Estimation of the parameters of Herschel-Bulkley fluid under wall slip using a combination of capillary and squeeze flow viscometers

Rheol Acta (2004) 43: 80 88 DOI 10.1007/s00397-003-0322-y ORIGINAL CONTRIBUTION Hansong S. Tang Dilhan M. Kalyon Estimation of the parameters of Herschel-Bulkley fluid under wall slip using a combination of capillary and squeeze flow viscometers Received: 28 March 2003 Accepted: 9 July 2003 Published online: 10 December 2003 Ó Springer-Verlag 2003 H. S. Tang Æ D. M. Kalyon (&) Stevens Institute of Technology, Hoboken, NJ 07030, USA E-mail: dkalyon@stevens.edu Abstract The determination of the parameters of viscoplastic fluids subject to wall slip is a special challenge and accurate results are generally obtained only when a number of viscometers are utilized concomitantly. Here the characterization of the parameters of the Herschel- Bulkley fluid and its non-linear wall slip behavior is formulated as an inverse problem which utilizes the data emanating from capillary and squeeze flow rheometers. A finite element method of the squeeze flow problem is employed in conjunction with the analytical solution of the capillary data collected following MooneyÕs procedure, which uses dies with differing surface to volume ratios. The uniqueness of the solution is recognized as a major problem which limits the accuracy of the solution, suggesting that the search methodology should be carefully selected. Keywords Viscoplasticity Æ Herschel-Bulkley fluid Æ Wall slip Æ Squeeze flow Æ Capillary flow Æ Viscometry Æ Parameters Introduction For the description of the shear viscosity material function of viscoplastic-type generalized Newtonian fluids, which exhibit a yield stress below which there is no deformation, generally a three parameter model described as the Hershel-Bulkley fluid is used (Herschel and Bulkley 1926 Bird et al. 1983). For instance, Garcia et al. (2002) showed that the Hershel-Bulkley model describes well the behavior of suspensions manufactured upon the ultra-fine wet grinding of calcite in a batch stirred bead mill. Zhou et al. (1997) compared the Bingham, the Power-Law, the Casson, and the Herschel- Bulkley models, and determined that the Hershel-Bulkley model adequately describes the shear viscosity data of a low-density oil-in-water emulsion mud. In the simulation of the processing operations of various highly filled materials the Hershel-Bulkley equation is widely used also (Lawal et al.1993 Lawal and Kalyon 1994a, 1994b, 1997 Kalyon et al. 1999). The investigation of the flow and deformation behavior of viscoplastic fluids cannot be undertaken without the analysis and incorporation of the wall slip condition (Mooney 1931 Yilmazer and Kalyon 1989 Aral and Kalyon 1994). For concentrated suspensions the wall slip occurs upon the formation of an apparent slip layer (the Vand layer) at the wall. The shear viscosity of the apparent slip layer is different (generally smaller) than the shear viscosity of the bulk of the suspension found outside of the apparent slip layer. If the shear viscosity of the fluid comprising the apparent slip layer is Newtonian than one obtains a linear relationship between the slip velocity and the wall shear stress. This linear relationship between the slip velocity and the shear stress is designated as the Navier slip condition (Navier and Sur 1827). However, if the material comprising the apparent slip layer is non-newtonian than the relationship between the slip velocity and the shear stress becomes nonlinear (a Power-Law type behavior is observed) (Yilmazer et al. 1989 Yilmazer and Kalyon 1991 Kalyon et al. 1993).

