Document1 / February 7, 2016
ONDŘEJ WEIN, VĚRA PĚNKAVOVÁ AND JAROMÍR HAVLICA
End effects in rotational viscometry II.
GNF at elevated Reynolds number
Institute of Chemical Process Fundamentals ASCR, Rozvojová 135, 16502 Prague,
Czech Republic
Ondřej Wein: Email: wein@icpf.cas.cz; Phone: +420 220 390 222; Fax: +420 220 920 661
Abstract
Neglect of end effects in rotational viscometry introduces 10-30% error in the estimate
of shear stress at surface of rotating cylindrical spindle. Actual values of the correction coefficient on end effects cL in a real sensor depend on pseudoplasticity level of a given sample
(measured by flow behavior index n = 1/m) and inertia level (measured by Reynolds number
Re). The correction coefficients in sensors with coaxial cylinders are calculated by solving the
related axisymmetric boundary-value problem for the power-law model of viscosity function.
In addition to the cylindrical sensors according to ISO 3219 standard, also some clones with
different geometry are considered.
Keywords
Rotational Couette viscometry; Pseudoplastic fluids; End effects;
Contents
Introduction ................................................................................................................................ 2
Correction on end effects ......................................................................................................... 2
Solving the base integral equation............................................................................................ 3
The problem statement ............................................................................................................. 4
The boundary-value problem ..................................................................................................... 5
Torque calculations .................................................................................................................. 6
Numerical simulation ................................................................................................................. 7
Results ........................................................................................................................................ 8
Ideal Plastic material ................................................................................................................ 8
Inertialess flow of pseudoplastic fluids .................................................................................... 9
Inertial flow of pseudoplastic fluids ....................................................................................... 10
Experimental test .................................................................................................................... 11
Discussion and conclusions ...................................................................................................... 14
List of symbols ......................................................................................................................... 17
References ................................................................................................................................ 18
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Introduction
According to the manufacturers’ prospects, the contemporary computerized rotational viscometers guarantee reading/adjusting of the primary data - rotation speed  and torque C - with
relative accuracy about 0.2%. On the other hand, the primary data for real microdisperse fluids
may be seriously distorted by various inherent but irreproducible effects: mechanical, chemical,
and biological degradation, surface evaporation, viscous heating and other uncontrollable temperature changes. Probably for this reason and in agreement with common expertise (Whorlow
1980), the existing standard ISO 3219 for rotational viscometry of non-Newtonian fluids assumes rather low reproducibility 2% of the viscosity data even for Newtonian fluids.
Unfortunately, the actual accuracy of a resulting information on viscosity function for
non-Newtonian fluids is often much worse due to improper treatment of the primary data
(, C). There are two main sources of such systematic errors in rotational viscometry of inelastic non-Newtonian fluids that should be clearly specified and cured to obtain a proper and
reliable information about the viscosity function.
Correction on end effects
The first source of errors lies in an inaccurate correction of primary data on the end effects, i.e. on difference between ideal Couette flow kinematics and the flow in a real sensor.
This correction can be expressed through a coefficient cL in the formula for calculating shear
stress  R from actual torque C for the finite cylindrical spindle of a real sensor with inner radius
R, outer radius R/, and annular gap of a finite length l, see Fig. 1:
R =
C
.
2R 2 l c L
(1)
The recommended experimental procedure to suppress this error, see Whorlow (1980),
consists in making two measurements of torque, C1 and C2, under same rotation speed with the
sensors of same design, differing only by the lengths of the annular gap between the coaxial
cylindrical walls, l1 and l2:
C1  C 2
R =
.
(2)
2R 2 (l1  l 2 )
The drawback of such an approach consists in (a) additional work: manipulation with
two sensors, (b) enhanced error due to differentiation of the primary data: subtraction of two
almost equal quantities, (c) uncertainty about exact identity of two fillings of a microdisperse
sample (polymer solution, colloidal suspension). Note that the correction factor cL can be calculated from these doubled measurements,
cL =
1  l 2 / l1
.
1  C 2 / C1
(3)
In general, the correction coefficient cL depends on the sensor geometry, rotation speed,
and material properties. However, for Newtonian fluids at low Reynolds numbers, it depends
only on the sensor geometry and, hence, it can be embedded in the working formula (1) as a
calibration constant. Whorlow (1980) summarizes the experimental data on the end effects for
Newtonian fluids, obtained by simultaneous measurements with cylindrical spindles of different l, see also Lindsley (1947). ISO 3219 (1994) recommends the estimate cL  1.1 for Newtonian fluids in the standardized sensors and warns that cL can be as high as 1.3 for viscoplastic
materials.
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Attempts at specifying the end effects in rotational viscometry for non-Newtonian fluids
are mostly limited to the differentiation method employing Eq. (2), see e.g. Gucuyener et al.
(2002), Kelessidis et al. (2010), and they are restricted to a specific sample.
The end/edge effects could be also analyzed by solving the related flow equations for a
given geometrical configuration and a specified rheological constitutive model. Roscoe (1962)
gives an approximate analytic formula for the torque contribution due to end effects at a face
of semi-infinite cylindrical spindle rotating at Re = 0 in coaxial infinite cylindrical vessel, filled
with Newtonian fluid. For the class of pseudoplastic fluids with power-law viscosity function,
some numerical data on cL(n, Re) for specific geometry configurations are given by Wein et al.
(2006) and Wein et al. (2007).
Solving the base integral equation
The second source of errors is related to functional transformation of the pretreated primary data  (R,) onto the viscosity (fluidity) function [ ] =  []. Theory of ideal rotational Couette flow for a non-Newtonian fluid with known viscosity function provides a simple
expression for the experimentally accessible function (R,):
 (R,) = 12 
R
 2
[ ] d =
R
1
2
R
 
