literature Cited
(1) Fisher, E. K., “Colloidal Dispersions,” p. 182, \Viley, New York, 1950. (2) Gabrysh, A. F., Cutler, I., Utsugi, H., Ree, T., Eyring, H., Division of Colloid Chemistry, 137th Meeting ACS, Cleveland, Ohio, April 6, 1960. (3) Gabrysh, A. F., Ree, T., Eyring, H., McKee, N.,Cutler, I., Trans. Soc. Rheol. 5 , 6:’ (1961). (4) Green, H., “Industrial Rheology and Rheological Structures,” p. 50, Wiley, New York: 1949. (5) Green, H.. IVeltmann: R., Znd. Eng. Chem., Anal. Ed. 15, 201 (1943). (6) Zbid., 18, 167 (1946:1.
(7) Green, H., \Yeltmann, R. N., J. Appl. Phys. 15, 414 (1944). (8) Hahn, S. J., Ree, T., Eyring, H., Znd. Eng. Chem. 51, 856 (1959). (9) Hauser, E. A., Reed, C. E., J . Phys. Chem. 41, 911 (1937). (10) Heinrich, H., Clements, J. E., PYOC.Sci. Sect. Toilet Goods Assoc.. No. 33 (1960). (11) Reiner, M.,‘Phys;cs 5 , 321 (1934). (12) Pryce-Jones, J . , J . 011 Colour Chemtsts Assoc. 19, 295 (1936). (13) Wkltmann, R., Znd. Eng. Chem., Anal. Ed. 15, 424 (1943). ’ (14) \\’eltmann, R., J . Appl. Phys. 14, 343 (1943). (15) Weltmann, R., Green, H., Zbid., p. 569.
RECEIVED for review August 24, 1964 ACCEPTED July 29, 1965
FLOW OF VISCOELASTIC LIQUIDS THROUGH SUDDEN ENLARGEMENTS G l A N N l A S T A R I T A I A N D L U l G l NICODEMO Zstituto di Chimica Industriale, Via Mezzocannone 16, ‘Yaples, Ztaly
A qualitative investigation of the macroscopic effects expected during flow of viscoelastic liquids through sudden enlairgements i s given: higher static heads, larger head losses, larger axial length of the turbulent region, and shift of the pressure distribution with flow rate. Experimental results confirm the predicted behavior.
of the phenomenon of drag reduction, which is known l(77, 74) to take place during turbulent flow of viscoelastic liquids through constant section tubes, has recently been proposed ( 3 ) , based on the hypothesis that turbulence is not suppressed even if friction factors lying on the extrapolated laminar 19ow curve are observed ( 7 7 , 76); drag reduction is supposed to be due to a reduced rate of energy dissipation in the turbulent flow field. The present work is conceived as a n indirect proof of the above interpretation. When a liquid flows through a sudden enlargement, the downstream pressure can be approximately calculated from an over-all momentum balance. This pressure is smaller than would be required in order to have a downstream energy flux equal to the upstream flux. Thus, a head loss results, which again can be approximately calculated. Energy is dissipated in a highly turbulent flow field immediately downstream of the enlargement; turbulence is caused by the inherent instability of a decelerating flow field. I t is clear that, if the structure of turbulence is anomalous in viscoelastic liquids, peculiar effects are to be expected during flow of a viscoelastic liquid through a sudden enlargement. The phenomenon ,which has been investigated has also engineering interest. I n fact, if the drag reduction characteristics of some macromolecular additives are to be used to improve pumpability ( 73), head losses in fittings, contractions, and so on should be studied in addition to distributed losses. All the data referred to have been obtained with dilute aqueous solutions of a vinyl polymer. Only representative plots are reported in this paper; original data are recorded elsewhere (7, 2, 7, 72)
A
N IKTERPRETATION
Theory
Consider the flow of an incompressible liquid through a sudden enlargement connecting two circular tubes of different diameter, such as in Figure 1. Let 1 denote the section immediately upstream of the enlargement, and let 2 denote that section downstream where the velocity profile characteristic of steady flow has been established. The continuity equation is simply written as:
Q
=
QI
=
(1)
Q2
If it is assumed that, in section 1, the pressure on the annular region where no flow takes place is equal to the pressure in the central region, the momentum balance reads:
where 7 is the shear stress a t the wall and S is the wall area of the cylindrical region between sections 1 and 2. The last term on the right-hand side of Equation 2 is presumably the minor one. Should the enlargement have no effect whatsoever on the pressure, the pressure a t section 2 would be: 2
p2‘
=
p1
-
A2
T ‘ d s
(3)
where T ’ is the shear stress a t the wall which would be observed in steady flow. Hence, the pressure rise due to the enlargement is :
Present address, Chemical Engineering Department, Univer-
sity of Delaware, Newark, Del.
