Some properties of general linear elastico viscous liquids

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ARCHIEF

Haifa, October

1, 1967

Lab. v. Scheepsbouwkunde

Technische Hogeschool

Delft

Technlon Israel Institute of Technology

Technlon Research and Development Foundation Ltd. Faculty of Civil Engineering

Hydraulics Laboratory

SOME PROPERTIES OF GENERAL LINEAR ELASTICO-VISCOUS LIQUIDS

by

H. Rubin and C. Elata

Office of Naval Research

Contract N 62558-4093

Task NR 062-338

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Technion - Israel Institute of Technology Technion Research and Development Foundation Ltd.

SOME PROPERTIES OF GENERAL LINEAR ELASTICO-VISCOUS LIQUIDS

by

H. Rubin and C. Elata

Office of Naval Research

Contract N 62558-4093

Task Er 062-338

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED Haifa, October 1, 1967

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Some Properties of General Linear Elastico-Viscous Liquids*

H. Rusnsi** AND C. ELATA***

Faculty of Civil Engineering, TechnionIsrael Institute of Technology, Haifa

Received April 2, 1967

ABSTRACT

An analysis of phase lag between stress and strain, normal stress differences, Weissenberg effect and non Newtonian viscosity phenomena, by use of several constitutive equations describing "Linear" elastico-viscous liquids, is presented. It is shown that new general constitutive equations of linear elastico-elastico-viscous liquids do describe all the above phenomena. The restrictions of the above models are also discussed in view of experimental results in flow of polymer solutions.

INTRODUCTION

The increased interest in flow of polymer solutions

and other non-Newtonian fluids has led several inves-tigators to examine constitutive equations which might

predict the non-Newtonian phenomena observed in

flows of such solutions [1, 4

Some complicated rheological models may

de-scribe all the anomalies observed, but it is desirable to

find much simpler constitutive equations for such

descriptions. The simplest possible rheological

equa-tions are the so-called linear elastico-viscous equaequa-tions.

Molecular considerations' top suggest that the

consti-tutive equations should be linear [3, 4, 5].

A new linear model proposed here describes

phe-nomena which previous models could not.

ISRAEL JOURNAL OF TECHNOLOGY, Vol. 5, No. 4, 1967, pp. 292-297

* This research forms a part of the D.Sc. thesis of the first author under the guidance of the latter. Lecturer

**' Associate Professor.

292

THE CONSTITUTIVE EQUATIONS

Conventional models

The simplest model of linear elastico-viscous liquid

is the Maxwell model, the constitutive equation of

which is

0

(1+

t

2rielP_J

at

where eil)

_j.

The tensor A; and el}) are the deviator stress tensor

and the rate of strain tensor respectively.

Another simple constitutive equation is that of the

Kelvin model,

21(1

+

.

at

e(J) rate

ofstrain tensor g gravitational acceleration

metric-tensor

K function defining the elasticityofliquids

N(T) distribution function of the relaxation times

hydrostatic pressure Pu stress tensor

deviator stress tensor radius

function related to the elasticityof liquids

NOTATION time velocity coordinate rateofshear - viscosity A retardation time specific mass relaxation time tfr relaxation function angular velocity

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Vol. 5, 1967 H. RUBIN AND C. ELATA 293

Whereas Eq. (1) describes relaxation of stresses, Eq.

(2) describes retardation of strains.

Oldroyd [6] has pointed out that the time derivative

in such equations should be Lagrangian. He has,

therefore, replaced the local derivative with a convec-tive one. Moreover, in order to describe the relaxation

and retardation as two properties of the liquid, he

has combined these two equations. As the convective

derivative for the covariant tensor is different from

that for the contravariant tensor, Oldroyd defined two

types of elastico-viscous liquids: liquid A, the

con-stitutive equation of which is

(1+

St

).

19' =

211(1+ A)e")

St

and liquid B, with the constitutive equation

(1+

=

271(1 ± A-L)e(1)ii

St

Walters [7] used the Boltzmann superposition

principle for convective integrals and generalized

Oldroyd's models to describe liquids having a more

Complicated dynamic response. He defined two types

of linear elastico-viscous liquids: Type A' liquid has

the constitutive equation

5 ax-

If-(X, t) = 2 0 (t t') 4,1)(x', Ode (5)

axi

ax` n

(Oldroyd's A model is a special case of this model).

Type B' liquid (of which Oldroyd's B liquid is a

special case) has the constitutive equation

Oxl

p!i(x, t)= 2f

(t e)

e")"" (x', Ode

(6) ax'm

- CO

The function tfr in Eqs. (5) and (6) is the stress relaxation function defined by

(r)

111(t e)

=f

exp (t

r] dr

. Jo T

The function N(-r) is the distribution function of the

relaxation times, T. X'i (X, t, e) is the position of a

fluid element at time e

. This

element passes point xl

at time t.