81 The simultaneous determination of the parameters of the shear viscosity vs the shear rate behavior of the viscoplastic fluid subject to wall slip is a challenge. If handled properly, capillary viscometry in conjunction with MooneyÕs procedure employing the systematic alteration of the surface/volume ratio of rheometer is sufficient to generate the wall slip vs the wall shear stress on one hand (Mooney 1931) and the wall shear stress vs the wall shear rate on the other hand (Yilmazer et al. 1989 Kalyon et al. 1993). However, the major problem faced for viscoplastic fluids is associated with the changes in the microstructure of the viscoplastic fluid during the viscometric characterization. Under such conditions it is advisable to utilize multiple rheometers to generate a realistic representation of the rheological behavior of the fluid. As shown by Kalyon et al. (1993) the use of steady torsional flow together with capillary flow, for example, presents such an approach. Another possibility is the use of a squeeze flow rheometer in conjunction with capillary or rectangular slit flows. Here this particular combination of viscometers will be probed further by formulating the problem of parameter estimation of the Herschel-Bulkley fluid and the slip velocity vs the shear stress relationship as an inverse problem. The analyses of the squeeze flow problem using various types of constitutive equations and wall boundary conditions are available (Scott 1931, 1935 Covey and Stanmore 1981 Adams et al. 1994 Zhang et al. 1995 Laun et al. 1999 Lawal and Kalyon 1998, 2000 Sherwood and Durban 1998 Meeten 2000). The squeeze flow has also been used to compute the parameters of the Herschel-Bulkley model and wall slip conditions (Mannheimer 1983). Ahmed and Alexandrou (1994) modeled numerically the squeeze flow and used it to determine the parameters of the Herschel-Bulkley model. In the following the squeeze flow is used in conjunction with the capillary flow data to provide the data necessary for the solution of the inverse problem to characterize the three parameters of the Hershel-Bulkley equation and the two parameters of the wall slip velocity vs wall shear stress relationship. Herschel-Bulkley fluid and wall slip condition Let i and j be two directions in an orthogonal coordinate system. The constitutive equation of the generalized Newtonian fluid can be written as s ij ¼ g_c ij ð1þ where s ij is the stress and _c ij is the rate of deformation tensor: _c ij ¼ @u j @x i þ @u i @x j ð2þ and g is the shear viscosity material function which is assumed to depend on the second invariant of the rate of deformation tensor. Here, x i and x j are the coordinates and u i and u j are the velocities in the i and j directions, respectively. Especially when numerical simulations are attempted the Herschel-Bulkley model can be modified as (Papanastasiou 1987) g ¼ mj_c ij j n 1 þ s y 1 exp ej_c ij j ð3þ j_c ij j in which j_c ij j 2 ¼ 2 X3 j¼1 2 @u j þ @u 2 þ @u 2 1 þ @u 3 þ @u 2 1 @x j @x 1 @x 2 @x 1 @x 3 þ @u 3 þ @u 2 2 : ð4þ @x 2 @x 3 In the modified Herschel-Bulkley model, m, the consistency index, n, the power law index, and s y, the yield stress, are parameters to be determined. e is the assumed growth parameter and its incorporation into the analysis facilitates the elimination of the necessity to determine fracture planes in the flow domain (at which the magnitude of the stress tensor is equal to the yield stress of the fluid). The use of the generalized Newtonian fluid constitutive equation in this manner is overall questionable. However, currently there is no constitutive equation that can describe accurately the rheological behavior of the viscoplastic fluids in such a way to be amenable to largescale computations of the processing flows of such fluids using two- and three-dimensional numerical analysis methods. Simplifications of Eq. (3) provide the solutions for other constitutive behavior including the Newtonian fluid (n=1, s y =0), the Ostwald-de-Waele power-law fluid (s y =0), and the Bingham fluid above the yield stress (n=1, e= ). In one-dimensional shear flow the Herschel-Bulkley model (Herschel and Bulkley 1926) above and below the yield stress becomes 8 < s 12 ¼ m @u 1 @x 2 n 1 @u 1 @x 2 s y js 12 j > s y ð5þ : @u 1 @x 2 ¼ 0 js 12 js y : Here, the minus sign is to be used when s 12 <0. For e= Eq. (3) becomes the 1-D Hershel-Bulkley equation for shear stress greater than the yield stress, i.e., Eq. (5). At the wall a non-linear relationship between the slip velocity, u i, and shear stress, s ij, is considered to exist (Yilmazer et al. 1989 Zhang et al. 1995): u i ¼ bs sb ij ð6þ for one-dimensional simple shear flow, where i is the flow direction. In the model, b, the slip coefficient, and sb, are