VOL. 5
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237
In Equation 4, the last term may reasonably be neglected, because the integrand (T - T‘) is the difference between two quantities of the same order of magnitude, both of which would yield an integral whose value is minor as compared to the first term on the right-hand side of Equation 4. Hence, approxima tely :
(5)
I
a
.
--
I
III
‘1 Figure ment
1.
z
:I ‘2
Flow through sudden enlarge-
1 ----7==-
The value of H.,{ can be easily measured by extrapolating up (down) to section 1 the linear pressure-axial position curve determined far downstream (upstream) (Figure 2). The momentum-average factor, a.\f, is defined as the ratio of the actual momentum flux to the flux which would be observed if the velocity profile were flat. There are two reasons for having nonunity values of cy.,[: the nonflat velocity profile itself, and the possible existence of deviatoric normal stresses in the direction of flow when the radial velocity gradients are not zero. The value of aM is given by:
t
I 1
where Z([) is the dimensionless steady-flow velocity profile
Figure 2.
and is a factor which takes into account deviatoric normal stresses in the direction of flow. For ordinary liquids, = 1; for viscoelastic liquids, a tension along the streamlines is to be < 1. expected, so that presumably By definition of the average velocity, Z, function E is subject to the condition :
Qualitative pressure distribution
fraction of the total inlet energy flux which is dissipated. For purely viscous fluids in turbulent flow, ( Y M , (YE, and x may be taken as unity, and Equation 11 degenerates into the familiar equation :
while H.v is given by: T h e energy balance for the system considered reads :
where hE is the head loss due to the enlargement, and the factor (YE is given by:
In Equation 10, @E takes into account the elastic energy (or “recoverable strain”) which is to be expected in viscoelastic liquids; for ordinary liquids, @E = 1. The value of hE is easily calculated by substituting Equations l and 2 into 9 ; neglecting again the wall-stress term yields :
where
-
f f ~ 2
ff22
ffE2
ffEl
x = 2 -
(1 3)
In Equation 11, the term in parentheses represents the 238
l&EC FUNDAMENTALS
The applicability of Equations 14 and 15 to purely viscous liquids is well established. Figure 3 is a plot of measured HM values us. G I , relative to runs with water through the test section described below. The agreement with Equation 15 is remarkable, which is an indication of the soundness of the only approximation involved-namely, the dropping of the wallstress term. In the following, the effects expected in the case of viscoelastic liquids are discussed. These effects are, in order of presumably decreasing importance: Anomalous values of the distance between sections 1 and 2 Influence of velocity profiles Influence of elastic relaxation Nonunity values of @M and @E
Z-Value. I t has been assumed, in the interpretation of the drag reduction phenomenon ( 3 ) , that the rate of energy dissipation in a turbulent flow field is, under assigned macroscopic boundary conditions, lower in a viscoelastic liquid than in a purely viscous liquid. For the process considered here, the total energy dissipation is fixed by the necessity of contemporary fulfillment of the over-all momentum and mass balances (see Equation 11); thus, when the dissipation rate is lower, the turbulent flow field where dissipation takes place is expected to be larger. Section 2 is expected to be located farther downstream in the case of viscoelastic liquids.
T o evaluate ( Y , ~ / @ . ~and a E / @ E , only the velocity distribution during steady flow-viz., the function E-needs to be known. I t has been shown (5) that, in the case of purely viscous liquids, the velocity distribution in turbulent flow does not depend on the particular shear stress-shear rate curve. The function 3 can be expressed as (75) :
E V
i:
I
E(()
=
- 5)llrn
a,(l
(1 9 )
where the constant a, (which equals the centerline-to-average velocity ratio) is obtained from Equation 8:
ii, ,cm/sec Figure 3.
