21

Journal of Non-Newtonian Fluid Mechanics, 6 (1979) 21-45 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

AN EXPERIMENTAL INVESTIGATION OF UPSTREAM FLOW CHARACTERISTICS OF VISCOELASTIC FLUIDS IN AN ANNULAR DIS ENTRY

K.L. TAN * and C. TIU

Department of Chemical Engineering, Monash University, Clayton, Victoria 3168, (Australia)

(Received September 1, 1978; accepted in revised form October 17,1978)

Flow behaviour of viscoelastic polymer solutions on the upstream side of an annular die entry has been experimentally investigated and compared with published results on entry flow in circular die. Stable and unstable flow patterns were observed depending on the magnitudes of Reynolds and elasti- city numbers. The latter number represents the relative effects of elastic and inertial forces. The stable flow region consists of an elasticity-controlled vor- tex growth regime and an inertia-controlled divergent flow regime. These two flow regimes have also been ohserved in circular entry flow. The criteria for the onset of flow instability in an annulus, based on the maximum vortex size, agree qualitatively with various criteria proposed for polymer melts and solutions in circular entry flow. The unstable flow patterns revealed a two- stage instability with a metastable region in between. The first-stage instabil- ity is character&d by low frequency disruption of the stationary vortex; while the second-stage instability, which occurs at high Reynolds numbers, is characterised by high frequency random distortion of the flow field.

1. Introduction

The polymer processing industry has been the most important area where viscoelastic fluids, and, in particular, viscoelastic polymer melts, sre encountered. It is not surprising then to find many publications in the litera-

* Present address: Esso Australia Ltd., Sale, Vie., Australia.

22

ture dealing with the flow behaviour of polymer melts and the mechanisms governing the onset of flow instability. Excellent review articles by White [ 11, Pearson [ 21, Astarita and Denn [ 31 and Cable [ 4) have discussed many of the important investigations in this area and their significance. Other review articles on polymer melt instability have been published by Dennison [ 51, Bialas and White [ 61, Pearson [ 71 and Petrie and Denn [ 8 J . Recently, it has been recognised that solutions of certain high molecular weight poly- mers exhibited many rheological similarities to polymer melts in both funda- mental fluid properties and flow behaviour in processing equipment. This, and the relative ease with which polymer solutions can be processed and character&d at ambient temperature and pressure have provided the incen- tive for using polymer solutions in place of melts in experimental investiga- tions of the entry region flow of viscoelastic fluids.

The flow geometry which has received the most attention is that of an accelerative entry region flow in circular ducts. This is motivated by the economic importance of such a flow geometry which is encountered in various industrial applications such as synthetic fibre spinning and extrusion moulding of plastic products. With the rapid expansion of the plastic indus- tries, the processing of viscoelastic fluids in geometries other than circular ducts has taken on added significance. One such geometry is the annulus. Annular entry flow is encountered in the extrusion of plastic tube, in wire coating, in film blowing and in annular rheometers.

This investigation is concerned with an experimental study of the laminar isothermal flow behaviour of viscoelastic polymer solutions in the entry region upstream of an abrupt 2 : 1 concentric annular contraction. The flow behaviour observed will be compared with the published results on circular entry flow.

2. Experiment and data analysis

Quantitative point velocity measurements and flow visualisation experi- ments were carried out in a closed loop flow system. The aspect ratios up- stream and downstream of the abrupt annular contraction were 0.21 and 0.42, respectively. Details of the experimental set-up and procedure are described elsewhere [S-11]. Visualisation of the flow field was made pos- sible through the use of an optical technique involving streak photography. The technique is similar to that described by Cable and Boger [ 121. Flow patterns were also recorded on motion pictures.

The test fluids were dilute aqueous solutions of Methocel SO-HG (hy- droxypropyl methyl cellulose), Separan AP30 and MG500 (partially hydro- lysed polyacrylamide) from Dow Chemicals. Varying concentrations were used to give a wide range of elastic properties. All tests fluids used and their respective nominal concentrations are tabulated in Table 1. The names are coded S for Separan and M for Methocel. The letters V and F preceding the

23

TABLE 1

Concentration of Test Fluids

Fluid Concentration

vs3 * 1.25% MG 500 vs4 0.50% MG 500

FM1 0.60% Methocel

FSl 0.31% AP30

FS2 0.40% AP30 F43 0.50% AP30

FS4 0.57% AP30 FS5 0.80% AP30 FS6 0.84% AP30 FS7 0.60% MG 500 FS8 * 1.25% MG 500

_____ __________~__. __.~ ~~-.- ____-

* Same test fluid

coded names referred to velocity measurements and flow visualisation studies respectively.

