Extensional flows

613

EXTENSIONAL FLOWS Christopher J S Petrie Department of Engineering Mathematics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK.

1. I N T R O D U C T I O N The simple, practical, definition of extensional flow is a flow in which the velocity vector can be expressed in the form u = elx ,

v = e2y ,

w = e3z

(1)

in a fixed rectangular Cartesian coordinate system, (x, y, z), where (u, v, w) are the components of the velocity vector and el, e2 and e3 are the principal rates of strain, (which may be functions of time). For an incompressible fluid, the sum of the principal rates of strain is zero. Uniaxial extension is obtained when el = k ;

e2 = e3 = - 8 9

(2)

For many fluids, including all Newtonian fluids, there is no more difference between shear and extension than there is in linear elasticity. Indeed, Trouton [1] introduced what we now call the elongational viscosity (his "coefficient of viscous traction") because he wished to obtain the conventional viscosity (the shear viscosity) of some highly viscous fluids. The elongational viscosity is the ratio of stress to rate of strain in the steady uniaxial extension of a uniform cylinder of material. For an incompressible Newtonian liquid Trouton proved theoretically (and demonstrated experimentally) that this is three times the shear viscosity, r/o, r/E = 37/0,

(3)

just as the Young's modulus is three times the shear modulus for an incompressible Hookean solid. We find that the behaviour of polymeric liquids and of suspensions of long slender particles (fibre suspensions) is markedly different in shear and extension. The ratio of elongational viscosity to shear viscosity, which we call the Trouton

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ratio, is no longer three, or even close to three. We associate this behaviour with the existence of some sort of structure within the fluid which can become markedly anisotropic during flow, for example when fibres or extended polymer molecules become aligned parallel to one another and to the flow direction. This aspect of the behaviour of the fluids mentioned is highly significant and, especially in the case of dilute polymer solutions, can give rise to Trouton ratios of 100 or 1000 and hence to stresses which are two or three orders of magnitude larger than in shear.

1.1. Definitions The Society of Rheology, in an effort to keep some sort of order in the use of terms by rheologists, has provided an unambiguous definition of "tensile viscosity" [2] - the quantity we have referred to as elongational viscosity, also known as uniaxial extensional viscosity. Tensile viscosity is defined as follows: A material is subjected to homogeneous simple extension, i.e. to a flow which is spatially uniform, with constant rate of strain, ~, in the x~-direction and - 89 in every direction perpendicular to the x 1-axis. The ratio of "net tensile stress", O'E ~ fill --flEE, to rate of strain is monitored as a function of time and the "tensile viscosity" is defined as

(4) This definition, of course, says nothing about methods of, or even the feasibility of, experimental realization. The notation r/T is sometimes used, instead of r/E, in honour of Trouton, and the elongational viscosity may be called the Trouton viscosity. If surface tension is significant, we need to be clear that the "net tensile stress" a E in our definition, Equation (4), is the measured stress corrected for surface tension according to Applied force O"E =

Coefficient of surface tension --

Area

Radius

.

(5)

We also need to be clear that the use of O E ~ O l l - - a 2 2 i s , in any case, valid only for incompressible fluids. This matter is discussed further elsewhere [3,4]. For simplicity, our discussion of other extensional flows, Equations (8), (10) and (11) below, refers to incompressible fluids in the absence of surface tension. As well as uniaxial extension, we may consider equibiaxial extension, el = e2 = ~ ;

e3 = - 2 ~

(6)

and planar extension, el = ~ ;

e2 = 0 ;

e3 = - + .

(7)

615

The equibiaxial extensional viscosity [2] is defined by r/B(~. ) = O'11 -- 0"33

(8)

and for a Newtonian fluid ~Tn = 6770.

(9)

Equibiaxial extension is kinematically the reverse of uniaxial extension, but significantly different in terms of the effect of the flow in tending to align long molecules or fibres. The idea that the functions r/E(~) and r/B(~) may be regarded as the same function, for positive and negative values of the rate of strain in the direction of the axis of symmetry, is not a particularly helpful one since there is no reason at all for supposing that the values of this function over the two ranges of values of its argument are in any way connected (except by continuity in a mathematical sense, which corresponds to Newtonian behaviour in the limiting case of small rate of strain). Planar extension is sometimes referred to as "pure shear" but it must be clearly understood that it is qualitatively different from simple shear, being irrotational (relative to the usual fixed axes). A particular point of interest for planar extension is that there are two extensional viscosities, the planar extensional viscosity ,p(g)

=

O'll -- 0"33

(10)

,

which refers to the tensile stress required to stretch the material in the x 1direction and a second "cross-viscosity" r](0)(~ -) = O"22 -- O"33

(11)

which refers to the tensile stress required to prevent deformation in the neutral direction (the x2-direction). The theoretical relations for a Newtonian fluid, with shear viscosity r/0, are Tip - -

477o ;

r/(~ = 2770.

(12)

The notation 7/(2~ is a simple example of the general notation [2,5-7] for general extensional flows. If el = ~ is the largest (positive) rate of strain, then we may define m such that e2 = m~ and then el = ~ ;

e2 = m~ ;

e3 = - ( 1 + m)~

(13)

for an incompressible fluid. The parameter m, which we take to be independent of time, describes the geometry of the extensional flow, with m = 1 for equibiaxial extension, m = 0 for planar extension, m = - 89 for uniaxial extension and

616

< m < 1 in general. The two extensional viscosities for this general flow

2 -are

~m)(~.)

= fill

-- 0"33

(14)

and Tl(m)(~.) = 0"22 - - 0"33 .