82 parameters to be determined. When sb=1, the model reduces to the Navier slip condition (Navier and Sur 1827). Inverse problem for parameter estimation The schematic representation of the squeeze flow is shown in Fig. 1. In the figure there is a fluid between the two circular plates with a radius of R s, the time-dependent distance between the plates is h, the top plate moves down at a speed of )h, the bottom plate is stationary, and the total force acting on the top plate as a function of time during the squeeze is f(t). On the other hand, the capillary flow is driven by a pressure gradient and is assumed to be occurring under the fully-developed conditions within a pipe of radius R c, the flow rate is Q, and the shear stress at the wall is s w. Let us consider that the experimental data from squeeze flow consist of the normal force acting on the top surface, which is the driving force for the flow, f1 e f 2 e...f M e s at gap (or distance of separation between the two plates) h 1 h 2...h Ms, respectively. The data from the capillary flow would be the flow rates or the corresponding apparent shear rates _c a1 _c a2... _c amc at the Bagley corrected wall shear stress values of s w1 s w2... s wmc, respectively. The parameter estimation problem can be formulated as the following inverse problem: Find m n s y b sb ¼ m n sy b sbjjðf C F e C e Þ ¼ min mnsy bsb JF ð C F e C e Þ ð7þ where the objective function is defined as JFCF ð e C e Þ 1 X Ms ¼j 1 f 2 i M s f e þ 1 XM s 1 1 f! 2 iþ1 f i i¼1 i M s 1 f e i¼1 iþ1 f i e 0 1 1 X Mc þð1 jþ 1 _c 2 ai M c _c e þ 1 XM c 1 1 _c 2 @ aiþ1 _c ai A: i¼1 ai M c 1 _c e i¼1 aiþ1 _ce ai ð8þ Here, 0 j 1, F ¼ ff 1 f 2... f Ms g, C ¼ f_c a1 _c a2... _c amc g, F e ¼ff1 e f 2 e... f M e s g, and C e ¼f_c e a1 _ce a2... _c e am c g. F values are the computed values of the forces acting on the top plate to drive the flow during squeeze flow and F e values are the experimental force values as a function of time during the squeeze flow. F values can be computed upon the integration of the pressure and normal stress distributions in the radial direction: fðþ¼ t Z Rs 0 2pðp þ s zz Þj z ¼ h rdr ð9þ where q and s zz are pressure and normal stress, respectively, and they can be computed numerically using the Finite Element Method. Our numerical method employs a Galerkin/penalty finite element approach with bilinear rectangular elements and the modified Herschel-Bulkley model (as of Eq. 3) with e=100. It should be noted that again the applicability of the generalized Newtonian fluid to the computation of the normal stress terms are questionable. G values are the apparent shear rates experienced by the viscoplastic fluid during capillary flow and are analytically determined from (Kalyon et al. 1993) _c a ¼ 4bssb w R c ð þ 4s1=n w ð Þ 1þ1=n m 1=n s 2 w s 2 y 1 s y =s w Þ 2þ1=n þ 2s y 1 s y =s w s w ð2þ1=nþ þ 1 s y=s w 3þ1=n ð Þ 3þ1=n ð10þ Fig. 1 Schematic representation of the squeeze flow and the comparisons of top plate forces obtained by computations and experiments (Zhang et al. 1995) for constant-velocity squeezing of HDPE. m=44900 PaÆs 0.25, n=0.25, s y =0, and b=0. Filled squares computation, open squares experiment where _c a =4Q/pR c 3 and wall slip condition at Eq. (6) is used. When 0<j<1, the objective function J is defined here as a weighted function of squeeze and capillary

