normal stress coefficient

Rheologica Acta Rheol Acta 27:567-574 (1988)

Original Contributions

Primary normal-stress coefficient prediction at high shear rates*)

Chr. Friedrich and L. Heymann

Academy of Sciences of the GDR, Institute of Mechanics, Karl-Marx-Stadt (GDR)

Abstract: On the basis of a brief analysis of well known normal-stress calculation methods, the necessity of improved models of prediction is elaborated. A modified form of the so-called mirror relation which meets these requirements is presented. In combination with the Carreau viscosity equation, an analytical solution is given which leads to a Carreau normal-stress coefficient equation and, thus, to a simple method of calculation. The comparison between measured normal stresses and those determined by experiments shows that the values calculated in accordance with the presented method agree well with the measured values, especially within the range of high shear rates. The parameters fi and K to be selected for this purpose are determined in dependence on the slope of the viscosity function el at high shear rates for each polymer individually, using empirical relations so that the global selection of parameters, which is common practice with other methods, is obviated. In an appendix a method for deriving the relations between material functions on the basis of operator calculation is given.

Key words: Material function, normal-stress prediction, mirror relation, K ramers- Kronig relation, polymer solution

1. Introduction

The calculation of the first normal-stress difference N1 and coefficient 01 (~) = N1 (~))/~2, where ~) is the shear rate, from the viscosity function r/(~)) is of importance in two respects. Firstly, the functional relations between these two functions is of interest in the evaluation of rheological constitutive equations. In [1, 2] it was pointed out that an operator Y which, in the most general cases, relates the two functions mentioned with each other is apt to make a statement on constitutive hypotheses. Second- ly, this connection may be utilized to predict the normal- stress function, respectively, its coefficient whose measurement is complicated, from the easier measurable viscosity function.

Knowledge of the first normal-stress difference is of great importance for a large number of technological processes and for understanding natural processes. Thus, for instance, the ratio of the first normal-stress coefficient to the viscosity function determines the effective relaxation time, while the ratio of the initial coefficients

*) Extended version of a paper read at the 2 nd Symposium on Rheology of the GDR in Tabarz/Thuringia, December 7 - l l , 1987 308

01 (0)/t/(0) determines the longest relaxation time. Fur- thermore, phenomena such as extrudate swell and sec- ondary flow which are of importance to a number of technological operations are described by means of normal stresses.

Following the second consideration, a number of papers dealing with different methods of normal-stress prediction [2-8] have been published. There, the question arose repeatedly whether such an approach is funda- mentally feasible.

In this connection, a brief explanation should be given. As the generally four different stress components for the description of a fluid in simple shear flow as well as the three equivalent material functions are of one and the same molecular origin, the three material functions can- not be independent of each other. This is evidenced by experimental facts such as the correlation between the time constant 2 v determined from the viscosity function and the time constant 2E determined via the first normal- stress difference [8, 14]

).v 0 . 3 < - - < 1 . 2E

An analysis of the papers dealing with normal-stress calculations shows on the one hand that, with Deborah

568 Rheologica Acta, Vol. 27, No. 6 (1988)

numbers De = 2 9 (2 being the characteristic time constant of the fluid) smaller than unity, the accuracy of calculation may be very high. On the other hand, the calculated 01-values considerably deviate from the measured values at high Deborah numbers. The reason for these deviations is that the calculated and measured functions decrease with different orders. Even in accessible measurement ranges, underdetermination of the normal stresses up to a power of 10 is possible. Extrapolation of these values may yield even greater deviations. Besides, the selection of the front or shift parameters required for the prediction seems to be very coarse (a distinction is drawn only between those for polymer solutions and those for melts); however, a material-specific selection really should be possible.

