Objective and generally applicable criteria for flow classification

69

Journal of Non-Newtonian Fluid Mechanics, 6 (1979) 69-76 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

Short Communication OBJECTIVE AND GENERALLY CLASSIFICATION

APPLICABLE CRITERIA

FOR FLOW

GIANNI ASTARITA Chemical Engineering

Department,

University

of Delaware,

Newark,

Delaware

19711 (U.S.A.}

(Received October 14, 1978; accepted January 8, 1978)

1. Introduction

The rheological behavior of elastic liquids is very different in extensional and viscometric flows, and rheological information obtained for the latter ones is of little if any help for predicting the behavior in the former ones. Furthermore, the kinematics of real flow fields such as may be of interest, say, in polymer processing, may well be neither viscometric nor extensional, and in fact may also not be somewhere in between these two categories. Therefore, the problem of classification of flow fields is an important one; an interesting analysis of the problems to be faced was presented in 1971 by Metzner

WI.

A semi-intuitive idea of the kind of flow types which may be encountered, and of the order in which they would need to be classified, is available. At one extreme lie the purely extensional flows, which constitute a well-defined class of flows which are known to result in particularly severe stresses in elastic liquids. There is then a class of not well-defined flows where some shearing is superimposed on the stretch; intuitively, viscometric flows constitute the other boundary of this class. Next comes a class of flows, again not well defined, which in some sense are even less severe, in terms of the stresses they generate in elastic liquids, than viscometric flows. The flow in a Maxwell rheometer (MRF) belongs to this class; some other specific examples have been considered in the literature [2,3]. The extreme case is that of rigid-body motions, which do not generate any extra stress in any fluid. It would be clearly useful to be able to calculate, for any given flow pattern, a crittirion the value of which identifies the flow considered in a classification of the type sketched above. In this paper, a proposal is presented for such a criterion. The criterion to be developed should enjoy, in my opinion, the following three properties.

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(A) It should be a local criterion, in the sense that one should be able to calculate its value at a point, rather than in the whole flow field. There are advantages and disadvantages in both local and global criteria, but for the problem at hand a local one is clearly preferable, since a flow field may well be, say, very close to viscometric at one point and very close to extensional at another one. (B) It should be objective (say, for a scalar-valued criterion, it should be invariant under a change of reference frame). Since the main reason for classifying flow fields is to decide which constitutive equation is more likely to produce useful results, the criterion should enjoy the same invariance properties that are required of the constitutive equation. (C) It should be generally applicable, in the sense that its value should be liable to calculation, and the result of such a calculation should make sense, for any conceivable local kinematics. A criterion applicable to only a restricted class of flow fields is of little utility, since within such a restricted class one generally understands the flow classification anyhow. A criterion which satisfies requirements (A), (B) and (C) may be needed for consideration of two somewhat distinct problems. Therefore, in fact perhaps two different criteria are needed, satisfying either one or the other of the following two requirements. (D) An entirely kinematic criterion, say one the value of which depends only on the kinematics of motion and not on the nature of the fluid. A (D)criterion is useful for kinematic classification, but does not yield information on the response of the fluid to the flow considered. (E) A criterion based on both the flow kinematics and some rheological parameter of the fluid, say typically some relaxation time A. For extensional flows, it is well known that simple constitutive equations such as the Maxwell model predict the existence of a critical stretching rate, the response of the material being quite different at stretching rates above and below the critical one. An (E)-criterion may be useful from this viewpoint. In this paper, both a (D)- and (E)-criterion enjoying properties (A), (B) and (C) are presented. 2. Literature survey The problem considered in this paper is an old one, which was identified as soon as the basic difference between rheological behavior of elastic liquids in extensional and viscometric flow became known. In the old literature, one may find statements which are (possibly implicitly) equivalent to the following one: (I) “The larger the diagonal components of the rate-of-strain tensor D are as compared to the off-diagonal ones, the closer the flow is to an extensional flow”. Since D is symmetric, the matrix of its orthogonal components can always be made diagonal by appropriate rigid rotation of the coordinate system.

