journal of applied polymer science volume 16 issue 2 1972 [doi 10.1002%2fapp.1972.070160213] b....

8/13/2019 Journal of Applied Polymer Science Volume 16 issue 2 1972 [doi 10.1002%2Fapp.1972.070160213] B. Hlavek; F. A. Seyer -- Determination of parameters

1/17

JOURNAL OF APPLIED POLYMER S C l E N C E VOL. 16, PP. 423-439 1972)

Determination of Parameters in Convected MaxwellModel from Linear Viscoelastic Parameters

I3. HLAVACEK AND F. A. SEYER, Department of Chemical andPetroleum Engineering, Un iversity of Alber ta, Edmonton 7 , Alberta, Canada

synopsisOwing to its simplicity and general ability to correctly portray a number of important

flow phenomena, the two-parameter Maxwell model has been employed in a number ofimportant engineering studies. The relation of this equation to linear viscoelasticityand to some molecular theories is considered. A new rule is developed which shows howthe shear-dependent relaxation time and viscosity of the Maxwell model can be deter-mined from linear viscoelastic parameters. It is thus shown that the two-parameterMaxwell model may be more general than earlier anticipated.

INTRODUCTIONThe convected Maxwell model,

SPP + 6 - = apd,S twas introduced some time ago by White and Metzner' utilizing ideas ofrubber rheology as well as the formalizations of Oldroyd.2 This equation,which has recently been shown to be quantitatively correct a t the large de-formation rates of interest in practical p r ~ b l e m s , ~ ~ ~as proved useful in anumber of other studies including turbulent drag reduction and flowthrough porous A number of equations similar to eq. (1) haveappeared (see, for example, refs. 8 ,9 , and 10)which are based on moleculararguments and which may be thought of as superposing the effects of alarge number of linear Maxwell elements. The possibilities are of courseunlimited. Although generalizations based on molecular arguments arecapable of defining the shear dependence of the functions 6 and p the in-hercnt simplicity of eq. (1) is lost, and, as will be shown later, there aresevere difficulties in a unique determination of a large number of physicalproperty parameters.

In eq. (I), he material parameters in general need to be functions of theinvariants of the deformation rate tensor d and are not restricted in form.From a molecular point of view, these functions may be viewed as repre-senting integral properties of a large number of molecules in solution.There has been no precise determination of the relation of t.hese functionsin the White-Metzner model to the behavior of individual molecules.

423@ 1972 by John Wiley Sons, Inc.


2/17

424 HLAVACEK AND SEYERWe consider first the possibility of reduction of a large number of linear

viscoelasticity parameters to a smaller number of parameters. Surpris-ingly, to the authors' knowledge, this has never been considered in detailbefore (usually the problem is considered from the opposite point of viewwherein an experimental function is used to generate a large number ofparameters). The results of the parameter reduction will then be appliedto show how the White-Metzner parameters B(I1,III) and p(I1,III) are r elated to simple constant Maxwell elements in both shear and elongationalflows.

PARAMETER REDUCTIONFor small gradients or deformations, we have 6 / 6 t equal to the ordinary

partial time derivative and require thatlim e(I1,III) = constlim ,A(II,III)= const

II,III--tO

II,III-+Oor that only two constants are necessary for defining the fluid. In simpleshear flow, the function e and p in eq. 1) are defined through the physicalcomponents of the stress tensor as

For a linear viscoelastic area, which is characterized for most polymersthrough many constant relaxation times (for example, the number of char-acteristic relaxation times p in the Rouse theorys is usually greater than 50)or through continuous spectra, it seems superficially that eqs. (1) and (2a,2b) are entirely unsatisfactory. In fact, we will show that excellent pre-dictions of the linear viscoelastic functions can be obtained over at leasttwo decades of independent variables. This property of parameter reduc-tion is a fundamental property of the basic viscoelastic functions andis not restricted in applicability to the nonlinear Maxwell model. Anequivalent problem is determination of a continuous box spectrum and itsrelation to e.