Values of Hw for water runs
Let pE be the energy dissipation rate per unit volume. T h e value of Z is given by the equation:
Definition of a n average dissipation rate, 4
Equation 19 suffers from two shortcomings: I t does not yield dE/d[ = 0 a t E = 0 (the predicted velocity profile is cuspidal a t the tube axis), and it cannot yield the laminar velocity profile, whatever the value of exponent m. Its main advantage lies in its powerfulness in representing the turbulent purely viscous velocity profile. Values of C Y ~ ~ / @and , , ~ of as calculated from Equation 19, are:
+E:
OZA.
aE and substitution of Equation 11 give: ff€,PziI3 z=X
/3[1
+ P2x - 2
P(a.wJffE,)I
(18)
2
(FE
In the case of purely viscous liquids, the energy dissipation rate in a turbulent flow field is known to be proportional to the third power of the mainstream velocity ( 9 , 70) ; this can be shown to be true by simple dimensional considerations when it is assumed that the liquid’s viscosity does not enter the problem (70). Thus, for the case a t hand, +E is proportional to cl3,and the value of Z is expected to be independent of the liquid flow rate. This is indeed the case (Table I). I n the case of viscoelastic liquids, a t least one second rheological parameter should be considered, say a natural time. This parameter cannot be excluded from consideration in a turbulent flow field: Indeed, drag reduction is probably caused just by the fact that the frequency of noninviscid eddies tends to be higher than the inverse of the natural time. Thus, + E may not be proportional to w13, and Z may depend on the liquid flow rate. This is the counterpart of the Toms effect during turbulent flow through constant-section tubes, where the friction factor is no longer a unique function of the Reynolds number but also depends on the tube diameter. Velocity Profile Effect. T h e velocity profile in steady turbulent flow of viscoelastic liquids is known (6, 76) to be steeper than in the case of purely viscous liquids. Hence, both aM/@.ir and a E / & are expected to be larger, and presumably cannot be taken as unity.
Table 1.
Values of G )or Water Runs
Pressure
til,
Cm./Sec.
Tap
590
649
729
804
890
3 4
0.98 0.945
0.98 0.99
1 .oo 1.01
5 6 7
0.53 0,065 0.021
0.5’5 0.065
1.01 1.02 0.56 0.061 0.020
0.99 1 .oo
0.0’20
0.55 0.057 0.019
0.53 0.056 0.019
+
(m q 3 ( 2m 4 m‘(3 m)(3
+
+
+ 2 m)
(22)
For purely viscous liquids a t moderate Reynolds numbers (Re = l o 4 + 105) the value m = 7 is recommended (75). Both Bv and are unity for such fluids, so that a, = 1.20; = 1.02 and cyE 1.06. At higher Reynolds numbers, higher values of m are required, so that a,, c y i f , and aE are even closer to unity. This justifies the usual assumption, a v = cyE = 1.0, which is made in deriving Equations 14 and 15. The velocity profiles which have been observed during turbulent flow of viscoelastic liquids are more nearly “laminar,” so that presumably a preferable correlating equation would be :
E ( € ) = a,(l - p)
(23)
where a4, which is again the centerline-to-average velocity ratio, is given by: aq
= (q
+ 2)/7
(24)
Equation 23 predicts a zero velocity gradient a t the tube axis, provided the velocity profile is convex, say q > 1. T h e laminar velocity profiles are obtained by a correct choice of exponent q, say q = 2.0 for Newtonian liquids, q = (l/n) 1 for power-law liquids. Values of a u / @ > r and of a € / @ €from Equation 23 are:
+
ffY/@U (YE/@€
= 3(q
+ 2)/(q + 1)
(25)
+ 2)2/(q f l ) ( 3 q + 2,
(26)
= (q
Whatever the velocity distribution, the values of CY^/@'.^ and of a E / a Eare restricted to a rather limited range, whose extremes are identified by.
