Extensional Flow - A Compilation

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Journal of Non-Newtonian Fluid Mechanics, 4 (1978) 23-38 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

CONVERGING FLOW AND STRETCHING FLOW: A COMPILATION *

F.N. COGSWELL

Imperial Chemical Industries Limited, Plastics Division, P.O. Box 6, Bessemer Road, Welwyn Garden City, Hertfordshire (Gt. Britain)

Summary

As fluids flow from a reservoir through a constriction they are subject to

a complex deformation history which may be considered to contain a strong stretching flow component. A number of workers have studied such flows with the objective of elucidating the stretching flow rheology of the material under test. A wide range of polymeric materials has been investigated in this way including solutions, melts and pastes. As yet there is no general accep- tance that converging flow may be interpreted as a dominantly stretching flow and each author who has taken this course has approached it from a dif- ferent route generating different equations which have, to some extent, ob- scured the picture. Thus there is, at present, no consensus view on the inter- pretation of converging flow : however, the differences between different authors are in fact relatively small compared to the magnitude of the prob- lem under investigation. This paper reviews the approaches that have been adopted, and compares the theories and experiences that have been record- ed. The objective is to provide an empirical basis upon which practical rheol- ogists may act, from which theoreticians may develop a more fundamental understanding, and as a result of which a more definitive method of measure- ment may be deduced. ____-_--__-~--.

1. Introduction

The shape of a body may be changed in three simple ways: bulk deforma- tion, simple shear and simple extension.

In classical mechanics extension can be considered as a combination of

* Presented at the British Society of Rheology Conference on General Rheology and Stretching Flows, Edinburgh, September 7-9, 1977.

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bulk deformation, simple shear and solid body rotation. Simple liquids are commonly assumed to be incompressible so that, since solid body rotation does not involve any dissipation, the rheology may be characterised by one parameter - the viscosity in simple shear. In practical studies of potentially anisotropic fluids and systems which are composed of more than one phase there is ample empirical evidence to suggest that a single dissipative function does not adquately describe the rheology and that inadequacy is most strongly evidenced in polymeric pastes, solutions and melts.

The bulk modulus of most liquids, and in particular most polymeric liq- uids is of the order 10’ N/m2 compared with shear and normal stress levels of practical interest which are seldom in excess of lo6 N/m’. In practice therefore the influence of an imposed flow field on the specific volume of a fluid is unlikely to exceed 0.001 and may be discounted as the source of a second major dissipative parameter. For practical purposes then we must look towards elongational flow to provide an additional independent dissipa- tive function. Indeed it is where the geometry of the flow departs from sim- ple shear, when streamlines cease to be parallel, that the simple concept of a single viscosity function is most frequently found to break down.

Many authors have studied converging flow with a view to obtaining a gen- eral fundamental understanding or to develop an empirical basis from which practical predictions can be made. Forsyth [l] has reviewed these approach- es and discussed converging flow in its broadest terms in a recent paper allowing this note to limit its consideration to those papers which explicitly consider the phenomenon of stretching flow.

Lodge [ 21 has demonstrated theoretically that materials with memory can exhibit a simple stretching flow response which is qualitatively different from its behaviour in simple shearing flows. To support this contention there are many examples in the literature of polymer solutions [3] and melts [4] which exhibit shear thinning and tension thickening. Unfortunately quantifi- cation of the rheological response in well defined extensional flows is at best difficult and at worst impossible (at present) and yet this information is urgently needed both for designing materials to meet particular needs and for designing flow processes to accommodate particular materials.

To be effective in an industrial environment the measurement of a prop- erty must be convenient and reproducible for a wide range of materials as well as being accurate. For studying viscosity in shearing flows, capillary rheometry based on the concept of Poiseuille flow is a widely used tech- nique. Several authors have suggested that the convergent flow from a wide cross section to a narrow cross section contains a significant stretching flow response and that that flow may be analysed to yield pertinent information on the stretching flow rheology. Different authors have approached the problem from different directions often simultaneously and this paper reviews those approaches and considers how the technique of converging flow may be developed to gain a wider acceptance as a method of studying rheology in stretching flow.

