Squeezing Flow between Parallel Disks. 11. Experimental Results Philip J. Leider’ Department of Chemical Engineering and Rheology Research Center, University ot Wisconsin, Madison, Wisconsin 53706
Squeezing flow results have been obtained for four different fluids having distinctly different properties in steady shear flow. The results are correlated using the following: fluid parameters obtained from steady shear flow, the applied force, geometric variables, and the half-time f 1 / 2 , or time required for the disk separation to reach one-half its initial value. The Scott solution is found to predict the force-halftime relationship provided the relaxation time of the fluid is less than f 1 / 2 . When t l 1 2 is small compared to A, the viscoelastic fluid squeezes out slower for a given force than the prediction of Scott. This result is contrary to the predictions of Tanner and Kramer given in part I ; it appears to be related to the stress overshoot phenomena. The excellent correlation of the data indicates that the squeezing flow experiment can be used to obtain power-law parameters and a relaxation time, provided one is in the power-law region. A detailed procedure for obtaining these parameters is given.
Introduction Squeezing flow between two circular disks has been used as a convenient experiment for testing the response of materials by a number of investigators. The materials that have been tested include: polymer melts, by Dienes and Klemm (1946) and by Dienes (1947); coal tar pitch, by Gent (1960); bitumen, by Dickenson and Witt (1969); suspensions, by Landel, et at. (1965); silicones, by Parlato (1969); and fresh mortar, by Reiner (1960). Usually, the parallel plate squeezing apparatus is used as a means of determining a viscosity for the material; the viscosity is obtained by assuming that a particular solution to the hydrodynamic problem is applicable. In many cases the solution for a Newtonian fluid (eq 6, part I with n = l) is used (e.g., by Dienes and Klemm (1946) and by Gent (1960)). Only rarely have independent rheological measurements of the fluid been undertaken in an attempt to evaluate the assumed theory over a range of conditions (cf. Gent, 1960; Parlato, 1969). One of the objectives of this investigation is to evaluate the existing solutions to the squeezing flow problem as presented in part I (Leider and Bird, 1974). For a proper evaluation, the fluids were characterized by testing their response in a separate experiment, namely, in steadystate shearing flow. A second objective is to determine if parameters defined in steady shear can be used to correlate the squeezing flow results in the hope that if a correlation exists, squeezing flow can be used with some degree of confidence to obtain these parameters. Further, the evaluation was performed on four different fluids, three of which exhibited a non-Newtonian viscosity and two of which were highly elastic. The determination of the role of elasticity in squeezing flow is a third objective. Although several theories for a viscoelavtic fluid undergoing squeezing flow have been set forth (part I), a complete set of experimental data has not been available to answer such questions as (i) do viscoelastic fluids make better lubricants than purely viscous fluids under squeezing flow conditions ( c f . Metzner, 1968; Tanner, 1965; Appeldoorn, 1965; Williams and Tanner, 1970), and (ii) how does elasticity influence the rapid squeezing flows which are encountered in plastic processing operations such as injection molding (cf. Dienes, 1947)?