(15)

These are equal to one another for equibiaxial extension and the second is zero for uniaxial extension while for all other extensional flows we have the two physical quantities which we may try to measure. As well as for the three standard cases, m = - ~1, m = 0 and m = 1, experiments have been carded out by Demarmels and Meissner [8] for m = +89 This flow has been referred to as "ellipsoidal extension" but may best be visualized as an unequal biaxial extension, with stretching at rates ~ and 1~ in two perpendicular directions. 2. T H E E X P E R I M E N T A L E V I D E N C E Experiments for determining extensional viscosity are difficult to carry out, and even where there is a consensus about results it is important to remember that this consensus is the fruit of much painstaking experiment. For polymer melts, it has been possible for some time to obtain reliable values for low and moderate rates of strain; the review by Meissner [9] is a useful source of information. Different polymers show different behaviour in the variation of Trouton ratio with rate of strain. As has been remarked above, the value of the Trouton ratio can be large, but for polymer melts it does not reach such extremely high values as for polymer solutions. The situation is much less clear for polymer solutions, as has been summarized by James and Waiters [10]. The experiment which corresponds to the definition of extensional viscosity has not been easy to perform for these mobile (low viscosity) fluids - in fact it had been judged impractical until the work of Matta [11] and Sridhar [12-14]. It is, however, generally accepted that the Trouton ratio for polymer solutions is large. Even with the considerable uncertainty surrounding interpretation, we believe that estimates of 100 or 1000 have genuine physical significance, whether or not the measured quantity is truly an extensional viscosity. Suspensions of long fibres generally behave in a similar manner to polymer solutions. In all cases, at least for fluids which are unoriented in their natural state (i.e. when at rest and relaxed), the limiting values of extensional viscosity for small rate of strain agree with the Newtonian predictions. There is no clear evidence

617 of what happens at high rates of strain, where experiments are difficult, but what evidence there is supports the theoretical conclusion [7] that uniaxial and planar extensional viscosities should be the same. This theoretical study shows two classes of asymptotic behaviour at large rate of strain in planar and uniaxial extension. When we use constitutive equations appropriate to polymer solutions or fibre suspensions, there is an "upper Newtonian" r6gime with a constant extensional viscosity corresponding to fully extended and fully aligned molecules or fibres. With the constitutive equations that were developed for polymer melts a decreasing extensional viscosity was found. The stress must grow with rate of strain, but can do so at a rate less than linear (e.g. (rE ~ In ~ for the Phan-ThienTanner model) [7]. This decrease sometimes follows a maximum in extensional viscosity at some intermediate rate of strain. It is extremely important to distinguish clearly between measurements which do give the extensional viscosity functions directly, i.e. experiments in which steady spatially homogeneous extensional flow is attained (to a reasonable approximation), and measurements which give a stress growth function or transient extensional viscosity (or something even less well-defined). This is not to say that the experiments in which the ideal extensional flow is realised are necessarily the best experiments. If we want measurements that will allow us to predict fluid flow and stress levels in geometrically complicated unsteady flows, we may well find that some sort of transient extensional viscosity is more appropriate. For fundamental understanding of fluid deformation and flow, however, the true extensional viscosity remains an important physical quantity which we should like to measure accurately and reliably.

2.1. Extensional viscosities of polymer melts The most extensively studied material, historically, is low-density polyethylene (LDPE). The most reliable experiments, of Meissner and co-workers (past and present) [15,16], show the elongational stress growth function (commonly called the transient extensional viscosity) for experiments at constant rate of strain. This function, for uniaxial extension, is compared with 3 times the analogous function in shear at low rates of strain (this is regarded as the linear viscoelastic response). Typical behaviour of LDPE is to show the linear viscoelastic response at very low rates of strain (such as 0.001 s -1) and a greater stress at moderate rates of strain (typically 0.01 s -1 up to 1 s -1). This behaviour is sometimes referred to as "strain hardening" or "deformation hardening" (although "strain rate hardening" might be a more appropriate term). This behaviour is also reported [17] for HDPE, which does show "strain hardening", although to a smaller extent. For other geometrical forms of deformation, there are Most work until recently has been on polyisobutylene be investigated at room temperature. PIB behaves in a in uniaxial extension. In biaxial extension the response

fewer results [5,8,18]. (PIB), because it can similar way to LDPE is closer to the linear

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viscoelastic response (multiplied by the expected factor of 6) at moderate rates of strain. In planar extension the first stress growth function (corresponding to ~Tp) lies between the uniaxial and biaxial responses (all suitably scaled), while the second stress growth function falls below the linear response after a fairly short time. Improved versions of the extensional rheometers have allowed different deformation histories (extension at constant stress or force as well as at constant rate of strain) and measurement have been made on polymethylmethacrylate [19], polystyrene [19] and polybutadiene [20]. Another recent development of considerable importance is the new multiaxial elongational rheometer from Meissner [21] which has been used for extensional flow investigations on polymer melts and blends at elevated temperatures [22].

2.2. Extensional viscosities of polymer solutions The history of extensional flow measurements for polymer solutions is full of apparently contradictory results [4,10]. The major source of this is the use of different experimental techniques, none of which, until recently, are even claimed to reproduce steady uniform uniaxial elongation. Among the techniques that have been used are (a) fibre spinning and open syphon (Fano) flow, (b) contraction flow (into an abrupt contraction or through an orifice), (c) converging flow (through a smoothly tapering channel), (d) stagnation-point flow (e.g. use of opposed jets), (e) falling weight and controlled extension. and each of these has its own advantages and drawbacks. James and Waiters [10] have collected results for one polymer solution and show, in their Figure 2.1, values claimed for the extensional viscosity over a range of three decades (10-10,000 Pa.s) for rates of strain in the range 1-100 s -1, with experimental results scattered over the whole of this region. The explanation of this is not that the experiments are all meaningless, but that, since we are dealing with a viscoelastic material, the differing flow histories and the fact that steady flow is not attained made considerable differences to results. The various collaborative testing programmes reviewed in [4] (one of which produced the results in [10] mentioned above) have made it clear that differences in the material being tested do not explain these differences. In short, the reported quantities are not "extensional viscosities". While the variety of experiments and corresponding predictions may not be diminished in the near future, recent work does suggest that reliable extensional viscosities (in the sense that experiments are reproducible in different laboratories) may be obtained for polymer solutions.