1. Flat velocity profile, 3 = 1, aw/aw = C Y E / @ E = 1. In this case, @ 1 ~and @E are necessarily equal to unity. 2. Triangular velocity profile, E = 3(1 - t ) ; a w / @ w = 1.5; ( Y E / @ € = 2.7. I n principle, the value of cyY could be calculated, without knowledge of the velocity profile, from Equation 5, if H.U is measured. Another possibility is the integration of the readings of a Pitot tube traveling across a diameter: In fact, the VOL. 5
NO. 2
M A Y 1966
239
nu
' 2
Figure 4. Energy average factors
and
momentum
Pitot reading would give the local value of the kinetic head minus the tension along the streamlines (73). Nonetheless, knowledge of a.lf does not imply knowledge of a E , and, in principle, the head loss, h E , cannot be calculated without knowing both the velocity distribution and the value of @ E . Figure 4 is a plot of a E / @ E cs. aJf/@.lfas calculated from Equations 19 and 2 3 . This figure shows that, to a first approximation, C Y E / @ E may be considered a unique function and QE are taken as unity, of a.,f/@.?r. This means that, if or if some knowledge of their value is available, the head loss can be calculated from the knowledge of Iflf. b'hen the flow is turbulent both upstream and downstream of the enlargement section, @.lf and are presumably not very different from unity. The main effect of elasticity is thus expected to be found in a.lf values larger than 1.0, particularly a t section 1. Hence, HJf values are expected, in the case of viscoelastic liquids which display drag reduction effects in tubes of the order of magnitude of the upstream tube, to be larger than predicted by Equation 15. Elastic Dissipation. In order to discuss stress-relaxation effects, it is preferable to refer to the usual splitting of the velocity into a time average ( T A ) and a fluctuating ( F ) component. The energy dissipation rate connected with the F component is, in the case of viscoelastic liquids, presumably low, because of the damping of any fluctuations of frequency higher than the inverse of the natural time. Thus, in viscoelastic liquids, the energy dissipation connected with the T A component cannot be neglected, and indeed nearly laminar velocity profiles have been reported (76). In flow through constant-section tubes, the T24 component is constant along the flow direction, so that only viscous dissipation needs to be considered, and there is no "elastic" contribution to the dissipation rate. A normal stress along the streamlines is possible, but its value is constant along the flow direction, and so is the elastic energy of the fluid. Of course, these considerations apply strictly only to laminar flow conditions, but a qualitative extension to the T A kinematics of turbulent flow does not seem unrealistic. I n floiv through sudden enlargement, the T A velocity is not constant along the flow direction, and the characteristic time for this phenomenon-viz., the ratio Z/d-may not be exceedingly large as compared to the natural time of the liquid. 240
l&EC FUNDAMENTALS
Hence, an elastic contribution to the rate of energy dissipation connected with the T A component should in principle be taken into account. Whether this contribution is positive or negative is open to discussion (77), for the flow field considered is decelerating; the state-of-the-art knowledge of both viscoelastic rheology and turbulence does not allow investigating this problem in any detail, though there are indications of its nontriviality (4) whenever the characteristic time is not exceedingly large as compared to the natural time. @ Values. Some qualitative comments concerning the value of @E have implicitly been made above. Further comare the following. ments on @.If and The value of is presumably always less than unity, inasmuch as a positive tension is expected along the streamlines. The technique proposed by White and Metzner (78) for determining normal stresses in viscometric floiz by measuring the jet swelling or Barus effect amounts to measuring the value of aJfin laminar flow. Indeed, there is no conceptual difficulty in admitting even negative values of (8); negative thrusts on a nozzle from which a viscoelastic liquid jet emerges have been reported (5, 8). 1Vhen the flow is turbulent and the velocity profile is not very steep? @31 values are expected to be only slightly less than unity; high values of the deviatoric stress are probable only in the boundary layer (73), and their effect on the over-all momentum balance should is expected to approach unity be minor. The value of \\hen the diameter of the tube is increased, so that, for the case considered, nonunity values of aJf are probably to be considered only upstream. FpE values are expected to be larger than unity. In fact, elastic energy may be transported by convection in addition to kinetic energy in the case of a viscoelastic liquid. The vaiue of @E again approaches unity with increasing tube diameter. This implies that the elastic energy flow downstream is lower than upstream, so that the released elastic energy also needs to be dissipated: This can be seen in Equation 11, which shoics that hE is increased when aiE, > aE?,as is to be expected even if the velocity profile at section 2 is the same as a t section 1. Experimental
A simple loop consisting of feed tank, centrifugal pump, flow-rate control valve, flowmeter, temperature reading, and test section was used for the experiments. In preliminary experiments on the drag reduction characteristics of the polymer solutions used, the test section was a straight circular tube, with length-diameter ratio of at least 100. Pressure taps were located 30 diameters from the inlet section, and a few diameters from the outlet section. The sudden enlargement test section is sketched in Figure 5. Besides the eight pressure taps whose location is indicated, two other taps were installed, one far upstream and one far downstream of the enlargement section. Pressure taps were connected with open-air piezometric tubes, filled with the same liquid which was circulated. Rheological properties of the solutions were determined as follows. The limiting viscosity (at s = 0) was measured with an Ostwald-Fenske viscometer. Standard capillary viscometer techniques were used to determine the shear stress-shear rate curve in the region of shear rates bet\veen 10' and 106 set.-' The jet expansion technique (78) was used to determine the normal-stress difference. Au, in viscometric flow. This technique proved to be satisfactory only over a limited range of polymer concentrations and shear rates. Rheological data are reported in Figure 6. At concentrations below O . l % , the solutions exhibit a constant (Newtonian) viscosity in viscometric flow. At higher concentrations, pseudoplastic behavior is observed, and an apparent flow index as low as 0.64 is observed at a concentration of 1%. Normal
2
1
I
'
34
5
I
6
7
8
I
I-LAMINAR
1of 2-0 08%.d=.7cm
riscous. jmooth pipes
1
n QJ
cc b
lo!