As noted above, eq. (1) reduced to a one-parameter Maxwell model fordifferentially small deformations and gradients. The measured quantities,for example, the relaxation modules G ( t ) , he real part of the dynamic vis-cosity ~ (o),r the dynamic modulus can be written in the form of aseries: 1-13


3/17

CONVECTED MAXWELL MODEL 425nG ( t ) = G, e-'/'P

p = 1 4)

where w is angular frequency, is the steady shear gradient, and T , and G ,may be viewed as constants for each Maxwell element. In eqs. (6) and (7)if constants Opand G , are the same, then the functions are the same and

In general, it appears that eq. (8) is incorrect and tha t better agreement isobtained without the 2 . l 4 v l 6 We account for this possibility by the non-equality of G , and Gp. We rewrite eq. 4) nto a discrete form with param-eter ti (noting that in matrix notation lower-case letters will refer to thefunction ~ t ) )

O r

The individual elasticity constants GI, Gzlaxation spectra H (log r through the relation16

. are connected with the r eGi = H(log TJAlOg 7. (10)

We will now show that the matrix 1 in eq. (9b) has such properties that asmall change in the measured matrix g is reflected in a large change in G.Equivalently, through eq. (lo), we note that a small numerical change inG( t ) is reflected as a large change in the function H(log T). Thus, if aseries of functions lLr(t)- ( t ) are possible whose shape and values are inthe short interval as, for example,


4/17

426 HLAVACEK AND SEYERwe can find a corresponding series of functions Hl(1og T ) , Hz (log T ) . . .Hm(log which differ significantly from one another. From this set, thefunction which can be characterized through the smallest number of param-eters will be chosen. Similar properties are exhibited by the matrices2 and E3 derived from eqs. ( 5 ) and (6) such that

For norm of a vector x, JIx I,we will assume the usual properties:(a) J/xII 0 if x + 0(b) l l 4 l = ~ . l l X l I( 4 IIX + Yll I lxll + Y

Thus, for the norm of g and G, we can use the functions

and

in which means absolute value.tions, we define in discrete form

Similarly, for differences in the func-

Ag = g 9 1and therefore

or for the relative value of the differences,n


5/17

CONVECTED MAXWELL MODEL 427

i = l

Let us suppose that the functions G t ) , H(1og T ) , and AG t) from the in-are given. The largest difference in H(1og T ) , aserval AG t)e -~

measured by the absolute value of the difference at all points, occurs ifthe quantity Y , defined as

3G t)100

is a maximum. Or, rearranging eq. (18),

and substituting eq. (9b), we obtain

According to Fadejev and Fadejeva, it is possible for every norm of avector I/xil to define the corresponding norm of a quadratic matrix A in sucha way that the following equation is valid:

Therefore, eq. 20) becomes4llAgll) = IlElll . maxllG-'ll. 22)

In eq. 22), the right-hand side is independent of the choice of IlAgll andtherefore Y is a constant given by

23)= lllll 111-11.For every eigenvalue X of a matrix A,

IAxI = 1x1 1x1 I IAl 1x1;therefore, in eq. 23),

in which max X t l and min ]All are the maximum and minimum eigenvaluesof E. It follows that the error interval AG is minimum if 1 is a unit matrix


6/17


7/17


responding response function G ( o ) given by eq. (6). The G , in eq. (9a)are determined from eq. (10)with Alog 7 = 0.1. Table I gives the assumedrelaxation spectra H(1og 7 numerically. HI, and H,, H are characterizedby two- and one-time constants, respectively, while H4 and H5 representcontinuous functions characterized by 10- and 20-time constants.

