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Journal of Non-Newtonian Fluid Mechanics, 3 (197711978) 25-40 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

ENTRY FLOW BEHAVIOUR OF VISCOELASTIC FLUIDS IN AN ANNULUS

K.L. TAN and C. TIU

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

(Received January 13, 1977; in revised form March, 24,1977)

Summary

Developing and fully developed velocity profiles in the entrance region of an abrupt Z-to-1 annular contraction were measured for a number of visco- elastic polymer solutions. Experimental results were obtained for Reynolds number and flow behaviour index in the range 9.8 < Re < 355 and 0.372 G n < 0.55 respectively. A momentum-ener~ integral technique was employed in the boundary layer analysis. The deviation from inelastic behaviour was indicated by the ratio of elastic to inertial forces, Ws/Re. Within the limits of confidence of the experimental results, good agreement with theoretical predictions was obtained and very little deviation from inelastic behaviour was observed for aware < 0.08. For the test fluids investigated, the entrance length was found to be longer than that predicted for the corresponding inelastic fluids of the same n.

1. Introduction

Boundary layer flows of viscoelastic fluids in complex geometries have been studied extensively. Most of the work reported in the literature is theo- retical. In view of the complexity of the kinematics of the flow field, most analyses have made use of approximate constitutive equations, and the resulting boundary layer equations have been solved by either a pe~urbation technique or the classical Von Karman momentum integral method. Very few experimental measurements with independent characterisation of fluid properties are available to substantiate the various approximate solutions. In the entry flow of viscoelastic fluids in ducts, conflicting experimental results

26

have been reported concerning the effects of fluid elasticity on entry length and excess pressure drop in the inlet region. To name a few, Sutterby [l] and Berman [2] have found no elastic effect in the laminar flow of dilute polymer solutions in the entry region of a conical and circular duct respec- tively . Sakiadis [ 3 ] observed experimentally larger pressure losses and longer entrance length than those predicted by inelastic theory for the entry flow of viscoelastic fluids in capillary tubes. Bilgen [4] measured higher pressure drops for dilute polyox solutions in pipe flow. Ramamurthy and Boger [ 51 reported a shorter entry length for viscoelastic fluids, but a substantial pre- development of velocity at the inlet contraction had been observed. Busby and MacSporran [ 191 also report shorter entry lengths for viscoelastic fluids in the case of flow into a re-entrant tube. Brocklebank and Smith [6] observed a longer entry length in their measurements of developing velocity profiles of viscoelastic fluids downstream of a distributor in a pipe. They found that elastic effects influenced the flow field at two different time scales giving rise to an elastic “solid-like” behaviour near the distributor and a “fluid-like” behaviour further downsteam. Their findings qualitatively sup- port the theoretical predictions of Metzner et al. [ 7,8]. It is obvious that more quantitative experimental work, in particular accurate measurements of velocity profiles, coupled with independent fluid property measurements is needed to establish the conditions under which elastic effects are con- sidered to be important.

Recently, theoretical and experimental work dealing with the flow of inelastic power-law fluids in the entrance region of an annulus have been presented [g-11]. The momentum integral and the momentum-energy inte- gral techniques have been employed to solve the boundary layer equations. The latter technique has been shown to give more conservative and accurate results. In the present study, the boundary-layer solution using the momen- tum-energy integral technique is extended to investigate the entry flow behaviour of viscoelastic fluids. The model proposed by White and Metzner [ 12,131 is used in the theoretical analysis. The validity and accuracy of the theoretical solution is established by comparing with experimental measure- ments of both developing and fully developed velocity profiles in an abrupt 2-to-1 annular contraction.