83 flows. When the weighting factor j=1 the objective function uses only the data from the squeeze flow. Finally, when j=0 the objective function uses only the data arising from the capillary flow. The objective function, J, defined here is different from the standard least square objective function that has only the first terms in the two brackets of Eq. (8). The following can be shown (for convenience, let j=1): jjðf F e Þ g jj 2 2 6jjF g0 þ df dh g 0 dh Fg e 0 df e 2 dh g 0 dhjj 2 6jjF g0 Fg e 0 jj 2 2 þjj df dh g 0 dh df e 2 dh g 0 dhjj 2 jjf g0 Fg e 0 jj 2 2 þjjf g F g0 Fg e þ F g e 2 0 jj 2 ¼ PM s i¼1 f i fi e M 2 Ps 1 þ f iþ1 f i fiþ1 e þ f i e 2 i¼1 6CJðF C F e C e Þ ð11þ where C ¼ M s max f PM s jfi ej2 MP s 1 jfiþ1 e f i ej2 g, being a i¼1 i¼1 constant, g 0 ¼ ðh 1 h 2 h Ms Þ, and g ¼ ðh 1 þ dh h 2 þ dh h Ms þ dhþ. The minimization of Eq. (7) involves nonlinear iterations that are carried out using the deepest descent method for the first few iterations and then adopting the conjugate gradient method for the rest of the iterations, following Press et al. (1992). The derivatives of the objective functions are evaluated by the use of central differences, i.e., @m ¼ J mþdmnsy bsb J m dmnsy bsb 2dm @n ¼ J mnþdnsy bsb J mn dnsy bsb 2dn @s ¼ J mnsy þdsy bsb J mnsy dsy bsb 2ds y @b ¼ J mnsy bþdbsb J mnsy b dbsb 2db @sb ¼ J mnsy bsbþdsb J mnsy bsb dsb 2dsb : ð12þ Here, dm, dn, ds y, db, anddsb are the step sizes for m, n, s y, b and sb, respectively. Numerical experimentation Validation of numerical methods From Eq. (7) it is seen that the parameter estimation is based on the calculations of forces F acting on the top plate to drive the squeeze flow and the apparent shear rates of capillary flow. Thus, it is crucial for the code to compute the force and shear rate values accurately. Figures 1 and 2 show the results of two validation runs for which the predictions are compared with the experimental data collected with a HDPE by Zhang et al. (1995) under two sets of conditions involving the no slip and wall slip following a nonlinear relationship between the slip velocity and the wall shear stress (Figs. 1 and 2, Fig. 2 Comparisons of top plate forces obtained by computations and experiments (Zhang et al. 1995) for constant-velocity squeezing of HDPE. m=44900 PaÆs 0.25, n=0.25, s y =0, b=5.127 10 )15 m s )1 ÆPa )2.345, sb=2.345. Filled squares computation, open squares experiment respectively). The agreement between the computed force values and the experimental forces is excellent. For capillary flow the validation of the source code is easier since the apparent shear rate is an explicit analytical function of the wall shear stress. The minimization program, on the other hand, was validated by employing objective functions with known solutions, an example of which is shown below: g ¼ sin px 1 2 2 þ ð x2 5Þ 2 þ x 2 3 2 þ x2 4 ð13þ This function, g, has its roots at {x 1, x 2, x 3, x 4 }={±2l,5, 0, 0}, l being any integer. Starting from an initial guess of {3.8, 0, )20, 2}, the program finds a solution at {4, 5, 0, 0}. This is a good example of a problem with multiple solutions, i.e., different values of integer l. The solution depends on the initial guess that is made to initiate the minimization problem. Convergence and uniqueness In this section, numerical experiments are made to test the capacity of the solution of the inverse problem, Eq. (7), for estimating parameters. Particularly, the uniqueness of the estimation is investigated. First, let us consider the estimation of parameters using the source codes upon their application to the squeeze flow only, i.e., j=1. For this problem the diameter of the plates 2R s =0.0572 m, the top velocity _ h=2.12 10 )5 m/s.

84 The set of gap vs the force data for the squeeze flow are given in Table 1. This set is obtained by the application of our FEM code using {m, n, s y, b, sb}={5066 PaÆs 0.45, 0.45, 1998 Pa, 2.14 10 )7 mæs )1 ÆPa )1, 1}. Obviously, this set of {m, n, s y, b, sb} is the exact solution of the inverse problem. In the following the units of the parameters are omitted for brevity. Let us first start with the estimation of two parameters. The following is an example to identify s y and b, while the other parameters are kept fixed. Starting from four different sets of initial guesses: m n s y b sb ¼ 5066 0:45 19:98 2:14 10 9 1 m n s y b sb ¼ 5066 0:45 19980 2:14 10 9 1 m n s y b sb ¼ 5066 0:45 19:98 2:14 10 5 1 m n s y b sb ¼ 5066 0:45 19980 2:14 10 5 1 the minimization indeed yields the exact solution. Here, m, n, and sb are fixed in the process of minimization. From this example it is seen that the solution of the inverse problem with two unknowns provides a unique solution over a broad range of s y and b