TABLE IAssumed R elaxation Spectralog 7 Hi H2 Ha H H5

1 . 21 . 11 . 00 . 90 . 80 . 70 . 60 . 50 . 40 . 30 . 20 . 10 . 0

- 0 . 1- 0 . 2- 0 . 3-0 .4-0 .5-0 .6-0 .7-0 . 8

0.00.00 00.00.00 . 00.00 . 0

10.00 .00 . 00 . 0

10.00 . 00.00 . 00.00.00.00.00.0

0.0 0.00.0 0.00.0 0.00.0 0.00.0 0.00.0 0.00.0 0.00.0 0.00.0 0.00.0 0.0

20.0 0 .00 . 0 20.00 . 0 0 . 00 . 0 0.00.0 0.00 0 0.00.0 0.00.0 0.00.0 0.00.0 0.00.0 0.0

0.00.00.00.00 .00 . 00.4832.663.482.481.672.503.492.660 .520 .00 .00 .00 . 00.00.0 -~

0 . 00.101

-0.004-0.207-0.216

0.1840.9541.842.593.063.223.082.621.850.930.15

-0.21-0.15

0.0350.066

-0.026

A comparison of the dynamic modulus functions determined from thevarious spectra is given in Table 11. Here, the square of the error relativeto G w ) , is given as a function of w. In all cases, it is clear that G (o ) ,atleast over approximately two decades, is insensitive to the form of theassumed relaxation spectra.

In the case of real spectra, extending over many decades, the exact re-production of the function G(w) is not possible by using only one Maxwellelement with constant parameters. For real polymers, To b o l ~ k y , ~ ~singhis X procedure, reduced a box spectrum extending over 6 decades into abox spectrum ranging over 2.2 decades with little effect in reproducing themeasured function G( t ) . In another case, T o b o l ~ k y ~ ~xplained the re-laxation properties of organic glasses using only one relaxation time.MeissneP has reduced a continuous spectrum over 4 decades for poly-ethylenes to four discrete Maxwell elements with constant parameters.We have shown that these cases of parameter reduction are not necessarilydependent on special properties of the materials (as Tobolsky b e l i e ~ e d ~ ~ . , ~ ) ,but reflect the general properties of operators governing memory effects in


8/17

43 HLAVACEK AND SEYERviscoelasticity. The data as well as the numerical example show that asingle time-constant is always adequate to replace at least one decade ofthe continuous spectral function.Similar comments apply to nonlinear continuum models which rctsin theessential features of eqs. (4)-(7). That is, from a continuum point ofview, we cannot distinguish the difference between models with very fewrelaxation times from those containing a large number (say 50) relaxation

TABLE I1Error in Real Part of Dynamic Modulus Predicted from Various Spectralog

w GH, (GH, - G H ~ ) ~G H ~ G H ~ ) ~G H ~- GH, ) (GH, - G H ~ )1.1 1.971.0 1.950.9 1.930.8 1.890.7 1.830.6 1.760.5 1.650.4 1.520.3 1.360.2 1.180.1 0.9980.0 0.811

-0.1 0.634-0.2 0.475-0.3 0.342-0.4 0.237-0.5 0.160-0.6 0.105-0.7 0.068-0.8 0.044___ -

0.5138 X0.127 X lo-'0.312 X lo-0.757 X lo-0.179 X0.413 X0.905 X0.183 X0.331 x 10-80.504 X lod20.593 x0.468 x0.173 X0.371 x0.143 X0.421 X0.5477 X0.470 X lo-*0.310 X0.1721 X

0.983 X0.2440 X lo- '0.601 X lo-'0.146 x0.351 X0.822 X0.184 X

-0.389 X0.752 X0.127 X 10-l0.181 x 10-l0.203 x 10-l0.168 X 10-l0.939 X0.293 X0.248 X0.873 X lo-*0.413 X0.468 X0.336 X

0.217 X lo-0.42.; X0.768 x lo-'1.23 x 10-41.65 X l o d 41.73 x 10-41.316 X0.656 X lo-'0.168 X0.0004 X l ov40.132 X lo-0.573 X lo-'1.18 x 10-41.58 x 10-41.51 x 10-41.11 x 10-40.370 x 10-40.68.5 X lo-0.183 X lo-'0.086 X l o -