619

2.3. Other experimental measurements The measurement of extensional viscosities and stress growth functions by no means exhausts the possibilities for useful experiments on the extensional flow behaviour of polymeric liquids. Meissner [9] has consistently urged that recoverable strain should be measured as well as total strain, and this has been done for a number of polymer melts. Recent measurements on polymer solutions [23-25] have demonstrated interesting behaviour in stress relaxation at the end of a period of steady uniaxial extension. Attempts have also been made to investigate further the fundamental difference between shear and extension by looking at flows in which the history of the principal extension ratio is the same for a shear flow and an extensional flow [26,27]. Experiments were carried out by Ztille et al. [26] both with ~ increasing exponentially and with ~ constant, where ~ is the principal extension ratio. In each case both shear and uniaxial extension were used. The results for LDPE showed that at large strains and large rates of strain the rheological behaviour is determined primarily by the history of the principal extension ratio, ~(t), and not by whether the flow is an extensional flow or a shear flow. This means that they were able to show deformation hardening and deformation thinning behaviour in extensional flow of LDPE, by choosing in the first case elongation at constant rate (~ = ~f = ~ exp(~t)) and in the second case, elongation with ~ constant. Similarly in shear the apparent viscosity (measured at constant shear rate) is a decreasing function of shear rate while the corresponding quantity for "exponential shear" increases with shear rate. Samurkas et al. [27] point out that to conclude from this that the material is deformation hardening in exponential shear may be misleading. Their point is that under exponential shear a Newtonian liquid will show a similar behaviour. Indeed Ztille et al. [26] show (in their Figure 4) that the measured stresses are generally lower than the predictions for exponential shear of linear viscoelasticity. Samurkas et al. [27] compare exponential shear with planar extension, and present their result in term of the damping function for the Wagner model (see below). Their conclusion is that the damping function obtained from steady shear flow measurements is better than that obtained from planar extension in predicting behaviour in exponential shear. This tends to the opposite conclusion to that reported above [26]. However neither set of results is conclusive, and direct comparison is difficult because of the different ways in which the data are presented. 3. BASIC P R O P E R T I E S OF SOME C O N S T I T U T I V E EQUATIONS IN EXTENSION The extensional viscosity (in uniaxial extension), as well as the planar and equibiaxial extensional viscosities have been tabulated for a number of simple constitutive equations [3]. We reproduce some of these results below (Table 1)

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and discuss two classes of constitutive equation which have become popular in recent years. These are the Wagner integral model [28] (often used for polymer melts) and the FENE dumbbell models [29] for polymer solutions. Larson [30] gives useful discussion of constitutive equations which is both more thorough and more up-to-date than [3]. Many of the results in [3] are for special cases of the Oldroyd eight-constant model, T + ,~1 '~ - / Z l (T- D + D . T) + vl tr(T- D) I + I~o tr(T)D i" = 27?0 [D + )~2

] - 2#2D" D + y2 tr(D. D) I] ,

(16)

in which the notation T denotes the corotational derivative of the extra-stress tensor, T, and D is the rate of strain tensor. The generalized convected Jeffreys model

is one useful example, with A = 0 giving the generalized convected Maxwell model. A generalized convected derivative D

V

T =T+{(T.

D + D . T) = T -

a ( T - D + D - T)

(18)

is sometimes defined, with a = 1 - ( being used in [3] and elsewhere, and Equation (16) could use the generalized convected derivative if a = #l/A1 = //2/,~2- The parameter ( takes values in the range 0 <_ ( _< 2 and correspondingly 1 >_ a >_ - 1 . The upper convected derivative corresponds to ( = 0 (i.e. a - 1) while ( - 1 (or a - 0 ) gives us the corotational derivative and ( = 2 (or a = - 1 ) gives the lower convected derivative. In terms of molecular (or micro-structural) models, only the upper and lower convected derivatives are obtained from so-called affine deformation models. In these models, hypothetical micro-structural elements in the fluid, (such as, for example, either network junctions or dumbbells) move affinely with the fluid. The parameter ( can be interpreted in terms of a non-affine deformation of elements in the fluid, i.e. a slip between micro-structural elements and the surrounding fluid. This leads to some predictions in shear flow which are regarded as undesirable and in extreme cases are contrary to physical intuition (e.g. negative stresses in response to a sudden large shear deformation). There are other consequences in extensional flows, such as non-existence of solutions to flows problems for some values of the parameter (, e.g. ( > 0.5 [3], but the whole range of non-affine convected derivatives are by no means ruled out

621

Model

tiE

Newtonian UCM LCM GCM Corotational

~B

3770 67/0 3r/ 60 (1+)~)(1-2,~) (1+4a~)(1-2)~) 3rl 67/ (1+2,~0(1- a0 (l+2aO(1-4aO 3r/ 67/ (l+a,~)(1-2a)~O (l+4a,~)(1-2aa~)

~p

r](0)

4r/o 47/ (1+2,~)(1-2,~) 4O _ 0+2aO(1-2aO 40 (l+2a)~)(1-2a,~)

27/0 2q (1+2)~) 27/ (1-2a~) 27/ (l+2aag)

Oldroyd B

3r/ 1-Ag(I+2M) 3r/(l+,~)(l_2M)

67/ ~_ l+2Ag(1-4ag) o~'](1+4)i~)(1-2~)

4r/ 4r/(1-4aAg 2) (1+2~)(1-2~)

27/ 27/ I+2Ag (l+2,~g)