The viscometric data were fitted with power-law models,

r=K?”

and

(1)

NI = PII -P22 =oy. (2)

For greater accuracy some of the flow curves were approximated with two power-law models in the low and high shear rate regions. Thus, depending on the shear rate encountered in the experimental runs, the appropriate model parameters were used in calculating the following dimensionless groups [ 91:

Reynolds number,

Re = 23-*r;;(u)2 -“p = inertial force

K[(eon + el)/n]” viscous force ’ (3)

Weissenberg number,

elastic force viscous force .

Elasticity number,

(4)

5 _ Ws _ elastic force Re inertial force .

(5)

Two other useful parameters related to fluid elasticity, viz. stress ratio S and the Maxwellian relaxation time 0, were calculated directly from the

24

measured viscometric data using the following definitions:

S = Ni/r (6)

and

8 =N,/27j. (7)

A summary of the power-law model parameters and the experimental con- ditions covered for all the test fluids used is given in Table 2. The values of the experimental conditions were based on fully developed flow in the downstream (smaller) annulus. All the viscoelastic test fluids were considered to have essentially similar shear thinning properties, but with different con- sistencies or apparent viscosities. The flow behaviour index, n, varied by only ?22% in the range 0.342 d n < 0.54. On the other hand, the consistency index, K, varied by more than a factor of ten between the highest and lowest values, 0.196 < K < 4.532. Test fluid FMI, which had a flow behaviour index n of 0.862 at -j< 57 s-l and 0.628 at 57 < 9 < 894 s-’ exhibited no measurable elastic properties under steadyshear measurements. In the present context, this particular fluid was considered to be inelastic and used for comparison between the flow behaviour of inelastic and viscoelastic fluids.

A wide range of Reynolds and elasticity numbers was covered in the experiments. For the flow visualisation study, Re varied from 1.33 to 522 and e from 0.0143 to 2.874. For the point velocity measurements, the cor- responding values were 0.0109 < t < 1.7610 and 66 G Re < 1308 respec- tively .

A large value of 4 indicates that fluid elasticity is the dominant factor and inertial effects are negligible. It has been shown [9] that the elasticity num- ber, .$, provided a better measure of fluid elasticity than Ws, 13 or S in analys- ing the downstream entry flow data. Furthermore, a theoretical study of the flow of a second-order fluid in plane channels with a step change [13] had found that the effect of fluid elasticity on the flow should not be dissociated from the Reynolds number. This suggests that t, which also arises as a conse- quence of making the equations of motion dimensionless using a modified Rivlin-Ericksen constitutive equation [ 141, is a logical parameter to describe the characteristics of the upstream annular entry region flow of viscoelastic fluids. Cable and Boger [ 121 did not use this parameter in analysing their tube flow results.

The upstream entry flow geometry is sketched in Fig. 1. The origin of the r-coordinate is at the contraction plane. L, represents the vortex detach- ment length, which is defined in the negative direction of the x-coordinate where the flow detaches from the outer annulus wall. R2 and r2 are the radii of the outer walls of the upstream and downstream annuli, respectively. R, is the radius from the centreline to the vortex boundary which varies with x within the vortex zone, -L, Q x < 0. In the subsequent figures to be presented the appropriate radius and average velocity in different regimes of

TA

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1.69

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0.66

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731

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26

Vortex Boundary

/ Sjationary Vortex

Vortex DetaahmmtJ Plane

LContraation Plane

Fig. 1. Schematic diagram of the entry flow geometry.

flow are chosen as characteristic parameters in normalising the co-ordinate positions and the measured point velocities respectively.

3. Results and discussion

3.1 Accuracy of measured fluid properties

It is important to establish the accuracy of the property measurements on the Weissenberg Rheogoniometer since considerable importance is placed on the measured fluid properties in analysing the experimental data. Sources of error involved with the rheogoniometer have been discussed previously [ 151. Boger and Cable [ 161 who used the same rheogoniometer, had reported excellent agreement between the measured viscometric functions, r and N,, for a non-linear test sample from the National Bureau of Standards (NBS) designed for standardization of normal force measurement techniques [ 171.

The reproducibility of the measurements was also verified by randomly characterising various samples of the test fluids first in the order of decreas- ing shear rate and then in the reversed order of increasing shear rate.

3.2. Comparison with polymer melts

Many previous investigations concerning flow instabilities of polymer fluids have studied melts rather than solutions for obvious industrial reasons. Since polymer solutions are used as test fluids in this study to elucidate the nature of the upstream entry flow behaviour of viscoelastic fluids, it is im- portant to establish the rheological similarities between these test fluids and polymer melts used in other studies. This has been established based on relaxation time data [ 121.

27

The only major difference is that polymer solutions have a considerably lower viscosity level. Hence, the experimental results for polymer solutions are not strictly comparable with those of melts, even at the same shear rate, on account of higher Reynolds numbers encountered in solutions. This prob- lem can be overcome by using the elasticity number t, to represent the rela- tive elasticity level. This is equivalent to comparing data with similar relative elastic to inertial effects, although different Reynolds numbers and hence different inertial effects may be encountered for different fluids.

