Existence of optimal cooling conditions in the film blowing process

J. Non-Newtonian Fluid Mech. 137 (2006) 24–30

Existence of optimal cooling conditions in the film blowing process Joo Sung Lee a , Dong Myeong Shin a , Hyun-Seob Song b , Hyun Wook Jung a , Jae Chun Hyun ∗ a

Department of Chemical and Biological Engineering, Applied Rheology Center, Korea University, Seoul 136-701, Republic of Korea b Corporate R&D, LG Chem Ltd. Research Park, Daejeon 305-380, Republic of Korea Received 2 June 2005; accepted 30 December 2005

Abstract The existence of optimal cooling conditions with respect to process stability has been experimentally found in film blowing, in a sharp contrast to other extensional deformation processes such as fiber spinning and film casting where maximum, not optimal, cooling always unequivocally enhances process stability. The same simulation model of film blowing of Hyun et al. [J.C. Hyun, H. Kim, J.S. Lee, H.-S. Song, H.W. Jung, Transient solutions of the dynamics in film blowing processes, J. Non-Newtonian Fluid Mech., 121 (2004) 157–162] proven quite accurate in portraying the transient dynamic behavior of the film blowing as compared with experimental data, has been used in this study to carry out a linear stability analysis, which then clearly confirms the experimentally found optimal cooling conditions. These optimal cooling conditions in film blowing stemming from the multiplicity of steady states of the process thus make the strategies for stabilization and optimization of the process quite a challenging task in both industry and academia because of their common relevance to the important process productivity issue. © 2006 Elsevier B.V. All rights reserved. Keywords: Draw resonance; Film blowing; Linear stability analysis; Optimal cooling; Stability; Transient solutions

1. Introduction Since the two seminal works laying the foundation for studying the film blowing process (Fig. 1), i.e., Petrie and Pearson [1,2] establishing the first modeling standards for the process, and Cain and Denn [3] reporting the first comprehensive stability and multiplicity analysis of the process, there have been many research efforts throughout the world enhancing both the theoretical understanding of and the concurrent experimental improvements on the process [4–16]. Lately the transient solutions of the film blowing dynamics have been obtained by Hyun et al. [13] for the first time in 30 years, employing new mathematical and numerical schemes like an orthogonal collocation on finite element and a coordinate transformation for solving the governing equations of the non-isothermal process of viscoelastic fluids. The results are very satisfying in that the dynamic behavior of the process is well portrayed even during the draw resonance oscillation, e.g., the temporal bubble radius profiles strikingly close to experimentally observed ones. In this study the existence of the optimal cooling conditions in terms of process stability in film blowing is reported with

∗

Corresponding author. Tel.: +82 2 3290 3295; fax: +82 2 926 6102. E-mail address: [email protected] (J.C. Hyun).

0377-0257/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.12.011

experimental data and then theoretically predicted, in a sharp contrast to other extensional deformation processes like fiber spinning and film casting where maximum, not optimal, cooling always brings about process stability [17,18]. These optimal cooling conditions stemming from the multiplicity of the steady states of the process as reported by Cain and Denn [3] represent an academically intriguing and industrially important subject because of their relevance to the process productivity. Historically the film blowing has been considered an engineering ingenuity in that a simple and robust design enables stretching of film in two directions simultaneously: the axial drawing of the film provided by the nip rolls whereas the circumferential drawing accomplished by the pressure difference between inside and outside the film bubble. Through the separate control of these two drawings, i.e., the drawdown ratio, Dr ≡ vL /v0 , and the blowup ratio, BUR ≡ rL /r0 , desired 2-D film properties such as the thickness reduction, TR ≡ w0 /wL are usually obtained the way we want. (The relationship Dr = TR/BUR holds among these three dimensionless ratios.) Due to the dynamically complicated nature of the process, stabilization and optimization strategies for the film blowing are engendered by the inherently intricate equipment design to perform the biaxial extension and the efficient surface cooling of the film. This, thus, provides additional degrees of freedom

J.S. Lee et al. / J. Non-Newtonian Fluid Mech. 137 (2006) 24–30

25

Table 1 Values of the process and material parameters used in the study Parameters