0.631 X0.615 x 10-40 . m x 10-4o . . ~ 10-40.476 X lo-'0.345 x 10-40.161 X0.254 X0.194 X0.353 X 10'0.060 X0.224 X lo-'0.263 X0.127 X lo-'0.272 X0.092 x0.071 X0.215 x 10-60.227 x

0.199 x 10-5

times. There needs to be, however, a spectral function which has a physi-cal basis, and this needs to be determined through optimum methods basedon unit operator. I6

APPLICATION TO WHITE-METZNER EQUATIONAlthough a formal procedure for separation of relaxation times is not

precisely defined, it is clear that if we wish to use a one-parameter relaxationspectrum, then separation of relaxation times by a t least one decade ispossible. Therefore, in the following we assume that T ~ + I s negligiblecompared to r t . The limit of zero gradient will be considered followed by :Igeneralization to larger gradients.

In the limit of zero gradient for steady shear flow, we can put from eys.5) and (7) (noting that the same series applies for p and 7 )


9/17

CONVECTED M AXW ELL MODEL 431nlimp(+) = C G p r .

Y-0 p = 1

Generally, since the G , vary only slightly (for a box spectrum they areconstant), comparison of eqs. ( 2 ) and ( 3 )with (27) and ( 28 ) gives

lim ~ ( O , I I , O )Z ~11-0

andlim p(O,II,O)11-0

since r2


10/17

432 HLAVACEK AND SEYERdifferent parts of the spectra such tha t 7 1- >>max e and e = j(11,III).Considering simple shear flow,

6 P S 6 P A 6 P B-=- stt 6t (33)/Is = /I + P B .

According to the stress-optical law,28-30f x is the birefringence angle,then

(34)

P l l A - B A2 P 1 z Acot ( 2 X A ) =

or, rearranging and using eq. (3),

Similarly, for component B and the whole system S,cot 2XBe = Y

35)

(37)

Because the principal stress of systems A and B are additive as vectors,3owe obtain for 0s

The geometric relation resulting in eq. (39) is shown in Figure 1, where thepoint S characterizes the stress state of the whole system and AS, AB,and AA are the principal stresses for S, B, and A, respectively. From eq.(35)-(39), it follows that

6 AP l Z A @ B P U Bes = P1zA PUBOr, in general for a system of n components,5 B , P ~ ~ ~

U ii = 1es =i = l

and


11/17

CONVECTED MAXWELL MODEL 433S

Fig. 1. Geometric relation for stress coutributions: AA, principal stress of system A ;AB, principal stress of system B; AS, principal stress of entire system S.

Equation (41), expressed through the White-Metzner parameters for dif-ferent noninteracting systems, is analogous to the Sadron optical rule30which was introduced by study of birefringent solutions. In general, thesystems are interacting or the friction coefficient of an individual monomerunit is changed, and the relaxation times of an individual component areshifted relative to those for pure components. The shift of relaxationtimes observed on blending different molecular weight fractions has beenstudied in some detail by Ferry and Ninomya11*31,32nd may be describedby a shift factor of order unity. The blending laws may be of especial im-portance in the estimation of the elastic properties of very dilute solutionswherein it is not possible to directly measure normal forces.

Equations (41) and (42) provide a general rule for formulating the shear-dependent White-Metzner relaxation time and viscosity as separate con-

TABLE I11Comparison of Predicted and Experimental Relaxation Times for

Polyisobutylene Solution71, Gn7w in sec poises sec-l e i . ) , sec Os -i), sec

1 10-1 6.1 10 2.5 x lO+ 3.9 x 10-22 10-2 4.0 108 4.6 x 10-3 7.1 x 10-33 10-8 1.9 103 6.1 x 10-4 14.4 x 10-4


