INSTABILITIES IN ELONGATION FLOWS OF POLYMERS
AT HIGH DEBORAH NUMBERS
A Dissertation Presented to The Graduate Faculty of The University of Akron
In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
Atanas Gagov December, 2007
INSTABILITIES IN ELONGATION FLOWS OF POLYMERS AT HIGH DEBORAH NUMBERS
Atanas Gagov Dissertation
Approved: Accepted:
______Advisor Department Chair Dr. Arkadii I. Leonov Dr. Sadhan C. Jana
______Committee Member Dean of the College Dr. James L. White Dr. Stephen Cheng
______Committee Member Dean of the Graduate School Dr. Avraam I. Isayev Dr. George R. Newkome
______Committee Member Date Dr. Alex Povitsky
______Committee Member Dr. Gerald W. Young
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ABSTRACT
The objective of this work is to study the instabilities of the contraction flows at
high Deborah numbers of various polymers. The following topics will be discussed:
1. Linear and nonlinear stability analysis of isothermal fiber spinning
2. Linear stability analysis of nonisothermal fiber spinning
3. Linear and nonlinear stability analysis of contraction flow
4. Propagation of reservoir instabilities in capillary
The analysis of these flow problems requires solution of the closed set of PDE’s
(or ODE’s), consisting of equations for conservation of mass and momentum, along with
an adequate viscoelastic constitutive equation, with appropriate initial/boundary
conditions.
The goal of this work is to demonstrate a procedure for determining the critical
regime beyond which the process becomes unstable and also to determine weather the process is stable when the disturbances grow to a finite size.
Linear and non-linear stability theories have been used to describe the
fluctuations of fiber spinning and contraction flow. Linear stability analysis determines
the onset of the instabilities of the process while nonlinear analysis establishes the
complete range of the stable and unstable conditions.
The melt fiber spinning is the most common of polymer fiber processing. Finding
critical process conditions and the stabilizing effect of the cooling is described in this
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work. The critical draw ratio is established using linear stability analysis and the effect of the finite size imposed disturbances is studied through nonlinear stability analysis.
Contraction flow is one of the benchmark problems in computational polymer fluid mechanics and polymer processing. In this modeling, the whole flow region is divided in naturally introduced sub-regions with well-known and highly simplified types of flow. Thus, the model analyzes the entire flow region in a simplified geometric manner with properly matched conditions between adjacent sub-regions.
The propagation of the disturbances formed in the reservoir region has been analyzed. Employing the isothermal “Jet approach” followed by linearized perturbation approximation of the governing equations for finding the onset of the instabilities supplies information about the stability of the contraction flow which has been used to describe the mechanism of propagation of the disturbances into capillary up to the die exit, and its numerical implementation.
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TABLE OF CONTENTS
Page LIST OF TABLES……………………………………………………………………..….x
LIST OF FIGURES………………………………………………………………………xi
CHAPTER
I. INTRODUCTION ...... 1
II. LITERATURE REVIEW ...... 4
2.1 Importance of Stability in Polymer Processing ...... 4
2.2 Isothermal Fiber Spinning...... 7
2.3 Fiber Spinning with Cooling Temperature Gradient ...... 11
2.4 Non-linear Dynamics of Isothermal Fiber Spinning...... 14
2.5 Flows of Polymer Melts in the Capillary Exit Region and the Die Swell ...... 15
2.6 Viscoelastic Constitutive Equations for Polymer Melts and Solutions ...... 17
2.7 Instabilities in Contraction Flow at High Deborah Numbers ...... 19
2.8 Numerical Studies of Polymer Flows in Channels and Dies ...... 26
2.9 Significance of Constitutive Equation ...... 28
III. REOLOGICAL BACKGROUND AND CONTRACTION FLOW MODEL...... 32
3.1 Rheological Characterizations of Polymers...... 32
3.2 The Choice of Constitutive Equation...... 33
3.3 Development of the Model ...... 34
v
3.4 Set of Equations for Solving Inherently Difficult Viscoelastic Flow Problems... 38
3.5 The Choice of the Numerical and Iteration Scheme...... 39
IV. ISOTHERMAL FIBER SPINNING...... 42
4.1 Linear Stability Analysis of Isothermal Fiber Spinning ...... 42
Process description ...... 42
Nondimensional Governing Equations...... 45
Initial and Boundary conditions...... 46
Steady State Solution...... 47
Linear Stability Analysis for Small Disturbances...... 49
Numerical Approach...... 51
Accuracy of the Numerical Scheme ...... 54
Results and Discussions...... 54
4.2 Nonlinear Stability Analysis of Isothermal Fiber Spinning...... 58
Governing Equations ...... 59
Initial and Boundary conditions...... 60
Numerical Approach...... 61
Accuracy of the Numerical Scheme ...... 62
Results and Discussions...... 63
V. FIBER SPINNING WITH COOLING TEMPERATURE GRADIENT...... 67
5.1 Linear Stability Analysis of Nonisothermal Fiber Spinning...... 67
Process description ...... 67
Viscosity Dependence on Temperature……………………………………...... 71
Nondimensional Governing Equations...... 72
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Boundary conditions...... 75
Steady State Solution...... 75
Linear Stability Analysis for Small Disturbances...... 78
Numerical Approach...... 80
Accuracy of the Numerical Scheme ...... 83
Results and Discussions...... 83
VI. INSTABILITIES IN HIGH DEBORAH NUMBER CONTRACTION FLOWS...... 91
6.1 Linear Stability Analysis of Contraction Flow ...... 91
Process description ...... 92
Flow in the Far Field Entrance Region...... 94
Flow in the Near Field Entrance Region - “Jet Approach” ...... 96
Matching Condition at the Boundary of two Flow Regions...... 99
Steady State Solution in Near Entrance Region ...... 100
Steady State Solution for Die Developing Flow...... 101
Linear Stability Analysis for Small Disturbances...... 105
Numerical Approach...... 107
Accuracy of the Numerical Scheme ...... 107
Results and Discussions...... 109
6.2 Non-linear Stability Analysis of Contraction Flow ...... 112
Governing Equations ...... 112
Initial and Boundary conditions...... 115
Numerical Approach...... 116
Results and Discussions...... 117
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VII. SUMMARY...... 127
7.1 Results...... 127
7.2 Recommendations for future work ...... 129
VIII. REFERENCES ...... 130
viii
LIST OF TABLES
Table Page
4.1. Simplified matrix (M) for solving linear system of equations……………...... ….53
4.2. Comparison between numerical results and experimental data for isothermal fiber
0 spinning T= 218 C ……....…..………………………………...….…….……..57
4.3. Comparison between numerical results and experimental data for three types of
o polyesters, T= 285 C ……………..………...…………………………….……58
5.1.1. Material parameters for non-isothermal fiber spinning 76
5.1.2. Simplified matrix (M) for solving linear system of equations……………...... 82
5.1.1. Comparison between numerical results and experimental data for PET at different
spinning lengths…………………………..……………………………….…….90
6.1.1. Material Parameters and Experimental Conditions of Contraction Flow
for PIB/PB/C14……...... ………………….………………………….…109
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LIST OF FIGURES
Figure Page
2.1. Schematic of fiber spinning process………………………………………………8
2.2. Schematic of contraction flow…………………………………………………...22
4.1. Sketch of the fiber spinning process……………………………………...……...43
4.2. Area change along spinline for HDPE 6009 for different draw ratios……...…...48
4.3. Velocity changes along spinline for HDPE 6009 for various draw ratios...... 49
4.4. Sketch of apparatus for isothermal fiber spinning….……………………………56
4.5. Sketch of apparatus for isothermal fiber spinning ………………………………57
4.6. Disturbance evolution for an unstable fiber spinning of HDPE 6009
with 1% introdused sinusoidal disturbance at DR=4.4 (DRcr=4.28)…...……….65
4.7. Disturbance evolution for a stable fiber spinning of PP 6523 with
20% introdused sinusoidal disturbance at DR=2.5 (DRcr=2.62)………...... 66
5.1.1. Sketch of the fiber spinning process……………………………………………..68
5.1.2. Temperature profiles for nonisothermal fiber spinning for PS…………...... 77
5.1.3. Temperature profiles for nonisothermal fiber spinning for PE ...……………...... 77
5.1.4. Schematic of experimental setup………………………………………..……….84
5.1.5. Fiber Diameter comparison between numerical results and experimental
data for PET-A-14 ………….…………………………………………………....86
x
5.1.6. Fiber Diameter comparison between numerical results and experimental
data for PP ………...……………………………………………….…………….86
5.1.7. Fiber diameter comparison between numerical results and experimental
data for PET without horizontal air…………..…………………………………..88
5.1.8. Fiber diameter comparison between numerical results and experimental
data for PET with horizontal air…………..……………………………………...88
5.1.9. Fiber diameter comparison between numerical results and experimental
data for PET with and without horizontal air………..…………………………...89
5.1.10. Comparison between numerical results and experimental data for PET
at various spinning lengths…………..………………………………………..…90
6.1.1. Sketch of the circular cross-section contraction flow process…………..……….93
6.1.2. Regions in contraction flow and velocity profiles in the jet…………..…………94
6.1.3. Steady state axial velocity profiles for different Deborah numbers…………....104
6.1.4. Axial velocity in reservoir and capillary for two mesh sizes…………...... 108
6.1.5. Schematic of die geometry for different contraction ratios...... …...110
6.1.6. Dependence of DE number on the maximum-minimum velocity ratio...... 111
6.2.1. Jet Radius fluctuation grow for an unstable contraction flow of
PIB/PB/C14 with .1% introduced sinusoidal disturbance at De=1.67...... 119
6.2.2. Jet Radius fluctuation grow for an unstable contraction flow of
PIB/PB/C14 with 4% introduced sinusoidal disturbance at De=1.67...... …120
6.2.3. Axial Velocity fluctuations for an unstable contraction flow of PIB/PB/C14
with 1% introduced sinusoidal disturbance at De=1.67………………………....122
xi
6.2.4. Axial velocity profile of disturbance growth and propagation for unstable
flow of PIB/PB/C14 with 5% introduced disturbance at De=1.67……………....123
6.2.5. Axial velocity profile of disturbance growth and propagation for an unstable flow
of PIB/PB/C14 with 5% introduced disturbance………………………...... 124
6.2.6. Disturbance evolution for a stable Contraction flow of PIB/PB/C14
with 20% introduced sinusoidal disturbance at DR=1.36…………....…………..126
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CHAPTER I
INTRODUCTION
In many polymer-processing operations, the productivity is limited by the onset of
flow instabilities. For example in extrusion the extrudate becomes distorted at a critical
throughput rate. This can be clearly seen at the exit of the die, where a regular defect may be observed on the surface of the extruded material. In film casting and fiber spinning, a periodic variation in diameter or thickness may occur beyond a critical take-up speed and
in these processing types the filament breakage is a common failure. Contraction flow in polymer extrusion applications often leads to increase in pressure drops and may initiate
flow instabilities at higher flow rates. The magnitude of the response is generally
determined by the rheological characteristics of the melt. It is the purpose of this work to
develop and implement a model, which can predict when the instabilities in different polymer processes will occur and compare the computational results with available
experimental data.
The understanding of the flow behavior of polymer melts requires the use of
reliable tools for measuring the shear and elongational properties of these materials.
These measurements can be relatively easily performed in shear, but determining the
elongational properties of polymers still remains problematic. These properties are
1
known to be very important in industrial processes such as fiber spinning, film blowing, blow molding or film casting.
There are a lot of attempts to simulate these processes using numerical methods
(e.g. finite difference, finite element and their typical viscoelastic extensions) and those related to some simplified (sometimes, oversimplified) mathematical models. Many numerical methods have been elaborated and applied to geometrically complex flows of viscoelastic liquids with non-linear constitutive equations, but still there are plenty of limitations on their implementations. For differential models, the limitations depend on the number of Maxwell modes in use, as well as on the computational abilities and costs
in computing fast flows.
Another way approaching the problem is the use of simplified mathematical
models that still could catch the major features of the flow. These methods may produce
crude enough results and also may need some theoretical assumptions, but their
computational cost is considerably less than the “exact” computational one, and they
might be used for much more sophisticated flow regimes, including non-isothermality,
wall slip, etc, and for materials with complicated rheological behavior.
The problem treated in this work can be formulated as follows. Given such processing conditions as the flow geometry, the flow rate, the take-up force etc, describe
the disturbances in polymer flow parameters as well as the domains of jet stability and
extensional rate. The structure of the work is as following:
Chapter 2 presents a short overview of some previous publications on the
contraction and melt spinning modeling, numerical studies of polymer flows and
discussion about the numerical techniques dealing with these flow problems.
2
Chapter 3 gives a detail picture of the rheological properties of the polymers, describes the approach that has been used for characterizing the flow and develops the rheological model.
Chapter 4 shows the numerical implementation of the model to the melt spinning.
Chapter 5 describes the stabilizing effect of the cooling on the critical spinning conditions using linear approach.
Chapter 6 investigates the instabilities in contraction flow and their propagation in the capillary.
3
CHAPTER II
LITERATURE REVIEW
2.1 Importance of Stability in Polymer Processing
The viscoelastic nature of polymer solutions and melts gives rise to instabilities
that are not seen in the flows of Newtonian liquids. In industrial applications such as
injection molding, extrusion film casting and film blowing, these so-called “elastic”
instabilities can impose a limitation on the throughput. Therefore, it is important to
understand, and if possible, suppress or limit these instabilities.
Stability is one of the major issues in both the science and engineering applied to polymer processing. It is important for those who run the polymer manufacturing
facilities, because beside the sensitivity of the process to outside disturbances the stability
is great factor in safety, productivity, quality of the product, and eventually to the
effectiveness of the process. Stability is always of the primary interest to theoreticians
who study the fundamental aspects of non-linear dynamic systems like existence and
uniqueness of the solutions, possibility of oscillation, bifurcation and chaos, etc. Thus the
stability of continuous processes of polymer processing like extrusion, fiber spinning,
film casting, calendaring, pultrusion, etc. has been an exciting topic for many researchers
around the world.
4
There are three classes of analyses: those that establish conditions under which a process is absolutely unstable to any disturbances, no matter how small they are; those
that determine the effects of small but finite disturbances near conditions corresponding
to absolute instability; and those that establish conditions under which a process is
absolutely stable regardless of the disturbance magnitude. Only the first two are of
interest here: little has been done in applying the third class of instability to polymeric
flows of processing significance.
The first attempt for modeling extensional flow was made in the middle of the
1960s when Kase and Matsuo studied the instabilities in fiber spinning process. [1] Even
in a simple process like fiber spinning the complexity of the problem was evident. Some
of these complications are coming from the ongoing crystallization, others from the
difficulty in modeling the stress variables in the amorphous/crystalline structure with
complex heat transfer, and also due to the 3D nature of the dynamics and the phase
changes occurring in the fiber. The non-linear constitutive equations and inertia effects in
high-speed fiber spinning complicate the predictions of the instabilities even more.
But none of these obstacles reduced the interest of the researches in this area, because a deep understanding of the fiber spinning stability can help of explaining the
dynamics in more complicated processes like film casting and film blowing and
contraction flow, where the extensional deformation dominates. Additionally the
knowledge of when and under which conditions the instabilities occur makes easier the
control of the process.
In fiber spinning there are two major mechanisms of instability. One, called draw-
resonance, a regular and sustained periodic variation of filament thickness due to
5
fluctuation of material properties (e.g., viscosity of the extruded polymer), and/or variation of external conditions (cooling air temperature, take-up speed), persisting over a long period of time with clearly defined and fixed period and amplitude. The draw resonance may be encountered when the point of solidification is carefully controlled. In the late 1960s and the beginning of the 1970s different authors [1-12] independently used linear stability analysis to describe the instabilities in fiber spinning of Newtonian and power law fluids. They suggested various techniques for controlling draw resonance and discussed the sensitivity of the spinning line to external perturbations at low and high speeds.
The second instability arises in the range of high Deborah numbers (i.e., at high
deformation rates and/or for highly viscoelastic materials), and is known as viscoelastic
failure. This phenomenon can occur either because of growing surface perturbations
(capillarity or necking) or because of cohesive failure. The appearance of the viscoelastic
failure in highly viscoelastic polyethylene was detected and described by Ishizuka and
Koyama. [13]
Stability with respect to infinitesimal disturbances is studied by solving a set of
linear PDE’s describing the transient behavior of the process near the steady state
[1,5,14,15]. There are two methods of solving of this set: the Fourier methods or the
method of separation of variables. The solution for the deviation from steady state
consists of infinite sums or the terms like: F( x ) e λt , there x is the space coordinates and
λ is the eigenvalue of the system. The process is unstable if Reλ > 0 , i.e. the real part of any eigenvalue is positive. This means that even a small disturbance will grow with time.
If the real parts of all eigenvalues are negative, then all infinitesimal disturbances will die 6
and the system is stable against small disturbances. However, this “stability in small” does not necessarily invoke the stability of the system to finite disturbances.
Finite stability theories are extensions of the infinitesimal theory. Here one needs to find a solution of a non-linear set of PDE. Such solutions are typically limited to weak non-linearity, or small enough amplitude of disturbances, where the general structure of the linear solution is expected to hold but the time dependence of disturbances will no longer be of the exponential form. Solutions obtained in unstable range, display the transient response of the unstable process, such as the magnitude of output perturbations.
Solutions in stable region establish the size of the finite disturbance, if any, which will cause the process to become unstable. [16]
2.2 Isothermal Fiber Spinning
The melt fiber spinning is the most common of polymer fiber processing, thus a
lot of papers have been published in this field [1-12]. A schematic of the process is
shown in Figure 1.1. The molten polymer is forced through small holes to form
continuous threads of polymer filaments, or synthetic fiber. Cooling gases decrease the
temperature of the fibers so that they solidify and an initial drive roll controls the initial
take-up speed. The fiber may undergo subsequent heating and stretching to increase the
molecular orientation. Finally, the fiber is taken up onto rolls at a constant speed, with a
special tension device to control the rate of rotation in order to maintain constant yarn
speed. Maintaining a constant speed of the take-up device is important factor for the
quality of the fiber and is the main parameter, which controls the diameter and the
evenness of the thread.
7
Figure 2.1. Schematic of fiber spinning process
Plenty of numerical simulations and mathematical models have been employed in
order to study profoundly the fiber spinning process. Kase and Matsuo [1] made the pioneering attempt in this research area as early as in 1965. They proposed an asymptotic
model that gave approximate results for the circular fiber spinning with a minimum period of calculation time. Because of the complexity of the 3D model this method
assumes that variations in the radial direction are negligible because the radius of the
filament is very small as compared to the spin-line length. They integrated a set of 8
equations over the cross-section of the filament and used in the following only this averaged-over-cross-section approach. Using some empirical relations for physical properties and heat transfer coefficients, they made numerical simulation for the
Newtonian fluid and found that the instabilities occur beyond a certain draw ratio (the ratio of the take-up velocity to extrusion velocity), which for Newtonian fluids is slightly over 20.
Later Pearson and Shah [17] developed a theory for power law fluids and
discovered that if the power law exponent n is less 1 then the critical draw ratio, it is then
less than 20 and vice versa. Other constitutive equations were also employed for
modeling of the melt spinning. Matovich and Pearson [18] used the second order
Coleman and Noll fluids, Han and Segal [19] worked with the three constant Oldroyd
model, Denn et al [20,21] employed Phan-Thien and Tanner model and White used
White-Metzner model [8] in an isothermal case. In recent studies these two models have been widely exploited in many publications [22-0].
The great effort of many theoretical and experimental researchers over the past
four decades resulted in well-advanced understanding of the instability phenomena
occurring in industrial fiber spinning processes.
Donnelly and Weinberger [26] examined a polysiloxane with small relaxation
times and shear viscosities, which are independent of the deformation rate. In a good
agreement with the theoretical predictions for Newtonian fluids, they found the critical
draw ratio being close to 20.
