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J. Non-Newtonian Fluid Mech. 148 (2008) 47–56

Primary and secondary normal stress differences of a magnetorheologicalfluid (MRF) up to magnetic flux densities of 1 T

Hans Martin Laun ∗, Claus Gabriel, Gerhard SchmidtPolymer Physics GKP, G 201, BASF Aktiengesellschaft, Carl Bosch Strasse, 67056 Ludwigshafen, Germany

bstract

First and second normal stress differences of a 50 vol.% magnetorheological fluid (MRF) are investigated by using a commercial plate–plateagneto-rheometer (Anton Paar GmbH) with plate–plate and cone–plate geometry. The manufacturer modified the instrument to achieve higher

ormal force (60 N) and torque (295 mNm) capacity. An additional modification by us allows an online determination of the true magnetic fluxensity B in the MRF by means of a Hall probe. FEM Maxwell 2D simulations quantitatively verify the Hall probe results and give detailed insightnto the radial flux density profile within the MRF sample.

Without shear, the static normal force FN for plate–plate increases as a power law: FN ∝ B2.4. A similar magnitude is found for cone–plateeometry, in contrast to the expectation. For steady shear at 10 s−1, the plate–plate normal force built-up limits the experiments at high fluxensities rather than the torque generated. The normal forces increase linearly with the shear stress at high flux density. The first normal stressifference N1 is positive and about five times larger than the shear stress. The second normal stress difference N2 is also positive. The experimentallyerived N2/N1 ratio of 1/4 distinctly deviates from theoretical predictions (N2/N1 = −1) for a semi-dilute MRF. Improvements of the radial fluxensity profiles are required to verify the normal stress difference ratio and to support the conjecture that the positive but small N2 is a consequence

f the densely packed MRF, which does not allow to create extended chain-like structures.

As shown in the appendix, the experimentally determined N2/N1 ratio is favorable to stabilize concentricity in concentric cylinder arrangements,elevant for the MRF application in clutches.

2007 Elsevier B.V. All rights reserved.

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eywords: Magnetorheological fluid (MRF); Normal forces; Plate–plate geomeorce in concentric cylinder

. Introduction

A magnetorheological fluid (MRF) is a concentrated suspen-ion of magnetizable particles in a low viscosity Newtonianiquid. Upon application of a magnetic field, the mostly fer-omagnetic particles form string-like structures parallel to theagnetic flux lines, thus creating a distinct yield stress in the

ystem [1–6]. Since the flowability of the MRF is reversibly con-rollable by an external magnetic field, it is used for mechanicalevices like dampers, clutches, actuators, etc. [7–9]. The mor-hology change of the MRF under magnetic field also causes aistinct anisotropy of the mechanical properties (e.g. [10]). Asresult, normal stresses occur also in simple shear flow [11,12].

ue to the strong magnetic dipole–dipole interaction, a normal

orce is observed in plate–plate rheometry even without flow11].

∗ Corresponding author. Tel.: +49 621 8280475; fax: +49 621 8282323.E-mail address: knowledge@launweb.de (H.M. Laun).

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377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2007.04.019

one–plate geometry; Ratio of first and second normal stress difference; Lateral

In plate–plate or cone–plate rheometry, the normal forces ofn MRF may become so large that they limit the range of acces-ible magnetic flux or of shear rates in a magneto-rheometer (seeelow). In devices like clutches with disk or concentric cylindereometry, undesirable lateral forces may be created. Thus, tostimate the magnitude of lateral forces, it is necessary to knowhe magnitude and sign of both the first (N1) and secondary nor-

al stress difference (N2) in simple shear and to compare theseo the magnitude of the shear stress.

Whereas several experimental techniques to determine both1 and N2 have been applied to polymer melts and solutions

see e.g. [13,14]), investigations on disperse systems are rare.all-Gleissle et al. [15] investigated normal stress differences

f concentrated suspensions with viscoelastic matrix by com-aring plate–plate and cone–plate results. Laun [16] determinedoth normal stress differences for an extremely dilatant poly-

er dispersion using the cone and partitioned plate technique.

n both publications representations of the normal stress dif-erences versus shear stress were beneficial to express normaltress differences by the shear stress magnitude. Based on the

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8 H.M. Laun et al. / J. Non-Newto

atio N2/N1 and sign of N1 an explanation of the tendency forxcentricity creation in concentric cylinder rheometry could beiven [16].