Proposed new models

Instead of using contravariant and covariant

expres-sions for the stress and rate of strain tensors we shall

use in this paper expressions of a more general mixed

form. Special cases of mixed models are the following:

+

6)p'jj

=

[g (1+

2-6)ek

&" + gkj

+21e")]

St St k T

(5)p;

[

_{it(5t) (1)k}

gk(+).(5)el')]

St gki ej St

If instead of one relaxation and one retardation time

there are many relaxation and retardation times, we

may, by following Walters' method, use the Boltzmann superposition principle for convective integrals. In this manner we get two new models of the so-called linear

elastico-viscous liquids, which we shall name C and

D' models. Their constitutive equations are

(x, t) =

f

(t e)

(7)

ki

ax-

axi k. aX'm aXi

g _{e214 (x; e) de}

and

6(x, t) =5

(t e) CO

[

Ski aXk ax"'_a

±

kj aa Xk lax'. x i.

4,1)..(x; e) de, ax

respectively.

NON-NEWTONIAN PHENOMENA IN LINEAR

ELASTICO-VISCOUS MODELS

In this section we shall examine the non-Newtonian

phenomena described by each of the above constitutive

equations. The results are summarized in Table I.

Phase lag between shear stress and rate of shear

For unsteady flows all the above constitutive

equa-tions show phase lag between shear stress and rate of

strain. This is, in fact, the only non-Newtontian

pro-perty of Maxwell's and Kelvin's models (Eq. (1) and (2)). Normal stress differences

In a simple shear flow (Figure 1) the first and second normal stress differences are usually defined as

PIA P22 ; P22 P33, (9)

where pu is the stress tensor.

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Like the Newtonian

mOdel, Maxwell's model does not exhibit any normal stress difference.

Walters [7] has shown that the normal stress

dif-ferences in liquid A' are

Pit P22 = 2Y2K;

P22

P33 = 2Y2 K

(10)

and in liquid B'

P1,P22 = 2Y2

P22 P33 = 0,

(11) where o3 Y 1ei 2(1) ,

K =

N (T) ch. . (12) Jo=

We shall now compute the normal stress differences for the C' and D' models. In simple shear the velocities are

2 .

_{= 0}

₍₁₃₎

yx , v2 -=- 0;

Where y is a constant. All the components of the rate

of strain tensor, except

,(1) ,(1) I

c12 c21 7

are equal to zero.

A fluid particle whose position at time t is X1 X2, X3

was at time t' at

x'l =

y x2(t t');

x'2 = x2;

x'3 = x3 (15)

From the Constitutive Eq. () it follows that for a

simple shear

- tfr(t

t')-2

_-

.

ax'l

axi

ax''

ax' ax'2 axi ax'2

[

ax'

+

axi

Ox'2

+

axi

ax" ±

oxi Using Eq. (15) we get from Eq. (16)

- ..

+ .2K

_

='

PT }' K ; P 33 =

Pi 1 = P T

;22

P

Pi 1 P22 = 2Y2 K; P22

P33 = Y2 K .

For liquid D' the values of the normal stresses are

computed in a similar manner. From Eq. (8) it follows

that in a simple shear flow of liquid D'

=

f

/(t

t')

2 _

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294 H. RUBIN AND C. ELATA

TABLE I

Israel J. Techn:,

NON-NEWTONIAN PHENOMENA IN RHEOLOGICAL MODELS AND EXPERIMENTS

_

Liquid Maxwell A' B' C' D' Experimental Results in flow of

polymer' solutions Constitutive

equation (i) (5) (6) (7) (8)

Property

Stress and strain X

phase lag exists exists exists exists

exists [8]

PitPaz

zero positiVe positive positive positive [9], [10], [11], [12] zero [9]

P22 P33 zero negative, zero negative positive [11], [12], [13]

negative [14] Viscosity . Weissenberg effect Newtonian f does not exist

Newtonian Newtonian non-Newtonian negative positive negative or positive

non-Newtonian positive negative [15]

Figure 1

Coordinate directionS in a simple shear flow.

ex'2

axi

ax' axi

ax''

[ax.'

_.

_i_

+

axi

ax''.., ax' 1 .0x, ax'1 ax` (3)e2 ax' ax'2 axi

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Vol. 5, 1967 H. RUBIN AND C. ELATA 295 We can see from Eq. (19) that in simple shear flow the

normal stress differences for liquids C' and D'- are the same.