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2. Experimental flow geometries for uniaxial extension, and notation

Four flow geometries are of particular importance:

DESCRIPTION

Free convergence

GEOMETRY MEASUREMENTS

Pressuredrop PO

X Half angle of convergence 8 Die radius r

i

Flow rate Q Swell ratio 6 Jet thrust

Constrained convergence (lubricated)

Constrained convergence (unlubricated)

Pressure drop PE Half angle of convergence 8 Die radius entry rc

exit rl

8, r,, q,O as above

Opposed jets Distance between jets 2s Dii radius r

4-2S4

For the purposes of this paper we define LJ = the axial velocity; s = dis- tance along the centre line; e = stretch rate, du/ds; u = tensile stress; X = elon- gational viscosity; y = notional shear rate at the die wall, 4Q/7rr3; u, 0: Y (0 < n <‘ 1) the shear stress at the die wall; 17 = a,/? the shear flow viscosity; the subscript *vR indicates an average over the cross section.

We assume that the relationship between shear stress and shear rate may be determined from a separate experiment and further assume that inertial forces may be neglected.

3. Genealogy

Figure 1 traces the ancestry of studies of convergent flow associated with stretching flow. The development of ideas around 1970 appears to have been very rapid and many of the approaches were probably developed without an appreciation of the work of others. The similarity of those approaches only becomes obvious in retrospect.

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FREECONVERGENCE CONSTRAINEDCONVERGENCE

Cwene 151 Hsrrlro” ,15)

There are two main lines. Free convergence from the reservoir to the die and constrained convergence. There is also one important side branch where the early observation of free convergence effects was interpreted as an elastic energy correction. This side branch is charted but sketchily - indeed far more authors have followed that course than the main stream presented here and since heresies are only perpetrated by minorities I will not seek to begin a dialectic argument on the incompatibility of the two concepts but rather note with pleasure that one of the most active protagonists of the elasticity school [22] appears to be reconciling that approach with the stretching flow hypothesis.

4. Studies of free convergence

4.1. Stretch rate

In free convergence the accelerating column of fluid approaches the die inlet between rotating vortices. For simplicity it is usually assumed that the peripheral velocity of the vortex is the same as the velocity of the acceler- ating column so that there is no shear in the converging stream. The calcula- tion of stretch rate depends on the iso-velocity surface which is assumed. Two alternatives have been propounded:

Hurlimann [12] and Cogswell [ll] suggest

6 = (y/2) tan 0

which denotes that the apparent stretch rate is controlled by the same geo-

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Free convergencewith vortices average stretch rate

I

1 3 10 30 0 CONE HALF ANGLE

Fig. 2.

tan 9 z

sin 8

sineg+cose)

4

0

metric and flow rate factors as determine the shear rate at the die wall plus the angle of convergence.

Metzner [ 91 followed by Balakrishnan [ 131 prefer

. 1 sin3 8 E=J(l-cose)

which for our purposes is more conveniently written

p = ~ sin 8 (’ + ~ e, 2

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for which Oliver [lo] has suggested the simplification

P = (v/Z) sin 8

for small values of 8.

Figure 2 shows the ratio of tensile strain rate to the notional shear rate as a function of the half angle 8. There is no significant difference in the three expressions at angles less than 10” and at 30” the difference is only 20%. Metzner’s expression,

.

C=ZsinB (1 + cos 0)

2 2

based on sink flow, would appear to have a better fundamental derivation and has the conceptual advantage that it does not predict an infinite stretch rate with a very wide convergence. However Metzner’s expression does have an awkward problem when it comes to the material actually exiting through the die plane there being no way of accounting for the flow in the shaded region without introducing shear.

