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Rheol Acta (2005) 44: 270–277 DOI 10.1007/s00397-004-0407-2

Y. Leong Yeow Dolly Chandra Albert A. Sardjono Hartono Wijaya Yee-Kwong Leong Ash Khan

Received: 6 February 2004 Accepted: 21 June 2004 Published online: 14 August 2004  Springer-Verlag 2004

Y. L. Yeow (&) Æ D. Chandra A. A. Sardjono Æ H. Wijaya Department of Chemical and Biomolecular Engineering, The University of Melbourne, Victoria, 3010, Australia E-mail: [email protected] Y.-K. Leong School of Engineering, James Cook University, Townsville, 4811, Queensland, Australia A. Khan Department of Chemical and Civil Engineering, RMIT University, Victoria, 3000, Australia

ORIGINAL CONTRIBUTION

A general method for obtaining shear stress and normal stress functions from parallel disk rheometry data

Abstract The problems of converting the torque and normal force versus rim shear rate data generated by parallel disk rheometers into shear stress and normal stress difference as functions of shear rate are formulated as two independent integral equations of the first kind. Tikhonov regularization is used to obtain approximate solutions of these equations. This way of handling parallel disk rheometer data has the advantage that it is independent of the rheological constitutive equation and noise amplification is kept under control by the userspecified parameter in Tikhonov regularization. If the fluid under test exhibits a yield stress, Tikhonov

Introduction Parallel disk and small-angle cone and plate rheometers are used to determine the shear stress versus shear rate and normal stress versus shear rate relationships of a wide range of fluids. The small-angle cone and plate rheometer has the advantage that experimental data, in the form of measured torque and normal force at different rotational speeds, can be converted directly into shear stress and first normal stress functions using very simple algebraic expressions that are independent of the rheological constitutive equation. These expressions are based on the small-angle approximation which means that the shear rate and consequently the shear and normal stresses are essentially uniform and known throughout the fluid sample under test. For the parallel

regularization computation will simultaneously give an estimate of the yield stress. The performance of this method is demonstrated by applying it to a number of data sets taken from the published literature and to laboratory measurements conducted specifically for this investigation.

Keywords Non-Newtonian viscosity Æ Normal stress Æ Inverse problem Æ Tikhonov regularization Æ Integral equation

disk the situation is quite different. The shear rate varies from zero at the centre of the disks to a maximum value at the rim. This variation can be significant and therefore using a representative average shear rate, as is often done in the case of narrow gap Couette viscometer, may not be a good approximation [1]. Variation in shear rate means that the shear and normal stresses also show significant radial variation across the disks. While the simple kinematics of the parallel disk flow field allows the radial variation of the shear rate to be determined, for a general non-Newtonian fluid the radial profiles of the shear and normal stresses are not known. Converting the measured torque and normal force into the corresponding shear and normal stresses is the main source of difficulty in processing parallel disk rheometry data.

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A number of procedures have been developed to overcome this difficulty. The simplest is to assume that the fluid under test can be approximated as a Newtonian fluid, which allows the shear stress and the normal stress prevailing at the rim of the disks and hence the material property functions at the corresponding shear rate to be determined. This simple procedure is used by many of the software that accompanies the present generation of parallel disk rheometers. Another approach is to start from the exact equations relating the shear stress and the normal stress at the rim of the disks to the shear rate at that location. These equations give the stresses in terms of the measured torque and normal force and their first derivatives with respect to the rim shear rate. The main difficulty in using these equations lies in the evaluation of these derivatives, as direct differentiation of experimental data often leads to unacceptably large noise amplification. In another approach, the power-law constitutive equation is assumed. It can then be shown that, for a fairly wide range of the power-law index, the shear and normal stresses in the test fluid at a carefully chosen radial position can be approximated to a high degree of accuracy by its Newtonian counterpart [2]. This method, often referred to as the single-point method, allows the stresses to be determined with relative ease and is particularly useful for the computation of material property functions in real time, as is often required in the control of industrial processes. The mathematical problem of converting capillary or Couette viscometry data into shear stress versus shear rate functions can be formulated as an integral equation of the first kind. This class of integral equations is illposed in that unless care is taken, the noise in the experimental data is likely to be amplified leading to unreliable results [3]. Tikhonov regularization is a general and reliable method for solving this class of integral equations [4, 5]. The aim of the present investigation is to show that this general approach, with appropriate modifications, can also be applied to parallel disk rheometry data. In this case, not only the steady shear stress function but also the function that relates the difference between the first and second normal stresses to shear rate can be obtained from the rheometry data. Besides the additional computation involving the normal force, the parallel disk rheometry problem also differs from the previously considered viscometry problems in that the roles of stress and shear rate are reversed. In capillary and Couette viscometry the shear stress at any radial position within the fluid under test is known and is the independent variable of the integral equation. In the parallel disk the independent variable is the shear rate which is known at any radial position. The dependent variables are the shear stress and the difference between the first and second normal stresses. This reversal of roles does not change the nature of the integral equations and therefore should not affect the