French scientists Demay and Agassant [27] observed the critical draw ratios in
fiber spinning in the range of 20. They made experiments with different types of
9
polyesters. For polyesters with vanishing Deborah numbers the critical draw-ratio was very close to the Newtonian computational results. The increase in the Deborah number results in a slight stabilization, which has been earlier predicted by Kase.
Weinberger and Cruz-Saenz [28] performed another set of experiments using five
different polymers. Three of them were shear thinning (shear viscosity decrease with
deformation rate increasing) – high and low-density polyethylene, polystyrene and polypropylene. The critical draw ratio for all of them was found below 20 as related to decrease in the stability with increasing relaxation time. The stability of isothermal spinning of viscoelastic fluids, which have strain-rate dependent relaxation time, has been investigated using the linear stability analysis. The draw resonance of the system was found to be dependent upon the material functions of the fluids like fluid relaxation time and the strain-rate dependency of the relaxation time.
Ishibara and Kase [29] found that the stability of fiber spinning process increases with decreasing the length of the fiber melt zone. For one of the melts the instabilities disappeared and for the other they achieved very high draw ratios of 50 which means that the same fiber diameter can be achieved at a greater throughput and less pressure drop.
Lamb [30] also reported of existence of upper stable region for polypropylene.
The speed ratio at the onset of draw resonance did not appear to depend upon the calculated tensile force, although the severity of oscillation at corresponding draw ratios was greater when the tensile force was higher in the spinning zone. Periods of oscillation could be correlated with residence time of a fluid element in the spinning zone. [31,32]
The analysis of fiber spinning flow consists of solving a system of coupled differential equations of very different quality. Some equations are exact, others are semi-
10
empirical, and some functions are unknown and formulated using different (sometimes non-reliable) assumptions. The initial and boundary conditions are formulated using given technological parameters of the melt spinning process, e.g. the initial fiber velocity, initial fiber temperature, velocity profile and temperature of quenching air, etc. In spite of many uncertainties and assumptions, the numerical realization of fiber spinning model allowed simulating the process of fiber formation for various technological conditions and compare different processes and materials. These simulations also resulted in optimizing the melt spinning process in regard of the best fiber quality and productivity.
2.3 Fiber Spinning with Cooling Temperature Gradient
Even though many researchers showed that the critical draw ratio is around 20 or
less many industrial processes (such as glass or polymer fiber spinning), which are
operated non-isothermally with melt cooling, are, however, found to be stable at much
larger extension ratios. Probably the main reason for this observed stability in the non-
isothermal industrial operation come up because thinner portions of the fibers cool more
quickly and so attain a larger viscosity; this in turn reduces the rate of extension of the
thinner portions and so induces greater stability in the process.
Kase and Matsuo [33] provided a general theoretical understating of the non-
isothermal spinning process. In order to analyze the effect of the cooling they used
linearized perturbation analysis to solve the set of equations. Similar research was made by Pearson and Shah [17] for the cases of melt spinning with temperature gradient, which
is probably more significant since, it is closer to the industrial case of cooling fluid
threadline.
11
Ishihara and Kase [29] found that the onset of draw resonance is delayed with
increasing the length of the melt zone. This is related to the Stanton number (St) (number
used in heat transfer in general and forced convection calculations in particular), which is proportional to the length and inversely proportional to flow rate and diameter of the fiber. If Stanton number is high then the heat transfer is important and increasing the melt zone has stabilizing effect on the process. Ishihara and Kase also established that if the solidification happens before the take-up device the draw resonance is not observed at all.
Reports on the effect of air-cooling to the stability are controversial. Han [35] stated that the airflow increased the severity of the disturbances. Kase [29] established that air-cooling is stabilizing the process at low St numbers and has destabilizing effect in high St numbers and compared this with his experimental results.
Later Fisher and Denn [36] observed the stabilization of very short and very long filaments. They used linear stability analysis to demonstrate that short spin line will be stabilized by fluid elasticity and long spin lines by cooling because for a short melt zone the elasticity will be dominating factor while for long filaments heat transfer will come to dominate.
Therefore the cooling in general was proved to have stabilizing effect on the stability but the reasons why were not yet explained. Sukanek et al. [37] theoretically investigated viscous heating in the flow of fluids with an exponential dependence of viscosity on temperature and show that for a given shear stress, two shear rates are possible. Above a critical value, the stress decreases as the shear rate increases. This seems the most likely thermal mechanism for instability of molten polymers, since the phenomenon of the double valued pressure-flow rate curve is well known when viscous
12
heating is important; two different throughputs can occur for the same pressure drop. The analysis of Sukanek shows a variety of modes of instability for different values of
Reynolds and Prandtl numbers, but no clear interpretation is offered in the context of melt flow instability.
Jung and Song [38] attempted to find the effect of cooling mechanism. They indicate that stabilizing effect of spinline cooling is that sensitivity of spinline tension to disturbances decreases with increased cooling. And as a consequence of the reduced spinline tension the spinning becomes more stable. In addition the high level of spinline tension caused by the increased cooling always minimizes the tension sensitivity to disturbances, which dominates the overall transmission linkage between disturbances to the final spinline area at the take-up. Cooling always has stabilizing effect no matter of the type of disturbances – step change or an impulse or whether disturbance is material property changes or process changes.
In interesting paper Barnett [39] considered free and forced convection boundary
conditions and obtained a temperature profile of a spinning filament by slightly
modifying the theories of flow along the surface of thin cylinders to find the required
heat-transfer coefficients. He assumed that heat is lost by forced convection, by free
convection, and by radiation. And while the loss due to radiation is small, the losses due
to free and forced convection are of a similar magnitude. Taking this into account he
calculated the temperature of the spinning filament under stable airflow conditions.
Copley and Chamberlain [40] employed both the radiative and convective heat
transfer conditions and showed an axial temperature distribution. They revealed that
attenuation of the polymer stream occurs immediately after it has left the spinneret. The
13
filament attenuates exponentially towards its final diameter, even when the draw ratio is as high as 300:1. After discussion about both radiated and convected heat loss from the spinning filament, they concluded that the spinning parameter with greatest sensitivity to the axial temperature gradient is the mass rate of extrusion.
Chung and Iyer [41] investigated a melt spinning process considering the radial conduction and radiation effects. They examined the process response to the external disturbances like sudden small change in the inlet area and the heat flow rate and measured the transient time need for the system to establish a new steady state value.
2.4 Non-linear Dynamics of Isothermal Fiber Spinning
Linear stability analysis of the fiber spinning process does not represent the full information of the stable and unstable domains. This is because in general, linear stability analysis does not provide information about the dynamical response of the process when infinitesimal disturbance grow to finite size. So it cannot be concluded that the stable region determined from linear stability analysis remains stable at finite disturbances. E.g. the system can be stable up to a critical perturbation and if the perturbation increases beyond critical value the process may become unstable. For that reason solving the non- linear system of equation is of great interest for complete understanding of the instabilities.
There have been plenty of researchers working on the non-linear analysis of the stability of the isothermal fiber spinning. Ishibara and Kase [42] used direct numerical simulations to solve the non-linear equations of fiber spinning for Newtonian fluids.
Later they used the same procedure for power law fluids [29]. The calculated results
14
agreed well for the case of Newtonian fluids but there was a large deviation of calculations from experimental data for PET, which suggests that the amplitude of draw resonance is very sensitive to the tensile rheology of polymers.
Fisher and Denn [16] used the methods of non-linear stability theory to show that the process is stable to finite disturbances below critical draw ratio defined by infinitesimal theory. Finite disturbances die in time unless they are big enough to make the fiber area go to zero. And for higher draw ratios they will be sustained and the amplitude and the period will be the same as calculated from stability theory. Their results are doubtful because any computation of this kind highly depends on the constitutive equation.
The stability of the fiber spinning process to suddenly imposed disturbances has been studied in the work by Iyer [41]. The reaction of the system to various types of disturbance was recorded. The take-up area was computed for 1% disturbance imposed at the spinneret area and the transient response is shown. The time required for the system to reach another steady state was observed.
2.5 Flows of Polymer Melts in the Capillary Exit Region and the Die Swell
Many important effects for high speed polymer processing happen in the flows of polymer melts and solutions at high Deborah (De)/ Weissenberg (We) numbers. These
two numbers, which characterize the elasticity effects in polymeric liquids are considered
the same (and called Deborah number) in this proposal. As the Deborah number becomes
greater than 1, the polymer does not have enough time to relax during the process,
resulting in possible extrudate dimension deviations or irregularities such as extrudate
15
swell, shark skin, and even melt fracture. In this section the effects occurring in polymer flows of channels and dies are described. The importance of these regions is usually different for different polymer liquids.
The die swell (or extrudate swell, or Barrus Effect) effect is related to elastic
memory and elasticity of the melt and is qualitatively described as the release of liquid
elastic energy stored in die flow, after a sudden reformation the flow type at the die exit.
The swell is usually observed in many polymer processing operations, such as fiber
spinning, extrusion, capillary/slit flow process, etc. Generally the swell is characterized by the parameter B (the ratio of extrudate to die cross sectional area). When flowing in
the exit region of a die, the polymer fluids reconstruct their flow to diminish axial force
to zero. The die swell is typically observed [43] (Ch.5) in this case. This effect increases
with the increase in flow rate. At low rates of extrusion the swell is observed at the pipe
exit. At high rates of extrusion the tendency of the jet to swell at the exit is suppressed
and there is a delay in the swell. The theory of this effect and comparison with data can be found in Refs. [44, 45] (Ch.13)
The process parameters highly affect the die swell influencing the stresses
imposed on the material. In fact, higher shear rate in the die increases of the die swell
ratio due to the growth of the recoverable deformation. And for a fixed shear rate, an
increase of the melt temperature corresponds to reduce of the die swell, attributed to a
decrease of the stresses generated into the die [46, 47, 48].
The geometrical parameters also affect the die swell magnitude. The use of long
dies, [49, 50] or small entrance angles [51, 52, 53] highly reduce die swell, due to
reducing the viscoelastic history of the polymer deformation.
16
In summary, it can be noted that there is a relation between the deformation history of polymeric liquids under certain processing conditions and the die swell. This history is very important when the flow contraction ratio and the Deborah number (De) are high, and the die is comparatively short. In this case, the flow in the die is not developed and extrudate keeps the memory of early deformation at the entrance region.
An interesting fact about the flows of molten polymers through rectangular ducts is that a non-uniform swelling of extrudate was detected. The swell was more pronounced on the long side of the rectangular duct than on the short side, with a maximum swell at the center of the long side.
Another type of instabilities called sharkskin can be observed when the flow is unstable. These disturbances usually form larger extrudate distortions at higher Deborah numbers. While one group believes that it is a result of surface fracture (e.g. see [55]), another considers that it results from a microscopic slip-stick [56]. Although, all the researchers worked in this area agree that the sharkskin is a local exit effect and it can be delayed or eliminated when using slippery surface coating of the die exit [57], they could not concur on the origin of the sharkskin.
2.6 Viscoelastic Constitutive Equations for Polymer Melts and Solutions
Constitutive equations are mathematical relations between kinematic and dynamic variables that allow computing the stresses in a liquid for a given flow history. They are often derived from constitutive models, which imply a set of assumptions and idealizations about the molecular or structural forces and motions producing stress.
Polymers, characterized by relatively long macromolecules, do not obey simple physical
17
laws because their behavior is intermediate between Newtonian liquids and finite elastic solids. As a consequence, constitutive equations are not simple for polymers when attempting to describe specific flow phenomena, such as die swell, elastic recoil, memory effects, flow instabilities, etc, which complicate the polymer processing.
The polymer processes can be viewed as transient (time-dependent flows and discontinuous processes as, for example, injection molding, thermoforming, blow molding process) or steady flows (continuous processes—extrusion or coextrusion flows in annular, flat or profile dies, fiber spinning, film casting, film blowing, etc.). A useful tool for understanding the polymer flow in these important industrial processes is modeling.
The main difficulty with formulation of proper viscoelastic constitutive equations
(CEs) for polymer melts and solutions is a need to describe strong non-linearities, which
occur because of a combination of elastic and viscous effects at high De numbers. The
CEs are generally classified into two main categories: differential and integral. An
extensive review of many CEs along with comparisons with experimental data and useful
discussions are given in [59-61]. To date, almost none of the existing CEs has been
demonstrated to consistently describe the entire set of available experimental data, even
in the restricted region of strain rates for common viscometric instruments.
Almost all the CEs proposed in the literature display various unphysical
instabilities of either Hadamard or dissipative types, or both. In the recent paper [62] it
was revealed that only three CEs (out of twenty examined) are globally stable. All of
them are differential constitutive equations presented as a set of non-linear PDE’s.
18
Therefore they seem to be more valuable than integral models, especially for the simulation of general complex flows in both Lagrangian and Eulerian representations.
Two of the stable CEs, the FENE dumbbell model (usually applied to dilute polymer solutions) and the upper convected version of PTT model, predict zero second normal stress difference in simple shearing. The stability and descriptive ability of the modified Leonov model was well tested under various types of flows in [63]. Pade-
Laplace method successfully regenerates all the experimental data without any prior assumptions, such as the number of Maxwell modes or certain artificial settings of
Hookean moduli and viscosities. Therefore the combination of Leonov model and Pade-
Laplace method becomes a powerful tool for simulating complex flows. Numerous data for five such different polymers as PS I, HDPE II, PIB P-20, and LDPE Melt I were successfully described in [63] for steady and unsteady simple shearing as well as for simple, planar and biaxial elongation flows, while satisfying all the stability constraints.
Only few (usually one or two) “non-linear” numerical parameters, additional to the linear viscoelastic spectrum, were involved in this modeling. One of them included in the dissipative term enabled to describe different patterns in elongation flows of polymers in both groups, i.e. hardening phenomena for the first and softening for the second group.
2.7 Instabilities in Contraction Flow at High Deborah Numbers
For numerical simulation of the flow of polymeric fluids, a purely viscous but
shear-thinning rheological behavior has been successfully used in many applications
involving shear-dominated flows such as in the injection and compression molding of polymers. However, in applications involving extensional flow, predictions from such a
19
simple formulation can be quite different from the real flow. Owing to their viscoelastic nature, polymers accumulate significant recoverable strain. This recoverable strain is the main reason for the poor prediction of extension-dominated polymeric flows by a purely viscous formulation.
The steady shear and elongational flows are used as model flow fields for testing
constitutive equations for polymers. Experimental measurements of the behavior of
specific polymer solutions and melts can be compared with the predictions of the
constitutive equations. The expectation is that constitutive equations that can capture the behavior of a polymer in these model flows might reasonably be expected to work in
more complicated flow fields as well. In addition, the model flows are used to obtain
rheological parameters needed in constitutive equations. For example, simple shear flow
can be used to obtain the zero shear viscosity and the relaxation time.
The reasons why the flow in abrupt contraction received more attention than any
other complex flow are the simplicity of the geometry and the fact that this flow is
frequently encountered in polymer-processing applications. The relatively simple base
flow kinematics in these flows, in combination with the fact that precise experimental
evaluation of geometrically related (say, corner) instabilities is possible [64,65,66] make
the contraction flow ideal for studying instabilities of viscoelastic fluids. In elongational
flows the velocity component varies only in the longitudinal direction. Fiber spinning and
tubular film blowing are closely approximated as uniaxial and unequal biaxial
elongational flows, respectively. Contraction flow has also been chosen as a benchmark problem to evaluate numerical methods and constitutive equations by comprising
numerical predictions with experimental results.
20
Flow fields which occur in processing are neither elongational nor shear, but it is
sometimes helpful to characterize more complex flows by the degree to which they are
extensional or shear flow types [67,68]. Many processing flows have a large elongational
component [69] and the stability of these complex flows may be related to the problem of
stability of elongational flow. In this work we consider the stability of truly elongational
flows, making also applications to the entrance flows.
In a typical polymer processing operation, viscoelastic fluids are forced to rapidly flow through complex and confining geometries under high stresses. This non-linear flow transitions coming from the elastic nature of the polymeric fluids have huge influence on the quality of the product and require strict limitations to the processing rate.
The lack of understanding the physical mechanisms of the elastic instabilities has led to eager activity in the 1980s and 1990s in this area. Most of the earlier studies were focused on viscoelastic instabilities in the complicated flow kinematics prior to the onset of the instability [70,71]. The difficulties come from the corner singularities in contraction flows that imposes high stress gradients and a thin stress boundary layer along the downstream channel wall, which complicates numerical calculations of the base flow. That is why the recent studies have concentrated their attention on elastic instabilities in simple shear flows such as those occurring in viscometric flows [72-75] and also the viscoelastic flow through sudden contraction in the geometry. These studies have recognized different mechanisms for viscoelastic instabilities, which may aid in better understanding of the instabilities encountered in more kinematically complicated flows.
21
The axisymmetric abrupt contraction flow geometry consists of two co-axially
connected tubes of different diameters. The diameter ratio of upstream (reservoir) to
downstream (capillary) cylinder is called the contraction ratio. Flow is assumed to have a
fully developed velocity profile in both the reservoir and capillary flows. Secondary flow
near upstream of the contraction plane is either in the form of steady Newtonian vortex or
unsteady elastic vortex and sometimes these vortices co-exist at the same time.
Newtonian vortex has concave boundary and diminishes in size as flow rate increases,
while elastic vortex has convex boundary and grows in size with increasing flow rate.
Evolution of these vortices is of interest since they strongly influence the dynamics and
the kinematics of the overall viscoelastic flow in this geometry [76-82].
Far-field reservoir entrance region
Near-field reservoir entrance region Real flow velocity profile Approximated jet vel. profile Die developing region
Die developed region
Figure 2.2. Schematic of contraction flow
22
As the flow rate increase a viscoelastic fluid changes its behavior from
Newtonian–like to levels, which may reveal significantly enhanced vortex size, or
streamlines that deviate from the centerline at some distance above the contraction plane.
Beyond some critical flow rate (which corresponds to a particular Deborah number) the
flow becomes unstable and as a result large disturbances in the flow field can be seen.
The vortex can pulse symmetrically or develop an asymmetry and spiral around the
upstream tube depending on the value of the Deborah number. Numerous publications
note that these transitions may proceed via the formation of independent elastic lip vortex
at the re-entrant corner where the upstream tube joins the downstream tube [80-82].
Many experimental works have investigated the flows of polymer melt and
solution in the entrance flow region. Quantitative flow visualization techniques, such as
Laser Doppler Velocimetry (LDV) [83,84], particle tracer methods [85,86,87], laser
speckle velocimetry [88,89], and the birefringence methods [90] have been employed to
analyze the development of flow patterns near the die entrance. LDV technique was used
for measuring the centerline velocity along the axial direction [83].
The axial centerline velocity profile displayed a maximum at the entrance cross-
section, which separates the accelerated reservoir flow from decelerating flow in the die
entrance region. The researchers [90] used the birefringence method in contraction flows
of polyisobuthylene (PIB) to establish the gap-wise stress profiles with different types of
the entrance geometries. Since the birefringence depends on both the first normal stress
difference and shear stress, it is hard to distinguish directly how much each stress
component contributes in the total stress. Nevertheless, these birefringence results allow
23
for the estimation of the flow pattern at a certain axial position. In addition to the flow visualization, the normal stress profile was measured at the wall by using pressure transducers placed normal to the wall along the die [43]. These measurements showed a rapid pressure drop in the entrance region.
Using the flow visualization experiments with Laser Doppler Velocimetry (LDV),
the authors of [83] measured the centerline axial velocity and the gap-wise velocity profiles of LDPE melt. The contraction flow was generated under a 10:1 reservoir to die
ratio in the piston driven slit rheometer (i.e. under a constant flow rate condition).
In the late 1970s Nguyen and Boger [91] successfully separated the shear and
inertia effects from purely viscoelastic effects, which allow them to monitor the
generation and the growth of the secondary flow in flows with different contraction ratio.