2
[ ] d ln  .
(4)
R
Mathematically, the treatment of primary data to desired information about the viscosity
function consists in solving the integral equation (4) with a known left-hand side. The problem
is further complicated by occurrence of a random noise, always included in the discrete primary
data on  (R,).
For treating the primary data according to this formula, the standard ISO 3219 recommends an approximation method of so called representative viscosity, see Giesekus et al.
(1977), calculated directly from the primary data with no additional information:
1  2
R,
r =
2
1  2
Ω  [ r ] ,
Dr =
1 2
r =
r
 [r].
Dr
(5abc)
There are many other ways of non-parametric approximate inversion of the integral equation (4), see e.g. Nguyen and Boger (1992), Yeow et al. (2000), Ancey (2005), which unfortunately start with additional tacit assumptions about the viscosity function. Several ones, e.g.
Mooney (1931), Code and Raal (1973), Wein et al. (2007), start with the well-known results
for the power-law viscosity function, [ ] = (/K)m :
(6)
Ω( R ,  )   R / K 1/ m / Φ(m) ,
Φ (m)  2m /(1   2m ) .
(7)
This parametric representation could be used for a non-parametric inversion method,
DR = (m) (R, )  [ R ]
(8)
where the effective value of m is estimated by differentiating primary data,
m  m(R, ) 
 ln Ω ( R ,  )
.
 ln  R
(9)
Another common approach to inversion of the functional (4) is based on representing the
viscosity function by a suitable empirical formula, which could be used for parametric leastsquare fitting and subsequent representing DR(R)  [ R ] through the applied model. The
accuracy and reliability of the parametric methods strongly depends on the choice of a suitable
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empirical model. No of the common wide-range models in the implicit form    [ ] – Carreau and Cross among others – does result in a simple formula or computational algorithm for
(R,  ; parameters) that would be suitable for a parameter least-square fitting. The form  =
[ ; parameters] should be preferred.
The local pseudosimilarity concept (Mitschka et al. 1976, Wein et al. 1981, Wein et al.
2007), suggests a combination of parametric and non-parametric methods that can be easily
implemented to PC programs for automated treatment of primary viscometric data. For a given
set of experimental points {, R, } and a given wide-range model [ ; parameters], the
estimates of apparent flow index m are calculated by differentiating analytically the model:
d ln [ R ]
d ln [ R ]
m  m*[R] 
= 1
.
(10)
d ln  R
d ln  R
With the known local values of m  m*[R], the non-parametric inversion formula (8) can be
used without numerically differentiating the experimental data and with preserving the smoothing power of parameter least-square fitting, necessary for distinguishing between random experimental errors and systematic errors, e.g. due to unsuitable empirical model.
Accuracy of the method depends basically on local pseudosimilarity approximation of
true radial profile of shear rate over the stress interval (2R, R). This possible source of errors
has been analyzed e.g. by Krieger and Elrod (1953), Code and Raal (1973) and Wein et al.
(2007). The accuracy of inversion is very good over the stress intervals, where the local flow
index m*[R] does not change too much, i.e. the viscosity (fluidity) function locally adheres
well to the power-law model.
The problem statement
Theory of rotational flow of non-Newtonian fluids, implemented in the software of contemporary PC-controlled viscometric instruments, should guarantee an accuracy of the mathematical treatment itself on the level, achievable for measurements with a calibration Newtonian
oil, i.e. about 0.2 - 0.5% in viscosities. In addition, such a software should allow to treat a
collection of primary data from several sensors. The local pseudosimilarity concept offers such
opportunity by including a broad choice of empirical models for parametric representing of the
viscosity (fluidity) function. Kernel of the approach is the solving of related boundary-value
problems on axisymmetric flow of power-law fluids in real sensors.
Results of the numerical simulation for geometrically similar sensors (e.g. the series of
HAAKE Z40-Z20-Z10 Couette sensors according to ISO 3219 standard) can be represented by
a function
C/CL = cL(n, Re).
(11)
In our foregoing report, Wein et al. (2007), this function has been presented only for the Newtonian fluids or zero Re (inertialess flow). The results are given in the present work, which
cover the complete domain of 1/m = n  1 and Re < 1000 for laminar flow of pseudoplastic
fluids. In addition, the domain of cL is extended for some clones of the standardized Couette
sensors with different H = h/R << 1, by considering the empirical formula
C/CL = cL(n, Re, H).
(12)
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The boundary-value problem
The correct torque C - at given rotation speed  in a real viscometric sensor with the
geometry specified by the wall contours  in meridional plane (r, z) for the fluid properties,
specified by density  and viscosity function [D], is determined by solving the related boundary-value problem. The considered geometry is sketched in Fig. 1.
Flow equations express the differential momentum balance for axisymmetric flow of an
incompressible Generalized Newtonian Fluid with the power-law viscosity function,
 = [D] = K Dn-1,
(13a)
where D stands for the second invariant of the rate-of-deformation tensor. The corresponding
local dissipation intensity:
 =  [D]   D = K Dn+1.
(13b)
The boundary conditions in cylindrical coordinates (, r, z) and corresponding velocity
components (v = r, vr, vz) are specified at the contours of the bounding surfaces, see Fig. 1:
S – Spindle rotating with a speed  = ; fluid adhering to the walls,
V - fixed Vessel,  = 0; fluid adhering to the walls,
F - planar Free surface; flow with zero traction, z(, vr, vz) = 0,
A- Axis (a collapsed singular surface); axial symmetry. r(, vr, vz) = 0,
L – part of spindle, related to ideal Couette flow in annular gap of Length l,
F
R/
V
Fig.1. Flow domain and bounding contours
(in a meridional plane r, z)
for a real sensor
S
R
S
l
L
A
z
r
V
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Torque calculations
The problem to be solved should provide the field of deformation stress components ij(r, z),
and the corresponding scalar field of the dissipation intensity, (r, z). It follows from the macroscopic momentum balance that the total torque C can be calculated in three various ways –
from the angular shear stresses at the spindle surface,
CS = 2

