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J. Non-Newtonian Fluid Mech. 157 (2009) 141–144

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

he changing face of rheology

oger I. Tanner ∗

chool of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney 2006 Australia

r t i c l e i n f o a b s t r a c t

eywords:heologyechanics

olymer scienceuture

By looking back to the formal beginning of rheology (1929) and studying more recent offerings at con-gresses, we see that there is a drift of content in the subject. In the beginning rheology was seen as anextension of classical continuum mechanics, but more recently one has seen the withering of offeringsin solids rheology and the start of some new branches of the subject. Some opinions on possible futuredirections are given.

. Overview

According to Bingham(1929)-see [1]-rheology is the science ofeformation and flow of matter, especially of non-classical mate-ials. It was viewed in those days as a descendant of elasticity,uid dynamics, linear viscoelasticity and plasticity theories—all ofhich comprise classical continuum mechanics. Rheology is now aiddle-aged science, but it has ever-evolving relevance to technol-

gy, and needs

1) To develop fundamental understanding of material behaviour atmultiple scales (I am avoiding the nano-word: organic chemistshave been doing nano-engineering for a long time);

2) To deduce realistic mesoscopic and macroscopic models, and toverify them experimentally;

3) To find economically viable methods of applying this knowl-edge, via a parsimonious set of experiments and easy, feasibleand reliable computations.

Looking at the 1929–1932 issues of the Journal of Rheologys a base for evidence of later changes, I found that about 32%f the articles were then devoted to fluids rheology, and 27% toolids rheology. Experimental methods was the only other largeategory—about 15%. It is interesting to see if this has changed overhe last 75 years.

Some 34 years after 1929, I went to the Fourth Internationalongress on Rheology – at Brown University in Providence –nd one can categorize the papers offered there as shown inable 1.

∗ Tel.: +61 2 9351 7153; fax: +61 2 9351 7060.E-mail address: rit@aeromech.usyd.edu.au.

377-0257/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2008.11.007

© 2008 Elsevier B.V. All rights reserved.

The large biorheology contribution has since separated frommainstream rheology; some of it may return, but the practitionershave their own show now, so it seems unlikely to reach 32% again. Aspresented in Table 1, it is noticeable that polymer engineering wasa small fraction (4%) of the total. These days many other technicalmeetings are available, but nevertheless in the 2000 congress about5% of the papers were of an industrial or applied nature, so somegrowth of this field has occurred, in conjunction with the generalgrowth of the subject of rheology.

Rheology, as defined by Bingham, is a general subject which ishedged about by some very large fields of activity—polymer science,materials science/solid mechanics, and applied mechanics (Fig. 1).One can ask if it will be swallowed by these fields.

In the 1963 International Congress, 10% of the papers weredevoted to polymer science, while in the 2000 congress thesepapers were largely absent or were subsumed under other head-ings. Despite the enormous increase in polymer science activity,the mainstream work in this area seems to have been siphoned offelsewhere. In the 2000 congress (Table 2) the fraction of papersdevoted to solid mechanics was negligible (∼5%), down from 17% in1963. Again there is a loss of a field, due to specialization. This areaI feel was an important loss; for example, much of the wranglingover the yield stress concept could have been avoided by studyingsolid mechanics more closely.

Fluids, melts, suspensions and solutions made up about 54% ofthe 2000 congress papers, up from 40% in 1963. Hence, from thisevidence, rheology is heading for a narrowing of interests. Is thisreally the case, since new subfields continue to appear?

Growth areas 1963–2000 include experimental methods (an

increase from 8% of papers in 1963 to 10% in 2000), and ofcourse computational rheology, which was a newborn subjectin 1963, made up 9% of the papers in 2000. Liquid crystals havealso increased (to 3% in 2000), and emulsions/foams is a newaddition.

142 R.I. Tanner / J. Non-Newtonian Flui

Table 14th International Congress on Rheology (1963). Papers by category.

Category No.

Continuum mechanics 22 (13%)Experimental methods 8 (5%)Solids 29 (17%)Polymer science 20 (10%)Fluid flow 18 (10%)Suspension mechanics 12 (7%)Biorheology 56 (32%)Crystallization 1 (1%)Polymer engineering 7 (4%)Powders 1 (1%)

174 (100%)

Fig. 1. Large fields encircling rheology.