values. Generally, this observation was determined to be true for any other two parameters also, as long as we were searching only two out of the five parameters within a given run. For the estimation of three parameters (with the other two kept fixed), multiple cases which generated unique solutions were also found to be possible, depending on the values of the initial guesses. The following is an example. Let us determine m, n, ands y, while fixing the values of the other two parameters by starting from: m n s y b sb ¼ 1013 0:8 2373 2:14 10 7 1 and the retrieval of the exact solutions is made. However, as the number of parameters to be estimated increases from three to four and then five, the Table 1 Gap vs force acting on the top plate during squeeze flow h i (mm) f i e (N) 5.7 13 3.4 16 2.3 22 1.5 34 1.19 50 0.95 71 0.69 123 0.59 164 0.43 298 0.31 564 0.25 863 0.21 1223 0.18 1668 0.16 2014 0.15 2413 probability for obtaining a unique solution decreases. For instance, let us fix sb and start the search from the initial guess of m n s y b sb ¼ 354 0:73 417 1:39 10 10 1 : Instead of reaching the exact solution, the minimization finds a solution at a local minimum which is different to the exact solution: m n s y b sb ¼ 4306 0:34 810 4:60 10 10 1 with J=6.7 10 )5. Starting from an initial guess which is closer to the exact solution: m n s y b sb ¼ 4964 0:50 2017 2:35 10 7 1 the minimization stops at m n s y b sb ¼ 5106 0:45 2000 2:12 10 7 1 with J=6.7 10 )7, which is still not the exact solution. This indicates that the objective function has multiple minima and the minimization generates more than one solution when the number of unknowns is four or five. With this solution method then, there is no guarantee that the minimization will find the global minimum. Which solution is generated depends on the initial guess used to initiate the solution of the inverse problem procedure. Also, the results indicate that the estimation procedure using squeeze flow only is unstable in the sense that the predicted parameters may change significantly upon relatively minor changes in the experimentally determined force values. Now, let j=0 and consider the estimation using capillary flow only. Let us use the same set of parameters {m, n, s y, b, sb}={5066 PaÆs 0.45, 0.45, 1998 Pa, 2.14 10 )7 ms )1 ÆPa )1, 1}, a set of data for _c a were generated knowing the exact solution and are shown in Table 2. The identification of two parameters has a unique solution over relatively large ranges of the initial guesses. For instance, starting from the following initial guesses (s y, b, and sb are fixed): m n s y b sb ¼ 2066 0:9 1998 2:14 10 7 1 m n s y b sb ¼ 9066 0:1 1998 2:14 10 7 1 m n s y b sb ¼ 2066 0:1 1998 2:14 10 7 1 m n s y b sb ¼ 9066 0:9 1998 2:14 10 7 1 exact solutions for m and n are obtained. For the estimation of the three parameters the situation is the same as that encountered in squeeze flow, that is, the exact solution is recovered in some cases depending on the values of the initial guess. An estimation for four parameters by fixing the fifth in capillary flow is also possible, and the following are

85 Table 2 Apparent shear rate and wall shear stress of capillary flow _c ai ðs 1 Þ s wi e (Pa) 5.5 6000 10.5 10000 18.6 15000 29.3 20000 153.9 50000 397.8 80000 932.4 120000 1503.3 150000 3446.1 220000 6790.1 300000 two initial guesses from which the exact solutions could be retrieved: m n s y b sb ¼ 4066 0:43 1998 2:34 10 7 1 m n s y b sb ¼ 8066 0:3 2998 2:14 10 7 1:6 : Here,sband b are respectively fixed. Starting the search for all the five parameters from either of m n s y b sb ¼ 3066 0:7 1277 7:14 10 7 0:6 m n s y b sb ¼ 3066 0:7 1277 7:14 10 7 1:4 the minimization finds the exact solution in capillary flow. However, like in the estimation using squeeze flow only, there is no guarantee for uniqueness in case of capillary flow. For instance, uniqueness can fail if the searching starts from m n s y b sb ¼ 4066 0:43 1987 2:34 10 7 1:1 since it ends up at a local minimum m n s y b sb ¼ 5060 0:45 1710 2:38 10 7 0:986 with J=4.9 10 )9. Using capillary and squeeze flows at same time is better for getting global minimum solutions. The following is an example. Let j=0.5 and sb be fixed, and start from m n s y b sb ¼ 4066 0:43 1987 2:34 