Metzner’s expression also suggests a maximum value of e/l at 8 = 60”. No physical significance has yet been attributed to that maximum. These prob- lems do not exist with the expression 8 = ($/2) tan 8 which exits parallel with the die plane and is monotonically increasing function of 0. Nevertheless that function is intuitively unacceptable for very wide angles because of the inference of an infinite stretch rate whereas we know that in practice many fluids will tend to sweep the reservoir and not generate vortices. It is interest- ing to note that Oliver’s expression, & = (q/2) sin 0, does not have any maxi- ma and remains finite.

For practical purposes it is unlikely that the peripheral velocity of the vor- tex will exactly match that of the accelerating core so that the flow will be partially constrained and the stretch rate is not uniform across the cross sec- tion. This will almost certainly be the case for very wide angles. In practice then computation of the stretch rate based on free convergence between vor- tices should be limited to those cases where the half angle of convergence is less than 30” and in that region any of the expressions given may be used.

There has also been one suggestion (11) for calculating the rate of exten- sion without a direct knowledge of the velocity profile but based instead on an assumption that the flow will adopt streamlines leading to a minimum pressure drop. This computation requires a knowledge of the pressure drop, PO, and the shear flow behaviour represented by u, 0: q” .

4% ti = 3(n + l)P() 1

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This expression implies a relationship between the function PO/a, and the approach angle 8. Shroff [ 261 has developed an empirical relationship be- tween these parameters qualitatively supporting that approach.

4.2 Tensile stress

There have been two approaches to determining the tensile stress based on jet thrust and on pressure drop. Balakrishnan and Gordon 1131 have recently reunited these diverging views in favour of the pressure drop as the major component for low Reynolds number and low swell ratio. There are some differences as to the precise relationship however:

Tensile stress Balakrishnan and Gordon [ 131 - PO Hurlimann and Knappe [ 121 “PO Cogswell [ 111 $(n + 1)Pe = 0.75 PO to 0.5 PO

for n = 1 to 0.3 Locati [ 3 51 PO

The relationship between the tensile stress and the energy dissipated as the fluid converges is clearly imperfectly understood however, at present, the differences are only of a factor of two.

There is an alternative expression for tensile stress if the approach half angle and shear flow behaviour are known [ll]

where us is the shear stress corresponding to the notional shear rate, Y = 4Q/nr3.

4.3 Elasticity effects

All the treatments described above assume a simple viscous response whereas we know that in practice stretching flows can be very highly elastic; The unease which this causes theoreticians has recently been summarised by Denn [ 381. However direct studies of the stretching response of polymer melts [4] suggest that the elastic deformation becomes saturated at relatively low stress levels. In converging flow the total deformation is large and, in general, the stress levels are high. We may thus justify ignoring the elasticity effects when estimating the viscous behaviour by postulating that the elastic response is saturated at low stresses and low deformations compared to those which will cause the major dissipation during the flow.

Nevertheless elastic deformation does occur and may be recorded from an observation of the swell ratio (B) through an orifice die as recoverable strain, ea=lnBgwh en using the Hencky measure of strain [ll], or as ea = Bg - 1 as preferred by Locati [35].

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4.4 Comparison between free convelrgence studies and other observations of stretching flow

Oliver [lo], Metzner [ 91 and Balakrishnan [ 131 carried out their studies with dilute polymer solutions of high molecular weight. In all cases the elon- gational viscosity of such solutions is observed to increase with stretch rate qualitatively confirming the predictions of Lodge [2] and in agreement with other methods of measurement like the tubeless siphon [23] and the triple jet [ 31. Cogswell’s technique [ll] has been used with one of the relatively concentrated solutions which Ferguson [24] has studied in a spin balance. Ferguson’s measurements on the spin balance indicated a complex response with viscosity decreasing and then increasing to a value above three times the zero stress value: the results for convergent flow were less complex showing a viscosity increasing slowly with increasing stretch rate. Hurlimann [ 121 made a detailed comparison with Meissner’s [ 251 measurements at constant elongation rate on low density polyethylene melt with quantitative agree- ment. Cogswell [4] has compared converging flow with constant stress elon- gation and obtained encouraging qualitative agreement for melts with both tension stiffening and tension thinning response. Shroff [ 261 has compared converging flow studies with Han’s [ 271 observations of an instrumented spin line with quantitative agreement for both tension stiffening and tension thinning melts. Finally White [ 221 compares converging flow studies with constant rate extension tests on a broad family of melts and concludes quali- tatively that large vortices are only generated in tension stiffening systems, a conclusion shared by the other authors.

