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Fibre spinning of a weakly elastic liquid
Journal of Non-Newtonian Fluid Mechanics, 27 (1988) 349-362 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
349
FIBRE SPINNING OF A WEAKLY ELASTIC LIQUID
T. SRIDHAR Department of Chemical Engineering, Monash University, Victoria, 3168 (Australia)
Wellington Road, Clayton,
and R.K. GUPTA Department of Chemical Engineering, State University of New York, Buffalo, N.Y. 14260 (U.S.A.) (Received October 28, 1987; In revised form December 15, 1987)
Summary This paper presents a study of a silicone oil (poly(dimethy1 siloxane)) in extensional deformation using an instrument developed recently by the authors. Data from steady shear and low amplitude sinusoidal deformation of this liquid clearly establish that it is weakly elastic. The viscometric data, for shear rates less than 100 s-l, are best represented by either the Maxwell model or the Jeffrey’s model, the latter being marginally superior. The extensional data show that at low deformation rates, this fluid exhibits a Newtonian behavior with an apparent extensional viscosity equal to three times the shear viscosity. Under these conditions the velocity profiles along the spinline are also well represented by the Newtonian model. However, at higher deformation rates better predictions of the velocity profiles are obtained from the Jeffrey’s and Maxwell models. At deformation rates above 100 s-l none of these simple models is adequate. Under the conditions used in these experiments, the fractional increase in tensile stress along the fiber is shown both theoretically and experimentally to be a unique function of the total strain. Furthermore, the apparent extensional viscosity at any point on the spinline can be calculated from steady state expressions if allowance is made for the variation of stretch rates by defining a time averaged stretch rate. The results obtained here show that elasticity must be considered if these model liquids are used to conduct rheological experiments at high deformation rates. Additionally, it is found that elastic effects in extension can be predicted using simple constitutive equations provided viscometric data can be represented properly in the deformation rate range of interest. Finally, 0377-0257/88/$03.50
0 1988 Elsevier Science Publishers B.V.
350 the present research further substantiates viscometer developed by the authors.
the utility
of the extensional
Introduction
A study of the extensional flow of polymeric liquids is important not only from the viewpoint of advancing our understanding of the constitutive behaviour of these materials but also for the purpose of polymer characterization. However, experimental difficulties associated both with stretching low viscosity liquids and with measuring the resulting extensional stress had slowed progress in this area. Recently, the authors [1,2] proposed a new instrument for measuring extensional viscosities of mobile liquids, which utilized a suction technique for extending a liquid filament. The new instrument has been used to investigate the behavior of ideal elastic liquids [3,4] and show that their behavior in extension is well represented by the Oldroyd model B; this confirmed the hypothesis of Prilutski et al. [5] which was based on shear data. In this paper, we use the same device to examine the behavior of another ideal liquid-silicone oil, a liquid that is generally used as a viscosity standard but which is considered to have a very small relaxation time (fluid elasticity). There are numerous similarities in the stress response of the ideal elastic liquids and silicone oils, and we draw heavily on our previous work [3,4]. For a discussion on the merits of various elongational rheometers, we refer the reader to a recent review [6]. Viscometric data
The fluid used in this work was a poly (dimethyl siloxane) or silicone oil obtained from the Dow Corning Company. Steady shear and dynamic data were obtained on a Weissenberg rheogoniometer model R-19 equipped with cone and plate fixtures. These data, which are shown in Fig. 1, clearly establish that the fluid viscosity is constant at 30 Pas and the first normal stress difference is quadratic in the shear rate up to a shear rate of approximately 50 s-l. Furthermore, the loss modulus, G”, is found to superpose with the shear stress while twice the storage modulus, 2G’, superposes with the first normal stress difference. The viscometric data can be represented by the upper convected Maxwell model [7]:
351
‘04C
10"
10-l
““.‘I
100
““‘.’
FREQUENCY,
10'
““‘.I
102
““‘C
103
SHEAR RATE (l/s)
Fig. 1. Steady shear data for silicone oil showing forces. 0 shear stress, A G”, H 2G’ and A NI.
or the upper convected Jeffrey’s element in parallel):
constant
viscosity
and quadratic
normal
model [7] (a Maxwell and Newtonian
(2) where T is the extra-stress tensor, D the deformation rate tensor, S/St the upper convected Oldroyd derivative, h or A,, the model relaxation time, A, the model retardation time and n the liquid viscosity. The relaxation time calculated according to eqn. (1) is 0.014 s while the relaxation and retardation times according to eqn. (2) are 0.0206 s and 0.0065 s, respectively. On comparing these relaxation times to a relaxation time of about 0.824 s for a constant viscosity elastic fluid of about the same shear viscosity [4], it is seen that the silicone oil used here is very weakly elastic.