Values

Radius of annual die (¯r0 ¯ 0 )a Die gap (w Extrusion velocity (¯v0 )b Cooling air velocity (¯vair )b Extrusion temperature (θ¯ 0 )b Solidification temperature (θ¯ F )c Cooling air temperature (θ¯ c )b Ambient temperature (θ¯ ∞ )b Nip-roll height (¯zL )b Material relaxation time (λ0 )d at 180 ◦ C Zero-shear viscosity (η0 )d at 180 ◦ C PTT model parameters, ε and ξ e ¯ d Activation energy (k) Density of fluid (ρ) at 180 ◦ Cd Heat capacity (Cp )d Emissivity (εm )c )a

a b c d e

Fig. 1. Schematic diagram of the film blowing process.

for controllability in operating the process as compared to other extensional deformation processes: film blowing has additional manipulating variables of the air amount and/or air pressure inside the bubble, not found in fiber spinning or film casting process. The fact that film blowing runs at much slower speeds than fiber spinning also allows larger process sensitivities toward these manipulating variables of the system, and hence greater controllability of the process in general. 2. Experimental findings The material, used for the experiments in this study, is a commercial low density polyethylene (LDPE coded FB2000) from LG Chem, Korea, with Mn = 17,170, Mw = 92,090 and MI = 2.0 g/10 min. Film blowing extrusion equipment, which has a 1.5 cm radius of annular die with gap widths of 0.08 cm, was manufactured by Collin. To find an optimal cooling condition, the different cooling air velocities (vair = 15–45 cm/s) issuing from the air ring surrounding die exit were employed onto the bubble under the fixed extrusion temperature (θ¯ 0 = 180 ◦ C). Detailed parameter values for material and equipment are summarized in Table 1. The process stability was determined measuring the variation of bubble radius using a real-time video camera system [21]. From the experimental observations, the stability diagrams with different cooling air velocities have been constructed in

1.5 cm 0.08 cm 0.166 cm/s 15–45 cm/s 180 ◦ C 135 ◦ C 25 ◦ C 25 ◦ C 105 cm 0.0723 s 11000 Pa s 0.015, 0.1 15.966 kJ/mol 765 g/cm3 1.910 J/g ◦ C 0.9

Specifications by the manufacturer, Collin Co. Experimental operating conditions. see Ref. [10]. Material data from LG Chem, Seoul, Korea. see Ref. [20].

BUR–TR plot of Fig. 2, revealing that the middle cooling level exhibits the smaller draw resonance instability region than either the higher or lower cooling level. Next, it is to be determined whether these interesting experimental findings can be predicted by the simulation model of Hyun et al. [13], which had demonstrated its capability of portraying the transient dynamic behavior of film blowing fairly well even during draw resonance oscillations. 3. Theoretical prediction The dimensionless governing equations of the nonisothermal film blowing of PTT fluids [19,20] are as follows: the same governing equations as in Hyun et al. [13] based on the seminal works by Pearson and Petrie [1,2] and Cain and Denn [3] who set up the modeling equations and the standards for all ensuing research efforts in this field. Equation of continuity:
∂ ∂ (rw 1 + (∂r/∂z)2 ) + (rwv) = 0, (1) ∂t ∂z where r = r¯r¯0 , w = w¯w¯0 , v = Axial force balance:

v¯ v¯ 0 ,

z=

z¯ r¯0 ,

t=

¯t v¯ 0 r¯0 .

2rw(τ11 − τ22 )  + B(rF2 − r 2 ) = Tz , 1 + (∂r/∂z)2 where Tz = τ¯ r¯

T¯ z ¯ 0 v¯ 0 , 2πη0 w

B=

r¯02 P ¯ 0 v¯ 0 , 2η0 w

(2)

P =  z¯ L A 2 0

π¯r d¯z

τij = 2ηij0 v¯00 . Circumferential force balance:   (τ33 − τ22 ) −(τ11 − τ22 )(∂2 r/∂z2 ) . +  B=w 3/2 r 1 + (∂r/∂z)2 (1 + (∂r/∂z)2 )

− Pa ,

(3)