10 7 1 the retrieval of the exact solution is made. Whereas, if we had used the same initial guess but employed the two flows individually, no retrieval could have been made. The minimization with squeeze flow only gives the solution m n s y b sb ¼ 5270 0:18 3470 2:15 10 7 1 with J=1.4 10 )3, and the minimization with capillary flow only yields m n s y b sb ¼ 9070 0:34 1190 6:82 10 7 1 with J=1.8. The conclusions discussed above were verified further by applying the analysis to different sets of conditions and exact solutions. From the above discussions it is seen that the estimation using squeeze and/or capillary flows may suffer from non-uniqueness, and the solution upon minimization is dependent on the initial guess when the number of parameters being sought exceeds two. To overcome the problem and obtain the solution pertaining to a global minimum, a new procedure is proposed here as follows. Suppose the smallest possible values for m, n, s y, b and sb are respectively m min, n min, s ymin, b min,andsb min, and the largest possible values are respectively m max, n max, s ymax, b max, and sb max. It is proposed that the whole possible domain of parameters [m min, m max ] [n min, m max ] [s ymin, s ymax ] [b min, b max ] [sb min, sb max ] be divided into many small subdomains by m ¼ m min þ ði 1ÞDm 16I6L m n ¼ n min þ ðj 1ÞDn 16J6L n s y ¼ s y min þ ðk 1ÞDs y 16K6L sy b ¼ b min þ ðm 1ÞDb 16M6L b sb ¼ sb min þ ðn 1ÞDsb 16N6L sb where Dm=(m max )m min )/Lm, Dn=(n max )n min )/L n, Ds y ¼ s y max s y min =Lsy,Db=(b max )b min )/L b,anddsb= (sb max )sb min )/L sb, and L m, L n, L sy, L b, and L sb are numbers of subdomains in directions of m, n, s y, b, and sb, respectively. Start the minimization from each {l, J, k, M, N}, and choose the solution for the parameters which gives rise to the lowest objective function J. The procedure works well provided that the subdomains are relatively small so that for any given subdomain the initial guess is never very far removed from the local minimum for each subdomain. The global minimum is then determined. Applications In this section the inverse problem of Eq. (7) is applied to estimate the parameters of a poly(dimethyl siloxane), PDMS, and its suspensions with rigid spherical particles. The PDMS was obtained from GE Silicones Company under the tradename of SE30 and has a specific gravity of 0.98. The particles were hollow glass spheres obtained from Potters Industries with a specific gravity of 1.09, arithmetic particle diameter of 12 lm and a maximum packing fraction of 0.65 (Aral and Kalyon 1997 Yaras 1995). The concentrations of the particles were 20 vol.% and 40 vol.% (22 wt% and 43 wt%). In the estimation, both squeeze and capillary flows are used, i.e., j=0.5. The new procedure given above is employed to deal with

86 the non-uniqueness problem. For the squeeze flow, a squeeze flow rheometer with a diameter, 2R s =0.0572 m, was used to collect the force values as a function of time for three different top plate velocities of _ h=2.1 10 )4 m/s, 4.2 10 )4 m/s, 8.4 10 )4 m/s. On the other hand, for capillary flows, capillaries with diameters of 2R c =2.5 10 )3 m and 1.5 10 )3 m were employed at a constant length over the diameter ratio of 40. All of the experiments were carried out under ambient temperature and using Instron Universal tester in conjunction with capillary and squeeze flow attachments. The materials of construction can have an effect on the wall slip behavior (Chen et al. 1993) and all the tools used were machined from stainless steel. Figures 3 and 4 show the squeeze and capillary flow results for the estimation of the set of five parameters for pure PDMS. Figure 3 shows the comparisons of the predictions of force for the squeeze flow obtained using the inverse problem solution vs the experimentally obtained forces, collected during the squeeze flow experiments. Figure 4 on the other hand shows the predicted and experimentally observed apparent shear rate vs wall shear stress behavior of pure PDMS in capillary flow. The estimation shows that the slip velocity over the average velocity is relatively small (in the order of 8 10%). The identified parameters upon the solution of the inverse problem using both flows simultaneously are m ¼ 17200Pa s 0:39 n ¼ 0:39 s y ¼ 0Pa b ¼ 8:3 10 15 m s 1 Pa 2:26 sb ¼ 2:26: and appear to agree with the experimental data as shown in Figs. 3 and 4. Figures 5, 6, and 7 give the results for parameter identification for the same PDMS but now incorporated with 20 vol.