352
FREQUENCY (l/s)
Fig. 2. Dynamic data for silicone oil. Predictions of the storage modulus using different constitutive equations. H 2G’, eqn. (3) and - - - eqn. (4).
The storage modulus for these two models can be calculated as G’ (Maxwell) =
G’ (Jeffrey) =
qho2
(3)
l+(hw)2’
77(X1-
h2b2
1 + (x,w)2
*
(4
The predictions of eqns. (3) and (4) are compared with experimental data in Fig. 2 and both models are found to be satisfactory. Since (h, - X2) in eqn. (4) is numerically identical to h in eqn. (3), the two equations predict the same results at low frequencies. Differences arise only when the frequency approaches 100 s-l and here the Jeffrey’s model seems to be superior. These two models are now used to analyze fiber spinning data for this liquid. Spinning experiments Spinning data were obtained using the extensional viscometer developed by the authors. A sketch of the set-up is shown in Fig. 3 while details of the
353
___________________. vlsCOMElER
i-t
: : :
’
: ;
:
;
SPINLINE
TAKE UP
+ VACUUM Fig. 3. Sketch of the Extensional Viscometer.
instrument are available elsewhere [2,4]. A suction device was used to generate the extensional flow field, and the tensile stress at the capillary exit was measured as an apparent decrease in pressure drop across the capillary. All the data were obtained at room temperature (around 20” C), and experiments were conducted over a wide range of conditions. Experimental measurements of diameter profiles were combined with measured flow rates to obtain velocity profiles and stretch rates along the spinline; the spinline was long enough that a consideration of end effects was not necessary. Figure 4 shows three stretch rate profiles; details corresponding to these experiments are given in Table 1. In each case the stretch rates increase to a maximum value, the maximum stretch rates encountered were above 100 Note that with such a viscous liquid, stretch rates of this magnitude are S -I. not accessible using the conventional rotating drum stretching technique. Tensile stresses along the spinline can be calculated with the help of a momentum balance which accounts for the effects of gravity, inertia and surface tension. The final result, in differential form, is [8]
0.0
0.2
0.4
0.6
0.8
1.0
DISTANCE(-) Fig. 4. Typical variation of stretch rates along the fibre. W run # 5, Cl run #6 and A run #7.
where y is the coefficient of surface tension, g the acceleration due to gravity and D the fiber diameter. In general, it was found that the net contribution to the force balance of terms due to inertia, surface tension and gravity was negligible. The net contribution of these terms was found to be less than 5% except for experiments 2 and 7 where the maximum contribution was 15% and 11% respectively. Consequently, the spinning takes place under conditions of constant force. Nonetheless, tensile stresses along the TABLE 1 Details to the experimental measurements in spinning experiments Run #
Flow rate cm3/s
Initial velocity cm/s
Filament length mm
AP Pa
Draw ratio
1 2 3 4 5 6 7
0.149 0.162 0.162 0.162 0.169 0.168 0.168
0.628 0.564 0.62 0.66 0.78 0.62 0.67
5.5 11 10.5 8.3 7.3 9.7 8.7
430.6 155.7 234.3 280.3 397.3 269.1 191.8
18.2 19.9 54.6 30.2 41.8 51.1 12.7
355
0’
0.0
a
0.2
a
0.4
’ 0.6
’ 0.0
m
1.0
DISTANCE (-)
n) calculated using eqn. Fig. 5. Tensile stress along the spinline for experiment #7 (W(5). Dotted line represents the tensile stress calculated by assuming the force to be constant. filament length were calculated using a finite difference analog of the above equation. A typical profile, corresponding to run #7, is shown in Fig. 5. Clearly the tensile stresses show no evidence of reaching a steady state, primarily due to the fact that the stretch rate increases for most of the spinline length. Figure 5 also shows the tensile stress calculated under the assumption that only the viscoelastic terms are dominant. It is evident that the net contribution of inertia, gravity and surface tension is indeed negligible. In the the absence of inertia, surface tension and gravity, the velocity profiles (or equivalenty the stress profiles) can be calculated with ease for any of the fluid models of interest-Newtonian, Maxwell or Jeffrey’s. For a Newtonian liquid, when viscous stresses are dominant, the dimensionless velocity, U, is related to the dimensionless distance, S, as 24= exp(S In DR), where Da is the draw ratio. Also, the stress at the spinneret is
(6)
356
0.0
0.2
0.4
0.6
0.8
1.0
1.2
DISTANCE (-)
Fig. 6. Prediction of velocity profiles at low deformation rates; experiment #7. Maximum stretch rate is 20 s- ‘. Solid line: Jeffrey, 0 -0 Maxwell, AA Newtonian and H experimental values.