Oldroyd A

l+Ag(1-2)~) 37/(1-~)(1+2~)

1-2Ag(l+4~) 6r/ (1-4~)(1+2~)

4r/(1-4~Ag 2) (1+2~)(1-2~)

,-, z r / ~1-2Ag

Table 1. Extensional viscosities for some simple constitutive equations. The abbreviations in Table 1 for rheological model names are " U C M " for the upper convected Maxwell model, "LCM" for the lower convected Maxwell model, "GCM" for the generalized convected Maxwell model. The corotational result applies to Maxwell and Jeffreys models. The Oldroyd fluids B and A are exactly equivalent to upper and lower convected Jeffreys models, respectively. by this behaviour (unless one adopts a very strict position on the mathematical requirements for a constitutive equation [31-33]). The material functions for the eight-constant Oldroyd model are 1 -/z2~" + (/Zlfl2 - 3T2)6"2 r/z = 37/0 1 -- #1~ + (#21 -- 371)~"2 1 + 2/z2~ + 4(/Zl/Z2 - 37-2)c2

r/a = 67/0 f + 2/Zl~ + 4(~u2

37-1)~2

1 + 47-2~2 r/p = 47/o 1 + 47-1~2

(20) (21)

1 - 2(#1 - ~ 2 ) c + 47-2c 2 + 8(,tt271 - ~ 1 7 2 ) ~'3

r/(2~ = 2r/o

(19)

(22)

1 + 4 T1 ~-2

where we have defined parameters 7-1 and 7-2: 7"1 = (/tO/2)(2/z1 -- 3Vl) -- ~1(/Zl -- 171) ,

(23)

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7-2 = (#0/2)(2#2 - 3y2) - / t l ( t t 2

-/,'2)

.

(24)

These are related to the parameters al = r~ + A2 and a2 = 7-2+ A1A2 which occur in the viscometric functions 1 + cr2k2 O(k) = ~o 1 + 0"1k 2 '

(25)

~t~l(k ) ---- 2 A l r / ( k )

(26)

- 2A2r/o,

~2(k) = -(A1 - / t l ) r / ( k ) + (A2-/t2)0o

(27)

for shear rate k. One interesting aspect of this is that all the parameter combinations which appear in the expressions for the extensional viscosities also appear in the expressions for the viscometric functions so that, in principle at least, the extensional flow behaviour of the eight-constant Oldroyd fluid can be predicted from complete data on shear flow. In fact, data for shear flow allow the calculation of the characteristic times A1 and A2, which do not appear in the extensional viscosity formulae, so that in a sense, for this model, the viscometric functions contain more information than the extensional viscosity functions. Note that the actual calculation of the parameters from viscometric data is not a trivial task and, for example, to obtain the parameters ttl and #2 which are essential for extensional viscosities one must have very good data on the second normal stress function, ~2(k).

3.1. Wagner and Kaye-BKZ constitutive equations The Wagner equation started out as a form of the Kaye-BKZ equation in which the nonlinear and time-dependent parts of the material behaviour are assumed to be independent in the sense that the kernel of the constitutive equation may be factorized. It is, at present, probably the most successful compromise between simplicity and generality in a constitutive equation for the quantitative description of the rheological behaviour of polymer melts. The model may be written

T =

M ( t - t')h(I~, I2)Ct~(t ') dr'

(28)

Oo

in which M ( t - t') is the memory function, c~-l(t ') the relative Finger deformation tensor and h(I1,12) the damping function, which introduces non-linearity of dependence on deformation through the invariants I1 and 12 of the Finger tensor. The product h(I1, I2)Ctl(t ') may be thought of as a non-linear measure of deformation [30,34]. If h = 1 and M ( s ) = ( G / A ) e x p ( - s / A ) we recover the integral equivalent of the upper convected Maxwell model. A sum of exponentials for M ( s ) gives the usual discrete relaxation spectrum. Equation (28) is not, in fact,

623

a Kaye-BKZ equation unless the damping function is independent of the second invariant, I2 - see below, Equation (35). In a general extensional flow, Equation (13), we may define the relative Hencky (or logarithmic) strain, e, and the relative extension ratio, g: e = +(t - t ' ) ,

g = e ~ = e eCt-t'~

and then we have

ctl(t ')=

00 )

0 0

0

(29)

(30)

(-2(re+l)

with invariants I1 = g2 + g2m + g-2(rn+l) ,

12 = g-2 + ( - 2 m + ~2(m+l) .

(31)

The damping function is chosen to fit data from shear or extensional viscometry (or preferably both). Early attempts used functions like h = e x p ( - n ~ ) for a shear strain of ~ = k ( t - t') (and we could write ~ = vql - 3, noting that I1 = 12 in simple shear). In order to fit data from different flows (shear and extension), the introduction of an invariant K = ~//311 +(1 -/3)I2 - 3

(32)

has been successful [35] and better fits to data have been obtained using a sum of exponentials h(I1,12) = f e - n l K + (1 - f)e -n2K .

