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**The physical meaning of
Weissenberg’s hypothesis**

**with regard to the second
normal-stress difference**

**PROFESSOR DR. H. GIESEKUS**

*Universitat*
*Dortmund,* *Bundesrepublic* *Deutschland*

**1. INTRODUCTION**

Weissenberg’s concept of an elastic fluid is
based on the notion of recoverable strain. But in contrast to the strain of an
elastic solid, this quantity is an inner variable depending on the deformation
history. It may be determined - subject to some restrictions - by the creep
recovery experiment or by flow birefringence. His ideas lead Weissenberg to
conclude that in steady shear flow the first normal-stress difference would be
positive, i.e. there would be a deviatoric tensile stress in the flow direction
while the second normal-stress difference would vanish^{1}.

This conclusion interested quite a few people
from the beginning, partly because the model of Reiner of a non-Newtonian fluid
- later termed Reiner-Rivlin fluid - which was developed shortly afterwards
predicted the equality of the two normal-stress differences^{2}.
Approximately at the same time Fromm^{3}, and independently a little
later DeWitt^{4}, rediscovered the model of Zaremba^{5}, which
had been developed about 40 years earlier. This model, a modification of a
Maxwell fluid obeying the principle of material frame indifference, predicted
that the normal-stress differences would be of equal amount but with opposite
sign.

The first experiments, designed by Roberts^{6}
and Philippoff^{7} to decide between these different hypotheses,
confirmed - within the limits of experimental error - the predictions of
Weissenberg and rejected those of Reiner and Zaremba-Fromm-DeWitt. Subsequently
performed experiments of various scientists indicated that these conclusions had
to be modified. Today it seems quite certain that in solutions of polymers, the
most typical representatives of elastic fluids, the second normal-stress
difference does not vanish completely. However it is about ten times smaller
than the first normal-stress difference and has the opposite sign, corresponding
to an additional pressure in the direction of the shear gradient. The behaviour
of such fluids lies between the elastic fluid of Weissenberg and that of
Zaremba-Fromm-DeWitt, although much closer to the former.

The physical background of Weissenberg’s hypothesis needs therefore more consideration: Is it conceivable that elastic fluids exhibit a second normal-stress difference which is small compared to the first one although it does not vanish completely? In view of this question we will outline the formulation of constitutive equations of a fluid with vanishing second normal-stress difference for all types of flow and the additional considerations needed to obtain those of more general fluids. This will then form the basis for the final interpretation of the fact that deviations from the behaviour predicted by Weissenberg do not exceed a certain limit.

**2. CONSTITUTIVE EQUATIONS OF A WEISSENBERG
FLUID**

In formulating the constitutive equations of
dumbbells and necklace particles ^{8},^{9} it is found that the
stress tensor s may be expressed as the contravariant convected derivative of a
“State tensor” S

**
(1)**

Here t is the density, su the gradient of the velocity field and the superscribed dagger means the transpose. Especially for necklace particles this tensor assumes the form

**
(2)**

where r* _{i}*
is the position vector of the

**3. WEISSENBERG FLUIDS AS ELASTIC FLUIDS**

According to Eq. (3) the state tensor S
describes the configuration of an elastic network.*^{1} This suggests to
relate these configurations to the equilibrium configuration by means of a
series of deformation tensors^{10},

**
(3)**

Here C* _{k}*
is the tensor transforming <r

As usual in elasticity it is appropriate to
replace the deformation tensor C* _{k},*
which is the Finger measure of deformation, by Green’s measure of strain

**
(4)**

since this is proportional to the change of configuration caused by the flow. Thus a series of differential equations

**
(5)**

results from Eq. (3). Here η^{o}* _{k}*
=

**
(6)**

The tensor

**
(7)**

is the usual rate of strain tensor.

At this stage the concept of recoverable strain
is readily introduced by assuming a one-to-one correspondence between the
stresses s* _{k}* and the
strains E

**
(8)**

with s (o) = o, E (o) = o *(11).*

This results in a decoupled system (5) of differential equations for the elastic strain tensors

**
(9)**

as well as for the tensors of partial stresses

**
(10)**

The last equation is, although nonlinear, closely related to the constitutive equation of a Maxwell fluid; the complete system is therefore analogous to a viscoelastic fluid as described by a generalized Maxwell model. For an incompressible fluid the assumption of the linear relation

**
(11)**

reduces Eq. (10) to the constitutive equation of a so-called Walters fluid B’

**
(12)**

an equation, which is readily integrated. The
case *k=*1 describes the Maxwell-Oldroyd fluid B. In all these cases s
denotes the deviatoric stress to which an isotropic pressure –pI
has to be added.

Without claiming this to be the only possible interpretation one may regard Eqs. (9) and (10) as constitutive equations of an elastic fluid in the sense of Weissenberg’s concept of recoverable strain. They do not describe the effects of aggregation and disruption like the entanglement and disentanglement effects in polymeric solutions; but the idea of recoverable strain seems to require the identity of the deformable structural units.

