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Rheol Acta (2014) 53:357371DOI 10.1007/s00397-014-0759-1

ORIGINAL CONTRIBUTION

Optimization of the vane geometryApplications to complex fluids

Aminallah Rabia Samir Yahiaoui Madeleine Djabourov Francois Feuillebois Thierry Lasuye

Received: 18 June 2013 / Revised: 14 January 2014 / Accepted: 15 January 2014 / Published online: 8 March 2014 Springer-Verlag Berlin Heidelberg 2014

Abstract The use of nonstandard geometries like the vaneis essential to measure the rheological characteristics ofcomplex fluids such as non-Newtonian fluids or particle dis-persions. For this geometry which is of Couette type, there isno analytical simple model defining the relation between theshear stress and the torque or relating the angular velocity tothe shear rate. This study consists on calibrating a nonstan-dard vane geometry using a finite volume method with theAnsys Fluent software. The influence of geometrical param-eters and rheological characteristics of the complex fluidsare considered. First, the Newtonian fluid flow in a rota-tive vane geometry was simulated and a parametric modelis derived therefrom. The results show an excellent agree-ment between the calculated torque and the measured one.They provide the possibility to define equivalent dimen-sions by reference to a standard geometry with concentriccylinders where the relationships between shear stress (resp.shear rate) and the torque (resp. the angular rotation) areclassical. Non-Newtonian fluid flows obeying a power lawrheology with different indices were then simulated. Theresults of these numerical simulations are in very good

A. Rabia () S. Yahiaoui M. DjabourovESPCI ParisTech - Laboratoire de Physique Thermique, 10 RueVauquelin, 75231 Paris Cedex 5, Francee-mail: aminallah.rabia@espci.fr

S. Yahiaouie-mail: samir.yahiaoui@espci.fr

F. FeuilleboisLIMSI, UPR 3251 - CNRS, BP 133, Bat 508, 91403 Orsay Cedex,France

T. LasuyeINEOS ChlorVinyls, Chemin des Soldats, 62670 Mazingarbe,France

agreement with the preceding Newtonian-based model insome ranges of indices. The absolute difference still under5 % provided the index is below 0.45. Finally, this study pro-vides a calibration protocol in order to use nonstandard vanegeometries with various heights, gaps, and distance to thecup bottom for measuring the rheology of complex fluidslike shear thinning fluids and concentrated suspensions.

Keywords Vane geometry Complex fluids Non-Newtonian Shear thinning fluids

Introduction

The vane in cylinder geometry has been used for threedecades in view of characterizing the rheology of complexfluids. The interest in this nonconventional geometry firstlycame from the possibility that the vane would avoid wallslip effects which are currently observed with some flu-ids in standard Couette geometry. For example, a wall slipmay disturb the characterization of the elastic properties andyield stress of concentrated dispersions. Earlier works, see,e.g., Nguyen and Boger (1983), Nguyen and Boger (1985),and Keentok (1982), were concerned with the procedure tocharacterize the yield stress using a vane geometry with acontrolled rotational speed and recording the torque versustime until a maximum torque was obtained. It was assumedthat the region of the suspension close to the edges of thevane blades deforms elastically (that is, the torque varies lin-early with time) due to the stretching of the bonds (Nguyenand Boger 1983; Sherwood and Meeten 1991). The vanegeometry gained an increasing interest in food research(Roos et al. 2006; Krulis and Rohm 2004) and concentratedparticle suspensions (Rabia et al. 2010), and it is generally afamiliar tool in soil mechanics. Apart from the reduction of

358 Rheol Acta (2014) 53:357371

wall slippage, the vane tool has other practical advantagesas follows: the insertion of the geometry inside a concen-trated suspension minimizes the disturbance of the internalstructure of the suspension which recovers its cohesion afterthe vane has been immersed in the medium (thixotropicfluids). A standard use of the vane geometry requires a pre-cise description of the stress and strain factors (Barnes andCarnali 1990; Derakhshandeh et al. 2010; Estelle and Lanos2012; Fisher et al. 2007; Boger 2013), like for the classicalgeometries (Couette, cone-plate . . . ) (Ovarlez et al. 2011).