3.3. Verification of fully developed condition upstream

In the interpretation of experimental results, it was assumed that any dis- turbances to the flow field well upstream of the contraction plane were com- pletely dissipated. To achieve this, a long calming section was built into the flow rig, far upstream of the contraction. A further check is to establish fullydeveloped flow condition in the upstream region before the contraction zone, by measuring velocity profiles far upstream of the contraction.

Figure 2 is a plot of dimensionless axial velocity against radial position at x = -16.9 cm (the zero origin being at the contraction plane) obtained for fluid VS4. The solid line represents the theoretical fully developed velocity profile given by Fredrickson and Bird [ 181 for the flow of power-law fluids

‘.O rL 0.6 -

E * 0.0109 r/R,

(16) Frrdrickron ond Bird

0.6 - x = -16.9 cm

0.4 -

Fig. 2. Fully-developed velocity profile for fluid VS4-4.

28

in annuli. The Reynolds number in the downstream annulus is 814. The measured point velocities in Fig. 2 are in excellent agreement with the theo- retical velocity profile, within a 25% confidence limit. This is a clear indica- tion that any disturbances to the upstream flow field have been completely dissipated. Since the Reynolds number reported is the highest in the entire experimental range considered, and any upstream disturbances are more likely to be present at high flow rate, it is reasonable to infer that fully devel- oped flow condition in the upstream annulus has been achieved for all the experimental runs in this investigation.

3.4. Stable entry flow

The vortex detachment length, L,, is represented in dimensionless form as

& = --L,/mH , (8)

where RH is the hydraulic radius of the upstream annulus. The vortex length was determined from streakline photographs of the entry flow pattern for each test fluid over the range of conditions given in Table 2. Typical streak- line photographs shown in Figs. 3 and 4 depict the flow patterns upstream of the contraction for fluid FSS at four different Reynolds and elasticity num- bers. The flow is from left to right.

The characteristic “funnel-shaped” or “wine-glass stem” flow region is clearly seen in Fig. 3. The bulk of the fluid accelerates smoothly into the downstream annulus through this region. Accompanying an increase in Rey- nolds number from 1.33 to 17.4, the size of the stationary vortex increases. The term vortex growth regime is used to describe this flow regime. This finding is consistent with circular tube results obtained by Cable and Boger [ 121. Although there is a substantial decrease in the elasticity number, E, from 2.27 to 0.33 in Figs. 3(a) and (b). the level is still sufficiently high to have a major influence on the flow field. This is manifested in the form of a stored elastic energy, which is thought to be responsible for the occurrence of the stationary vortex at the corners of the contraction plane. In the vor- tex growth regime, the stored elastic energy increases with increasing shear rate or Reynolds number. To accommodate this increase requires an enlarge- ment of the vortex size. In addition, there is a faster recirculation rate with- in the vortex cell. This is evident from the motion pictures taken in connec- tion with this study. (The tine film is available for loan from the Department of Chemical Engineering, Monash University).

Further increase in Reynolds number results in an abrupt change in the nature of the flow pattern, as shown in Fig. 4. At these conditions, Re = 53.5 to 159 and E = 0.116 to 0.05, the streaklines near the outer walls diverge out towards the solid boundary as the vortex detachment plane is approached. This flow regime is termed divergent flow regime. The severe divergent flow of the bulk fluid is a result of significant inertial effects, as indicated by the low values of t. The stationary vortex is seen to be

29

Fig. 3. Flow patterns upstream of the contraction for fluid FS8 showing vortex growth : (a) Re = 1.33, g = 2.874 (0.5 s at f/5.6); (b) Re = 17.4, .$ = 0.334 (0.2 s at f/4).

Fig. 4. Flow patterns upstream of the contraction for fluid FS8 showing I divergent i (a) Re = 53.5, g = 0.116 (0.2 s at f/5.6); (b) Re = 159, g = 0.050 (0.04 s a It f/2).

3ow:

31

“pushed” towards the contraction plane by the inertia of the bulk diverging stream. Corresponding to this change in the flow field from the vortex growth to the divergent flow regime, the shape of the vortex cell becomes more conical.

Figure 4(b) illustrates the effect of higher inertial effects. There is a more pronounced bending of the streaklines and a deceleration occurs near the inner wall of the annulus. It is significant to point out that divergent flow is only observed in polymer solutions and not in melts. This is an indication that this flow regime is predominantly due to inertial effects, as polymer melts are characterised by inertialess or creeping flow conditions. The diver- gent flow regime has also been observed in tube flow [ 121.