2. Analysis

For a two-dimensional velocity field of the form

u = u(x, r), u = (x, r), w = 0, (1)

the boundary layer equations describing the flow in the entry region of an annulus are given, in cylindrical coordinates, by

Continuity :

(2)

27

Motion:

x-camp. P~+U~)=-~+=(ITn)+~, (3)

r-camp. (-j f !$ + “,“1+ kZ_!?! . (4) r

Since the growth of boundary layers in an annulus is asymmetric, the axis of maximum velocity being closer to the inner core, each boundary layer must be considered separately. Integrating eq. (4) from the walls r = rl or r2 to any radial position I” = r, and assuming the Weissenberg hypothesis, t,, - toe = 0, gives

-~(x, r) = rrrw - t&x, r), (5)

where the normal stress rji is the sum of the hydrostatic pressure p and the corresponding deviatoric stress tit,

rii = -p + tii (i = x, r, 0) (6)

Eliminating the pressure gradient in eq. (3) with the aid of eq. (5) results in

(7)

Contrary to the Von Karman momentum integral method normally used in solving the boundary layer equations, a momentum-ener~ integral method first introduced by Campbell and Slattery [ 141 for Newtonian flow in pipes in used in this analysis.

A mechanical energy balance over a differential length dx is obtained by multiplying eqn. (7) by the axial velocity U, and integrating the resulting equation over the entire flow cross-section,

dr,,. Q w s

’ d r2 &.dr _ ,:’ a urdr =- - dx 2 dx s J u ar (rcx M-

‘1 ‘1 ‘1

The radial velocity u and the term dr,,w /dx in eq. (7) are eliminated with the aid of eqs. (2) and (8) respectively. Integrating the resulting equation over the cross-sections of the two boundary layers yields the following pair of boundary layer equations:

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Inner boundary layer:

r1+61 pu2rdr - UzJ pur dr

‘1

t- rlT1 - s r1

;f);x (t,, - W r dr] -

Outer boundary layer:

dT,rw _ 2 ~ - ___._~

dx (2r26, - 63)

p2

+ r2r2 - f a (txx - t,,)rdr . r2-62 ax I

(9)

In order to solve the two boundary layer equations simultaneously, it is necessary to specify velocity profiles inside the boundary layers and a con- situtive equation relating shear stress and first normal stress difference in terms of kinematic variables.

A second-order velocity profile of the form

u/u=2y-ya, (11)

where

y = (r-r,)/& (inner boundary layer) and

y = (r2 - rW2 (outer boundary layer) (12)

is assumed, which satisfies the following boundary conditions:

u/U=Oaty=O and

UlU = I, a(du)/ar = 0 at y = 1. (13)

White and Metzner’s [ 12,131 constitutive equation is adopted in the pre- sent analysis. Following the usual boundary layer approximations, it can be shown that the stress components reduce to

(Trx)w = (L), =+y ,

t 5% -tr~=2+~)+-=o(-~)e.

(14)

(15)

The final boundary layer equations are recast in dimensionless form by in-

29

traducing

Reynolds number

Z3-” rk ( u)~-% inertial force Re =

K[ eonn+ cl] n = viscous force ’ (16)

Weissenberg number

a (u) s--n

ws=z, [ 1 elastic force = __ _.__- (17)

H viscous force *

The hydraulic radius of the annulus is given by rH = r2 (1 - h)/2. Reynolds number is defined in such a way that f = 16/Re for any aspect ratio of the annulus characterised by the geometric parameters ee and c1 [15]. The boundary layer equations were solved numerically by the Adams-Moulton technique with the aid of a computer. The entrance length, defined as the distance from the inlet contraction to the point where the maximum veloc- ity reaches 98% of its fully developed value, was determined from the know- ledge of boundary layer thicknesses. The complete theoretical analysis will be presented elsewhere.

3. Experimental details

Developing and fully developed velocity profiles were measured in the entrance region of a 2-to-1 annular contraction. The flow loop used in this study for flow visualisation measurements has been described in detail else- where [ 11,161. Only minor modifications were required for the present experimental investigation.