It is, therefore, the purpose of this paper to derive a normal-stress calculation algorithm which predicts the correct order of magnitude of the function of the first normal-stress coefficient and, in addition, is simple enough to be governed by elementary calculations. Fur- thermore, the selection of the parameters required for the prediction shall be material-specific and simple. The usefulness of the procedure will be tested by the quality of reproduction of the normal-stress coefficient in case of high shear rates but also by its initial value.

2. Deve lopment of a prediction model

2.1 General relations

Consider a form of material function as it is obtained for most of the single-integral models (see, for instance, [1, 21)

tl(g = y ~b(s) F~(9s) ds (1) o

with t/(O)= t/o representing the Newtonian zero-shear viscosity and

0o 01 (9) = ~ (~ (s) F 2 (9 s) ds (2)

0

with 0~ (0) = 010 designating the initial value of the first normal-stress coefficient.

~b (s) is a distribution function which, in most cases, can be identified by the memory or relaxation function. Func- tions F 1 and F2 are influence functions which, in accordance with the constitutive hypotheses, describe a change in the distribution function in dependence on shear conditions.

Corresponding to the theory of linear operators [11, 12], Eqs. (1) and (2) may also be written as

= d l (~), (1')

01 = ~ 2 (~), (2')

where sd 1 and d 2 are linear, homogenous operators the inverses of which are postulated to exist. Inverting Eq. (1') and substituting in (2'), we obtain the desired general relation between the material functions

01 = d2 [ d l 1 (t/)] (3)

or, in generalized notation [11],

01 = 3- (q) with ~-- = d2" d , - 1 (3')

where the product of operators is not commutative.

2.2 Special cases

On the basis of Eqs. (3), we obtain for concrete rheological constitutive equations (i. e., concrete F 1 and F2) operators 3- by means of which the normal-stress coefficient can be calculated. Wagner's model [4], for instance, yields

~'~1 = -- ~ " ~ , ~ 2 = 9 2 . ~ , (4)

where ~ is the differential operator ~ = d/dp, p = m 9 and 5¢ the Laplace operator (see Appendix). Equation (4) yields for Y

and, consequently,

1 dq 01 -- (5)

m dg"

On the basis of the Laplace operator, also the equation for the mirror behavior, i.e.,

dO~ (g 9) 2 drl

d9 9 d9

and, in integrated form,

i l dr/ 01 (K ~) = - 2 . 9-- ~7 d9 (6)

can be derived (see Appendix). In terms of this equation and the Carreau equation

t/(9) = t/~o + (go - 700) [1 + (2 v • gz]-N (7)

results of normal-stress calculation were presented in [7, 8]. The shift factor K was selected such that, within the range of 2 v 9 ~" 1, the measured normal stresses agree with those calculated. Consequently, normal-stress calculation with lower Deborah numbers is sufficiently accu- rate as already mentioned. With high Deborah numbers, however, the function values for the coefficient decrease faster than the experimental values. This behavior is con- firmed by a consideration of the order of growth for Eq. (6) [1]. Accordingly,

~2m = 1 + ~ > c~2 (8)

Friedrich and Heymann, Primary normal-stress coefficient prediction at high shear rates 569

applies, i.e., that, in case of mirror behavior, the order 0{2m of the decrease of the normal-stress coefficient calculated through the order of the viscosity function 0{1 is greater then the actual value % observed in experiments.

A way of overcoming this incorrect behavior was presented in [2] where shear-rate dependent elastic defor- mation was introduced. The power term was normalized by means of a time constant which was obtained from the description of viscous properties (lower index V). In combination with Eq. (6), this modification yields an improved prediction model

~1 (K 9)= - 2 ~ ( 2 ; ! ')p dr/dg' (9) dg'

which, in case of high shear rates, will behave as

01 (9) = o {9-(1 +~,-e~} (10)

so that it admits the setting of 0{2 through ft. Having described a feature of Eq. (9) for any viscosity

function by means of Eq. (10), let us continue by finding the exact solution of Eq. (9) in combination with Eq. (7). By substituting Eq. (7) in Eq. (9) and with the substitution of z = [1 + (2v9) 2] i we obtain a tabulated integral [12] having the following solution

s B~ (., .) is here the incomplete Beta function. By convert- ing the incomplete Beta function into the hypergeometric function 2F1 (.,. ;. ;z) [12], we obtain a useful solution

2(~o- ~ ) 2v 0{1 01(K 9) = [1 + (~v 9)21 (~ +~-e~/2 0{~ + 1 - / 3 (12)

'2/ :1[ ~1+1-fl2 , 1-2 fl ' % + 3 - / ~ . 2 , z ] .