effectiveness of Tikhonov regularization as a method of solution. The advantages of this method compared to other methods of processing rheometry and viscometry data should therefore also be retained. These will be verified by applying the method to a number of parallel disk data sets taken from the literature and to data obtained specifically for this investigation.

Equations of parallel disk rheometer Parallel disk rheometry data are in the form of a set of measured torque G and normal force F versus rotational speed x data points. It is common practice to express the rotational speed in the form of shear rate at the rim c_ R ¼ xR=h: R is the radius of the disks and h is the gap between the disks. The data thus take the form (_cR1 ¼ min ðc_ R Þ; G1m,F1m), (_cR2 ; G2m, F2m), ...(_cRi ; Gim, m Fim),... ...(_cRND ¼ Maxðc_ R Þ; Cm ND ; FND ),.... ND is the number of data points. Typically ND is of the order of 10–20. Superscript m is used to denote experimentally measured quantities. The equations relating these measured quantities to the material property functions are [1] 2pR3 C ¼ 3 c_ R c

pR2 F ¼ 2 c_ R c

Zc_ R

sðc_ Þ_c2 d_c:

ð1Þ

0

Zc_ R

½N1 ðc_ Þ  N2 ðc_ Þ_cd_c

ð2Þ

0

Superscript c is used to distinguish the computed torque and normal force from their experimentally measured counterparts. sð_cÞ is the unknown shear stress function and ND ð_cÞ ¼ N1 ð_cÞ  N2 ð_cÞ is the unknown difference between the first and second normal stress functions to be extracted from the rheometry data. Both unknowns are regarded as well-behaved smooth functions of the local shear rate c_ . Both Eq. 1 and Eq. 2 are integral equations of the first kind and reveal the ill-posed nature of the problem [3].

Methods To apply Tikhonov regularization the shear rate interval, from 0 at the centre of the disk to the maximum rim shear rate max ðc_ R Þ; is divided into NK uniformly spaced points. NK is usually much larger than the number of data points ND. Typically NK is of the order of 201 to 601. For data sets where the spread of c_ R is large, typically when the ratio c_ RND : c_ R1 > 103 , an even larger NK becomes necessary if a high degree of accuracy is required at low c_ . In discretized form, the unknown functions sð_cÞ and ND ð_cÞ are represented by the

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unknown column vectors s ¼ fs1 ; s2 ; s3 ; . . . ; sj ; . . . sNK gT and N D ¼ fND1 ; ND2 ; ND3 ; . . . ; NDj ; . . . NDNK gT . In terms of these discretized variables Eq. 1 and Eq. 2 become CC i ¼

NK 2pR3 X aij sj c_ 2j D_c c_ 3Ri j¼1

ð3Þ

FiC ¼

NK pR2 X aij NDj c_ j D_c c_ 2Ri j¼1

ð4Þ

where aij are the numerical coefficients arising from the discretization of the integrals in Eq. 1 and Eq. 2. For example if Simpson’s 1/3 rule is used in the numerical integration then ai1=1/3, aij=2/3 for odd j, aij=4/3 for even j and aij=0 for c_ j > c_ i : If c_ i does not coincide with a discretization point, then the last three of the non-zero aij will have to be evaluated by interpolation to correct for fractional step size. In matrix notation, Eq. 3 and Eq. 4 take the form Cc ¼ As

ð5Þ

minimized. It can be shown that the s and ND that minimize the linear combinations are [3]  1 s ¼ AT A þ kbT b AT CM ð7Þ  1 N D ¼ BT B þ kbT b BT F M