In their experiments they used semi-dilute polymer solutions known as Boger fluids and
established that there is a significant vortex growth only if the contraction ratio is high
enough. Later McKinley [80] used Laser Doppler velocimetry and flow visualization to
confirm these results.
There are two large groups of polymer melts that display different behaviors in
the entrance contraction flows [92]. Belonging to the first group that clearly displays
entrance vortices are such polymer melts as low-density polyethylene (LDPE), polystyrene (PS), polymethylacrylate (PMMA), branched polydimethylsiloxane (PDMS)
and polypropylene (PP). Belonging to the second group are such polymers as high-
density polyethylene (HDPE), polybutadiene (PB), polyisoprene (PI), linear PDMS and
isotactic polypropylene (IPP), which generally do not display entrance vortices. The
extensive studies of entrance effects for many polymer fluids are discussed in [93,94].
24
Researchers reported that for certain polymer melts [93,94] and some polymer solutions
[95], the vortices grow with increasing flow rate. The uneven vortex growth leads to a spiral (sometimes unsteady) flow at the die entrance, which propagates downstream and creates grossly distorted extrudate [43]. It is known [59] that the polymers from the first group show the more severe extrudate distortions with shorter capillary lengths.
However, the vortices were small and did not grow with an increasing flow rate for the other molten polymers. Contrary to the first group of polymers, the extrudate distortions of polymers from the second group increase with growing capillary length [59].
In addition to the secondary flow problem, the secondary circulatory flow problem in long tubes with complicated cross-sectional geometry has also been investigated. In this case, all the experimental data, including [96], confirmed that the secondary flows are very weak, with a typical ratio of cross to longitudinal velocity of
1%. Therefore they are insignificant for such fast polymer processing as injection molding and extrusion. The paper [67], which was mostly concerned with approximate analytical evaluations of entrance flows, was the first attempt to indicate the importance of elongation flow for many aspects of the entrance flow. The importance of the extension effect at the entrance region [85,92] was experimentally confirmed using birefringence and streak photographic techniques [43]. Furthermore, similar behavior was also found for a concentrated butyl rubber solution in transformer oil [83] with the LDV technique, and for carbon black filled LDPE.
25
2.8 Numerical Studies of Polymer Flows in Channels and Dies
Analytical solutions to partial differential equations exist only in a limited number
of cases, whereas numerical methods can be used to obtain an approximate solution. The
two most common numerical methods are finite difference and finite element methods.
The main idea of the numerical methods is to calculate the values of the unknown
variables only at limited (finite) number of points, which are usually called nodes.
Analytical solution, if exists, can be used to calculate values of the unknowns at any point
in the flow domain. The purpose of one of these methods is to solve practical problems
using finite difference method, which is much easier to program and is very useful in the process of "learning-by-doing". The idea of finite difference method is to replace the partial derivatives by finite differences.
The numerical simulation of an entry region is a great challenge, because of the
high dependency of flow behavior on viscoelastic properties. Viscoelastic fluids develop
instabilities, which cannot be seen for purely Newtonian fluids. They are result of the
elastic behavior of the fluid and can be observed at very low Reynolds numbers. The biggest problems in this field are the accuracy and the inability to provide convergence
for the numerical methods at high elasticity. Several researchers have been noted that the
numerical techniques implemented play an important role in the stability of the solving
scheme.
Bush et al [97] connected the computational instability at high Deborah number to
the formulation of viscoelastic constitutive equations (CE’s) and discovered that
discrepancy of direct computations with the data is not only because of the unstable
numerical algorithm but also from the instability of CE’s used. It has been established
26
that the computed results with mPTT and modified Leonov models are in good agreement with experimental data. Another two types of CE’s has been studied by
Wesson et al [98] in their numerical study emphasizing on the instability of CE’s. In the recent publication a general theory of stability has been established and all of the above concerns were clarified [99].
During the last decade, significant progress has been made in the development of
numerical algorithms for the stable and accurate solution of viscoelastic flow problems.
An important issue that needs to be taken into account is the apparent loss of temporal
stability for many numerical schemes for viscoelastic flow computations. The problems
come from the presence of convective terms in the constitutive equation whose relative
importance grows with increasing Weissenberg number and the choice of discretization
spaces of the independent variables (velocity, pressure, extra stresses est.). Especially
for moderate to high Weissenberg number flows this can become a real trouble when purely numerical instabilities occur while the flow itself is physically stable. The
numerical computation of viscoelastic flows involves strongly non-linear, coupled
equations of a mixed elliptic-hyperbolic type. The use of conventional numerical
techniques has been proven unsuccessful due to a loss of convergence, even at low levels
of melt elasticity. Conventional finite element techniques based on the application of the
Galerkin method to the mixed formulation have failed to converge even at low levels of
elasticity. It has been proven that the computational problems are arising from the
hyperbolic nature of the constitutive equations, the existence of stress singularities, and
the method of resolution of the coupled equations.
27
Since viscoelastic constitutive equations typically represent a set of first-order hyperbolic equations, an upwind scheme is usually referred in order to obtain stable numerical results. Various upwind schemes have been proposed in the literature.
Upadhyay and Isayev [100,90] integrated the Leonov constitutive equation along streamlines to compute the extra stress due to the recoverable portion of the Finger tensor. Velocity and pressure in this scheme were determined by solving the Stokes-flow equations with the extra-stress term treated as a body force. The same method has been used more recently by Hulsen and Zanden [101] and Isayev and Huang [102] for simulations with Giesekus (axisymmetric) and Leonov (planar) models, respectively, where results (up to 256 by Hulsen and Zanden and 846 by Isayev and Huang) at high
Deborah numbers have been obtained for a 4:1 entrance flow.
2.9 Significance of Constitutive Equation
The major requirement for successive modeling is using reliable constitutive equations in combination with sophisticated numerical tools. The constitutive models should be able to quantitatively describe the complex viscoelastic behavior of polymer melts in both well-defined rheological experiments as well as inhomogeneous prototype industrial flows, while the numerical techniques should be sufficiently robust, efficient and accurate. The numerical scheme used is an important factor for precise results but the constitutive equation used is evenly essential, as the solution is only as accurate as the equation used.
Typically, a numerical simulation involves solving the set of momentum, continuity and viscoelastic CEs. For the steady, incompressible flows, the momentum
28
equation is predominantly elliptic, and the CE is hyperbolic. At high De numbers the strong hyperbolicity of the CEs can lead to a numerical instability. Hence, the high De number limit is an indicator of a physically stable CE as well as the robustness of the computational scheme.
In the late 1980s a lot of researchers realized that an appropriate constitutive
model for the stress should be selected. Apart from providing accurate predictions for the
steady base flows, this model should also be able to capture the essential dynamics of the polymer melt. They concentrated their attention on the modeling of viscoelastic flow and
came up with non-linear differential models like FENE-P and Chilcott-Rallison serving
as a generalization of Maxwell and Oldroyd-B constitutive equations [103,110]. These
equations showed good agreement with the experimental data despite of their simple
formulation.
Several numerical studies [105,106] of fast viscoelastic flows near sharp corners
revealed non-convergence with mesh refinement when using the upper convected
Maxwell model (UCM). They demonstrate that the numerical inaccuracies initiated by
singularities lead to artificial changes in equation type, which can result in loss of positive definiteness and therefore loss of convergence. These phenomena have been well
studied for Newtonian viscous liquids and linear elastic solids, using 2D simulations
[107]. For relatively fast corner flows of viscoelastic liquids, the singularities have been
extensively studied only in the case of UCM model [108]. Several attempts have been
made to study corner singularities for differential viscoelastic CEs [105,109]. It was
established that the nature of CE’s so highly affects the behavior of the solution near the
corner that a uniform behavior for all the models is hardly expected. Several research
29
papers compared different CE’s and several benchmark problems, including the contraction flow, and demonstrated that the Leonov type of CE is the most stable and the corner singularities do not affect computations for this CE up to the value of Deborah number approaching 700 [90].
A number of researches tried to develop a simplified flow model for the entrance converging flow. The methods developed by Cogswell [67,68] or Binding [110] for a simplified analysis of some converging flows and for determining the elongational viscosity from entrance pressure drops have received a lot of attention. These techniques are generally based on an analytical calculation of the shear and elongational contributions in the contraction flow that is generated during the flow of a viscoelastic fluid from a reservoir into a capillary. However, because of attempt to treat the problem purely analytically, the constitutive equation remains very simple and poorly represents the behavior of polymer melts, even including some refinements of the technique such as a power law variation of the elongational viscosity or normal stresses. Moreover, for the sake of simplicity, some kinematic approximations are generally needed. On the other hand, the use of more realistic constitutive equation to take into account the features of the viscoelastic behavior of polymer melts requires sophisticated numerical methods.
The present work will use a recently developed model [111] for high De number
contraction flow with abrupt change in entrance geometry. The development of the model
and its advantages are explained in details in the following chapter.
Along with a reliable constitutive equation some adequate parameters have to be
chosen for correct rheological characterization of polymer. Linear viscoelastic parameters
are determined from dynamic shear experiments. For this purpose the modified Pade-
30
Laplace method has been used to calculate the relaxation spectra [63]. They form the basis for the non-linear behavior. The chosen relaxation times determine where the non- linear start-up curves will deviate from the linear viscoelastic viscosity, depending on the strain and shear rates imposed. Highly non-linear parameters are especially important for description of elongational behavior of the material, and are significantly less important for description of shear. Although materials are mostly characterized by using linear viscoelastic and viscometric tests, elongational data includes more information about the non-linear material behavior. Contrary to the relatively easy viscometric, i.e. simple shear, experiments, it is very difficult to obtain reliable, accurate and reproducible shear- free data [112]. Most elongational experiments are inhomogeneous and time dependent in a Lagrangian sense. Furthermore, it is questionable if a real constant strain rate is applied and if true steady state is reached [113].
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CHAPTER III
REOLOGICAL BACKGROUND AND CONTRACTION FLOW MODEL
3.1 Rheological Characterizations of Polymers The most frequent techniques for rheological characterization of polymer melt and solutions include:
- Linear dynamic tests with determining real and imaginary parts of complex dynamic moduli, G’( ω) and G”( ω);
- Imposing a step strain in linear region with following relaxation;
- Start up shearing flow and relaxation after cessation of steady flow in non-linear
region, with measurements of steady shearing characteristics, σ12 ( γ ɺ ) and N1(γɺ ) ;
- Steady and unsteady experiments in the simple elongation with the free recovery and relaxation;
- Steady capillary flows at relatively high De numbers.
These isothermal experiments are usually carried out at different temperatures to obtain “temperature independent” plots. We extract the relaxation spectra using the data from linear dynamic test and the imposed step strain and then implementing the modified
Pade-Laplace algorithm [63]. The above experiments are performed in stable region of deformations using RMS mechanical spectrometers, elongation rheometer of Meissner type, and capillary rheometers.
32
3.2 The Choice of Constitutive Equation
Important aspects of the modeling of viscoelastic flows are the choice for the
numerical algorithm and rheological model. It is well known that the numerical algorithm
is essential for the correct prediction of the stability behavior of complex flows since
most numerical schemes produce approximate solutions that are not solutions of the
original problem. Also, if the numerical analysis is correct and approximate solutions are
solutions of the viscoelastic operator, the outcome of the stability analysis will depend on
the capability of the non-linear constitutive model to describe real polymer melts. For that
reason the choice of the constitutive equation is a crucial point in modeling of complex
flows. The suitable CEs have a capability to correctly represent non-linear behavior of the
melts in both elongation and shear.
In recent years, many different constitutive equations have been proposed in the
literature in other to capture the viscoelastic behavior of polymers, for example, the
extended Pom–Pom model (XPP), a few-parametric modified White–Metzner model
(mWM), and the modified Leonov model (mLeonov). An extensive review of these and
other constitutive equations can be found in the book by Larson. [60]
In this work we will use modified Leonov model for simulating the elongational
flow. Some reasons why Leonov model is preferred are as follows.
The model, developed by Leonov is based on the principles of irreversible
thermodynamics. It can be considered a non-linear form of the generalized upper
convected Maxwell model with multi-mode relaxation times, and a viscous retardation
term that includes strain hardening. It has the advantage of being used in non-isothermal
33
and compressible problems and yields additional information about the energetics of viscoelastic flow and recoverable strain tensor.
This model, being thermodynamically consistent, is also capable to describe the
non-isothermal effects of viscoelastic liquids due to the dissipative heat generation [59]
(Ch.9) under certain non-isothermal conditions. The coupling between the heat and
constitutive equations is possible owing to the relationship between the dissipative term
in the heat equation and sharp dependencies of relaxation times (or viscosities) on
temperature. Some effects of thermal instability and thermal blow up in extensive flows
can be found in [59] (Ch.9).
The model is intended to save computational cost by simplifying the contraction
flow using two different characteristics of the flow in different geometrical regions. It
demonstrates a satisfying comparison with existing experimental data and direct
numerical simulations for shear and elongational flows. Furthermore, in this approach
there are no adjustable parameters involved in the flow modeling and the calculations
demonstrate better agreement with experimental data or/and direct calculations when De
number and contraction ratio are high enough.
Additionally, in the Leonov model, the Finger tensor is positive definite by
definition which was used to prove that Leonov model based on the rubber-elasticity
work potential is globally evolutionary [61].
3.3 Development of the Model
The general form of the CEs for the extended Leonov model written for simplicity
for a single non-linear Maxwell mode is:
34
∇ ∇ T ccT+ϕθ()/()0; = cc ≡+⋅∇−⋅∇−∇⋅ ∂t (v) cc v(v); c σρ∂∂ = 2 cFc ⋅ /. (3.1)
Here c is the elastic (Finger) strain, a symmetric and positive definite tensor. The Finger tensor represents the portion of total strain, which will be recovered if all the stresses are suddenly released; v is the velocity vector, ∇v is the velocity gradient tensor; ϕ(c ) is
the dissipative term; θ (T ) is the relaxation time depending on temperature T; σ is the stress tensor related to the free energy function F per mass unit and ρ is the density. The first equation in (3.1) is the evolution equation for the tensor c , the second one is the
definition of upper convected tensor time derivative, and the third one defines the elastic
strain-stress relation. In the isotropic case under consideration, F= FTII(,1 , 2 , I 3 ) . Here
Ik are the basic invariants of tensor c :
2 2 2 2 Itrc1=; I 21 =− 1/2( Itrc ); I 3 = det c ( I 30 = ρ /) ρ . (3.2)
Here ρo is the density at the rest with the same temperature T In incompressible
case, I3 =1 (ρ ≡ ρ o ) and ∇⋅v = 0 , when ρo F≡ WTI(,1 , I 2 ) , the elastic strain-stress
relation in (3.1) is of the form:
σ=−+p δσσ; =⋅ 2 cWccWIcWI ∂∂ /2/2 = ∂∂ − −1 ∂∂ /. (3.3) e e 1 2
Here σ is the extra stress tensor, p is the isotropic pressure and δ is the unit e
tensor. It should be noted that in general, the elastic potential could be scaled as:
35
W= GTwI()(,1 I 2 ;) a (3.4)
where G( T ) is the Hookean modulus slightly depended on temperature, and a is the set
of numerical parameters (usually one or two) which characterize the non-dimensional
function w .
In incompressible case, the dissipative term ϕ(c ) in Eq (3.1) has the general
structure:
2 ϕ()c= bIImc (,;)[12 +−− cI ( 21 I )/3 δ ] (3.5)
where m is the set of numerical parameters (usually one or two), which characterize the
non-dimensional function b . Equation (3.5) can easily be extended on compressible case.
In order to complete formulating the model for one Maxwell mode, we need to
specify two scalar functions: the non-dimensional elastic potential w in Eq.(3.4) and the function b in Eq.(3.5). The proper choice of those functions has been suggested and
tested in [61, 63]. Here we use the fairly general case of elastic potential W for the further
formulations:
3G ( T ) WIIT( , , )= {(1 −β )[( I / 3)n+1 −+ 1] β [( I / 3) n + 1 − 1]} (3.6) 1 2 2(n + 1) 1 2
with the specifications for testing materials used in this proposal:
bI(1 , I 2 ; m )= 1, β = 0 and n ≈ 0.1 : (3.7)
36
CEs (1)-(5) are extended in [59,63] on multi-mode case with linear discrete
relaxation spectrum, {G ,θ } , when using a set {c } of elastic strains associated with k k k
each k th non-linear relaxation mode. Multiple relaxation times within the relaxation
spectrum are necessary to satisfactorily describe the linear and non-linear behavior of
commercial polymers. Single-mode models can qualitatively describe the behavior of polymer melts and therefore contribute to understanding polymer processing. However, a
quantitative description with single-mode models is generally impossible. They predict
incorrect slopes in both transient start-up as well as steady state behavior.
The model is formulated in such a way that the evolution equation (3.1) holds for
each mode with different parameters θk , and the stress tensor is represented with the aid of Eq. (3.6) as:
σδσσ=−+p; = 2 Gcwc ⋅ ∂∂ / = GIc (/3)n . (3.8) ee∑k kk k ∑ k 1 k k k
It is important that in all the multi-modal treatments of the above CEs the sets of non-linear numerical parameters α and m were kept the same for all the relaxation modes. In this case, the functions w and b are the same for each relaxation mode, i.e.
wc( )≡ wc ( ); bc ( ) ≡ bc ( ). (3.9) kc k k kk
This gives rise the opportunity to employ the scaling approach in computations of
complex flows.
37
3.4 Set of Equations for Solving Inherently Difficult Viscoelastic Flow Problems
The simulation of the flow of viscoelastic materials involves the solution of a set of coupled partial differential (or integral-differential) constitutive equations, including the equations for conservation of mass and momentum. If the flow is compressible or non-isothermal, conservation of energy and an equation of state for density are also required. Here the constitutive equation is needed to relate the extra stress to the deformation field. If the density is pressure dependent, then a constitutive equation relating pressure and density is required. In what follows we shall generally take the density to be independent of the isotropic pressure, although polymer melt compressibility may sometimes be important.
The extra stress tensor is sometimes split into viscous and viscoelastic
contributions. This splitting is often arbitrary but it has a strong stabilizing effect in most
numerical schemes. In the present work we use the rheological equation described by the
modified Leonov model.
The mathematical type of a system of equations can be described in terms of characteristics. Ellipticity is represented by complex characteristics, while hyperbolicity is represented by real characteristics. Ellipticity and hyperbolicity are the mathematical concepts associated with diffusion and propagation. Ellipticity has a regularizing effect on a singularity while hyperbolicity destabilizes the flow by propagating the singularity.
For example, long distance effects controlled by an elliptic set of equations depend only on averaged quantities, whereas phenomena controlled by a hyperbolic system of equations, like shocks or singularities, are transported. The set of equations for a purely viscous fluid is elliptic, but a viscoelastic constitutive equation like Leonov model is
38
hyperbolical. The combination of ellipticity and hyperbolicity complicates the singular behavior near abrupt contraction. It induces a stress singularity and results in both mathematical and numerical difficulties, since the rheological equation transports the effect of the singularity and pollutes the velocity field. In the next section we will discuss some numerical approaches that have been proposed to overcome these difficulties.
3.5 The Choice of the Numerical and Iteration Scheme
Several discretization techniques have been used to solve the conservation and constitutive equations, and these include finite element, boundary element, finite difference and spectral methods. The majority of published simulations have been carried out using finite difference methods because of its relatively easy programming and low computational time.
According to their properties the upwind formulations of the finite difference
methods produce the most stable results for solving the system of differential equations.
The stability of the numerical scheme is important since the origin of the disturbances
must be demonstrated and the numerical instabilities must be avoided.
Two iteration schemes have typically been used to solve the set of governing equations - root finding technique and matrix solver.
