On the nonparallelism effect in thin film plate–plate rheometryEfrén Andablo-Reyes, Juan de Vicente, and Roque Hidalgo-Alvarez

Citation: Journal of Rheology (1978-present) 55, 981 (2011); doi: 10.1122/1.3606633 View online: http://dx.doi.org/10.1122/1.3606633 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/55/5?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Inertial effects at moderate Reynolds number in thin-film rimming flows driven by surfaceshear Phys. Fluids 25, 102108 (2013); 10.1063/1.4825134 Rheometry of dense granular materials: The crucial effects of gravity and confining walls AIP Conf. Proc. 1542, 49 (2013); 10.1063/1.4811866 Non-isothermal flow of a thin film of fluid with temperature-dependent viscosity on astationary horizontal cylinder Phys. Fluids 23, 062101 (2011); 10.1063/1.3593393 Pressure and temperature effects in slit rheometry J. Rheol. 43, 1099 (1999); 10.1122/1.551043 Wall slip in the molecular dynamics simulation of thin films of hexadecane J. Chem. Phys. 110, 2612 (1999); 10.1063/1.477982

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On the nonparallelism effect in thin film plate–platerheometry

Efren Andablo-Reyes, Juan de Vicente,a) and Roque Hidalgo-Alvarez

Biocolloid and Fluid Physics Group, Department of Applied Physics,Faculty of Sciences, University of Granada, C= Fuentenueva s=n,

18071 – Granada, Spain

(Received 25 May 2010; final revision received 31 May 2011; published July 8, 2011)

Synopsis

The effect of nonparallelism in a plate–plate torsional geometry is investigated for confined

Newtonian fluids by directly solving Reynolds equation. A nondimensionalization is proposed, and

theoretical results are compared to triborheological experiments by Kavehpour and McKinley

[Tribol. Lett. 17, 327–335 (2004)] in the form of a frictional Stribeck curve. VC 2011 The Societyof Rheology. [DOI: 10.1122/1.3606633]

I. INTRODUCTION AND BACKGROUND

Today, there is a great interest in better understanding of the behavior of complex flu-

ids in a mesoscopic scale of the order of microns to bridge the gap between the rheologi-

cal and tribological domains [Kavehpour and McKinley (2004); Clasen et al. (2010);

Pipe et al. (2008); Andablo-Reyes et al. (2010)]. A direct possibility to explore this

region is by using parallel-plate rheometry where the flat surfaces can be progressively

approached in a controlled way. Actually this route has been explored in several papers

in the past arguing that a well defined shear or squeeze flow kinematics exists in this case

in contrast to what happens in conventional tribological setups.

As the gap height decreases in a parallel-plate geometry, the measured viscosity is

found to decrease as well, especially at very narrow gaps well in the mesoscopic region

[Walters (1975)]. This viscosity-gap dependence has been easily accounted for, in the

case of Newtonian fluids, by introducing a so-called “gap error” to correct the shear rate

and, hence, the viscosity by using a gap distance, which is equal to the commanded gap

plus the gap error [Connelly and Greener (1985)]. The physical origin of this gap error

has been traditionally ascribed to nonparallelism, nonconcentricity, nonflatness, wall slip,

edge effects, and the gap-zeroing procedure that arises due to the squeeze flow of air

[Connelly and Greener (1985); Kramer et al. (1987); Davies and Stokes (2005); Henson

and MacKay (1995); Kalika et al. (1989)]. In spite of their practical importance, the indi-

vidual contribution of any of these features to the gap error has been scarcely investigated

in the literature from a theoretical point of view. Among all, the major source of gap error

comes from nonparallelism of the plates, which can typically be of the order of 5–50 lm

[Davies and Stokes (2008); Andablo-Reyes et al. (2010)].

a)Author to whom correspondence should be addressed. Fax: 0034 958 243214. Electronic mail: jvicente@ugr.es

VC 2011 by The Society of Rheology, Inc.J. Rheol. 55(5), 981-985 September/October (2011) 0148-6055/2011/55(5)/981/6/$30.00 981