Now, Eq. (12) consists of two terms; a first term which we may describe as Carreau equation for the coefficient of the first normal-stress difference, and a second one, which is a correcting term• Actually, the latter term

0{ 1 y - 0{1+1-/3

. 2 F l [ ~ l + l - f l 1 - / 3 . ° q + 3 - f l . 1 ] 2 ' 2 ' 2 ' 1 + (2v 9) ~

with

Y =

for 2 v 9 = 0,

0{1 for 2v 9 ~ oe 0{1 + 1 - fl practically for

2v~) > 10,

nearly shows no change over the entire ranges of shear rates. For a set of reasonable parameters (see Table 1) 0{1 = 0.6 and fi = 0.3, this function increases monotonical from its initial value of 0.641 to a final value of 0.923.

On this basis, it is now possible to use equation

01(K9) = 2(t/o- t/~) 2v 0{1 0{~+l - f l

• [1 + (2v" 9) 2] -(al + I -/~3/2 (14)

for the calculation of normal stresses in case of high shear rates and the equation

01o = 2(t /o- t/o~) 2v (15)

for determining the initial coefficient 01o. Thus, we have an analytical expression which allows us

to simply calculate the normal-stress coefficient or its initial value with the aid of other information from viscosity function (see section 3). The usefulness of Eqs. (14) and (15), as well as the selection of the parameters still missing, will be described in the following section.

3. Results and discussion

For the fluids given in Table 1, the parameters for the Carreau viscosity equation are given. The fluids are ex- clusively polymer solutions which allow the experimental determination of the normal-stress coefficient up to Deborah numbers of 104 . The stresses measured for these fluids are given in figures 1-3. The parameters 0{1, fi and K contained in the table were determined so that the calculated coefficients of the first normal-stress difference agree with those measured at high Deborah numbers (De > 10).

The parameters K and fi, however, must be known previously in order to be able to predict the normal stresses. To enable the determination of these two constants in dependence on the given parameters of the given viscosity function, a correlation between fl and 0{1 (see figure 4) as well as K and 0{1 (see figure 5) was established. A linear dependency was selected for the first of the two relations and the following equation was obtained

fl = 0.80 - 0.67.0{ 1 . (16)

For the second relation, a step-function type dependency was selected as a first approximation. This relation is expressed as

1 for 0{1 < 0.65, K = 0.6 for 0{1 > 0.65. (17)


Table t. Data of the fluids investigated as well as the determined parameters cq, fl and K

Fluid Refer- Carreau viscosity Determined parameters Symbol ence

alo a~ 2,, N ~1 fl K (Pa s) (Pa s) (s) . . . .

i) 2% PIB in Primol 355 [3] 2) 0.75% PAA in 95/5 water/glycerin [3] 3) 7% Al soap in decalin and m-kresol [3] 4) 5%polystyrene in Aroclor 1242 [3] 5) 1% PAA in 50/50 water/glycerine [9] 6) 2% PAA, like 5 [9] 7) 3 % PAA, like 5 [9] 8) 1% polystyrene in Aroclor 1248 [9] 9) 2.5% polystyrene, like 8 [9]

10) 4% polystyrene, like 8 [9] t t) 8 % polystyrene in tricresyl [t 3]

phosphate 12) 12% polystyrene, like 11 [13] 13) 6.76% linear polybutadiene [15] 14) 23.9% radial polybutadiene [15] 15) Silicone oil BW 400 [15] 16) 2% polyethylene oxide in water [16] 17) 3% polyethylene oxide in water [16]