In the simpler approach, the use of the Newton-Raphson method linearizes the complete set of equations including the constitutive one, which are solved simultaneously. This method is preferred because it has better convergence rate. It has been used by Marchal and Crochet [114] and Rao and Finlayson [115]. However, simultaneous solution of the velocity, pressure and stress equations, which can easily be
39
of the order of 10,000 equations, even with only one mode in the constitutive equation, makes the scheme computationally inefficient. Furthermore, due to the non-linear nature of the constitutive equation, the matrix as well as the forcing vector for the simultaneous linear equations needs to be updated on every iteration step.
The Picard iteration scheme solves the problem in two stages. Starting with an
initial approximation of the velocity and stress fields, the momentum and mass-
conservation equations are solved in the first step, with treating the extra stress from the
viscoelastic constitutive equation as a known body force. With known velocity field, the
constitutive equation is then solved to update the extra stress tensor. The two steps are
then repeated successively until it converges. Besides breaking the problem into two parts, there are two reasons why this scheme is especially desirable for solving the
viscoelastic flow problems. Firstly, since the momentum and mass conservation
equations are linear, the matrix for the system of linear equations for the Stokes-flow problem is formed and triangulated using LU-decomposition only in the first iteration. In
subsequent iterations, as the extra stress changes, only the forcing vector on the right-
hand side of the linear equations changes, so that the solution requires only back
substitution. Secondly, in the case of multiple modes in the constitutive equation, the
equations for each mode are decoupled and can be solved separately in the Picard
scheme. However, as noted by many researchers in the literature, such scheme tends to
have poor convergence.
In our research we might need to use either of the two iterative methods based on
the problem we are dealing. The equations for isothermal fiber spinning are been solved by root finding technique since the time needed to obtain the solution of the PDE set is in
40
within couple of minutes. The computing time increases when the grid is refined over
300 nodal points, but the accuracy of the solution is not much improved. Thus root finding technique will be preferred unless the time for computing the solution is too large.
41
CHAPTER IV
ISOTHERMAL FIBER SPINNING
4.1 Linear Stability Analysis of Isothermal Fiber Spinning
Linear stability analysis is a technique used to describe the response of the fiber spinning process when infinitesimal disturbances are imposed and also to determine the critical conditions of the process.
4.1.1 Process description
The fiber spinning process is shown schematically in Figure 4.1. An axisymmetric stream of viscoelastic liquid emerging from a die of diameter D is taken up by a drawing
force F with a speed uL larger than the extrusion velocity, at the position downstream with the distance L from the die. The extruded fiber initially swells owing to the relaxation of the normal stresses, but beyond a distance of approximately two diameters downstream it slims continuously. Near the location of the maximum diameter, the axial velocity and the viscoelastic stress became virtually uniform over the cross section of the fiber i.e. flow becomes fully developed there. If the fiber is long enough i.e. L / D >>1, the die swell becomes unimportant and the fiber diameter, drawing force, and the draw ratio can be deduced by analyzing the flow between 0< z < L as shown in Figure 1. For that reason the computations has been made choosing as an initial point where the
42
diameter of the fiber after the die swell becomes equal to the diameter of the die. The distance L is taken from initial point till the point of solidification.
Figure 4.1. Sketch of the fiber spinning process
The analysis of any flow problem requires solution of the closed set of PDE’s,
consisting of equations for conservation of mass and momentum, along with an adequate
viscoelastic constitutive equation, with appropriate initial/boundary conditions.
Our goal here is to demonstrate a procedure for determining the critical draw ratio beyond which there is onset of instability in isothermal fiber spinning. The same technique will be used to simulate contraction flow instabilities and predict the critical
Deborah number beyond which the instabilities occur.
The governing equations for isothermal melt spinning are described as follows:
Neglecting all the secondary forces on the thread-line like inertia, air draw resistance,
gravity and surface forces, and the variations of variables across the spinline cross-
section, results in the equations describing 1D model. Here the origin of the longitudinal
coordinate starts at the die swell region thus ignoring the pre-spinneret conditions on the
spinline. It is also assumed that no crystallization occurs inside the spinline. In non-
inertial approach the flow rate must be constant along the fiber and thus equal to the flow
43
rate in the spinneret upstream. It can be assumed that the uniaxial deformations and stresses in long fibers are close to those in homogeneous elongation, so the shear stresses could be neglected in this 1D model.
The simplified for of continuity equation for one dimensional flow is represented as following:
∂a ∂ ( v a ) + = 0 (4.1.1) ∂t ∂ x
As discussed in the previous chapter, the Leonov model for simple elongation flow can be presented in slightly inhomogeneous fibers spinning situation as:
1∂λd λ 111 k k 2 ɺ +v + (λk −+= )(1 ) ε . (4.1.2) λk ∂t dx 6 θλλk k
I1, k n 2− 1 σext=∑G k( ) ⋅ ( λ k − λ k ), (4.1.3) k 3
The condition of constant spinning force acting on the fiber along the spinline is
represented as:
I1, k n 2− 1 2− 1 FaaGext≡⋅=⋅σ ext∑ k()( ⋅−= λ k λ k ) Ft ext (), I1k=λ k + 2 λ k . (4.1.4) k 3
In equations (4.1.2-4.1.4) a is the spinline cross-section area, v is the (averaged over cross-section) spinline velocity, λ is the elastic extension ratio, σ is the spinline axial stress, x is the distance from spinneret, t is the time and F is the spinning tension which
44
is in general time dependent, and the index k indicates that there are several relaxation
modes. The parameter n is same for most of the polymers and it is equal to 0.1.
Thus the set of equations consists of the equation of continuity, the condition for
constant tension and a number of evolution equations that must be solved simultaneously.
4.1.2 Non-dimensional Governing Equations
The linear system of equations is non-dimensionalized in order to ensure for the different magnitude of the variables. The continuity equation, the momentum balance
equation, and the dimensionless form of the evolution equation for elastic strain λk are
represented below:
∂A ∂ ( V A ) + = 0 (4.1.5) ∂t' ∂ X
∂ (A .σ ') = 0 ∂ X (4.1.6)
∂∂λ λλλλ(2 −− 1 )( − 1 + 1) ∂V k+V kkkk + = λ (4.1.7) ∂tX' ∂ 6 Dek ∂ X
Here we introduced the following dimensionless variables and parameters:
V t t ' = 0 - non-dimensional time, (4.1.8) L
x X = - non-dimensional spinline distance from a spinneret, (4.1.9) L
a A = - non-dimensional cross-sectional area, (4.1.10) A0
45
v V = - non-dimensional velocity of the filament, (4.1.11) V0
σ .A σ ' = 0 - non-dimensional stress, (4.1.12) F
θ.V De = 0 - Deborah number. (4.1.13) L
In definitions (4.1.8)-(4.1.13), L , A0 and V0 are the distance from a spinneret to a take-up device, cross sectional area and the velocity at the die exit, respectively. Additional dimensionless parameter, the Deborah number, characterizes the dynamic behavior of this viscoelastic fluid in melt spinning.
4.1.3 Initial and Boundary conditions
The question of boundary conditions to be imposed on the 1D melt spinning flow
is by no means clear from a physical point of view. Except for the downstream condition,
all the boundary conditions can be questioned. Even the location of the upstream boundary is not obvious. The sensitivity of stability to the choice of boundary conditions
has been observed for multimode models at low Deborah numbers. Although the critical
draw ratio for the onset of the draw resonance instability was in agreement with previous
analyses, at higher De it was found significantly depending on the stress boundary condition imposed at the inflow boundary. For a certain boundary condition and beyond a certain value of De , a new mode of instability with half the temporal and spatial
frequency of draw resonance was found at small draw ratios.
46
In these calculations the prescribed velocity boundary conditions were carefully chosen as:
t > 0 : A= A 0 = 1 , V= V 0 = 1 at x = 0 (4.1.15)
V= VL = r at x = 1
4.1.4 Steady State Solution
Equations for the steady state are obtained when the time derivatives in Eqs
(4.1.5)-(4.1.7) are equal to zero. Then the set is presented in the form of first-order ordinary differential equations:
∂ (V A ) = 0 (4.1.16) ∂ X
∂(A .σ ') = 0 (4.1.17) ∂X
∂λ( λ2 − λ− 1 )( λ − 1 + 1) ∂V V k+ k k k = λ (4.1.18) ∂X6 Dek ∂ X
A numerical procedure has been employed in the calculations using root-finding routine,
Newton-Raphson method, for solving Eq. (4.1.16), (4.1.17) and (4.1.18) until the boundary condition (4.1.15) is satisfied. The following figures illustrate the calculated
dimensionless area and dimensionless velocity profiles on example of HDPE, for
different values of the draw ratio. The same numerical computations are performed for
the rest of the polymers as well using different draw ratios.
47
1
0.9
0.8 DDR=1.2 0.7 DDR=2.5 DDR=6.2.5 7 0.6 DDR=20 A 0.5
0.4
0.3
0.2
0.1
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance from the die
Figure 4.2. Area change along spinline for HDPE 6009 for different draw ratios
20 DDR=1.2 18 DDR=2.5 DDR=6.72.5 16 DDR=20
14
V 12
10
8
6
4
2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance fom the die
Figure 4.3. Velocity change along spinline for HDPE 6009 for various draw ratios
48
4.1.5 Linear Stability Analysis for Small Disturbances
As mentioned before, many researchers have successfully applied conventional linear stability analysis to a fiber spinning process. The procedure assumes that all variables can be considered as sums of finite, reference components that describe the main flow and infinitesimally small perturbations about those values.
Substitution those sums for each dynamic variable into the equation of continuity
and motion, and holding only first order perturbation terms results in a series of linear
equations that can be solved under physical constraints imposed on the system under
investigation.
The small perturbations were introduced as follows:
Atx(,)= Ax ().(1 + Axe (). .t ) (4.1.19) 0 1
Vtx(,)= Vx ().(1 + Vxe (). .t ) (4.1.20) 0 1
λ(,)tx= λ ()(1 x + λ (). xe .t ) (4.1.21) 0 1
Here the subscript 0 indicates steady state, A1 , V1 and λ1 are the perturbed
amplitudes, and is a complex eigenvalue that accounts for the growth rate of the perturbations. For simplicity, the way of introducing disturbances for elastic strains is
demonstrated in (4.1.21) for a single relaxation mode. The value of is determined by solving an eigenvalue problem. If the real part of is positive, then any arbitrarily small deviation from steady state will grow in magnitude with time. Since infinitesimal disturbances cannot be kept out of any real process, it follows that such a steady state
49
flow cannot be maintained in practice despite the fact that it is a solution of equations of motion. Conversely, if the real part of is negative, then all infinitesimal disturbances
will decay to zero, and the system is stable. This infinitesimal stability does not
necessitate, however, the stability relative to finite disturbances, which an experimenter
may see as small.
By substituting Eqs (4.1.19) through (4.1.21) into PDE’s (4.1.16) through (4.1.18)
and neglecting the second and higher-order terms the perturbation, the following linear
homogeneous ordinary differential equations are obtained, which for simplicity are
demonstrated only for one relaxation mode:
∂A ∂ V +.A V1 + V 1 = 0 (4.1.22) 1 0∂X 0 ∂ X
VVV10100.λ . ∂ λ ∂λ1 λ 1 ∂ V 0 ∂ V1 ++λ1 () ++V0 g 1 − ()0λ 110 +− VV = (4.1.23) λ0 λ 0 ∂∂x xDe6 ∂ x ∂ x
σ 1+A 1 = 0 (4.1.24)
Here σ1 is dimensionless stress perturbation and σ1= Gg 21 λ/ σ 0 , and the functions g1 and
g2 here dependent only on λ0 , are presented as follows:
−2 2 − 1 g1=(2λλλ 000 + )( ++− 1) λλ 00 (4.1.25)
λ2+ 2 λ − 1 2n g =+(2λλ−2 )(0 0 )n ++ ( λλλλλλ 21 2− )( −− − 221 )( − ) (4.1.26) 2003 3 n 000000
50
These functions gk (λ0 ) are found from the steady-state solution and considered below as known functions of x .
For the spinning processes the take-up speed is taken as constant and the related boundary conditions are:
A1(0)= V 1 (0) =λ 1 (0) = V 1 (1) = 0 (4.1.27)
4.1.6 Numerical Approach
Equations (4.1.22)-(4.1.24) with boundary conditions (4.27) represent an eigenvalue problem whose nontrivial solution exists only for a certain value of . Since
this problem cannot be solved analytically, we have to use a numerical approach.
The finite difference method, discretizing the set of ODE’s, has been proven to be most
stable and accurate. For simplicity we illustrate below the employed numerical procedure
on the example of a single relaxation mode. We first define the values of A0 , V0 and
λ0 in the finite number of points solving the steady flow problem. The second step is to
insert the perturbation terms into the system of equation for steady flow. After
discretizing and rearranging these equations, we obtain linear algebraic system of
equations. If n is the number of grid points, this algebraic system consists of
n −1equations of continuity, n −1 constitutive equations and n − 2 equations of constant
force. The unknown number of variables 3n − 4 is same as the total number of equations.
This set of equations is presented in the matrix form as:
51
i i A1 A 1 i i MV.1= . NV . 1 (4.1.28) i i λ1 λ 1
Here M is the “grid” matrix, whose form is demonstrated for a simplified case in the
following table. N is a matrix which takes care of the missing time dependent term in
i the constant force equation and makes all V1 from the right hand side equal to zero.
Given the boundary conditions the eigenvalues of the system can be easily obtained using
QZ algorithm. This is a generalization of the QR algorithm, which solves standard
eigenvalue problems. The matrixes M and N are first simultaneously reduced to
condensed forms by unitary equivalence transformations. If applied iteratively, it reduces
M to triangular or quasi-triangular form, while preserving the triangular structure of N .
Eigenvalues can be computed from the diagonals of the triangular form. All the
eigenvalues are sorted on the basis of sign of the largest eigenvalue the real part. The process is marked as stable if this eigenvalue is negative, or unstable otherwise. The
critical draw ratio is defined as that where the real part of the largest eigenvalue is equal
to zero.
For following example the number of equations is taken to be 11 and the number
of grid points is 5.
52
Table 4.1. Simplified matrix ( M) for solving linear system of equations
Coefficients in front of A1 Coefficients in front of V1 Coefficients in front of λ1
Ai=1 Ai=2 Ai=3 Ai=4 Vi=1 Vi=2 Vi=3 λi=1 λi=2 λi=3 λi=4 A 0 0 0 A 0 0 0 0 0 0
B A 0 0 B A 0 0 0 0 0
0 B A 0 0 B A 0 0 0 0
0 0 B A 0 0 B 0 0 0 0
0 C 0 0 0 0 0 0 D 0 0
0 0 C 0 0 0 0 0 0 D 0
0 0 0 C 0 0 0 0 0 0 D
0 0 0 0 E 0 0 F 0 0 0
0 0 0 0 A E 0 B F 0 0
0 0 0 0 0 A E 0 B F 0
0 0 0 0 0 0 A 0 0 B F
V0 ( i ) V0 ( i ) Gg 2λ 1 A = − B = C =1 D =σ1 = x x σ 0
λ0 (i − 1) g1( i ) ViVi0() 0 (− 1) Vi 00 ()λ ( i − 1) EVi=0( ) − Vi 0 ( −+ 1) Vi 0 ( ).( ) F =− + + − . λ0 (i ) 6De x x xλ0 ( i )
The period T of the draw resonance oscillation can be obtained from the imaginary part of the eigenvalue as:
2. π i = (4.1.29) T
The above relation is valid only at the critical draw ratio because harmonic oscillations are possible only at the onset of draw resonance, while at higher ratios the oscillations become non-harmonic.
53
4.1.7 Accuracy of the Numerical Scheme
The accuracy of the numerical scheme has been checked. Since the time for computation is very small we can calculate the eigenvalues with precision much higher then the desired one. If the computation time increases for the more complicated cases the precision might be reduced.
All the computations are made for different grid size. Starting from 50 grid points up to 2000 and it was concluded that for a grid size (number of grid points) larger than
200 the accuracy of the result is not worth making lengthier computations. Therefore a grid size of 200 was chosen for the given example.
Computer software used has been elaborated for multi-mode computations. The experimental data available did not include polymer melts or solutions with more then one relaxation mode. For future applications the more sophisticated version of the computer code can be used.
4.1.8 Results and Discussions
In order to test the numerical scheme and the reliability of constitutive equation
we compare the results of numerical simulations to the literature experiments. We choose
two separate experimental papers [27, 28] where the fiber spinning process was performed for different polymer melts. The numerical simulations are carried out for all polymers and the results are compared to the experimental data available.
The first set of experiments was conducted with a melt spinning apparatus
consisting of extruder, an isothermal chamber, a quench bath and a variable speed take-
54
up device (Figure 4.2). The materials were pressured through a 1.59 mm circular die
(L/D=3). The isothermal chamber was 165 mm long, extended from the die to the quench bath; so all the drawing took place within the chamber and at a constant temperature
(218 o C). Constant throughput was maintained at 6.0 cm 3/s corresponding to die velocity
of 5 cm/s. The speed of the take-up system was increased till the draw resonance was
observed. The materials used in the paper are polystyrene (melt index 4.5g/10 min),
atactic polypropylene and high and low-density polyethylene. Mixtures containing each
of HDPE 6009 10%, LDPE NA107 30%, PP 6523 10%, PS HF77 10%, HDPE 900N
10%, were prepared with xylene and water, stirred for 45 min at 120 oC and 300 rpm, and settled for approximately 10 min. Filtering the hot solution, cooling the filtrate to approximately 50 oC gave a clear, particle free solution of all polymers. Several test have been performed in order to describe fully the rheological properties of these polymers.
Shear and dynamic experiments have been conducted for the materials and the following
relationship for the relaxation time has been suggested:
G1 λ0 = lim (4.1.29) ω→0 2 η1 ω
The relaxation times for all the polymers are shown in table 4.2. The same values are used for performing the calculations and the results are compared in the table.
55
Pressure Transducer
Extruder Die Tempo couple Isothermal Chamber Take Up Rolls
Water bath
Figure 4.4 Sketch of apparatus for isothermal fiber spinning
A computer code using Matlab was created for solving the set of ODE implementing the procedure described above. The computer program was able to work with multimode constitutive equations and generally took less then a minute to compute the eigenvalues (for 200 grid points) for a given processing conditions.
The results of the numerical simulation are presented in table 4.2. The numerical result for LDPE deviated from the experimental data but the experiment was not carried out under isothermal conditions, which increases the stability of the process. For the case of HDPE we can see the draw ratio is close to the predictions for Newtonian fluid, because of the small Deborah number.
56
Table 4.2. Comparison between numerical results and experimental data for isothermal fiber spinning, T= 218 0 C
Polymer Relaxation time (s) Experimental [27] DDR cr Computed DDR cr
LDPE NA107 1.31 4.6 3.64 non-isothermal
HDPE 6009 1.64 4.5 4.28
PS HF77 0.68 3.6 3.46
PP 6523 0.65 2.7 2.62
HDPE 900N 0.358 18.5 16.88
Another set of experiments were made for three different polyesters: linear, branched and obtained by post condensation. The experimental setup is shown on the following figure.
Barrel
Spinneret
Water Take-up Pulley
Guide Pulley
Figure 4.5. Sketch of apparatus for isothermal fiber spinning
57
The temperature of experiment was 285 o C. The die velocity was 4.18 cm/s. The length of the isothermal chamber was 100-mm. The take-up speed varied from 3 up to
300 m/min. The mean relaxation times are deducted for different values of elongational strain rate from the shear tests performed experimentally. These relaxation times are used in calculating the critical draw ratio. The computed and experimental draw ratios are compared in Table 4.3.
Table 4.3. Comparison between numerical results and experimental data for three types of polyesters, T= 285 o C .
Polymer Relaxation time (s) Experimental [28] DDR cr Computed DDR cr
Linear polyester 0.0066 17.2 16.3
Branched polyester 0.0176 15.9 15.28
Post condensed 0.0007 19 18.22
There is a fair agreement between the numerical predictions and experimental
data, which gives us good staring point for solving more complicated problems like
temperature dependent fiber spinning and contraction flow.