Ilg et al. [12] extract a positive N1, which increases with mag-etic flux density, as well as a constant ratio N2/N1 = −1 fromon-equilibrium molecular dynamics (NEMD) simulations ofemi-dilute ferrofluids, whereas the dynamic mean field (DMF)heory yields N2/N1 = −1.17. A negative sign of N2 of a fer-ofluid, has been found experimentally by Odenbach et al. [17],hereas the absolute value was only one-fourth of the NEMD

esult. For a concentrated MRF, See and Tanner [11] report a pos-tive N1 which decreases with increasing shear rate. De Vicentet al. [18] derive expressions for the magnitude of the normalorce in plate–plate geometry and report a positive and strain-ependent normal force going through a maximum at start-upf shear.

This paper reports measurements of the first and second nor-al stress difference in steady shear of a 50 vol.% MRF. The

nalysis is based on a comparison of plate–plate and cone–plateesults. Since the radial profiles of true magnetic flux density inhe sample cannot be expected to be identical for both geome-ries, we have carefully analyzed the flux density profiles bothy Hall probe measurements and Maxwell 2D FEM simula-ions. In addition, we address the normal stresses effect on theoncentricity in coaxial cylinder geometry.

. General conjecture

Fig. 1 schematically depicts the general conjecture: causedy strong magnetic dipole–dipole interactions which try to alignhe magnetized spherical particles parallel to the magnetic fluxensity vector, the net effect are lateral forces of (a priory unde-ned) magnitude A, pushing the spheres into the already existinghain of spheres. This lateral squeezing tries to extend an existinghain parallel to the flux density direction.

The normal forces acting on a volume element have the

pposite sign. Since the normal forces in 1-direction (directionf shear) and 3-direction (vorticity direction) are equal, the 2-irection (magnetic flux) experiences a twice as high normalorce but with opposite sign, in order to keep the hydrostatic

ig. 1. General conjecture for the origin of a positive normal force due toqueezing of magnetized spheres into existing chains.

wt(

3

3

eLicuFot

ot

Fluid Mech. 148 (2008) 47–56

ressure zero. The static stress tensor thus reads:

stat=

⎛⎜⎝

A 0 0

0 −2A 0

0 0 A

⎞⎟⎠ . (1)

he first and second normal stress differences follow as

1 = σ11 − σ22 = 3A (2a)

2 = σ22 − σ33 = −3A (2b)

hile N1 is positive, the normal stress ratio is

N2

N1= −1. (3)

he measured normal stresses for plate–plate and cone–plateeometry, respectively, are related to the normal force FN [20]y

N,CP = πR2

2N1 (4)

nd

N,PP = πR2

3[N1 − N2]

3

2 + m, (5)

here m stands for the normal force power law exponent

= d ln FN,PP

d ln γ. (6)

or the investigated MRF, the exponent m is typically close toero (see below). One would expect thus from Eq. (3) in thetatic case

N,PP ≈ 2FN,CP. (7)

pon shear, the strings of spheres get tilted: they partially breaknd reform again. This may cause deviations in the normal stressifferences, to be investigated. Furthermore, for a dense systemith concentration in the range of 50 vol.%, one could anticipate

hat the morphology picture using isolated strings or columnsmulti-chains) of particles may no longer be adequate.

. Experimental

.1. Rheometer with on-line flux density measurement

Measurements were made in a Physica MCR501 rheometerquipped with a Physica MRD180 magneto-cell as described byaeuger et al. [22], and manufactured by Anton Paar GmbH. The

nstrument is torque controlled but also allows imposition of aonstant shear rate. The standard geometry supplied by the man-facturer is plate–plate with radius RG = 10 mm. As depicted inig. 2, the vector of flux density B is perpendicular to the planesf shear. In this geometry, a flux density of ≥1 T is obtained for

he maximum allowed coil current.