(33)

Wagner and Laun [36] found that values f = 0.57, n l = 0.310 and n 2 = 0.106 gave a good fit to shear data for the LDPE Melt known as "IUPAC A", and /3 = 0.032 gave a good fit for uniaxial extension also. Another form of damping function which is popular is due to Papanastasiou [37] 1 h(I1,12) = 1 + a K 2 "

(34)

Larson [30] notes that neither of these functions, Equations (33) and (34), fits data for biaxial extension well and he proposes a model which is strictly of the Kaye-BKZ form, with a potential function, U, and the Cauchy deformation tensor, Ct(t~), T = ft__~ M ( t -

t') 2

0U(I1, I2) 0U(I1, I2)ct(~,)] d r ' . 0-]-i C t l ( t ') - 2 012

(35)

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The potential function used here is U(Ii,12)=~In

1+

(I-3)

(36)

with both c~ and I depending on the invariants I1 and 12: a = CO+ C2 tan -1

[ Cl (I2 -- I1)3 ] 1 +(12--I1)2 ,

(37)

I = (1 - 3)11 + X/1 + 2fli2 - 1

(38)

and, for IUPAC A, co = 0.20, cl - 0.05, c2 - 0.121 and 3 = 0.1. This does a better job of fitting all the data than the Wagner equation without the Cauchy deformation tensor, Equation (28); Equation (28) also has the defect of predicting a zero second normal stress difference in simple shear. Similarly, Demarmels found [8,38] that the Wagner equation does not give a consistently good fit to data from shear and several different extensional flows for PIB and that a better fit can be obtained with a model which introduces the Cauchy deformation tensor, Ct(tt), as well as the Finger tensor. This gives us an equation of the form proposed by Rivlin and Sawyers, which we write here T = f t_ o M ( t

- t ' ) [ h l ( I i , I 2 ) C t l ( t t ) + h2(Ii,I2)Ct(tt)]

dt t

(39)

and Wagner and Demarmels [38] propose a form for the two damping functions, for one particular PIB melt, hl(I1, I2) = (1 +/3)h,

1

h2(II,I2) =/3h,

h = 1 + a'~*lv'k~- 3)(12 - 3)

(40)

with constant values of the parameters a = 0.11 and 3 = - 0 . 2 7 for the particular PIB melt studied in [8]. The parameter/3 in this model, which gives the ratio of second to first normal stress differences in simple shear, is important in general extensional flows but unimportant in uniaxial extension. This choice of functions h l and h2 is not obtainable from a potential, so Equations (39) and (40) do not give a Kaye-BKZ equation. A variation in use of the Kaye-BKZ model [39] seeks to express the dependence of the kernel on the invariants I1 and 12 through dependence on the principal stretches (or their squares, which are the eigenvalues of the Finger tensor). This is motivated by the success of some strain-energy functions for rubbers, but has not been found to be successful so far. It has not proved possible to pick a simple dependence on the principal stretches which gives a good fit to shear and extensional data with the same values of parameters.

625

One final, well-known, defect of these models has been considered by Wagner [40]. This is the fact that even the best of the models with a damping function or non-linear deformation measure has difficulty in fitting data obtained in flows where the fluid experiences flow reversal. The simple example of this is one step strain followed by a second step strain in the opposite direction. The problem can be explained in terms of a temporary network model in terms of irreversible loss of entanglements. This suggests a damping functional as a replacement for the damping function. The damping functional proposed is the smallest value over the time interval of the conventional damping function. For a motion with a "nondecreasing deformation", defined as a deformation for which the damping function is a nonincreasing function of time, the damping function is correct. If the motion involves a "decreasing deformation", a smaller damping factor is used. This approach has some success in predicting recovery (elastic recoil) after uniaxial extension [40].

3.2. FENE constitutive equations The FENE (finitely extensible nonlinear elastic) dumbbell model is found to be useful for polymer solutions, though there are many questions, both about the theoretical foundations of the model and about quantitative agreement with data on polymer solutions. There are a number of variants on the model, which we shall discuss briefly. First we outline the basic features of the model, while avoiding a formal derivation of the equations; for details see, for example, [29,30]. The simplest FENE dumbbell model represents a polymer molecule in solution as an isolated dumbbell (two beads connected by a spring) whose motion is governed by a balance between a spring force tending to contract or coil the molecule and hydrodynamic drag on the dumbbell ends due to the solvent, which tends to stretch the molecule and align it with the flow. A Hookean spring has the disadvantage of allowing the molecule to extend indefinitely, so a non-linear spring law, commonly the Warner spring law,

HR Fa = 1 - ( R 2 / L 2) '

(41)

is used. Here R is the end-to-end vector for the dumbbell (the molecule), H is the spring constant, R = IR] is the end-to-end length and L is the maximum permitted end-to-end length for the dumbbell, so that R < L. The hydrodynamic drag force on a bead is given by a drag coefficient (d multiplying the velocity of the bead relative to the solvent. This gives us the relaxation time, A = ( d / ( 4 H ) , which is a characteristic time for an individual dumbbell to come to equilibrium under the competing action of the drag and spring forces. There is a second characteristic time, 0 = L 2 ( d / ( 1 2 k T ) which is associated with the balance between hydrodynamic drag and Brownian diffusion and a modulus,

626

G = n k T , (as in rubber elasticity, where n is the number of dumbbells per unit volume). We define the "FENE parameter", b = 30/A = n H L 2 / G

(42)

and can also introduce a characteristic molecular dimension a by

a 2 = 3L2/b = 3 k T / H .

(43)

This gives the equilibrium length of a dumbbell (when the solution is at rest), possibly multiplied by a factor like b/(b + 5), depending on the precise details of the FENE model. These ideas are used to obtain a configuration tensor, A = (RR) and the polymer contribution to the extra-stress tensor. The mean, ( ) , is an ensemble average involving the distribution function, g,(R), and the trick is to obtain equations without having to calculate if,. This is most usually done by making the Peterlin approximation, which involves pre-averaging the end-to-end length, so that Fd =

HR 1 -- ( ( R 2 ) / L 2)

(44)

instead of using the true ensemble average of Equation (41). This leads to the FENE-P model, with configuration evolution equation v XA +

1

L2 A = ~ I 1 - ( R 2 / L 2) (b + 2)

(45)

and the extra-stress T=

G(b/L2) A 1 -- ( R 2 / L 2)

Gb I + 2r/~D (b + 2)

(46)

in which r/~ is the solvent viscosity. The Chilcott-Rallison (or FENE-CR) model [41,42] is

v AA + T=

1 (L2/b) AI 1 - ( R 2 / L 2) 1 - ( R 2 / L 2) '