’ Westvaco. Covington Research Center, Covington. V a . 24426. 342
Ind. Eng. Chem., Fundam., Vol. 13, No. 4,
1974
Experimental Equipment (a) Apparatus. Figures 1 and 2 in part I describe the experiment under consideration. The fluid to be tested completely fills the gap between two disks a t an initial separation of 2ho. A constant force F is applied to both disks at time t = 0 and the separation 2h is recorded as a function of time. A schematic diagram of the apparatus used in the investigation is shown in Figure 1. The apparatus contained essentially the same components as that used by Dienes and Klemm (1946) with several unique features: (i) the position of the top plate as a function of time could be measured at three equally spaced points on the disk circumference; (ii) very sensitive linear transducers (LVDT’s) were used to measure h ( t ) ,and runs for ho down to 0.0015 cm were made; (iii) the weight of the top plate was balanced by three springs as shown in Figure 1 so that recovery experiments could be performed; and (iv) the apparatus could be easily dismantled for cleaning by sliding off the spring support ring and the LVDT mounting ring. The force was applied by a plunger which acted through a linear bearing. The bearing was mounted in a frame, and the plunger shaft was connected to a lever arm which multiplied the force (not shown in Figure 1).It was also possible to place the apparatus shown in Figure 1 into a tensile-compression testing device to obtain higher forces, although this was not done in this series of experiments. Glass disks, in. thick and either 2 or 11, in. in diameter were used. A detailed description of the procedure that was used for each run can be found elsewhere (Leider, 1973). (b) Fluids. The following four fluids were used in the investigation: (1) a Dow-Corning silicone oil No. 100,000, molecular weight not available, (2) a 1.0% by weight solution of hydroxyethyl cellulose, HEC, dissolved in 50% water, 50% glycerine; the HEC was obtained from Union Carbide, molecular weight not available, (3) a 0.5% solution of Separan 273 (polyacrylamide) in glycerine; the Separan 273 was obtained from the Dow Chemical Co., molecular weight not available, and (4) a polyisobutylene, PIB, solution which contained 20.3% Vistanex LM-MS (Enjay Chemical Co., molecular weight of 2.5 x lo4), 5.2% Oppanol B-200 (Badische Anilin-und-Soda-Fabrik, BASF, molecular weight of 4.5 x 106), and 74.5% Oppanol B-1 (BASF, molecular weight of 400). The HEC and Separan solutions were easy to prepare; the PIB solution was prepared as outlined by Kaye, et al. (1968).
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Figure 2. Viscosity r~ and first normal stress difference N I = - 722 for silicone as functions of shear rate.
711
Characterization of the Fluids (a) Fluid Parameters. The viscosity q = - r X y / y and primary normal stress difference iV1 = T~~ - T~~ were measured as a function of shear rate using a Weissenberg R-16 rheogoniometer. The results for the four fluids obtained a t 23.7 h 0.1"C and for gap angles of 1" and 2" are shown in Figures 2-5. Note that the silicone oil showed a constant viscosity, whereas the other fluids exhibited extended power-law regions. Also, the HEC solution and the silicone fluid were relatively inelastic as seen from the fact that their measured values of iV1 were considerably lower than those for the Separan and PIB solutions a t corresponding shear rates. For the purpose of correlating the squeezing flow data, it was convenient to obtain some well defined parameters from the steady shear data, Figures 2-5. The power-law constants m and n for the viscosity function are fairly well established parameters describing the fluid. In addition to these, power-law constants rn' and n' can be defined for the iV1 us. i. curve. Thus we have for steady-state shearing flow
From the set of constants m, m', n, n' it is possible to define a relaxation time for the fluid as = (m'/2m)l/(n'-n) (2 Although other definitions for X have been given (see, e.g.,
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Bird, 1965; Truesdell, 1964; Slattery, 1968), eq 2 was used for the reasons given in part I. Values of m, m', n, and n' obtained from Figures 2-6 are listed in Table I along with computed values of A. Also included in Table I are the zero shear viscosities 70 for the fluids since these could be easily determined from the 7 us. i. plots. (b) Dimensionless Groups. The experimental squeezing flow data to be presented consist of measurements of the half-time t l l z , the time required for the disk separation to reach one-half its initial value, obtained for the four fluids using various initial gap separations, forces, and radii. For this problem the geometric variables are R and ho, the fluid parameters are taken to be m, n, A, and n', and the dynamic variables are F and t l l z . A characteristic velocity is not necessary since ho/tl,z is a measure of velocity. Applying the Buckingham Pi theorem (cf. Bird, et al., 1960), where the fundamental units are mass, length, and time, yields the result that only five groups are needed for an adequate correlation. Motivated by the Ind. Eng. Chern., Fundam., Vol. 13, No. 4 , 1974
343
Table I. Material Parameters from Steady Shear Flow ~~~
~
m',
Fluid
g/(cm sec)"
n
g/(cm sec)"'
n'
X, sec (eq 2)
Silicone HEC Separan PIB
1.06 x 103 2.1 x 102 2.5 x 10' 1.4 x 103
1.00 0.Q33 0.400 0.350
80 300 4100 1.7 x 104
I.50 0.567 0.830 0.677
0.00143 0.238 129 247
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100
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101
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theoretical analysis presented in part I, the following groups are suggested
Note that if density were included, although it was not varied significantly in the experiments, then a Reynolds number would have to be included with the above groups. If a correlation cannot be obtained using the groups of eq 3, then it can be concluded that either one is not relating the groups in the proper manner, something has been excluded like inertia effects, or else the fluid parameters in steady shear flow are insufficient to charact,erize the fluid and its behavior in squeezing flow, a transient experiment.