In order to establish that fluid elasticity is the dominant factor behind the presence of a large stationary vortex, the flow pattern upstream of the con- traction is investigated for an inelastic fluid, FMI, in the low Reynolds num- ber region. Comparing with the viscoelastic test fluids, fluid FMI, with power-law model parameters of n = 0.862 and K = 0.196N sn/m2 at 7 < 57 s-l, has a relatively high flow behaviour index and low viscosity level. It is extremely difficult to obtain an inelastic fluid with IZ and K of similar mag- nitudes to the viscoelastic test fluids. Despite these differences, the flow pat- terns for fluid FM1 can be used as an indication of the relative effects of the upstream axial diffusion of momentum and the fluid elasticity on the growth of the stationary vortex.

A typical photograph of the flow pattern just upstream of the contrac- tions for fluid FM1 is shown in Fig. 5. The Reynolds number is 6.0, where the influence of axial diffusion of momentum is expected to be significant. It is evident from the streaklines that the flow is purely radial at the 90“ comer of the contraction plane. Except for a very small “dead space” (top half of the annulus) at the extreme corner, the flow pattern is in complete contrast to Figs. 3 and 4.

In a theoretical study Perera and Walters [ 191 found the effect of shear thinning to be of secondary importance. Hence, based on the evidence at hand, it can be said that the occurrence of a large stationary vortex in an abrupt annular geometry is a manifestation of fluid elasticity. This does not rule out the influence of fluid inertia on the secondary cell size. The inertia effect is expected to be significant since all experimental Reynolds numbers exceed the creeping flow limit. In the present context, a large value of t simply indicates that the influence of fluid elasticity is more pronounced than fluid inertia. Hence, in the vortex growth regime, the rate of growth of the secondary cell as a result of fluid elasticity is greater than the rate of decrease in cell size as a result of fluid inertia. Whereas in the divergent flow regime, the opposite is true.

All the stationary vortex data for the eight test fluids are presented in Fig. 6 as Ws/X, versus Re. The ordinate, Ws/X,, is chosen to correlate the vortex size data because it has been shown to represent a macroscopic Deborah number for the stationary vortex [ 121. A simple correlation appears to exist

32

Fig. 5. Flow pattern upstream of the contraction for fluid FM1 at Re = 6.0 (1 s at f/l 1).

between Ws/X, and Re, as seen in Fig. 6. The solid line represents the line of best fit. The vortex growth regime is characterised by a constant limiting value of Ws/X,, independent of Reynolds number. The regime extends up to a Reynolds number of about 10. This suggests that the vortex growth regime is not affected by fluid inertia and depends only on rheological forces with fluid elasticity or normal force being more important than viscous force. The vortex growth regime is described by the limiting equation

Lim Ws/X, = 8.0 . (9) Re-+O

At higher Reynolds numbers (Re > 10) and lower elasticity numbers, fluid inertia begins to influence the flow behaviour. This marks the onset of the divergent flow regime where Ws/X, becomes an increasing function of Re.

As a comparison, the pipe flow data of Cable [4] for contraction ratio of 2 was recalculated and plotted in Fig. 6. For the sake of clarity, only his data points for a 1.0% Separan AP30 solution have been shown. The results of the

33

60

40

WI/X,

20

0

v FSI

q FS2

0 FS3

A FS4

n FS5

A FS6

+ FS7

. FS6

0 Pip5 Flo#)

WX, =6

/ _ -40 go0

0. a l A

I I .̂ .__

I I”

Re

1””

Fig. 6. Macroscopic Deborah number versus Reynolds number for all test fluids.

other test fluids fell in a band about the data points for this fluid. Within the level of scatter of the experimental results, there is agreement between the two sets of data. This is unexpected in view of the different flow geometries. It may therefore be deduced that for the same contraction ratio of the flow conduits, the functional relationship between the Deborah number and Rey- nolds number in the stable flow region is unique regardless of the shape of the flow geometry, provided the hydraulic radius is used to characterise the flow geometry. However, further experimental work using different contrac- tion ratios is necessary to substantiate this statement.

The nature of the velocity profile in the vortex growth regime is shown in Fig. 7 for fluid VS3-4. Experimental conditions based on the downstream annulus are Re = 2.53 and t = 1.761. In Fig. 7(a), the axial position is -3.73 cm which is just upstream of the vortex detachment length, L, = -3.61. The good agreement between the measured point velocities and the theoretical fully developed velocity profile indicates that a fully developed condition exists at this position. Within the vortex zone there is a rearrangement of flow patterns which causes a change in the velocity profile as illustrated in Fig. 7(b). The axial position is -1.39 cm. The velocity profile becomes more pointed and skew towards the inner wall than the fully developed profile. This is to be expected, since flow pattern observations have shown a severe funneling effect in the vicinity of the inner wall within the vortex zone (Fig. 3). Furthermore, the dimensionless maximum velocity u,/<u>, is about

I

(a)

l/R,

I - Fredrickson and Bird (‘6’

15.6

t

-3.73 cm .

x =

L” = -3.61 cm 8

& -

. P

l

:i_ ,__g: ,

0 1.0

0.6 - - - Exprrimrntal

- Frrdriokron and Bird”*) l

LLY DEVELOPED PROFILE

RI = 2.53 b .\ f - 1.761

0.6 I = -1.39 cm

L”= -3.61 om

0 I.0

IllGUS ”

Fig. 7. Velocity profile in the vortex growth regime for fluid VS3-4. (a) Outside the vor tex zone; (b) inside the vortex zone.