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J.S. Lee et al. / J. Non-Newtonian Fluid Mech. 137 (2006) 24–30

where θ = U¯ 0 r¯01.24 ¯ 0 v¯ 1.76 ρCp w 0

θ¯ , θ¯ 0

,E=

θ¯ c , θ∞ θ¯ 0 εm σSB θ¯ 04 r¯0 . ¯ 0 v¯ 0 θ¯ 0 ρCp w

θc =

=

θ¯ ∞ , θ¯ 0

U = U0 z−0.24 v0.76 air , U0 =

Boundary conditions:

v=w=r=θ=1

at z = z0 and t ≥ 0,

(6a)

∂r = 0, ∂z

θ = θF

(6b)

v = Dr ,

at z = zF and t = 0,

∂r ∂r v = 0, +  ∂t ∂z 1 + (∂r/∂z)2 v



1 + (∂r/∂z)2

Fig. 2. Stability diagram by experimental observations for three different cooling air velocities: (a) vair = 15 cm/s, (b) vair = 30 cm/s, and (c) vair = 45 cm/s.

Constitutive equation (PTT model):  Kτ + De0

∂τ + v · ∇τ − L · τ − τ · LT ∂t

 =2

De D, De0

(4)

where K = exp [ε De tr τ], L = ∇v −  ξD, 2D = (∇v + ∇vT ), ¯ λ0 v¯ 0 1 De0 = r¯0 , De = De0 exp k θ − 1 , k = Rkθ¯ . 0 Equation of energy: v U ∂θ E ∂θ 4 + + (θ − θc ) + (θ 4 − θ∞ ) = 0, 2 ∂z ∂t w w 1 + (∂r/∂z)

= Dr (1 + δ),

θ = θF at z = zF and t > 0. (6c)

where r, w, v, t, z, θ, τ, and D denote the dimensionless variables of bubble radius, film thickness, fluid velocity, time, distance coordinate, film temperature, film extra stress tensor and strain rate tensor, respectively; P, Pa , B, A, and Tz denote the air pressure difference between inside and outside the bubble, atmospheric pressure, dimensionless air pressure difference, air amount inside the bubble, and dimensionless axial tension, respectively; ε, ξ PTT model parameters, De Deborah number, λ0 material relaxation time, η0 zero-shear viscosity; k, U, E, θ c , θ ∞ , zL , and zF denote the dimensionless quantities of activation energy, heat transfer coefficient, radiation coefficient, cooling air temperature, ambient temperature, distance between the die exit and the nip rolls and the freeze-line height, respectively; R gas constant, εm film emissivity, σ SB StefanBoltzmann constant, ρ fluid density, Cp heat capacity, Dr drawdown ratio, and δ a step disturbance at take-up velocity initiating dynamic response of the system. Boundary conditions at the freeze-line height represent no deformation beyond this position on the convected-coordinate. Overbars denote dimensional variables. Subscripts 0 and F denote the die exit and the freezeline conditions, respectively. Subscripts 1, 2 and 3 denote the flow direction, normal direction, and circumferential direction, respectively. The only difference between the above equations and the ones by Hyun et al. [13] is the expression of the heat transfer coefficient that has the dependency on the spinning distance and cooling air velocity shown in Eq. (5) in this study patterned after the literature [10] as opposed to the constant values before [13]. To elucidate the effect of the cooling air conditions on the process dynamics an expression like this is needed. As in simulations by Hyun et al. [13], on top of the orthogonal collocation on finite element (OCFE) and a coordination transformation [15,16], the actual shape of the film bubble is traced in calculating the bubble volume and pressure, instead of using the so-called cylindrical approximation used by previous researchers. 4. Linear stability analysis

(5)

As the additional boundary conditions to analyze the process stability using Eqs. (1)–(6), the set of the air amount inside the