Squeezing Flow Results and Discussion In order to establish that the experimental results are independent of geometry, half-times were measured using different diameter disks and different applied forces (in practice it is easier to vary the initial gap separation, 2h0, instead of the force). The results for the silicone fluid are shown in Figure 6. The straight line on the plot is the prediction of eq 7, part I, for n = 1. Keep in mind that on this and the following plot each point represents a separate run. Similar independence on geometry was observed for the other three fluids tested. The data for all of the runs taken a t 22.0 f 0.5"C and plotted as Knf-lln(R/ho)l+(lfn)us. tl12/nX are shown in Figure 7 where f = FXn/irR2rn. This plot represents the main result of the investigation and includes a total of 181 runs for four different fluids. These results cover a wide range of conditions including values of R/h from 25 to 3000, t1/2 from 0.5 to 4000 sec, ho from 0.0015 to 0.10 cm, X from 0.00143 to 247 sec, n from 0.333 to 1.0, forces of 4, 8, and 16 kg, and disks of 2- and ll,-in. diameter. Figure 7 indicates that for tli2/nX > 1.0, eq 7, part I, accurately predicts the half-time. Thus, only two material properties, m and n, are needed to describe squeezing flow 344
Ind. Eng. Chem.,
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13, No. 4, 1974
Figure 7. Dimensionless plot of squeezing flow data representing 181 runs for the four fluids described in Table I. The dimensionless force f is defined as FXn/rR2m, where X is defined by eq 2. The solid curves, labeled with various R/ho values, are from the crude theoretical calculation of eq 24, part I, with a = 0.1, b = 0.24. and n = 0.38.
in this region. When tl12/nX < 1.0, the fluid squeezes out ) the corresponding power-law fluid slower (higher t l , ~ than would for a given force. This effect is not a minor one. At the lower end of the time scale there is a factor of 10 difference between the observed value of tl12 and the prediction of Scott. In order to determine if one is operating in this region, a third parameter X is necessary. For tl/2/nX < 0.1, the data for PIB and Separan separate slightly. This could be because (1)more parameters are needed to describe the material (in this region the time for the process tl12 is much less than the relaxation time of the fluid X and steady-state properties might be insufficient) or (2) the data should be correlated in a different manner in this region, and perhaps a different dependence on A, n, or n' exists here. The author feels that the latter is a more plausible explanation since constitutive equations with four or fewer constants, all of which can be determined from steady shear flow, have been demonstrated to perform fairly well in the transient experiments (cf. Leider and Lilleleht, 1973; Huppler, et al., 1967). Presently, the form of the correlation that will put all of the data on a single curve for tl12/nX < 0.1 is not known. Hopefully, a complete solution to the squeezing flow problem using a realistic constitutive equation will become available and indicate the exact dependence on t1/2in this region. The deviation of the data in Figure 7 from the Scott solution is considered to be a manifestation of elastic effects. In part I the effect of inertia was discussed and it was concluded that when inertia is important, more force would be required to produce the same tll2. Thus, inertia would tend to make the data deviate from the power-law solution in the same manner as observed in Figure 7. That, in fact, inertia is insignificant is demonstrated in
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111
Figure 8. Shear stress growth a t inception of steady shear flow for A+ = 455,compared with the expression in eq 19 of part I.