4.2 Non-linear Stability Analysis of Isothermal Fiber Spinning
Linear stability analysis of the fiber spinning process does not represent the full
information of the stable and unstable domains. This is because in general, linear stability
analysis does not provide information about the dynamical response of the process when
infinitesimal disturbance grow to finite size. So it cannot be concluded that the stable
58
region determined from linear stability analysis remains stable at finite disturbances. E.g. the system can be stable up to a critical perturbation and if the perturbation increases beyond critical value the process may become unstable. For that reason solving the non- linear system of equation is of high interest for complete understanding of the fiber spinning instabilities.
4.2.1 Governing Equations
In order to obtain the solution for the non-linear case we have to solve a closed set
of PDE’s, consisting of equations for conservation of mass and momentum, along with
viscoelastic constitutive equation, with appropriate initial/boundary conditions.
The main objective of this analysis is to determine the behavior of the process around the critical draw ratio and to determine the response of the system if finite disturbances are introduced. The same procedure will be used later for describing the non-linear instabilities in contraction flow.
The problem here is no longer the eigenvalue but initial boundary value problem with Dirichlet boundary conditions.
For simplicity again all the secondary forces on the thread-line like inertia, gravity and surface forces, and the variations of variables across the spinline cross-section, are neglected. So the 1D model is used to predict the evolution of the disturbances. It is also assumed once again that no crystallization occurs within the spinline.
We rewrite the non-dimensional equations already used before - the continuity equation, the momentum balance equation and the evolution equation for elastic strain:
59
∂A ∂ ( V A ) + = 0 (4.2.1) ∂t' ∂ X
∂ (A .σ ') = 0 (4.2.2) ∂X
∂∂λ λλλλ(2 −− 1 )( − 1 + 1) ∂V k+V kkkk + = λ (4.2.3) ∂tX' ∂ 6 Dek ∂ X
Here the dimensionless variables and parameters are the same as described in the previous section (4.1.2).
4.2.2 Initial and Boundary conditions
The non-linear viscoelastic fiber-spinning problem is separated in two parts. Once
again the steady state case is solved first and the base flow is determined. Then the
disturbances are introduced as an initial condition and the boundary value problem has been solved.
All the assumptions regarding die swell and the sensitivity of the boundary
conditions discussed in the previous section are also implied here. The boundary
conditions for the steady state part are the same as before:
t > 0 : A= A 0 = 1 , V= V 0 = 1 at x = 0 (4.2.5)
V= VL = r at x = 1
When solving the problem for disturbances, the initial conditions are important parameters and by varying them the limits of the process can be determined. Sinusoidal
60
disturbance has been introduced in the fiber area at a certain time, which has been taken as starting point for the computations. It means that the simulation started when the disturbance appears and this moment. To preserves the continuity at the system the harmonic disturbance has been chosen as shown below:
An e w = AA0 − 1 .sin(2.π . Nx . ) (4.2.6)
Here Anew is the disturbed value of flowing filament area, A0 is the initial value of the
area calculated from the steady state, A1 is the amplitude of the disturbance and N is the
wave number of initial disturbance. By changing the amplitude and the period of the
introduced disturbance the full range of the stable and unstable region can be found.
4.2.3 Numerical Approach
Equations (4.2.1) through (4.2.4) represent an initial boundary problem. The finite difference scheme has been chosen to obtain the numerical solution of the problem.
First the steady flow problem has been solved using finite difference formulas for derivatives in perturbation equations described in the previous section. The critical draw ratio is defined using the linear stability analysis. Then the system of PDE’s has been solved for draw ratios near the critical one. The reason for this was establishing whether the process is unstable relative to finite disturbances in the stable region. For numerical discretization forward in time and central in space of second order scheme has been used.
It is worth mentioning the fact that one of the equations does not contain the time dependent term. The system has been solved by establishing the extensional rate at the
61
next time step, then calculating the area from the equation for constant force, and finally using the continuity equation to find the velocity at the next point employing backward difference in time. Here is the finite difference discretization of the system of equations for a single relaxation mode:
dt. g n (λ n ) dt dt λλnn+1 =+i i +.(V nnn −− V ). λλλ .( nnn − ). V (4.1.7) ii6.De 2. dx iii+−11 2. dx iii +− 11
n+1G 0.11 nnn + 12 +−+ 11 12 2 0.1 Ai=() .[(λ iii ) − ( λ )].[( λ ) − n+1 ] (4.1.8) 3 σ0 λ i
nn+1 + 1dx n + 1 n AVii.−1 − ( A i + A i ) n+1 dt Vi = n+1 n + 1 (4.1.9) 2. Ai− A i −1
n n2 n− 1 n n Where gi=[(λ i ) − ( λ i ) ][ λ i + 1] is a function of λi only.
4.2.4 Accuracy of the Numerical Scheme
The accuracy of the numerical scheme has been checked again. All the computations were made for different grid size from (50 to 5000). As usual the accuracy drastically improved with coarser grid but the computational time increased as well. For that reason the grid size of 2000 has been taken in order to satisfy both factors: accuracy and computational cost. Increasing the grid size more make the computations too lengthy and the accuracy improves with less then 2%. The computations can take even longer time if the relaxation times are more then one. For the chosen grid size the computation time is around thirty minute for a single relaxation mode.
62
4.2.5 Results and Discussions
The simulations have been performed for various amplitudes of the introduced sinusoidal disturbance (1%, 5% and 20%) and several periods of the disturbance in order to find the complete range of the stable and unstable regions. Two separate cases have been examined: nearly below and nearly over the critical draw ratio for four different polymers.
Starting with 1% and then increasing the amplitude of the introduced disturbance the respond of the system has been examined. For all amplitudes the period (i.e. frequency) of the disturbance has been changed from 2 to 15. For all the polymer solutions described in the pervious section (5.5) the results showed that independently of the amplitude of the introduced disturbance the process continue to be stable in linear stable area.
The general conclusion is that the process is stable to finite amplitude disturbances if the draw ratio is less then the critical draw ratio. This means that the sign of real parts of all the eigenvalues are negative.
Another set of calculations has been made in order to find whether the finite disturbances imposed in non-steady region would stabilize the flow. All the results demonstrated the negative answer to this question. So the process remained unstable with disturbances growing until the fiber breaks.
The following figures demonstrate the time evolution of the initial disturbance for two polymer solutions with different draw ratios (HDPE 6009 and PP 6523). The figures are created every couple of hundred iterations showing the increase (or decrease for
63
stable case) of the amplitude of the initial disturbance. The same numerical experiment has been performed for the rest of the polymers used in the previous section. The results acquired are similar. The first figure shows stable conditions, and then even 1% amplitude of initial disturbances grow and eventually break the fiber after a certain period of time, independent of the wavelength of the disturbances. While the space periodicity of the initial disturbances does not change, their shape was getting a non-sinusoidal character during their growth. On the other side when the process is stable even the 20% of initial disturbance amplitude decreases with time and the process remains stable again independently of the disturbance wavelength.
Unfortunately the numerical results cannot be compared to the experimental data since the publications on isothermal fiber spinning are scarce.
64
PO SITIVE EIG ENVALUE 1
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5 Area 0 . 4
0 . 3
0 . 2
0 . 1
0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 POSITIVE EIGENVALUEL 1
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5 Area 0 . 4
0 . 3
0 . 2
0 . 1
0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5 Area 0 . 4
0 . 3
0 . 2
0 . 1
0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 L Dimensionless length
Figure 4.6. Disturbance evolution for a unstable fiber spinning of HDPE 6009 with 1% introduced sinusoidal disturbance at DR=4.4 (DR CR =4.28)
65
NEGATIVE EIGENVALUE 1
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5 Area 0 . 4
0 . 3
0 . 2
0 . 1
0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 NEGATIVE EIGENVALUE 1 L
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5 Area 0 . 4
0 . 3
0 . 2
0 . 1
0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 NEGATIVE EIGENVALUE 1 L
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5 Area 0 . 4
0 . 3
0 . 2
0 . 1
0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 DimensionlessL length
Figure 4.7. Disturbance evolution for a stable fiber spinning of PP 6523 with 20% introdused sinusoidal disturbance at DR=2.5 (DR CR =4.28)
66
CHAPTER V
FIBER SPINNING WITH COOLING TEMPERATURE GRADIENT
5.1 Linear Stability Analysis of Non-isothermal Fiber Spinning
Linear stability analysis is a technique used to describe the response of the non- isothermal fiber spinning process when infinitesimal disturbances are imposed.
5.1.1 Process description
Thermal effects can contribute to the stability of flows of molten polymers.
Coupling between the energy and momentum equations needed for non-isothermal analyses, requires a temperature dependence of viscoelastic parameters (or density, if gravitational or inertial effects are likely to be significant) and taking in account either significant dissipative heating or an externally imposed temperature difference.
In this part of the research it will be considered, the formulation of the non- isothermal fiber spinning. Using the similarity approach due to the time-temperature superposition, we introduced the same temperature dependence of viscosities (relaxation times) in each relaxation mode, neglecting the temperature dependence of mode’s moduli, and employ in our calculations the coupling of the energy and evolution equation under the similarity condition. Thus the final form of the energy equation with ignoring dependence of thermal conductivity on temperature and taking into account the absence of heat source is:
67
∂T 2α ρCv=+ ()() TTtrD −+ σ ⋅ (5.1.1) p∂t R a
Here T, ρ , C p , and α , are the temperature, density, specific heat capacity at constant pressure, and convective heat transfer coefficient, respectfully, D is strain rate
tensor and tr(σ ⋅ D ) is the density of mechanical power.
In this section for simplicity we will assume that variations in the radial direction
are negligible, because the radius of the filament is so small compared with the spin-line
length, which means that none of the parameters will have dependence in radial direction.
We will perform a linear stability analysis of the non-isothermal fiber spinning and find
the conditions under which the draw resonance occurs. The non-isothermal fiber spinning process is shown schematically on Figure 5.1.1:
Figure 5.1.1. Sketch of the fiber spinning process 68
The device is not much different from the one for isothermal case. A molten polymer is pushed from a die of diameter D is taken up by a drawing force F with a
speed uL considerable larger than the extrusion velocity, at the position downstream with the distance L from the die. Once again following the assumptions for isothermal case
the die swell will be neglected and the computations has been made choosing as an initial point where the diameter of the fiber after the die swell becomes equal to the diameter of
the die. The distance L once again is taken from initial point till the point of solidification.
The analysis of this flow problem requires solution of the closed set of PDE’s,
consisting of equations for conservation of mass, momentum and energy, along with an
adequate viscoelastic constitutive equation, with appropriate initial and boundary
conditions.
The goal here is to develop a reliable model to determine the critical draw ratio beyond which there is onset of instability in non-isothermal fiber spinning.
The assumptions for simplifying the system of equations for non-isothermal fiber spinning are the same as cited in the previous chapter (Neglecting all the secondary forces on the thread-line like inertia, air draw resistance, gravity and surface forces). It is also assumed that no crystallization occurs inside the spinline, so the cooling of the fiber is done by convection and heating from crystallization is not taken into account.
The same system of equations is used as described in details in previous chapter.
We start with continuity equation, which does not defer from the isothermal case.
69
∂a ∂ ( v a ) + = 0 (5.1.2) ∂t ∂ x
As discussed in the previous chapter, the Leonov model for simple elongation flow can be presented in slightly inhomogeneous fibers spinning situation as:
1∂λd λ 111 k k 2 ɺ +v + (λk −+= )(1 ) ε . (5.1.3) λk ∂t dx 6 θλλk k
I1, k n 2− 1 σext=∑G( ) ⋅ ( λ k − λ k ), (5.1.4) k 3
The condition of spinning tension constancy along the spinline is:
I1, k n 2− 1 2− 1 FaaGext≡⋅=⋅σ ext∑ k()( ⋅−= λ k λ k ) Ft ext (), I1k=λ k + 2 λ k . (5.1.5) k 3
In equations (5.1.3-5.1.5) a is the spinline cross-section area, v is the (averaged over
cross-section) spinline velocity, λ is the elastic extension ratio, σ is the spinline axial stress, x is the distance from spinneret, t is the time and F is the spinning tension which
is in general time dependent, and the index k indicates that there are several relaxation modes. The parameter n is same for most of the polymers and it is equal to 0.1.
Energy balance equation describes the temperature changes due to the viscous
dissipation and convective heat transfer between polymer and air respectively. Heat
conduction within the fiber is neglected, since it is very small in comparison with the heat
exchange between polymer and ambient air. In the current work we do not consider
crystallization; therefore we also neglect the heat sources. It is assumed that the density is
70
constant, hence approximated internal energy is Cp T . Then the energy conservation equation can then be rewritten as:
2.α ∂V ρC(v)∂+⋅∇ T =− ( TT − ) + σ (5.1.6) p t Ra ∂ x
The first term on the right hand side is taking care of the convective heat transfer between the filament and the surrounding air and the second term is the viscous dissipation.
Thus the set of equations consists of the equation of continuity, the condition for
constant tension, and a number of evolution equations, which has to be solved
simultaneously.
5.1.2 Viscosity Dependence on Temperature
Viscosity is assumed to be dependent of temperature according the time
temperature superposition. The following equation is employed in order to calculate the
temperature dependence of liquid viscosity.
E T0 (T )= 0 exp( ( − 1)) (5.1.7) RT0 T
here 0 is the elongational viscosity at temperature T0 , E represent material activation energy and R is a the gas constant.
Along with the above data, there is a common similarity assumption (or time- temperature superposition principle) of temperature dependence of rheological
71
θ G parameters, which states that, the ratios, k and k , are temperature <θo >
η when the weak temperature dependence of the elastic moduli is neglected, the ratios k ηo
are also temperature independent.
Heat transfer coefficient at any point is calculated based on the model developed by Kase and Matsuo [29] of heat transfer for air flowing past a fine cylindrical filament:
V 8v α=0.388(k ν −0.334 )( ) 0.333 (1( + a )) 20.167 (5.1.8) F F A V
−0.334 The factor (kFν F ) depends on the physical properties of air and is approximately
constant for all temperatures, so equation (5.1.8) simplifies to:
V 8v α =0.473*10−4 ( ) 0.333 (1 + (a )) 20.167 (5.1.9) A V
This equation is valid for any polymer as long as the cross-section is
approximately circular.
5.1.3 Non-dimensional Governing Equations
The linear system of equations is non-dimensionalized in order to ensure for the different magnitude of the variables. The continuity equation, the momentum balance
equation, the evolution equation for elastic strain λk and energy equation are represented below:
72
∂A ∂ ( V A ) + = 0 (5.1.10) ∂t' ∂ X
∂ (A .σ ') = 0 ∂ X (5.1.11)
∂∂λ λλλλ(2 −− 1 )( − 1 + 1) ∂V k+V kkkk + = λ . (5.1.12) ∂tX' ∂ 6 Dek ∂ X
∂TTL ∂2π α 1 σ 0 ∂ V AAV+ =( TTa −+ ) σ A (5.1.13) ∂t ∂ XCAVρp0 0A ρ CT p 0 ∂ X
As already derived for the heat transfer coefficient the following expression will be used:
V 8v α =0.473*10−4 ( ) 0.333 (1 + (a )) 20.167 (5.1.14) A V
Rearranging the terms in the energy equation it leads to the final form used later in the numerical simulations:
V ∂TT ∂ 0.333− 0.833y 20.167 σ 0 ∂ V +=−V StV A( TTa −+ )(1 (8 ) ) + σ ' (5.1.15) ∂tX' ∂ Vρ CTXp 0 ∂
Here we introduced the following dimensionless variables and parameters:
V t t ' = 0 - non-dimensional time, (5.1.16) L
73
x X = - non-dimensional spinline distance from a spinneret, (5.1.17) L
a A = - non-dimensional cross-sectional area, (5.1.18) A0
v V = - non-dimensional velocity of the filament, (5.1.19) V0
σ .A σ ' = 0 - non-dimensional stress, (5.1.20) F
T T ' = - non-dimensional temperature (5.1.21) T0
' T Ta = - non-dimensional air temperature (5.1.22) T0
θ.V De = 0 - Deborah number. (5.1.23) L
1.67*10 −4 L St = 0.833 .333 - Stanton number (5.1.24) ρCp A0 V 0
σ δ = 0 - Viscous dissipation (5.1.25) ρCp T 0
In previous definitions, L , A0 , V0 and T0 are the distance from a spinneret to a take-up device, cross sectional area, the velocity at the die exit and the temperature at the die, respectively. Additional dimensionless parameters here are: Deborah number characterizes the dynamic behavior of this viscoelastic fluid in melt spinning, Stanton number – dimensionless heat transfer coefficient and δ viscous dissipation.
74
5.1.4 Boundary conditions
In these calculations the prescribed velocity boundary conditions were chosen as:
t > 0 : A= A 0 = 1 , V= V 0 = 1 at x = 0 (5.1.26)
V= VL = r at x = 1
The following boundary condition for temperature distribution will be used in the calculations:
t > 0 : T= T 0 = 1 at x = 0 and T= T a at x = 1 (5.1.27)
meaning that the constant temperature is kept constant on the outside surface. Here T 0 is
the constant temperature at the die exit and Ta is the cooling air temperature.
5.1.5 Steady State Solution
Equations for the steady state are obtained when the time derivatives in Eqs
(5.1.11)-(5.1.15) are equal to zero. Then the set is presented in the form of first-order ordinary differential equations:
∂ (V A ) = 0 (5.1.28) ∂ X
∂(A .σ ') = 0 (5.1.29) ∂X
∂λ( λ2 − λ− 1 )( λ − 1 + 1) ∂V V k+ k k k = λ (5.1.30) ∂X6 Dek ∂ X
75
V ∂T 0.333− 0.833y 20.167 σ 0 ∂ V V=− StV A( TTa −+ )(1 (8 ) ) + σ ' (5.1.31) ∂X VCTXρ p 0 ∂
A numerical procedure has been employed in the calculations using root-finding routine,
Newton-Raphson method, for solving Eq. (5.1.29) through (5.1.31) until the boundary
conditions (5.1.26) and (5.1.27) are satisfied.
A comparison between experimental and computed temperature profiles is shown on the Figures (5.1.2) and (5.1.3) for polystyrene (Dow Styron 666) and polyethylene
(Dow 560 E). Detailed rheological characteristics for these polymers can be seen in
Table 5.1.1. The fiber was drawn from the Modern Plastics Machinery screw extruder through stagnant air at 20-25 oC. Temperature measurements were made along the length of the fiber using thermodot infrared pyrometer. The temperature at the die for both the polymers was 200 oC and flow rate 0.58 g/min.
There is an excellent agreement between the numerical predictions and
experimental data, which gives us good staring point for describing the more complicated problem - unsteady non-isothermal fiber spinning.
Table 5.1.1: Material Parameters
Manufacturer Relaxation Melt index Temperature Polymer o Reference Commercial grade time (s) [g/10 min ] ( C) DOW HDPE 1.04 1.6 200 560 E 19 Dow Polystyrene 0.86 2.5 200 Styrene 666 19
76
200
190 Computed + Experiment 180 Temperature[C] 170
160
150
140
130
120 0 10 20 30 40 50 60 70 80 90 100 Distance [mm]
Figure 5.1.2. Temperature profiles for non-isothermal fiber spinning for PS (---- Computed + Experimental [19] )
2 0 0
1 8 0 Computed Experiment + Temperature[C] 1 6 0
1 4 0
1 2 0
1 0 0
8 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Distance [mm]
Figure 5.1.3. Temperature profiles for non-isothermal fiber spinning for PE (---- Computed + Experimental [19] )
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5.1.6 Linear Stability Analysis for Small Disturbances
Conventional linear stability analysis has been used in order to study the
instabilities of non-isothermal fiber spinning process. The procedure assumes that all
variables can be considered as sums of finite, reference components that describe the
main flow, and infinitesimally small perturbations about those values. Substitution of
those sums for each dynamic variable into the equation of continuity and motion, and
holding only first order perturbation terms results in a series of linear equations that can be solved under physical constraints imposed on the system under investigation.