Figure 5. Location of pressure taps on test section
'0
/
/
10' 10'
3
Figure 7.
3
/
i
Re
10'
Drag reduction effects
Dimensionless pressure drop vs. dimensionless flow rate
B 0
,
0 5
:
10'
7.0
concentration, % b y weight
Figure
6.
Rheological properties of polymer solutions
stress differences could be measured with an acceptable degree of confidence only a t concentrations higher than 0.27,; values of &J a t a shear of 104sec.-l are reported in Figure 6. Results and Discussion
Drag Reduction. Yalues of the fraction factor, A, have been measured for turbulent flow of dilute solutions through circular tubes 01 different diameters (2). Plots of the dimensionless pressure drop, (X/8)Re2 LS. the Reynolds number have been obtained for three polymer concentrations and three tube diameters. Some typical curves are reported in Figure 7. This figure clearly shoi+s the occurrence of the drug reduction effect, accompanied by a marked shift \+ith diameter (Toms effect). Water through Enlargement. A preliminary series of runs with the test section sketched in Figure 5 was made with water. Measured values of H,f are reported in Figure 3, which shous that, in the case of water, Equation 15 holds. These results have also been used to test the conclusion that, in purely viscous liquids, Z is independent of Q . In order to do so, the pressure distribution along the flow axis has been made dimensionless by definition of the dimensionless pressure :
which, \\hen 2 does not depend on Q, is expected to be a unique function of the axial position. This is confirmed by the data reported in Table I, where values of G recorded a t the different pressure taps for several values of Q are tabulated. Polymer Solutions through Enlargement. The increased length of the turbulent region where energy is dissipated, which has been predicted in the case of viscoelastic liquids, is clearly shown by the data reported in Figure 8, where the dimensionless pressure c istribution observed a t the same up-
0
2
4
6
downstream position, diameters Figure 8. Influence of polymer concentration on pressure distribution
stream velocity is reported for water and for three polymer solutions. The axial position is identified by the number of (large) diameters downstream of the enlargement. Values of G larger than 1 are observed a t the first pressure taps for the polymer solutions, which is due to the fact that Hli values larger than predicted by Equation 15 are exhibited by these solutions. This is to be expected, because the solutions used display drag reduction effects, which imply steeper velocity profiles, and hence higher values of aXi. T h e dependency of 2 on flow rate, predicted for viscoelastic liquids, is clearly sho\vn in Figure 9, \+here the dimensionless pressure distribution observed a t several flow rates for the 0.2% solution is plotted. The effect was observed, although less markedly, also a t lower polymer concentrations. T h e data in Figure 9 should be compared with the data in Table I. The results in Figures 8 and 9 allow evaluation of the possible importance of stress-relaxation effects. For the more concentrated solution (0.2%) the value of Z is seen from Figure 9 to be about 4 diameters, say 8 cm. This corresponds, for the upstream velocities used, to characteristic times, Z / U , , of the order of 1 second. The natural time of the liquid can be evaluated from the data reported in Figure 6. At a shear rate of IO4 set.-', the normal-stress difference for the 0.2y0 solution can be evaluVOL. 5
NO. 2
MAY 1966
241
----
= Reynolds number, =
5
= = =
S T 0.2 .R solution
I .o
2
0
downstream Figure 9.