where L is the spinline length. From a knowledge of the measured stress at the spinneret, one can calculate the draw ratio from eqn. (7); insertion of this quantity into eqn. (6) then yields the predicted velocity profile. The equations for the computation of the velocity profiles for the fiber spinning of a Maxwell liquid have been published by Denn et al. [9] while those for the Jeffrey’s liquid (also called the Oldroyd fluid B) have been provided by Gupta et al. [3]. Consequently, these will not be repeated here. We point out, though, that the Jeffrey’s model is a linear combination of the Newtonian and the Maxwell models so that it can exhibit both extremes in behavior. The equations for the Maxwell and Jeffrey’s models were solved numerically using the Gear method, and the results for the dimensionless velocity as a function of the dimensionless distance are shown in Figs. 6-8. Also shown are the experimental data points corresponding to experiments 5-7. These runs were chosen so as to span a wide rang of stretch rates. In Fig. 6, maximum stretch rate only 20 s-i and, not surprisingly, at these low deformation rates, the Maxwell, Jeffrey’s and Newtonian models give essen-
357 70
c-i.00
0.20
0.40
0.60
0.60
1.00
1.20
DISTANCE(-)
Fig. 7. Prediction of velocity profiles to medium deformation; experiment #6. Maximum stretch rate is 90 s-l. Solid line: Jeffrey, dashed line: Maxwell, semi-dashed line: Newtonian and n experimental values.
tially the same predictions which are in good agreement with experimental data. The Newtonian model, though, does show some deviation from the experimental data near the end of the spinline. In Fig. 7 the maximum stretch rate is about 90 s-l, and one begins to notice considerable deviations from the predictions using the Newtonian model. However, the Jeffrey’s and Maxwell models still lead to a good match with data. Presented in Fig. 8 are the results for the case for which the maximum stretch rate is about 140 s-r. Here both the viscoelastic models fail to represent data in a satisfactory manner. The major reason for this appears to be the failure of these models to portray even the viscometric data correctly. If one uses the second invariant of the rate of deformation tensor to relate the stretch rate, L, in the fiber spinning experiments to the shear rate, 9, in the viscometer experiments, then [lo] ?=fiS.
(8) Consequently, a stretch rate of 140 s-i implies a shear rate of 242 s-l. From an examination of Fig. 1, it is evident that the predictions of the Maxwell
358
0.0
0.2
0.4
0.6
0.8
1.0
1.2
DISTANCE (-)
Fig. 8. Prediction of velocity profiles at large deformations; experiment #5. Maximum stretch rate is 140 s-l . Solid line: Jeffrey, dashed line: Maxwell, semi-dashed line: Newtonian and W experimental values.
and Jeffrey’s models begin to diverge from experimental of about 100 s-l. Thus, the lack of agreement in Fig. 8 In addition to predicting the velocity profiles using models, one may also compute an apparent elongational
data at a shear rate is not surprising. specific rheological viscosity defined as
For Newtonian liquids, qIE is exactly three times the shear viscosity while for viscoelastic liquids it usually exceeds three times the zero shear viscosity; the ratio V&Q,, called the Trouton’s ratio, is therefore a measure of fluid elasticity. Shown in Fig. 9 is the variation of Trouton’s ratio with distance along the spinline for run #7. One finds that during the initial part of the spinline qn/~ indeed equals three, indicating Newtonian behavior. However, the Trouton’s ratio progressively increases to higher values further down the spinline, indicating viscoelastic behavior. This agrees with the results previously presented in Fig. 6 where the velocity profile matched with the Newtonian model for the initial portion of the spinline.
359
1 t
0’ 0.0
0.2
0.4
0.6
0.6
1
.o
DISTANCE (-)
Fig. 9. Variation of Trouton’s ratio along spinline, experiment #7; dashed line; predicted values.
W experimental values,
Had a steady in the stress been obtained, the Maxwell model would have predicted the steady extensional viscosity to be given by (see Reference for example [lo])
(1+X$-2AE).
