23-35 (1985).Koehler E. P., D. W. Fowler, C. F. Ferraris, and S. A. Amziane, “New, portable rheometer forfresh self-consolidating concrete,” ACI Mater J , 97-116 (2006).Leighton, D., and A. Acrivos, “Measurement of shear-induced self-diffusion in concentratedsuspensions of spheres,” J. Fluid Mech. , 109-131 (1987a).Leighton, D., and A. Acrivos, “The shear-induced migration of particles in concentrated sus-pensions,” J. Fluid Mech. , 415-439 (1987b).Lhuillier, D., “Migration of rigid particles in non-Brownian viscous suspensions,” Phys. Fluids , 023302 (2009).Lormand, B. M., and R. J. Phillips, “Sphere migration in oscillatory Couette flow of a viscoelasticfluid,” J. Rheol. , 551-570 (2004).Mahaut, F., X. Chateau, P. Coussot, and G. Ovarlez, “Yield stress and elastic modulus ofsuspensions of noncolloidal particles in yield stress fluids,” J. Rheol. , 287-313 (2008).Mahaut, F., S. Mok´eddem, X. Chateau, N. Roussel, and G. Ovarlez, “Effect of coarse particlevolume fraction on the yield stress and thixotropy of cementitious materials,” Cem. Concr.Res. , 1276-1285 (2008).Mart´ınez-Padilla, L. P., and C. Rivera-Vargas, “Flow behavior of Mexican sauces using a vane-in-a-large cup rheometer,” Journal of Food Engineering , 189-196 (2006).25art´ınez-Padilla, L. P., and D. Quemada, “Baffled cup and end-effects of a vane-in-a-large cuprheometer for Newtonian fluids,” Journal of Food Engineering , 24-32 (2007).Mills, P., and P. Snabre, “Rheology and Structure of Concentrated Suspensions of Hard Spheres.Shear Induced Particle Migration,” J. Phys. II France , 1597-1608 (1995).Moffatt, H. K., “Viscous and resistive eddies near a sharp corner,” J. Fluid Mech. , 1-18(1964).Morris, J. F., and F. Boulay, “Curvilinear flows of noncolloidal suspensions: The role of normalstresses,” J. Rheol. , 1213-1237 (1999).Nguyen, Q. D., and D. V. Boger, “Measuring the flow properties of yield stress fluids,” Annu.Rev. Fluid Mech. , 47-88 (1992)Nott, P., and J. F. Brady, “Pressure-driven flow of suspensions: Simulation and theory,” J. FluidMech. , 157-199 (1994).Ovarlez, G., F. Bertrand, and S. Rodts, “Local determination of the constitutive law of a densesuspension of noncolloidal particles through magnetic resonance imaging,” J. Rheol. , 259-292 (2006).Ovarlez, G., S. Rodts, A. Ragouilliaux, P. Coussot, J. Goyon, and A. Colin, “Wide-gap Cou-ette flows of dense emulsions: Local concentration measurements, and comparison betweenmacroscopic and local constitutive law measurements through magnetic resonance imaging,”Phys. Rev. E , 036307 (2008).Ovarlez, G., Q. Barral, and P. Coussot, “Three-dimensional jamming and flows of soft glassymaterials,” Nature Materials , 115-119 (2010).Parsi, F., and F. Gadala-Maria, “Fore-and-aft asymmetry in a concentrated suspension of solidspheres,” J. Rheol. , 725-732 (1987).Phillips, R. J., R. C. Armstrong, R. A. Brown, A. L. Graham, and J. R. Abbott, “A constitutiveequation for concentrated suspensions that accounts for shear-induced particle migration,”Phys. Fluids , 30-40 (1992).Potanin, A., “3D simulations of the flow of thixotropic fluids, in large-gap Couette and vane-cupgeometries,” J. Non-Newtonian Fluid Mech. , 299-312 (2010).Putz, A. M. V., T. I. Burghelea, I. A. Frigaard, and D. M. Martinez, “Settling of an isolatedspherical particle in a yield stress shear thinning fluid,” Phys. Fluids , 033102 (2008).Raynaud, J. S., P. Moucheront, J. C. Baudez, F. Bertrand, J. P. Guilbaud, and P. Coussot,“Direct determination by NMR of the thixotropic and yielding behavior of suspensions,” J.Rheol. , 709-732 (2002).Richards, A. F. (ed.), Vane Shear Strength Testing in Soils: Field and Laboratory Studies (ASTM, Philadelphia, 1988).Rodts, S. , F. Bertrand, S. Jarny, P. Poullain, and P. Moucheront, “D´eveloppements r´ecentsdans l’application de l’IRM `a la rh´eologie et `a la m´ecanique des fluides,” Comptes RendusChimie , 275-282 (2004).Saak, A. W., H. M. Jennings, and S. P. Shah, “The influence of wall slip on yield stress andviscoelastic measurements of cement paste,” Cem. Concr. Res. , 205-212 (2001).26alen¸con, J., Handbook of continuum mechanics. General concepts, thermoelasticity (Springer-Verlag, Berlin, Heildeberg, 2001).Savarmand, S., M. Heniche, V. Bechard, F. Bertrand, and P. J. Carreau, “Analysis of the vanerheometer using 3D finite element simulation,” J. Rheol , 161-177 (2007).Shapley, N. C., R. A. Brown, and R. C. Armstrong, “Evaluation of particle migration modelsbased on laser Doppler velocimetry measurements in concentrated suspensions,” J. Rheol. ,255-279 (2004).Sherwood, J. D., and G. H. Meeten, “The use of the vane to measure the shear modulus of linearelastic solids,” J. Non-Newtonian Fluid Mech. , 101-118 (1991).Stokes, J. R., and J. H. Telford, “Measuring the yield behaviour of structured fluids,” J. Non-Newtonian Fluid Mech. , 137-146 (2004).Tehrani, M. A.,“An experimental study of particle migration in pipe flow of viscoelastic fluids,”J. Rheol. , 1057-1077 (1996).Wallevik, J. E., “Minimizing end-effects in the coaxial cylinders viscometer: Viscoplastic flowinside the ConTec BML Viscometer 3 ,” J. Non-Newtonian Fluid Mech. , 116-123 (2008).Wyss, H. M., D. L. Blair, J. F. Morris, H. A. Stone, and D. A. Weitz, “Mechanism for cloggingof microchannels,” Phys. Rev. E , 061402 (2006).Yan, J., and A. E. James, “The yield surface of viscoelastic and plastic fluids in a vane viscome-ter,” J. Non-Newtonian Fluid Mech. , 237-253 (1997).Zhu, H., N. S. Martys, C. ferraris, and D. De Kee, “A numerical study of the flow of Bingham-likefluids in two-dimensional vane and cylinder rheometers using a smoothed particle hydrody-namics (SPH) based method,” J. Non-Newtonian Fluid Mech. , 362-375 (2010).Zhang, X. D., D. W. Giles, V. H. Barocas, K. Yasunaga, and C. W. Macosko, “Measurement offoam modulus via a vane rheometer,” J. Rheol.42