Fig. 1 shows the details of the entrance section of the annulus following a 2-to-1 contraction. The outer and inner tubes of the test section upstream and downstream of the contraction were made of precision-bore glass of 76 mm and 38 mm inside diameter, respectively. The inner core was a 16 mm o.d. stainless steel tube extended throughout the entire test section. The aspect ratio downstream of the contraction was 0.42. A smooth 74 mm long conical perspex section with a converging angle of 15.9” and a flow straightener made up of a bundle of small copper tubes were inserted upstream of the contraction in order to generate a uniform entry velocity profile. A flat entry profile was essential if experimental results were to be used for comparison with the boundary layer solution.

The 2-to-1 abrupt change in diameters of the annulus from 76 mm to 38 mm was achieved with the aid of a 295 mm long cast perspex rod. A hole of 38 mm diameter and 141 mm long was machined in the front section of the rod. The rear of the rod was machined to fit the smaller glass tube so that a continuous smooth tube of 38 mm inside diameter was obtained. A circular

30

6mm

STRAIGHTENER

Fig. 1. Schematic diagram of the annular entry.

butt of length 6 mm was retained near the centre of the perspex rod to help in positioning the two outer sections of precision-bore glass tube. The outer surface of the perspex rod was sealed to the glass tube with O-rings. The small gap spaces between the glass tubes and the perspex rod were filled with a viscous silicone oil to maintain optical clarity. The annular space down- stream of the perspex section, between the 76 mm and the 38 mm glass tubes was filled with distilled water.

A technique employing streak photography was used for point velocity measurements. Details of the optical system have been described earlier [ll, 161. Basically, the motion of tracer particles dispersed in the test fluid was photographically recorded using a double flash technique. With a knowledge of the time interval between flashes, the system dimensions and magnifica- tion, point velocities were determined using an X-Y data reader.

Dilute aqueous solutions of Methocel90-HG (hydroxypropyl-methyl-cel- lulose), Separan MG-500 and AP-30 (partially hydrolyzed polyacrylamide), all from Dow Chemical, and Cellulose Gum (sodium carboxymethyl cellu- lose) from Hercules Co. were used as the test fluids. Fundamental fluid properties, shear stress and first normal stress difference as functions of shear rate, were measured on an R-16 Weissenberg rheogoniometer fitted with a cone-and-plate assembly.

4. Experimental results

Shear stress-shear rate and first normal stress difference-hear rate data were measured for all test fluids over a shear rate range of 4 set-’ to 1112 set-’ . All the experimental runs conducted in the annular test section were within this range of shear rates. The normal stress data were all corrected for inertial effects [ 171. All test fluids exhibited two distinct regions of flow

31

IO3 II, I II,, I, I II‘

- 1.0% METHOCEL

s = 0.54

to*- S’l.01

“E ‘; n = 0.49

bZ /-1 ; .

IO’ - n = 0.72

Fig. 2. Rheological properties for the 1.0% Methocel solution.

IO3

IO2

El .- _I

T IO’ 3bJ

IO0

curves, each of which could be approximately fitted with a power-law model. A typical plot of the flow curves obtained for the 1% Methocel solu- tion is shown in Fig. 2. The value of n changed from 0.72 in the low shear rate region to 0.49 in the high shear rate region. The variation of K over the same regions was 1.45 Ns”/m2 to 3.88 Ns”/m2. Similarly, s changed from 1.01 to 0.54, and u from 0.68 Ns”/m2 to 7.82 Ns”/m2 between the low and

L_ r2

0.6

0.28% SEPARAN AP-30

- - THEORY

---- EXPT

Fig. 3. Velocity distributions for the 0.28% Separan AP-30, without the flow straightener

and the conical entry section.

32

Q8

..........- 5.0% ERROR SAND

Q6 - VISCOELASTIC

--- INELASTIC

Fig. 4. Velocity distributions for the 0.50% Separan MG-500 solution.

the high shear rate regions. The power-law parameters were obtained from a least squares fit of the rheological data. During the course of each experi- ment no significant shear degradation was observed for all fluids as indicated by the viscometric measurements before and after each experimental run. Each fluid was characterised at the same temperature as encountered in the flow experiment. The temperature varied from 15°C to 23 + 0.5”C.