VIE=
In the present case, although a steady state is not attained, one can predict an approximate value of the apparent extensional viscosity simply by using a time averaged value of the stretch rate in eqn. (lo), i.e. one replaces k by
where the residence time 0 is given by @= f - dz (12) / 0 u(t) * Figure 10 presents a comparison of the predictions of eqns. (10-12) with all
360
140 -
. . 120 -
. . .
f
.
0 g g
. 100
p
.
:i
.
i .*
% I K ::
. .
60-
60 -
40
0
20
10
AVERAGE
STRETCH
RATE (l/s)
Fig. 10. Variation of extensional viscosity with time averaged stretch rate. The solid line shows predictions based on steady state Maxwell model; + data from all experiments.
the experimental data collected in this work. The agreement is fair. The reason why such a procedure seems to work is that the behaviour of a weakly elastic liquid is just a perturbation about the Newtonian fluid behavior. Consequently, any reasonable viscoelastic model suffices. Before concluding this paper, we digress somewhat to point out that some previous investigators [ll] have attempted to account for varying stretch rates along the spinline by plotting the results as a function of the total strain, E, defined as
The tensile stress at any point along the spinline can be correlated against E. Such a plot is shown in Fig. 11. The excellent correlation, however, does not result from a particular choice of variables but, as shown below, is a logical consequence of the fact that the force remains constant along the spinline. Under these conditions eqn. (5) simplifies to’
361 40
.
.
30 -
2 0 $ 20 !i &
10 -
0
_I
0
1
2
3
4
STRAIN (-)
Fig. 11. Variation of tensile stress with total strain. The solid line represents predictions based on eqn. (16); n data from all experiments.
or
dbzz- 7,x) du (r,,-r.J = U’
(15)
As u is dz/dt, the righthand side of the above equation can be written as t dt. Integrating both sides of eqn. (15) then yields
(16) where the denominator represents the tensile stress at the spinneret. Equation (16) is shown as the solid line in Fig. 11, and it becomes clear why the data must collapse onto a single curve as they do. Conclusion The extensional flow behavior of weakly elastic constant viscosity liquid is shown to exhibit a smooth transition from Newtonian inelastic to non-New-
362
tonian viscoelastic behavior. The apparent extensional viscosity is found to gradually increase from the Newtonian value as the stretch rate is increased; this can be predicted with the help of simple viscoelastic models. The results of this work are likely to be of use to those interested in carrying out model rheological experiments using these constant viscosity liquids. In addition, the present research further substantiates the utility of the extensional viscometer proposed by the authors. Acknowledgements
The viscometric data on the silicone oil were obtained by Mr. R.J. Binnington. Mr. P. Papp collected some of the extensional data presented in the paper. The continuing collaboration between the authors is supported by the U.S.-Australia co-operative science program. T. Sridhar also acknowledges support from the Australian Research Grants Scheme. References 1 R.K. Gupta and T. Sridhar, in: B. Mena, A. Garcia-Rejon and C. Rangel-Nafaile @Is.), Advances in Rheology, Vol. 4: Applications, Universidad National Autonoma de Mexico, Mexico, 1984, p. 71. 2 T. Sridhar and R.K. Gupta, Rheol. Acta, 24 (1985) 207. 3 R.K. Gupta, J. Puszynski and T. Sridhar, J. Non-Newtonian FIuid Mech., 21 (1986) 99. 4 T. Sridhar, R.K. Gupta, D.V. Boger and R.J. Binnington, J. Non-Newtonian Fluid Mech., 21 (1986) 115. 5 G. Prihttski, R.K. Gupta, T. Sridhar and M.E. Ryan, J. Non-Newtonian Fluid Mech., 12 (1983) 233. 6 R.K. Gupta and T. Sridhar, in: D.W. Clegg and A.A. Collyer (Eds.), The Physical Principles of Rheological Measurement, Elsevier Applied Science Publishers, 1988, Ch. 8. 7 C.J.S. Petri, Elongational FIows, Pitman, London, 1979. 8 K.M. Baid and A.B. Metzner, Trans. ‘Sot. Rhkol., 21 (1977) 237. 9 M.M. Denn, C.J.S. Petrie and P. Avenas, AIChE J., 21 (1975) 791. 10 P.K. AgrawaI, W.K. Lee, J.M., Lomtson, C.J. Richardson, K.F. Wissbrun and A.B. Metzner, Trans. Sot. Rheol., 21 (1977) 355. 11 N.E. Hudson and J. Ferguson, Trans. Sot. Rheol., 20 (1976) 265.