One of the challenges of magneto-rheology is the knowledgef the magnetic flux in the gap with sample, which depends onhe magnetization of the MRF. An on-line measurement of the

H.M. Laun et al. / J. Non-Newtonian

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ig. 2. Schematic of the magneto-cell modified for an online measurement ofhe magnetic flux density.

rue flux density in the sample was achieved by a Hall probeF.W. Bell probe 1X and Model 9500 Gaussmeter), attached tohe magneto-cell: the probe is located in the rectangular horizon-al channel of a non-magnetic disk. The disk is located directlybove the bottom yoke of the magneto-cell and plays the rolef the stationary plate of the gap. The thickness of this addi-ional plate is 1.4 mm. The Hall probe strip with cross sectionf 1 mm × 4 mm may be placed at various radial positions toetermine the radial flux density profile. The flux-sensitive partf the probe has a diameter of about 3 mm. At fixed position, theall probe monitors the flux density for various coil currents.Due to the reduced remaining gap between upper yoke and

all probe housing plate, a new non-magnetic plate rotor withG = 10 mm and thickness 0.8 mm was machined. For a standardap height of hG = 0.3 mm, there remained sufficient space toount a PVC plate of 0.3 mm thickness on top of the rotor

late, acting as a guard ring (compare Fig. 11). In addition, a non-agnetic rotor with a cone angle of α = 3◦, RG = 10 mm and total

eight of 0.8 mm with additional PVC guard ring was machined,o allow also cone–plate measurements. All measurements were

ade at 25 ◦C.The motor of our rheometer is a special version allowing a

aximum torque of 295 mNm (without duration limit). For theeasurement of normal force, the air bearing of the rheometer

cts as an elastic spring, the displacement of which (being linearn the specified range to the normal force) is determined. The

aximum normal force has been increased from 50 to 60 N.bove that value the rheometer drive is automatically switchedff.

The shear rate given in the figures is the constant shear rate γ

or a cone and plate gap or the rim shear rate γR of the plate–plateap, respectively:

˙R = ϕRG

hG, (8)

here ϕ stands for the angular speed of rotation. Whileone–plate geometry directly yields the true shear stress τ,

Newtonian (or apparent) shear stress τN is obtained forlate–plate geometry [20] by assuming Newtonian liquid in the

nums

Fluid Mech. 148 (2008) 47–56 49

ap (M being the torque):

N = 2M

πR3G

. (9)

he true shear stress follows from the Newtonian shear stress as

= τN3 + n

4, (10)

ith a power law index n defined by

≡ d log τN

d log γR. (11)

hen a new MRF is brought into the gap, the rheometer isemagnetized using the manufacturer’s software. Before the rhe-logy investigation at a given magnetic flux density, the samples first sheared for 20 s at a shear rate of 10 s−1 without coilurrent, and subsequently kept at rest for 10 s. Then the field iswitched on and the constant shear rate imposed for 15 s. Thisime was chosen as a compromise to achieve steady-state con-itions at low shear rate ≥0.1 s−1 and to minimize dissipativeeating at the large shear rates.

.2. Sample

The magnetorheological fluid investigated is a MRF with lowiscosity at high volume concentration. Fifty percent by vol-me of magnetizable spherical carbonyl iron powder particlessize several �m) are suspended in hydrocarbon oil. Additivesrevent sedimentation occurring due to the large density mis-atch of particles and base oil and ensure easy redispersibility

fter long-term rest. Fig. 3 top shows flow curves (steady-stateewtonian shear stress versus rim shear rate for various lev-

ls of magnetic flux density). In the high flux density regime,nly a weak increase of the shear stress with growing shearate is observed. Thus, a positive power law index n (Eq. (11))lose to zero is obtained. Since the investigations to be pre-ented below concentrate on a shear rate of 10 s−1, we set n = 0or simplicity. Fig. 3 bottom shows the corresponding normalorces for plate–plate geometry. Similar as the shear stress, theormal force FN increases with growing flux density. However,he values slightly decrease as the shear rate increases. Thisields a negative power law index m (Eq. (6)) close to zero. Forimplicity, we also set m = 0 in the following.

.3. Maxwell 2D FEM simulation

Finite element simulations were performed to get insight intoetails of the magnetic flux density distribution in the rheometerap with sample. Details of the yoke geometry of the MRD180agneto-cell were kindly provided by Dr. Huck of Anton PaarmbH Germany. The only adjustable parameter was the effec-

ive gap thickness of the channel for the thermostat liquid inhe bottom yoke. This was assumed to be a concentric chan-

el with flat walls, whereas in reality a screw-like channel issed. Fig. 4 shows the magnetization characteristics of yokeaterial (machining steel) and MRF used in the Maxwell 2D

imulations.