J.S. Lee et al. / J. Non-Newtonian Fluid Mech. 137 (2006) 24–30

bubble, A, and drawdown ratio, Dr , among the four possible sets of boundary conditions explained by Cain and Denn [3], has been chosen to be held constant as determined from each steady state calculation, since most film blowing processes control the air amount inside the bubble, not the air pressure difference. The air amount inside the bubble is then calculated by integrating the bubble radius along the flow-direction using the ideal gas law, not using the cylindrical approximation. The PTT model parameters were selected as ε = 0.015, ξ = 0.1 for low-density polyethylene (LDPE) film [20] to represent the extensional characteristics of the polymer melt. Under the same operating conditions as in Fig. 2, the linear stability analysis for the non-isothermal film blowing process was carried out, which generally entails the procedure of introduction of perturbed variables on the state variables, linearization of the governing equations, formulation of an eigenvalue problem using Jacobian and Mass matrices, and then solving for the largest eigenvalues with the real parts being equated to zero to find the onset of the instability. Appendix A shows the details of this linear stability analysis of the system of Eqs. (1)–(6). In Fig. 3 showing the results of the linear stability analysis that can also be obtained through transient solutions, albeit a tedious process, it can be readily recognized that optimal cooling conditions maximizing the stability window in the diagram exist just as experimentally found in Fig. 2. Fig. 4 displays some typical results of the transient solutions portraying different temporal pictures of the system depending on the different states in the stability diagram. Specifically, using the case depicted in the stability diagram of Fig. 3b as an example system, the temporal pictures of the blow-up ratio for four different states located on the same constant drawdown ratio line (Dr = 35) are shown in Fig. 4b. Points A and C denote stable cases whereas points B and D represent a draw resonance and a helical instability, respectively. It is thus confirmed that the stability results obtained by the linear stability analysis in Fig. 3 are corroborated by the transient solutions in Fig. 4. One comment worth mentioning here is that although in Figs. 2 and 3 only two variables of BUR and TR are shown on the x- and y-axis, other state variables of the system in both experiments and simulations are necessarily different depending on the particular state of the system. For example, the freeze-line height and the air amount inside the bubble take on different values as BUR and TR change their values, even though the drawdown ratio can be held constant. The air amount and the freeze-line height in general increases and decreases, respectively, as the BUR increases. Regarding the cooling air effect on the system as shown in Fig. 3, it is clearly seen that too much or too little cooling both can aggravate the process stability. This is a sharp contrast to other extensional deformation processes like fiber spinning and film casting where cooling unequivocally stabilizes the system [17,18]. Despite the fact that many assumptions were incorporated into the simulation model and many values of the parameters were assumed from the literature, the simulation and experimental results show qualitatively good agreement

27

Fig. 3. Stability diagram by linear stability analysis for three different cooling air velocity: (a) vair = 15 cm/s, (b) vair = 30 cm/s, and (c) vair = 45 cm/s.

as far as the general portraying of the dynamic pictures of the bubble shape is concerned. The only exception is the existence of a narrow stable region between the two instability regions of draw resonance and helical instability which appears in the simulation results in Fig. 3, while in experiments they are hard to detect as shown in Fig. 2. This may be related to the inherent limitation of the 1-D model of the 3-D process, because the helical instability requires a 3-D model for its full investigation. This point certainly warrants further investigation in both simulation and experiments to get a clear answer for the matter. To clarify more the existence of the optimal cooling conditions in film blowing process, the critical BUR curves are plotted

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J.S. Lee et al. / J. Non-Newtonian Fluid Mech. 137 (2006) 24–30

Fig. 5. Critical blowup ratio curves as a function of cooling air velocity at TR = 60 in Figs. 2 and 3: (a) experimental observations and (b) theoretical results from linear stability analysis.

5. Conclusions Fig. 4. Transient solutions by simulation at four different states in Fig. 3b along the constant draw ratio line (Dr = 35). (BURss denotes the mean value of BUR and λr and λi denote real, imaginary parts of the leading eigenvalues, respectively, determined by the linear stability analysis.)

against the cooling air velocity in Fig. 5 using the results of experiments (Fig. 5a) and of simulations (Fig. 5b) under the same film thickness reduction of TR = 60 as in Figs. 2 and 3. The most stable case was found with the case of cooling air velocity, vair = 30 cm/s, in both diagrams. One comment is worth here referring to the nature of the helical instability occurring in this process. The cases with overcooling produce the helical instability, where unlike in draw resonance, the film bubble not only resonates but also rotates around the axis. This instability mode that could not be described by 1-D model of this study obviously requires more elaborated models such as a 2- or a 3-D model. The 1-D model of this study, however, gives some insights about this interesting instability feature: the linear stability analysis in the helical instability region reveals that the imaginary part of the dominant eigenvalue exhibits extremely large value, as compared to that in undercooling cases, meaning the fast oscillation in the helical instability region (the last diagram (point D) in Fig. 4b). This point should be further investigated later on when a 3-D model is employed to simulate and analyze the rotational nature of the helical instability.

