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Radial flow of viscoelastic liquids
343
Journal of Non-Newtonian Fluid Mechanics, l(l976) 0 Elsevier Scientific Publishing Company, Amsterdam
343-355 - Printed in The Netherlands
RADIAL FLOW OF VISCOELASTIC LIQUIDS PART II: EXPERIMENTAL
MICHAEL C.H. LEE and MICHAEL C. WILLIAMS Chemical Engineering Department, California 94720 (USA) (Received
University of California, Berkeley, Berkeley,
April 15,1976)
Summary Pressure profiles P(r) are measured for a highly non-Newtonian polymer solution undergoing divergent radial flow between circular disks. The liquid is designed to be highly elastic, unlike others examined in radial flow, with (rl f - ~a~)/ 721 > 1 in viscometric shear. Also unlike earlier results, pressure data are found to lie above the predictions of the power-law viscosity model. This may be attributed to elastic effects, as predicted with an Oldroyd model in Part I, providing the fluid time spectrum obeys certain restrictions. The Oldroyd model is incapable of quantitative accuracy, but good curve-fits of P(r) data are possible with empirical modification of the rigorous analysis. Introduction Diverging radial flow between parallel disks may be taken as a hydrodynamic prototype for molding flows encountered in the polymer processing industry. Many important complexities of real molding processes [l] are deliberately avoided here for the purpose of studying the most elementary rheologieal phenomena associated with this nonviscometric flow. Thus, we are concerned only with isothermal steady-state motion of incompressible liquids which are viscoelastic and non-Newtonian (shear-thinning, in this case). In the preceding companion paper [ 11, the general strategies for obtaining theoretical predictions of velocity and pressure in this.flow were outlined and developed specifically for a five-constant Oldroyd fluid. Results reduced to those of several other workers using other fluid models of low-order viscoelasticity. However, it was clear that comparison of model predictions with experimental data could not be qu~ti~tively valid; deficiencies in the model and t~ncation of the series solutions meant that realistic ~e~dependence could not emerge. The primary importance of that development was qualitative, insofar as accom-
344
odating viscoelastic character of wide generality .as welI as inertial effects in predicting dimensionless pressure profiles AP(r) and velocities rrdr, z) and u,(r, z). One of the major results of the preceding analysis ]I] was that APpelas, the contribution of viscoelasticity to the pressure profile AP(r), depended on fluid time constants in such a way that various special cases (corresponding to several wellknown models, among others) predicted different signs as well as grossly different magnitudes for AP,,,. This suggested that radial flow offered the potential for assessing the models themselves, and that experimentation would have relevance to model-building as well as to processing. It appears that Generalized Newtonian Fluid (GNF) models are adequate for predicting Afir) and flow rate & when fluid elasticity - as measured, for example, by N& in simple shear - is small. Laurencena and W~iams [2] found that the power-law CNF, ral = RZ$~, was entireIy adequate for predicting $ for two shear-thinning polymer solutions with 0.07 < N,(+ ) < 0.6; systematic deviations of AP(r) predictions from the data could be attributed to applying the model beyond its range of characterization in simple shear. Such an explanation could not clearly be made for a third solution which showed greater deviations and for which Nw --f 0.8. In all cases [ 21 the power-law prediction correctly indicated that pressure was reduced (when n < 1) below that predicted for a Newtonian fluid with viscosity ne comparable to the low-shear limit of the non-Newtonian viscosity ~(9). However, it was also shown that strain rate gradients in the radial direction were not large enough to make the flow appear substantially different from ordinary shear flow (at least, for fluids of low elasticity), The role of normal stress and other hid-el~tici~ behavior (Nw > 1) simply could not be determined with those data. Data presented by Schwarz and Bruce [ 33 for dilute polymer solutions also showed pressure reductions relative to the Newtonian predictions, but no conclusions could be drawn as to whether this was caused by non-Newtonian viscosity or normal stresses. The latter were not characterized independently, so no Nw is available, and no elastic parameters were evolved by comparing the data with their sophisticated analysis of third-order Rivlin-Ericksen fluid behavior. Most recently, Winter f4] described an approximate analysis of the problem which invoked power-law expressions for ~(9 ), N,(+ ), and Nz(+ )_ He concluded that pressure should be reduced beiow GNF predictions by virtue of the existence of normal stresses. Tests on several polymer melts supported this qu~i~tively but no data on NX and NZ were available, so Winter’s power-law model remains unevaluated. There are really three separate questions involved when evaluating AP(r) data for radial flow: (1) What is the magnitude and type of normal stress contribution, even for lowest-order hydrodynamics [u, = f(z)/r, u, =2O]? (2) Is it important to consider more complex hydrodynamics, such as secondary flows, which could be enhanced by elasticity and shear-dependence of fluid parameters? (3) Is some portion of the data a manifestation of inlet effects associated with high Deborah numbers? t Ng? = Iv1 1721 in shear flow normal stress functionand
of the form c3l = 7)x2, where is the shear stress.
~~1
iV1 =
711-
722
is the
primary
345
Schwartz and Bruce [ 3f grappled with (2) for one viseoelastic model but had inadequate data. Winter [4f confronted (1) with power-law rheology but also had no normal stress data. In Part I [ 11, we discussed (2) in a general fashion, and here we accompany it with data which bear on (l), (Z), and (3). These measurements contain, for the first time, results for a demonstrably high-elasticity fluid (N, > 1). It is shown that AP(r) is increased relative to the GNF predictions, in accord with predictions of an Oldroyd model in the companion paper [ 11, despite the shear-thinning nature of q(y). Quantitative agreement between the model and data, however, is possible only with empirical modification of the rigorous prediction. Experiment
The apparatus of Laurencena and Williams [Z] was employed with only minor changes [ 51. As shown in the schematic flow loop of Fig. 1, fluid was held in a reservoir and forced under air pressure through a calibrated rotameter, into a vertical approach tube, and thereafter into the intradisk space. The radial flow exited into a trough for return to a holding tank, but recirculation was avoided in order to minimize mechanical degradation of the polymer (none was noted). The intradisk separation, 2 h, was varied between 0.15 and 0.05 cm. Previous workers [ 2,3] have reported difficulty in ~ontroll~g and even measuring h because the pressure imposed during testing causes the disks to be forced apart7 and this has sometimes been noted too late to avoid error. This severely limited the utility of the Schwarz and Bruce data ] 31, and led Laurencena and Williams [ 2]
Fig. 1. Row loop hr radial flow apparatus. (‘I) Resew&r tank, high pressure. (2) Receivef tank, atmospheric pressure when recebing. (3) Air pressure. (4) Vacuum line, for recycfing between tanks if needed. (5) Rotameter. (6) Lucite approach tufie. (7) Trough around radial flow section.