35

16% larger than the fully developed value. The “overdevelopment” of the velocity profile has also been observed in tube flow [ 121. While in tube flow situation the shape of the profile tends to round out as the contraction plane is approached and attains fully-developed shape at the contraction; this over- development of velocity profile is retained up to the contraction plane in the annular case.

The nature of the velocity profile changes drastically in the divergent flow regime. Figures 8 and 9 show the velocity profiles for fluids VS3-5 and VS3-6, respectively, at Re = 23.3 and 148.8 and t = 0.296 and 0.051. The axial position for the velocity profile shown in Fig. 8 is -3.02 cm, which is just inside the vortex zone, since L, = -3.91 cm. A small maximum is seen to occur in the vicinity of the outer wall and the velocity profile is fairly blunt. The shifting of the maximum velocity position towards the outer annular wall is evident from an inspection of the flow patterns in the diver- gent flow regime as shown in Fig. 4, which indicates a deceleration of the flow field near the inner wall of the annulus.

At higher Reynolds numbers, inertial forces become dominant and there is a more pronounced bending of the streamliners near the vortex detachment plane. The nature of the velocity profile at Re = 149 is shown in Fig. 9. The axial position is -2.38 cm, which is just upstream of the vortex detachment plane, L, = -2.25 cm. There is a large maximum in the velocity profile near

1.0

0.3

r/R,

0.4

0.23 0

-- Expwimtntol

Ra = 23.3

x . -3.02 cm

FULLY MVELOPED

PROFILE

Fig. 8. Velocity profile in the divergent growth regime for fluid VS3-5 inside the vortex zone.

36

r/R,

0.2

0.4

- - Exprrimantal

1 = -2.30 am .

L”= -2.25 am e@ / . . l -

b f /: l I FULLY DEVELOPED

PROFILE

Fig. 9. Velocity profile in the divergent growth regime for fluid VS3-6 outside the vor- tex zone.

the outer wall. The velocity profile is quite different from that observed in the vortex growth regime (see Fig. 7(a)). Because of the difficulties involved in measurements of point velocities when there is severe bending of the streamlines, the data shown in Fig. 9 reveals a great amount of scatter. Nevertheless, the essential feature of the velocity profile is clearly discern- ible.

Although a complete set of developing velocity profiles was not obtained in the divergent flow regime, a qualitative description of flow behaviour is possible from the results of Figs. 8 and 9. Near the vortex detachment plane, the velocity profile is quite pointed and exhibits a maximum near the outer wall. The maximum becomes more pronounced at higher Reynolds numbers when there is a more severe bending of the streamlines (Fig. 9). When the fluid enters the vortex zone, the velocity profile becomes more blunt and the large off-centered maximum diminishes in size (Fig. 8). It is possible that at sufficiently high Reynolds and low elasticity numbers, the entrance velo- city profile would approach a uniform or flat shape, as expected for inelastic fluids.

There is no resemblance of velocity profiles in both vortex growth and divergent flow regimes between the annular and circular flow. This is

37

expected since annular velocity profiles are asymmetric due to the presence of the central core.

3.5 Unstable entry flow

A common feature of all the viscoelastic test fluids studied in the abrupt contraction geometry is that increasing the flow rate beyond the divergent flow regime, a transition point is reached where a first stage disturbance to the stable entry flow patterns occurs. This transition point may be taken as the limit which marks the onset of unstable flow. At this point there is a low frequency random disruption of the stationary vortex and fluid elements from the vortex flow into the main stream. The stationary vortex is observed to break and then reform in a random manner. Unlike the grossly unstable flow behaviour observed for LDPE melts [20,21], where the central filament entering a circular die breaks and retracts while material from the comer vor- tex surges into the die, there is only a partial disruption of the central flow in this case.