ΓS
r 2 ( r dr   z dz ) ,
(14a)
from the angular shear stresses at the vessel surface,
CV = 2

ΓV
r 2 ( r dr   z dz ) ,
(14b)
or from the dissipation intensity throughout the fluid volume,
2
Ψ (r , z ) rdr dz .
CD =
(14c)
Ω domain
Comparison of these three expressions for the total torques is applied in the following for the
accuracy estimates. In addition, torque for the ideal Couette flow of a power-law GNF fluid is
known:
CL = 2

ΓL
r 2 dz = 2 K  n R3 L  1/ m(m).
(14d)
In a close analogy to the hydrodynamics of Newtonian fluids, the inertia effect can be
characterized by a single similarity criterion, generalized Reynolds number:
Re =
 R 2 Ω 2 n
K
 R 2 Ω 2 1/ m
Φ ( m) .
=
R
(15ab)
Some general conclusions about the torque at elevated Re can be drawn from the similarity theory for the rotational flows of power-law GNF under various rheodynamic regimes,
see e.g. Wein et al. 1981. In particular, for any inertialess rotational flows in geometrically
similar situations:
C/(K nR3) = k0(n),
Re = 0.
(16a)
For the rotational flows with secondary inertia effects:
C/(K nR3) = k0(n) +k2(n) Re2,
Re << 1.
(16b)
For the rotational laminar boundary-layer flows:
C/(K nR3) = k(n) Ren/(1+n), Re >> 1.
(16c)
By combining these asymptotic estimates, an empirical formula for wide-range correlation
could be suggested,