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Therefore, statements of Type I are not invariant even with respect to a change of coordinate system, let alone with respect to a change of reference frame. In 1967, I proposed [4] a criterion for flow classification. Only flow fields “between” viscometric and extensional were being considered; for the sake of the present discussion, it may be useful to consider the square of the criterion proposed in 1967, namely: RIIE = A2(trD2 + tr W2) = A2 tr L2

(1)

where L is the velocity gradient tensor and W is the vorticity tensor W = 2 (L - LT). RIIE is clearly an (E)-type criterion; a (D)-type one is easily constructed from it, however, say: &ID

=

-

tr W2/tr D2 .

(2)

However, RI1 (both in the (D) and (E) form) is not objective, as was pointed out in the published literature by Huilgol [ 51. The requirement of objectivity cannot of course be relaxed, and therefore both RII’s are unacceptable criteria. In 1974, I and Marrucci [6, pp. 178 and 2681 proposed a criterion which satisfies requirements (A) and (B), but not requirement (C), since it is only applicable to motions with constant stretch history (MWCSH). For MWCSH, the deformation gradient Ff carrying the configuration at time t - s into the one at time t is given by: Ff = Q(t -s)

- [exp(-ksN)]

* QT(t)

(4)

(here and in the following, the notation in [6] is used). Tensor N is of unit magnitude, which can always be obtained by selecting appropriately the constant k. In one particular frame of reference, and only in one, kN coincides with the velocity gradient L; the orthogonal tensor-valued function Q( - ) simply describes the transformation from that particular frame to the one which is in fact being considered. Tensor N is, quite obviously, objective, and therefore so are all its invariants. The criterion proposed is: RIIID = tr Ns

(4)

and is an objective, but restricted, variation of RIID. Again, an (E)-form is easily constructed: RIIIE = A2k2 tr Ns

(5)

Since most flow fields of interest are not MWCSH, the RIII criteria are of limited usefulness. In 1975, Tanner and Huilgol [7] proposed a criterion for distinguishing between ‘weak’ and ‘strong’ flows. Although this is not immediately apparent, their analysis is also restricted to MWCSH, since they explicitly state that they do not “seek to discuss the problem of flow classification when the stress state at a particle varies in time”. Since in any flow which is not a MWCSH the stress state at a particle does vary in time, the restriction is obvious. Tanner

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and Huilgol criterion may be written as follows:

R IVD

= max[Re(akIV)l

where Re( ) is the real part, and akN are the eigenvalues of kN. Tanner and Huilgol choose to express their equations in terms of L rather than kN, but if their transformations are followed closely one realizes that the eigenvalues of IzN are in fact being considered. Flows where RIvD > 0 are classified as strong; flows where RIvD < 0 a~ weak. A comparatively straightforward application of Cayley-Hamilton’s theorem shows that, for a flow to be weak the RIIID criterion must be non-positive. Tanner and Huilgol state that the advantage of their criterion lies in the fact that it will classify MRF as a weak flow. This is true, but it also can be said for RIIID, which is negative for a MRF and positive in extensional flow. Indeed, the Astarita-Marrucci and the TannerHuilgol criteria appear to be equivalent, with the first one being slightly easier to calculate. The (E)-form of Tanner and Huilgol’s criterion has been proposed by Tanner [8] in 1976: R IVE

= m=[RehkN-;

111 *

(7)

By considering the eigenvalues of AkN - & 1, rather than those of kN, the (E)-requirement is fulfilled. Both forms of the RIv criterion are of as limited usefulness as the RIII criteria. The latter point, however, has not been fully appreciated in the literature, Denn [9] states that Tanner and Huilgol’s “definition is not adequate to enable one to identify extensional flows with strong flows, however, without the further requirement that the positive real part of an eigenvalue of L must be of the same order of magnitude as the largest element of L. The reason for this restriction is to eliminate lubrication-type flows, which are shearing flows with a slowly changing cross-sectional area, and yet which can be strong flows according to the Tanner-Huilgol definition”. This statement implies that Denn is considering the eigenvalues of L, rather than those of kN, and in fact that he is willing to do so for any flow: lubrication-type flows with a slowly changing cross-sectional area are not MWCSH, and the stress state at a particle for such flows does change (albeit slowly) in time - of course, for flows which are not MWCSH tensor N is simply not defined. One could regard Denn’s statement as an implicit proposal of a new criterion, say: R VD