Figure 6 . The data for silicone, an inelastic fluid, show no tendency to depart from Scott's prediction even for very short t 1 / 2 (high forces). However, the results for PIB and Separan, obtained for the same values of t l ! z , show a significant deviation. Further, applying the first-order correction of eq 14, part I, shows that inertia effects should not be important. Taking h = ho, !i = ho/tl/z, h = ho/ ( t l , ~ )and ~ , p = T O as an estimate, the highest value of 3ph%/5ph for all of the runs was the term 5 p h ( - h ) / 7 p 2.6 x 10-7. This is indeed negligible in comparison to 1 and even a decrease in the viscosity by 104 would not produce a significant correction in eq 14,part I. Also shown in Figure 7 is the prediction of eq 24 of part I for several R/ho ratios. The two parameters a and b were obtained by fitting, approximately, stress overshoot data for the PIB and Separan solutions as shown in Figure 8. The R/ho ratios represented in Figure 7 are reasonable values; for tli2/nX < 0.2, R/ho was between 25 and 200. The introduction of two additional parameters a and b suggests that steady-state properties may be insufficient to predict squeezing flow. At the same time, it is equally likely that a properly formulated constitutive equation, containing parameters all of which can be determined from steady shear, can predict the results shown in Figure 7 . Equation 19, part I, is not intended to be a proper rheological equation; it was introduced merely as a means of estimating the effect of the stress overshoot phenomena. The dependence on R/ho for small values of tllz/nX predicted by eq 24, part I, is probably a result of the many approximations that were made in obtaining the equation. In particular, the substitution r = R and z = ho/2 to obtain eq 23 may have introduced the R/ho dependence which was not experimentally observed. The predictions of Tanner and Kramer were not included in Figure 7 since they both departed from the Scott prediction (1931) in the opposite direction from the data. The apparent success of eq 24, part I, lies in its ability, unlike the Lodge rubberlike liquid and the contravariant convected Maxwell model, to account for the stress overshoot phenomena (see Figure 5, part I).
+
Determining Fluid P a r a m e t e r s from Half-Time Measurements The excellent correlation over seven of the eight decades of tlla/nX shown in Figure 7 suggests that some of the parameters appearing in Table I can be obtained directly from the squeezing flow experiment. By rearranging
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101
102
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Figure 10. Determining power-law parameters for Separan; n is determined from the slope and m from eq 4.
eq 4 we see that plotting FholR3 us. (R/ho)/t1,2on a loglog plot yields a straight line with slope n for an inelastic fluid (one with a very small A). This has been done for the HEC solution as shown in Figure 9. A slope of 0.333 is obtained; the rheogoniometer data give the same value. Further, by taking the ordinate and abscissa corresponding to any point on the straight line, the value of m can be obtained from
This procedure demonstrates that for an inelastic fluid only two points are needed to determine m and n provided one is in the power law region. Obtaining these points is simply a matter of measuring the time required for the disk separation to reach one-half of its initial value. A manipulation of the data as suggested by Oka (1960) requiring the complete h ( t )curve is not necessary. When elastic effects 'are present, the above procedure is valid as long as the data used to construct the plot are for tl,z/nX > 1.0. However, since X is not known, a two point determination is no longer possible. Plotting FholR3 us. (R/ho)/tl12over a range of half-times should result in a curve similar to that shown for Separan in Figure 10. The long half-time data indicate the correct slope of 0.40 is obtained; the data for tl12/nh < 1.0 show a pronounced deviation attributed to elastic effects. When one is dealing with an elastic fluid, at least three points obtained at long half-times are required to determine rn and n confidently. Of course, it is recommended that a complete set of data be taken that covers a large range of t 1 / 2 whether or not the fluid is elastic. The plot in Figure 7 indicates that the relaxation time X Ind. Eng. Chem., Fundam.,Vol. 13,No.4, 1974
345
can also be obtained using half-time measurements. A log-log plot of (R2/F)lIn (R/ho)l+l:nus. tl12/n for a given fluid should result in a curve that exhibits a distinct break at tl12/n = A. This has been demonstrated for only two solutions (having different solvents, polymer-solvent interactions, molecular weights, and molecular weight distributions). In the absence of a complete and thorough theoretical solution to the problem or experimental data to the contrary, this procedure is proposed as a method for obtaining the relaxation time defined in eq 2.