The small perturbations were introduced in the steady state values of area,
velocity extensional rate and temperature as follows:
Atx(,)= Ax ().(1 + Axe (). .t ) (5.1.32) 0 1
Vtx(,)= Vx ()(1 + Vxe (). .t ) (5.1.33) 0 1
λ(,)tx= λ ()(1 x + λ (). xe .t ) (5.1.34) 0 1
Ttx(,)= Tx ()(1 + Txe (). .t ) (5.1.35) 0 1
Here the subscript 0 indicates steady state, A1 , V1 , T1 and λ1 are the perturbed amplitudes, and is a complex eigenvalue that accounts for the growth rate of the perturbations. For simplicity, the way of introducing disturbances for elastic strains is demonstrated in (5.1.34) for a single relaxation mode. The value of is determined by solving an eigenvalue problem. If the real part of is positive, then any arbitrarily small
deviation from steady state will grow in magnitude with time.
78
By substituting Eqs (5.1.32)-(5.1.35) into PDE’s (5.1.28)-(5.1.31) and neglecting the second and higher-order terms the perturbation, the following linear homogeneous ordinary differential equations are obtained, which for simplicity are demonstrated only for one relaxation mode:
∂A ∂ V +.A V1 + V 1 = 0 (5.1.36) 1 0∂X 0 ∂ X
VVV10100.λ . ∂ λ ∂λ1 λ 1 ∂ V 0 ∂ V 1 ++λ1() ++V 0 g 1110 − ()0λ +− VV = (5.1.37) λ0 λ 0 ∂∂x x6 De ∂ x ∂ x
σ 1+A 1 = 0 (5.1.38)
∂T1 ∂T0 +TVAT1000( + ( TAV 111 ++− )) gATTT 3001 ( −+−−a ATT 10 ( a )) ∂x ∂ x (5.1.39)
∂V0 ∂V1 −gA400σ'(( σ 111 +++ VAV ) 0 )0 = ∂x ∂ x
Gg 2λ 1 Here σ1 is dimensionless stress perturbation and σ1 = , the functions g1 and σ 0
g2 here dependent only on λ0 , are presented as follows:
−2 2 − 1 g1=(2λλλ 000 + )( ++− 1) λλ 00 (5.1.40)
λ2+ 2 λ − 1 2n g =+(2λλ−2 )(0 0 )n ++ ( λλλλλλ 21 2− )( ++ − 221 )( − ) (5.1.41) 2003 3 n 000000
These functions gk (λ0 ) are found from the steady-state solution and considered below as
known functions of X. Functions g3 and g4 are respectively equal to:
79
2αL π g3 = (5.1.42) ρCp V0 A 0
σ 0 g4 = (5.1.43) ρCp T 0
For the non-isothermal spinning processes the take-up speed is taken as constant and the related boundary conditions are:
AV1(0)= 1 (0) =λ 1 (0) = TV 1 (0) = 1 (1) = 0 (5.1.44)
5.1.7 Numerical Approach
Equations (5.1.36) - 5.1.39) and boundary conditions (5.1.44) represent an
eigenvalue problem whose nontrivial solution exists only for a certain value of . Since this problem cannot be solved analytically, we have to use a numerical approach.
Once again finite difference discretization is used to solve the problem numerically. For simplicity below is illustrated employed numerical procedure on the
example of a single relaxation mode. First the values of A0 , V0 and λ0 are defined in the
finite number of points in the interval [0,1] solving the steady flow problem using finite
difference formulas for derivatives in the perturbed equations (5.1.28)-(5.1.31). After
discretizing and rearranging these equations, we obtain linear algebraic system of
equations. If n is the number of grid points, this algebraic system consists of
n −1equations of continuity, n −1 constitutive equations (for multimode the constitutive
equations are k times more), n − 2 equations of constant force and n-1 energy equations.
80
The unknown number of variables 4n − 5 is same as the total number of equations. This set of equations is presented in the matrix form is follows:
i i A1 A 1 i i V1 V 1 M.= . N . (5.1.45) λi λ i 1 1 i i T1 T 1
Here M is the “grid” matrix, whose form is demonstrated for a simplified case in the
following table. N is a matrix which takes care of the missing time dependent term in
i the constant force equation and makes all V1 from the right hand side equal to zero.
Given the boundary conditions the eigenvalues of the system can be obtained using QZ algorithm. Eigenvalues are computed from the diagonals of the triangular form.
All the eigenvalues are sorted on the basis of sign of the largest eigenvalue the real part.
The process is marked as stable if this eigenvalue is negative, or unstable otherwise. The critical draw ratio is defined as that where the real part of the largest eigenvalue is equal to zero.
For following example the number of equations is taken to be 15 and the number of grid points is 5.
81
Table 5.1.2. Simplified matrix ( M) for solving linear system of equations
Coefficients in front of Coefficient in front of Coefficients in front of Coefficients in front of T1 A1 V1 λ1
Ai=0 Ai=1 Ai=2 Ai=3 Vi=0 Vi=1 Vi=2 λi=0 λi=1 λi=2 λi=3 Ti=0 Ti=1 Ti=2 Ti=3
A 0 0 0 A 0 0 0 0 0 0 0 0 0 0
B A 0 0 B A 0 0 0 0 0 0 0 0 0
0 B A 0 0 B A 0 0 0 0 0 0 0 0
0 0 B A 0 0 B 0 0 0 0 0 0 0 0
0 C 0 0 0 0 0 0 D 0 0 0 0 0 0
0 0 C 0 0 0 0 0 0 D 0 0 0 0 0
0 0 0 C 0 0 0 0 0 0 D 0 0 0 0
0 0 0 0 E 0 0 F 0 0 0 0 0 0 0
0 0 0 0 A E 0 B F 0 0 0 0 0 0
0 0 0 0 0 A E 0 B F 0 0 0 0 0
0 0 0 0 0 0 A 0 0 B F 0 0 0 0
G 0 0 0 I 0 0 H 0 0 0 K 0 0 0
0 G 0 0 H I 0 0 H 0 0 L K 0 0
0 0 G 0 0 H I 0 0 H 0 0 L K 0
0 0 0 G 0 0 H 0 0 0 H 0 0 L K
82
V0 ( i ) V0 ( i ) Gg 2 A = − B = C =1 D =σ1 = λ 1 x x σ 0
λ0 (i − 1) g1( i ) ViVi0() 0 (− 1) Vi 00 ()λ ( i − 1) EVi=0( ) − Vi 0 ( −+ 1) Vi 0 ( ).( ) F =− + + − . λ0 (i ) 6De x x xλ0 ( i ) gi()∂ Vi () Ti () G=−3 (() TiTgii' −− ' )) ()()σ ' 0 + Vi () 0 2 0a 40 x 0 ∂ x Gg giVi() ()σ '() i V() iT () i V() iT () i H = − 2 4 0 0 I = − 0 0 K = 0 0 σ 0 x x x ViTi() () ViTi ()( (+ 1) − Ti ()) L=+00 00 0 −− gTT( ) x x 3 0 a
5.1.8 Accuracy of the Numerical Scheme
The accuracy of the numerical scheme has been checked. All the computations are made for different grid size. Starting from 50 grid points up to 2000 and it was concluded that for a grid size (number of grid points) larger than 300 the accuracy of the result is not worth making lengthier computations. Therefore a grid size of 300 was chosen for further computations.
Computer software used has been elaborated for multi-mode computations. The experimental data available did not include polymer melts or solutions with more then one relaxation mode. For future applications the more sophisticated version of the computer code can be used.
5.1.9 Results and Discussions
In other to test the numerical scheme and the reliability of system of equations for
non-isothermal fiber spinning the results of numerical simulations are compared to the
83
literature experiments. The numerical simulations are carried out for various polymers and the results are compared to the experimental data available.
In series of publications Kase and Ishihara accomplished a comprehensive experimental research on non-isothermal fiber spinning [29]. They examined two different polymers: conventional fiber-grade PET and a fiber-grade PP resin. The polymers were spun at different down draw ratios beyond the point where draw
resonance occurs. The experimental setup can be seen on Figure 5.1.4 - a melt spinning
device consisting of extruder a quench bath and a variable speed take-up device. The
temperature of all the experiment was 284 oC and the spinneret orifice diameter was 0.3 mm. The waveform of draw resonance was observed by measuring the diameter of the filament at the take-up device with a high-precision dial gauge.
Spinneret
L
Water Take-up Pulley Bath
Guide Pulley Figure 5.1.4. Schematic of experimental setup
84
A computer code using Matlab was created for solving the set of ODE implementing the procedures described in the previous sections. The computer program was able to work with multimode constitutive equations and generally took less then five minutes to compute the eigenvalues (for 300 grid points) for a given processing conditions. With computed eigenvalues the curves for area along the spinline are drawn.
All the parameters used in computations are taken from the experimental papers provided by the authors.
Comparison with experimental data shows fair agreement. For both cases linear stability analysis describes well the instabilities occurring in the fiber beyond the point of draw resonance. The differences are rather small and it is likely that a more complete representation of the relaxation spectrum would result in closer agreement between the simulation and experiments. It is to be emphasized again that this model contains no adjustable parameters. On this basis the agreement must be considered quite well. The results of the numerical simulation are presented in the following figures:
Table 5.1.3. Material Parameters for used polymers
Manufacturer intrinsic Melt index Temperature Polymer o Reference Commercial grade viscosity [g/10 min ] ( C) DOW PET-A- 14 0.563 3.4 284 Cleartuf 8006 29 Dow Polypropylene 0.254 5.1 284 Epolene G-3003 29
85
2
D [mm] 1.5
1
0.5
Computed Experimental 0 20 40 60 80 100 120 140 160 180 200 Distance from the die [mm]
Figure 5.1.5 Fiber diameter comparison between numerical results and experimental data for PET-A- 14
1.4
1.2
1
D 0.8 [mm]
0.6
0.4
0.2 Computed Experimental
0 50 100 150 200 250 300 Distance from the die [mm]
Figure 5.1.6 Fiber diameter comparison between numerical results and experimental data for PP
86
Two other sets of experiments have been performed in order to prove the
stabilization effect of the cooling. [29]. A water-quenched melt spinning of PET was
carried out under draw resonance conditions. The polymer was quenched 20 mm below
in cold water. Keeping the same spinning conditions, horizontally blown air was
introduced to the fiber spinning system and the reading for the fiber diameter was taken
again.
As expected the cooling has stabilizing effect on the fiber spinning process. The simulation done with spinning conditions taken from the same paper also confirms this result. It can be seen from the following figures that after introducing the horizontal air the draw resonance have been suppressed and the process in on a verge of being stable.
The computations have been made for both cases: with and without horizontal air. For the case where no air is blown the spinning is unstable and the eventually the fiber will break. Cooling the fiber with air reduces the fluctuations in area and helps to improve the stability of the process. The draw resonance is not as severe as in the case without air. Both simulated cases showed good agreement with the experimental results.
87
4 Computed Experimental
3.5
3
D 2.5 [mm] 2
1.5
1
0.5
0 50 100 150 200 250 300 350 400 Distance from the die [mm]
Figure 5.1.7. Fiber diameter comparison between numerical results and experimental data for PET without horizontal air
2 Computed Experimental 1.8
1.6
D 1.4 [mm] 1.2
1
0.8
0.6
0.4 0 50 100 150 200 250 300 350 400 Distance from die [mm]
Figure 5.1.8. Fiber diameter comparison between numerical results and experimental data for PET with horizontal air
88
W ith horizontal air Computed 4 With horizontal air Experimental W ithout horizontal air Computed Without horizontal air Experimantal 3.5
3
2.5
2 Diameter
1.5
1
0.5
0 50 100 150 200 250 300 350 Distance from the die Figure 5.1.9. Fiber diameter comparison between numerical results and experimental data for PET with and without horizontal air
Ishihara and Kase [1975b] conducted an interesting experiment. They draw fiber spinning of PET at 280 oC at the draw ratio 48.6 for various spinning lengths. The intrinsic viscosity of the PET is 0.563 and relaxation time of approximately 0.001 s. The experiment was made non-isothermally and the stabilization of the fiber diameter occurs due to cooling. They recorded diameter of the fiber and computed the ratio of maximum and minimum diameter.
In the following table comparison with the experimental data and computed fiber diameter is presented. The values for the relaxation time and heat transfer coefficients are taken from the same paper.
89
Table 5.1.3 Comparison between numerical results and experimental data for PET at various spinning lengths
Length (cm) Dmax /D min experimental [30] Dmax /D min computed 0.5 1 1 1 2.51 2.25 2 7.27 7.31 5 7.93 7.81 10 3.69 3.74 15 N/A 1.53
The experiment shows that stabilization can be achieved at short length of the
fiber as well as at longer lengths. The reason for attaining stabilization at longer length is
the effect of cooling which correspond to an upper stable region reported by other
scientist [30] as well. The computed values of the ratio of maximum and minimum
diameter are close to the experimentally obtained by Ishihara and Kase.
8
7
6
5
Dmax/Dmin 4
3
2
1 0 1 2 3 4 5 6 7 8 9 10 Lenthg [cm] Figure 5.1.10. Comparison between numerical results and experimental data for PET for different spinning lengths 90
CHAPTER VI
INSTABILITIES IN HIGH DEBORAH NUMBER CONTRACTION FLOWS
6.1 Linear Stability Analysis of Contraction Flow
The presence of flow contractions in polymer extrusion applications often leads to
added pressure drops and may initiate flow instabilities at higher flow rates. In the present study, the contraction flow behavior has been investigated using abrupt
axisymmetric contraction geometry.
The modeling used here to simulate the contraction flow is similar to fiber spinning process and the flow is expected to have instabilities after reaching certain
Deborah number similar to the critical draw ratio in the spinning process.
The aim of this work is to develop and implement a numerical technique for simulating the flow instabilities by employing finite difference method. A critical
Deborah number beyond, which the instabilities occur, will be established. A systematic investigation of the role of contraction ratio on viscoelastic flow through contractions will be performed. This task is carried out effectively by numerical simulations and in the current study we consider the effect of contraction ratio upon the main overall features of this type of flow. In particular, we look at the variation with critical Deborah number.
Employing the isothermal “Jet approach” followed by linearized perturbation
approximation of the governing equations for finding the onset of the instabilities will
91
supply information about the stability of the contraction flow which will be used to describe the mechanism of propagation of the disturbances into capillary up to the die exit, and its numerical implementation.
6.1.1 Process description
Contraction (converging) flow is a type of channel flow that has a lot of industrial applications. The flow out of relatively larger cross-sectional channel (reservoir) to smaller cross-sectional one (die) with certain contraction ratio of reservoir to die is called converging. While entering the die, the flow undergoes transitional zone (pre-entrance region), where polymeric melt reformulates its flow type. Some polymer melts demonstrate secondary flow (vortices) at the corner. Based on the various experimental results, it is viewed that the reformulating of flow and vortices in the pre-entrance region is related to dominant transient extensional deformation over simple shear deformation.
When the flow enters the die section, it turns back to simple shearing flow and starts to release elastic deformation at the entrance region. During these transition it produces a severe pressure drop and to keep the same flow rate as in the reservoir section, the flow in the smaller cross-sectional area has usually much higher De number than in the reservoir.
The contraction flow in the near field entrance region is idealized as an inhomogeneous extensional flow similar to the one occurring in polymer fiber spinning .
It is assumed that such an approach is approximately valid when the flow De number is high enough. For low and medium De number flows, the birefringence experimental data
92
confirmed that the contribution of the secondary flow might be important for the initial development of the jet. However, this contribution is insignificant for flows with higher
De numbers such as injection molding and extrusion.
The reservoir geometry used for all computer simulations is circular, with the
reservoir radius RR (Figure 6.1.1).
P ressu re
2R R 1
2
L z
2R D
Figure 6.1.1. Sketch of the circular cross-section contraction flow process 1 – reservoir, 2- die
93
6.1.2 Flow in the Far Field Entrance Region
A necessary condition is the reservoir to be long enough so the initial flow in the
reservoir can be assumed as developed. This region flow is approximated as the steady
shear flow. (Figure 6.1.2)
Far-field Reservoir Entrance Flow
Near-field Real velocity profile Reservoir Entrance Flow Approximated Jet velocity profile Die Developing flow
Die Developed flow
Figure 6.1.2. Regions in contraction flow and velocity profiles in the jet
The matrices formed by the velocity gradient tensor, ∇v , the extra stress, σ , and k the elastic strain, c in matrices are: k
94
000 σ11,kk σ 12, 0 c 11, kk c 12, 0 ∇=v100,γɺ σσσ = 0, c = c c 0 (6.1.1) k 12,kk 22, k 12, kk 22, 000 00σ 33, k 001
and the steady state solution is
2Z 2 Z 2 − 1 c=, c = , c = , c = 1 11,kZ+1 22, k Z + 1 12, kZ +1 33, k
n σije,=∑ σ ijk , = ∑ G kk( I / 3) c ijk , (6.1.2) k k
2 2 Z()γɺ≡ ( Ik − 1)/21 −≡+ 1(2 θ k γɺ /) b
th Here θk is the relaxation time in k mode obtained in linear region, γɺ is the shear rate, b
is the non-dimensional function scaling the relaxation time in non-linear region, and Ik is
the invariant of c in simple shearing. k
Well above the die entrance, the flow is of steady simple shearing type. Then around a certain distance l from the die entrance, the flow is approximately changes its
type from shearing to elongation. In this simplified scheme, the switch in the flow type is
assumed as jump-like change with the unknown parameter l searched for.
Along with the solution of constitutive equation for steady incompressible simple
shearing, the inertialess momentum balance equation with neglecting the gravity term
will result in:
dP x2 σ12,e (x 2 ) = . (6.1.3) dx1 H
95
Here P is pressure and σ12, e is shear stress. Here after we employ for the sake of
convenience the coordinate system {,x1 x 2 , x 3 } and the corresponding velocity
components {v1 ,v 2 ,v 3 } , where 1 stands for axial ( z ), 2 – for transversal (y ) , and 3 – for
neutral (x ) directions. The expression for flow rate Q in the “upper” shearing reservoir flow is:
R R Q=2πR v x dx = − π R x2 γɺ dx . (6.1.4) ∫0122 ∫ 0 22
The flow rate in the steady flow Q is constant and is considered as given. When Q is
specified, the shear rate, γɺ (x2 ) , can be found from equations (6.1.2), (6.1.3) and (6.1.4) which allows to compute the complete profiles of rheological variables for the steady shear flow up to the distance l from the entrance. This distance is the location of the borderline between the far and the near field reservoir entrance flows.
For a known value of flow rate Q, while satisfying equations (6.1.2)- (6.1.4) a
root finding method (Newton-Rhapson) has been used to calculate the gap-wise shear rate profile. The numerical integration of Eq.(6.1.4) was performed using the trapezoidal rule.
6.1.3 Flow in the Near Field Entrance Region - “Jet Approach”
The basic assumption here is that the “jet flow” is dominant in complicated
converging near field entrance flow with abrupt geometry. This assumption is based on
the Cogswell idea and is confirmed by the flow visualization experiments [67], where the
measured stress distribution for high enough De number flows becomes more jet-like as
the flow get closer to the die.
96
Although this “jet” approach simplifies the converging flow with ignoring the
drag from secondary vortices at the entrance corners and also neglects the shear effects
near the reservoir walls, it is getting more and more realistic in describing the flow when
it is approaching the entrance.
In the pre-reservoir region, the flow is modeled as developed simple shearing flow. Then following the Cogswell theory [67,68], the main converging stream of polymer liquid in the reservoir region is modeled as an inhomogeneous extensional flow
similar to that employed in the fiber spinning. In the region where the entrance die flow is
developing, it is used once again the CE’s for unsteady simple shearing, with evolution of
elastic strains originated from the entrance reservoir flow. The time derivative is
simplistically treated in this steady capillary flow model as space convective derivative.