ð8Þ

b is the tri-diagonal matrix with rows of the general form (0, 0,..... 0, 1, )2, 1, 0,..... 0, 0). This arises from the standard finite difference approximation of second derivatives. k is an adjustable weighting/regularization factor. A large k favors 2, giving smooth material property functions at the expense of accurate representation of the experimental data. On the other hand a small k favors 1, leading to a close match with the experimental data but the resulting property functions may exhibit excessive and physically unreal fluctuations. The present investigation adopts a simple practical approach in locating k. It is chosen so that the average deviation between the measured and computed G and F are of the same order of magnitude as the estimated error bars of the experimental data.

F c ¼ BN D

ð6Þ   3 2 3 2 where  2  Aij ¼ 2pR aij c_ j D_c= c_ Ri and Bij ¼ pR aij c_ j D_c= c_ Ri : The unknown column vectors s and ND are chosen so that they 1. Minimize the sum of squares of the difference between the measured and computed torque in Eq. 5 and that of the difference between the measured and computed normal force in Eq. 6 to ensure accurate representation of the experimental data. 2. Minimize the sum of squares of the second derivatives of sð_cÞ and of ND ð_cÞ at the internal discretization points to ensure that the resulting material property functions are smooth. In Tikhonov regularization, instead of minimizing 1 or 2, a linear combination of these two requirements is

Fig. 1a, b Data and results of hydroxypropyl methylcellulose solution. a Torque versus rim shear rate data. Points are the data of Steffe [6] and the curve is back-calculated from the results of Eq. 7. b Shear stress shear rate relationship. The continuous curve is given by Eq. 7 and the points are from Steffe [6] based on a power-law model

Results and discussion Aqueous solution of hydroxypropyl methylcellulose Steffe [6] obtained the torque versus rim shear rate data for a 3% aqueous solution of hydroxypropyl methylcellulose at 24.2 C in a parallel disk viscometer with R=25 mm and h=0.70 mm. His data, for 0:0127 s1 6 c_ R 6 12:55 s1 ; are reproduced as discrete points in Fig. 1a. These are presented in logarithmic format to reveal the spread in c_ R and G. Equation 7 is applied directly to the data in Fig. 1a and the resulting sð_cÞ, for 0 6 c_ 6 12:55 s1 ; is shown as a continuous curve in Fig. 1b. This represents the maximum amount of information that can be extracted from the data. According to Eq. 7, the sð_cÞ curve has an

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intercept of )0.0168 Pa on the vertical axis. This small negative value, probably smaller than the resolution limit of the Tikhonov regularization computation, will be ignored and that means this fluid does not exhibit a yield stress. As a quick check of the reliability of the sð_cÞ given by Eq. 7, it is used to back-calculate the torque versus rim shear rate relationship for the parallel disks used in the measurement. The back-calculated relationship, obtained by computer software independent of that used to set up and solve Eq. 7, is shown as a continuous curve in Fig. 1a. There is very good agreement with the original experimental data for the entire range of the data. The average deviation is around 2.8% with a maximum of 12.1%. The number of discretization points NK=2,001 in this case. Such a large NK is needed because of the large spread in c_ R of the data points. If a smaller number, such as NK=601, is used then the resulting sð_cÞ for small c_ ; in the neighbourhood of 10)2, will be affected and this is reflected by a larger difference between the experimental data and the back-calculated G at low c_ R . However the sð_cÞ curve and the computed G for larger c_ and c_ R respectively are not affected. As a further check, the sð_cÞ given by Eq. 7 is compared against that reported by Steffe [6]. The sð_cÞ given by Steffe is shown as discrete points in Fig. 1b. There is very good agreement between the points and the curve at low shear rate, even when they are compared in a logarithmic plot (not shown). However, at high shear rates the result given by Eq. 7 is consistently smaller than that reported by Steffe. Instead of using the local slope of the G versus c_ R data to convert the G into rim shear stress, Steffe followed the standard practice of using the global slope given by the best-fit straight line drawn through the logarithmic plot of the data points. It can be seen from Fig. 1a that the logarithmic plot deviated significantly from a straight line. Thus the use of the global slope derived from a linear plot is the likely cause of the differences observed in Fig. 1b. Guar gum solution at high shear rates Krammer et al. [7] developed a method of using the parallel disk viscometer, with gap as small as 50 lm, as an instrument for measuring viscosity at very high shear rates. Special precautions were taken to ensure that viscous heating was kept at a negligible level. These authors presented their data, with R=50 mm and h=50 lm, for a 0.7% guar gum solution in the form of c_ R versus sRApp plots. sRApp is the apparent rim shear stress and is obtained directly from the measured G using the simple expression sRApp=2G/pR3. In general it is not the true rim shear stress and only becomes the true rim shear stress for Newtonian fluids. The data of Krammer et al. are reproduced in Fig. 2a as two continuous curves that exhibit oscillations with frequency that increases

