Reprinted from
THE PHYSICS OF FLUIDS
Stability of Couette Flow of Dilute Polymer
Solutimpellt:
INTRODUCTION
IT
in smooth turbulent flow may be appreciablyhas lately been shown that the friction coefficient reduced by dissolving high molecular-weight poly-mers in the fluid.' Several investigators havesug-gested that this phenomenon is connected with
non-Newtonian properties of the solutions, such as a shear dependent viscosity or visco-elasticity. It has thus become important to determine the material properties of the drag reducing solutions. At the extremely low concentrations needed to obtain large reductions of the friction coefficient, the viscosity of these solutions often hardly differs from that of the solvents. No dependence of the viscosity on the shear has been found in these dilute solutions.
The visco-elasticity of fluids is usually determined by measuring the normal stresses in concentric flows between plate and cone, parallel plates and rotating cylinders, and in rectilinear flow between stationary concentric cylinders. Such stresses have also been determined from the rate of spreading of free laminar jets or from the inpact of such jets on perpendicular plates. Solutions of drag-reducing polymers at high concentrations exhibit normal-stress effects. No method is available to measure such effects in dilute solutions, because of their relatively low viscosity. Thus it is not clear to which extent the dilute solu-tions are actually viscoelastic.
It has been shown lately, in the analytical
in-vestigations of several theological models describing viscoelastic fluids, that the critical Taylor number and wavelength of the vortex cells depend heavily on the additional parameters appeasing in the rhe-ological equations. This has led the authors to an
* Based on a thesis submitted in partial fulfillment of the requirements for a M.Sc. degree bSi the first author under supervision of the latter.
1 J. H. Hoyt and A. G. Fabula, in Proceedings of the Fifth Symposium on Naval Hydrodynamics (United States
Govern-ment Printing Office, Washington, D. C., 1966).
Lab.
v.
ScheepsbauvAmiI
Printed in. U-SA.VOLUME 9., NUM B Telelinische
littinckv.c19
_1_ 66H. RuBIN AND C. ELATA
Technion-Israel Institute of Technology, Haifa, Israel (Received 9 March 1966)
The critical Taylor numbers Tc and corresponding nondimensional wavelengths of the vortex cells
E, were experimentally determined for various polymer solutions. The measurements show that for the investigated dilute solutions I', increases with concentration while' E. remains constant. These results, together with normal stress differences measured previously in more concentrated solutions of other polymers, are compared with characteristics displayed by several theological models of viscoelastic fluids. It is shown that, of the models considered here, only Ericksen's anisotropic fluid may be suitable to describe the investigated solutions.
experimental investigation of the stability of drag-reducing solutions in Couette flow between
con-centric cylinders. The stability criteria for such
solutions were measured and compared with those exhibited by various rheological models.
BACKGROUND
The stability of the flow of a Newtonian fluid between coaxial rotating cylinders was investigated
for the first time by Taylor.' He predicted the
appearance of torous shaped vortex cellswhich
were later named Taylor vorticesat critical flow conditions. Chandraseldiae defined two nondimen-sional parameters, the Taylor number and the non-dimensional wavelength, which serve as criteria for flow stability. For a system in which only the inner cylinder is rotating and the gap width is relatively small, the Taylor number is given by
[4 n2/0 2)] d4(04/02
T
where d = r2 7'1; = r,/r2; 7-2 and r2 are the radius
of inner and outer cylinder, respectively; co, is the angular velocity of the inner cylinder; and v the kinematic viscosity of the fluid at zero shear rate. The dimensionless wavelength of the vortex cells is defined by e = X d, where X is the wavenumber of the cells. It was found from Chandrasekhar's cal-culations that for Newtonian fluids under the above mentioned conditions the critical Taylor number at which the flow becomes unstable is T = 3390. The corresponding wavelength is E. = 3.12. Num-erous experimental investigations have confirmed these results.
Lately, several investigators have analysed the stability criteria for non-Newtonian fluid models. The following is a review of three such models, their
2 G. I. Taylor, Phil. Trans. Roy. Soc. (London) A223, 289
(1923).
S. Chandrasekhar, Mathematika 1, 5 (1954).
1929
Utica of U.S. Naval
Research
normal stress differences in shear flow and the results
of their stability analysis.