9.23.102 0.15 191 0.321 0.642 0.320 1 ©

1.06.101 0.01 8.04 0.318 0.636 0.360 1 [] 8.96.101 0.01 1.41 0.400 0.800 0 .360 0.625 1.01 - 102 0.059 0.84 0.3tO 0.620 0.440 1 + 8,07-101 O. 24.88 0.323 0.646 0.280 0.58 0 4.79.102 O. 59.9 0.340 0.680 0.210 0.55 L1 7.71 • 102 O. 36.3 0.372 0.744 0 .250 0.62 • 3.30.100 0.3 0.189 0.114 0.228 0.640 t • 3.23.101 0.3 0.77t 0.211 0.422 0.578 1.2 [] 2 . 4 3 . 1 0 2 0.3 3.05 0.276 0,552 0.466 0.9 A 3.90.102 0.07 2.11 0.325 0.650 0.400 1 0

6,70.103 0,07 15.0 0.390 0,780 0.360 0.85 0 1.75.103 O. 1.78 0.365 0.730 0.316 - 2.10.103 O. 0.155 0.415 0.830 0.471 1.64.10'* O. 17.5"*) 0.41 0.810 0.180 - 2.91 • 101 O. 8.17 0.287 0.574 0.482 - 1.42. tO z O. 22.88 0.294 0.588 0.397

-2

-3

-4

• ,- \ .i.

÷

d

. d o - I o . g~

\ %X \ \ "\,'% ,

Mi)'ror - re lot ion /n ~ h e

~iYeD p r o c e d u r e

- - " - M i r r o r - r e / o # O n

2 ,3 4 /9 '~, ~*,4

With the aid of these coefficients, which are to be individually selected for each fluid in dependence on e l , the normal-stress coefficients were calculated and represented in figures 1 and 2 by full continuous lines. I t can be seen that predeterminat ions of normal stresses within the range of high Deborah numbers are very possible on the basis of the equation given here. It is clearly shown that the slope of the calculated normal stresses corresponds to that actually measured. Thus, a more realistic calculation of these values within the range of high Deborah numbers and a secure extrapolat ion into the range of shear rates, which are hardly accessible to the experiment, are possible. A comparison of the method presented here with other methods of calculation, such as the mirror relation (figures I and 2) or the Wagner relation (figure 2) shows clearly that, at high shear rates, a better predict ion is possible.

As the method presented here is based only on infor- mat ion obtained from the viscosity function which is in- dividual for each fluid, in contrast to other methods (where shift factors are distinguished al together only for

Fig. t. Representation of the measured (symbols) and calculated (lines) normal-stress coefficients for four polymer solutions in dependence on the dimensionsless shear rate (Deborah number). A is a shift factor for the sake of a clearer representation of the results


solutions and melts), deviations for different polymers can be minimized.

Let us now briefly remark on the initial coefficient. By means of Eq. (15), the initial coefficients for some fluids were calculated and compared with experimental data (see Table 2). The latter were either directly determined or

-1

~ - 2

c3~

- 3

- 5

9

i.

Q~

'b,, II I •

2,5N from[O.]

72,0g \ \+ "~ 6

Mirror - rel~bn in fhe given " ~ , k p d u r e . . . . \ . ~ 6

\ \ \

i

~9 ~2+A

Fig. 2. As in figure 1, but for five polystyrene solutions with different concentrations

obtained from extrapolations of the first normal-stress difference or the storage modulus at low shear rates or angular frequencies. The comparison shows (and this ver- ifies the determination of 01o by Eq. (12), cf. [15]) that satisfactory calculation is possible. The influence of the coefficients K and /~ on these values is unimportant. While K is irrelevant, it was estimated that consideration of J~ will result in an increase of the 01o-values of approx- imately 6% to 12%. This suggests that the equation for

l A

A . o

-1 I

\ . \

Fig. 3. As in figure 1, but for three polyacrylamide solutions with different concentrations