**4. GENERALIZED ELASTIC FLUIDS**

Disregarding those effects makes it still
possible to generalize the foregoing model^{11}. One has just to recall
the very special assumption that the motion of the particles of the elastic
network takes place in an, at least in an average sense, isotropic medium. In
more concentrated systems one has to allow for an anisotropy caused by the
orientation of the network structure. Consequently, the proportionality of the
force k* _{i}* acting on the

**
(13)**

where B* _{i}* is a symmetric
tensor.

Since B* _{i}* will be isotropic in
the undeformed medium the decomposition

**
(14)**

has the property that b* _{i}* will
vanish in the limit of no flow. Eqs. (9) and (10) will therefore be unchanged in
form, provided the additional term

**
(15)**

will be added to the l.h.s.. Since the b* _{k}*
will be functions of the complete state of strain or stress, i.e. will depend on
the complete set of E

**5 .
ELASTIC FLUIDS IN SLOW FLOW**

Some qualitative insight into the behaviour of the fluids defined above can already be obtained for the special case where there is only one mechanism of relaxation. This is completely analogous to the transition from the general Walters fluid B’ to the Oldroyd fluid B. In this case the stress tensor is decomposed into its isotropic pressure part, a Newtonian part and a viscoelastic part

**
(16)**

where the last part obeys a relation like Eq. (10) with the additional term (15). For this case it seems reasonable to assume that the mobility term b depends solely on s’. This implies that b commutes with s’ and the expression (15) simply becomes b(s’).s’. Consequently the constitutive equation for e’ takes the form

**
(17)**

Thus, the material behaviour is determined not
only by η_{o} and η’ but also by the material functions E(s’) and b(s’).

To get some insight into the character of this
fluid we specialize to the case of slow flow. This makes it possible to develop
s’ in a series of kinematic tensors, c.f. (*12*). Due to the appearance of
the contravariant convected derivative *in* Eq. (17) the series of the
contravariant convected kinematic tensors

**
(18)**

is best suited:

**
(19)**

The choice of the factor of e^{(1)} is
done in the light of results to follow. Likewise the functions E and b are
developed as power series in s’

**
(20) (21)**

Strictly speaking the coefficients in Eqs. (19)
to (21) should be functions of the invariants of the kinematic tensors and s’,
respectively. In the last two series one should, using the Cayley-Hamilton
equation, add then the coefficients ε^{(0)} and b^{(0)} as
factors of the unit tensor, both of which, however, contain no absolute term but
vanish for s’→0. For a second-order approximation the terms ε^{(1)} and
b^{(1)} as well as *K*^{(1)} and K^{(11)} may be
taken as constants.

Utilizing Eq. (19) in (20) and (21) and inserting these later two series into Eq. (17) permits comparison of like terms. The lowest order terms yield the second-order coefficients

**
(22)**

In principle the higher-order coefficients of the expansion (19) can be determined successively, but we will not go into detail at this point.

**6. NORMAL-STRESS DIFFERENCES IN THE STEADY
VISCOMETRIC FLOW**

To answer the basic question we will return to the starting point of this analysis, the normal-stress behaviour in steady viscometric flows.

In general the behaviour of such flows can be described by three functions, which may be defined as follows

**
(23)**

Here the s_{ik} are the components of
the stress tensor, q is the rate of shear and the indices in their natural order
correspond to the direction of flow, the direction of the shear gradient and the
neutral direction. Thus, the first normal stress difference is given by

**
(24)**

while the second one is *^{2}

**
(25)**

In a second-order approximation for slow flow
the two functions *F(q ^{2})* and

**
(26)**

It is readily verified - c.f.^{12} -
that these expressions and the coefficients of the last chapter are related by

**
(27)**

To lowest order the normal-stress differences are thus

**
(28)**

and the ratio of the two differences is

**
(29)**

Since 2 ε^{(1)} is the compliance of
the recoverable strain the statement *m _{o}* << 1 is equivalent to
the statement that in fluids under shear the effect of the elastic compliance
dominates the induced anisotropy of the structure.

For highly elastic fluids the range of validity of the approximation for slow flow is usually restricted to a very narrow range of the rate of shear, and it might well be that this range is out of reach for practical purposes. Thus the question is appealing whether readily understood conclusions can be drawn from our constitutive equation independent of this approximation.

This is, indeed, the case. To this end it seems advisable to express the shear not by E but by Finger’s measure of deformation C - c.f. Eq. (4) - and, correspondingly, the anisotropy of the mobility by B = I + b so that the defining equation becomes

**
(30)**

We merely assume now that β and C as functions
of s’ have the same principal axes as s’ so that the principal values β_{I},
β_{II}, β_{III}
and C_{I}, C_{II},
C_{III} correspond to the
principal stresses s’_{I}, s’_{II},
s’_{III}. To profit from this
coaxiality one has to write Eq. (30) in the components of the system of
principla axes and subsequently transform back to the system introduced above.
It is readily seen that the principal values with index III play no role at all
so that the problem may be treated as a two-dimensional one.