346
Section AA’ Fig. 2. Test region. (1) Lucite upper disk, l-in. thick. (2) Aluminum rim. (3) Aluminum spokes (trapezoidal) to brace the Lucite. (4) Pressure transducer, movable horizontally along the dashed line as the lower plate moves. (5) Tracer injector.
to calibrate h as a function of total thrust on the disks. In the present study, dial indicator readings were taken on relative displacement of the disks throughout each run. A transient of l-3 s duration was always noted, after which h became stabilized. The dial indicator was also used to detect imperfections in disk smoothness (kO.004 cm) and parallelism (gradient of 0.002 cm/ft). Because of the smallness of h, the sum of these minor defects can be quite large in a relative sense, but here the local uncertainties in local spacing bloc were always within 5-1576 at most and probably the average value of bloc was well within 5% of that which was recorded during flow. A cross-section of the disks is seen in Fig. 2, showing the strain gauge pressure transducer embedded in the lower disk. Since the lower “disk” was actually a large pan mounted on a milling machine table, it could be moved laterally to any position for pressure measurement. Ten standard positions were generally used, with the inner nine pressures referred differentially to the outermost one beyond the rim. No significant excess pressures were ever noted at the rim position, r* = R = 12 in. +. Pressure readings were recorded only after pressure transients had subsided, within 2-10 s. A total of 30 runs were made with the single test fluid, with AP* = P*(O.l R) P*(R) across the disks ranging from 1.4 to 12.9 psig. Variation of pressure and h t The variables r, z, P, 7, Tij, and time coefficients A, K are dimensionless when used without asterisk, and bear their usual dimensions when superscripted with (*). All other symbols here are dimensional in every usage. Note that r = r*/h and z = z*lh.
led to a range of flow rates from 0.25 to 4.35 cm3/s. Dye tracer observations were made (see injector in Fig. 2) through the upper disk, which was 1 in. Lucite and transparent, Temperature was kept at 70” * 0.5”F by a room thermostat. Test liquid The objective was to create a fluid w@h Nw > 1. A nonionic polyacrylamide (Separan NP 10, Dow Chemical) with Mw 2 lo6 was dissolved with gentle stirring in a mixed glycerin/water solvent (equal-weight proportions). The glycerin increases solvent viscosity and, hence, also the solution relaxation times, and, moreover, suppresses water evaporation so that concentration gradients and even gradual bulk changes can be virtually eliminated. Solution concentration was 1.0% polymer, density was 1.14 gfcm3, and limiting viscosity as measured by fallingsphere methods was estimated [ 51 at no z 237 poise. Shear-dependent viscosity and normal stress N1 were determined with a Weissenberg Rheogoniometer R-17, using a cone-and-plate geometry of cone angle 0.5” and diameter 10 cm. Shear rate and all stresses are uniform throughout the fluid in this geometry; no secondary motion or fluid ejection was seen. From these data, N,(q) could be computed as well as the Weissenberg relaxation time A,($) = Nw /y. Results
Rheogoniometer data are displayed in Fig. 3 and near-power-law dependence is observed. Using 721 = my” and NI = n~‘+~‘, we find n = 0.474 and n’ = 0.897. The solid curves, representing r) - yn-’ and Nw - q”‘-“, are read from the lefthand abscissa. Since the range of Nw is 0.7-7, the objective of creating a high-
Cone-ond-Plate Rhe~oni~etef
IO
-_10-l
9, poise NW
-
XW’ set
Fig. 3. Shear-dependence of viscosity q. Weissenberg number NW = (711- 722)/712; and Weissenberg relaxation time hw = NW/~ for homogeneous cone-and-plate flow in the Rheogoniometer. Cone angle was 0.5”, platen diameter 10 cm.
348 0(h)-3,
s-x-’
IO5
Fig. 4. Pressure drops with the highly elastic fluid of Fig. 3, compared with the Newtonian prediction (upper left) and the power law prediction (lower right), as appropriate functions of flow rate Q and intradisk half-spacing h. Ten representative runs from among the thirty are shown and identified by number.
elasticity fluid has been achieved. Also shown is hw, read from the right-hand ordinate, which behaves as yn’--n-l - i,-“*577 which is very close to ~(9) i,-o.526. This has been noted previously for many fluids; in terms of the usual Maxwell mechanical analogue (A z q/G) this suggests that the elastic shear modulus G is nearly shear independent. The range of S& represented here, to 1083 dynes/cm2, easily encompasses that encountered for wall shear stress 7,(r) in the radial flow data (as estimated with power-law viscosity assumptions [ 51). Thus, the data in Fig. 3 may be used to characterize the fluid in the experiments to be described next. Radial flow
only the total pressure drop, over the span 0.1 R to R terms), corrected for what are presumed to be inlet effects. Corrections were made by extrapolating the P*(r*) profiles * smoothly to the innermost position rl, i gn oring the data point at rl which was always high. The extrapolated intercept P,* (rl) was used to compute AF’,*= P,* -P*(R). Typical results, a selection of ten representative runs, are displayed in Fig. 4 and compared there with two theoretical predictions. One of these is the Newtonian viscosity prediction [ 21: We initially present
(rl to r2 in dimensionless
3Q AP:
=r)o
[
4n
1
W-2/r1) h3
and the other is the power-law
(1) viscosity
prediction
[ 21
t Transducers measure the total stress, P* = (p* - r&)lz*=a.
349
(2) Equations (1) and (2) are both derived from lowest-order hydrodynamics. Clearly, eqn. (1) with q. = 237 poise overpredicts badly, but this is expected because the fluid is known to be shear-thinning. However, eqn. (2) accomodates a very realistic non-Newtonian representation and still badly underpredicts; thus, the technique found by Laurencena and Williams [ 21 to be successful for fluids with Nw < 1 does not prevail here, when Nw > 1. This result also disagrees with Winter [ 41, whose analysis and data on melts showed AP* < AP& . However, as discussed in the companion paper [ 11, the present data are qualitatively consistent with predictions of the 5-constant Oldroyd model and some others, providing parameters are chosen in certain ways. Discussion The implication of Fig. 4, together with our preceding pressure prediction [ 11, is that normal stresses are responsible for AP* > AP& with this particular fluid. However, a direct test of this contention cannot be realistically made, because eqn. (32) of ref. 1 for AP does not accomodate the high-order shear dependence of properties which is present in this fluid. Still, an empirical approach will be taken here to use the framework of eqn. (32) as a vehicle for representing elastic effects, but allowing fluid parameters to vary as a function of shear. This permits P(r) data to be fitted quite successfully, as seen in Fig. 5.
14
Model Run No. 24:
t
Co&orisons Q = 3.13 cckec 2h = 0.100 cm
Newtonion Viscosity, “IO 237 p q
A?", Psig
\ Extended Powerlow
Jeffreys With Coefficients
0.3 r/r2 = r*/R
0.6
I
Fig. 5. Measured pressure profiles compared with predictions for Newtonian (eqn. (l)), viscometric power law (eqn. (2)), and shear-dependent Oldroyd fluid models (eqns. (3b), (5), (8), and (14)). Assumptions made for the latter are discussed in the text.