The transition between flow types in these different flow regions is modeled by specific
matching boundary conditions (described in details below).
The modeling of this region consists of a system of general and stable CE’s. It
simplifies the flow complexity using the quasi-uniaxial, i.e. inhomogeneous elongation
approach. The corresponding approximate expressions for matrices of jet elastic strain
tensors c , velocity gradient tensor ∇v , and the extra stress tensor σ are: k e
2 λk 00 100 σ 11 00 −1 c =0 λ 0, ∇=ε−vɺ 01/20, σ= 0 σ 0 (6.1.5) k k e 22 −1 00λk 001/2 − 00 σ 33
97
Where εɺ = dv1 / dx 1 is the elongation rate. Inserting equation (6.1.5) into (3.1) leads to:
∂λk1d λ k 12 11 +⋅v1 + (λk − )(1 += ). εɺ (6.1.6) ∂t'λk dx 1 6 θλλ k k
Here v1 is the value of axial velocity averaged over the jet cross-section. The expression
for the elongation stress (a total axial stress in jet cross-sections), σ ext in the multi-mode case is presented as follows:
n 2− 1 2− 1 σext=∑G( I 1, k / 3) ⋅ ( λ k − λ k ), I1k=1 +λ k + λ k . (6.1.7) k
Since the flow rate is the same at every position the continuity equation is expressed as
follows, where a( x 1 ) is cross-sectional area of jet flow.
∂a ∂ ( v a ) + = 0 (6.1.8) ∂t ∂ x
To make the system a complete the assumption that the force acting on the jet is constant
is being used: Fext=σ ext a = const This condition is expecting to work well enough for high Deborah number flows where the jet flow dominates. Using this condition, equation
(6.1.7) is represented in the final form:
n 2 2 Faext≡=⋅σ ext aGI∑ kk(1 /3)( ⋅−= λλ k 1/) k constI ; 1 kk =+ λλ 2/. k (6.1.9) k
98
Thus equations (6.1.6), (6.1.8) and (6.1.9) represent a closed system of partial
differential equations, which is ready for numerical discretization.
6.1.4 Matching Condition at the Boundary of two Flow Regions
The jet flow model has not yet been completed because the initial elastic stretches
l λk (k = 1,2,...) in Eq. (6.1.9) are unknown. In order to find them, the matching conditions
th for each k non-linear Maxwell mode are employed at the unknown boundary x1 = − l , where the flow changes from shear to extensional.
The following energetic condition is used to match asymptotically various flow parameters (homogeneously or inhomogeneously distributed) in different regions of flow at an effective “interface” with discontinuities in the values of these parameters.
n+1 3G I [()WIsh− WI ()] jet = 0 ; W( I )=k1 k − 1 (k=1,2,) (6.1.10) kk1 kkxl 1 1 =− k1 k 2(n + 1) 3
sh th Here Wk are averaged over cross-section elastic potentials for each k non-linear
Maxwell mode in the “upper” shearing flow. Using the formulae known for simple
shearing and planar elongation,
sh jet 2− 2 I1k= c 11, k + c 22, k + 1; I1k=()λ k + () λ k + 1 (k=1,2,…) (6.1.11)
At x1 = − l , the jet cross sectional area coincides with that for the reservoir. Equation
l (6.1.11) allows to find the initial jet elastic stretches λk from the known elastic strain
tensor profile for far field shear flow, cij, k , which are found in equations (6.1.2)-(6.1.4).
99
2π RR x= − l : x⋅v(sh c ++ c 1) n+1 dx =+ [()λ l 2 2()],( λ ln − 11 + k = 1,2,.) (6.1.12) 1 ∫ 2 1 11,kk 22, 2 kk Q 0
Employing this procedure, the total longitudinal jet force can be calculated from Eq.
l (6.1.7) by inserting the values of λk and using total area rheometry reservoir. As a result
A( x 1 ) can be expressed through σ ext under constant value of Fext .
Equations (6.1.9) and (6.1.10) along with (6.1.12) represent a well-defined and
simple model for steady entrance flow calculations. These calculations will provide the
developed flow in the die with appropriate boundary conditions. It is remarkable that this
approach has no fitting constants.
6.1.5 Steady State Solution in Near Entrance Region
Obtaining the numerical simulations for the base flow is the first necessary step
for calculating the linear instabilities. For the steady state calculations in reservoir and
capillary all time derivatives in Eqs (6.1.5)-(6.1.8) are equal to zero.
Then the system is presented in the form of first-order ordinary differential
equations:
∂ (V A ) = 0 (6.1.13) ∂ X
A.σ = F (6.1.14)
∂λ( λ2 − λ− 1 )( λ − 1 + 1) ∂V V k+ k k k = λ (6.1.15) ∂X6 Dek ∂ X
100
The appropriate boundary conditions for the system of equations are:
(see Fig 6.1.1):
2 l x1 = − l : A= Al = π ( R R ) , λk= λ k . (6.1.16) 2 0 x1=0: AA == 0Dkπ ( R ), λ = λ k .
Here the known geometric parameter R D is the radius of circular die and R R is
the radius of the reservoir.
A numerical procedure has been employed in the calculations using root-finding
routine, Newton-Raphson method, for solving Eq. (6.1.13)-(6.1.15) until the boundary
condition (6.1.16) is satisfied.
6.1.6 Steady State Solution for Die Developing Flow
The developing flow in the circular die is treated as a non-steady viscoelastic
Poiseuille shearing flow, where the time derivative term was changed to a simplified
convective derivative term as: d/ dt≈ v1 ∂ / ∂ x 1 . This modeling treats the viscoelastic
developing flow in the entrance region as a transitional flow between the near field
entrance, elongation jet flow, and the developed shear Poiseuille flow, asymptotically
achieved as x1 → ∞ . It means that the developing flow in the die reflects the memory of the reservoir elongation entrance flow.
All the “initial” variables at the beginning ( x1 = 0 ) of the developing die flow are assumed to be are distributed homogeneously across the entrance cross-section. The well-
101
known fact that after the sudden imposition of shear velocity, there is no instant shear stress response leads to the assumption that the shear stress at the entrance cross-section
x1 = 0 is zero.
With the structure of elastic strain tensors c shown in Eq. (6.1.1), the evolution k
equation for each kth mode (the index k is omitted) has the form:
∂c b ∂ v v12 +cc ( + c ) = c 1 ("12" component) 1∂x2θ 1211 12 22 ∂ x 1 2 ∂c22 b 2 2 v1+ (c 12 +−= c 22 1) 0 ("22" component) (6.1.17) ∂x1 2θ 2 c11 c 22− c 12 = 1 incompressib ility condition
The shear and longitudinal normal stress components for the extra stress tensor are
represented as:
n n σ12=∑GIck(/3) 12, k ; σ 11 = ∑ GIc k (/3) 11, k . (6.1.18) k k
As the consequence of the formulae (6.1.17) and (6.1.18), only the longitudinal
component of the momentum balance equation is used further,
∂p ∂σ1 ∂ ( x σ ) =11 + 212 (6.1.19) ∂x1 ∂ xx 1 2 ∂ x 2
102
The continuity equation and the condition for the constant flow rate are written as follows:
∂v 1 ∂ (v)x 1+ 2 2 = 0 (6.1.20) ∂x1 x 2 ∂ x 2
R R D D 2 Q=2π x212 v dx =− π x 22 γɺ dx = const . (6.1.21) ∫0 ∫ 0
And the non-slip boundary conditions for the axial and radial components of velocities
are represented as:
x2=± L D : v 1 == v 2 0, (6.1.22)
In order to find the boundary conditions at the entrance, x1 = 0 , the matching condition is used once again to implement the change of flow type between the jet flow in the reservoir and entrance shearing flow in the die. This leads to the following matching conditions for any k th relaxation mode:
x==0: v0 v (v=0); ccccJJ 0 = 0, 000020 =1, ==+− 1/ (λ ) ( λ ) 1; 1 1 12 12 33 11 22 (6.1.23) J (λ0 )= 1/2[( λ 02 ) + 2/ λ 0 ]
0 Using the matching conditions to determine the initial elastic stretches λk gives complete
form of the system of equations of developing die flow. Discretizing the equations using
finite difference numerical procedure and using a matrix solver for finding shear rate in
each cross-sectional grid gives the first step of this numerical simulation. Then from the
0 0 0 boundary conditions (6.1.23), the initial values of variables, {v1 ,c 12 , c 22 } at x1 = 0 are
103
considered as known, and the values {,c12 c 22 , P } at the first step x1 are found using the initial values. The same procedure is applied for calculations of all variables in cross- sections along the die.
Illustration of the calculations for steady state flow is presented in Figure (6.1.4).
Axial velocity profile is calculated for various Deborah numbers (calculated in the capillary). The contraction ratio, fluid parameters (relaxation time and modulus) and die geometry are kept constant and only the flow rate was increased to achieve higher
Deborah number. As the flow elasticity is increased the distance required to reach a fully- developed flow in the down-stream channel becomes longer.
1.8 De=25 De=100 De=500 1.6
1.4
1.2
Axial velocity Axial 1
0.8
0.6
-40 -30 -20 -10 0 10 20 30 40 Axial Distance
Figure 6.1.3. Steady state axial velocity profiles for different Deborah numbers
104
6.1.7 Linear Stability Analysis for Small Disturbances
Linear stability analysis in contraction flow does not defer much from the one for fiber spinning. The same small disturbance procedure is used here in order to calculate the stability of the contraction flow.
The small perturbations were introduced in the steady state values of area, velocity extensional rate and temperature as follows:
Atx(,)= Ax ().(1 + Axe (). .t ) (6.1.24) 0 1
Vtx(,)= Vx ().(1 + Vxe (). .t ) (6.1.25) 0 1
λ(,)tx= λ ()(1 x + λ (). xe .t ) (6.1.26) 0 1
Ft()= F .(1 + Fxe (). .t ) (6.1.27) 0 1
Here the subscript 0 indicates steady state, A1 , V1 , λ1 and F1 are the perturbed amplitudes, and is a complex eigenvalue that accounts for the growth rate of the perturbations. The value of is determined by solving an eigenvalue problem. If the real part of is positive, then any arbitrarily small deviation from steady state will grow in magnitude with time. Since infinitesimal disturbances cannot be kept out of any real process, it follows that such a steady state flow cannot be maintained in practice despite the fact that it is a solution of equations of motion. Conversely, if the real part of is
negative, then all infinitesimal disturbances will decay to zero, and the system is stable.
105
By substituting Eqs (6.1.24)-(6.1.27) into PDE’s (6.1.6) through (6.1.9) and neglecting the second and higher-order terms the perturbation, the following linear homogeneous ordinary differential equations are obtained, which for simplicity are demonstrated only for one relaxation mode:
∂A ∂ V +.A V1 + V 1 = 0 (6.1.28) 1 0∂X 0 ∂ X
VVV10100.λ . ∂ λ ∂λ1 λ 1 ∂ V 0 ∂ V1 ++λ1 () ++V0 g 1 − ()0λ 110 +− VV = (6.1.29) λ0 λ 0 ∂∂x xDe6 ∂ x ∂ x
σ 1+A 1 = F 1 (6.1.30)
Gg 2λ 1 Here σ1 is dimensionless stress perturbation and σ1 = , and the functions g1 and σ 0
g2 here dependent only on λ0 , are presented as follows:
−2 2 − 1 g1=(2λλλ 000 + )( ++− 1) λλ 00 (6.1.31)
λ2+ 2 λ − 1 2n g =+(2λλ−2 )(0 0 )n ++ ( λλλλλλ 21 2− )( −− − 221 )( − ) (6.1.32) 2003 3 n 000000
These functions gk (λ0 ) are found from the steady-state solution and considered below as
known functions of x .
The force acting on the jet is constant but there is a disturbance in the force which is only dependent on time. This creates the need for additional boundary condition in
order to find the variable F1
106
Considering the geometry of contraction flow the following boundary conditions
are imposed in order to solve the system of ordinary differential equations:
x1 = − l : A1=0, V 1 = 0,λ 1 = 0 (6.1.33) ∂V x=0: A = 0,1 = 0 1 1 ∂x
The boundary condition at x1 = 0 is chosen to satisfy the rigid of the jet at the end of the reservoir and also to assure that the disturbances can propagate in the capillary region.
6.1.8 Numerical Approach
Equations (6.1.28)-(6.1.30) with boundary conditions (6.1.33) represent an eigenvalue problem whose nontrivial solution exists only for a certain value of . Since this problem cannot be solved analytically, we have to use a numerical approach.
First the values of A0 , V0 and λ0 in the finite number of points in the interval [0,1]
are defined, solving the steady flow problem. After discretizing and rearranging the
equations with already inserted disturbances a linear algebraic system of equations is
obtained. The real part of the largest eigenvalue of the system is characterizing the flow.
The process is marked as stable if this eigenvalue is negative or unstable otherwise. The
critical Deborah number is defined as that where the real part of the largest eigenvalue is
equal to zero.
6.1.9 Accuracy of the Numerical Scheme
Computer software used has been elaborated for multi-mode computations for obtaining the eigenvalues of the system of ordinary differential equations for contraction
107
flow. Using the symmetry of the problem, only the flow on one side of the axis of symmetry has been simulated. Besides the symmetry condition along the axis and no-slip condition on the solid walls, the velocity and stresses for fully-developed channel flow have been specified at the entrance whereas fully-developed channel flow has been imposed on the velocity alone at the exit.
The accuracy of the numerical scheme has been checked. All the computations are made for different grid size. Starting from 500 grid points up to 10000 and it was concluded that for a grid size (number of grid points) larger than 1000 the accuracy of the result is not worth making lengthier computations. Therefore a grid size of 1000 was chosen. Following is a comparison of the calculations between a coarse mesh of 1000 grid points and fine mesh 10000 grid points for a random polymer.
1.8 N=1000 1.6 N=10000
1.4
1.2
1 Velocity 0.8
0.6
0.4
0.2 -40 -30 -20 -10 0 10 20 30 40 Distance Figure 6.1.4. Axial velocity in reservoir and capillary for two mesh sizes 108
6.1.10 Results and Discussions
It was found that increasing flow rate beyond a certain limit resulted in unstable disturbances to the steady entry flow. The purpose of the linear stability analysis is to determine the critical Deborah number of the contraction flow at different geometries and flow conditions.
A numerical study of a fluid through planar sudden contractions was carried out to investigate the effect of contraction ratio (the ratio of reservoir to capillary diameters) for the critical Deborah number. Computational experiments were performed for a polymer – 0.31% PIB/PB/C14 – with known rheological parameters (See Table 6.1.1)
(116). The computed critical Deborah number is compared to the experimentally achieved one by Laser-Doppler velocimetry.
Table 6.1.1. Material Parameters and Experimental Conditions of Contractional Flow for PIB/PB/C14 Parameters Polymer Melts at 323K
θk (sec) Gk, (KPa)
.0098 123.571
Polyisobutylene/ .1094 15.146 Polybutylene/ C14 .7361 2.278
2.755 0.4022
109
The geometry of the reservoir was kept the same for all calculations, but the diameter of the capillary was changed to achieve various contraction ratios (Fig 6.1.5).
The flow rate was gradually increased until growing disturbances appeared in the capillary. Flow rate was increased even more beyond the critical Deborah number and the ratio of the maximum and minimum diameter of the jet was computed. Deborah number
3 was calculated using the equation De=θγ ⋅ɺ App = 32. θ . Q / π D R . Here θ is the average
2 ɺ relaxation time θ= ∑Gkk θ/ ∑ G kk θ , γ App is the apparent shear rate, Q is flow rate and k k
RR is the radius of the reservoir.
Fully developed flow
D D D
D/2 D/8 D/4
Figure 6.1.5. Schematic of die geometry for different contraction ratios
110
The onset of the instability in polymer flow appears to depend on the ratio of reservoir to capillary diameters. The computed critical Deborah number is around 0.91 for 1:8 contraction ratio and increases up to 1.43 with decreasing the contraction ratio up to 1:4. It is equal to 2.93 for ratio 1:2. For 1:4 contraction, the computed critical Deborah number 1:43 is very close to the experimentally value 1.5.
The simulation shows that after increasing the flow rate beyond the critical one, the polymer behaves differently for every contraction ratio. The ratio of maximum and minimum diameters of the jet grows much faster at high contraction ratios and slower at low contraction ratios. (Figure 6.1.6)
Propagation of these jet diameter fluctuations in the capillary will be studied in
the next section.
4 1:8 3.5 1:4 1:2
3
2.5
2
1.5 Maximum/Minimum Axial Velocity Axial Maximum/Minimum
1
0.5 0 1 2 3 4 5 6 7 Deborah Number
Figure 6.1.6. Dependence of Deborah number on the Maximum-Minimum velocity ratio 111
6.2 Non-linear Stability Analysis of Contraction Flow
Non-linear stability analysis in contraction flows is a complicated problem with various branches and alternatives. In this work we will demonstrate the propagation of the disturbances occurred in the reservoir and also investigate the comprehensiveness of the linear stability analysis.
In general, linear stability analysis does not provide information about the dynamical response of the process when infinitesimal disturbance grows to a finite size.
The system can be stable up to a critical finite perturbation and if the perturbation increases beyond critical value the process may become unstable. For that reason solving the non-linear system of equations is of high interest for complete understanding of the instabilities in contraction flow.
6.2.1 Governing Equations
The analysis of non-linear instabilities of viscoelastic contraction flow involves the solution of a coupled set of partial differential equations, consisting of equations for conservation of mass and momentum, along with viscoelastic constitutive equation, with appropriate initial and boundary conditions.
For simplicity again all the secondary forces on the thread-line like inertia, gravity and surface forces, and the variations of variables across the spinline cross-section, are neglected. We now rewrite the non-dimensional equations already used before in reservoir and capillary keeping all the time dependent terms.
112
The continuity equation, the momentum balance equation and the dimensionless
form of the evolution equation for elastic strain are:
∂a ∂ ( v a ) + = 0 (6.2.1) ∂t ∂ x
∂ (a .σ ) = 0 (6.2.2) ∂x
1∂λd λ 111 k k 2 ɺ +v + (λk −+= )(1 ) ε . (6.2.3) λk ∂t dx 6 θλλk k
In order to find the boundary conditions at the capillary entrance, x1 = 0 , the matching
condition technique is used to implement the change of flow type between the jet flow in
the reservoir and entrance shearing flow in the die.
x==0: v0 v (v=0); ccccJJ 0 = 0, 000020 =1, ==+− 1/ (λ ) ( λ ) 1; 1 1 12 12 33 11 22 (6.2.4) J (λ0 )= 1/2[( λ 02 ) + 2/ λ 0 ]
The evolution equation for each kth mode (the index k is omitted) has the form:
∂c ∂ c b ∂ v 12+v 12 +cc ( + c ) = c 1 ("12" component) ∂t1 ∂ x2θ 1211 12 22 ∂ x 1 2 ∂c22 ∂ c 22 b 2 2 +v1+ (c 1222 +−= c 1) 0 ("22" component) (6.2.5) ∂t ∂ x 1 2θ 2 c11 c 22− c 12 = 1 incompressib ility condition
113
The shear and longitudinal normal stress components for the extra stress tensor are represented as:
n n σ12=∑GIck(/3) 12, k ; σ 11 = ∑ GIc k (/3) 11, k . (6.2.6) k k
As the consequence of the formulae (6.2.5) and (6.2.6), only the longitudinal component of the momentum balance equation is used below along with the continuity equation:
∂p ∂σ1 ∂ ( x σ ) =11 + 212 (6.2.7) ∂x1 ∂ xx 1 2 ∂ x 2
∂v 1 ∂ (v)x 1+ 2 2 = 0 (6.2.8) ∂x1 x 2 ∂ x 2
Due to incompressibility of the flow the output at any cross-section is the same at a given
moment of time:
R R Q=2πD x v dx = − π D x2 γɺ dx (6.2.9) ∫0212 ∫ 0 22
And the non-slip boundary conditions for the axial and radial components of velocities are represented as:
x2=± L D : v 1 == v 2 0, (6.2.10)
The main objective of this analysis is studying the propagation of the disturbances formed in the reservoir region, to determine the behavior of the process around the critical Deborah number and to determine the response of the system on finite
114
disturbances. The problem here is no longer the eigenvalue but initial boundary value problem with Dirichlet boundary conditions.