The nature of the first stage instability is illustrated in Fig. 10 for fluid FS8 Four still photographs are shown to indicate the flow patterns at different time intervals in sequence. The experimental conditions are Re = 312 and 5 = 0.0297. In sequence 1, the flow field is symmetric with two stationary vor- tices in the 90’ comer. At a short time later, (sequence 2) the vortex in the top half breaks and the main flow is disturbed by the radial flow of material from the comer. In sequence 3, the flow pattern near the 90’ comer illus- trates a purely radial flow. The streamlines indicate that as the fluid approaches the contraction plane, it diverges towards the 90” comer and the stationary vortex is either very small or has completely disappeared. There is a rapid acceleration of the fluid from the comer into the smaller annulus. Sequence 4 shows that the flow field is about to revert to that in the first photograph. The vortex in the top half has just reformed while the flow is still in the radial direction in the lower half of the annulus.

Only qualitative comparisons can be made between the flow patterns reported in this work and those of other investigations in view of the differ- ent flow geometries used. For the 2 to 1 annular contraction used in this work, no spiralling or pulsating flow patterns were observed as in circular entry flow. Divergent flow may be regarded as stable since the first stage instability was obtained at the upper flow rate limit of this flow regime. In the case of polymer melts flowing through circular dies, Bagley and Birks [21] and Tordella [22] observed only a stable entry flow and a grossly dis- rupted flow for LDPE, whereas Ballenger and White [20] observed a gradual spiralling motion into the die inlet which marks an intermediate region between the stable flow and the grossly disrupted region. This spiralling behaviour was found to be not as pronounced as in HDPE, polystyrene or polypropylene and only existed over a limited range of extrusion rates.

(3) (41 Fig. 10. Flow patterns upstream of the contraction for fluid FS8 showing low frequency random distortions - first stage instability, at Re = 312, E = 0.0297 (l/32 s at f/2.8).

Hence, this phenomenon could easily be missed as one changed from one extrusion rate to another [ 11.

Geometric factors may also play an important part in influencing the flow patterns. At the same flow condition, different metastable regions including spiralling [ 12,23,24], diverging and pulsating [ 121 flow patterns were ob- served for polymer solutions at the entry to an abrupt circular contraction for contraction ratios greater than two. For a 2 to 1 circular contraction, Cable and Boger [ 121 observed only a stable flow region, both vortex growth and divergent flow at low flow rates, and a grossly disrupted flow region at higher flow rates. The presence of an intermediate metastable region at higher contraction ratios for the same polymer solution, could be attributed mainly to geometric effects. The size of the stationary vortex in the 2 to 1

39

contraction was considerably smaller than those in the higher contraction ratios. Consequently, any disturbance to the flow field in the 2 to 1 geom- etry was of a very mild nature. It is possible that spiralling behaviour is present, but not easily detectable because of low intensity. In view of the similarities in the flow behaviour between the annular and circular geom- etries in the stable flow region, the first stage disturbance observed in this work is likely to be related to the spiralling motion seen in circular geom- etries.

Althought the first stage disturbance was found to occur at different flow rates for different test fluids, it was observed, in all cases, in a very narrow elasticity number range, 0.03 < t < 0.035. At slightly lower elasticity num- bers, a metastable flow region was observed. This region occurred over a small range of t, between 0.025 and 0.03. Four different multiple flow pat- terns were observed at similar flow conditions. Each flow pattern was found to persist for as long as half an hour. Two of the flow patterns were symme- tric while the other two were asymmetric.

Still photographs for the two symmetric flow patterns are shown in Figs. 11(a) and (b). The fluid used, FS7, is the same in both figures. The experi- mental conditions are nearly identical, t = 0.028 and 0.0285 and Re = 157 and 152, respectively. The flow pattern illustrated in Fig. 11(a) shows no sig- nificant stationary vortex, although there may be a small dead space along the wall of the contraction plane. The fluid near the central core flows smoothly into the smaller annulus, while in the vicinity of the 90” comer the streamlines show a severe diverging flow. Fluid elements arriving at the extreme comer accelerate rapidly, along the walls of the contraction plane, into the smaller annulus. No disruption of the streamlines at the entry is evi- dent. This flow pattern is termed rudiul flow regime. In comparison, the flow pattern shown in Fig. llb has a distinct region of recirculating flow at the 90” comer. The bulk of the fluid converges smoothly into the smaller annu- lus and there is only a mildly divergent flow phenomenon near the entry of the contraction. It is interesting to observe that in the region between the bulk flow and the stationary vortex, fluid elements near the outer walls appear to accelerate rapidly into the smaller annulus. The boundary of this region is clearly indicated by the tracer particles. This flow pattern is termed accelerating flow regime.