350
Rationale for power-law Oldroyd model The curvature of P(r) data such as is shown in Fig. 5 could, in principle, be predicted by the strategies described earlier [l] if (a) the Oldroyd model were capable of fitting quantitatively the characterization data of Fig. 3, and (b) all terms of the series in eqn. (32) were available. This series, in powers of l/r, was originally written : AP = P(r) - P(rz) = AP,, + AP,,, (In r) (r-2)
+ AP,,,
+ .. .
(r-2)
(r-2k)
+- (order of r)
(3a)
where A Pvi, and A Ph,, contained the lowest-order rheological information (no) and AP,,, the first-order corrections (quadratic normal stress Psi: and deviations from Newtonian viscosity pG$) due to fluid elasticity. With all higher terms, the series would contain all rheological information so that P(r) curvature could be predicted as accurately as the model allows. Interpreted on a different physical basis, terms in the series can be decomposed and regrouped as: AP(r) = [ AP,, + A@::: +
+ higher viscosity terms] +
[APiner+ higher inertial terms] +
+ [Ape::
+ higher normal stress terms]
(3b)
= APv + AP, + AP,, where the viscosity, inertial, and normal stress contributions each contain all model parameters (no, X1, X2, K~, K~). Furthermore, each of the three contributions reflects the shear dependence of material properties. However, the Oldroyd model would still be inadequate even if the series could be collapsed as in eqn. (3b); its deficiencies in fitting the y-dependence of real data are well known [ 61. We will therefore use power-law expressions to represent APv, AP,, and APNs, as justified by the data of Fig. 3. For APv we will employ AP& from eqn. (2). First, however, we remove dimensional objections to the power law by using: r)
=
#yy
*/yoy
(4)
so that q” and q” are material parameters with units of poise and s-l, respectively, and m = q~~(y~)‘-~. In this notation, the dimensionless version of eqn. (2) becomes:
““=(l
LB”-l [@n-l -n)NO,,
_ (IL)“-11
where 1+2n
R
p12= (-4x)%
(64
=p”
jl&
(6~)”
v”
In obtaining eqn. (5), we have used the power-law expression for shear rate [when u, = f(z)/r] at the disk wall: (7) and the definition of dimensionless pressure, P = P*/pp. Equation (5) is known ]2] to be sufficient for power-law-viscosity fluids in radial flow when iVW< 1 and Deborah numbers are small. For AP, we use APi=-
$
KJ$)2
-($)‘I
(8)
where (9) which is derived in the Appendix. This contribution is invariably small in real polymer processing operations, being dominated by APv and AP,,, and was almost negligible for experimental conditions here. The shear-dependent normal stress contribution is more difficult to resolve, since eqn. (32) [l] does not indicate how the functions N,(q) = rll - 722 and Nz(+i) = 722 - ~23 measured in viscometric shear can be accomodated in radial flow. Our initial approach is simply to endow the viscosity and time “constants” in eqn. (32) with +-dependency and evaluate them at the wall, n(qz), xF($$), . .. First, from eqn. (32) we obtain Ape:: and can then write for APns in eqn. (3b): AP,,
= &- ae [($I2
-($-r]
+ higher terms
w-0
where : c = 4(X1 -X,)
- 3(K1 - Ks)
(W
t The characteristicvelocity is defined as V = Q/Rh. * The conventional “power-law Reynolds number” is NE, = A&B’“. * Note that Winter’s data [4], showing AP < APm, can be qu~i~tively representedif C < 0. This is entirely possible for a shear-thinningfluid; one such case is h2 = 0.6 hl , K1 = 0.7 hl , ~2 = 0.1 hl, not extremely different from eqn. (15) which gives C > 0.
352
in terms of dimensionless time constants h1 = hf V/h, etc. A plausible generalization of the lead term for a power law fluid would be to insert ~(9:) from eqn. (4) into the Reynolds number and, in evaluating C(y), employ: AT(r*) = h~(~~/~“)“-l
(12)
or, in dimensionless form: (13) with similar expressions for Xs, K~, /t2 having identical exponents v. However, this insertion of NRe(r) and C(r) into eqn. (10) does not give satisfactory r-dependence to represent our data. It does indeed come close to Winter’s prediction [ 41 ( APNs - l/r”‘) * but this functionality is too severe at small r. In reference to eqn. (lo), the defect is related to retention of the binomial rv2 - rz2, which becomes excessively large at small r. We shall therefore suppose that the hydrodynamics hidden in the “higher terms” of eqn. (10) would modify this, so that r-’ - rF2 may be replaced by r-& - rF” where 1 < (Y< 2. The expectation that Q!< 2 is based on the observation that terms in this series alternate in sign and, thus, the net result is a suppression of the r-dependence suggested by the first term. The final empirical expression to be used here is: APNs
=
COB”+-’ 5 fie
KT1
(4)“’
[(Y
-(q-J
(14)
where i? is identical in form to C in eqn. (11) but contains time constants such as Xy, rather than h:. This expression retains the original [ 11 result that the time-constant spectrum affects the sign and magnitude of AP,s rather sensitively, through C. This is not at all accomodated by the power-law treatment [ 41 employing N1 and N2. One defect in the latter seems to be the identification of the uiscometric functions N, and N2 with the radial-flow functions r,, - r,, and r,, - 7e0 ; even for the limiting case of Newtonian (nonelastic) fluids this leads to substantial error [ 11. Model predictions and data Evaluation of eqn. (14) requires knowledge of the time-constant spectrum. General arguments [l] restrict the Oldroyd spectrum in a number of ways when these parameters are true constants. We shall choose the power-law constants to satisfy the same criteria and, specifically, to agree with the companion paper [l]. X0,= 0.5 hy ,
IC; = 0.6 h;,
K”Z= 0.15 A:
* The dominant term in Winter’s result is APN~ - (r-“’ - rz”’ ) which, for r < r2, is approximately r-“‘. Our modification of eqn. (10) to this point gives APNs - PA (rm2 - rF2) with y - r-l. For power-law properties, and again neglecting ry2, this becomes r- (n+u). Using the definition of hw to identify v = n’ - n, we also obtain AP,, - r-“‘.