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To the best of our knowledge, only the flow in a misaligned cone-and-plate rheometer

where the axis is not perpendicular to the surface has been investigated in detail by the

method of domain perturbation [Dudgeon and Wedgewood (1993)]. However, cone plate

geometries are not as relevant as plate–plate ones since gap thickness must remain con-

stant in the former geometry to get a viscometric flow. For Newtonian fluids in a parallel-

plate system, Taylor and Saffman (1957) found theoretically and Greensmith and Rivlin

(1953) experimentally that an extra pressure is exerted at the plate when the axis of rota-

tion is not perfectly perpendicular to the fixed plate and that the pressure on the plate

varies with position. In the aligned parallel plates, of course, there should be no such nor-

mal force for a Newtonian fluid [Walters (1975)]. It is our experience, however, that in

most parallel plate arrangements the rotation axis is very accurately perpendicular to the

surface and, on the contrary, suffer from imperfections when closely looking the parallel-

ism between the plates [Bernzen (personal communication)]. It has been experimentally

evidenced in the past that a small degree nonparallelism can give rise to a large positive

pressure in the converging flow region and a large negative pressure in the diverging flow

region superimposed on the pressure due to normal stress effects [Adams and Lodge

(1964); Greensmith and Rivlin (1953); Walters (1975)]. In this work, we will show theo-

retical evidences on the effect of nonparallelism on thin film rheological measurements

using plate–plate geometries. Theoretical results will be compared to triborheological

experiments reported in the literature.

II. THEORY AND NUMERICAL SOLUTION

The aim of this work is to determine the thickness of the fluid film separating two rigid

flat slightly tilted surfaces that are pressed together at a fixed load (load capacity FN) as

shown in Fig. 1. The fluid film thickness in the gap h was then modeled as a one-wave

waviness hðr; hÞ ¼ er cos hþ h0, where e ¼ ðh0 � hminÞ=R, hmin is the minimum film

thickness, R is the plate radius, h0 is the height at the midpoint in the contact, and r and hare the plate coordinates measured from the midpoint. The upper plate was made to rotate

at a commanded angular speed X while the lower one was kept stationary. Here, we con-

sidered the case of a purely viscous Newtonian fluid sheared between the plates.

A solution to the problem was obtained by numerical solution of Navier–Stokes equa-

tion under the lubrication approximation. Hydrodynamic lubrication by a Newtonian

FIG. 1. Schematic representation of the nonparallelism in a plate–plate torsional geometry.

982 ANDABLO-REYES, DE VICENTE, and HIDALGO-ALVAREZ

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fluid is mathematically described by the so-called Reynolds equation. In polar coordi-

nates, Reynolds equation is written as

@

@rrh3 @pðr; hÞ

@r

� �þ 1

r

@

@hh3 @pðr; hÞ

@h

� �¼ 6gXr

@hðr; hÞ@h

; (1)

where pðr; hÞ is the hydrodynamic pressure, h r; hð Þ is the fluid film thickness, and g is the

Newtonian viscosity. Reynolds equation was solved here using a finite differences nu-

merical method where the resulting matrix is solved by Gauss–Seidel iteration using the

Reynolds boundary condition [Khonsari and Booser (2001)].

III. STRIBECK CURVE

A convenient way to discuss the hydrodynamics in the problem is by using the friction

coefficient—Gumbel number representation in the form of a “Stribeck curve” [Kaveh-

pour and McKinley (2004); de Vicente et al. (2005)]. On the one hand, the friction coeffi-

cient is defined here as l � 2T=RFN , where T ¼Ð 2p

0

Ð R0

s r; hð Þr2drdh is the total torque

acting on the rotating plate and s r; hð Þ is the shear stress. On the other hand, the Gumbel

number is defined here by Gu � gX=r, where r ¼ FN

�pR2 is the average or nominal

normal stress on the plates. With this, a nondimensionalized Reynolds equation is

obtained as follows:

@

@r0r0h03

@p0ðr0; hÞ@r0

� �þ 1

r0@

@hh03@p0ðr0; hÞ

@h

� �¼ 6

Gu

h002

r0@h0ðr0; hÞ

@h; (2)

where r0 ¼ r

R, p0 ¼ p

r, h0 ¼ h

h0

¼ eh00

r0 cos hþ 1, and h00 ¼h0

R.