5. Constrained convergence

5.1 No slip at the wail

In constrained convergence the angle of convergence is restricted by the walls of the die and when the die is unlubricated the boundary layer next to the wall is assumed to have zero velocity. This produces the velocity profile

which includes telescopic shear and elongation. Defining the stretch rate, & = dv/ds, where v is the velocity and s is the distance along the streamline it can be shown that at the wall the velocity and so the stretch rate is zero and that the maximum velocity and stretch rate occurs along the centre line. Everage and Ballman [16] follow Harrison [15] and show that for a Newtonian liq-

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uid the maximum stretch rate

6 2-L 1 i-cos2e

m*x I 8 cots8 (1 - c0s e)a(i + 2 cos e) 3

while for the same system Cogswell [ll] suggests that the maximum stretch rate

Lix =ytme

and the average stretch rate

Lerage = ‘2Y tan f3

By rewriting the equation of Everage and Bzillman using l/cot 0 = tan 8 and COSze =2COS2 e - 1 and cos 8 = d/(1 - sin28), we obtain

= gv tde 2 h2e

Gnax (I-cos~)~(I + 2 case)

In the limit as B --f 0, tan 8 = sin 8 = 8 and cos e = 1 - ie2, SO that

limit emax = ye 0-0

(Everage and Ballman [16])

limit f,,, = ye e-+0

(Cogswell [ 111)

Indeed plotting the two functions (Fig. 3) there is no significant differ- ence up to half angles of 10” and at 30” the difference is only of the order 25%. For practical design purposes we are unlikely to utilise wider angles.

The more exact expression of Harrison is, of course, limited to Newtonian flow. Cogswell [ 111 suggests a more versatile expression to include the possi- bility of a shear thinning response

t,,= - ( 1 3n+1 jhe n+l 2

where II is the power-law index shear stress a (shear rate)“. Even when large recirculating vortices are formed in free convergence

there may still be significant contributions from shearing flow. Detractors from the concept of using convergent flow as a rheometric technique point out that it is impossible to separate the shearing and extensional flows and that it is also necessary to know how these flow fields interact. In con- strained convergence, where shearing flow is deliberately included, it is of interest to note that the maximum transverse velocity gradient, which is responsible for shearing flow, occurs at the die wall where the longitudinal velocity gradient, responsible for stretching flow, is zero, and that along the centre line, where the stretch rate is a maximum, the shear rate is zero. In the intermediate regions the interaction wilI undoubtedly be very complex but the fortuitous blending of the velocity profiles allows us to assume that

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Constrained convergence assuming Newtonian flow, maximum stretch rate

1

Fig. 3.

I I I

3 lo 30 !

0 CONE HALF ANGLE

3

those interactions are only of secondary importance. Han and Drexler [14] have made a detailed study of velocity gradient and flow birefringence for constrained convergent flow of molten polymers. They deduce that over two thirds of the cross sectional area of the die the primary normal stress differ- ence varies less than 10% and that over the same area the longitudinal veloc- ity gradient is also approximately constant giving qualitative support to the inference of stretching flow rheology from constrahred convergent flow.

5.2 Lubricated walls

A special case of constrained convergence is obtained when the wall is lubricated so that no shear flow occurs. Three authors have studied this and they obtain :

Stretch rate at exit Tensile stress at exit

Snelling and Lontz [ 171 i f sin 8 (1 +cos e) 13

2 BP/[l -(:)“I

Cogswell [ 111

Shaw [18]

.