(15)
353
This leaves $O, $, and CYas the remaining parameters. We will choose To = 1 s-l so that no will be numerically equal to m, but Xt and (Ymust be evaluated as a pair to optimize curve-fits of P(r) data. For a given (Y(and h), a plot of APT uersus Q predictions can be prepared and h! selected to agree with pressure-drop data for that h. This is ihustrated in Fig. 6 for values 0 < Ai < 50 s; the case hy = 0 represents the power-law viscosity prediction alone, without AP&. The data points superimposed on Fig. 6 suggest that hy = 14.5 s provides a reasonable fit for all runs with 2 h = 0.100 cm. In this way, various pairs of a! and hy can be selected, but not all are successful in fitting pressure profiles. If cuis too close to 2 the original defects are retained, with strongly concave-up curvature to P(r) at small r. The highly successful curve-fit shown in Fig. 5 was obtained with LY= 1.2, corresponding to Xy = 14.5 s as obtained from Fig. 6, and (Y= 1.2 leads to such reasonable shapes that it was adopted for all further efforts as well. Results for several runs, representing the extremes of h values, are shown in Fig. 7. It is seen that very good fitting of P(r) is possible with a! = 1.2; the principal deviating points are the spuriously high innermost ones which apparently represent inlet effects. Since they were taken at r* = 0.1 R while the actual flow inlet was at R. = 1 in. = 0.083 R, some anomalies could have been expected. For the narrowest channel, this means the first point represents a dimensionless downstream position of only Or*/2 k! = 0.01’7 R/2 h = 7.85, but the second point is downstream at 26.3 and most inlet effects have died away. The fact that the innermost points deviate more severely with decreasing h indicates a Deborah number phenomenon rather than mere velocity profile re~~gement.
15.
I 2h = 0.100 a= 1.20
I
I
I
Extended Jeffreys Model (Powerlaw Coefficients) a = 1.20
I
cm
0 0.
I
0.3 r/r2
0.6
1
I
= r*/R
Fig. 6. Theoretical predictions of overall pressure drop versus volumetric flow rate, for selected values of hp in the Oldroyd model and 2 h = 0.100 cm. Data from three runs are superimposed. Fig. 7. Oldroyd model predictions compared with data for five runs, using ff = 1.2 in eqn. (14).
The magnitudes of key dynamical parameters are of special interest; we shall comment on Run 20, the upper curve in Fig. 7. These conditions gave Nk, = 0.02 and, if an r-dependent Reynolds number were defined as pVh/q(+,), it would not exceed 0.05 even at R0 according to power law y,-estimates. However, elasticity effects are large for all r, with Nw at the wall ranging from 1 (at R) to 2 (at 0.1 R). It is clear that an important shortcoming exists in the empirical curve-fitting technique employed here. Figure 7 shows that different values of hy are needed to characterize the fluid as h is varied, when (Yis held constant. From a large number of trials, it was found that approximately hyh = 0.75 when a! = 1.2. This difficulty could be circumvented in some measure if CY= a(h) - for example, all the data in Fig. 7 could be fitted by Xy = 19 s if we choose CY= 1.28 when 2 h = 0.1 cm and 01= 1.15 when 2 h = 0.066 cm. But the curve-fits would be slightly less good, the choice of 1ywould still be uncertain in general, and the range of CY available for this sort of variation very limited for proper curve-shape prediction. There is probably no simple way to avoid this unsatisfactory predicament, in view of the rather naive attempt here to force an extension of the lead term in an infinite series for APNs. This has ignored the kinematics of flow away from the wall and higher-order couplings between the rheological properties and the hydrodynamics leading to complex velocity fields. The latter criticism could also be leveled at the use of A PpL to replace the lead term in AP, . With all these shortcomings, it is not surprising that optimum curve-fits of data cannot be achieved a priori. It is likely that numerical analysis, using a realistic fluid model, would be required to achieve accurate predictions in general. Tracer observations A 10% mixture of blue-black ink in the test fluid was used as a tracer. It was injected into the fluid layer immediately adjacent to the Lucite column wall at a rate of 0.1 cm3/s. Visual observation suggested that this perturbation to the bulk flow was insubstantial, and the 5-7 seconds required for the tracer to enter between the disks was deemed long enough for typical hydrodynamics to be restored, even if mild distortions had occurred upon injection. Photographs [ 51 of the tracer pattern were taken for conditions of 0.85 < Q < 2.85 cm3/s and 2 h z 0.1 cm, but their quality is not adequate for reproduction here. Ripple patterns of undetermined depth were again noted, similar to those reported by Laurencena and Williams [2], with the ripple spacing larger at higher Q. Purely Newtonian fluids do not reproduce this pattern [ 21. The ripple may be merely a manifestation of inlet effects characteristic of highly elastic fluids. It might also stem from secondary motions akin to those predicted for certain Oldroyd-type models with constant properties [ 11, although the shear-dependent version of the Oldroyd model presented here does not make such predictions [ 51. There also seemed to be a slight asymmetry to the tracer pattern, with a tendency for streamlines to curve laterally. This might be due to geometrical irregularities, but the length of the curve rules out local imperfections as the cause. The presence of a ve component is at variance with the postulate [l] that ve = 0, but the possibility of flow cells with periodic e-spacing cannot be ruled out entirely.
355
References 1 C.H. Lee and M.C. Williams, J. Non-Newtonian Fluid Mech., 1 (1976) 323-341. B.R. Laurencena and M.C. Williams, Trans. Sot. Rheol., 18 i1974) 331. W.H. Schwarz and C. Bruce, Chem. Eng. Sci., 24 (1969) 399. H.H. Winter, Polym. Eng. Sci., 15 (1975) 460. C.H. Lee, MS. Thesis, University of California, Berkeley, 1974. R.B. Bird, RX. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, Wiley, New York, 1976. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, Chap. 7. J.H. McGinn, Appl. Sci. Res. Sect. A, 5 (1955) 255. H.W. Woolard, J. Appl. Mech., 24 (1957) 9,644.
Appendix Inertial contributions to the total pressure drop are influenced by fluid rheology only to the extent that velocity profiles are affected. This suggests a simple approximate method for estimating APi with the Bernoulli Equation which gives exact results in limiting cases. If friction loss is neglected, the pressure variation due to inertia alone is [ 71: AFT = P*(r*) -P*(R)
= -$
p
--
(Al)
where ( 1 represents a z-average across the channel and, of course, is still r-dependent. When the velocity profile u:(T*, z”) is known, eqn. (Al) can be evaluated for any fluid in laminar flow (or turbulent flow, if u,* represents a time-averaged profile). Using the power-law solution for u, = f(z)/r presented elsewhere [ 21 for creeping flow, which led also to AP, in eqn. (5), we obtain eqn. (8). Despite this being only an approximation because u,* had its origin in a total neglect of inertia, eqn. (8) is exact as the lowest-order perturbation to Newtonian (n = 1) creeping flow [ I]. This reinforces our confidence in it as an inertial perturbation when nonNewtonian viscosity is involved. Furthermore, the sum APv + A& for Newtonian fluid has been confirmed expe~men~ly [8,9] as an excellent predictor of AP even at high Na,. This was found true also in the turbulent case, where APi dominates AP and flat velocity profiles can be assumed; the latter corresponds exactly to eqn. (8) in the limit n --f 0.