It is important to use the fluid parameters evaluated at the same shear rates as those encountered in each experiment. Table 1 lists the relevant rheological constants for the test fluids together with experimental condi- tions under which the fluid parameters were evaluated.

1.0

0.8 -

_L r2

0.6 - -~~~~~~~.~ 50% ERROR EAND

- VISCOELASTIC

-- INELASTIC

0 u3 o 1.0 1.0

Fig. 5. Velocity distributions for the 0.75% Separan MG-500 solution.

33

0.25% SEPPRAN MG-500

.

al3 - x= Q2cm x= 2.35an

r

r2 x = 0.001 : ! x= 0.0119 * 1.

I:

a6 - . .Z

- VISCOELASTIC b

--- INELASTIC 1

1

a42- _ __2--

0 a5 1.0 0 05 ‘1.0

u <u>

Fig. 6. Velocity distributions for the 0.25% Separan MG-500 solution.

The fluid relaxation time 19, is calculated from the equation

Ll - Lx - 67 “f -

2i7, ’

which is derived from the Maxwell fluid model. The Deborah number is defined as the ratio of fluid relaxation time to process time. All quantities tabulated in Table 1 were based on the fully developed downstream wall conditions.

Figure 3 shows developing and fully developed velocity profiles.

(18)

0.8

L ‘2

CI6

a42

2.0% CELLULOSE GUM

- - VISCOELASTIC

INELASTIC . -_--

a5 0

1.0 a5 1.0 1.5

Fig. 7. Velocity distributions for the 2.0% Cellulose Gum solution.

TA

BL

E

1

Flu

id

prop

erti

es

at e

xper

imen

tal

con

diti

ons

Flu

id

n K

s

( Nsn

/m2)

yN

ss/m

2)

fmse

c)

(U)

Re

Wsl

Re

De

cm/s

2.0%

C

ellu

lose

G

um

1.

0%

Met

hoc

el

0.28

%

Sep

aran

A

P-3

0 0.

25%

S

epar

an

MG

-500

0.

5%

Sep

aran

R

IG-5

00

0.75

%

Sep

aran

M

G-5

00

1.0%

S

epar

an

MG

-500

0.55

2.

862

0.47

4.

449

0.51

0.

448

0.54

7 0.

344

0.47

6 0.

941

0.42

3 2.

043

0.37

2 3.

878

--

0.69

8 1.

837

1.2

104.

0 82

.0

0.01

69

0.01

62

0.63

5 4.

485

6.5

25.1

9.

8 0.

193

0.02

12

0.85

5 0.

54

- 70

.1

342.

0 0.

017

- 0.

889

0.42

1 11

65

.4

355.

0 0.

0176

0.

094

0.66

4 3.

488

11

74.5

24

1.0

0.03

87

0.10

8 0.

703

4.76

14

.2

59.5

10

4.0

0.03

29

0.11

0 0.

806

3.31

1 14

.6

47.3

50

.0

0.11

75

0.09

35

1.0% SEPARAN MG-500

cl8 -

0.6 -

- VISCOELASTIC

---- INELASTIC

0 0.5 1.0

&

Fig. 8. Velocity distributions for the 1.0% Separan MG-500 solution.

obtained for a 0.28% Separan AP-30 solution without the flow straightener and the conical entry section upstream of the contraction. A representative set of developing velocity profiles in the annulus with flow straightener and conical entry upstream is shown in Figs. 4 to 9. The solid curves in the devel- oping region represent the theoretical predictions from the viscoelastic boundary layer analysis. The fully developed profiles are those obtained by Fredrickson and Bird [ 181. The broken-line profiles represent the inelastic solutions for power-law fluids of the same n values [9]. Dotted line profiles

1.0

1.0% METHOCEL

Q8 -

k x*o.2cm .