Newtonian and non-Newtonian viscosities

The Maxwell, Oldroyd and Walters liquids have a

Newtonian viscosity, namely, they exhibit a linear de-pendence between the rate of shear and the shear stress:

PI2 = 2?1 e(1.12 = Y (20)

The viscosity ri of Walters' liquids A' and B' is defined as

'1=

=1

N (r) dr . (21)

Jo

The relationship between the shear stress and the rate

of shear in liquids C' and D' follows from Eqs. (16)

and (19):

P12 = FiY-/SY3 = (F1 1SY2)7 (22)

where is defined according to Eq. (21). The function S is

N (r) dr .. (23)

We see that in liquids C' and D' the ratio between the

shear stress and rate of shear is not a constant. The

viscosity thus consists of two terms. The first one is a

constant and the second one is proportional to the

term Sy2, where S is a scalar and y2 an invariant.

Weissenberg effect

We shall now examine the flow of the above liquids

between coaxial cylinders (Couette flow) and see

whether they exhibit the Weissenberg effect (climbing

of the liquid on the inner rotating cylinder).

This Weissenberg effect is not shown by the Maxwell model.

Walters [7] has already demonstrated that liquid

B' shows a positive Weissenberg effect while liquid A'

shows a negative Weissenberg effect, i.e. the liquid's

climbing on the outer, stationary cylinder when the

inner cylinder is rotating.

hi a Couette flow of liquids C' and D' the physical

components of the stress are according to Eqs. (17)

and (22): c/f2 12

dr

(24)

fl

df23

60) =

+

ddr

H

dr

where f2 is the angular velocity.

From the equations of motion in cylindrical

co-,

ordinates we get

Op 1

prf22 =

+ - (r p'

)

ar

r

(") 1 a

0 =

(r p))

r

Or ap

0 =

+ pg .

az Solving Eq. (26) we obtain

M N

SI = L +

+ -7 ,

(28)

r

where

L

=fl ,. 2

2 II.2 2S r? (r/2

nir 01- et)

₍₂₉₎

(r3

_ri)

_{3 (pi}

_r-)4 (02 ) 2S (f22 fli)2 (r1 r7) 114 = 1 (r3

ri)

ri)3 (30) and

N=

2S rt

(f22 (r3 r?)-3 (31) are constants, and 1.1, Di and r2, S22 are the radii and angular velocities of the inner and the outer cylinders, respectively.

The solution of Eqs. (25) and (27) gives

The Weissenberg effect is determined by the second

term, that depends on K, in Eq. (32), which can be

P =

P(zz)= .A42

2ar In

P

+

[

' L2 r2

gz + 2 +

N2

IN

MN

(32)

r

27'2 Ii

[2M2

K r4

10r"

77MN '2r4 3r6 4r8

30N2]

+

r1 2

(7)

296 H. RUBIN AND C ELATA Israeli. Techn.,

either an increasing or a decreasing function of r

(according to the sign of the product MN, which can

be positive or negative).

COMPARISON WITH EXPERIMENTAL RESULTS

Several investigators have attempted to measure

non-Newtonian phenomena using polymer solutions.

Un-fortunately results of measurements of all the effects

in any one of these fluids are not available.

Rouse et al. [8] have measured the phase lag

be-tween stress and strain in unsteady flows of solutions

of polystyrene in toluene and polyisobutylene in oil.

Their results can be described by any of the models

A', B', C' and D'.

Normal stress differences have been measured in

several polymer solutions. Roberts [9], using a cone

and a plate rheometer, has shown pii

p22 > 0 and

P22 - P33 = 0 to be consistent with model

B'.

Positive values of pil

p22 were measure also by

Markovitz and Brown [10], Ginn and Metzer [1]

and others. This fact is compatible with the A', B',

C' and D' models.

The measurements of the second normal stress

dif-ference show more diverse results.

While Roberts obtains p22

p33 = 0, experiments

by Adams and Lodge [11] with polyisobutylene

solu-tions in dekalin (which is one of the solusolu-tions used by Roberts) as well as experiments of Markovitz [12] with

solutions of polyisobutylene in ceta.ne show that

P22 - P33 >

Hayes and Tanner [13], investigating the flow of

polymethylmethacrylate solution in toluene, also found

that the second normal stress difference is positive.

These results (p22

p33 > 0) cannot be described by

any of the linear elastico-viscous models investigated

so far, for they give only '722 - p33

0.