For homogeneous steady flow the contravariant convective derivative of C is given by

**
(31)**

To transform this to the system of principal axes of s’ one utilizes the defining equation, which in matrix notation can be formulated as follows:

**
(32)**

where φ is the angle between the first principal axis and the flow direction. This transformation yields

**
(33)**

Inserting this in Eq. (30) gives

**
(34)**

as well as the defining equation for the orientation of the principal axes

**
(35)**

Elementary calculation and backtransformation produces the components of stress

**
(36)**

and consequently the following normal-stress differences and their ratio

**
(37)**

Quite generally, we find the same situation we
already encountered in the approximation for slow flow, which means that we can
answer the question of this topic without restriction*^{3} as follows:
Weissenberg’s hypothesis that the second normal-stress difference vanishes in
steady viscometric flows of elastic fluids, is equivalent to the assumption that
the anisotropy of the mobility, induced in such a flow, is negligible. The
experimental results that the ratio of the second normal-stress difference and
the first one is small compared to one indicates the predomination of the effect
of elastic strain over this anisotropy. As can be seen from Table 1 a ratio *m*
approx. 1/10 by a medium recoverable strain *(C*_{I}/*C*_{II}
approx. 2) corresponds to an anisotropy ratio of β_{I}/β_{II
}approx. 1.2, by a strong strain *(C*_{I}/*C*_{II}
approx. 5) of β_{I}/β_{II}
approx. *2.5* and by a very strong strain *(C*_{I}/*C*_{II}
approx. 10) even of β_{I}/β_{II}
approx. 8. The observation, sometimes found, that *m* decreases rapidly by
exceeding a critical rate of shear is fathomable since the recoverable strain
can increase substantially by increasing the rate of shear even though the
anisotropy has already reached its final value.

*Table*
*1.* The ratio of normal-stress differences as a function of recoverable
strain and anisotropy

*^{1}
Elasticity may be caused by elastic connections between the mass centres (c.f.
in a bead-spring model) as well as by the thermal motion of the mass centres
themselves.

*^{2} The last definition refers to the
experimental fact that *s*_{33} > *s*_{22}. In the
introduction the original definition *s*_{22}*--s*_{33}*
*was used for the second normal-stress difference

*^{3} This result is valid - in the
steady case - also if aggregation and disruption effects are included.

**REFERENCES**

1. Weissenberg K., Conference of British Rheologist’s Club, London 1946; Proc. 1st. Intern. Congr. Rheology, Amsterdam 1948, 1, 29-46 (1949).

2. Reiner, M., Amer. J*.* Math. 67, 350-362 (1945).

3. Fromm, H., Ing. Arch. 4, 432-466 (1933); ZAMM 25/27, 146-150 (1947); 28, 43-54 (1948).

4. DeWitt, T. W., J. Appl. phys. 26, 889-894 (1955).

*5.*
Zaremba, S., Bull. Int. Acad. Sci. Cracovie 594-614, 614-621 (1903).

6. Roberts, J. E., Proc. 2nd. Intern. Congr. Rheology, Oxford 1953 (London 1954) pp. 91-98.

7. Philippoff, W., Rheol. Acta 1, 371-375 (1961).

8. Giesekus, H., Rheol. Acta 1, 404-413 (1961).

9. Giesekus, H., Rheol. Acta 2, 50-62 (1962).

10. Giesekus, H., ZAMM 42, 259-262 (1962); in: *Reiner-Abir* (eds.),
Proc. Intern. Symp. Second-Order Effects in Elasticity, Plasticity and Fluid
Dynamics, Haifa 1962 (New York 1964) pp. 553-584.

11. Giesekus, H., Rheol. Acta 5, 29-35 (1966).

12. Giesekus, H., ZAMM 42, 32-61 (1962).

**
Preface /
Acknowledgements /
Biographical Notes**

**
Weissenberg’s
Influence on Crystallography**

**
Karl Weissenberg
and the Development of X-Ray Crystallography**

**
The Isolation of, and the Initial Measurements of the Weissenberg
Effect**

**
The Role of Similitude in Continuum Mechanics**

**
Elasticity in Incompressible Liquids**

** The Physical Meaning of Weissenberg's Hypothesis with Regard to
the Second Normal-Stress Difference**

**
A Study of Weissenberg's Holistic Approach to Biorheology**

**
The Weissenberg Rheogoniometer Adapted for Biorheological Studies**

**
Weissenberg’s Contributions to Rheology**

**
The Early Development of
the Rheogoniometer**

**
Some of Weissenberg's More Important Contributions to Rheology:
An Appreciation**

**
Publications of
Karl Weissenberg and Collaborators /
List of
Contributors**

**© Copyright John Harris**