50 H.M. Laun et al. / J. Non-Newtonian Fluid Mech. 148 (2008) 47–56

F(i

pds(l

Fa

Ff

(dclt

pmopaeaAHtdistinctly decreases with decreasing radius. The quality of the

ig. 3. Shear rate dependence (decimal spacing) of the Newtonian shear stresstop) and the normal force (bottom) for various magnetic flux densities measuredn plate–plate geometry at 25 ◦C.

Details of the flux density field for an empty gap andlate–plate geometry including the Hall probe housing plate are

epicted in Fig. 5 for a coil current of 3 A. The picture alsohows a detail of the non-magnetic plate rotor with guard ringas for the effect of the guard ring see below). From such simu-ations we can derive the flux density at the hall probe position

ig. 4. Magnetic flux density vs. magnetic field strength of the MRF (50 vol.%)nd of the yoke (machining steel) used for the Maxwell 2D simulations.

flmm

Frd

ig. 5. Detail of the plate–plate gap geometry with magnetic flux density linesor 3 A coil current.

fixed radius r = 7.5 mm). A comparison of the FEM result andirect Hall probe measurement of magnetic flux density versusoil current is shown in Fig. 6. Noteworthy, the fit of the abso-ute values was achieved by adjusting the channel width for thehermostat liquid in the bottom yoke.

Fig. 7 compares radial flux density profiles, again at the hallrobe level without MRF in the gap, from FEM and Hall probeeasurements. The FEM simulation yields a distinct maximum

f flux density close to the rim, which is not seen in the Hallrobe data. Here one needs to take into account that the probeverages over a certain range of radius due to the 3 mm diam-ter of the sensitive area. After averaging the FEM results overrange of 2 mm, the maximum becomes significantly smaller.t radii < 8 mm, we find an excellent agreement of the FEM andall probe data, indicating that the FEM predictions are realis-

ic. It is also obvious, however, that the radial flux density profile

ux density profile is thus not satisfactory and needs improve-ent [21]. Flux density profiles with MRF at the level of theeasurement gap will be discussed in Section 4.

ig. 6. Comparison of Hall probe data (squares) and Maxwell 2D simulationesults (diamonds) for the empty gap and Hall probe position r = 7.5 mm (fluxensity plateau).

H.M. Laun et al. / J. Non-Newtonian Fluid Mech. 148 (2008) 47–56 51

FHr

4

4

ategfpI0

F

tTwttrss

Fcwm

Fagwptsgo

Fpr

rmsimulation demonstrates in addition, however, that the radialflux density profiles for the two geometries differ significantly.As expected, cone–plate gives rise to a stronger flux density

ig. 7. Comparison of radial flux density profiles measured for an empty gap:all probe data (unfilled squares); FEM prediction (broken line); FEM prediction

adially averaged over 2 mm (full line).

. Result

.1. Static normal force

As expected from the general conjecture and reported by Seend Tanner [11], strong positive normal forces (trying to increasehe gap) are measured both for plate–plate and cone–plate geom-try (Fig. 8). In the linear representation (top), both geometriesive about the same magnitude of normal force. The normalorce is negligible at B = 0 and increases much stronger thanroportional with increasing flux density (Hall probe data).n the log–log representation (bottom) the plate–plate data for.1 < B < 0.8 follow a power law:

N,PP ∝ B2.4, (12)

he exponent being close to the value 2.6 reported by See andanner [11]. The cone–plate data seem to start at lower level butith a larger exponent in the range of 3.4. As a consequence,

he cone–plate data are higher at high flux density. In summary,he expectation of Eq. (7) is not verified at high flux density,ather both normal forces have about the same magnitude. Thistatement is only true, however, if the flux density profile in theample is comparable for both geometries.

The comparison of plate–plate and cone–plate geometry inig. 9 indicates that the radial flux density profiles in the MRFannot be equal, even if the flux density is constant versus radiusithout MRF. This is due to the fact that the thickness of theagnetizable MRF increases with radius.This is verified quantitatively by the FEM simulations in

ig. 10, which compare the radial flux density profiles 0.15 mmbove the Hall probe housing (i.e. in the middle of the plate–plateap) for cone–plate and plate–plate geometry, both with andithout MRF in the gap, for a constant coil current of 3 A. Com-ared to the empty gap (independent of the type of geometry),

he MRF in the gap gives rise to an increase of the flux den-ity, in the order 6% for plate–plate and 10% for cone–plateeometry, respectively. This effect is verified and monitorednline by the Hall probe. This allows to adjust the coil cur-

Fet

ig. 8. Static normal force vs. true magnetic flux density in the MRF (Hallrobe): (top) linear representation; (bottom) power law behavior in log–logepresentation. Each symbol type represents one sample.

ent such that the same value of flux density in the sample isaintained for r = 7.5 mm (fixed radial Hall probe position). The

ig. 9. Magnetic flux in plate–plate and cone–plate geometry and direction ofxpected string morphology (schematic). Due to the radially differing MRFhickness in cone–plate, the radial flux density profile is different, too.