In a sharp contrast to other extensional deformational processes like fiber spinning and film casting where the cooling unequivocally brings about process stability, it has been experimentally found that non-isothermal film blowing process exhibits optimal cooling conditions with respect to process stability [22]. The linear stability analysis technique was then brought to check this experimental finding. Employing the same 1-D simulation model that had proved to portray the dynamic behavior of film blowing fairly accurately [13], the linear stability analysis has successfully confirmed the fact that there do exist optimal cooling conditions in film blowing guaranteeing maximum process stability, proving once again the utility of the simulation model [13] for its ability to portray the dynamics of film blowing process, not only the transient solutions but also the stability and multiplicity of the system. There is one unresolved point regarding the stability diagram: the narrow stable region obtained by simulation in the stability diagram between two instabilities of draw resonance and helical instability is hard yet to detect by experiments. This point certainly warrants further investigation in both simulation modeling and experiments. Acknowledgement This study was supported by research grants from the Korea Science and Engineering Foundation (KOSEF) through the

J.S. Lee et al. / J. Non-Newtonian Fluid Mech. 137 (2006) 24–30

Applied Rheology Center (ARC), an official KOSEF-created engineering research center at Korea University, Seoul, Korea. Appendix A To effectively portray the periodic motion of the freezeline height as well as other state variables during draw resonance instability, the original time–space coordinate set in the governing equations of Eqs. (1)–(6) was transformed into the time–temperature coordinate as before [13]. Then, perturbation variables are introduced on the state variables in the governing equations for the linear stability analysis. Φ(ζ, t) = Φs (ζ, t) + Φ,

(A1) , z]T , Φ

where Φ = [r, w, v, τ11 , τ22 , τ33 = k(ζ) exp(Ωt), k = T [α, β, γ, δ11 , δ22 , δ33 , χ] ; Φ - denotes the state variables, Φ represents the perturbation variables, k- means small perturbations on the system dependent on the distance only, ζ is the dimensionless temperature coordinate defined as (θ 0 − θ)/(θ 0 − θ F ), and Ω is the eigenvalue to determine process stability. The reduced governing equations and their linearized governing equations are as follows. ˙ = 0. R(Φ, Φ) ˙ s) + R(Φs , Φ

∂R ∂R ˙ = 0. ( Φ) + ( Φ) ˙ s ∂Φ s ∂Φ

∂R1 ⎜ ∂r ⎜ ⎜ ∂R2 ⎜ ⎜ ∂r ⎜ ⎜ ∂R3 ⎜ ⎜ ⎜ ∂r ⎜ ∂R ⎜ 4 J =⎜ ⎜ ∂r ⎜ ⎜ ∂R5 ⎜ ⎜ ∂r ⎜ ⎜ ∂R6 ⎜ ⎜ ∂r ⎜ ⎝ ∂R7 ∂r

(A3)

(A5)

˙ are Jacobian and Mass where J ≡ ∂R/∂Φ and M ≡ −∂R/∂Φ matrices, respectively. For the non-isothermal film blowing system, the Jacobian and Mass matrices are generated as follows. ⎛ ⎞ ∂R1 ∂R1 0 0 0 0 0 ⎜ ∂˙r ⎟ ˙ ∂w ⎜ ⎟ ⎜0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ∂R4 ⎟ ∂R4 ⎜ ∂R4 ∂R4 ⎜ ⎟ 0 0 0 ⎜ ˙ ∂w ∂˙τ11 ∂˙z ⎟ M = − ⎜ ∂˙r ⎟, ⎜ ∂R5 ∂R5 ∂R5 ⎟ ⎜0 ⎟ 0 0 0 ⎜ ˙ ∂w ∂˙τ22 ∂˙z ⎟ ⎜ ⎟ ⎜ ∂R6 ∂R6 ∂R6 ∂R6 ⎟ ⎜ ⎟ 0 0 0 ⎜ ∂˙r ˙ ∂w ∂˙τ33 ∂˙z ⎟ ⎜ ⎟ ⎝ ∂R7 ⎠ 0 0 0 0 0 0 ∂˙z