The analogy between the vane geometry and a Couettedevice has been examined in several papers. The numberof blades, arranged at equal angles, is not the same in alldevices: Savarmand et al. (2007) used a six-blades vane,while Nguyen and Boger (1983) and Ait-Kadi et al. (2002)used four blades and Sherwood and Meeten (1991) usedtwo. The importance of the number of blades was usuallynot examined. The blades were considered to be equiva-lent to an inner cylindrical surface, and some authors simplyassumed that the diameter of this inner cylinder is equal tothe vane diameter. This is in contradiction with the experi-mental determination made by Keentok (1982) who foundcorrections as large as 10 % of the radius, depending onthe material properties. The shear rates of the liquid atthe bottom and the top were taken into account by addinga corrected length to the height, assuming a power lawdependence of the stress with the radial position r (Nguyenand Boger 1985) for the yield stress measurement. Their

Fig. 1 A 3D view of the vane in cup geometry with six blades. H vaneheight, z vane immersion, S1 wall surface of the cup, S2 bottom surfaceof the cup, d distance to the cup bottom, R1 vane radius, R2 cup radius,and e gap

Fig. 2 Isometric view of the used mesh for Geo20-30-5

corrections are only valid for their specific vane and cupdimensions. In the recent publication by Savarmand et al.(2007), a numerical method clearly showed the end effectsof the finite length of the vane and results were testedagainst experimental data. The correction to the vane lengthderived from the simulations and experiments depends onmaterial properties, like the index for a power law fluid orthe yield stress for a Bingham fluid (Potanin 2010).

Savarmand et al. (2007) performed numerical simula-tions on a vane geometry in a cylindrical cup, with anarrow gap. This choice of the gap does not always corre-spond to the experimental conditions used for characterizingcomplex fluids such as particulate suspensions. They simu-lated experiments on various types of fluids (silica particlesuspensions, Xanthan solutions, or a Newtonian oil) and

Fig. 3 (Color online) Middle hight cutting plane (z = 15 mm) show-ing isovalues of the velocity magnitude for e = 5 mm with =1rad/s

Rheol Acta (2014) 53:357371 359

Fig. 4 (Color online) Top and side cutting planes showing isovaluesof the velocity magnitude for a gap e = 5 mm and a distance to thebottom of geometry d = 5 mm and = 1rad/s

introduced end-effect corrections of the effective height ofthe vane which depend on the fluidity index of the solutionsto be characterized.

The aim of this paper is first to provide a practicalparametrization and optimization of the vane geometry,based on a rigorous numerical calibration for Newtonianfluids. We then demonstrate that the wide gap parametrizedvane geometry provides a reliable tool for the measurementof the rheology of non-Newtonian liquids and power lawfluids under certain conditions.

It is natural to start with a Newtonian fluid. We performboth a systematic set of experiments and for each case theflow field is calculated with 3D computer fluid dynamicssimulation using the finite volume method (FVM).

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

140

Fig. 5 Dimensionless torque versus immersed height for differentgaps e, and for d = 0.5

0 0.2 0.4 0.6 0.8 140

60

80

100

120

140

Fig. 6 Dimensionless torque versus gap in total immersion (H = 3)for various values of the distance d

The free surface and end effects due to the bottom ofthe blades are taken into account. The influence of the dis-tance between the blade and the bottom of the cup is studiedsystematically. By analogy with a Couette geometry, wethen deduce the equivalent height Heq and the equivalentradius Req of the vane normalized by the geometrical radiusof the vane. Recommendations for optimization of proce-dures are derived from the comparison between theory andexperiments.

We then consider non-Newtonian fluids. The analogyestablished with a Couette device first allows to obtain thestress and the strain factors for any configuration of the vaneand the cup. We then analyze the discrepancy obtained whenusing the vane with a non-Newtonian power-law fluid andspecify the range of validity of the Newtonian approxima-tion using a suitable definition of the average distance insidethe gap.

0 0.5 1 1.5 2 2.5 30

50

100

150

Fig. 7 Interpolation of the total torque versus immersion for a gape = 0.1

360 Rheol Acta (2014) 53:357371

Table 1 Linear interpolations of the numerical values of the torque versus immersion for various gaps and for d = 0.5, allowing 1/z to becalculated

Linear fit (Slope=1/z) R square SSE

e = 0.1 45 z + 4.978 1 0.2817e = 0.2 29.51 z + 5.283 1 0.05873e = 0.3 23.19 z + 5.046 1 0.06491e = 0.4 19.83 z + 4.947 1 0.06979e = 0.5 17.78 z + 4.936 1 0.0871e = 0.6 16.42 z + 4.984 0.9999 0.1196e = 0.7 15.41 z + 5.061 0.9999 0.1537e = 0.8 14.74 z + 5.134 0.9999 0.1925e = 0.9 14.16 z + 5.23 0.9999 0.2575e = 1 13.61 z + 5.348 0.9998 0.3678

R square and sum of squares due to error (SSE) indicate the precision of these interpolations

Model of the vane for newtonian fluids

Definition of the equivalent Couette device

The purpose of the modeling is to provide a calibration ofthe vane geometry. By analogy with a Couette device, theequivalent inner radius R1eqv and height Heqv are calcu-lated.