Journal of Non-Newtonian Fluid Mechanics, l(l976) 0 Elsevier Scientific Publishing Company, Amsterdam
343-355 - Printed in The Netherlands
RADIAL FLOW OF VISCOELASTIC LIQUIDS PART II: EXPERIMENTAL
MICHAEL C.H. LEE and MICHAEL C. WILLIAMS Chemical Engineering Department, California 94720 (USA) (Received
University of California, Berkeley, Berkeley,
April 15,1976)
Summary Pressure profiles P(r) are measured for a highly non-Newtonian polymer solution undergoing divergent radial flow between circular disks. The liquid is designed to be highly elastic, unlike others examined in radial flow, with (rl f - ~a~)/ 721 > 1 in viscometric shear. Also unlike earlier results, pressure data are found to lie above the predictions of the power-law viscosity model. This may be attributed to elastic effects, as predicted with an Oldroyd model in Part I, providing the fluid time spectrum obeys certain restrictions. The Oldroyd model is incapable of quantitative accuracy, but good curve-fits of P(r) data are possible with empirical modification of the rigorous analysis. Introduction Diverging radial flow between parallel disks may be taken as a hydrodynamic prototype for molding flows encountered in the polymer processing industry. Many important complexities of real molding processes [l] are deliberately avoided here for the purpose of studying the most elementary rheologieal phenomena associated with this nonviscometric flow. Thus, we are concerned only with isothermal steady-state motion of incompressible liquids which are viscoelastic and non-Newtonian (shear-thinning, in this case). In the preceding companion paper [ 11, the general strategies for obtaining theoretical predictions of velocity and pressure in this.flow were outlined and developed specifically for a five-constant Oldroyd fluid. Results reduced to those of several other workers using other fluid models of low-order viscoelasticity. However, it was clear that comparison of model predictions with experimental data could not be qu~ti~tively valid; deficiencies in the model and t~ncation of the series solutions meant that realistic ~e~dependence could not emerge. The primary importance of that development was qualitative, insofar as accom-
344
odating viscoelastic character of wide generality .as welI as inertial effects in predicting dimensionless pressure profiles AP(r) and velocities rrdr, z) and u,(r, z). One of the major results of the preceding analysis ]I] was that APpelas, the contribution of viscoelasticity to the pressure profile AP(r), depended on fluid time constants in such a way that various special cases (corresponding to several wellknown models, among others) predicted different signs as well as grossly different magnitudes for AP,,,. This suggested that radial flow offered the potential for assessing the models themselves, and that experimentation would have relevance to model-building as well as to processing. It appears that Generalized Newtonian Fluid (GNF) models are adequate for predicting Afir) and flow rate & when fluid elasticity - as measured, for example, by N& in simple shear - is small. Laurencena and W~iams [2] found that the power-law CNF, ral = RZ$~, was entireIy adequate for predicting $ for two shear-thinning polymer solutions with 0.07 < N,(+ ) < 0.6; systematic deviations of AP(r) predictions from the data could be attributed to applying the model beyond its range of characterization in simple shear. Such an explanation could not clearly be made for a third solution which showed greater deviations and for which Nw --f 0.8. In all cases [ 21 the power-law prediction correctly indicated that pressure was reduced (when n < 1) below that predicted for a Newtonian fluid with viscosity ne comparable to the low-shear limit of the non-Newtonian viscosity ~(9). However, it was also shown that strain rate gradients in the radial direction were not large enough to make the flow appear substantially different from ordinary shear flow (at least, for fluids of low elasticity), The role of normal stress and other hid-el~tici~ behavior (Nw > 1) simply could not be determined with those data. Data presented by Schwarz and Bruce [ 33 for dilute polymer solutions also showed pressure reductions relative to the Newtonian predictions, but no conclusions could be drawn as to whether this was caused by non-Newtonian viscosity or normal stresses. The latter were not characterized independently, so no Nw is available, and no elastic parameters were evolved by comparing the data with their sophisticated analysis of third-order Rivlin-Ericksen fluid behavior. Most recently, Winter f4] described an approximate analysis of the problem which invoked power-law expressions for ~(9 ), N,(+ ), and Nz(+ )_ He concluded that pressure should be reduced beiow GNF predictions by virtue of the existence of normal stresses. Tests on several polymer melts supported this qu~i~tively but no data on NX and NZ were available, so Winter’s power-law model remains unevaluated. There are really three separate questions involved when evaluating AP(r) data for radial flow: (1) What is the magnitude and type of normal stress contribution, even for lowest-order hydrodynamics [u, = f(z)/r, u, =2O]? (2) Is it important to consider more complex hydrodynamics, such as secondary flows, which could be enhanced by elasticity and shear-dependence of fluid parameters? (3) Is some portion of the data a manifestation of inlet effects associated with high Deborah numbers? t Ng? = Iv1 1721 in shear flow normal stress functionand
of the form c3l = 7)x2, where is the shear stress.
~~1
iV1 =
711-
722
is the
primary
345
Schwartz and Bruce [ 3f grappled with (2) for one viseoelastic model but had inadequate data. Winter [4f confronted (1) with power-law rheology but also had no normal stress data. In Part I [ 11, we discussed (2) in a general fashion, and here we accompany it with data which bear on (l), (Z), and (3). These measurements contain, for the first time, results for a demonstrably high-elasticity fluid (N, > 1). It is shown that AP(r) is increased relative to the GNF predictions, in accord with predictions of an Oldroyd model in the companion paper [ 11, despite the shear-thinning nature of q(y). Quantitative agreement between the model and data, however, is possible only with empirical modification of the rigorous prediction. Experiment
The apparatus of Laurencena and Williams [Z] was employed with only minor changes [ 51. As shown in the schematic flow loop of Fig. 1, fluid was held in a reservoir and forced under air pressure through a calibrated rotameter, into a vertical approach tube, and thereafter into the intradisk space. The radial flow exited into a trough for return to a holding tank, but recirculation was avoided in order to minimize mechanical degradation of the polymer (none was noted). The intradisk separation, 2 h, was varied between 0.15 and 0.05 cm. Previous workers [ 2,3] have reported difficulty in ~ontroll~g and even measuring h because the pressure imposed during testing causes the disks to be forced apart7 and this has sometimes been noted too late to avoid error. This severely limited the utility of the Schwarz and Bruce data ] 31, and led Laurencena and Williams [ 2]
Fig. 1. Row loop hr radial flow apparatus. (‘I) Resew&r tank, high pressure. (2) Receivef tank, atmospheric pressure when recebing. (3) Air pressure. (4) Vacuum line, for recycfing between tanks if needed. (5) Rotameter. (6) Lucite approach tufie. (7) Trough around radial flow section.