1. The Linear Models of Walters and Oldroyd
Walters' defined a so-called A' fluid by the
equa-tion,
1311.,t) = 2 f
#(tt t') aa?e( (xT) dt', (1)az' axk mr
where Plk is the covariant deviator stress tensor; G a stress relaxation function defined by
t') = f°
N(r)exp [ (t
t')/r] dr;
0 7
N is the distribution,function of the relaxation times
T; x'' (x, t, t') is the position at time t' of a -fluid
ele-ment which is instantaneously at x' at time
t; e(,", = U,.,.); and U is the velocity. This model converts to Oldroyirs A fluid whenN(r) = A(X2/X1) (5(7) A[(X1 X2)/X1] Xi), (2)
where is, Xi, and X2 are constant -fluid parameters: the dynamic viscosity, the relaxation, and the re-tardation time, respectively; is the Dirac function. The relation between the contravariant stress and rate of strain tensors can be expressed in a similar way. The fluid thus defined is indicated by B' by Walters and converts to Oldroyd's B fluid with the same restricting assumption. In cases of steady shear
flow, where P12 = 127 and P13 -= P23 = 0, the normal
stress differences which arise in these fluids can be expressed as follows. For the A' fluid,
P11 P22 = 2721(0; P22 P33 = -'272K0, (3)
where Pil P22 and P22 P33 are the first and second normal stress differences, respectively, with directions defined in Fig. 1, 7 is the rate of strain, and
K2 = f
N(r) dr.0
x2
FIG. 1. Definition sketch of coordinate
directions for flow under steady shear.
K. Walters, in IUT AM International Symposium on
Second Order Effects in Elasticity, Plasticity and Fluid Dy-namics, M. Reiner and D. Abir, Eds. (Pergamon Press, Inc., New York, 1964), p. 507.
5 J. G. Oldroyd, Proc. Roy. Soc. (London) A200, 523(1950).
FIG. 2. Comparison of experimental 5000
results with the critical Taylor
num-bers and dimensionless wavelengths for: (1) Walters' 'A' fluid; (2) Walters'
'B' fluid; (3) Rivlin-Ericksen's
second-order fluid; (4) Ericksen's anisotropic
fluid.
Similarly for the B' fluid,
Pii P22 = 272K0 P22 P83 = 0, (4)
In the case of an Oldroyd fluid, K0 = AL(Xi X2).
The stability problem for the B' fluid was solved
by Thomas and Walters' and for the A' fluid by
Chan Man Fong' in a similar way. The results
indicate that both T, and e, depend on a positive dimensionless number kb defined by k, = Ko/p d2.
In the special case of the Oldroyd fluids, k, = 1.1(X1
X2)/p d2. For the A' fluid an increase in ki will be accompanied by an increase of T. and a decrease of ,. For the B' fluid, an increase in ki will Cause the opposite effect. The changes Of T, and e, with ki are shown in Fig. 2, for both fluids.
2. The Second-Order Rivlin-Ericksen Fluid
This 'fluid nander is defined by
Pii = --Pgii
2cl1e2) 4a2e(i1)41)k 2a8eT, (5)where Pi, is the total stress tensor, p the pressure, g,, the metric tensor, 4) is the convective derivative
of a, and at, a2 and a3 are constant parameters
of the "fluid. The normal stress differences for this
fluid can be expressed as follows:
P11 P22 = 272%; P22 P88 = 72(22 H- 2a3), (6)
while P12 = i.ey and P13 =. P23 = 0.
The stability problem was solved by Datta,9 who showed that the stability criteria depend on a
di.-mensionless number,
- k. = (a2 2a3)r1/2p d3.
The value of k2 may be larger or smaller then zero; for k. = 0, T. and E. Will have the same values as for a Newtonian fluid. An increase in k2 will be accompanied by a decrease of T, and an increase of
ec.,- and vice versa, as shown in Fig,. 2.
5 R. H. Thomas and K. Walters, J. Fluid Mech. 18, 33 (1964).
7 C. F. Chan Man Fong, Rheol. Acta 4, 37 (-1965). 8 R. S. Rivlin, J. Ratl. Mech. Anal. 5, 179 (1956). 9 S. K. Datta Phys. Fluids 7, 1915 (1964).