% and 40 vol.% of glass spheres. According to our current understanding of the development mechanism of apparent wall slip the slip layer consists solely of pure PDMS (Yilmazer and Kalyon 1989 Aral and Kalyon 1994), and thus, the exponent in the slip Fig. 3 Predicted and experimentally obtained gaps and forces of squeeze flow for pure PDMS. Filled squares computation, open squares experiment Fig. 5 Predicted and experimentally obtained gaps and forces of squeeze flow for 20 vol.% and 40 vol.% PDMS. Filled symbols computation, open symbols experiment Fig. 4 Predicted and experimentally obtained apparent shear rates and wall stresses of capillary flow for pure PDMS. Filled squares computation, open squares experiment Fig. 6 Predicted and experimentally obtained apparent shear rates and wall stresses of capillary flow for 20 vol.% and 40 vol.% PDMS. Filled symbols computation, open symbols experiment

87 Fig. 7 Predicted slip velocity for 20 vol.% and 40 vol.% PDMS velocity vs the wall shear stress relationship (Eq. 7) is determined a priori as sb=2.165. The set of the other unknown four parameters were obtained upon the solution of the inverse problem by employing both squeeze and capillary flows. The parameters for the PDMS with 20 vol.% and 40 vol.% glass spheres are, respectively: m ¼ 33900 Pa s 0:32 n ¼ 0:32 s y ¼ 10 Pa b ¼ 3:6 10 14 m s 1 Pa 2:165 and m ¼ 50000 Pa s 0:5 n ¼ 0:50 s y ¼ 403 Pa b ¼ 2:1 10 14 m s 1 Pa 2:165 : The agreement between the predictions upon the solution of the inverse problem and the experimental results for squeeze and capillary flows are shown in Figs. 5 and 6. The corresponding computed slip velocity vs the wall shear stress relationships are given in Fig. 7. For both suspensions the predictions using the parameters obtained with the solutions of the inverse problem using both flows simultaneously agree well with the experimentally collected data reported in Figs. 5 and 6. Overall, computations also suggest that the wall slip effects are significant for filled PDMS. For example, the ratio of the slip velocity over the mean velocity in the capillary flow is about 0.5 for both 20% and 40% filled PDMS. The typical slip velocity values in capillary flow obtained with MooneyÕs method are u s =1.3 10 )3 m/s at s w =80,000 Pa for 20% filled PDMS and u s =1.8 10 )3 m/s at s w =105,000 Pa for 40% filled PDMS. It is seen from Fig. 7 that these values are not far away from the values provided by the solution of the inverse problem using both flows. Concluding remarks This paper proposes an inverse-problem-solution based procedure for the determination of the parameters of the Herschel-Bulkley fluid and the wall slip velocity vs wall shear stress relationship under isothermal conditions. The inverse problem is first formulated as the modified least square estimation using experimental data generated by squeeze and capillary flows. The capacity of the method to identify the parameters is investigated. It is shown that it is easier to obtain a unique solution when the number of unknowns is 2 or 3, but it is difficult to determine all of the five parameters. The combination of the data emanating from squeeze and capillary flows vs the use of data from either the capillary or the squeeze flow provides a more reliable estimation. The division of the solution domain into a large number of subdomains, over which the local minima are computed to allow the global minimum to be determined, is an important part of the procedure. Acknowledgements We thank Ms. Elvan Birinci of Highly Filled Materials Institute of Stevens for the experimental characterization of the silicone oil and its suspensions using squeeze and capillary flows. References Adams MJ, Edmondson B, Caughey DG, Yahya R (1994) An experimental and theoretical study of squeeze-film deformation and flow of elastoplastical fluids. J Non-Newtonian Fluid Mech 51:61 78 Ahmed A, Alexandrou AN (1994) Processing of semi-solid materials using a shearthickening Bingham fluid model. FED vol 179, Numerical methods for non-newtonian