346
Section AA’ Fig. 2. Test region. (1) Lucite upper disk, l-in. thick. (2) Aluminum rim. (3) Aluminum spokes (trapezoidal) to brace the Lucite. (4) Pressure transducer, movable horizontally along the dashed line as the lower plate moves. (5) Tracer injector.
to calibrate h as a function of total thrust on the disks. In the present study, dial indicator readings were taken on relative displacement of the disks throughout each run. A transient of l-3 s duration was always noted, after which h became stabilized. The dial indicator was also used to detect imperfections in disk smoothness (kO.004 cm) and parallelism (gradient of 0.002 cm/ft). Because of the smallness of h, the sum of these minor defects can be quite large in a relative sense, but here the local uncertainties in local spacing bloc were always within 5-1576 at most and probably the average value of bloc was well within 5% of that which was recorded during flow. A cross-section of the disks is seen in Fig. 2, showing the strain gauge pressure transducer embedded in the lower disk. Since the lower “disk” was actually a large pan mounted on a milling machine table, it could be moved laterally to any position for pressure measurement. Ten standard positions were generally used, with the inner nine pressures referred differentially to the outermost one beyond the rim. No significant excess pressures were ever noted at the rim position, r* = R = 12 in. +. Pressure readings were recorded only after pressure transients had subsided, within 2-10 s. A total of 30 runs were made with the single test fluid, with AP* = P*(O.l R) P*(R) across the disks ranging from 1.4 to 12.9 psig. Variation of pressure and h t The variables r, z, P, 7, Tij, and time coefficients A, K are dimensionless when used without asterisk, and bear their usual dimensions when superscripted with (*). All other symbols here are dimensional in every usage. Note that r = r*/h and z = z*lh.
led to a range of flow rates from 0.25 to 4.35 cm3/s. Dye tracer observations were made (see injector in Fig. 2) through the upper disk, which was 1 in. Lucite and transparent, Temperature was kept at 70” * 0.5”F by a room thermostat. Test liquid The objective was to create a fluid w@h Nw > 1. A nonionic polyacrylamide (Separan NP 10, Dow Chemical) with Mw 2 lo6 was dissolved with gentle stirring in a mixed glycerin/water solvent (equal-weight proportions). The glycerin increases solvent viscosity and, hence, also the solution relaxation times, and, moreover, suppresses water evaporation so that concentration gradients and even gradual bulk changes can be virtually eliminated. Solution concentration was 1.0% polymer, density was 1.14 gfcm3, and limiting viscosity as measured by fallingsphere methods was estimated [ 51 at no z 237 poise. Shear-dependent viscosity and normal stress N1 were determined with a Weissenberg Rheogoniometer R-17, using a cone-and-plate geometry of cone angle 0.5” and diameter 10 cm. Shear rate and all stresses are uniform throughout the fluid in this geometry; no secondary motion or fluid ejection was seen. From these data, N,(q) could be computed as well as the Weissenberg relaxation time A,($) = Nw /y. Results
Rheogoniometer data are displayed in Fig. 3 and near-power-law dependence is observed. Using 721 = my” and NI = n~‘+~‘, we find n = 0.474 and n’ = 0.897. The solid curves, representing r) - yn-’ and Nw - q”‘-“, are read from the lefthand abscissa. Since the range of Nw is 0.7-7, the objective of creating a high-
Cone-ond-Plate Rhe~oni~etef
IO
-_10-l
9, poise NW
-
XW’ set
Fig. 3. Shear-dependence of viscosity q. Weissenberg number NW = (711- 722)/712; and Weissenberg relaxation time hw = NW/~ for homogeneous cone-and-plate flow in the Rheogoniometer. Cone angle was 0.5”, platen diameter 10 cm.
348 0(h)-3,
s-x-’
IO5
Fig. 4. Pressure drops with the highly elastic fluid of Fig. 3, compared with the Newtonian prediction (upper left) and the power law prediction (lower right), as appropriate functions of flow rate Q and intradisk half-spacing h. Ten representative runs from among the thirty are shown and identified by number.
elasticity fluid has been achieved. Also shown is hw, read from the right-hand ordinate, which behaves as yn’--n-l - i,-“*577 which is very close to ~(9) i,-o.526. This has been noted previously for many fluids; in terms of the usual Maxwell mechanical analogue (A z q/G) this suggests that the elastic shear modulus G is nearly shear independent. The range of S& represented here, to 1083 dynes/cm2, easily encompasses that encountered for wall shear stress 7,(r) in the radial flow data (as estimated with power-law viscosity assumptions [ 51). Thus, the data in Fig. 3 may be used to characterize the fluid in the experiments to be described next. Radial flow
only the total pressure drop, over the span 0.1 R to R terms), corrected for what are presumed to be inlet effects. Corrections were made by extrapolating the P*(r*) profiles * smoothly to the innermost position rl, i gn oring the data point at rl which was always high. The extrapolated intercept P,* (rl) was used to compute AF’,*= P,* -P*(R). Typical results, a selection of ten representative runs, are displayed in Fig. 4 and compared there with two theoretical predictions. One of these is the Newtonian viscosity prediction [ 21: We initially present
(rl to r2 in dimensionless
3Q AP:
=r)o
[
4n
1
W-2/r1) h3
and the other is the power-law
(1) viscosity
prediction
[ 21
t Transducers measure the total stress, P* = (p* - r&)lz*=a.
349
(2) Equations (1) and (2) are both derived from lowest-order hydrodynamics. Clearly, eqn. (1) with q. = 237 poise overpredicts badly, but this is expected because the fluid is known to be shear-thinning. However, eqn. (2) accomodates a very realistic non-Newtonian representation and still badly underpredicts; thus, the technique found by Laurencena and Williams [ 21 to be successful for fluids with Nw < 1 does not prevail here, when Nw > 1. This result also disagrees with Winter [ 41, whose analysis and data on melts showed AP* < AP& . However, as discussed in the companion paper [ 11, the present data are qualitatively consistent with predictions of the 5-constant Oldroyd model and some others, providing parameters are chosen in certain ways. Discussion The implication of Fig. 4, together with our preceding pressure prediction [ 11, is that normal stresses are responsible for AP* > AP& with this particular fluid. However, a direct test of this contention cannot be realistically made, because eqn. (32) of ref. 1 for AP does not accomodate the high-order shear dependence of properties which is present in this fluid. Still, an empirical approach will be taken here to use the framework of eqn. (32) as a vehicle for representing elastic effects, but allowing fluid parameters to vary as a function of shear. This permits P(r) data to be fitted quite successfully, as seen in Fig. 5.
14
Model Run No. 24:
t
Co&orisons Q = 3.13 cckec 2h = 0.100 cm
Newtonion Viscosity, “IO 237 p q
A?", Psig
\ Extended Powerlow
Jeffreys With Coefficients
0.3 r/r2 = r*/R
0.6
I
Fig. 5. Measured pressure profiles compared with predictions for Newtonian (eqn. (l)), viscometric power law (eqn. (2)), and shear-dependent Oldroyd fluid models (eqns. (3b), (5), (8), and (14)). Assumptions made for the latter are discussed in the text.