349
FIBRE SPINNING OF A WEAKLY ELASTIC LIQUID
T. SRIDHAR Department of Chemical Engineering, Monash University, Victoria, 3168 (Australia)
Wellington Road, Clayton,
and R.K. GUPTA Department of Chemical Engineering, State University of New York, Buffalo, N.Y. 14260 (U.S.A.) (Received October 28, 1987; In revised form December 15, 1987)
Summary This paper presents a study of a silicone oil (poly(dimethy1 siloxane)) in extensional deformation using an instrument developed recently by the authors. Data from steady shear and low amplitude sinusoidal deformation of this liquid clearly establish that it is weakly elastic. The viscometric data, for shear rates less than 100 s-l, are best represented by either the Maxwell model or the Jeffrey’s model, the latter being marginally superior. The extensional data show that at low deformation rates, this fluid exhibits a Newtonian behavior with an apparent extensional viscosity equal to three times the shear viscosity. Under these conditions the velocity profiles along the spinline are also well represented by the Newtonian model. However, at higher deformation rates better predictions of the velocity profiles are obtained from the Jeffrey’s and Maxwell models. At deformation rates above 100 s-l none of these simple models is adequate. Under the conditions used in these experiments, the fractional increase in tensile stress along the fiber is shown both theoretically and experimentally to be a unique function of the total strain. Furthermore, the apparent extensional viscosity at any point on the spinline can be calculated from steady state expressions if allowance is made for the variation of stretch rates by defining a time averaged stretch rate. The results obtained here show that elasticity must be considered if these model liquids are used to conduct rheological experiments at high deformation rates. Additionally, it is found that elastic effects in extension can be predicted using simple constitutive equations provided viscometric data can be represented properly in the deformation rate range of interest. Finally, 0377-0257/88/$03.50
0 1988 Elsevier Science Publishers B.V.
350 the present research further substantiates viscometer developed by the authors.
the utility
of the extensional
Introduction
A study of the extensional flow of polymeric liquids is important not only from the viewpoint of advancing our understanding of the constitutive behaviour of these materials but also for the purpose of polymer characterization. However, experimental difficulties associated both with stretching low viscosity liquids and with measuring the resulting extensional stress had slowed progress in this area. Recently, the authors [1,2] proposed a new instrument for measuring extensional viscosities of mobile liquids, which utilized a suction technique for extending a liquid filament. The new instrument has been used to investigate the behavior of ideal elastic liquids [3,4] and show that their behavior in extension is well represented by the Oldroyd model B; this confirmed the hypothesis of Prilutski et al. [5] which was based on shear data. In this paper, we use the same device to examine the behavior of another ideal liquid-silicone oil, a liquid that is generally used as a viscosity standard but which is considered to have a very small relaxation time (fluid elasticity). There are numerous similarities in the stress response of the ideal elastic liquids and silicone oils, and we draw heavily on our previous work [3,4]. For a discussion on the merits of various elongational rheometers, we refer the reader to a recent review [6]. Viscometric data
The fluid used in this work was a poly (dimethyl siloxane) or silicone oil obtained from the Dow Corning Company. Steady shear and dynamic data were obtained on a Weissenberg rheogoniometer model R-19 equipped with cone and plate fixtures. These data, which are shown in Fig. 1, clearly establish that the fluid viscosity is constant at 30 Pas and the first normal stress difference is quadratic in the shear rate up to a shear rate of approximately 50 s-l. Furthermore, the loss modulus, G”, is found to superpose with the shear stress while twice the storage modulus, 2G’, superposes with the first normal stress difference. The viscometric data can be represented by the upper convected Maxwell model [7]:
351
‘04C
10"
10-l
““.‘I
100
““‘.’
FREQUENCY,
10'
““‘.I
102
““‘C
103
SHEAR RATE (l/s)
Fig. 1. Steady shear data for silicone oil showing forces. 0 shear stress, A G”, H 2G’ and A NI.
or the upper convected Jeffrey’s element in parallel):
constant
viscosity
and quadratic
normal
model [7] (a Maxwell and Newtonian
(2) where T is the extra-stress tensor, D the deformation rate tensor, S/St the upper convected Oldroyd derivative, h or A,, the model relaxation time, A, the model retardation time and n the liquid viscosity. The relaxation time calculated according to eqn. (1) is 0.014 s while the relaxation and retardation times according to eqn. (2) are 0.0206 s and 0.0065 s, respectively. On comparing these relaxation times to a relaxation time of about 0.824 s for a constant viscosity elastic fluid of about the same shear viscosity [4], it is seen that the silicone oil used here is very weakly elastic.