a2 (n) Re 2
,
C/(K nR3)  k0 (n)1 
( 2 n ) /(1 n ) 
1

a
(
n
)
Re



(16d)
where the coefficients k0, k2, k, a2 = k2/k0, a = k2/k for a fixed n are either constants or functions of geometry similarity simplexes, like H = h/R in the case under consideration.
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Numerical simulation
The subject of the present study is the flow of power-law fluids in a class of real sensors
according to ISO 3219 and some clones. Their dimensions are specified in Table 1.
Table 1. Real geometry of the viscometric sensors according to ISO 3219
(HDIN = 0.08470.0004) and its clones, see Fig. 1 and Fig. 2
DIN sensors
Clones Computation
Z40-Z20-Z10
2H
3/2 H
1H
½H
R [mm]
20-10-05
20-10-05 20-10-05
20-10-05
20-10-05
H = h/R
HDIN
2 HDIN
1.5 HDIN
HDIN
0.5 HDIN
L = l/R
3
3
3
3
3
Q = RS/R
0.3
0.3
0.3
0.3
0.3
LF = lF/R
1
1
1
1
1
LB = lB/R
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
sin 
, C
(a)
, C
(b)
lF
RS
RS
l
lB
60
60
Fig. 2. Effect of pseudoplasticity on the flow
kinematics in a real viscometric sensor.
Thin lines – contours of viscometric surfaces
graded by /100;
thick lines – contours of viscometric surfaces
graded by /10;
horizontal dashed straight lines – splitting the
total sensor into three parts,
with the middle part of length l,
corresponding to the ideal Couette flow.
(a) Clone 2HDIN. n = 1, Re = 0.
(b) Clone 2HDIN. n = 1/6, Re = 0.
R
R
h
h
Because of some doubts about actual accuracy of the previous solutions, Wein et al.
(2007), two independent method of numerical solution were used:
(a) The home-made software Prog_cL_m, using the standard finite-difference method
with homogeneous grids, is applied for the inertialess 2D axisymmetric flow, Re = 0, and
(b) The commercial CFD (Computational Fluid Dynamics) software ANSYS FLUENT
14.0 is applied for solving the flow equations of 2D axisymmetric flow also at finite Re.
Details of the numerical algorithm in Prog_cL_m have been already mentioned (Wein et
al. 2007) and the software is freely available (Wein 2012). An essential feature of the program
is the splitting of the overall sensor onto three parts, as indicated in Fig. 2 by the dashed horizontal lines. The upper and lower parts (semi-sensors) include only a short entrance region of
the gap, usually about triple of the gap thickness. In the middle part, including just a finite
section of the ideal endless annular gap, no numerical solving is needed. Density of the grid
with square meshes is chosen by selecting MH, the number of intervals across the gap.
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The operating of CFD software, on the contrary, uses a grid, covering the overall flow
domain including the entire gap. The typical features of the optimized inhomogeneous grid:
(a) large longitudinal and small radial steps inside the gap, (b) the smallest steps in proximity
of the walls, (c) gradual rising steps (diluting grid) far from the walls. Grid at the edge singularities corresponds to MH = 80 of the Prog_cL_m.
For the inertialess rotational flow, Re = 0, which exactly keeps assumptions of the viscometric flow kinematics (Pipkin and Tanner 1972), the velocity field is completely specified by
angular speeds,  = (r, z), i.e. the only non-zero deformation stress components are
r = rr z = rz
(17ab)
where  = [D], D = r ((r)2 +(z)2)1/2,  = ((r)2+(z)2)1/2. The local momentum balance
under the assumed symmetries reduces to a non-linear elliptic 2D equation:
r(r2r) +z(r2z) = 0.
(18)
The related boundary-value problem has been solved using the both numerical methods, which
provide the resulting torques CS, CV, CD, calculated according to the equations (14abc).
Centrifugal effects generate radial pressure gradient. If not completely compensated by
reactions of the solid walls, it gives rise to a secondary meridional flow at lower Re and to
laminar boundary-layer flow at higher Re. The corresponding 2D axisymmetric flow has been
tackled using the CFD software with the same grid as for Re = 0. Again, this method provides
the three independent estimates CS, CV, CD of the torque for estimating the accuracy.
Results
Ideal Plastic material
Kinematics of the rotational viscometric flow is strongly affected by the level of pseudoplastic
behavior, characterized by the flow-behavior index n = 1/m. An example of the resulting viscometric shear surfaces for the Newtonian behavior, n = 1, and strongly pseudoplastic behavior,
n = 1/6, is shown in Fig. 2. The viscometric shear surfaces, (r, z) = const, are concentrating
closer to the spindle with decreasing flow behavior index n. The asymptotic case of n = 0 corresponds to motion of the Ideal Plastic material, which stays undeformed in the bulk and creates
a slip surface at location with the maximum shear stress, which cannot exceed the critical value
y, specific for a given material. In the cases under consideration, the slip surface is located
very close to the spindle surface and adheres to all its convex parts (Wichterle and Wein 1978).
For the sensors under consideration, the slip surface deviates from the spindle wall only slightly
in a close proximity to connection of the cylinder to the shaft. The resulting torque estimate
according to (14a) is
CS /2y =
Γ
S
r 2 ((dr ) 2  (dz ) 2 )1/ 2  RS3 (0.5407  0.5 RS / R) ,
(19)
where the additional term in parentheses corresponds to a slight deviation of the slip surface
from the solid wall due to the spindle non-convexity. Simple calculations for the Z40-Z20-Z10
sensors give cL = 1.263 in a reasonable agreement with the upper estimate of cL for viscoplastic
materials, noted in ISO 3219. The correction on the non-convexity corresponds to less than
0.4% of the total torque.
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Inertialess flow of pseudoplastic fluids
The numerical simulations were taken for all the sensors listed in Table 1 and for six
values of the flow index, n = {16 , 2 6 , 3 6 , 4 6 , 5 6 , 6 6} . The individual estimates of the total torque
are expressed in the normalized form
cL(S)  CS/CL -1, cL(V)  CV/CL -1, cL(D)  CD/CL -1.
(20abc)
Accuracy of the torque calculations in Prog_cL_m was estimated (a) by comparing results (20abc), (b) by halving intervals, i.e. by doubling MH. The grid density was adjusted by
selecting very small initial MH = 4 and then step-by-step doubling it to MH = 256. Note that for
the standard gap, H = HDIN, this last grid contains as many as 4 106 knots for each semi-sensor.
The results are shown in Fig. 3.
(a)
Z40, n = 6/6, Re = 0
(b)
0.092
Z40, n = 2/6, Re = 0
0.200
 cL
(V)
(D)
(S)
(V)FLUENT
(D)FLUENT
(S)FLUENT
0.090
0.088
0.086
 cL
(V)
(D)
(S)
(V)FLUENT
(D)FLUENT
(S)FLUENT
0.196
0.192
0.188
0.084
10
100
(c)
0.184
MH
10
MH
Z40, n = 1/6, Re = 0
0.244
 cL
(V)
(D)
(S)
(V)FLUENT
(D)FLUENT
(S)FLUENT
0.240
0.236
0.232
0.228
0.224
10
100
100
Fig. 3. Comparing various estimates of cL for
inertialess flow in Z40 sensor.
Circles – by Prog_cL_m;
Squares – by CFD software;
(V), (D), (S) – calculations
according to Eqs. (20abc);
(a) n = 1; (b) n = 1/3; (c) n = 1/6.
MH
There are several particular conclusions about the accuracy, which are illustrated by the data
shown in Fig. 3:
 Differences between the individual estimates of cL grow with decreasing flow behavior
index n, the best accuracies are reached for Newtonian fluids, n = 1.
 The differences do not exceed 0.004 for n = 1 and 0.015 for n = 1/6.
 The differences among (V), (D) and (S) estimates according to Eqs. (20abc) are larger than
the differences due to halving steps (increasing MH).
 The (V) estimate differs extremely from the other estimates for very low n. This is understandable, as this estimate loses any sense for n = 0.
 Accuracy of any estimate is not markedly improved by halving steps above MH = 64.
 The differences among (V), (D) and (S) estimates are larger for the CFD software than for
Prog_cL_m.
 The best agreement between the both methods is found for (D) version, i.e. for integration
of local dissipation intensities over the total domain in (r, z) plane.
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In general, the both numerical methods leave unsolved some uncertainty of cL, probably due to uncured singularity at the edge of cylindrical spindle. The deviations of the individual (D) estimates from the average do not exceed 0.002 (0.2% of total torque).
The final (D) estimates of cL0(n, H), shown in Fig. 4 by solid points, were fitted to polynomial formula,
cL0(n,H) = a00 +n (a10 + a11 H+ a12 H2)+ n2 (a20 + a21 H+ a12 H2)
(21)
where
 a00 a10 a20   0.2629  0.4929 0.2501 