350
Rationale for power-law Oldroyd model The curvature of P(r) data such as is shown in Fig. 5 could, in principle, be predicted by the strategies described earlier [l] if (a) the Oldroyd model were capable of fitting quantitatively the characterization data of Fig. 3, and (b) all terms of the series in eqn. (32) were available. This series, in powers of l/r, was originally written : AP = P(r) - P(rz) = AP,, + AP,,, (In r) (r-2)
+ AP,,,
+ .. .
(r-2)
(r-2k)
+- (order of r)
(3a)
where A Pvi, and A Ph,, contained the lowest-order rheological information (no) and AP,,, the first-order corrections (quadratic normal stress Psi: and deviations from Newtonian viscosity pG$) due to fluid elasticity. With all higher terms, the series would contain all rheological information so that P(r) curvature could be predicted as accurately as the model allows. Interpreted on a different physical basis, terms in the series can be decomposed and regrouped as: AP(r) = [ AP,, + A@::: +
+ higher viscosity terms] +
[APiner+ higher inertial terms] +
+ [Ape::
+ higher normal stress terms]
(3b)
= APv + AP, + AP,, where the viscosity, inertial, and normal stress contributions each contain all model parameters (no, X1, X2, K~, K~). Furthermore, each of the three contributions reflects the shear dependence of material properties. However, the Oldroyd model would still be inadequate even if the series could be collapsed as in eqn. (3b); its deficiencies in fitting the y-dependence of real data are well known [ 61. We will therefore use power-law expressions to represent APv, AP,, and APNs, as justified by the data of Fig. 3. For APv we will employ AP& from eqn. (2). First, however, we remove dimensional objections to the power law by using: r)
=
#yy
*/yoy
(4)
so that q” and q” are material parameters with units of poise and s-l, respectively, and m = q~~(y~)‘-~. In this notation, the dimensionless version of eqn. (2) becomes:
““=(l
LB”-l [@n-l -n)NO,,
_ (IL)“-11
where 1+2n
R
p12= (-4x)%
(64
=p”
jl&
(6~)”
v”
In obtaining eqn. (5), we have used the power-law expression for shear rate [when u, = f(z)/r] at the disk wall: (7) and the definition of dimensionless pressure, P = P*/pp. Equation (5) is known ]2] to be sufficient for power-law-viscosity fluids in radial flow when iVW< 1 and Deborah numbers are small. For AP, we use APi=-
$
KJ$)2
-($)‘I
(8)
where (9) which is derived in the Appendix. This contribution is invariably small in real polymer processing operations, being dominated by APv and AP,,, and was almost negligible for experimental conditions here. The shear-dependent normal stress contribution is more difficult to resolve, since eqn. (32) [l] does not indicate how the functions N,(q) = rll - 722 and Nz(+i) = 722 - ~23 measured in viscometric shear can be accomodated in radial flow. Our initial approach is simply to endow the viscosity and time “constants” in eqn. (32) with +-dependency and evaluate them at the wall, n(qz), xF($$), . .. First, from eqn. (32) we obtain Ape:: and can then write for APns in eqn. (3b): AP,,
= &- ae [($I2
-($-r]
+ higher terms
w-0
where : c = 4(X1 -X,)
- 3(K1 - Ks)
(W
t The characteristicvelocity is defined as V = Q/Rh. * The conventional “power-law Reynolds number” is NE, = A&B’“. * Note that Winter’s data [4], showing AP < APm, can be qu~i~tively representedif C < 0. This is entirely possible for a shear-thinningfluid; one such case is h2 = 0.6 hl , K1 = 0.7 hl , ~2 = 0.1 hl, not extremely different from eqn. (15) which gives C > 0.
352
in terms of dimensionless time constants h1 = hf V/h, etc. A plausible generalization of the lead term for a power law fluid would be to insert ~(9:) from eqn. (4) into the Reynolds number and, in evaluating C(y), employ: AT(r*) = h~(~~/~“)“-l
(12)
or, in dimensionless form: (13) with similar expressions for Xs, K~, /t2 having identical exponents v. However, this insertion of NRe(r) and C(r) into eqn. (10) does not give satisfactory r-dependence to represent our data. It does indeed come close to Winter’s prediction [ 41 ( APNs - l/r”‘) * but this functionality is too severe at small r. In reference to eqn. (lo), the defect is related to retention of the binomial rv2 - rz2, which becomes excessively large at small r. We shall therefore suppose that the hydrodynamics hidden in the “higher terms” of eqn. (10) would modify this, so that r-’ - rF2 may be replaced by r-& - rF” where 1 < (Y< 2. The expectation that Q!< 2 is based on the observation that terms in this series alternate in sign and, thus, the net result is a suppression of the r-dependence suggested by the first term. The final empirical expression to be used here is: APNs
=
COB”+-’ 5 fie
KT1
(4)“’
[(Y
-(q-J
(14)
where i? is identical in form to C in eqn. (11) but contains time constants such as Xy, rather than h:. This expression retains the original [ 11 result that the time-constant spectrum affects the sign and magnitude of AP,s rather sensitively, through C. This is not at all accomodated by the power-law treatment [ 41 employing N1 and N2. One defect in the latter seems to be the identification of the uiscometric functions N, and N2 with the radial-flow functions r,, - r,, and r,, - 7e0 ; even for the limiting case of Newtonian (nonelastic) fluids this leads to substantial error [ 11. Model predictions and data Evaluation of eqn. (14) requires knowledge of the time-constant spectrum. General arguments [l] restrict the Oldroyd spectrum in a number of ways when these parameters are true constants. We shall choose the power-law constants to satisfy the same criteria and, specifically, to agree with the companion paper [l]. X0,= 0.5 hy ,
IC; = 0.6 h;,
K”Z= 0.15 A:
* The dominant term in Winter’s result is APN~ - (r-“’ - rz”’ ) which, for r < r2, is approximately r-“‘. Our modification of eqn. (10) to this point gives APNs - PA (rm2 - rF2) with y - r-l. For power-law properties, and again neglecting ry2, this becomes r- (n+u). Using the definition of hw to identify v = n’ - n, we also obtain AP,, - r-“‘.