fluid dynamics. ASME Aral B, Kalyon DM (1994) Effects of temperature and surface roughness on timedependent development of wall slip in steady torsional flow of concentrated suspensions. J Rheol 38:957 972 Aral B, Kalyon DM (1997) Viscoelastic material functions of noncolloidal suspensions with spherical particles. J Rheol 41:599 620 Bird BR, Dai GC, Yarusso BJ (1983) The rheology and flow of viscoplastic materials, Rev Chem Eng 1:1 71 Chen Y, Kalyon DM, Bayramli E (1993) Effects of surface roughness and the chemical structure of materials of construction on wall slip behavior of linear low density polyethylene in capillary flow. J Appl Polym Sci 50:1169 1177 Covey GH, Stanmore BR (1981) Use of the parallel-plate plastometer for the characterisation of viscous fluids with a yield stress. J Non-Newtonian Fluid Mech 8:249 260

88 Garcia F, Le Bolay N, Frances C (2002) Changes of surfaces and volume properties of calcite during a batch wet grinding process. Chem Eng J 85:177 187 Herschel WH, Bulkley R (1926) Measurement of consistency as applied to rubber-benzene solutions. Am Soc Test Proc 26:621 633 Kalyon DM, Yaras P, Aral B, Yilmazer U (1993) Rheological behavior of a concentrated suspension: a solid rocket fuel simulant. J Rheol 37:35 53 Kalyon DM, Lawal A, Yazici A, Yaras P, Railkar S (1999) Mathematical modeling and experimental studies of twin screw extrusion of filled polymers. Polym Eng Sci 39:1139 1151 Laun HM, Rady M, Hassager O (1999) Analytical solutions for squeeze flow with partial wall slip. J. Non-Newtonian Fluid Mech 81:1 15 Lawal A, Kalyon DM (1994a) Single screw extrusion of viscoplastic fluids subject to different slip coefficients at screw and barrel surfaces. Polym Eng Sci 34:1471 1479 Lawal A, Kalyon DM (1994b) Nonisothermal model of single screw extrusion of generalized Newtonian fluids. Numer Heat Transfer A 26:103 121 Lawal A, Kalyon DM (1997) Viscous heating in nonisothermal die flows of viscoplastic fluids with wall slip, Chem Eng Sci 52:1323 1337 Lawal A, Kalyon DM (1998) Squeeze flow of viscoplastic fluids subject to wall slip. Polym Eng Sci 38:1793 1804 Lawal A, Kalyon DM (2000) Compressive squeeze flow of generalized Newtonian fluids with apparent wall slip. Intern. Polym Process XV:63 71 Lawal A, Kalyon DM, Yilmazer U, (1993) Extrusion and lubrication flows of viscoplastic fluids with wall slip. Chem Eng Comm 122:127 151 Mannheimer RJ (1983) Effect of slip on flow properties of cement slurries can flaw resistance calculations. Oil Gas J 81:144 147 Meeten GH (2000) Yield stress of structured fluids measured by squeeze flow. Rheol Acta 40:279 288 Mooney M (1931) Explicit formulas for slip and fluidity. J Rheol 2:210 222 Navier CL, Sur MH (1827) Les lois du mouvement des fluides. Mem Acad R Sci Inst Fr 6:389 440 Papanastasiou TC (1987) Flows of materials with yield. J Rheol 31:385 404 Press H, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran. Cambridge University Press Scott JR (1931) Theory and application of the parallel-plate plastometer. Trans Inst Rubber Ind 7:169 175 Scott JR (1935) Theory and application of the parallel-plate plastometer, part 2. Trans Inst Rubber Ind 10:418 493 Sherwood JD, Durban D (1998) Squeezeflow of a Herschel-Bulkley fluid. J Non- Newtonian Fluid Mech 77:115 121 Yaras P (1995) Rheological behavior and twin screw extrusion of flow of noncolloidal concentrated suspensions. PhD Thesis, Stevens Institute of Technology Yilmazer U, Kalyon DM (1989) Slip effects in capillary and parallel disk torsional flows of highly filled suspensions. J Rheol 33:1197 1212 Yilmazer U, Kalyon DM (1991) Dilatancy of concentrated suspensions with Newtonian matrices. Polym Compos 12:226 232 Yilmazer U, Gogos CG, Kalyon DM (1989) Mat formation and unstable flows of highly filled suspensions in capillaries and continuous processors. Polym Compos 4:242 248 Zhang W, Silvi N, Vlachopoulos J (1995) Modeling and experiments of squeezing flow of polymer melts. Int Polym Process X:155 164 Zhou F, Wang H, Yang X, Liu Y (1997) Study on rheological properties of an oil-in-water emulsion at high temperature and high pressure. Proceedings of 1997 International Symposium on Multiphase Fluid, Non-Newtonian Fluid and Physico-Chemical Fluid Flows, ISMNP 97. International Academic, pp 5.63 5.68