Table 213th International Congress on Rheology (2000). Papers by category; growth/declinefigures refer to the [%2000]–[%1963].

Growth (+)/decline (−)since 1963

Solids 5% −12%Computational 9% +9%Experimental 10% +5%Food/bio 3% −29%Continuum mechanics – −13%Melts 17% )Solutions/fluids 21% ) +18%Suspensions 16% +9%Thermal science 1.1% +1%Crystallization 0.3% −0.7%Polymer science 1% −9%Liquid crystals 3% +3%IE

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ndustrial 5% +1%mulsions/foams 9% +9%

100% (517 papers)

Thus there has been a change in emphasis in rheology over 75ears, I believe, and some notes on this follow.

. The decline and fall of continuum mechanics

I begin by briefly discussing the world as seen by the ‘rationalechanics’ school propelled by Truesdell and Noll [2]. This book,

d Mech. 157 (2009) 141–144

first published in 1965, refers to (p. viii) “The volume of grand andenlightening discoveries since 1955” and implies that (p. 8) morewas known about non-linear mechanics in 1895 than in 1945. Thereare many results presented in this work, including the principle ofmaterial objectivity; which is sometimes appropriate and some-times is not. I believe this formal and over-hyped approach haslong since run its course and cannot be expected to be important towhere rheology is going. Many of its approaches have proved to bedead ends or raised false hopes (e.g. the order-fluid concept and theidea that a fluid must be isotropic) and one can still find more sensein the less formal works of Oldroyd [3]. There remain some Trues-dell devotees and survivors, and they continue to develop complex“frameworks”, but it is clear that not many are listening.

Truesdell did a very useful job in diffusing many old results inmechanics [4] and some of his historical writing on mechanics is ofinterest, but sometimes one needs to beware of the spin.

The decline of this style of work has perhaps been acceleratedby our ability to do large-scale computing, but is probably mostlydue to the impotence of the methods to deliver concrete, practicallyuseful results.

3. Experimental rheology

The experimental area continues to develop with great improve-ments in, e.g. electron and other microscopes, NMR, AFM, opticsand video recording. In short, the ability to see what is occurringis now much more available. I am sure that this trend will con-tinue and that dramatic improvements in understanding will occurvia these improvements in “seeing”. Opaque materials still posesome problems; the work on wall slip also benefits from theseimprovements—one can “see” wall slip in action.

The amount of rheological work on melts, solutions and fluidflow continues to increase, and now dominates the subject; betterknowledge of structure helps us with need (2) mentioned above inthe overview-that is, improved modelling.

4. Microstructural approaches

The Vol. 2 of Bird et al. [5] set the tone here, although somework predates this; an improved account was given in the sec-ond edition (1987). Given a view of the microstructure perhapsfound from chemical physics or materials science, or perhapssimply invented, then it has proved fruitful to construct macro-scopic stress-deformation laws using these ideas. We have seen theGreen-Tobolsky-Lodge network ideas increase and then decreasein influence. More recently the de Gennes-Doi-Edwards reptationconcepts have captured the imagination and no doubt work in thisfield will persist. A vast reptative industry has sprung up, and moreand more complexity is being heaped on the original reptation con-cept, with some, but not complete, success.

For example, the so-called pom–pom models give reasonableresults in simple shearing and elongation (as do the older PTTmodels), but, except for the XPP variant [6] do not exhibit a sec-ond normal stress difference, do not permit overshoot after steadyelongation [7] and exhibit too much recoil after stretching [8]. Inaddition [9] they fail to exhibit the stress-time superposition aftera step shear which has been so well documented for low-densitypolyethylene by Laun [10]. The pom–pom and PTT models are infact extremely similar and both can be regarded as general networkmodels [9]. Thus some more work remains to be done to improve

models in this mainstream area.

More formal micro–macro approaches have been available forsome time. Bird et al. [5] used very formal micro-models, but itis clear that this approach is not attracting practitioners, and willdie. Beris and Edwards [11] have another formal scheme, as does

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ttinger [12], but the results from these approaches are not yet, iteems to me, inspiring a lot of followers.