(47)

G(b/L2) [A - (L2/b)I] + 2~/,D. 1 - ( R 2 / L 2)

(48)

This model has qualitatively the same behaviour in extensional flows as the FENE-P model, as far as is known from investigations to date [4]. It has the property of a constant viscosity in shear, unlike the FENE-P and FENE models

627

which are shear-thinning [41], which is both a simplification and is desirable for modelling the behaviour of Boger fluids. One shortcoming of all the FENE models discussed above, as far as fitting data on polymer solutions is concerned, is that they have only a single relaxation time. A recent discussion of multimode models (i.e. bead-spring chains) by Wedgewood, Ostrov and Bird [43] points out that a straightforward generalization of the simple dumbbell leads to a complicated set of coupled nonlinear differential equations. They point out that some earlier attempts to analyse this contain serious errors and go on to propose a further approximation, the FENEPM model. The model consists of a set of N - 1 nonlinear springs joining N beads and the end-to-end vector for the i-th spring is denoted by Ri. The FENE-PM force law is taken to be Fi =

HRi 1 --(iV -- I) -I ~N~'((R2)/L2)

(49)

and the M in FENE-PM stands for the mean value taken in the denominator of Equation (49). The M is also used to denote "multimode", but it is necessar3, to remember that the FENE-PM model is not just a multimode FENE-P model, for the reasons of complexity to which we have alluded. Even so, the model obviously remains more difficult to use than the single mode FENE-P and FENE-CR models and the questions to be settled are whether the extra effort is adequately rewarded and whether the approximation introduces any undesirable side-effects. As examples of unwanted side-effects, the fact that the FENE-P model leads to equations for steady extension with multiple solutions may be instanced. This is a comparatively minor matter, which can be resolved [4] by analysis of the full equations, as is discussed below. A more interesting matter is the demonstration by Keunings [44] that the dumbbells in the FENE-P model do not actually behave as finitely extensible dumbbells. A simulation shows that a noticeable fraction of the dumbbells exceed the supposed maximum length L. This arises from the fact that the approximation leads to a distribution function for end-to-end lengths which is Gaussian (with an infinite tail), while for a true FENE model the distribution must have a cut-off at L and hence must be nonGaussian. A comparison between the FENE-P and FENE model predictions (using a stochastic simulation) [44] shows that the rheological effect of the Peterlin pre-averaging approximation is seen in a much more rapid increase in the tensile stress during the start-up of an extensional flow. The stress growth is smoother and somewhat slower for the tree FENE model. The steady state stresses are the same. None of the FENE models yields explicit formulae for the extensional viscosities (unless one wishes to write down formally the algebraic solution to a

628

cubic equation). It is therefore less easy to make simple statements about their properties. In the limit of small extension rate, the Newtonian ratios between the extensional viscosities and the shear viscosity are recovered. In uniaxial extension the viscosity curve is S-shaped and at large extension rate, an "upper Newtonian" rrgime is obtained [45], tiE = 3q~ + 6 G ~ = 3rl~ + 2b~Tp

(50)

where r/p = AG is the polymer contribution to the viscosity. As far as the FENEPM model is concemed, the model can be expected to improve the quantitative fit to data. No surprises in the qualitative behaviour have come to light so far. 4. S O M E M A T H E M A T I C A L ASPECTS OF E X T E N S I O N A L F L O W Once we have a constitutive equation and a particular flow to analyse, whether exactly or approximately, we can address mathematical questions of existence and uniqueness of solutions. These are by no means trivial or unnecessary exercises for the nonlinear systems with which we are faced, and there are particular problems in analysing boundary-value problems, even for simple onedimensional systems (and ordinary differential equations). In this section we discuss one existence problem in detail, recognizing that rheologically it is rather simple, as an illustration of the sort of problem that may have to be faced. It is an open question as to whether the limitations that are uncovered are seen as defects in the rheological model used, defects in the fluid dynamical approximations or merely as a warning that predictions obtained with the model must always be treated with a modicum of scepticism. After this, we shall discuss an example of lack of uniqueness of solutions to the steady-state equations. This has been mentioned above, and is a likely consequence of non-linearities of the sort we are introducing. The stability of such simple solutions, which is described by the full, time-dependent, dynamical equations, can be used to make a choice between the possibilities. In some cases, such as the example below, we can in fact go further than this and make some very general claims about the global behaviour of the system, which rule out some physically unrealistic solutions even though they do, formally, satisfy the steady-state equations.

4.1. Existence of solutions to a boundary-value problem Steady fibre spinning of a convected Maxwell model can be posed as a boundary-value problem which only has solutions for a limited range of values of one boundary condition. We consider the steady axisymmetric extensional flow given by Equation (2) with the spatially varying rate of strain k = U ' ( X ) in which U is the velocity at distance X from the start of the fibre (as it emerges from a spinnerette or die). The fibre take-up is at X = Ls where the velocity is

629

set to be U~, while the initial velocity is Uo at the spinnerette. If the volumetric flow rate, Q, is given, we can relate the cross-sectional area, A ( X ) to the velocity since we make the basic assumption that, to a first approximation, all quantities are uniform across the fibre. In the absence of gravity, inertia, surface tension and air drag, the five equations governing this flow for an isothermal incompressible upper convected Maxwell model, Equation (17) with ( = 0 and A = 0, are mass conservation, equilibrium of forces axially and radially and axial and radial components of the constitutive equation:

U(X)A(X) = Q,

(51)

{Txx(X) - P(X)} A(X) = F,

(52) (53) (54)

TRR(X) -- P ( X ) = O, T x x ( X ) + A {U(X)TJcx(X) - 2 U ' ( X ) T x x C u and

= 2r/U'(X)

TRR(X) + A {U(X)T~RR(X)+ U'(X)TRR(X)} = - r l U ' ( X )