The asymmetric flow patterns, which may be regarded as mirror images of one another, seem to arise from a combination of the two previous sym- metric flow patterns. One of them exhibits radial flow in the lower half of the flow field with the top half in accelerating flow. The mirror image of this asymmetric flow pattern has the radial and accelerating flows occurring in the top and lower halves respectively. However, it should be noted that as only a two-dimensional view of the flow field is visualised, the four metas- table flow patterns may possibly be just different two-dimensional planar views of the same three dimensional flow pattern. Nevertheless, these metas- table flow regimes have not been reported previously for flow of polymer

. 11. Flow patterm upstream of the contraction for fluid FS7 aho1 at Re = 157, 5 = 0.028 (l/32 s at f/2.8); (b) accelerating flow at Re = (l/32 s at f/2.8).

wing (a) radial flow

152, E = 0.021 85

41

melts and solutions in circular tubes or dies. Radial flow patterns at the entry to a circular die were observed in HDPE and polypropylene, but they only occurred in the stable flow region. Hence, they are probably unrelated to the metastable radial flow regime observed in this study.

Proceeding to higher Reynolds numbers and hence lower elasticity num- bers t < 0.025, a second stage flow instability was obtained. This second stage instability is character&d by high frequency random distortions of the flow field just upstream of the contraction. The flow can be described as chaotic and the frequency of disturbance increases with increasing Reynolds number. In view of the high Reynolds numbers and low elasticity numbers encountered, the chaotic flow regime is clearly an inertia-induced and not an elastic induced flow instability. This is significantly differemt from the grossly disrupted flow patterns reported in polymer melts at very low Reynolds num- bers, where inertial effects are negligible.

3.6 Criteria for the onset of flow instability

Many criteria have been proposed for the onset of flow instability in poly- mer melts and solutions and it is not the intention here to propose yet another one. Instead, an attempt will be made to point out the similarities between the different criteria and the results of this investigation.

An obvious critical condition for the onset of flow disturbance is that which corresponds to the condition at which the first stage instability occurs. However, this critical condition is difficult to identify and therefore it is not a useful criterion. A more easily established critical condition is the maximum vortex size. The divergent flow regime, which occurs following the maximum vortex size, may be considered to be the initial stage of the un- stable flow region since the first stage instability is observed at the upper limit of this flow regime. Thus this criterion represents a conservative limit above which the flow is potentially unstable. This criterion was first sug- gested by Cable and Boger [12] for tube flow.

Of the eight viscoelastic test fluids employed in the flow visualisation study, the critical condition could not be determined for the three least con- centrated solutions on account of the relatively small vortex size. For the other five test fluids, the experimental conditions for the maximum vortex size are summarized in Table 3. Many similarities exist between these critical experimental conditions and those of other investigators involving circular entry geometry. For the majority of polymer melts, the critical wall shear rate which marks the onset of unstable flow is in the range 102-10’ s-i [ 61, whereas the present critical wall shear rates for polymer solutions vary from 66.5 s-i to 220.2 s-l. Results of other investigations using polymer solutions showed a comparable range of critical wall shear rates. For example, Cable and Boger [ 121 using the same Separan solutions as in this study found the critical wall shear rate to be in the range 46.9-187 s-l. Ballenger et al. [25] obtained critical values in the range of 400-600 s-l for a 21% PIB solution,

42

TABEL 3

Summary of experimental conditions for maximum vortex size

Fluid (u) ’ cm/s L-l ) &/m2) N1

X s Re Rem E Wm2)

“.ltUX

FS4 16.8 126.5 14.3 94.5 0.41 6.61 15.80 6.82 0.2722 FS5 13.3 104.5 18.0 103.9 0.56 5.76 7.80 3.11 0.5051 FS6 15.7 124.4 22.7 146.3 0.58 6.45 8.70 3.43 0.4752 FS-7 8.6 66.5 8.8 52.4 0.44 5.94 6.77 2.80 0.6204 FS8 26.5 220.2 32.3 281.9 0.66 8.73 17.40 6.34 0.3340

while Southern and Paul [26] observed a range of about 15-30 s-’ for poly- styrene/benzene solutions. Rama Murthy [24] obtained critical values in the range 40 < 9 G 100 s-l for a 1.49% PAA/glycerine-water solution.

Using a modified Reynolds number criterion for flow stability, Tordella [ 271 found that for polymer melts, the value was much less than unity. With considerably less viscous polymer solutions, the critical values were much higher. Values of the modified Reynolds number obtained in this study varied from 2.80 to 6.82. Vinogradov and Manin [28] found Re, = 7 for alu- minium napthenate solutions, while Rama Murthy [ 241 obtained Re, = 0.76 for his PAA solution. Cable and Boger’s [ 121 critical values were in the range 18.2 < Re, < 49.9 and 1.41 < Re, < 11.6 for 4 to 1 and 2 to 1 circu- lar contractions, respectively.