.’ %

x= 00367’ ** , I

x’23an. .

a6- - VISCOELASTIC

*; /

--- INELASTIC

Fig. 9. Velocity distributions for the 1 .O% Methocel solution.

36

in Figs, 4 and 5 indicate the 55% confidence band around the viscoelastic theoretical profiles.

5. Discussion

Accuracy of point velocity measurements

In order to establish the accuracy of point velocity measurements, fully developed velocity profiles were measured for all test fluids over a wide range of flow rates under laminar flow conditions. Since deviator&z normal stresses play no part in fully developed flow, the velocity profile for visco- elastic fluids is identical to that for inelastic fluids. Thus, the inelastic power- law velocity expression of Fredrickson and Bird, which had been shown to give good agreement with experiments [ll], was used for comparison with the present experimental data. Theoretical and experimental fully developed profiles are shown in all the figures except Fig. 5. It can be seen from Figs. 4 and 5 that almost all the experimental data points fit within a 5% confidence band (dotted profiles) around the theoretical fully developed profiles. The same level of confidence is also applicable to experimental data points in the developing region. without exception, the agreement between experimental data points and theoretical profiles for all other fluids was as good or better than the +5% confidence limit. The scattering of data points is attributed to experimental errors involved in the data analysis of the film.

Another independent check on the accuracy of experimental data was achieved by graphically integrating the measured velocity profiles and com- paring the result with the separately measured average velocity, Again the ~~~rne~t was within +5% for all fluids. Thus, the accuracy of the point velocity measurements has been justified and the 2 5% confidence limit serves to establish the level of errors involved in the comparison between the theoretical and experimental results,

The effect of the entry geometry on the velocity profile in the annular entrance region is clearly illustrated in Figs. 3 and 6. Ramamurthy and Boger [ 51 has reported that certain viscoelastie fluids exhibited a substantial veloc- ity predevelopment at the inlet position through an abrupt circular eontrac- tion. Consequently, a shorter entrance length was obtained. Figure 3 shows the velocity profiles in the entrance region of the annulus for a 0.28% Separan AR-30 solution at Re = 342, without the flow straightener and the conical entry section upstream of the contraction. A significant predevelop- ment of velocity is seen to occur at a distance of 0.9 cm from the inlet con- traction. At this location, the predicted velocity profile from the boundary layer analysis is still rather flat. The experimental velocity profiles develops much faster because of the pre-development occurring at the inlet. Hence, a

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shorter distance is required for it to attain the fully developed value. This is in qualitative agreement with Ramamurthy and Boger’s results. At x = 9.2 or x = 0.0484, both theoretical and experimental results indicate that the fully developed flow region has been reached.

On the other hand, Fig. 6 shows two developing velocity profiles for a 0.25% Separan MG-500 solution at Re = 355. In this case, the flow straight- ener and the conical entry section have been inserted on the upstream side of the annular contraction, as shown in Fig. 1. The two fluids presented in Figs. 3 and 6 have very similar viscoelastic properties, and almost identical flow conditions (see Table 1). For the latter fluid, the velocity profile very close to the inlet position, x = 0.2 cm, is nearly flat. This condition satisfies the assumption imposed in the boundary layer analysis. Even at a distance of x = 2.35 cm, the profile for the latter is more blunt than the former mea- sured at x = 0.9 cm. It can be concluded that the present experimental set- up achieves its aim of generating a uniform entry profile which is critical for a valid comparison between theoretical predictions and experimental results to be made.