(15)
353
This leaves $O, $, and CYas the remaining parameters. We will choose To = 1 s-l so that no will be numerically equal to m, but Xt and (Ymust be evaluated as a pair to optimize curve-fits of P(r) data. For a given (Y(and h), a plot of APT uersus Q predictions can be prepared and h! selected to agree with pressure-drop data for that h. This is ihustrated in Fig. 6 for values 0 < Ai < 50 s; the case hy = 0 represents the power-law viscosity prediction alone, without AP&. The data points superimposed on Fig. 6 suggest that hy = 14.5 s provides a reasonable fit for all runs with 2 h = 0.100 cm. In this way, various pairs of a! and hy can be selected, but not all are successful in fitting pressure profiles. If cuis too close to 2 the original defects are retained, with strongly concave-up curvature to P(r) at small r. The highly successful curve-fit shown in Fig. 5 was obtained with LY= 1.2, corresponding to Xy = 14.5 s as obtained from Fig. 6, and (Y= 1.2 leads to such reasonable shapes that it was adopted for all further efforts as well. Results for several runs, representing the extremes of h values, are shown in Fig. 7. It is seen that very good fitting of P(r) is possible with a! = 1.2; the principal deviating points are the spuriously high innermost ones which apparently represent inlet effects. Since they were taken at r* = 0.1 R while the actual flow inlet was at R. = 1 in. = 0.083 R, some anomalies could have been expected. For the narrowest channel, this means the first point represents a dimensionless downstream position of only Or*/2 k! = 0.01’7 R/2 h = 7.85, but the second point is downstream at 26.3 and most inlet effects have died away. The fact that the innermost points deviate more severely with decreasing h indicates a Deborah number phenomenon rather than mere velocity profile re~~gement.
15.
I 2h = 0.100 a= 1.20
I
I
I
Extended Jeffreys Model (Powerlaw Coefficients) a = 1.20
I
cm
0 0.
I
0.3 r/r2
0.6
1
I
= r*/R
Fig. 6. Theoretical predictions of overall pressure drop versus volumetric flow rate, for selected values of hp in the Oldroyd model and 2 h = 0.100 cm. Data from three runs are superimposed. Fig. 7. Oldroyd model predictions compared with data for five runs, using ff = 1.2 in eqn. (14).
The magnitudes of key dynamical parameters are of special interest; we shall comment on Run 20, the upper curve in Fig. 7. These conditions gave Nk, = 0.02 and, if an r-dependent Reynolds number were defined as pVh/q(+,), it would not exceed 0.05 even at R0 according to power law y,-estimates. However, elasticity effects are large for all r, with Nw at the wall ranging from 1 (at R) to 2 (at 0.1 R). It is clear that an important shortcoming exists in the empirical curve-fitting technique employed here. Figure 7 shows that different values of hy are needed to characterize the fluid as h is varied, when (Yis held constant. From a large number of trials, it was found that approximately hyh = 0.75 when a! = 1.2. This difficulty could be circumvented in some measure if CY= a(h) - for example, all the data in Fig. 7 could be fitted by Xy = 19 s if we choose CY= 1.28 when 2 h = 0.1 cm and 01= 1.15 when 2 h = 0.066 cm. But the curve-fits would be slightly less good, the choice of 1ywould still be uncertain in general, and the range of CY available for this sort of variation very limited for proper curve-shape prediction. There is probably no simple way to avoid this unsatisfactory predicament, in view of the rather naive attempt here to force an extension of the lead term in an infinite series for APNs. This has ignored the kinematics of flow away from the wall and higher-order couplings between the rheological properties and the hydrodynamics leading to complex velocity fields. The latter criticism could also be leveled at the use of A PpL to replace the lead term in AP, . With all these shortcomings, it is not surprising that optimum curve-fits of data cannot be achieved a priori. It is likely that numerical analysis, using a realistic fluid model, would be required to achieve accurate predictions in general. Tracer observations A 10% mixture of blue-black ink in the test fluid was used as a tracer. It was injected into the fluid layer immediately adjacent to the Lucite column wall at a rate of 0.1 cm3/s. Visual observation suggested that this perturbation to the bulk flow was insubstantial, and the 5-7 seconds required for the tracer to enter between the disks was deemed long enough for typical hydrodynamics to be restored, even if mild distortions had occurred upon injection. Photographs [ 51 of the tracer pattern were taken for conditions of 0.85 < Q < 2.85 cm3/s and 2 h z 0.1 cm, but their quality is not adequate for reproduction here. Ripple patterns of undetermined depth were again noted, similar to those reported by Laurencena and Williams [2], with the ripple spacing larger at higher Q. Purely Newtonian fluids do not reproduce this pattern [ 21. The ripple may be merely a manifestation of inlet effects characteristic of highly elastic fluids. It might also stem from secondary motions akin to those predicted for certain Oldroyd-type models with constant properties [ 11, although the shear-dependent version of the Oldroyd model presented here does not make such predictions [ 51. There also seemed to be a slight asymmetry to the tracer pattern, with a tendency for streamlines to curve laterally. This might be due to geometrical irregularities, but the length of the curve rules out local imperfections as the cause. The presence of a ve component is at variance with the postulate [l] that ve = 0, but the possibility of flow cells with periodic e-spacing cannot be ruled out entirely.
355
References 1 C.H. Lee and M.C. Williams, J. Non-Newtonian Fluid Mech., 1 (1976) 323-341. B.R. Laurencena and M.C. Williams, Trans. Sot. Rheol., 18 i1974) 331. W.H. Schwarz and C. Bruce, Chem. Eng. Sci., 24 (1969) 399. H.H. Winter, Polym. Eng. Sci., 15 (1975) 460. C.H. Lee, MS. Thesis, University of California, Berkeley, 1974. R.B. Bird, RX. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, Wiley, New York, 1976. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1960, Chap. 7. J.H. McGinn, Appl. Sci. Res. Sect. A, 5 (1955) 255. H.W. Woolard, J. Appl. Mech., 24 (1957) 9,644.
Appendix Inertial contributions to the total pressure drop are influenced by fluid rheology only to the extent that velocity profiles are affected. This suggests a simple approximate method for estimating APi with the Bernoulli Equation which gives exact results in limiting cases. If friction loss is neglected, the pressure variation due to inertia alone is [ 71: AFT = P*(r*) -P*(R)
= -$
p
--
(Al)
where ( 1 represents a z-average across the channel and, of course, is still r-dependent. When the velocity profile u:(T*, z”) is known, eqn. (Al) can be evaluated for any fluid in laminar flow (or turbulent flow, if u,* represents a time-averaged profile). Using the power-law solution for u, = f(z)/r presented elsewhere [ 21 for creeping flow, which led also to AP, in eqn. (5), we obtain eqn. (8). Despite this being only an approximation because u,* had its origin in a total neglect of inertia, eqn. (8) is exact as the lowest-order perturbation to Newtonian (n = 1) creeping flow [ I]. This reinforces our confidence in it as an inertial perturbation when nonNewtonian viscosity is involved. Furthermore, the sum APv + A& for Newtonian fluid has been confirmed expe~men~ly [8,9] as an excellent predictor of AP even at high Na,. This was found true also in the turbulent case, where APi dominates AP and flat velocity profiles can be assumed; the latter corresponds exactly to eqn. (8) in the limit n --f 0.