52 H.M. Laun et al. / J. Non-Newtonian Fluid Mech. 148 (2008) 47–56

Fig. 10. Radial flux density profiles 0.15 mm above the Hall probe housing plateaM

iflrgwodgwmst

Table 1Normal force in plate–plate geometry for subsequent cycles with stop and fieldoff, stop and 0.5 T, and steady shear at 10 s−1 and 0.5 T (demagnetisation onlybefore first cycle)

Cycle FN (N) stopand field off

FN (N) stopand 0.5 T

FN (N) 10 s−1

and 0.5 T

1 −0.03 5.3 19.62 −0.24 8.3 18.43 −0.34 8.7 18.64 −0.33 9.8 18.65 −0.35 11.0 18.66 −0.26 10.2 18.778

4

ca1o(m(at

Fe

t 3 A coil current: without MRF (lower full line); plate–plate gap filled withRF (broken line); cone–plate gap filled with MRF (upper full line).

ncrease with growing radius. Furthermore, one gets pronouncedux density maxima at the rim (caused by the vertical MRFim in the simulation), which does not show up for the emptyap. In summary, the quality of the radial flux density profilesith MRF is even less satisfying compared to Fig. 7. Becausef the distinct radial flux density gradients, one creates pon-eromotive magnetic forces on the CIP particles, which mayive rise to radial concentration gradients in the MRF. Note-

orthy, the distinct flux density maximum at the rim tries toove CIP particles to the edge of the gap. The poor radial con-

tancy of the flux density field in the sample is thus by far notrivial!

c

bs

ig. 11. Climbing effect of MRF without guard ring: (left) after measurement at 0.1ffect).

−0.28 10.8 18.6−0.27 13.1 18.6

.2. Comparison of static and dynamic normal force

A comparison of Fig. 3 bottom and Fig. 8 bottom indi-ates that the static normal force in plate–plate geometry isbout 1/3 of the normal force measured in steady shear at0 s−1. The question remains what value is obtained after stopf steady shear. Table 1 compares the initial static normal forcecycle 1) with a series of subsequent cycles where the nor-al force was first measured at stop without magnetic field

coil current zero), at stop but with 0.5 T, and subsequentlyt 10 s−1 and 0.5 T. Then the coil current was switched offo control the residual bias normal force and to start a new

ycle.

After demagnetization and for coil current zero, a negligi-le negative static normal force is measured at rest (cycle 1). Inubsequent cycles a larger negative bias normal force remains,

T (sample remains in gap); (right) after 0.5 T measurement (strong climbing

H.M. Laun et al. / J. Non-Newtonian Fluid Mech. 148 (2008) 47–56 53

Table 2Torque and normal force of a high shear stress MRF (also 50 vol.%) in plate–plate geometry at 10 s−1

Flux density, B (T) Torque, M (mNm) Shear stress, τN (kPa) Normal force, FN (N) NSD, (N1 − N2)a (kPa)

0 0.24 0.2 −0.4 −40.1 6.6 4.2 0.7 70.2 18.5 11.8 4.9 470.3 35.9 22.9 12.7 1210.4 55.5 35.3 24.5 2340.5 76.6 48.7 37.3 3560.6 98.1 62.4 48.8 4660

0

pmwprips

4

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mraitps

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4

flssurements are compared. A new sample was used for each run.The maximum scatter below 0.8 T reaches about ±12% in theshear stress, whereas the scatter in the normal force appears dis-tinctly smaller! Above 0.8 T a crossover of curves is observed,

.7 116.0 73.9

.8 Torque < 295 mNM

resumably due to the remanence of the yoke. The static nor-al force in the first cycle reproduces the value from Fig. 8ithin experimental scatter. After imposition of steady shear, theositive normal force jumps to a distinctly higher value, whichemains nicely reproducible in the following cycles. Interest-ngly, the static normal force after shear gradually reaches alateau value of about 11 N, which is twice as high as the initialtatic normal force without any shear.