with c_ R . One of these curves shows the sRApp collected during the upward sweep of the viscometer as c_ R is increased from 0 to the maximum value of 50,000 s)1, and the other curve shows the sRApp collected on the return sweep. The two curves essentially overlap with one another indicating that viscous heating is negligible. The oscillation exhibited by the two sRApp curves is a consequence of small imperfections on the disk surfaces and/or minor misalignment of the disks [7]. The oscillations of the data in Fig. 2a need to be corrected prior to the computation for sð_cÞ using Eq. 7. This is done following a procedure similar to that outlined by Krammer et al. [7]. First the upper and lower envelopes of the oscillating sRApp curves are constructed and these are shown as broken curves in Fig. 2a. The averages of these envelopes are then computed and shown as a dotted curve in Fig. 2a. This average sRApp is used to compute the torque G=pR3sRApp/2 for 0 \ c_ R 6 50; 000 s1 : The resulting monotonically increasing G is shown as discrete points in Fig. 2b. Finally Eq. 7 is applied to these data points to give the sð_cÞ curve. In order to compare against the results of Krammer et al. [7], the sð_cÞ of the guar gum solution from Eq. 7 is presented as a continuous viscosity curve gð_cÞ ¼ sð_cÞ=_c in Fig. 2c. The filled points in Fig. 2c are the results of Krammer et al. [7] based on the parallel disk data. In processing the parallel disk data Krammer et al. divided the logarithmic plot G versus c_ R into four linear sections and used the slopes of these linear sections to convert the torque G into actual rim shear stress. These authors also measured the viscosity of the guar gum solution using a capillary viscometer. Their capillary results are shown as open squares in Fig. 2c. There is very good agreement in the three sets of viscosity for 103 6 c_ 6 105 : For c_ 6 103 s1 deviation becomes apparent. To verify the reliability of the sð_cÞ given by Eq. 7 for 0 6 c_ 6 105 it is used to back-calculate the G versus c_ R relationship. The outcome is shown as a continuous curve in Fig. 2b. The average deviation between the back-calculated curve and the experimental data is 0.9% with a maximum of around 7.5%. According to Eq. 7, s(0)=372 Pa. This can be taken as an estimate of the yield stress of the guar gum solution. The presence of a significant sY is evident from the unlimited growth of gð_cÞ in Fig. 2c as c_ ! 0: Torque and normal force data of LDPE Carvalho et al. [2] developed a single-point technique for processing parallel disk rheometry data. To test the performance of their technique these authors measured the torque and normal force versus rim shear rate, for 0:025 s1 6 c_ R 6 1:0 s1 ; of a LDPE melt at 120 C and 140 C. Their data are shown as discrete points in Fig. 3a,b. R=12.5 mm and h=1.4 mm for these data.

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Fig. 2a–c Data and results of guar gum solution. a Apparent rim shear stress versus rim shear rate. The two grey curves are the data of Krammer et al. [7]. The dashed curves are the upper and lower envelopes of the oscillating sRApp. The dotted curve is the average of the envelopes. b Torque versus rim shear rate. Discrete points are converted from the dotted curve in a, and the continuous curve is back-calculated from the results of Eq. 7. c Viscosity of guar gum. The continuous curve is from Eq. 7 and the discrete points are the parallel disk (filled circle) and capillary (empty square) viscometry results of Kramer et al. [7]