Lodge [14], however, reports recent investigations

showing p22

p33 <0. It is worth noting here that

the normal stress differences at low rates of shear

re-ported so far are proportional to the square of the rate

of shear. This was predicted by the linear models of

the elastico-viscous liquids:

The viscosity of polymer solutions is usually

non-Newtonian. They behave like pseudoplastic or dilatant

liquids. This property is described by the constitutive

equations of liquids C' and D'. These constitutive

equations suggest that the non-Newtonian viscosity of

the liquid is caused by its elasticity.

The positive Weissenberg effect is a well known pro-perty of polymer solutions. It is described by model B' as well as by models C' and D'. Negative Weissenberg

effect was also observed experimentally in flows of

some dilatant pastes [15].

CONCLUSIONS

The so-called linear elastico-viscous models can

de-scribe many of the non-Newtonian properties of

polymer solutions. This description is summarized in

Table I. The general elastico-viscous models in this

work can also describe the non-Newtonian viscosity

of these liquids. None of these models, however, can

describe a positive, second normal stress difference.

ACKNOWLEDGMENTS

This research was sponsored by the U.S. Office of

Naval Research, under Contract N62558-4093 and

by the Technion Research Fund. The authors wish to

thanks Dr. M. Poreh for his kind assistance in this

work. The help of Eng. A. Cohen, who reviewed the

manuscript, is also acknowledged.

REFERENCES

[I] GINN, R. F. AND METZNER, A. B., 1965, Normal stress in polymeric solutions, Proc. 4th Int. Congress on Rheology, ed. by E. H. Lee, pt. 2, 583-601, Interscience Publ.,. .

RUETN, H. AND ELATA, C., 1966, Stability of Couette flow of dilute polymer solutions, Phys. Fluids, 9, 1929-1933. ROUSE, P. E., 1953, A theory of the linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys., 21,

1272-'1280.

Zimm, B. H. J., 1956, Dynamics of polymer Molecules in dilute solutions: visco-elasticity, flow birefrigence, and dielectric loss, J. Chem. Phys., 24, 269-278.

Zitvim, 13.- H., AND ROE, G. M. AND .EpsrErN, L. F., 1956,

Solu-tion of a characteristic problem from the theory of chain

molecules, J. Chem. Phys., 24, 279-280..

[6] OLDROYD, J. a-, 1950, On the formulation of theological equation's of state, Proc. Ray. Soc. (London), A200, 523-541. WALrERs, K., 1964, Non-Newtonian effects in some general elastico-viscous liquids; IUTAM Int. Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, ed. by M. Reiner and D. Abir, 507-519, Pergamon Press.

-Sirra., K., ROUSE, P. E., BALLEY, E..D., 1954, Method for determining the Visco-elastic properties of dilute polymer solutions at audio-frequencies, J. app!. Phys.. 25, 1312-1320.

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Vol. 5, 1967 H. RUBIN AND C. ROBERTS, J. E., 1957, Normal stress effects in tetralin soluticin of polyisobutylene, Nature, 179, 487-488.

MARKOVITZ, H. AND BROWN, D. R., 1964, Normal stress meas- [131

urements on a polyisobutylene-cetane solution in parallel plate and cone-plate instruments, IUTAM Mt. Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, ed. by M. Reiner and D. Abir, 585-602, Pergamon Press. - [14]

ADAMS, N. AND LODGE, A. S., 1964, Rheological propertids of [15]

concentrated polymer solutions, Phil. Trans., A256, 149-184. MARKOVITZ, H., 1965, Normal stress measurements on polymer

ELATA 297

solutions, Proc. 4th Int. Congress Rheology, ed. by E. H. Lee, Pt. 1, 1-89-212, Interscience PUN.

HAYS, J. W. AND TANNER, R. I., 1965, Measurements of the second normal stress difference in polymer solutions, Proc. 4th Mt. Congress Rheology, ed. by E. H. Lee, Pt. 3, 389-399, Interscience PUbi.

LODGE, A. S., 1964, Elastic liquids. Academic Press. BANTOFf, E., 1959, Some measurements on pigment-plasticizer dispersions, Rheology of disperse systems, ed. by C. C. Mill, 105-126, Pergamon Press.

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SOME PROPERTIES OF GENERAL LINEAR ELASTICOVISCOUS LIQUIDS

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Rubin, H. ; Elata, C.

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13. ABSTRACT

An analysis of phase lag between stress and strain, normal stress

differences, Weissenberg effect and nonNewtonian viscosity phenomena,

by use of several constitutive equations describing "Linear" elastico-viscous liquids, is presented. It is shown that new general constitutive equations of linear elasticoviscous liquids do describe all the above

phenomena. The restrictions of Ihe above models are also discussed in

view of experimental results in flow of polymer solutions.

D DI F.1."4641473

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