 

a11 a21   
3.7056  2.7607  .


a12 a22  
 9.0813 7.3859 

(22)
0.3
cL0
0.2
H ==22 HDIN
H
HDIN
H ==1.5
H
3/2HDIN
HDIN
H ==11 HDIN
H
HDIN
H
1/2HDIN
HDIN
H ==0.5
0.1
0.0
0
0.5
n
1
Fig. 4. Final corrections on end effect for the inertialess flow, Re = 0, of pseudoplastic fluids,
0 < n ≤ 1, in the all sensors, listed in Table 1.
Points – numerical data on cL0(n, H); Lines – polynomial fit (21).
Inertial flow of pseudoplastic fluids
Using CFD software, the related boundary-value problem has been solved over the full range
of parameters n, Re only for the DIN (Z40-Z20-Z10) sensors, H = HDIN, which are geometrically similar and, hence, they differ only by Re at a given rotational speed.
In addition, the clones H = 2 HDIN and H = ½ HDIN over the same range of
Re = (11000) were considered to study the effect of varying H (Wein et al. 2007). Statistical
treatment of the obtained numerical data has shown that the effect of varying H can be included
by modifying the correlation formula (16d) to the form
cL = 1 + cL(n, Re, H) = 1 + cL0(n, H) E(n, Re, H)).
(23)
Nature of the differences among individual (V), (D), (S) estimates of cL, suggested in
Fig. 3 was not affected by changing Re. Therefore, the (D) estimates were used in further data
treatment. The full set of numerical results for the DIN sensors, shown in Fig. 5, has been statistically treated using the empirical formula (23) with
E(n, Re, H) = 1 
(a21n  a22n 2  a23n3  a24n 4 ) H Re 2
,
1  (a1n  a 2 n 2  a3n3 ) Re ( 2n) /(1n)
10
(24)
Document1 / February 7, 2016
where cL0(n, H) has been specified in Eq. (21) and
 a 21