Respectively, Eq. 7 and Eq. 8 are used to convert these data to sð_cÞ and ND ð_cÞ. Following Carvalho et al. [2], results are presented as viscosity function gð_cÞ ¼ sð_cÞ=_c and normal stress difference coefficient w1 ð_cÞ  w2 ð_cÞ ¼ ND ð_cÞ=_c2 : These are shown as continuous curves in Fig. 3c,d. As in the previous examples, these material property functions were used to backcalculate the torque and normal force versus rim shear rate curves. The back-calculated curves (Fig. 3a, b), are in good agreement with the original data. Typically the average deviation in these curves is of the order of 8% and the largest of the maximum deviation is around 20%. Carvalho et al. [2] applied several methods to process their LDPE data . In the method referred to by them as the exact method, these authors differentiated the experimental data and used the resulting slopes in the differentiated form of Eq. 1 and Eq. 2 to compute the gð_cÞ and w1 ð_cÞ  w2 ð_cÞ functions. Their results, after allowing for typographical error in tabulation, are shown as discrete points in Fig. 3c,d. There is consistently good agreement between these results of Carvalho et al. [2] and that of the present investigation. The results

of Carvalho et al. [2] obtained from their one-point method are also in good agreement with the results of the present investigation. This is not surprising as the range of c_ R covered by the experimental data is relatively narrow. Aqueous solution of polyacrylamide The marked points in Fig. 4a are the G verus c_ R data for a 0.25 wt% aqueous solution of a polyacrylamide with sulphonated groups (Ciba Magnafloc 358, Mw  20 million) using a parallel disk viscometer with R=12 mm. Measurements were performed for three different disk gaps, h=0.7 mm, 1.4 mm and 2.85 mm. These data were obtained using a relatively basic viscometer and are typical of the data encountered in routine rheological tests. The fact that the data for all three gaps fall essentially on a single curve is an indication that there is no slippage at the disks and consequently the three sets of data can be processed as a single set [8]. Equation 7 is now used to convert this combined data set into a single sð_cÞ function. This is shown as a

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Fig. 3a–d Data and results of LDPE melts a Comparison of torque. b Comparison of normal force. c Comparison of viscosity. d Comparison of normal stress coefficient difference. In each plot the marked points are the experimental data or results of Carvalho et al. [2] and the curves are results of Eq. 7 and Eq. 8 or backcalculated from these results

continuous curve in Fig. 4b. The range of c_ covered is again the maximum possible, from 0 to 220 s)1. In this case s(0)=0.7088 Pa and is taken as the yield stress of this polyacrylamide solution. The torque versus rim shear rate curve back-calculated from the sð_cÞ function is shown in Fig. 4a. Taking into consideration the scatters exhibited by the three data sets, the agreement between the back-calculated curve and the original data is considered very satisfactory. Here the average deviation is 7.8% with a maximum of 22.5% (at one of the low rim shear rate data points). As an independent verification of the sð_cÞ given by Eq. 7, a cone and plate viscometer (cone angle=3, radius=24 mm) was used to measure the steady-shear property of the polyacrylamide solution. The shear stress versus shear rate relationship given directly by the cone and plate is shown as marked points in Fig. 4b. In view of the different origins of the data, the entirely different ways these data were processed and the basic nature of the viscometers used, the results in Fig. 4b are in surprisingly good agreement. In particular it is interesting to note that the cone and plate data, while not giving a yield stress without data extrapolation, is consistent with the yield stress of 0.7088 Pa given by Eq. 7.

Discussion As a rheometer the parallel disk has a number of practical advantages over the cone and plate. These include ease of setting up and operation and the extended range of shear rates covered [1]. However, as mentioned above, it does not have the equivalent of the small gap or small cone angle approximation, and data processing is therefore more complicated. Because of this, the parallel disk is not as popular as the cone and plate with most rheologists. The simplicity and generality of Eq. 7 and Eq. 8 should go a long way to change this situation, thereby making the parallel disk a viscometer/rheometer for routine use. The change in role between shear rate and shear stress in parallel disk rheometry has not changed the mathematical nature of the integral equations that relate the material property functions to the experimentally measured quantities. It is therefore not surprising that the general procedure based on Tikhonov regularization developed for capillary and Couette viscometry has, with the appropriate modifications, performed well in parallel disk rheometry. All the advantages of Tikhonov regularization reported for capillary and Couette viscometry are also retained. These include the ability to