12/17

434 H L A V A C E K A N D S E Y E Rtributions from a number of components. It is worthwhile to emphasizethat these relations must be satisfied for any choice (for which (Pll -P22 s = F , ( j ) + . . F, i.) and 6* are determined from er = Fi/2ptf2) ofthe component functions, including those which arise from theoretical con-siderations. The addition rule provides a direct relation between the vis-cosity function and shear dependence of the relaxation time. Again, fordilute solutions, it should be noted th at cqs. (41) and (42) apply rigorouslyonly to the polymer contribution of the stress.The addition law can be illustrated from Tanners dataI5 or a 5.4 poly-isobutylene cetane solution. Tanners spectrum has been reduced(crudely) to three characteristic times T,, as shown in Table 111. The6 f) is estimated directly from the published normal stresses using eq. (3)or equivalently eq. (1))and 6 s has been determined using eq. (41). In viewof the crude reduction of parameters, the agreement with 6 determinedfrom the normal stress measurements is rather good. Better agreement canbe obtained by a slight shift of the relaxation times to coincide with thepeaks in the spectra determined by Tanner.

NONVISCOMETRIC FLOWOwing to the possibility of parameter reduction, we can comment brieflyon the form of material functions for elongational flow bya consideration ofmolecular arguments. In elongation flow, the kinematics are defined by

VI = di i*xiVz = (-d11/2)xzV3 = (-d11/2)Z3

and the second and third invariants of d are3I1 - - - j 2

d - 4

(43)

1I11 - -73 .d - 4For the elongational viscosity defined as the total stress in the axial di-rection divided by the deformation rate, eq. (1) predicts

3p(II,III)[ l+ 6(11,1II)dii][l - 26(1I,III)dii]s = (45)

I n eq. (45) no information is given as to how the parameters 6 and p de-pend on the second and third invariants in elongational flow. As a firstapproximation, these have been assumed equal to the function determinedin simple shear, and dependence on the third invaiiant therefore was ne-g le~ ted .~JAlternatively, if we assume contributions from a number of


13/17

CONVECTED MAXWELL MODEL 435Maxwell elements (characterized through constant parameters and built onmolecular notions), then the elongational viscosity may be written

n(46)

in which ri and qf are constants. In shear flow, the most elementary de-pendence on the second invariant of the contribution to solution viscosityof each Maxwell element has been shown in a number of instances to be(see, for example, refs. 11 and 34)

311iG e = ci = l (1 + Tid l l ) l - 27idll)

q i .1 + 4 r i 2 - I Ii = (47)Thus, in the viscosity function of the White-Metzner equation, the relevantviscosity function which is consistent with simple molecular arguments is

n np(I1,III) = c pi = c q i fi(II1)i 1 i = 1 1 + 48,211

where we insert f(II1) to account for the possibility of p depending in thethird invariant. The function f i (III) has not been considered experimen-tally; however, we requirefi(0) = 1. Furthermore, it is clear from eq. (46)that the dominant contribution to qe is always from the longest relaxationtime. Previous calculations based on a molecular mode13j where the sud-den increase in elongation viscosity was observed to be in the area of thefirst relaxation time are in agreement with this. In the area of longest re-laxation time (in fact, for at least the first decade), we have shown that asingle constant e in the White-Riletzner equation is sufficient to portray theviscoelastic functions. Therefore, applying eqs. (30) and (31), say as anapproximation, where A reflects longest relaxation time and using eq. (48)for the viscosity function,

As before, if we assume that @A>> eB, the first term in eq. (49) is dominant.Thus, if we compare with eq. (46), it is seen tha t

e A 1 = constand

or that the proper parameters in the White-Metzner equation for elonga-tional flow are simply the zero shear viscosity and first relaxation time ofthe material. Therefore, using eq. 44),we find

fi (III) = 1 + 3 ~ 1 ~ ( 4 1 I I ) ~ ' ~ . (50)


14/17

436 HLAVACEK AND SEYEREquation 50) gives an explicit dependence of viscosity on the third in-variant which is consistent with molecular arguments and would appear tobe valid for use with the White-Metzner equation. From a molecularpoint of view, we are in effect noting that if the elongation viscosity of oneelement increases indefinitely, then the elongation viscosity of the entiresolution must increase indefinitely.