3. The Anisofropic Fluid of Ericksen
This fluid'° is defined in a Cartesian form by
T
Poi;
2iLe!r ± (32ea'nkn,)nini21.32(eTn2n; el'nkni), (7)
where n; is a unit vector indicating a preferred
direction, which can be found from Ai = in; -I- 00(e;rni
where Di, = U,,,), and $o, SI, S2 and 03 are fluid parameters. In the special case of Couette
flow, investigated by Leslie" where 0, = 0 and
ISol > 1, n,
0 and the following normal stressdifferences were found
P11 P22 = 27711n202(n21 n22),
(8)
P22 P33 = 2.7nin2(82n22 203),
while /313 = 2-y(11 + 03 + S2n2inD, P13 = P23 = 0. The Taylor stability of the anisotropic fluid was investigated by Leslie for the same case. Here T. and ec depend on n n2, and the rheological param-eters. As example Leslie assumed that AL = 02 = 03 and fio = 2, then 94 =
and n: = I. For the case
n1 > 0 and n2 > 0, his calculations gave T.
20.000, while ec came out similar to its value for
the newtonian case; for n, > 0 and n2 > 0, T,
5000, while e,== 5. On the basis of Leslie's analysis,
T- E, were also calculated for different values of
1302 02) and /33 in the case of n, > 0 and n2> 0. The results show that T, increases with any-of these fluid parameters while the'variations of ec were found to be- relatively small. For instance with $o = 1.5,
02 = and 133 0, T, 4.600 and ec 3.1; with
= 1.2 and /32 -= 133 = T, 16.000 and e..=
3.0. The trend in the variation of T. with ee for
increasing $o is shown in Fig. 2.The normal-stress differences have been 'deter-mined from experimental investigations of several polymer solutions. From measurements in a cone and plate rheometer Roberts" has shown that P,1
P,> 0 and P22
P33 = 0, for various solutions.The first result, P,,
11 P22 > 0, is confirmed in general by measurements of Markovitz and Brown,"Ginn and Metzner," and others. Cone and plate
18J. L. Ericksen, Kolloid-Z. 173, 117 (1960).
11 F. M. Leslie, Proc. Cambridge Phil. Soc. 60, 949 (1964).
12 J. E. Roberts, Nature 179, 487 (1957).
"H. .Markovitz and D. R. Brown, in IUTAM
Inter-national Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, M. Reiner and D. Abir, Eds. (Pergamon Press, Inc., New York, 1964), p. 585.
14 R. F. Ginn and A. B. Metzner, in Proceedings of the Fourth International Congress on Rheology, E. H. Lee, Ed.
(Interscience Publishers, Inc., New York, 1965), Pt. 2, p. 583.
measurements with a 3% solution of polymethyl-methacrylate in dimethyl phthalate by Adams and Lodge" indicate, however, that though P,111 P22
at first increases, then decreases with strain rate, eventually reaching negative values. While Roberts
obtained /323 P33 = 0, experiments of Greensmith and Rivlin" and Adams and Lodge with a polyiso-butylene solution in dekalin (which is one of the solutions investigated by Roberts) show that P22 P. > 0. Hayes and Tanner' investigating the flow of polymethylmethacrylate in toluene also found the same trend. The results of measurements by Markovitz" on a 5.39% solution of polyisobutylene in cetane can be expressed by (PI, P22)/72 =
(P22 P22)/72 = 0.6 g/cm, for limiting small rates
of strain. Lodge" reports that recent investigations
have yielded negative values of P.
P33. Theabove sometimes conflicting results were all obtained
with concentrated, viscous solutions.
The stability of Couette flow of several polymer solutions was measured by Merrill et al." by deter-mining the deflection point in the rate of strain-shear
relation in a cylinder system with an extremely
narrow gap at high rates of strain. Their data are scattered however and do not point to significant changes from Newtonian behavior.
EXPERIMENTAL INVESTIGATION
-
The experiments were conducted in a coaxial
cylinder system, one meter high, shownschematic-ally in Fig.
3. The cadmium-plated brass inner cylinder, having an outer diameter of 10.80 cm, was rotated concentrically inside a transparent cylinder,FIG. 3. Schematic drawing of rotating cylinder apparatus; (1) variable speed unit;
(2) outer cylinder; (3) inner cylinder;
(4) dye injector.
15 N. Adams and A. S. Lodge, Phil. Trans. A256, 149, (1964).
18H. W. Greensmith and R. S. Rivlin, Phil. Trans. Roy.
Soc. (London) A245, 399 (1953).
17J. W. Hayes and R. I. Tanner, in Proceedings of the
Fourth International Congress on Rheology, E. H. Lee, Ed.
(Interscience Publishers, Inc., New York, 1965), Pt. 3, p. 389. 18 H. Markovitz, in Proceedings of the Fourth International
Congress on Rheology, E. H. Lee and A. L. Copley, Eds.
(Interscience Publishers, Inc., New York, 1965), Pt. 1, p. 189.