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Fig. 4a, b Data and results of 0.25 wt% polyacrylamide solution. a Torque versus rim shear rate. Points are parallel disk data for different gaps: 0.7 mm (filled square), 1.4 mm (filled triangle) and 2.85 mm (empty diamond) and the curve is backcalculated from the results of Eq. 7. b Shear stress shear rate relationship. The continuous curve is from Eq. 7 and the discrete points are independent cone and plate measurements

extract the maximum amount of information from a given data set, independence of any assumed rheological constitutive equation and the ability to keep noise amplification under control by the appropriate choice of the regularization parameter k. For all the examples reported above, the choice of k was particularly simple. It was chosen so that the average deviation between the computed and measured torque or normal force is comparable with the estimated error bars of the experimental data and the resulting material function does not show physically unreal oscillations. It was observed that the resulting material functions are not very sensitive to small changes in k and hence fine-tuning of k is not required. Because of this simplicity alternative ways of choosing k have not been examined [3]. For fluids with yield stress it has been demonstrated that Tikhonov regularization when applied to capillary and Couette viscometry data allows the yield stress sY to be determined. Since in these two viscometers the shear stress is the independent variable, determination  of  sY requires the iterative solution of the equation c_ sc ¼ 0 which in turn requires repeated solutions of the capillary or Couette equivalent of Eq. 7 [4, 5]. In parallel disk viscometry c_ becomes the independent variable and Eq. 7 gives the numerical value of sð_cÞ for 0 6 c_ 6 max ðc_ R Þ: This means that sY=s(0) is given automatically by Eq. 7 without any iterative computation. This greatly simplifies the determination of the yield stress. It is important to note that sY is not obtained by extrapolation of the G versus c_ R data but as an integral part of the solution of Eq. 1. Numerical experimentation shows that the reliability of the sY obtained depends strongly on the distribution of data points at low c_ R and the noise level in these data points. The LDPE example shows that Eq. 8 allows the normal stress function ND ð_cÞ ¼ N1 ð_cÞ  N2 ð_cÞ of viscoelastic fluids to be determined with relative ease. This will greatly enhance the usefulness of the parallel disk as a rheometer. The ND ð_cÞ from a parallel disk rheometer

can be combined with the N1 ð_cÞ from a cone and plate rheometer to give N2 ð_cÞ [9]. It is generally agreed that N2 ð_cÞ is a viscoelastic material property that is difficult to determine accurately. The improved reliability of ND ð_cÞ given by Eq. 8 will therefore provides a way to more reliable N2 ð_cÞ. In Fig. 4b, the data for the polyacrylamide solution for different gaps were processed as a single set leading to a single s(c). Equation 7 has also been applied to the data for each gap separately leading to three sð_cÞ curves. However, these individual sð_cÞ curves, each covering a slightly different range of shear rates, are sufficiently close together that their differences, particularly at low shear rates where the noise level is high, can be ignored. If the G versus c_ R data for different gaps do not coincide, indicating that significant wall slip is present, Eq. 7 can still be used to convert the individual G versus c_ R data each into a separate sð_cÞ curve. In this case the resulting independent variable c_ is no longer the true shear rate and is referred to as the apparent shear rate, c_ a , which includes the effects of wall slip [8]. Additional computation will need to be performed to convert these apparent sðc_ a Þ curves into a true shear stress versus shear rate function and a slip velocity versus wall shear stress function. Yoshimura and Prud’homme [8] have developed a procedure for doing this conversion for pairs of sðc_ a Þ curves based on data from two disk gaps. A more general procedure based on the sðc_ a Þ curves given by Eq. 7 and applicable to multiple sets of data with different disk gaps has been developed. This will be reported separately.

Conclusion The problem of converting the steady shear data of parallel disk rheometry into material property functions can be treated as integral equations of the first kind. The procedure, based on Tikhonov regularization, provides a computationally simple way of solving these equations

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and results are independent of rheological models. For fluids that have a yield stress the procedure will provide an estimate of the yield stress. Noise amplification is kept in check by the appropriately chosen regularization parameter.

Acknowledgements YKL thanks Ciba Specialty Chemicals (Australia) for providing the samples of polyacrylamide solutions used in this investigation.

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