.3. Experimental limitations at steady shear

If the flux density outside the gap is much smaller comparedo that in the gap, the resulting ponderomotive magnetic forceselp to keep the MRF within the gap, even against centrifugalorces at elevated shear rates. Yet a guard ring is required to pre-ent the MRF from climbing around the rim of the rotor. Thiss demonstrated in Fig. 11 for a rotor without guard ring. Sub-equent to shearing the MRF at 100 s−1 and 0.1 T flux density,he front part of the upper yoke was removed (Fig. 11 left). The

RF remained in the gap. However, the same test at 0.5 T causeshe MRF to climb around the rotor (Fig. 11 right), thus leavingpartially filled gap. The guard ring avoids the MRF climbingut makes it impossible to visually inspect the sample rim aftermeasurement.

For an MRF exhibiting very high yield stresses at elevatedagnetic flux density, the range of accessible flux density is

ather limited by the maximum normal force the air bearing canccept, and not so much by the torque limit. This is demonstratedn Table 2 for another MRF with higher shear stress (comparedo the sample that has been used for all the other measurementsresented) for a given flux density. Plate–plate geometry andhear rate 10 s−1 was used.

At 0.7 T the torque has reached 116 mNm, which is muchess than half the maximum torque. The normal forces, except forero flux density, are positive and reach 58.5 N at 0.7 T. For 0.8 The normal force would exceed the 60 N normal force limit. Thus,measurement at 0.8 T is immediately stopped by the software.comparison of the Newtonian shear stress and the apparent

ormal stress difference [N1 − N2]a (NSD by short), obtained ifhe correction factor 3/(2 + m) in Eq. (5) is ignored, shows thatSD is much larger than the shear stress. This indicates that

lso under steady shear, the normal stresses dominate the statef stress.

Ff1s

58.5 558

Normal force limit reached!

.4. Reproducibility of shear stress and normal force

The reproducibility of shear stress and normal force versusux density (Hall probe data) for plate–plate geometry and ahear rate of 10 s−1 is demonstrated in Fig. 12. Several mea-

ig. 12. Reproducibility of respectively Newtonian shear stress (top) and normalorce (bottom) vs. magnetic flux density (Hall probe); plate–plate geometry and0 s−1. Each curve has been measured on a new sample, represented by variousymbols.

54 H.M. Laun et al. / J. Non-Newtonian Fluid Mech. 148 (2008) 47–56

F(

pr

(aaflsw(ifn

4

ysfl

Fsn

Fig. 15. N1 from cone–plate geometry (data from Fig. 14) and N1 − N2 frompca

prb

N

ss

Ntcp

N

Ot

ig. 13. Normal force in plate–plate geometry plotted vs. Newtonian shear stressdata from Fig. 12). Each symbol type represents one sample.

resumably indicating geometry changes at the sampleim.

If the same data are plotted as normal force versus shear stressFig. 13), the flux density only enters as an indirect parameternd the scatter of the data points remains below ±10%. This isdvantageous to reduce the influence of uncertainties in the trueux density profile (see above). Interestingly, the data for sheartresses >10 kPa (>0.2 T) fall on a straight line! A straight lineill also be obtained if [N1 − N2]a is plotted versus shear stress

see below). To stress the broad validity of the linear relation,t should be mentioned that an analogous behaviour was foundor various temperatures and concentrations of CIP, albeit we doot intend to further discuss these measurements here.

.5. Normal stress versus shear stress

The normal force measured in cone–plate geometry directlyields the first normal stress difference. If plotted versus thehear stress (Fig. 14), a straight line is appropriate to fit the highux density regime, too. The data points stem from various inde-

ig. 14. First normal stress difference in cone–plate geometry plotted vs. (true)hear stress. Different symbols represent various reproduction measurements onew samples.

Bs

N

Tn

Naticbt

5

Mt

late–plate geometry (data from Fig. 13) plotted vs. shear stress. In the latterase, both power law exponents m and n were set to zero and Eqs. (5) and (10)pplied. Each symbol type represents one sample.

endent measurements and thus give an indication of the goodeproducibility. The straight line in the figure may be expressedy

1 = 7.2|τ| − 42 kPa (±10%), (13)

ince the normal force is independent of the sign of the sheartress or direction of rotation.