(55)

in which F is the constant force at any cross-section and P ( X ) is the constitutively undetermined hydrostatic pressure (which is, in effect, calculated from the radial force balance, Equation (53)). Equations (51)-(53) can be combined to give

Txx(X)-

(56)

TRR(X)= F U ( X ) / Q

and so we have a second-order differential-algebraic system, Equations (54)-(56), for the velocity and two extra-stress components, with three conditions, Txx(O) =

To,

U(O) = Uo ,

(57)

U(L~) = U1,

two of which are needed for the differential equations and the third to determine the force, F, which has to be applied to maintain the specified take-up velocity. The extra-stress value, To, is all that is required from the flow history, corresponding to the internal structure of the fluid at X = 0, i.e. to that which has been determined by the flow of the material upstream of the spinnerette. This formulation shows the nature of the problem as a two-point boundaryvalue problem, and the question we can address is whether there are solutions for any draw ratio, DR = U1/U0, and how the answer to this may be affected by the value of To, i.e. by the flow history of the material. In dimensionless form, with u

u , Uo

x

x=-;--,

~s

c~=

~Uo L~

t,=

,1Q AUoF

,

7-o=

ToQ FUo

(58)

630 the system of equations may be reduced to c~2u" = (1 + 2c~u')(1 -

om')o~u'/u - 3it (cru'/u) 2

(59)

with # undetermined, so that the three conditions 1 u(0) = 1 , u'(0) = go = 3 a ( # + 70) - 2c~ ' u(1) = may be satisfied. It can easily be shown that, if 90>0.

DR

(60)

DR > 1, it is necessary that

What we can prove is that the value, u(1), of the solution of Equation (59) at the take-up, x = 1, is bounded above by a function of the Deborah number, a, whatever the values of To and #. This means that there is an upper limit to the draw ratio, DR, for any given Deborah number, regardless of the flow history and regardless of how great a take-up force is applied. The proof involves considering the comparison equation cr2v" = (1 + 2crv')(1 - a,v')crv'/v which can be integrated, with initial values v(0) = 1, v~(0) = 90 to give cry'= (v 3 + K)/(v 3 - 2 K ) where K = (cr9o - 1)/(2c~9o + 1) and then x =

ce

fv y3

_

2K dy.

y3+K

Equation (64) gives v implicitly as a function of

y3 9~(v, K ) =

2K

flv y3 + K dy .

(61) (62) (63) (64)

x/o~ and K ; we may write (65)

and the solution to 9~(v, K ) = x/o: is written v(x; a, 90) = ~ ( x / a , K ) . (66) We note that the condition ago > 0 implies that - 1 < K < 0.5 and can establish that dv/dK > 0 under this condition. Hence, for all admissible values of 90, v(1; c~, go) < ~(1/a,, 0.5) (67) and we can calculate the fight-hand side of Inequality (67) numerically. For example, if (x = 1/19, ~,(19,0.5)= 20.6328. Finally a comparison theorem applied to Equations (59) and (61) shows that DR = u(1; c~,#,go) < v(1; c~,go) < g,(1/~,0.5) (68) which gives us an upper bound on DR for any chosen c~, irrespective of the values of # and go.

631

4.2. Multiple steady state solutions We consider uniform uniaxial extension, Equation (2), which starts at time t = 0. We will need to specify an initial configuration in order to obtain a specific solution to the evolution equations for the configuration tensor (and hence the stress). We consider the FENE-P model, Equations (45) and (46) and define the dimensionless configuration variables y=

A~ - A22 L2 ,

tr A All + 2A22 z = L2 = L2 .

(69)

The first of these, V, may be interpreted as describing the degree of alignment of the dumbbells with the direction of elongation while the second, z, gives the mean end-to-end length of the dumbbells in the flow (i.e. the extent to which the dumbbells are fully stretched). It is obvious that 0 < z < 1

(70)

and also fairly easy to establish that

-z/2

<_ y <_ z

(71)

(see [4,45] where an argument suggested by 0ttinger is set out). If y = z the dumbbells are fully aligned with the flow, while if y = - z / 2 the dumbbells are all aligned perpendicular to the flow. The result of substituting the flow field given by Equation (2) in Equation (45) and expressing the result in terms of the variables defined in Equation (69) is the pair of evolution equations A~=sz+

(1) s

1-z

y

(72)

m,

~ = 2sy

+c (73) 1-z in which s = ~k is the dimensionless rate of strain (which may be regarded as a Deborah number) and c = 3/(b + 2) is a constant depending on the finite extensibility parameter, b. For large values of b, c ~ a 2 / L 2, the equilibrium mean square end-to-end length of the dumbbells expressed as a fraction of their maximum length (and this is small). We may examine the steady-state solutions of this pair of equations and discover that there are three, of which only one satisfies Inequalities (70) and (71). It is relatively straightforward, if tedious, show that the unphysical steady-state solutions are unstable. A more important question is whether, if y and z have initial values which satisfy Inequalities (70) and (71), i.e. are physically reasonable, the subsequent solution, (y(t), z(t)), remains physically reasonable. We answer the question by considering the behaviour of solutions represented by curves in the phase-plane, (z(t), y(t)). We shall prove that solutions which start

632

in the triangle 0 < z < 1, - z / 2 all subsequent time, t > 0.