 a 22
a
 23
a
 24
a1 
 11.2


a 2 
 4   14.2
 10 
a3 
2.00


 2.62



5.35 

 7.20 
.
4.4 



(25)
The final results of numerical simulations for the sensors according to ISO 3219 are compared with the empirical formula (23) in Fig. 5.
0.3
 cL
n=0
n = 1/24
n = 1/6
n = 2/6
n = 3/6
n = 4/6
n = 5/6
n = 6/6
0.25
0.2
0.15
0.1
0.05
1
10
100
Re
1000
Fig. 5. Correction on the end effects for pseudoplastic inertial flow in real sensor Z40
according to ISO 3219 standard.
Points - numerical data on cL0(D) obtained using CFD software;
Lines – fit by the formulas (21), (23), (24).
Experimental test
An experimental testing of the theory has been conducted on the HAAKE RheoStress RS600
device with three DIN sensors Z40, Z20, Z10. The Newtonian silicone calibration oils by
Brookfield Inc., see Table 2, were used in a way, analogous to a standard calibration protocol.
Table 2. Silicone calibration oils at 25C by Brookfield Inc.
Name
 [ kg/m3]  [mPas] by
 [mPas] by
capillary viscometry
rotational viscometry
Oil 10
933.9
9.33
9.30  0.1
Oil 100
964.3
99.2  0.2
99.6
11
Document1 / February 7, 2016
The process control and primary data acquisition were accomplished using the software
HAAKE RheoWin (2007) within two operating modes CS/CR, programmed to go roughly
through the sequence of rotation speeds {10-20-50-100-200} RPM upward and then downward.
The obtained time series at directly controlled C (CS series) or under feedback control for 
(CR series) were treated using the homemade software AWSWork_2013_5 to provide the individual points {, C} under steady flow condition. Report on this treatment is shown on selected examples in Table 3 and Fig. 6.
Table 3. Report on statistical treatment of runs (time series), shown in Fig. 6
Fig. 6 Job (data segment)
Torque r.m.s.
Speed
r.m.s.
[RPM] [%]
[ N.m] [%]
Mode
Oil 10, Z20(CR), Seg. 1
2.80
31.3
10
Y/AR
(a)
Oil 10, Z20(CR), Seg. 5
55.2
0.6
200
Y/AR
(b)
Oil 10, Z20(CS), Seg. 2
5
17.7
4.6
Y/ES
(c)
Oil 10, Z20(CS), Seg. 4
25
87.0
0.3
Y/ES
(d)
Y/AR - the controlled shear rate (CR) time series, treated by averaging procedure
Y/ES - the controlled shear stress (CS) time series, fitted by exponential extrapolation
The relative errors of individual jobs are characterized in Table 3 by relative root mean
squares (r.m.s.) of the resulting C -  points. Rather bad statistical results were selected here to
show limited accuracy of the measuring of low-viscosity samples at low rotation speed, i.e. at
very low torque reading. This problem becomes critical for the devices like HAAKE RS600 in
CR automated regime, because the torque is being adjusted via a feedback loop until PC control
in effort to keep the rotation speed, prescribed in the software. Such oscillations of torque are
apparent e.g. in Fig. 6a. The typical feature in CS regime is an exponential start of the rotation
speed after the programmed step change of torque. In most of the cases, not shown here, the
estimated errors of individual points were lower than 1 %.
(a) Oil 10, Z20(CR)
(b) Oil 10, Z20(CR)
(c) Oil 10, Z20(CS)
(d) Oil 10, Z20(CS)
Fig. 6. Time course of selected runs during viscosity measurements of Oil 10 (see Table 2 and
Table 3) on HAAKE RS 600 in automated regimes with controlled staircase course of torque
(CS) or speed (CR).
12
Document1 / February 7, 2016
The resulting statistically pretreated primary data C = C() for the both calibration oils
are presented in the form cL – Re, see Fig. 7. The corresponding estimate of CL is calculated
according to Eq. (14d) for Newtonian fluids, with experimental values of  taken from capillary
viscometry, see Table 2:
CL = 2  ΩR3 L
1 2
.
2
(26)
Z40(CS) Oil100
Z20(CS) Oil100
1.2
Z10(CS) Oil100
C/CL
Z40(CS) Oil100
Z40(CS) Oil10
Z40(CR) Oil10
1.1
Z20(CR) Oil10
Z10(CR) Oil10
theory
ISO 3219
1.0
1
10
100
Re
1000
Fig. 7. Comparison of actual C- data from rotational viscometry with theoretical prediction
of the correction on the end effects for Newtonian fluids (n = 1). The used standard sensors
are specified in Table 1.
As shown in Table 3 for a few selected samples, each point of the pretreated experimental
data is characterized by its own estimated error. With a few exception in the series Z10(CR)
Oil10 and Z20(CR) Oil10, these errors were below 1 %. For these points, the deviation between
theoretical curve and experiment are lower than the estimated experimental error. The data with
low-viscosity oil and the smaller sensor, i.e. with larger error, are included in Fig. 7 to demonstrate the effect of primary data uncertainty on resulted estimate of cL.
13
Document1 / February 7, 2016
Discussion and conclusions
An improved computer-aided method of treating the primary data in rotational viscometry has
been suggested. In comparing with the common software for on-line treatment of primary viscometric data, e.g. RheoWin of HAAKE inc. , it provides several novel features:
(a) The local pseudosimilarity approach approximates viscosity function for a single experimental point, i.e. over a narrow interval (2R, R), by local power-law (logarithmically
linear) course, characterized by local flow-behavior index m according to Eq. (10).
(b) End-effect correction cL = cL(m, Re, H) according to formula (1) is taken from a numerical
solution to the mathematical model of pseudoplastic flow in the real sensor under real flow
condition.
(c) Inversion operator, i.e. solution of the base integral equation (4), uses the concept of local
pseudosimilarity: Each particular point of primary data (, C; sensor) is converted according to formula (8) to the corresponding pair {DR, R}.
(d) Parameters of the chosen broad-range model of fluidity function, [; parameters], are determined by non-linear least-square fitting aimed at minimizing the weighted root-mean
square of relative deviations, RMS(parameters) = (i m2(DR,i, R,i)/(ii)1/2. The difference
m(D,) is redefined to characterize the relative distance of an individual point {D, } from
the assumed viscosity function,
m(D, ) 
ln ([ ] / D)
ln ([ ] /( D /  ))
=
.
*
2 1/ 2
(1  (m [ ]  1) )
(1  (m*[ ]  1) 2 )1/ 2
(27)
For Newtonian fluids, m = 1, [] = const, it reduces to commonly known relative error for
the viscosity,
1(D, )  ln ([ ] / D) = ln(/[]).
(28)
(e) Two six-parameter broad-range models with power-law intermediate asymptotes, [] 
mm-1, were designed to fit the fluidity data about purely viscous fluids, [  0]  0,