. Anisothermal rheology and two-phase flows

In many polymer processing operations large changes of tem-erature, ultimately resulting in a solid product, are seen. Focussingn injection moulding, one sees the rapid chilling of the materialext to the wall, resulting in a variation of structure across the finalroduct. Several problems arise:

1) How to account for the temperature changes;2) How to predict structural changes;3) What happens after the product is removed from the mould.

Beginning with (1) above, one could go back to 1969 and lookt Truesdell’s book “Rational Thermodynamics” [13]. Whilst it isntertaining as a polemical tract, it will never help many of us, Iear.

In 1993 a IUTAM Symposium was held in Kerkrade in theetherlands entitled “Numerical Solution of Non-Isothermal Flowf Viscoelastic Liquids” [14]. It had some interesting papers goingeyond the classical time-temperature shift of properties which isommonly assumed.

The paper by Wapperom and Hulsen-see ref [14]- led on to aurther paper by them in J. Rheol. [15]. Using the concept of internalariables they produced some results for the temperature equationf the form

cpT − T˛p + �T.

�s = D − ∇ · � (1)

here D is the dissipation term; the superposed dot denotes a mate-ial (“hydrodynamic”) derivative with respect to time. Eq. (1) differs

y the term �T ·.

�s (�s = change in entropy due to change in thenternal variables)—the other terms are well-known [16]. The workas tied to specific models of the constitutive equation, but I think

his kind of approach can be made to work generally. The evalua-ion of �s has to be found by experiment or otherwise, which mayot be easy. Experimental work to find the extra term in Eq. (1) isery difficult. One possible escape path was suggested by Astaritand Sarti [17] who assumed that the free energy was governed onlyy entropy, but since free energy in polymers is known not to beompletely entropic [18,19], this approach is not very helpful.

Van den Brule -see ref [14]- touched on an importantoint—what happens to thermal conductivity in a flowing polymer?bviously this is very important for predicting temperatures in

njection moulding and many practitioners swear that considerablehanges occur. Van den Brule suggested that thermal conductivitys enhanced along lines of flow and diminished across the flow,easoning from network and dumbbell theories. Experimental evi-ence is very hard to find. Undoubtedly the best experiments (ontep strain tests) were done by Venerus et al. in Chicago [20]. Theirethod is complex (infra-red decay), but it does suggest that afterstep strain the work of van den Brule holds. What happens incontinuously flowing polymer is still a mystery. We only found

mall effects with sheared polypropylene [21] and I think muchore needs to be done here. In summary, this area needs attention,

ut it is difficult.

.1. Two-phase problems

There are a host of unsolved problems when more than onehase is present in the material. Liquid/solid and gas/liquid prob-

ems are very hard. Confining ourselves to solid/fluid systemssuspensions) it is interesting to find that no satisfactory constitu-ive equations are known for large deformations of material made

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d Mech. 157 (2009) 141–144 143

of solid particles in viscoelastic matrices. (For small deformationswe can use the correspondence principle.) There are numerouspapers using Newtonian matrices, but even there the matter is com-plex, due to particle drift. In processing rheology, perhaps this slowdrift can be ignored, and then we can go some way to describeviscoelastic-matrix solutions [22].

There is a lot of experimental work on viscosity, much less onnormal stresses. Various writers have seen not only shear-thinningbut also shear-thickening at high shear rates. Many suspensionmodels, of various degrees of complexity, have been proposed, butmany of them do not accurately reflect the behaviour observed. Asimple suggestion of Zarraga et al. [23] and Mall-Gleissle et al. [24]proposes the simple addition of two kinds of stress term:

� = �N + �v (2)

where the suffix N denotes the bead-generated or “Newtonian”behaviour, and � denotes the ‘viscoelastic’ behaviour of the matrix;� is the total extra-stress term. Calculated results of the secondnormal stress difference are a reasonably good fit to the limitednumber of experimental measurements [22]. Attempts to calcu-late the response of these suspensions continue—see, for example,Huang and Hulsen [25].

6. Conclusion

In 2000 in the BSR Gold Medal lecture, I nominated 10 problemsthat seemed to me to need solution. Some of these are progressingwell-thin film lubrication via non-equilibrium molecular dynamics(NEMD), slip at the wall, others (thermal problems, filled viscoelas-tic matrices) I have alluded to above. A useful new attack on flowstability is still wanting, and the old questions associated with dieswell and behaviour at exit corners are still with us. The behaviourof the UCM model around a confined cylinder is still a puzzle—butpersonally I do not want to do any more work on this problem.