F = magnitude of the force, dyn h = one-half of the disk separation, cm
Summary Remarks The effect of elasticity in the squeezing flow experiment is to resist the applied force, resulting in larger half-times than would be obtained for a comparable purely viscous fluid. This confirms the observation of Appeldoorn (1965) that viscoelastic lubricants perform better than inelastic non-Newtonian lubricants in squeezing flow applications. The theoretical predictions of Kramer and Tanner are in direct contradiction to this experimental fact, suggesting that the rheological equations they use do not describe the fluid properly. By accounting for the effect of stress overshoot in an approximate manner, the correct result, namely, that half-times larger than those predicted by the Scott equation will be observed for rapid squeezing. While a theoretical explanation of elastic effects based on the hydrodynamic solution to the initial and boundary value problem is lacking, Scott's solution assuming quasi-steady state and applying the lubrication approximation to a power-law fluid appears to describe the half-time-force relationship adequately for inelastic fluids.
Greek Symbols i. = shear rate, sec-I
Acknowledgments The author gratefully acknowledges financial assistance provided by the National Science Foundation via Grant GK-24749 and the Vilas Trust Fund of the University of Wisconsin administered by Professor R. B. Bird. Discussions with Professors R. B. Bird, A . s. Lodge, and J. M. Kramer during the course of the investigation were most helpful.
ho = one-half of the initial disk separation, cm m = power-law parameter in eq 1Jyn cm - 2 sec- n m' = power-law parameter in eq 1,dyn cm-2 sec-n' N I = primary normal stress difference = 7 x x - 7 y y , dyn cm-2 n,n' = power-law slopes in eq 1 t = time, sec t l l z = time for the disk separation to reach one-half its initial value, sec
7 = viscosity, g cm-1 sec-1 70 = zero shear viscosity, g cm-1
sec-1
A= relaxation time defined in eq 2, sec g = Newtonian viscosity, P T~~
= shear stress, dyn cm-2
7 ~ x , T y y = normal
stresses, dyn cm-2
Literature Cited Appeldoorn, J. K., J. Lubrlc. Tech., Trans. ASME, 18, 182 (1965). Bird, R. B.. Chem. Eng. Progr. Symp. Ser. No. 58, 61, 86 (1965). Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena," Wiley, New York, N. Y., 1960. Dickinson, E. J., Witt, H. P., Trans. SOC. Rheol., 13, 484 (1969). Dienes, G. J., J. Colloid Sci., 2, 131 (1947). Dienes, G. J., Klemm, H. F., J. Appi. Phys., 17, 458 (1946). Gent, A. N., Brit. J. Appi. Phys., 11, 85 (1960). Huppier, J. D., Macdonald, i. F., Ashare, E., Spriggs. T. W.. Bird, R. B., Trans. SOC. Rheol., 11, 181 (1967). Kaye, A,, Lodge, A. S.,Vale, D. G., Rheol. Acta, 7, 368 (1968). Landei, R. F., Moser, B. G., Bauman, A. J., "Proceedings of the Fourth International Congress on Rheology," Part i I, pp 663-692, Wiley, New York. N. Y., 1965. Leider, P. J.. "Squeezing Flow Between Parallel Disks," Rheology Research Center Report No. 22, University of Wisconsin, Aug 1973. Leider, P. J., Bird, R. B., Ind. Eng. Chem., Fundam., 13, 336 (1974). Leider, P. J., Lilleleht, L. U.. Trans. SOC. Rheol., 17, 501 (1973). Metzner, A. B., J. Lubric. Tech., Trans. ASME, 90, 531 (1968). Oka, S.,"Rheology," F. R. Eirich, Ed., Vol. 3, Chapter 2, pp 73-75, Academic Press, New York, N. Y., 1960. Parlato, P., M.S. Thesis, University of Delaware, 1969. Reiner, M., "Rheology," F. R. Eirich, Ed., Vol. 3, Chapter 9, pp 431-432, Academic Press, New York, N. Y., 1960. Scott, J. R.. Trans. lnsf. Rubber ind., 7, 169 (1931). Slattery, J. C., AIChEJ., 14, 516 (1968). Tanner, R. I., ASLE Trans., 6, 179 (1965). Truesdeil, C., Phys. Fluids, 7, 1134 (1964). Williams, G., Tanner, R. I., J. Lubric. Tech., Trans. ASME, 92, 216 (1970).
Nomenclature a, b = constants defined in part I f- = dimensionless force = FAn/rR2m
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Ind. Eng. Chern., Fundarn., Vol. 13, No.4, 1974
Received for review October 23, 1973 Accepted May 24,1974