=

m~[W~dl

enjoying property (C). However, RVD is not objective, since L is not objective. Indeed, it is easy to show that, by superimposing a sufficiently rapid rigid-body rotation, the eigenvalues of L can be made to have non-positive real parts for any flow field. In conclusion, one may say that the criteria proposed in the literature are either generally applicable but not objective (RI1 and R,), or objective but

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restricted to MWCSH (Rm and Rrv). The problem considered in this paper is still an open one. 3. The criteria The essential physical character of the problem under consideration has been identified since 1967 [4]. If the largest rate of stretching is always applied to the same material line, an elastic liquid may accumulate stresses due to progressive strains at a rate competitive with the rate of stress relaxation. If, however, the material rotates, the same material line is not always exposed to the highest rate of stretch, and in fact it will occasionally undergo negative stretches, so that stress accumulation becomes less severe. The difficulty in formalizing this simple concept is only related to the fact that the vorticity tensor which measures the rate of rotation is not objective. This difficulty has led, as seen in the preceding section, to the formulation of nonobjective criteria. In hindsight, the difficulty is a trivial one. What is relevant is not the rate of rotation of the material with respect to an arbitrary reference frame (which is not objective), but the rate of rotation with respect to the direction of maximum stretch, i.e., with respect to the principal axes of D. The latter rate of rotation is, quite obviously, objective, since it is not related to an arbitrary choice of the frame of reference. In 1976, Drouot and Lucius [lO,ll] have introduced two new kinematic tensors. The first one, a, is the rate of rotation of the tensor D at a particle. If ei are the proper vectors of D (i.e., unit vectors along the principal axes of D), the definition of 0 is: DeJDt = fi - ei where D/Dt is the substantial time derivative. Should two of the eigenvalues of D coincide (the case where all three coincide is trivial for incompressible fluids), say the 1 and 2 ones, the definition (9) is completed by requiring a2,, to coincide with the corresponding component of the vorticity, !&z = Wis. The second tensor introduced is the tensor w defined by: FV= w-n

(10)

which clearly measures the rate of rotation of a particle with respect to the rate of strain’s principal axes at that particle. Although both W and R are not objective, w is, as shown in the following proof. Consider a change of frame as follows: X*=Y(t)+Q(t)-(X-Z).

(11)

Since D is objective, so are the ei’s, say: eT = Q(t)-ei .

(12)

Notice that DQ/Dt = dQ/dt = Q. Differentiation of Eq. 12 gives:

DC__ -

Q *ei + Q

Dt

.?$&QT.ef+Q.fi.QT

.&

(13)

and hence: a*=Q-n-QT+&QT.

(14)

Equation (14) shows that a is not objective. This is not surprising, since the definition in eqn. (9) is based on the operator D/Dt which is known not to be objective. Indeed, since R measures the rate of rotation of the principal axes of D with respect to some arbitrarily chosen reference frame, one does not expect it to be objective. The rule of transformation of the vorticity tensor is well known: j+‘*=Q+-QT+&QT

(15)

Subtracting eqn. (15) from eqn. (14) yields: ~*GQ.w.QT

(16)

i.e., w is objective, and so are all of its invariants. Since an objective measure of the relevant rate of rotation is now available, the RI1 criteria can now be formulated in an appropriately objective form, say: Rn=-tr~IT2/trD2

(17)

RE = A2(tr D2 + tr w2)

(18)

Equations (17) and (18) give the criteria proposed here; they clearly enjoy properties (A), (B) and (C). It remains to show that they indeed classify flow fields in the desirable way. In order to show this, attention is focused on the following restricted class of flow fields, which are described in a Cartesian coordinate system X, y, .z by: u, = TY VY = ayx u,

w

=o

Since the x and y axes can be interchanged and/or inverted, no generality is lost by restricting attention to the case where: a2 < 1,

y>

0.