∂R1 ∂w ∂R2 ∂w ∂R3 ∂w 0 ∂R5 ∂w 0 ∂R7 ∂w

∂R1 ∂v 0 0 ∂R4 ∂v ∂R5 ∂v ∂R6 ∂v ∂R7 ∂v

29

0 ∂R2 ∂τ11 ∂R3 ∂τ11 ∂R4 ∂τ11 ∂R5 ∂τ11 ∂R6 ∂τ11 0

0

0

0

0

∂R4 ∂τ22 ∂R5 ∂τ22 ∂R6 ∂τ22

∂R3 ∂τ33 ∂R4 ∂τ33 ∂R5 ∂τ33 ∂R6 ∂τ33

0

0

0

∂R1 ∂z ∂R2 ∂z ∂R3 ∂z ∂R4 ∂z ∂R5 ∂z ∂R6 ∂z ∂R7 ∂z

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (A6)

When the mass matrix M is singular and the Jacobian J is nonsingular, as in this system, eigenvalues of equations become infinitely large. To overcome this singularity problem, it is required to remove in advance infinite eigenvalues in the eigenproblem using a shift-invert transformation method as follows. (J − νM)−1 M k =

(A2)

where R - i representing each governing equation. The temporal perturbations are next introduced onto this linearized equation of Eq. (A3) and then governing equations for ˙ s ) = 0. All the governing steady state vanishing, i.e., R(Φs , Φ equations are now compactly represented in a matrix-vector form. ∂R ∂R [k(ζ) exp(Ωt)] + [Ωk(ζ) exp(Ωt)] = 0. (A4) ˙ s ∂Φ s ∂Φ ΩM k = J k.

⎛

1 k. (Ω − ν)

(A7)

where the shift ν is real. With this transformation method, the generalized eigenproblem is modified to a simple eigenproblem without any difficulty obtaining eigenvalues. The details of this linear stability analysis of non-isothermal film blowing process are provided by Lee [21]. References [1] J.R.A. Pearson, C.J.S. Petrie, The flow of a tubular film. Part I. Formal mathematical representation, J. Fluid Mech. 40 (1970) 1–19. [2] J.R.A. Pearson, C.J.S. Petrie, The flow of a tubular film. Part II. Interpretation of the model and discussion of solutions, J. Fluid Mech. 42 (1970) 609–625. [3] J.J. Cain, M.M. Denn, Multiplicities and instabilities in flow blowing, Polym. Eng. Sci. 28 (1988) 1527–1540. [4] Y.L. Yeow, Stability of tubular film flow: a model of the film blowing process, J. Fluid Mech. 75 (1976) 577–591. [5] X.-L. Luo, R.I. Tanner, A computer study of film blowing, Polym. Eng. Sci. 25 (1985) 620–629. [6] T. Kanai, J.L. White, Kinematics, dynamics and stability of the tubular film extrusion of various polyethylenes, Polym. Eng. Sci. 24 (1984) 1185–1201. [7] W. Minoshima, J.L. White, Instability phenomena in tubular film, and melt spinning of rheologically characterized high density, low density and linear low density polyethylenes, J. Non-Newtonian Fluid Mech. 19 (1986) 275–302. [8] A. Ghaneh-Fard, P.J. Carreau, P.G. Lafleur, Study of instabilities in flow blowing, AIChE J. 42 (1996) 1388–1396. [9] K.-S. Yoon, C.-W. Park, Stability of a blown film extrusion process, Int. Polym. Proc. 14 (1999) 342–349. [10] T. Kanai, G.A. Campbell, Film Processing, Hanser, Munich, 1999. [11] A.K. Doufas, A.J. McHugh, Simulation of film blowing including flowinduced crystallization, J. Rheol. 45 (2001) 1085–1104. [12] J. Laffargue, L. Parent, P.G. Lafleur, P.J. Carreau, Y. Demay, J.-F. Agassant, Investigation of bubble instabilities in film blowing process, Intern. Polym. Proc. 17 (2002) 347–353.

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