352
FREQUENCY (l/s)
Fig. 2. Dynamic data for silicone oil. Predictions of the storage modulus using different constitutive equations. H 2G’, eqn. (3) and - - - eqn. (4).
The storage modulus for these two models can be calculated as G’ (Maxwell) =
G’ (Jeffrey) =
qho2
(3)
l+(hw)2’
77(X1-
h2b2
1 + (x,w)2
*
(4
The predictions of eqns. (3) and (4) are compared with experimental data in Fig. 2 and both models are found to be satisfactory. Since (h, - X2) in eqn. (4) is numerically identical to h in eqn. (3), the two equations predict the same results at low frequencies. Differences arise only when the frequency approaches 100 s-l and here the Jeffrey’s model seems to be superior. These two models are now used to analyze fiber spinning data for this liquid. Spinning experiments Spinning data were obtained using the extensional viscometer developed by the authors. A sketch of the set-up is shown in Fig. 3 while details of the
353
___________________. vlsCOMElER
i-t
: : :
’
: ;
:
;
SPINLINE
TAKE UP
+ VACUUM Fig. 3. Sketch of the Extensional Viscometer.
instrument are available elsewhere [2,4]. A suction device was used to generate the extensional flow field, and the tensile stress at the capillary exit was measured as an apparent decrease in pressure drop across the capillary. All the data were obtained at room temperature (around 20” C), and experiments were conducted over a wide range of conditions. Experimental measurements of diameter profiles were combined with measured flow rates to obtain velocity profiles and stretch rates along the spinline; the spinline was long enough that a consideration of end effects was not necessary. Figure 4 shows three stretch rate profiles; details corresponding to these experiments are given in Table 1. In each case the stretch rates increase to a maximum value, the maximum stretch rates encountered were above 100 Note that with such a viscous liquid, stretch rates of this magnitude are S -I. not accessible using the conventional rotating drum stretching technique. Tensile stresses along the spinline can be calculated with the help of a momentum balance which accounts for the effects of gravity, inertia and surface tension. The final result, in differential form, is [8]
0.0
0.2
0.4
0.6
0.8
1.0
DISTANCE(-) Fig. 4. Typical variation of stretch rates along the fibre. W run # 5, Cl run #6 and A run #7.
where y is the coefficient of surface tension, g the acceleration due to gravity and D the fiber diameter. In general, it was found that the net contribution to the force balance of terms due to inertia, surface tension and gravity was negligible. The net contribution of these terms was found to be less than 5% except for experiments 2 and 7 where the maximum contribution was 15% and 11% respectively. Consequently, the spinning takes place under conditions of constant force. Nonetheless, tensile stresses along the TABLE 1 Details to the experimental measurements in spinning experiments Run #
Flow rate cm3/s
Initial velocity cm/s
Filament length mm
AP Pa
Draw ratio
1 2 3 4 5 6 7
0.149 0.162 0.162 0.162 0.169 0.168 0.168
0.628 0.564 0.62 0.66 0.78 0.62 0.67
5.5 11 10.5 8.3 7.3 9.7 8.7
430.6 155.7 234.3 280.3 397.3 269.1 191.8
18.2 19.9 54.6 30.2 41.8 51.1 12.7
355
0’
0.0
a
0.2
a
0.4
’ 0.6
’ 0.0
m
1.0
DISTANCE (-)
n) calculated using eqn. Fig. 5. Tensile stress along the spinline for experiment #7 (W(5). Dotted line represents the tensile stress calculated by assuming the force to be constant. filament length were calculated using a finite difference analog of the above equation. A typical profile, corresponding to run #7, is shown in Fig. 5. Clearly the tensile stresses show no evidence of reaching a steady state, primarily due to the fact that the stretch rate increases for most of the spinline length. Figure 5 also shows the tensile stress calculated under the assumption that only the viscoelastic terms are dominant. It is evident that the net contribution of inertia, gravity and surface tension is indeed negligible. In the the absence of inertia, surface tension and gravity, the velocity profiles (or equivalenty the stress profiles) can be calculated with ease for any of the fluid models of interest-Newtonian, Maxwell or Jeffrey’s. For a Newtonian liquid, when viscous stresses are dominant, the dimensionless velocity, U, is related to the dimensionless distance, S, as 24= exp(S In DR), where Da is the draw ratio. Also, the stress at the spinneret is
(6)
356
0.0
0.2
0.4
0.6
0.8
1.0
1.2
DISTANCE (-)
Fig. 6. Prediction of velocity profiles at low deformation rates; experiment #7. Maximum stretch rate is 20 s- ‘. Solid line: Jeffrey, 0 -0 Maxwell, AA Newtonian and H experimental values.