Comparisons between the theoretical predictions and the experiment are shown in Figs. 4 and 9. Both inelastic and viscoelastic solutions, represented by broken and solid lines respectively, are included in these figures. It can be observed that, as the ratio of elastic to inertial forces Ws/Re increases, the influence of fluid elasticity becomes more pronounced. The theoretical vis- coelastic profile deviates increasingly from the inelastic profile. For all the test fluids employed in this study, the parameters n and s are less than unity. The experimental results indicate that all the profiles develop slower than the inelastic prediction, and hence a longer entrance length for these fluids is required using this comparison. This agrees qualitatively with the theoretical prediction for viscoelastic fluids. It is also interesting to compare the degree of velocity development shown in Figs. 6 and 7. At a distance of about 2.3 cm from the inlet, the profile for the 2.0% Cellulose Gum is already fully developed, whereas that for the 0.25% Separan MG-500 is still in the early stage of development. Table 1 shows that both test fluids have almost iden- tical n and Ws/Re values but the consistency index K for the 2% Cellulose Gum solution is about 8 times larger than the 0.25% Separan MG-500 solu- tion. Consequently, the former fluid has a smaller Reynolds number than the latter. Both fluids should yield the same dimensionless entrance length x or x/r,Re, but the actual distance x for the 0.25% Separan solution to attain the fully-development would be more than 4 times longer than the Cel- lulose Gum solution in view of the difference in Reynolds numbers.

The theoretical analysis predicts a shorter entry length for viscoelastic fluids with s > 1.0 even when IZ is less than 1.0. Unfortunately, this behav- iour could not be substantiated experimentally in the present study. Under the present experimental conditions, it is difficult to find a viscoelastic fluid which falls within that range of n and s.

Figures 4,6 and 7 indicate that there is very little difference between the

38

predicted viscoelastic and inelastic velocity profiles, at low values of Wsl~e. The difference is well within the level of scatter in the experimental data as shown by the 5% confidence band for the developing velocity profiles in Fig. 4. This suggests that both solutions can be adequately used to describe the entry behaviour of very dilute polymer solutions in an annulus. For engineering purposes, the simpler inelastic solution could be used to describe the behaviour of very weakly viscoelastic fluids, provided that the inlet velocity profile is uniform. However, this is not the case for the more con- centrated polymer solutions, as seen from Figs. 5, 8 and 9. For most visco- elastic fluids, the fluid elasticity generally increases with increasing concen- tration. This is manifested with a higher value of Ws/Re at a particular flow rate. It can be observed from Fig. 5 that the predicted inelastic velocity pro- file at x = 0.0035 for the 0.75% Separan solution lies outside the 5% eonfi- dence band drawn around the predicted viscoelastic profile. Almost all the experimental data fall within the 5% confidence region. Hence, there is a significant difference between the predicted inelastic and viscoleastic veloc- ity profiles for higher value of Ws/Re. Figures 5,8 and 9 show better agree- ment between the experimental results (+5% confidence limit) and the pre- dicted viscoelastic theory at higher values of WsfRe (0.0829,0.1175 and 0.193 respectively). Note that both Ws and Re change with flow rate, although not to the same extent depending on the values of n and s. To achieve higher values of Ws/Re or to increase the effect of elasticity would mean to decrease Re substantially without altering Ws too much. One must be careful in doing this because of the assumptions involved in the boundary layer analysis. The boundary layer solution is not expected to be valid for low Reynolds number flow when the axial diffusion of momentum becomes significant. One interesting result was found with the 1.0% Methocel solution at Re = 9.8 as shown in Fig. 9. This Reynolds number is in the range where the validity of the theoretical solution becomes questionable. Nevertheless, good agreement is obtained between the predicted viscoelastic solution and experimental results. This suggests that the present boundary layer solution may be valid for Reynolds numbers as low as 10.

Quantitative comparisons of experimental and theoretical entrance length could not be made here. In any experimental run, only two to three devel- oping velocity profiles were measured in the entrance region because of the short distance involved. To establish the experimental entrance length requires an extrapolation of the experimental profiles to the 98% fully devel- oped value, This would involve a least squares fitting of data points. In view of the scattering of experimental data around the theoretical curve and the degree of accuracy involved, representing the experimental data with a best- fitted line is not warranted.