[] = 0r  [1  (0 /  ) r ] /[1 / q  1 /(m m1 ) q ]r / q
1/ r ,
(29a)
and viscoplastic materials, [  y]  0,
q
[] = ([m m1 (1   y /  ) ]r  r ) 1/ r ,
(29b)
see Wein et al. (2006), Wein and Pěnkavová (2014).
The method is freely available as the EXE file AWSWork_2013_5 for downstream treatment of primary viscometric data on the TXT files, generated by HAAKE RheoWin.
Traditional estimates of relative errors, based on the ratio of viscosity values, 1(D, ),
are rather overrated. In particular for pseudoplastic materials, the systematic error of viscometric results due to method of representative viscosity according to ISO 3219 approaches infinity
for n  0, while m(Dr,r )  ln(r/R)  -(1 - ) for this asymptotic case, see Fig. 8. The
systematic error of viscometric results due to pseudosimilarity method is much lower. In particular, it is zero for materials with the power-law viscosity function, 0  n  1.
The both error estimates for simulations with the Bingham model are compared in Fig. 9
for representative method and in Fig. 10 for pseudosimilarity method. Even the estimate
1(DR,R) remains finite at R/y  1 and m(DR,R) is smaller than 1%, excepting a narrow
interval of R/y close to 1.
14
Document1 / February 7, 2016
Relative difference
0.1
1-
0.08
1(Dr,r)
0.06
m(Dr,r)
0.04
0.02
0
power-law
-0.02
0.001
0.01
0.1
n
1
Fig. 8. Systematic errors due to “representative viscosity” approximation of the inverse
operator for power-law fluids in the standard ISO 3219 sensor.
Relative difference
0.04
1(Dr,r)
0.02
0.00
-0.02
m(Dr,r)
Bingham
-0.04
0.01
0.1
1
r/y -1
10
Fig.9. Systematic errors due to “representative viscosity” approximation of the inverse
operator for Bingham viscoplastic material in the standard ISO 3219 sensor
.
15
Document1 / February 7, 2016
Relative difference
1.E+0
1(DR,R)
1.E-1
m(DR,R)
1.E-2
1.E-3
Bingham
1.E-4
0.01
0.1
1
R/y-1
10
Fig. 10. Systematic errors due to “pseudosimilarity” approximation of the inverse operator for Bingham viscoplastic material in the standard ISO 3219 sensor
The second important source of systematic errors is a wrong estimate of cL(m, Re), which
results in the wrong estimate of R. This error corresponds to ln(cL,correct/cL,wrong), i.e. to
ln(1.3/1.1)  17% for ISO 3219 treatment of the viscoplastic materials under conditions
R/y  1, i.e. r/y  2/(1+2). As it has been discussed in our previous work, Wein et al.
(2007), the maximum deviation of cL estimates for the power-law model from the estimates by
simulations for the Cross model do not exceed 0.5 % when 1/m* > 0.5 and 5 % when
1/m* > 0.05. The origin of such a large uncertainty is a fast local change of m*[R], not its high
value. In most cases, encountered in viscometric practice, the error m(DR,R) due to incorrect
estimate of cL(1/m*, Re) does not exceed 0.5%.
In final conclusions:
(a) Pseudosimilarity method of downstream treating the primary viscometric data in
combination with proper correction on the end effects guarantees better accuracy of
the resulting viscosity function than other existing methods.
(b) Freely available software AWSWork_2013_5.exe supports the pseudosimilarity
method in rotational viscometry for cylinder-cylinder, disk-disk and cone-plate, but
its input is still limited to text files generated by HAAKE RheoWin.
(c) The pseudosimilarity correction coefficients, cL(m, Re) were presented here for the
sensors according to ISO 3219 with clones, but cL can be calculated also for other
geometries using the software Prog_cL_m.exe.
Acknowledgements
This work was supported by the Grant Agency of the Czech Republic under contract No
P105/12/0664.
16
Document1 / February 7, 2016
List of symbols
total torque for a real sensor, Pa m3
numerical estimates of total torque; Eq. (14abc)
total torque for ideal Couette flow, Pa m3
correction on end/edge effects; Eq. (1)
deformation rate, experimental estimate to  [], s-1
correction on end effects at Re = 0
correction on end effects at a finite Re
gap thickness, m
normalized gap thickness
consistency coefficient for the power-law viscosity function, Pa sn
length of the gap between coaxial cylinders; Fig. 1, m
normalized length of the gap
flow index for the power-law viscosity function
apparent flow index; Eq. (9)
local flow index; Eq. (10)
radius of the spindle (inner rotating cylinder), m
shaft radius, m
weighted root mean square of deviations
Reynolds number; Eq. (15)