0.7

0,6

0,5

0.3

0,2 0,2

Oo- 1

O,8

Fig. 4. Dependence of the coefficient p on the order of the viscosity function at high shear rates s 1 (Dots: measured values, full line: regression line)


Table 2. Comparison of the initial values of the first normal-stress coefficient obtained by experiment and calculated as well as of the three time constants 2 v (viscous time constant), 2 E (elastic time constant), and relaxation time according to rouse 2R

__ e x p - - ,~V/,~ E Fluid as in a~o v ~al~ "~V "~E -- a 2 o / a l o 2g 6a1° Mw a2° rc z c R T

Table 1 (Pa s 2) (Pa s 2) (s) (s) (s)

1) - 2.35- 105 191 255") 170 0.749 3) 3.30.102 1.92.102 1.41 3.68 - 0.383 4) - 1.13.102 0.84 1.11 *) 0.89 0.757

11) 1.31'103 1.11.103 2.11 3.36 1.34 0.628 12) 1.80.10 s 1.51.105 15.0 26.87 24.66 0.558 13) 3.74"103 4.51.103 1.78 2.14 2.22 0.832 14) 5.38-10 z 5.04"]02 0.155 0.256 0.427 0.605 15) 2.20 '106 4.61 "105 17.5 **) 134 - 0.131 16) - 3.06 "102 8.17 10.5") 1.62 0.778 17) - 4.15 "103 22.88 29.22*) 5.26 0.783

t a l c *) Defined as a2o /alo s **) Defined from series statement for a 1 with 2 v = Z wi,~, where w~ are weight factors and 21 the appertaining time constants

i = 1

L2

t.t

0,5

0,5 O.2

--O ....... +--QO

O.9

%

O.O

O,7

%

g

..2.

i F i i r

~3 ~ 0.5 O.6 0.7 0.0

6 1

. / o ".//

o

/ ' '7 ' - I 0 2

/ 1 y , S

Fig. 5. Dependence of the shift factor K on the order of the viscosity function at high shear rates cq (Dots: measured values, full line:first approximation for an analytical description of this correlation)

Ol0 derived in [7, 8] is a l ready sufficient for the prede- t e rmina t ion of this quant i ty .

Final ly, we should like to re turn to the p rob lem of the re la t ion between "viscous" and "elastic" t ime cons tan ts which was raised in the in t roduct ion . Table 2 also shows a compar i son of these quant i t ies and their re la t ion to a re laxat ion t ime 2 R (according to Rouse) calculated from molecular data. These da ta confirm the re la t ion between these quant i t ies as given in the in t roduc t ion and set up elsewhere. The corre la t ion between these re laxat ion times is d iagrammat ica l ly shown in figure 6. This figure con- firms indirect ly that it is possible to calculate no rma l stresses from viscosity data.

. , d

Fig. 6. Correlation of the time constant 2~ determined from the first normal-stress difference N t and of the relaxation time according to Rouse 2R with the time constant 2 v determined from the viscosity function t/


4. A p p e n d i x

4.1 Definition of some operators and operations

Let us define the operator ~-~(integral operator) by the following

t

~ - * [f] = 5f(t ' ) dt'. (A1) o

The function f (t) is assumed to have no poles on the positive semi-axis, to vanish at infinity and to be finitely summable. The inverse of the integral operator is assumed to exist and is described as differential operator

@ [f] = ~ f (t). (A2)

Furthermore, a second operator is necessary. It is defined by the following

co

~ , 1 [f] = ~ f ( t ' )d t ' . (A3) t

The product of two operators ~¢ and N is understood to be the operator ¢~ = s¢. N given by the equation

[f] = ~¢ [N [f]] (A4)