6. Conclusions

(1) In order that the theoretical analysis could be experimentally substan- tiated, it was necessary to generate a flat entry velocity profile with the aid

39

of a flow straightener followed by a smooth conical entry section on the upstream side of the contraction. An abrupt entry caused a significant pre- development of velocity at the inlet plane due to the upstream axial diffu- sion of momentum, even for Reynolds number as high as 342.

(2) At very low Ws/Re, the entrance behaviour of dilute solutions in an abrupt 2-to-1 annular contraction could be adequately predicted from the inelastic theory; while at higher Ws/Re (>0.08), the experimental velocity profiles were in good agreement with the predicted viscoelastic profiles to within an error band of +5%.

(3) The experimental results indicated that the entrance length for visco- elastic fluid was longer than predicted for the corresponding inelastic fluid of the same n.

Nomenclature

De Deborah number

f friction factor k aspect ratio K, m, n, s power-law parameters defined in eqns. (14) and (15)

P 5, r2

I;Ek

tii

t

u <U>

u, v, w ws X, r, e Y

613 62

Eo, El

P u

7, X

Y

0,

rii

7 rrw

pressure inner and outer radii of an annulus hydraulic radius Reynolds number defined by eqn. (16) deviatoric stress component; i = x, r, 8 time free-stream velocity outside the boundary layers average velocity velocity components Weissenberg number defined by eqn. (17) cylindrical coordinates dimensionless distance defined by eqn. (13) inner and outer boundary layers geometric parameters of the annulus density parameter given in eqn. (15) shear stress; subscripts 1 and 2 denote inner and outer walls x/r,Re, dimensionless axial distance du/dr, shear rate fluid relaxation time normal stress components, i = X, r, e wall normal stress

Acknowledgement

The authors wish to acknowledge the financial support received from the Australian Research Grant Committee.

40

References

1 J.L. Sutterby, Trans. Sot. Rheol., 9 (1965) 227. 2 N.S. Berman, A.1.Ch.E. J., 15 (1969) 137. 3 B.C. Sakiadis, A.1.Ch.E. J., 8 (1962) 317;9 (1963) 706. 4 E. Bilgen, Trans. ASME, J. Appl. Mech., 40 (1973) 381. 5 A.V. Ramamurthy and D.V. Boger, Trans. Sot. Rheol., 15 (1971) 709. 6 M.P. Brockiebank and J.M. Smith, Rheol. Acta, 9 (1970) 396. 7 A.B. Metzner and J.L. White, A.1.Ch.E. J., 11 (1965) 989. 8 A.B. Metzner, J.L. White and M.M. Denn, A.1.Ch.E. J., 12 (1966) 863. 9 C. Tiu and S. Bhattacharyya, Can. J. Ch. E., 51 (19’73) 47.

10 S. Bhattacharyya and C. Tiu, A.1.Ch.E. J., 20 (1974) 154. 11 C. Tiu and S. Bhattacharryya, A.1.Ch.E. J., 20 (1974) 1140. 12 J.L. White and A.B. Metzner, J. Appl. Polym. Sci., 7 (1963) 1867. 13 J.L. White and A.B. Metzner, A.1.Ch.E. J., 11 (1965) 324. 14 W.D. Campbell and J.C. Slattery, Trans. ASME., J. Basic Eng., 85D (1963) 41. 15 W. Kozicki and C. Tiu, Can. J. Ch. E., 49 (1971) 562. 16 S. Bhattacharrya, Ph.D. Thesis, Monash Univ., Australia (1973). 17 K. Walters and ND. Waters, Polymer Systems: Deformation and Flow, McMillan,

London, 1968. 18 A.G. Fredrickson and R.B. Bird, Ind. Eng. Chem., 50 (1958) 347. 19 E.T. Busby and W.C. MacSporran, J. Non-Newtonian Fluid Mechanics, 1 (1976) 71.