Hence, given Gu and e, Eq. (2) can be solved for h0ðr0; hÞ and p0ðr0; hÞ on the disk. For

example, in Fig. 2 it is shown the dependence of the central film thickness, h00, on the

Gumbel number, Gu, for e values in the range from 0:5� 10�3 to 2� 10�3. As observed

h00 increases with Gu. In agreement with the definition of Gu, this implies that h00

FIG. 2. Effect of the Gumbel number Gu on the normalized central film thickness h00.

983NON-PARALLELISM IN PLATE-PLATE RHEOMETRY

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increases with the value of the parameter gX and decreases with the load capacity, FN .

For a given Gu value, the larger values for h00 are found for the larger values of e. In other

words, the lubrication forces are enhanced by increasing the nonparallelism of the plates.

Once the pressure and film thickness are numerically computed using Eq. (2), the nor-

malized shear stress can be obtained as follows:

s0ðr0; hÞ ¼ srh00¼ Gu

h020

r0

h0ðr0; hÞ þh0ðr0; hÞ

2r0@p0ðr0; hÞ

@h(3)

and from this, the normalized total torque acting on the plate gives

T0 ¼ T

R3rh00

¼ð2p

0

ð1

0

s0r02dr0dh: (4)

Finally, the friction coefficient can be found from the minimum film thickness and total

torque by using

l ¼ 2h00T0

p: (5)

Results obtained by simply giving values to Gu and computing the friction coefficient,

l, according to Eq. (5) are shown in Fig. 3 for e values between 0:5� 10�3 and

2� 10�3. As observed, the Stribeck curve only depends on the nonparallelism, e, existing

in the geometry. For completeness, experimental data by Kavehpour and McKinley

(2004) are included in Fig. 3. These correspond to a Newtonian Pennzoil lubricant (700

mPa s) operating at 10 N normal load with e ¼ 1:25� 10�3. As observed, the theoretical

calculations for e ¼ 1� 10�3 are in good agreement with the experimental data by

Kavehpour and McKinley (2004) suggesting that the appearance of a normal force on the

plate is actually due to the nonparallelism of the geometry. It is worth noting here that

bridging forces between the plates due to surface tension were not taken into account in

FIG. 3. Stribeck curves for sheared Newtonian fluids. Circles correspond to experimental results for a plate–

plate contact [Kavehpour and McKinley (2004), Fig. 4 ].

984 ANDABLO-REYES, DE VICENTE, and HIDALGO-ALVAREZ

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this analysis as they were found to be negligible. For a liquid of volume V and surface

tension c, bridging forces are proportional to 2pR4c=V and increase as the gap between

the plates increases until a force plateau value is reached [Fortes (1982)]. The gap dis-

tance at which the plateau is achieved and the force plateau value itself depend on both

the fluid volume V and the plate’s radius R. For instance, for c ¼ 20� 10�3 N=m, R¼ 1

cm, and V¼ 0.0157 cm3, the maximum value for the bridging forces is about 0.125 N

and is reached at a gap distance� 53 lm. These bridging forces are negligible compared

to the load capacity of the contact FN � 20 N found for h00 ¼ 5� 10�4, g ¼ 0:7 Pa�s,

e ¼ 1� 10�3, and X ¼ 50 rad=s.

IV. CONCLUSIONS

In agreement with Reynolds lubrication theory, the nonparallelism in plate–plate rhe-

ometers provokes the appearance of lubrication forces in the fluid film between the

plates. The magnitude of the normal force appearing on the rotating plate is directly pro-

portional to the angular speed and the fluid viscosity. The proportionality constant

increases with relative tilt of the plates and decreases with the fluid film thickness. The

Stribeck curve model presented here successfully explains the results of triborheological

experiments available in the literature.

ACKNOWLEDGMENTS

The authors acknowledge Professor Gareth McKinley and Professor Pirouz Kaveh-

pour for fruitful discussions. This work was supported by MICINN MAT 2010-15101

project (Spain), by the European Regional Development Fund (ERDF), and by Junta de

Andalucıa P07-FQM-02496 project (Spain).

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