Obviously the subject of rheology is changing, often going downfrom the continuum to a micro-level, and we have plenty of rheolog-ical work to do—hopefully we will not be swallowed up by polymeror materials science or applied mechanics, as I think rheologistshave a special broadly based point of view to offer.

References

[1] R.I. Tanner, K. Walters, Rheology: An Historical Perspective, Elsevier, Amster-dam, 1998.

[2] C. Truesdell, W. Noll, The Nonlinear Field Theories of Mechanics, Springer-Verlag, 1965.

[3] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc.London Ser. A 200 (1950) 523–541.

[4] C. Truesdell, The mechanical foundations of elasticity and fluid dynamics, J.Rational Mech. Anal. 1 (1952) 125–300.

[5] R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Fluids.Vol. 2: Kinetic Theory, 2nd ed. 1987, Wiley, New York, 1977.

[6] W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Viscoelastic analysis of com-plex polymer melt flows using the eXtended Pom–Pom model, J. Non-Newt.Fluid Mech. 108 (2002) 301–326.

[7] H.K. Rasmussen, J.K. Nielsen, A. Bach, O. Hassager, Viscosity overshoot in thestart-up of uniaxial elongation of low-density polyethylene melts, J. Rheol. 49(2005) 360–381.

[8] R.I. Tanner, A.M. Zdilar, S. Nasseri, Recoil from elongation using general networkmodels, Rheol. Acta 44 (2005) 513–520.

[9] R.I. Tanner, On the congruence of some network and pom-pom models,Korea–Aust. Rheol. J. 18 (2006) 9–14.

10] H.M. Laun, Description of non-linear shear behavior of a low-density polyethy-lene melt by means of an experimentally determined strain dependent memoryfunction, Rheol. Acta 17 (1978) 1–15.

[11] A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems, Oxford U. Press,1994.

12] H.C. Öttinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken, NJ, 2005.13] C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969.14] J.F. Dijksman, G.D.C. Kuiken (Eds.), IUTAM Symposium on Numerical Simulation

of Non-Isothermal Flow of Viscoelastic Liquids, Kluwer, Dordrecht, 1995.15] P. Wapperom, M.A. Hulsen, Thermodynamics of viscoelastic fluids: the temper-

ature equation, J. Rheol. 42 (1998) 999–1019.

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Rheol. 45 (2001) 1065–1084.[24] S.E. Mall-Gleissle, W. Gleissle, G.H. McKinley, H. Buggisch, The normal stress

44 R.I. Tanner / J. Non-Newtonia

16] R.I. Tanner, Engineering Rheology, 2nd ed., Oxford U. Press, 2000.17] G. Astarita, G.C. Sarti, The dissipative mechanism in flowing polymers: theory

and experiments, J. Non-Newt. Fluid Mech. 1 (1976) 39–50.18] L.R.G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Clarendon Press, Oxford,

1975.19] J.P. Gao, J.H. Weiner, Nature of stress on the atomic-level in dense polymer

systems, Science 266 (1994) 748–752.20] D.C. Venerus, J.D. Schieber, V. Balasubramanian, K. Bush, S. Smoukov,

Anisotropic thermal conduction in a polymer subjected to shear flow, Phys.Rev. Lett. 93 (2004) 098301-4.

21] S.C. Dai, R.I. Tanner, Anisotropic thermal conductivity in sheared polypropylene,Rheol. Acta 45 (2006) 228–238.

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d Mech. 157 (2009) 141–144

22] R.I. Tanner, F. Qi, A phenomenological approach to suspensions with viscoelasticmatrices, Korea–Aust. Rheol. J. 17 (2005) 149–156.

23] I.E. Zarraga, D.A. Hill, D.T. Leighton, Normal stress and free surface deformationin concentrated suspensions of noncolloidal spheres in a viscoelastic fluid, J.

behaviour of suspensions with viscoelastic matrix fluids, Rheol. Acta 41 (2002)61–76.

25] W.R. Huang, M.A. Hulsen, Towards the computational rheometry of filled poly-meric fluids, Korea–Aust. Rheol. J. 18 (2006) 171–181.