The importance of separating the polymer contribution to the shearingand normal stresses in determination of e and p can be emphasized by con-sidering eqs. (38) and (49). In eq. (49) it is clear that large elongationviscosities occur if 2 e ~ d ~ 1pproaches unity. Thus, in eq. (38) the limitingbirefringence angle is approximately 31". Available data28.38based onthe total stress of the solution, not just the polymer contribution, indicatethat at high shear rates the birefringence angle becomes a constant, x owhich is characteristic of the solution, or equivalently that 20y = constant.Now, if xE > 31", the limit of 20-i - nity would be exceeded at the ap-propriate deformation rate in elongation flow while, if the constant xc


15/17


character of t,hese two flows expressed through dependences on the secondand third invariants, therefore, wll probably depend on the rotation orvorticity of the flow.

The estimation of es from birefringence measurements is possible througheqs. (33)-(41) and, since these are connected directly to structural char-acteristics of the material, may provide an important tool in connectingcontinuum and molecular models.

AppendixEquation (26) is a more general form of eq. (18) in which the possible error of matrixEl xpressed in eq. (26b) and provides a means for estimation of how Ag or CF1 con-

tribute to AG.We, in fact, solve the equation

i*G* = g* (A-1)where

g = g * + AgG = AG + G*.

From eqs. (9b) and (A-1) we have( I - C . LG * = g* (A-4)

FIG* = ( I - C -g* = ( I + C + C*+ . g + ( I + C + C2 + . . Og, (A-5)and through rearrangement of eq. (A-5) in terms of th e norms,

which is eq. (26).List of Symbols

empirically determined constant (approx. unity)constantdeformation rate tensordiscrete (matrix) representation of relaxation modulus andL error, 3r change, in relaxation modulusnorm of matrix Ctimevelocitycoordinate directionsnorm of matrix xgeneral quadratic matrixerror matrix

matrix of exponential functions (see 9a)

real part of dynamic modulus, respectively

i


16/17

438 H L A V A C E K A N D S E Y E H

2, 3 = corresponding matrices for relation of real part of dynamicmodulus and real part of dynamical viscosity to discreterelaxation spectraG ( t ) = linear relaxation modulus

G(o)AGG , coelastic Maxwell modelH(1og r = continuous logarithmic relaxation spectrum

= unitmatrixP = deviatoric stress tensorPI1 P = first normal stress difference 1 indicates flow direction, 2 in-11,111 = second and third invariants of shear rate tensor-3 = velocity gradient66t6 t r ) = Dirac delta function~ ( w )T e l??el

exP = the viscosity parameter of White-Metzner equationY7 = continuous relaxation timeTP6X = birefringence angle6J = angular frequency

= real part of dynamic modulus- 6 error, or possible change, in discrete relaxation spectra= modulus which characterizes spring stiffness of linear vis-

dicates direction of shear)

= Oldroyd convected derivative= real part of dynamic viscosity= elongation viscosity of White-Metzner model, eq. 45)= elongation viscosity of model which is built on molecular= relaxation time of White-Metzner equation= eigenvalue of matrix A= parameter defined through eq. (18)= constant relakation time of pth Maxwell element= functions that are possible representations of the relaxation

-

motion

modulus G( t )