$ane

i 2sin 8 (1 + cos 8)

2 \ p/ bin(%) + 1. (theory)

L *P (as recorded)

where P is the pressure drop, rl is the radius at the die exit and r. that at the die entry. The stretch rate expressions of Shaw and Cogswell are essentially the same as shown in Section 4.1 above. Those. expressions differ from that of Snelling and Lontz by a factor of $ and this difference can be attributed to a different definition of stretch rate between the different authors, Snel- ling and Lontz defining stretch rate as u/s whereas Cogswell and Shaw use du/ds where u is the velocity towards, and s is the distance from, the cone apex. The tensile stress expression of Snelling and Lontz and Cogswell agree apart from the factor i though Shaw gives a very different expression the derivation and application of which I have not been able to understand, but although Shaw’s theoretical expression is very different the actual value which he computes for tensile stress ~2 pressure drop concur with the other authors as to magnitude.

In passing it is interesting to note that Snelling and Lontz, having derived an extensional flow field, went on to calculate the apparent shearing flow response on the assumption shear rate = 3 X stretch rate. The utility of their approach in characterising lubricated pastes of polytetrafluoroethylene is well attested by their own results as well as those of Couzens [28] and Cole- man [29].

5.3 Forced convergence

In the examples of constrained convergence noted above the fluid is, in general, constrained in such a way as to reduce the rate of extension.

Mackley [ 191 has used a specially constrained convergent flow for devel- oping very high stretch rates by opposing two jets

where the stretch rate is given by

P = iy r/s ,

where r is the radius of the jet and 2s is the distance between them.

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6. Convergent flow and biaxial extension

The results compiled here have only considered conicylindrical geometry in its analogy to simple elongational flow. It is possible to extend these anal- yses to wedge flow and radial flow and for the general case of an annular gap between two cones we follow the analytical approach of ref. [ll] to obtain:

shear rate at wall

. 69 3Q ‘Y = 2rR(2h)’ = 4?rRh2

average stretch rate

e= && -$ (2rR2h) = - Idh+lm\ hdt Rxj

Then if we measure distance in the primary flow direction as s we may write

dR -=-sinp, dR ds ds z=-zsinfl,

!!L-tancu dh ds , -=-$tanG”,

dt

whence the average stretch rate

. dsl E=dt htanu+isin/3,

( )

but ds/dt is the average velocity towards the cone apex at 0, i.e.

ds Q Th -=-=- dt 2?rR2h 3

whence

p=l 3 tam++lp I 1 There are two special cases of this general form.

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(1) Wedge flow approximating to pure shear flow given by R = infinity or 6=O,inwhichcase&=~+tancr.

(2) If the flow is reversed to flow outward from the centre then if R tan a = h sin 0, the flow approximates to equal biaxial extension.

Following the same analytical procedure as ref. [ 111 we can calculate the relationship between pressure drop and maximum extensional stress to be

P=+s],

when R,, and he, and RI and hi are the radii and half gap at the entrance and exit respectively.

While there are no detailed comparisons between such measurements and independent measurements of biaxial extensional flow fields their impor- tance has been attested in practical polymer melt processing situations including tube extrusion [ 301 and injection moulding [ 311.

7. Convergent flow and non-laminar flow

The industrial problem of non-laminar flow leading to extrudate distor- tion - sometimes termed “melt fracture” - has given impetus to many studies of converging flow [6,7,11,12,16,18,32-34,371. Up to the present time it has been common practice to assign a critical shear rate or shear stress to the onset of non-laminar flow. It is the opinion of Hurlimann, Cogswell, Everage and Shaw that rather than a critical shear condition it is, in fact, a critical tensile flow which induces non-laminar flow. Hurlimann, Cogswell and Shaw all suggest a critical tensile stress of about lo6 N/m2 (dependent on material) as being responsible for flow defects of this kind in converging flow systems and this estimate agrees well with independent observations of failure stresses in simple elongational flows.