vertex angle of the spindle cone; Fig. 2
 =  []
viscometric shear rate, viscosity function, s-1

spindle contours in a meridional plane; Fig. 1
m,
error of viscosity function for non-Newtonian fluids; Eq. (27)
1
error of viscosity value for Newtonian fluids; Eq. (28)
  ]/  viscosity, Pa s
 =  []/ fluidity, Pa-1s-1
 = 1/(1+ H), ratio of radii of the inner to outer cylinder; Fig. 1

shear stress, Pa
y
yield stress of viscoplastic materials, Pa
R
shear stress at the spindle surface for ideal Couette flow, Pa
r
representative shear stress; Eq. (5a)
ij
Cartesian components of the stress tensor
{, r, z}
polar cylindrical coordinates
(r,z)
meridional field of angular speed in fluid, rad s-1

angular speed of the spindle, rad s-1

dissipation intensity, J m-3s-1; Eq. (13b)
(m) = 2m/(1-2m), shear rate factor in local pseudosimilarity; Eq. (8)
C
CS, CV, CD
CL = C/cL
cL = 1 + cL
D, Dr, DR
cL0(n,)
cL(n, Re)
h
H = h/R
K
l
L = l /R
m, n = 1/m
m(R)
m*[]
R
RS
r.m.s.
Re
[..]
representation of a material function
Subscripts
r
R
S, V, D
representative values r, Dr, r = r/Dr (the optimized consistency variables)
correct values R, DR, R = R/DR (a point on the viscosity function)
numerical estimates of torque (Spindle, Vessel, Dissipation)
17
Document1 / February 7, 2016
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Document1 / February 7, 2016
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