"A. S. Lodge, Elastic Liquids (Academic Press Inc.,
New York, 1964).
28 E. W. Merrill, H. S. Mickley, and A. Ram., J. Fluid
1932
FIG. 4. Variation of relative
viscosity with concentration for several polymer solutions.
having an inner diameter of 11.42 cm. Dye was
injected in the gap between the cylinders through small holes, symmetrically distributed along the circumference of the inner cylinder. The dye, flu-orescein, would cover the surface of the inner cyl-inder under stable flow conditions. The onset of instability could be determined from the Appearance of heavy-colored concentric dye-lines indicating the formation of vortex cells. The critical Taylor number was calculated from the rotational velocity at which the dye streaks appeared. The wavelength of the cells was measured at the surface of the outer cylinder, by taking the average over a large number of distances between the heavy-colored lines: For calibration of the cylinder system glycerol solutions in water at various concentrations were used. The polymer solutions under investigation were known to show appreciable friction reduction in turbulent flow. Aqueous solutions of two industrial polymers, polyethylene oxide (Polyox WSR-301) and poly-acryls.mide (PAM-250), and one natural polymer, guar-gum (Jaguar),' were used in the experiments, at concentrations up to 0.2%. Viscosities were
meas-ured in the Epprecht Rheomat-15, a rotational
viscometer. The relation between shear and rate of strain of each solution was measured over a range of strain rates of 30-1000 1/sec. For the dilute solutions the viscosity was found to be independent of shear; at higher concentrations pseudoplasticity was
ob-Te
FIG. 5. Variation cif etiticAl
Taylor number with concentra-tion, for several polymer
solu-tions.
21 These polymers were supplied by the Union Carbide Corporation, American Cyanamid Company, and SteinHall,
respectively.
served, but only at the higher strain rates. The
viscosity at zero shear rate, used in the calculation of T, could be determined with confidence in each case. In Fig. 4 the resulting relative viscosities of the polymer solutions are shown as a function of their concentration. The results of the stability study are presented in Figs. 5 and 6. The critical Taylor num-ber as a function of concentration for each of the investigated solutions is plotted in Fig. 5. The de-pendence on concentration of the corresponding critical dimensionless wavelengths is presented in
Fig. 6. As can be seen from those figures, T. increases
with concentration in all three polymer solutions tested. The measure of increase is smallest for the guar-gum and largest for the Polyox solution; the effectiveness of the solutions as friction-reducers in turbulent flow is in the same order, Polyox being the most effective. The wavelengths remained the same in all experiments. Comparing these data with the rheological models considered in the preceding sec-tion as was done in Fig. 2, it can be seen that neither
FIG. 6. Variation of critical dimensionless wavelength with concentration, for several polymer solutions.
Walters' nor Rivlin-Ericksen's fluid would give the right results. In both fluids an increase of T, would be accompanied by a decrease of ec, which was not the case. Even if the observed change in T, is con-sidered without taking ec into account, there are deviations with these models. From Walters' fluids only A' would show 7', exceeding the value for a Newtonian fluid. The existence of such a fluid has, however, never been confirmed from normal stress measurements, which would have to follow Eq. (3). A Rivlin-Ericksen fluid would have k2 < 0 for an increase in 17,, and thus a, ± 2a2 <0; from Eq. (6) it follows that P22 - P33 < 0, which is contrary to most of the observed data reviewed in the preceding section. It should be remembered that these data refer to different more- concentrated solutions than
the dilute ones investigated here. Though it
isusually assumed that different polymer solutions behave in a qualitatively similar manner, there is no actual proof of such an assumption.
Ericksen's anisotropic fluid for n1 > 0 and n2> 0 might be a suitable model for the investigated fluids. Not only can this model predict an increase of"71,;
without appriciable changes in but both P1111 7 P22
and P22 - P33 are positive, which is in accordance with most of the experimental results.
CONCLUSIONS
The stability of Couette flow was investigated for three polymer solutions, which are known to have highly reduced friction coefficient in turbulent flow. The experimental data show that the critical Taylor number increases with concentration while the
wave-length of the vortex cells -remains unaltered. A
comparison of these results with the stability criteria
and other characteristics
of several theologicalmodels indicate that neither Walters' fluids nor
Rivlin-Ericksen's second-order fluid are suitable. Only Ericksen's ardsotropic fluid May be an ap-propriate model for the investigated fluids.
ACKNOWLEDGMENTS
This research was sponsored by the Office of Naval
Research, under Contract N62558-4093, and by the Technion Reseatch Fund.