where L is the spinline length. From a knowledge of the measured stress at the spinneret, one can calculate the draw ratio from eqn. (7); insertion of this quantity into eqn. (6) then yields the predicted velocity profile. The equations for the computation of the velocity profiles for the fiber spinning of a Maxwell liquid have been published by Denn et al. [9] while those for the Jeffrey’s liquid (also called the Oldroyd fluid B) have been provided by Gupta et al. [3]. Consequently, these will not be repeated here. We point out, though, that the Jeffrey’s model is a linear combination of the Newtonian and the Maxwell models so that it can exhibit both extremes in behavior. The equations for the Maxwell and Jeffrey’s models were solved numerically using the Gear method, and the results for the dimensionless velocity as a function of the dimensionless distance are shown in Figs. 6-8. Also shown are the experimental data points corresponding to experiments 5-7. These runs were chosen so as to span a wide rang of stretch rates. In Fig. 6, maximum stretch rate only 20 s-i and, not surprisingly, at these low deformation rates, the Maxwell, Jeffrey’s and Newtonian models give essen-
357 70
c-i.00
0.20
0.40
0.60
0.60
1.00
1.20
DISTANCE(-)
Fig. 7. Prediction of velocity profiles to medium deformation; experiment #6. Maximum stretch rate is 90 s-l. Solid line: Jeffrey, dashed line: Maxwell, semi-dashed line: Newtonian and n experimental values.
tially the same predictions which are in good agreement with experimental data. The Newtonian model, though, does show some deviation from the experimental data near the end of the spinline. In Fig. 7 the maximum stretch rate is about 90 s-l, and one begins to notice considerable deviations from the predictions using the Newtonian model. However, the Jeffrey’s and Maxwell models still lead to a good match with data. Presented in Fig. 8 are the results for the case for which the maximum stretch rate is about 140 s-r. Here both the viscoelastic models fail to represent data in a satisfactory manner. The major reason for this appears to be the failure of these models to portray even the viscometric data correctly. If one uses the second invariant of the rate of deformation tensor to relate the stretch rate, L, in the fiber spinning experiments to the shear rate, 9, in the viscometer experiments, then [lo] ?=fiS.
(8) Consequently, a stretch rate of 140 s-i implies a shear rate of 242 s-l. From an examination of Fig. 1, it is evident that the predictions of the Maxwell
358
0.0
0.2
0.4
0.6
0.8
1.0
1.2
DISTANCE (-)
Fig. 8. Prediction of velocity profiles at large deformations; experiment #5. Maximum stretch rate is 140 s-l . Solid line: Jeffrey, dashed line: Maxwell, semi-dashed line: Newtonian and W experimental values.
and Jeffrey’s models begin to diverge from experimental of about 100 s-l. Thus, the lack of agreement in Fig. 8 In addition to predicting the velocity profiles using models, one may also compute an apparent elongational
data at a shear rate is not surprising. specific rheological viscosity defined as
For Newtonian liquids, qIE is exactly three times the shear viscosity while for viscoelastic liquids it usually exceeds three times the zero shear viscosity; the ratio V&Q,, called the Trouton’s ratio, is therefore a measure of fluid elasticity. Shown in Fig. 9 is the variation of Trouton’s ratio with distance along the spinline for run #7. One finds that during the initial part of the spinline qn/~ indeed equals three, indicating Newtonian behavior. However, the Trouton’s ratio progressively increases to higher values further down the spinline, indicating viscoelastic behavior. This agrees with the results previously presented in Fig. 6 where the velocity profile matched with the Newtonian model for the initial portion of the spinline.
359
1 t
0’ 0.0
0.2
0.4
0.6
0.6
1
.o
DISTANCE (-)
Fig. 9. Variation of Trouton’s ratio along spinline, experiment #7; dashed line; predicted values.
W experimental values,
Had a steady in the stress been obtained, the Maxwell model would have predicted the steady extensional viscosity to be given by (see Reference for example [lo])
(1+X$-2AE).