(20)

The flows described by eqns. (19) are all MWCSH (hence criteria RIII and RIv can be used for comparison). Furthermore, they span the whole range of inter est, from rigid-body motions (a = -1) to viscometric flows (a = 0) to extensional flows (a = 1). That a = 1 corresponds to pure extension can be seen by considering the deformation of a material line located initially at x = y. The flow-type scanning is reflected by the calculation of stresses for a contravari-

75

ant Maxwell model which yields:

+ a)/(1 - 4 ayzhz) 7xY = /Ly(l

(21)

Equation (21) shows that the apparent viscosity r,,,/y increases with increasing y whenever a > 0 (i.e., in all ‘strong’ flows according to Rive), is constant for viscometric flows (a = 0) and decreases with increasing y for the ellyptical flows discussed in [3], i.e., when -1 < u < 0. The rigid-body limit leaves of course the viscosity undefined. The criteria Ro and RE take the following values: R

(22)

D

RE = q2A2.

(23) These should be discussed in some detail. RD is a measure of how fast the material protects itself from severe stretching by rotating; its value is indeed infinity for rigid-body rotation, and decreases continuously to unity for viscometric flow and to zero for extensional flow: in the latter limit no stressrelieving rotation at all takes place. When RD > 1, the Maxwell viscosity decreases with increasing 7; when RD < 1, it increases with increasing 7. For MRF, RD > 1, hence in Tanner and Huilgol’s terminology MRF is predicted to be weak. The value of the criterion RE is negative for any flow between rigid-body rotation and viscometric flow, including MRF. Conversely, RE > 0 for any flow between viscometric and extensional. The RE criterion correctly predicts the existence of a critical stretching rate for the Maxwell model, as can be seen by comparing eqns. (21) and (23). Furthermore, it predicts that the critical value of the magnitude of the rate of strain decreases as the flow character proceeds from viscometric to extensional: the condition for the right-hand side of eqn. (21) to be finite is, for u > 0: yz < (4 ahz)-l

(24) which shows that the critical stretching rate decreases from infinity in viscometric flow to l/2 A in extensional flow. The lubrication-type flows considered by Denn [9] would correspond to 0 < a << 1, and would therefore be characterized by a very large critical stretching rate.

Acknowledgements -.I am indebted to Mme R. Drouot for the proof of objectivity for W, and to her as well as to M.M. Denn, A.B. Metzner, J.R.A. Pearson and C.J.S. Petrie for helpful discussions. References 1 A.B. Metzner, Extensional Acta, 10 (1971) 434.

primary

field approximations

for viscoelastic

media,

Rheol.

76 2 H. Giesekus, Strijmungen mit konstantem Geschwindigkeitgradienten und die Bewegung von darin suspendierten Teilchen, Rheol. Acta, 2 (1962) 101,112. 3 G. Marrucci and G. Astarita, Elliptical flows of elastic liquids, Proc. 1st Int. Congr. Theor. Appl. Mech., AIMETA, Udine, 1971. 4 G. Astarita, Two dimensionless groups relevant in the analysis of steady flows of viscoelastic materials, Ind. Eng. Chem. Fundam., 6 (1967) 257. 5 R.R. Huilgol, On the concept of the Deborah number, Trans. Sot. Rheol., 19 (1975) 297. 6 G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGrawHill, Maidenhead, 1974. 7 R.I. Tanner and R.R. Huilgol, On a classification scheme for flow fields, Rheol. Acta, 14 (1975) 959. 8 R.I. Tanner, A test particle approach to flow classification for viscoelastic liquids, AIChE J., 22 (1976) 910. 9 M.M. Denn, Extensional flows: experiment and theory. In R.S. Rivlin (Ed.), The Mechanics of Viscoelastic Flow, A.S.M.E., AMD Vol. 22,1977, pp. 101-124. 10 R. Drouot and M. Lucius, Approximation du second ordre de la loi de comportement des fluides simples. Lois classiques deduites de l’introduction d’un nouveau tenseur objectif, Arch. Mech. Stasow., 28 (1976) 189. 11 R. Drouot, Definition du transport associe P un modele de fluide de deuxieme ordre, C.R. Acad. Sci. Paris, Serie A, 282 (1976) 923.