The plate–plate data from Fig. 13 were transferred into1 − N2 versus true shear stress by using m = n = 0. In Fig. 15

hese data are compared with N1 versus shear stress fromone–plate geometry. The straight line fit gives the followingarameters:

1 − N2 = 5.4|τ| − 20 kPa (±10%). (14)

bviously, N1 and N1 − N2 are not so much different in magni-ude, in contrast to the expectation from the general conjecture.y subtracting Eq. (14) from Eq. (13) one gets an explicit expres-

ion for N2:

2 = 1.8|τ| − 22 kPa (larger error) (15)

hus, the second normal stress difference is positive and theormal stress ratio for high flux density is in the range of

N2

N1≈ 1

4. (15)

oteworthy, the relatively small difference between plate–platend cone–plate results gives rise to a larger error. In addition,he unsatisfactory radial flux density profiles certainly needmprovement [21] to verify this result. In the Appendix, we dis-uss how the positive first normal stress difference and the smallut also positive second normal stress difference are of advantageo maintain concentricity in coaxial cylinder geometry.

. Conclusions

The combination of online Hall probe measurements andaxwell 2D simulations yield valuable insight into details of

he true radial magnetic flux density profile in the MRF under

nian

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mfstdnmaopdsmtaisrtfi

hd

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A

pfiht

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wc

πrr(Ri) = σrr(Ri) − ph(Ri) − p. (A.2)

Here, ph stands for the hoop stress and p for the hydrostaticpressure. It is possible to replace the terms on the right-hand

H.M. Laun et al. / J. Non-Newto

nvestigation. While a very good agreement between Hall probeata and simulation prediction could be achieved, it is also obvi-us that the quality of the radial profiles with MRF, realized forhe present non-magnetic rotors and yoke design, is not at all sat-sfactory to quantitatively compare cone–plate and plate–plateeometry.

For a high shear stress MRF, it could be shown that the exper-mental access to the high flux density regime is rather limitedy too large normal forces and not by the rheometer’s torqueimit. As a consequence, we have selected an MRF with lowerhear stress to allow a wide range of flux density (up to 1 T)or the normal force measurements using both plate–plate andone–plate geometry, in order to determine both the first andecond normal stress difference.

The formation of ordered structures under a magnetic fieldives rise to a static normal force without shear. For plate–plateeometry, the normal force increases proportional to B2.4, in lineith the findings by See and Tanner [11]. For cone–plate geome-

ry, a similar magnitude is obtained, in contrast to the expectationor isolated strings (columns), which would give a twice as highormal force for plate–plate compared with cone–plate. Pre-umably, this effect is a consequence of the high density ofagnetizable particles, which do not allow for the formation

f strings at 50 vol.%. This hypothesis needs clarification byEMD simulations of dense systems.Applying steady shear at the same magnetic flux density,

ields a distinctly higher normal force. After stop of shear, theormal force decreases rapidly to a lower static value again.or repeated cycles, however, the static normal force tends toeach a plateau, which is about a factor of two higher than thenitial static value without any pre-shear. This indicates that theuality of structure achieved without any shear is less perfecthan that after stop of shear. In addition, there is obviously aistinct normal force contribution by the continuous structurereak-up and rebuilt during steady shear.

While both the shear stress and normal force exhibit a sig-oidal shape versus magnetic flux density, a plot of normal

orce versus shear stress yields a straight line in the high sheartress or flux density regime, respectively. The latter represen-ation appears to be less sensitive to calibration errors of fluxensity or discrepancies of the radial flux density profile. Theormal stress differences N1 and N1 − N2 derived from the nor-al forces for cone–plate and plate–plate geometry, respectively,

lso yield straight lines versus shear stress, except near therigin. N1 is positive and by a factor of about 5 larger com-ared to the shear stress. This means that the state of stress isominated by the normal stress components. Similar as for thetatic normal force, both N1 and N1 − N2 are rather similar inagnitude, in contrast to the expectation from NEMD simula-

ions for a semi-dilute MRF (i.e., N2 = −N1). The quantitativenalysis for the linear regime versus shear stress yields a pos-tive N2 and a normal stress difference ratio of N2/N1 = 1/4. Inpite of experimental uncertainties, like the poor quality of the

adial flux density profiles addressed above, we tend to believehat the detected N2 behavior predominantly stems from theact that we investigate a dense system. Measurements withmproved flux density profiles are on the way to support this

Fsr

Fluid Mech. 148 (2008) 47–56 55

ypothesis. We also encourage further simulation work for theense MFR.