VIE=
In the present case, although a steady state is not attained, one can predict an approximate value of the apparent extensional viscosity simply by using a time averaged value of the stretch rate in eqn. (lo), i.e. one replaces k by
where the residence time 0 is given by @= f - dz (12) / 0 u(t) * Figure 10 presents a comparison of the predictions of eqns. (10-12) with all
360
140 -
. . 120 -
. . .
f
.
0 g g
. 100
p
.
:i
.
i .*
% I K ::
. .
60-
60 -
40
0
20
10
AVERAGE
STRETCH
RATE (l/s)
Fig. 10. Variation of extensional viscosity with time averaged stretch rate. The solid line shows predictions based on steady state Maxwell model; + data from all experiments.
the experimental data collected in this work. The agreement is fair. The reason why such a procedure seems to work is that the behaviour of a weakly elastic liquid is just a perturbation about the Newtonian fluid behavior. Consequently, any reasonable viscoelastic model suffices. Before concluding this paper, we digress somewhat to point out that some previous investigators [ll] have attempted to account for varying stretch rates along the spinline by plotting the results as a function of the total strain, E, defined as
The tensile stress at any point along the spinline can be correlated against E. Such a plot is shown in Fig. 11. The excellent correlation, however, does not result from a particular choice of variables but, as shown below, is a logical consequence of the fact that the force remains constant along the spinline. Under these conditions eqn. (5) simplifies to’
361 40
.
.
30 -
2 0 $ 20 !i &
10 -
0
_I
0
1
2
3
4
STRAIN (-)
Fig. 11. Variation of tensile stress with total strain. The solid line represents predictions based on eqn. (16); n data from all experiments.
or
dbzz- 7,x) du (r,,-r.J = U’
(15)
As u is dz/dt, the righthand side of the above equation can be written as t dt. Integrating both sides of eqn. (15) then yields
(16) where the denominator represents the tensile stress at the spinneret. Equation (16) is shown as the solid line in Fig. 11, and it becomes clear why the data must collapse onto a single curve as they do. Conclusion The extensional flow behavior of weakly elastic constant viscosity liquid is shown to exhibit a smooth transition from Newtonian inelastic to non-New-
362
tonian viscoelastic behavior. The apparent extensional viscosity is found to gradually increase from the Newtonian value as the stretch rate is increased; this can be predicted with the help of simple viscoelastic models. The results of this work are likely to be of use to those interested in carrying out model rheological experiments using these constant viscosity liquids. In addition, the present research further substantiates the utility of the extensional viscometer proposed by the authors. Acknowledgements
The viscometric data on the silicone oil were obtained by Mr. R.J. Binnington. Mr. P. Papp collected some of the extensional data presented in the paper. The continuing collaboration between the authors is supported by the U.S.-Australia co-operative science program. T. Sridhar also acknowledges support from the Australian Research Grants Scheme. References 1 R.K. Gupta and T. Sridhar, in: B. Mena, A. Garcia-Rejon and C. Rangel-Nafaile @Is.), Advances in Rheology, Vol. 4: Applications, Universidad National Autonoma de Mexico, Mexico, 1984, p. 71. 2 T. Sridhar and R.K. Gupta, Rheol. Acta, 24 (1985) 207. 3 R.K. Gupta, J. Puszynski and T. Sridhar, J. Non-Newtonian FIuid Mech., 21 (1986) 99. 4 T. Sridhar, R.K. Gupta, D.V. Boger and R.J. Binnington, J. Non-Newtonian Fluid Mech., 21 (1986) 115. 5 G. Prihttski, R.K. Gupta, T. Sridhar and M.E. Ryan, J. Non-Newtonian Fluid Mech., 12 (1983) 233. 6 R.K. Gupta and T. Sridhar, in: D.W. Clegg and A.A. Collyer (Eds.), The Physical Principles of Rheological Measurement, Elsevier Applied Science Publishers, 1988, Ch. 8. 7 C.J.S. Petri, Elongational FIows, Pitman, London, 1979. 8 K.M. Baid and A.B. Metzner, Trans. ‘Sot. Rhkol., 21 (1977) 237. 9 M.M. Denn, C.J.S. Petrie and P. Avenas, AIChE J., 21 (1975) 791. 10 P.K. AgrawaI, W.K. Lee, J.M., Lomtson, C.J. Richardson, K.F. Wissbrun and A.B. Metzner, Trans. Sot. Rheol., 21 (1977) 355. 11 N.E. Hudson and J. Ferguson, Trans. Sot. Rheol., 20 (1976) 265.