[11]. This product is not commutative, i.e., s¢- N =~ N . ~¢. Pow- ers of s¢ (e.g., s¢") are understood to be the n-fold product of the operator by itself. The equations

~, , = s¢~ - 1 . sO', ~,o = g (A5)

apply, g is the unit operator which submits an identical mapp- ing. Accordingly, the following relations between the differential operator and the integral operator can be derived

~ - ~ = - ~.i = g , (A6)

@, = - ~ . (A7)

In addition, integral transformations (corresponding to operators) are required. These are, in detail, the Laplace transformation

L(s) : ~ [f, s] = i f (t) e -~t dt (A8) 0

where s is a complex variable and the function f (t) must satisfy such conditions as are given, for instance, in [17].

Furthermore, the Fourier sine transformation

F~ (r) = ~ [f, r] = f (t) sin (r t) dt, (A9)

the Fourier cosine transformation

F~ (r) = ~ [f, r] = ! f (t) cos (r t) dt, (A10)

and the Hilbert transformation

1 oo H ( r ) = J ~ [ f , r ] = - ~ f ( t ) ( r - t ) ~dt (All)

gC - o o

are introduced. The bar in the integral is to be understood in the sense of its principal value. The brackets of the operator may be omitted, if this does not cause any abstrusness.

4.2 Derivation of the operator of the mirror relation

The mirror relation is based on the apparent similarity of the functions t/(~)) and r/+ (t) as well as ~p~ (~)) and ~+ (t) by the follow-

ing relations co

rl(~) = ~ N ( z ) K 111 - e x p ( - ~t)] dzlt=+ 1, (112) 0

01(7) = 2 ~ N ( × ) z - Z [ l - (1+ ~t) e x p ( - zt)] dzlt=K~ o (A13)

where r/+ and ~ ( are the viscosity function and the first normal- stress coefficient, respectively,, in a start-up flow and N (z) is the density function of the relaxation frequency spectrum.

The Eqs. (A12) and (A13) can be represented by the variable p = ~- 1 by means of (A3) and (A8) as follows

o~

tl - ~o = - ~ , ~ ~ [N,p], ~o = ~ N (~) ~- ~ dz, (A14) 0

~ - Or0 { ~ . 2 ~[N,p] +p~.~ ~ [ N , p ] } , (A15a) 2

co

I~10 = 2 ~ N(z)x 2 dz. (A15b) 0

Thus, the reduced operators SJ1R and ~2R (see Eqs. (Y) and (2') reduced because a constant appears on the left - can be written

a s

S¢IR = -- ~ , ~ S , (A16)

The operator J - is obtained by inversion of (A16)

j - = ( ~ , 2 + p ~ , t ) Af 5~- 1 ~ , (A18)

By factoring-out the operator ~ , 1 and making use of the relation (N,- 1 + p) = ~ , 1 [p N] as well as of Eqs. (A6) and (A7), Eq. (A18) can be simplified as

j - = ~ , l p ~ . (A19)

This form corresponds to the form of the mirror relation

~ - 01o = i 1 d~ 2 o SZ ~ d~;' (A20)

given in [20].

4.3 Derivation of the operator of the Kramers-Kronig relation in terms of the example of the eonstitutive equation by Goddard-Miller

In [3], the Kramers-Kronig relation for normal-stress calculation, which was derived on the basis of the Goddard-Miller model, was used. Let us now develop this relation on the basis of the approach proposed in this paper. For this purpose, new variables have to be introduced for the viscosity function and the normal-stress coefficient: a 1 = t/(~) and a 2 = ~/2 ~1 (~). Thus, for the Goddard-Miller model (see, e.g., [3])

a 1 = ~ G(s) cos3)s ds, i.e. sJ 1 = ~ [ G , ~], (A21) 0

a 2 = ~ G(s) sin')s ds, i.e. d z = -~ [G, ~]. (A22) o

The operator ~- appears in the form

~- = ,~ ~,~ - ~. (A23)