4
6
position, diameters
Influence of flow rate on pressure distribution
ated a t IO4 dynes per sq. cm., while the tangential stress is 400 dynes per sq. cm. Thus, a natural time of the order of 2.5 X second is observed a t a shear rate of IO4 sec.-l; a t higher shear rates, slightly higher values are calculated. The order of magnitude of the natural time is not much smaller than that of the characteristic time, so that stressrelaxation effects cannot be dismissed. The macroscopic results obtained do not give an insight on the actual values of and @ E . Acknowledgment
T h e authors are indebted to Domenico Acierno and Enrico Astarita, who carried out most of the experiments. Nomenclature
A
= cross-section area, L2 a p = centerline-to-average velocity ratio, dimensionless d = tube diameter, L G = dimensionless pressure, dimensionless g = gravity acceleration, L T d 2 H.$f = static head difference, L hE = head loss, L L = length n, q = exponents in velocity distribution laws, dimensionless M = mass n = power-law exponent, dimehsionless n’ = apparent flow index, slope of log-log plot of shear stress us. shear rate, dimensionless p = pressure, ML-1T-2 = pressure which would be observed without enlargep’ ment effect, M L - l T Q = flow rate, L 3 T - l = distance from tube axis, L = tube radius, L a,
k
242
=
U Q
c3
l&EC FUNDAMENTALS
2 Rzip/p, with p evaluated a t wall shear stress shear rate, T-’ wall area, L2 time velocity, LT-1 average velocity, LT-1 volume, L3 distance between sections 1 and 2, L
Re -4
= = =
V Z
GREEKLETTERS (YE = energy-average factor, dimensionless LY.~ = momentum-average factor, dimensionless /3 = A l / A 2 , dimensionless X = friction factor, kinetic head loss over a one-diameter length of pipe, dimensionless p = viscosity, M L - l T - l = viscosity of water, ML-lT-l po 3 = u/U, dimensionless [ = r/R, dimensionless p = density, ML-3 = normal stress difference, ML-lT” ha = wall shear stress, M L - i T - 2 = wall shear stress which would be observed in steady r‘ flow, ML-‘T+ = energy dissipation rate, ML-ITd3 YE = average energy dissipation rate, ML-1T-3 (oE = elastic-energy correction factor, dimensionless @E = elastic-stress correction factor, dimensionless (2 cydM2/ a‘,, ( Y E * ) - ( c x E ~ / c Y E ~ )dimensionless , = see Equation 13, dimensionless x M
literature Cited
(1) Alfano, G., chemical engineering thesis, University of Naples, 1965. (2) Astarita, E., chemical engineering thesis, University of Naples, 1965. (3) Astarita, G., IND.ENG.CHEM.FUNDAMENTALS 4, 354 (1965). (4) Astarita, G., “Motion of a Gas Bubble through a Viscoelastic Liquid,” unpublished. (5) Astarita, G., Alfano, G., Greco, R., Zng. Chim. Ztal. 1, 101 (1965). (6)’-Bogue, D. C., Metzner, A. B., IND.ENG.CHEM.FUNDAMENTALS 2, 143 (1963). (7) Greco, R., Chemical engineering thesis, University of Naples,
-,--.
1065
(8) Harris, J., A‘atature 190, 993 (1961).
I
9) Hinze, J. O., “Turbulence,” McGraw-Hill, New York, 1959. 10) Levich, V. G., “Physico-Chemical Hydrodynamics,” Prentice-Hall, Englewood Cliffs, N. J., 1964. 11) Metzner, A. B., Park, M. G., J . Fluid Mech. 20, 291 (1964). 112) sandulli, E., chemical engineering thesis, University of Naples, 1965. (13) Savins, J. G., A.Z.Ch.E. J . 11, 715 (1965). (14) Savins, J. G., SOG.Petrol. Eng. J . 4, 203 (1964). (15) Schlichting, H., “Boundary Layer Theory,” McGraw-Hill, New York, 1960. (16) Shaver, R. G., Merrill, E. W., A.Z.Ch.E. J . 5 , 181 (1959). (17) White, J. L., Metzner, A. B., “Thermodynamics and Heat Transport Considerations for Viscoelastic Liquids,” unpublished. (18) White, J. L., Metzner, A. B., Trans. Soc. Rheol. 7, 295 (1963). RECEIVED for review May 14, 1965 ACCEPTEDSeptember 24, 1965