In the Appendix, the total normal force acting on the bobf a Searle-type geometry is derived and expressed by N1 andhe normal stress difference ratio. Deviations from concentricityreate a net force, which drives the bob back to the desiredosition, thus stabilizing the concentricity of the arrangement.he result is of relevance for MRF clutches using concentricylinder geometry.

cknowledgements

We are grateful to Dr. G. Oetter for providing the MRF sam-les. We acknowledge the help of Dr. Huck of Anton Paar GmbHor kindly providing details of the MRD180 magneto-cell asnput for the Maxwell 2D simulations. P. Schuler is thanked foris help in modifying the magneto cell, M. Bach for performinghe measurements.

ppendix

Of interest is the question whether in concentric cylindereometry the normal forces of an MRF under magnetic fieldend to stabilize or destabilize concentricity. A similar taskas been treated for an extremely dilatant dispersion [16]. Wessume polar coordinates with the z-direction parallel to the fieldFig. A1). The radial shear stress profile is given by

(r) = τi

(Ri

r

)2

, (A.1)

here the subscript i denotes the outer surface of the innerylinder. The total normal stress at this surface is

ig. A1. Searle-type geometry (bob rotating in a cylinder) with coordinateystem and total normal stresses acting on the bob at the right and left side,espectively, for off-centricity (schematic).

5 nian

s

N

N

a

σ

σ

p

Tm

N

T

p

TwE

p

wf

π

TodabsFghFlIititr

t1tti

π

Tt

R

[

[

[

[

[

[

[

[

[

[

[of radial flux density profiles in a magneto-rheometer. Appl. Rheol., in

6 H.M. Laun et al. / J. Non-Newto

ide by normal tress differences. Since

1 ≡ σθθ − σrr (A.3)

2 ≡ σrr − σzz (A.4)

nd σzz = 0 (neutral axes), one gets

rr = N2 (A.5)

θθ = N1 + N2 (A.6)

= σrr + σθθ + σzz

3= N1 + 2N2

3. (A.7)

o derive the hoop pressure term, we make use of the highagnetic flux density approximation of the normal stresses

1 + N2 ∼= Aτ − B. (A.8)

his yields

(Ri) =∫ Ri

R0

−σθθ(r)d ln r=−AτiR2i

∫ Ri

R0

dr

r3 + B

∫ Ri

R0

d ln r

= Aτi

2

[1 −

(Ri

R0

)2]

+ B ln

(Ri

R0

)c. (A.9)

o further simplify the result, we use the approximation (whichould also be reproduced by neglecting the constant term B inq. (A.8)):

(Ri) ≈ Aτi − B

2

[1 −

(Ri

R0

)2]

= (N1 + N2)1 − β2

2.

(A.10)

here β stands for the radius ratio Ri/R0. For the total normalorce we finally get

rr(Ri) ≈ N2 − (N1 + N2)1 − β2

2− N1 + 2N2

3

= −N1

[1

3

(1 − N2

N1

)+

(1 + N2

N1

)1 − β2

2

](A.11)

he last expression represents the total normal force as functionf the first normal stress difference and of the normal stressifference ratio. If the normal stress difference ratio is positivend smaller than 1, as in the experimental results, the term inrackets is positive. Since N1 is always positive, the total normaltress vector points towards the axis of rotation as depicted inig. A1. For a slightly eccentric position of the inner cylinder, theap on the right-hand side being more narrow and that on the left-and side wider, there are two different effects to be considered.irst, the shear stress in the narrower gap will increase due to the

ocally increased shear rate for a positive power law exponent.n addition, one could imagine an increase of the flux densityn the narrower gap. In line with the shear stress increase on

he right-hand side, the positive normal stress difference N1 willncrease in magnitude. Second, the geometry change will modifyhe value of β in the expression. A narrower gap causes theadius ratio to increase and thus reduces the second term in

[

Fluid Mech. 148 (2008) 47–56

he brackets. However, for narrow gaps the contribution of the− β2 term is relatively small. It is thus reasonable to conclude

hat the increase of N1 for narrowing gap by Δ is the dominatingerm. As a consequence the resulting force on the inner cylinders such to push it back to its concentric position:

rr(Ri + Δ) − πrr(−Ri − Δ) < 0. (A.12)

he double cylinder arrangement thus experiences a stabiliza-ion of concentricity by the sheared MRF under magnetic field!

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