By means of the Laplace transformation, this operator can be represented in another way. If the following relations

~ - 1 = (1/2n)1/2 { y [.;ip] + L~a[.; _ i~]}, (A24)

= (1/2~) 1/z i { ~ [. ;i9] -- 5f [.; -- i~]} (A25)


are substituted in (A23), we obtain

Y = ( 1 / 2 ~ z ) { i ~ [ S f [ . ; i ~ ) ' ] ; i ~ ] + i~[~,°[. ; i~)];- i~]

- i2~[~°[.;i~']; - i7'] - i5¢ [•[.; - i?)]; - i7 ' ]} . (A26)

The sum of the first and third terms as well as of the second and fourth terms in the brace brackets yield in each case the ~-fold of the the Hilbert transformation so that, in the case of the Goddard-Miller model

Y = ~ (A27)

and thus

2 7 al(~)' ) , . , 7z y -Y~ ~ a2. (A28)

This is the form of the Kramers-Kronig relation given in [18]. The form that is better known in rheological literature will be obtained when the integral is divided into two parts, the proper- ty of a 1 to be an even function is used and the integral relation

:~ ( ~ _ ~,~)-1 d~' = o 0

is utilized:

2 ~ ~ (9') - ~ (9) . . , (A29)

The three relations between the viscosity function and the first normal-stress coefficient derived in this paper for a particular rheological constitutive equation each apply to other constitutive equations. An appropriate list is contained in [1].

References

1. Friedrich Chr (1984) Paper published in the Proceedings of the 1 st Symposium on Rheology of the GDR, January 9-13, 1984, Berga/Elster; Institute of Mechanics of the Academy of Sciences of the GDR, Report-No. R-IMech-07/85, pp 20-31

2. Friedrich Chr, Heymann L (1988) In: Giesekus H, Hibberd MF (ed), Progress and Trends in Rheology II. Steinkopff Darmstadt, pp 102-104

3. Abdel-Khalik SI, Hassager O, Bird RB (1974) Polym Eng Sci 14:859-867

4. Wagner MH (1977) Rheol Acta 16:43-50 5. Gleissle W (1982) Rheol Acta 21:484-489 6. Stastna J, DeKee D (1982) J Rheol 26:565-570 7. Friedrich Chr (1984) Plaste und Kautschuk 31:12-15 8. Friedrich Chr (1985) Acta Polymerica 36:509-514 9. Stastna J, DeKee D (1986) J Rheol 30:207-230

i0. Gelfand IM, Shilov GE (1967) ,,Verallgemeinerte Funktio- nen (Distributionen)" Deutscher Verlag der Wissenschaften, Berlin

11. Kantarowitsch LW, Akilov GP (1964) Funktionalanalysis in normierten Rfiumen. Akademie-Verlag, Berlin

12. Abramowitz A, Stegun IA, "Pocketbook of Mathematical Functions" Verlag Harri Deutsch, Thun and Frankfurt/ Main 1984

13. Graessley WM, Park WS, Crawley RI (1977) Rheol Acta 16:291-301

14. Dealy JM (1970) Trans Soc Rheol 14:461 481 15. Friedrich Chr, Schnabel R (1985) J Non-Newtonian Fluid

Mech 17:185-191 16. Martischius F-D (1982) Rheol Acta 21:288 310 i7. Korn GA, Korn TM, "Mathematical Handbook for Scien-

tists and Engineers" McGraw Hill, New York 1968 18. Landau LD, Lifschitz EM, ,,Statistische Physik, Teil 1"

Akademie-Verlag, Berlin 1984

(Received January 25, 1988)

Authors' address:

Dr. Chr. Friedrich, Dr. L Heymann Institut ffir Mechanik der Akademie der Wissenschaften der DDR Postfach 408 DDR-9010 Karl-Marx-Stadt

normal stress coefficient

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