< y < z, for t = 0, remain in that triangle for

It is easy to see, from Equation (73), that ~. < 0 for z ~ 1 - , so that z cannot reach the line z = 1 provided that z(0) < 1. Next we consider the angle between the solution curve as it crosses the line y = z and the normal the that line which points into the region V < z. If we denote this normal by the vector n, we have n = ( 1 , - 1 ) . The tangent to the solution curve (in the direction of increasing t) is parallel to the vector v, given by v = (~,, #). If the angle between these two vectors is between - 7 r / 2 and 7r/2, i.e. if the scalar product n . v is positive, then the solution curves all cross the line y = z from the region y > z to the region Y < z. When Y = z, ~, = y + c so the scalar product is n. v = (#+c)

- (#) = c > 0

(74)

and therefore solutions cannot leave the physically sensible region across the line y = z. The same argument may be applied to the line y = - z / 2 , (1/2, 1). Using this value of y gives il = s z - s z / 2 + z/2(1 - z) iz = - s z - z / ( 1 - z) + c and hence the scalar product n . v = ~ / 2 + fl = c / 2

> 0

with normal n = (75) (76) (77)

which shows that solution curves cross the line V = - z / 2 from the region y < - z / 2 into the region y > - z / 2 , as required. We may note, further, that for s > 0, i.e. for uniaxial extension (rather than for uniaxial compression or its equivalent, equibiaxial extension), # > 0 on y = 0 for 0 < z < 1 so that in extension alignment parallel to the flow is favoured (in the sense that if V(0) > 0 we have y(t) > 0 for all t > 0 while if y(0) < 0 we may expect v(t) > 0 at some subsequent time). Similarly if s < 0 alignment perpendicular to the flow is favoured, as we would expect in uniaxial compression. The same result may be proved for the FENE-CR model, Equations (47) and (48). The differences from the FENE-P model are only in the coefficient of the isotropic tensor, I, and the effect on the configuration evolution equations is that c is replaced by c/(1 - z) in Equation (73). Equation (72) remains unaltered and the arguments used above hold for the direction in which solution curves cross the lines Y = z and Y = - z / 2 . The behaviour as z ~ 1 is the same, provided that c < 1 and it is easy to see that this will be the case, since r > 1 would require b <_ 1 and the model only makes sense for b > 3 (otherwise a 2 > L2). Hence the result that solutions which start off in the physically sensible region remain there for all time is established for this model too.

633

5. SOME OUTSTANDING P R O B L E M S IN EXTENSIONAL F L O W

There are many unsolved problems in extensional flow, and so plenty of work for rheologists to do. The choice of the best constitutive equation is still an open question- to which the safest answer is often that it depends on the purpose for which the constitutive equation is required. Good quantitative agreement with extensional flow data, and data on linear viscoelastic and viscometric properties, may usually be obtained only at the price of introducing a complicated equation which may present too great a challenge in the computational simulation of complex flows. There is, of course, the question of whether the need to choose a constitutive equation may be avoided by computational methods involving direct simulation of behaviour at a molecular or micro-structural level. My view on this is that we still have a need for constitutive equations, as an aid to our understanding, even if not for our use of computers to solve problems. A useful perspective on the importance of extensional flows may be gained from the list of key challenges in polymer processing given by Kurtz [46]. Of the five key challenges listed, two ("Blown film bubble stability" and "Draw resonance") quite clearly involve extensional flow. The other three ("Screw wear", "Sharkskin melt fracture" and "Scale up problems") may involve extensional flow as a more or less important feature. Some comments on this and other issues have recently appeared [47].

5.1. Interpretation of experimental results Two pieces of work may usefully be mentioned here. The first seemed at first sight to be a good idea, but probably will not stand up to scrutiny. That is the idea of a three-dimensional plot of transient extensional viscosity as a function of strain and time [48-50]. While this appears promising for limited data, its theoretical foundations are questionable [51 ] and the effort involved in preparing three-dimensional plots, let alone using them, does not seem to be justified by any increase in our understanding of extensional flows, or by any practical application of the three-dimensional plots. Certainly it is true that a transient extensional viscosity for a viscoelastic material should never be regarded as a function of instantaneous rate of strain alone and the emphasis of that point is a useful outcome of this discussion. It is doubtful whether, in general, one more parameter can carry all the information about the state of the material (as influenced by its flow history) that is needed to give a well-determined value for the transient extensional viscosity. The second set of results is more convincing, showing how results from the "Rheotens" apparatus (in effect a melt spinning device) may be presented in a set of "mastercurves" and "super-mastercurves" [52,53]. This presents a challenge to the theorist to explain why such behaviour is observed - does it tell us something about the family of experiments or about the appropriate constitutive

634

equation which we have not yet appreciated? A final comment on experimental methods is perhaps worth making. While the analysis of complex flows, such as converging flow or flow into a contraction, does not give a reliable extensional viscosity in the sense of a fundamental material property, such flows have a large component of extensional flow. Hence if the Trouton ratio of a fluid is large, the stresses in the fluid will be predominantly those associated with the extensional part of the rate of strain. A more practical point is that, if a consistent analysis can be made, measurements on such flows should give data which can be used for reliable predictions of stresses for similar flows in industrial processes. The test of a fluid property derived from converging flow is then not so much "Does it give a true extensional viscosity?" as "Does its use give reliable predictions in related practical flows?"

5.2. Stability of extensional flows This topic has considerable relevance to polymer processing, where the prediction of flow instability gives an understanding of limitations to production rates for artefacts made from polymeric materials. There are a variety of instabilities and failures in extensional flow which the author has discussed elsewhere [54]. One recurrent theme is the need for clear distinction between different instabilities such as "draw resonance" and the classical instability of a filament due to capillarity. Similar clarity of thought and of description is needed when filament rupture is considered [55]. Instabilities in extrusion ("melt fracture", "sharkskin") are not immediately associated with extensional flow, but there are strong links according to some of the mechanisms for these two distinct flow defects [46,56]. The topic of approximations for nearly extensional flows [3,4,57] has several connections both with stability and with the analysis of industrial processes like fibre spinning and film manufacture, both by tubular film blowing and by film casting. It is therefore not merely a topic of interest to theorists as a matter of scientific curiosity, but a topic which could shed much light on our analysis of a number of industrial processes.

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