Many previous investigations of flow instability in polymer melts have cor- related the onset of unstable flow using a recoverable shear criterion. For recovery after steady shear, this parameter has been defined as half of the stress ratio S [ 291. Careful examination of Table 3 reveals that if one ignores fluid FS8, the critical S is nearly constant at around 6. This corresponds to a critical recoverable shear of around 3. Cable and Boger [12] showed that S increased slightly with Reynolds number, and obtained a creeping flow limit (Re + 0) of 5.0 for the critical stress ratio in circular tube flow. Thus, despite the differences in flow geometry and in unstable flow patterns, both circular and annular systems yield similar critical recoverable shear of the order of 2.5-3.0. A comparison with existing tube flow data for polymer melts shows that the present critical value is conservative, i.e., the onset of flow instability for polymer solutions occurs at a much lower value of recoverable shear than for melts. For example, Bagley [ 301 reported critical values to be around 6.4-6.9 for HDPE and 14.5 for LDPE at 190°C. Assum- ing Hooke’s law in shear, Bagley re-analysed Spencer and Dillon’s poly- styrene data [ 311 and obtained a critical recoverable shear of about 7.1-7.7. Using the same assumption, Schott [ 321 estimated critical values of 3.5-6.3 for LDPE and 7.4-8.0 for HDPE; while using Bagley end correction method, he obtained values in the range 5.0-7.0 for LDPE. It should, however, be remembered that the onset of flow instability in this present study has been

43

based on the condition of maximum vortex size and not on the first-stage instability. The critical value would have been larger and more compatible with the polymer melt results if the condition of first-stage instability could be identified precisely.

4. Summary and conclusions

Several flow patterns on the upstream side of the annular contraction were observed. They could be broadly classified into stable and unstable flow regions. The stable region consisted of a vortex growth regime at low Re or high E and a divergent fZow regime at intermediate Reynolds numbers. In the vortex growth regime, the large stationary vortex at the 90” comer of the contraction was found to increase in size with increasing Re. The regime was characterised by a constant macroscopic Deborah number, Ws/X, = 8.0, and was primarily an elasticity-controlled regime. The vortex size reached a maximum at certain critical conditions, beyond which it began to diminish in size with increasing Re. This was the start of the divergent flow regime. In this regime, an increasing dependence of Ws/X, on Re indicated an inertia- controlled process. The flow patterns observed in the stable flow region in the annulus are similar to those of circular flow.

Critical conditions for the onset of flow instability, based on the maxi- mum vortex size, were in qualitative agreement with various proposed crite- ria for polymer melts and solutions in tubular flow. The critical shear rate was of the order of lo2 s-‘, modified Reynolds number of less than 10, and the critical stress ratio of around 6 or critical recoverable shear of 3. It appeared that similar mechanisms governed the flow instability phenomenon for polymeric materials in both annular and circular entry geometries.

The unstable flow patterns in the upstream region showed a two-stage flow instability with a metastable region in between. The first-stage instabil- ity, which was observed over a limited range of elasticity numbers 0.03 < E < 0.035, was marked by a low frequency random disruption of the station- ary vortex. At lower fluid elasticity, two symmetric and two asymmetric metastable flow patterns were obtained at similar flow conditions in a nar- row range, 0.025 < [ < 0.03. A radial flow and an accelerating flow regime, which were symmetric, were identified. The asymmetric flow regimes were mirror images of one another, with radial flow in one half and accelerating flow in the other half of the annulus. These metastable flow regimes have not been observed previously for flows of polymer melts and solutions in circular dies. The second-stage instability was obtained at higher Reynolds numbers and lower elasticity numbers (t < 0.025). In view of the high Rey- nolds numbers encountered, fluid inertia is likely to be predominant in this flow region. As distinguished from the first-stage instability, the stationary vortex in this flow region had completely disappeared and the flow was chaotic with high frequency random distortions in the flow field, just up-

44

stream of the contraction. In contrast, the flow field for an inelastic fluid at similar Reynolds numbers was found to be stable.

Acknowledgements

The authors wish to acknowledge the financial support received from the Australian Research Grants Committee. One of us (K.L. Tan) also wishes to thank Monash University for the award of Graduate Scholarships during the course of his postgraduate study.

Notation

K, n, s, c L” N1 =Pll -Pz2

pii

rl r2

R2

Rv

rH

RH

S U

(24)

“x

Rl?

Rem ws 60, El ?

;

P

References

power law parameters vortex detachment length first normal stress difference deviatoric normal stresses, i = 1,2 inner radius of downstream annulus outer radius of downstream annulus outer radius of upstream annulus radius of the core region in vortex zone (see Fig. 1) (r2 - r,)/2, hydraulic radius of downstream annulus (R2 - r,)/2, hydraulic radius of upstream annulus stress ratio defined in eqn. (6) local velocity average velocity in downstream annulus; subscript u and ZJ refer to upstream and vortex zone, respectively. axial distance dimensionless detachment length defined in eqn. (8) Reynolds number defined in eqn. (3) modified Reynolds number, p( u j2* (4rH)“/K Weissenberg number defined in eqn. (4) geometric parameters steady shear rate steady shear stress relaxation time defined in eqn. (7) density

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