If we note the shear stress form factor F and theshear rate factor F of the geometry, they are, respectively,defined as the shear rate over the angular velocity and theshear stress over the torque by the following relations:{

= F : shear stress = F : shear rate

As the viscosity is given by the ratio / , the relationshipbetween the vane rotational velocity and measured torquemay be written as follows:

= F

F(1)

This relationship is exact for a Newtonian fluid. For a coax-ial cylinders geometry with a large gap, these factors arederived as for instance, Couarraze and Grossiord (2000):1

F = R22 + R21

R22 R21and F = R

22 + R21

4R22R21H

(2)

where R1 and R2 stand here for the inner and outer cylinderradius, and H is the height of the cylinders.

End effect are not considered in these relationships. Inthe case of a vane geometry (see Fig. 1 where the usualnotations are displayed), the immersed height is replaced byan equivalent immersion (effective immersion) zeqv and theradius of the mobile cylinder is replaced by the equivalent

1The shear stress factor is calculated for a mean stress between theouter and inner radius: = (R1)+(R2)2

radius R1eqv , so that Eq. (1) may be written in the followingform:

= 4R

22R

21eqv

R22 R21eqvzeqv (3)

The equivalent immersion may also be written as zeqv =Heqv H + z with Heqv the equivalent height of the vane.

Equation (3) is made dimensionless by normalizing thetorque with R3ref , where the reference length Rref istaken as the geometrical radius of the blades (Rref =Rvane). Equation (3) then becomes:

= 4R22R

21eqv

R2ref

(R22 R21eqv

)zeqvCCC (4)

where the overbar (, z, . . . ) denotes dimensionlessquantities.

Model equations

Let us consider a Newtonian fluid with a viscosity . Forthe fluid velocities are very slow or the viscosities arevery large, inertial forces are small compared with vis-cous forces (i.e., the Reynolds number2 is low compared tounity). Stokes equations apply with appropriate boundaryconditions for the fluid flow:{ p = 2u

u = 0 (5)

2In the experiments presented in section Experimental validation ofthe numerical model for a Newtonian fluid, the Reynolds numberrange is 0.0025 Re 0.04 and the Froude number is Fr = 103

Rheol Acta (2014) 53:357371 361

The boundary conditions for the velocity are as follows:

uwall = 0 : No-slip on the outer cylinderur = uz = 0; u = r : No-slip on the surface of the

bladesuz = 0 : At the free surfacez = H

(6)where u and uz denote the components of the fluid veloc-ity u in a cylindrical coordinates system attached to theouter cylinder and p is the pressure. A representation of theboundary conditions is displayed in Fig. 1. The torque onthe blades has two main contributions:

1. Normal stress (essentially the pressure on blades) on themain part of the blade surface

2. Tangential viscous stresses on the tip of the blades

For our configuration, the torques on the set of blades andon the cup have opposite values. It is then more convenientto obtain the torque by surface integration on the cup S sincethe viscous stresses there are more regular, so that a standardmesh is sufficient:

=S

r v ndS (7)

where r = rer is the position vector from the axis, n is aunit vector normal to the surface and pointing here into thefluid, and v is the deviatoric (viz. the viscous) part of thestress tensor for a Newtonian fluid:

= 2D (8)with D = 12

(u + T u) the rate of strain tensor. The sur-face S on which the viscous stresses apply is composed oftwo parts: the lateral surface S1 and the one at the bottomof the cup, S2. At its top, the fluid is in contact with air.The interface is assumed to be horizontal since the capillarynumber (ratio of viscous to surface tension forces) is usu-ally low. From the zero tangential stress condition, there isthen no contribution to the torque.

The total torque is evaluated by its axial component (zcomponent) and can be written as follows:

=S1

rvdS +S2

rvdS (9)

It will be seen later (from the data in Fig. 5) that thetorque varies linearly with the immersion height z. In thelimit z = 0, the only contribution to the torque arises fromthe end-effect due to the bottom (surface S2), which is thesecond term in Eq. (9): 2 =

S2

rvdS.

The first term is the lateral contribution (surface S1) dueto the shearing inside the gap. It can be written as follows:

1 =S1

rvdS = zs1

R2vdl = z1/z

1/z is the derivative of the torque versus immersed height,or torque per unit of immersed height. Here, dl is a linearelement of the circumference delimiting the flow field.

The normalized total torque can thus be written as fol-lows:

= z1/z + 2 (10)

Numerical results and discussion for newtonian fluids

The Stokes equations and associated boundary conditionsfor the steady flow field were solved by the finite vol-ume method (FVM) using a commercial software ANSYS-Fluent. The used spacial discretization scheme is theUPWIND scheme, whereas the computation was performedusing the iterative scheme: semi-implicit method for pres-sure linked equations.

0 0.2 0.4 0.6 0.8 10.92

0.925

0.93

0.935

0.94

0.945

0.95

0.955

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

a

b

Fig. 8 Equivalent dimensions of vane geometries compared to cylin-drical Couette. a Shows the equivalent radius. b [Heqv H ], where His the dimensionless geometrical height of the vane. Tables 10 and...