References1. J. L. White and A. B. Metzner, J . App l . Po lym. Sci. 7,1867 (1963).2. J. G. Oldroyd, Proc. Roy. SOC., er . A , 200,523 (1950);ibid., 245,278 (1958).3. B. J . Meister and R. D. Biggs, A.1.Ch.E. J . , 15,643 (1969).4. M. M. Denn, and G. Marrucci, A.I .Ch.E. J . , in press.5. F. A. Seyer, and A. B. Metzner, Can. J . Chem. Eng., 45,121 (1967).6. F. A . Seyer and A. B. Metzner, A.I .Ch.E. J., 15, 426 (1969).7. R . J. Marshall and A . B . Metzner, In d. Eng. Chem., Fundam., 6,393 (1967).8. F. Bueche, J . Chem. Phys., 2 2 , 6 0 3 and 1570 (1954).9. P. E. Rouse, J . Chem. Phys., 21, 1272 (1953).10. B. H . Zimm, J . Chem. Phys., 24, 269 (1956).

11. J. D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1970.12. R . B. Bird and P. J. Carreau, Chem. Eng. Sci. , 23,427 (1968).13. T. W. Spriggs and R. B. Bird, Zng. Eng. Chem ., Fund am., 4, 182 (196.5).14. R . I. Tanner, Trans. SOC.R h w l . , 12, 155 (1968).


17/17

CONVECTED MAXWELL MODEL 43915. R. I. Tanner and J. M . Simmons, C .E .S . , 22, 1803 (1967).16. B. Hlav Eekand V. Kotrba, Rheol. Acta, 6, 288 (1967).17. D. K. Fadejev and V. N. Fadejeva, The Numerical M ethods of Linear Algebra,18. J. F. Clauser and W . G. Knauss, Trans. Soe. Rheol., 10, 191 (1966).19. S. Twoomey, J . Assn. Computing Machinery, 10, January (1963).20. I).L. Phillips, J . Assn. Computing Machinery, 9, January (1962).21. V. Sinevic and B. HlavAEek, to be published.22. J. S. Berezin and N. P. Zidkov, The Methods of Calculus, Vol. I, Nauka, MOSCOW,23. J. Todd, Quart. J.Mech. App l. M ath., 2,469 (1949); Von Neumann, J. and Gold-24. A. V. Tobolsky and J. J. Alkonis, J. Phys. Chem., 68,1970 (1964).25. A. V. Tobolsky, J. J. Alkonis, and G. Akovali, J. Chem. Phys., 42, 723 (1965);26. J. Meissner, Kunststofle, 5,.57, 397 (1967); ibid., 9,57, 702 (1967).27. B. HlavAEek and V. Sinevic, Rheol. Acta , 9, 312 (1970).28. A. S. Lodge, Nature, 176, 383 (1955).29. W. Philippoff, Polymer Letters, 8,107 (1970).30. W. Philippoff, Trans. Soe. Rheol., 5 , 163 (1961).31. K. Ninomiya, J. Colloid Sci., 14,49 (1959); ibid., 17, 759 (1962).32. K. Ninomiya, and J. D. Ferry, J.Colloid Sci ., 18, 421 (1963).33. R. B. Bird, M . W. Johnson and J. F. Stevenson, Proceedin gs of the Fifth Inter na-34. P. J. Carreau, I. F. MacDonald, and R. B. Bird, Chem. Eng. Sci ., 23,901 (1968).35. F. A. Seyer and B. HlavAEek, Kol lo iAZ. , in press.36. A. Peterlin, in Rheology, Vol. I, F. R. Eirich, Ed., Academic Press, New York,37. R. J. Gordon, J . App l . Po lym. Sci., 14, 2097 (1970); A. E. Everage, and R. J.

Fizmatgiz, MOSCOW,960.

1966.stine, H. H., Bull. Amer . Math . SOC. , 3, 1021 (1947).

A. V. Tobolsky and R. B. Taylor, J. Phys. Chem., 67,2439 (1963).

tional Congress of Rheology, 4, 159 (1970).

19.56.Gordon, A.1.Ch.E. J., 17, 1257 (1971).Received June 14, 1971Revised September 21, 1971

journal of applied polymer science volume 16 issue 2 1972 [doi 10.1002%2fapp.1972.070160213] b....

Documents