Shaw’s observations [18] in this context are of particular interest. He demonstratedtmif wide angle dies are lubricated the distort~o~p~tern is exaggerated despite a reduction in the extrusion pressure. There have also been reports [ 32,331 that extrudate distortion may be worse in dies which have a half angle of convergence of about 45” than in the more extreme “square entry” die. The implication here is that dies with an angle of conver- gence which is slightly wider than their natural convergence restrict the melt from adopting that natural cone by suppressing its ability to form a vortex.

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This phenomenon is exaggerated if the conical die is lubricated. To be effective in reducing non-laminar flow it is necessary that the taper angle should be less than that which the flow would naturally adopt.

Converging flow with vortices may sometimes be complicated by a secon- dary expansion as the fluid approaches the die entry. For dilute polymer solutions the velocity profile may take the onion form indicated in sketch A

This defect may be partially relieved by finely dividing the vortex from the accelerating core as indicated in sketch B. A second contributory factor may be the sudden decelleration of the surface layer at the entry to the die.

8. Towards a converging flow rheometer

The rheometric techniques which have thus far utilised converging flow have taken an existing simple flow system and analysed it to obtain approxi- mate data. Valuable as these techniques are they can only ever give qualita- tive and comparative rather than definitive data. To obtain definitive data it is necessary to have control or precise measurement of the stress and rate of deformation history; preferably keeping one of these constant and measuring the other. It is also necessary that the sample should start from a predeter- mined, preferably rest, state. It is also desirable that the experiment should yield a constant measurement so that recording the results is not compli- cated by time dependent factors. Such a system could be based on converg- ing flow.

Shaw [18] lubricated his die to eliminate the shearing effects at the wall and Snelling and Lontz [ 171 found that their pastes were fortuitously lubricated. The use of a lubricating layer should eliminate most of the complexity of adventitious shearing effects and the die could then be precisely profiled to give a uniform extension rate. Stress history could be monitored by a series of pressure taps along the die and/or by flow birefringence. By exiting into a neutral buoyancy medium, strain recovery after extension could be readily measured. Alternatively the flow could be continued along a parallel die and the relaxation of normal stress recorded. The system could be developed for

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MAINSTREAM

LOW VlSCOSlTY LUBRICANT 4 ; l- :r LUBRICATING LAYER --- ~~

VISUALISATION

ME PROFILED \C

TAPS

TO GIVE UNIFORM EXTENSION RATE

RELAXATION IN A NEUTRAL BUOYANCY MEDIUM

the study of uniaxial, pure shear or biaxial extension though initially pure shear extension would probably be the most convenient with respect to monitoring and interpreting flow birefringence and flow visualisation. Suit- able variants of this concept could be appropriate to a very wide range of materials including gels, solutions, pastes and melts. The theory of this type of instrument is currently being considered [36] and the practicalities of design are being studied by the Rutherford Laboratory of the Science Re- search Council. Such a development, if successful, will give a versatile mea- surement of stretching flow which is steady in both the Lagrangian and Eulerian senses.

References

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Processing, Pisa, (1977); see also Rheol. Acta, 16 (1977) 655. 30 F.N. Cogswell, S.G. Maakell, P.D.R. Rice, J.C. Weeks and P.C. Webb, Plastics and

Polymers, (1971) 340. 31 I.T. Barrie,S.P.E. J. 27 (1971) 64. 32 D. Poller and D.L. Reedy, Mod. Plast., (1964) 133. 33 A.B. Metzner, E.C. Carley and T.K, Park, Mod. Plast., (1960) 133. 34 F.N. Cogswell and P. Lamb, Plastics and Polymers, (1970) 331. 35 G. Locati, Rheol. Acta, 15 (1976) 525-532. 36 K. Walters and C. Coleman, University College of Wales, Aberystwyth, private com-

munication. 37 J.R.A. Pearson and T.J.F. Pickup; Polymer, 14 (1973) 209. 38 M.M. Denn, in R.S. Rivlin (ed.), The Mechanics of Viscoelastic Fluids, AMD-Volume

22, American Society of Mechanical Engineers, New York, 1977, p. 101.