Karl Weissenberg - The 80th Birthday Celebration Essays
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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 vanish1.

 

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 differences2. Approximately at the same time Fromm3, and independently a little later DeWitt4, rediscovered the model of Zaremba5, 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 Roberts6 and Philippoff7 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 ri is the position vector of the i-th mass-centre relative to the centre of mass of the entire particle, Ni their number per unit mass and Bi their mobility. The angular brackets denote the expectation value taken over the particle ensemble. Thus, S is a symmetric tensor having non-negative eigenvalues. It is easy to show that Eq. (1) causes the second normal-stress difference to vanish in steady viscometric flows. As has been proposed earlier9, those fluids will therefore be termed Weissenberg fluids.

 

 

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 tensors10,

 

  (3)

 

Here Ck is the tensor transforming <rok rok> to <rk rk>, uok denotes the kinematic viscosity of the k-th class of mass centres in the equilibrium state and S is the state tensor corresponding to this equilibrium state, which is described by an isotropic tensor.

 

As usual in elasticity it is appropriate to replace the deformation tensor Ck, 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 ηok = ρeuok denotes the dynamic viscosity and sk the partial stresses corresponding to the k-th class of particles so that

 

  (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 sk and the strains Ek

 

  (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 model11. 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 ki acting on the i-th mass-centre and its velocity wi relative to the surrounding medium has to have tensorial character

 

  (13)

 

where Bi is a symmetric tensor.

 

Since Bi will be isotropic in the undeformed medium the decomposition

 

  (14)

 

has the property that bi 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 bk will be functions of the complete state of strain or stress, i.e. will depend on the complete set of Ek and sk respectively, the system of equations will no longer be decoupled. It is this additional term (15) which causes the non-vanishing of the second normal-stress difference in steady viscometric flows.

 

 

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 ηo 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 sik 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(q2) and G(q2) may be replaced by their values at q = 0,

 

  (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 mo << 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 βI, βII, βIII and CI, CII, CIII 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 (CI/CII approx. 2) corresponds to an anisotropy ratio of βIII approx. 1.2, by a strong strain (CI/CII approx. 5) of βIII approx. 2.5 and by a very strong strain (CI/CII approx. 10) even of βIII 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 s33 > s22. In the introduction the original definition s22--s33 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).

 

 


Contents

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

The Effect of Molecular Weight and Concentration of Polymers in Solutions on the Normal Stress Coefficient

        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

        Dr. Karl Weissenberg, 1922-28

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

Index

 

 

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