6.2.2 Initial and Boundary conditions
The non-linear viscoelastic contraction flow problem is separated in two parts.
First the steady state case is solved and the base flow is determined. Then the disturbances are introduced as an initial condition and the boundary value problem has been solved.
The boundary conditions for the steady state solution are the same as used in previous chapter:
2 l x1 = − l : A= Al = π ( R R ) , λk= λ k (6.2.11) 2 0 x1=0: AA == 0Dkπ ( R ), λ = λ k
When solving the problem for disturbances, the initial conditions are important parameters and by varying them the limits of the process can be determined. Sinusoidal disturbance has been introduced in the jet area at a certain time, which has been taken as starting point for the computations. To preserve the continuity in the system the harmonic disturbance has been chosen as shown below:
A1n e w = AA 0 − 1 .sin(2.π . Nx . ) (6.2.12)
Here Anew is the disturbed value of flowing filament area, A0 is the initial value of
the area calculated from the steady state, A1 is the amplitude of the disturbance and N is
115
the wave number of initial disturbance. By changing the amplitude and the period of the introduced disturbance the full range of the stable and unstable region can be found.
The initial conditions for velocity in capillary flow are the final values of the
velocity in the reservoir simulation. The final values for the elastic stretches λk are transformed to the elastic strain tensor and used as initial condition in the capillary flow.
This technique allows examining the propagation of the disturbances generated in the reservoir and track their decay or growing downstream.
6.2.3 Numerical Approach
Equations (6.2.1) through (6.2.4) represent an initial boundary problem and finite difference scheme has been chosen to obtain the numerical solution.
First the steady flow problem in reservoir and capillary has been solved using finite difference formulas for derivatives. The critical Deborah number is found for specific flow conditions employing the linear stability analysis in near entrance region.
Then the system of PDE’s (6.2.1)-(6.2.3) with boundary conditions (6.2.11) has been solved for Deborah numbers close the critical one. For numerical discretization second order scheme forward in time and central in space has been used. The final values for velocity and elastic stretches are used as initial conditions in capillary region.
Another system of PDE’s (6.2.5)-(6.2.9) with boundary conditions (6.2.10) has been solved using forward discretization, and the propagation of disturbances has been analyzed. The same critical Deborah number has been used to investigate the propagation of the disturbances in the capillary.
116
6.2.4 Results and Discussions
The numerical scheme presented above has been applied to a planar 4:1 abrupt contraction, which has been extensively used as a bench-mark problem in the literature.
Two separate cases have been examined: nearly below and nearly over the critical
Deborah number of the contraction flow. The numerical calculations have been performed for various amplitudes of the introduced sinusoidal disturbance (0.1%, 1%, 5% and 10%) and several periods of the disturbance in order to investigate the complete range of the stable and unstable flow regions.
The following figures demonstrate the time evolution of the initial disturbance for a rheologically known polymer (Table 6.1.1) at different Deborah numbers. The figures are created every couple of hundred iterations showing the increase (or decrease for stable case) of the amplitude of the initial disturbance. The region from 0 to 1 on each figure represents the dimensionless length of the reservoir.
The Deborah number of the process is carefully selected so the real part of the largest eigenvalue is less then critical one (more then critical one for unstable process) and the imaginary part of the eigenvalue is equal to zero.
Initial disturbance is imposed on the jet radius and effect on the velocity and propagation is followed till the end of the capillary. The jet radius disturbances are only in the reservoir region and it is becoming zero in the capillary. As expected the disturbances in velocity decay downstream due to stabilizing nature of the process, but their growth in axial direction highly depends on the amplitude of the initial disturbance imposed.
117
The first group of numerical simulations shows the disturbance propagation under unstable conditions. Generally there are two separate cases observed during the numerical simulation. The system behaves differently depending on the size of the initially introduced disturbance.
At very small introduced initial disturbances the instabilities grow in size for a short period of time. The instabilities propagate to the capillary with time. The converging flow moves up and down like waves on elastic string. The flow is pulsing in the reservoir, with symmetrical, cyclic distortions. The top and bottom part of the die distortions are in phase (Figure 6.2.1). These fluctuations can be seen for a long period of time which can be classified as a metastable state of the contraction flow.
Depending on how close is the used Deborah number to the critical one the amplitude at which the stabilization of the growth of the disturbances occurs is different.
If the Deborah number used is close to the critical one the initial amplitude can reach up to 4%. (Figure 6.2.2)
118
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0 Jet Profile -0.2 Jet Profile -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 -1 0 0.5 1 1.5 0 0.5 1 1.5 Reservoir Capillary Reservoir Capillary
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0 Jet Profile -0.2 Jet Profile -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 -1 0 0.5 1 1.5 0 0.5 1 1.5 Reservoir Capillary Reservoir Capillary
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0 Jet Profile -0.2 Jet Profile -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 -1 0 0.5 1 1.5 0 0.5 1 1.5 Reservoir Capillary Reservoir Capillary
Figure 6.2.1. Jet Radius fluctuation grow for an unstable contraction flow of PIB/PB/C14 with 0.1% introduced sinusoidal disturbance at De=1.67 (De CR =1.51)
119
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0 Jet Profile Jet Profile -0.2 -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 -1 0 0.5 1 1.5 0 0.5 1 1.5 Reservoir Capillary Reservoir Capillary
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
Jet Profile -0.2 Jet Profile -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 -1 0 0.5 1 1.5 0 0.5 1 1.5 Reservoir Capillary Reservoir Capillary
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
Jet Profile -0.2 Jet Profile -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 -1 0 0.5 1 1.5 0 0.5 1 1.5 Reservoir Capillary Reservoir Capillary
Figure 6.2.2. Jet Radius fluctuation grow for an unstable contraction flow of PIB/PB/C14 with 4% introduced sinusoidal disturbance at De=1.67 (De CR =1.51)
120
The instabilities in velocity are similar to those calculated for the jet radius. They are regular pulsations which propagate in the capillary. Their propagation is simulated solving the system of PDE’s for shear flow and using the final conditions from reservoir as initial condition in capillary.
The imposed disturbances in reservoir propagate in capillary with almost the same frequency and decreasing amplitude in space. Using the previous unstable conditions and
1% initial disturbance the axial velocity profile is shown in Figure 6.2.3. The figures are created every couple of hundred iterations showing the increase of the amplitude of the initial disturbance. The region from 0 to 1 on each figure represents the dimensionless length of the reservoir and from 1 to 2 is the dimensionless length of capillary region.
At higher amplitudes (>5%) of initial disturbance the flow becomes irreversibly unstable after a short period of time. The disturbances of jet radius increase till they become larger then the size of the reservoir and then the computation becomes meaningless. On Figure 6.2.4 the average axial velocity of the flow is illustrated for an unstable flow and initial disturbance larger then the critical. The propagation of the velocity disturbances is calculated and shown up to the exit of the die.
When inserting initial disturbance with higher frequencies N=5 the disturbances are reaching their peak for a shorter period of time compared to the case where N=2, where all other parameters are kept same (Figure 6.2.5).
121
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
Figure 6.2.3. Axial Velocity fluctuations for an unstable contraction flow of PIB/PB/C14 with 1% introduced sinusoidal disturbance at De=1.67 (De CR =1.51) 122
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
Figure 6.2.4. Axial velocity profile of disturbance growth and propagation for an unstable flow of PIB/PB/C14 with 5% introduced initial disturbance at De=1.67 (De CR =1.51)
123
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
Figure 6.2.5. Axial velocity profile of disturbance growth and propagation for an unstable flow of PIB/PB/C14 with 5% and N=5 introduced initial disturbance at De=1.67
124
Another set of numerical simulations has been made in order to find whether the finite disturbances imposed in a contraction flow with Deborah number under the critical one could destabilize the flow.
Starting with 1% and then increasing the amplitude of the introduced disturbance the respond of the system has been examined. For all amplitudes, the period (i.e. frequency) of the disturbance has been changed from 2 to 8. The results showed that independently of the amplitude of the introduced disturbance the process continue to be stable as in the case of linear stability. Figure (6.2.6) demonstrates an example of the axial velocity for reservoir and capillary at the large initial imposed disturbance (20%).
The time needed for the complete decay of the fluctuations increases with the initial amplitude of the disturbance.
The general conclusion is that the process remains stable to finite amplitude disturbances if the draw ratio is less then the critical draw ratio. This means that the signs of real parts of all the eigenvalues are negative.
125
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary
18 18
16 16
14 14
12 12
10 10
8 8 Axial Velocity Axial Velocity 6 6
4 4
2 2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reservoir Capillary Reservoir Capillary Figure 6.2.6. Disturbance evolution for a stable Contraction flow of PIB with 20% introduced sinusoidal disturbance at DR=1.36 (De CR =1.51) 126
CHAPTER VII
SUMMARY
7.1 Results
The objective of this work was to describe the instabilities occurring in contraction flows at high Deborah numbers. A linear and non-linear stability analysis has been used to find the stable and unstable processing conditions in fiber spinning and contraction flow.
The model used in this work has the following general features and advantages: improved description of polymer flows at higher contraction ratios and De numbers, the model is completely free from any artificial fitting parameters, it considerably saves computational costs compared to direct computations, it has the flexibility to be applicable to any viscoelastic constitutive equation and can be easily extended to describe other phenomena, such as wall slip, stress-induced crystallization, etc.
Linear stability with respect to infinitesimal disturbances is studied by solving a set of linear PDE’s describing the transient viscoelastic flow near the steady state. A methodology has been established for finding the critical draw ratio (critical Deborah number for contraction flow). The onset of the instability is determined by solving an eigenvalue problem. If the real part of the largest eigenvalue is positive, then any arbitrarily small deviation from steady state will grow in magnitude with time, i.e. the
127
flow is unstable. Conversely, if the real part of largest eigenvalue is negative, then all infinitesimal disturbances will decay to zero, and the system is stable. The agreement between the numerical predictions and experimental data was found to be very good.
Non-linear stability analysis has been used to determine the behavior of the flow near the critical draw ratio and to establish the response of the system on finite disturbances. In order to obtain the solution for the non-linear case we solved a closed set of PDE’s, consisting of equations for conservation of mass and momentum, along with viscoelastic constitutive equation, with appropriate initial/boundary conditions.
Initial sinusoidal disturbance has been introduced, which has been taken as starting point for the computations and by changing the amplitude and the period of the introduced disturbance the full range of the stable and unstable behavior is found.
The stabilizing effect of the cooling has been investigated. It has been found that the critical draw ratio can reach much higher values compared to the isothermal fiber spinning. Using the similarity approach due to the time-temperature superposition, we introduced the same temperature dependence of viscosities (relaxation times) in each relaxation mode, neglecting the temperature dependence of mode’s moduli, and employed in our calculations the coupling of the energy and evolution equation under the similarity condition.
Cooling the fiber reduces the fluctuations in area and helps to improve the stability of the process. All simulated cases showed good agreement with the experimental results.
128
The propagation of the introduced disturbances in the capillary has been investigated. At unstable regimes and very small introduced initial disturbances the instabilities grow in size for a short period of time. The converging flow moves up and down like waves on elastic string. The flow is pulsing in the reservoir, with symmetrical, cyclic distortions.
The instabilities in velocity are similar to the one calculated for the jet radius.
They are regular pulsations which propagate in the capillary. Their propagation is simulated solving the system of PDE’s for shear flow and using the final conditions from reservoir as initial condition in capillary.
7.2 Recommendations for future work
The main purpose of this work was to obtain an understanding of the instabilities in reservoir and their propagation into capillary. Further studies can investigate the propagation of the capillary instabilities outside the die and include the oscillatory die swell phenomenon. Using matching conditions for the boundary between capillary and the free surface, the final result from capillary calculations can be used as initial condition for calculating the instabilities propagation after the polymer leaves the die.
Additional work can be done for investigating the thermal effect in contraction flow and also the effect of the secondary flow. But this work requires a solid experimental effort as well.
129
REFERENCES
1. Kase, S; Matsuo, T. “Melt spinning. I. Fundamental equations on the dynamics of melt spinning” Journal of Polymer Science, Part A: General Papers (1965), 3(7)
2. Han, C.D.; Lamonte, R. R.; Shah, Y. T. Melt spinning. III. Flow instabilities in melt spinning. Melt fracture and draw resonance” Journal of Applied Polymer Science (1972), 16(12), 3307-23.
3. Kase, S. “Melt spinning. IV. Stability of melt spinning” Journal of Applied Polymer Science (1974), 18(11), 3279-304
4. Denn, M. M.; Petrie, C. J. S.; Avenas, P. “Mechanics of steady spinning of a viscoelastic liquid” AIChE Journal (1975), 21(4), 791-9.
5. Fisher, R J.; Denn, M. M. “Finite-amplitude stability and draw resonance in isothermal melt spinning” Chemical Engineering Science (1975), 30(9), 1129-34.
6. Petrie, C. J. S., Denn, M. M. “Instabilities in polymer processing” AIChE Journal (1976), 22(2), 209-36.
7. Ide, Y.; White, J. L. “The spinnability of polymer fluid filaments” Journal of Applied Polymer Science (1976), 20(9), 2511-31
8. White, J. L.; Ide, Y. “Instabilities and failure in elongational flow and melt spinning of fibers” Journal of Applied Polymer Science (1978), 22(11), 3057-74.
9. Hyun, J. C. “Theory of draw resonance: Part I. Newtonian fluids” AIChE Journal (1978), 24(3), 418-22
10. Hyun, J. C. “Theory of draw resonance: Part II. Power-law and Maxwell fluids. AIChE Journal (1978), 24(3), 423-6
11. Tsou, J. D., Bogue, D. C. “The effect of die flow on the dynamics of isothermal melt spinning” Journal of Non-Newtonian Fluid Mechanics (1985), 17(3), 331-47
12. Anturkar, N. R.; Co, A. “Draw resonance in film casting of viscoelastic fluids: a linear stability analysis” Journal of Non-Newtonian Fluid Mechanics (1988), 28(3), 287-307 130
13. Ishizuka, O; Koyama, K. “Extensional viscosity and breaking in melt spinning” Sen’i Gakkaishi (1975), 31(9), 377-380.
14. Ishihara, H, Hayashi, S. “Spinning rheology. 1. Instability analysis” Nihon Reoroji Gakkaishi (1989), 17(1), 19-26
15. Ishihara, H., Hayashi, S., Yasuda, H. “Spinning rheology. (2). Analysis of melt spinning behaviors and its application”Nihon Reoroji Gakkaishi(1992), 20(3), 109-17
16. Fisher, R. J.; Denn, M. M. “A theory of isothermal melt spinning and draw resonance” AIChE Journal (1976), 22(2), 236-46.
17. Pearson, J. R. A.; Shah, Y. “Stability of isothermal and non-isothermal fiber spinning of power-law fluids” Industrial Engineering Chemistry Fundamentals (1974), 13(2), 134-8
18. Matovich, M. A.; Pearson, J. R. A. “Spinning a molten threadline; I. Steady-state isothermal viscous flows” Industrial & Engineering Chemistry Fundamentals (1969), 8(3), 512-20
19. Han, C. D; Segal, L. “Fiber extrusion in wet spinning. I. Experimental determination of elongational viscosity” Journal of Applied Polymer Science (1970), 14(11), 2973-98
20. Denn, M. M.; Marrucci, G. “Effect of a relaxation time spectrum on mechanics of polymer melt spinning” Journal of Non-Newtonian Fluid Mechanics (1977), 2(2),159-67.
21. Keunings, R.; Crochet, M. J.; Denn, M. M. “Profile development in continuous drawing of viscoelastic liquids” Industrial & Engineering Chemistry Fundamentals (1983), 22(3), 347-55.
22. Lee, J. S.; Jung, H. W “Effect of fluid viscoelasticity on the draw resonance dynamics of melt spinning” Journal of Non-Newtonian Fluid Mechanics (2001), 99(2), 159-66.
23. Jung, H. W.; Hyun, J. C. “Stability of isothermal spinning of viscoelastic fluids” Korean Journal of Chemical Engineering (1999), 16(3), 325-330.
24. Ching-China; Yang, Jyh-Chau. “Stability of Newtonian-PTT coextrusion fiber spinning” Polymer Engineering and Science (1996), 36(22), 2685-2693
25. Hagen, T; Renardy, M. “Non-adiabatic elongational flows of viscoelastic melts” ZAMP (2000), 51(6), 845-866.
131
26. Donnelly, G. J.; Weinberger, C. B. “Stability of isothermal fiber spinning of a Newtonian fluid” Industrial & Engineering Chemistry Fundamentals (1975), 14(4)
27. Demay, Y.; Agassant, J. F. “Experimental study of the draw resonance in fiber spinning” Journal of Non-Newtonian Fluid Mechanics (1985), 18(2), 187-98.
28. Weinberger, C. B.; Cruz-Saenz, G. F.; Donnelly, G. J. “Onset of draw resonance during isothermal melt spinning: a comparison between measurements and predictions” AIChE Journal (1976), 22(3), 441-8
29. Ishihara, H; Kase, S. “Studies on melt spinning. VI. Simulation of draw resonance using Newtonian and power law viscosities” Journal of Applied Polymer Science (1976), 20(1), 169-91.
30. Lamb, P. “Analysis of fabrication processes [for polymers]” Advance Polymer Science Technology (1967), 26, 296-312
31. Fisher, R. J.; Denn, M. M.; Tanner, R I. “Initial profile development in melt spinning” Industrial & Engineering Chemistry Fundamentals (1980), 19(2), 195-7.
32. Young, D. G.; Denn, M. M. “Disturbance propagation in melt spinning” Chemical Engineering Science (1989), 44(9), 1807-18
33. Matsuo, T. Kase, S. “Studies on melt spinning. VII. Temperature profile within the filament” Journal of Applied Polymer Science (1976), 20(2), 367-76.
34. Shah, Y. T.; Pearson, J. R. A. “Stability of non-isothermal fiber spinning. General case. “Industrial & Engineering Chemistry Fundamentals (1972), 11(2), 150-3.
35. Lamonte, R. R.; Han, C.D. “Melt spinning. II. Analysis of the deformation and heat transfer processes” Journal of Applied Polymer Science (1972), 16(12), 3285-306.
36. Fisher, R. J.; Denn, M. M. “Mechanics of non-isothermal polymer melt spinning” AlChE Journal (1977), 23(1), 23-8.
37. Sukanek, P. C.; Laurence, R. L. “Experimental investigation of viscous heating in some simple shear flows” AIChE Journal (1974), 20(3), 474-84.
38. Jung, H. W.; Song, H. “Analysis of the stabilizing effect of spinline cooling in melt spinning” Journal of Non-Newtonian Fluid Mechanics (1999), 87(2-3), 165-174.
39. Barnett, Thomas R. “Calculation of the temperature of filaments in melt spinning” Polymer Preprints (American Chemical Society, Division of Polymer Chemistry) (1966), 7(2), 691-8
132
40. Copley, Malcolm; Chamberlain, Norman H. “Filament attenuation in melt spinning and its effect on axial temperature gradient” Applied Polymer Symposia (1967)
41. Iyer, V. R “Thermal modeling and parametric evaluations of polymer melt spinning” Master thesis. University of Akron
42. Ishihara, H.; Kase, S. Melt spinning. “V. Draw resonance as a limit cycle” Journal of Applied Polymer Science (1975), 19(2), 557-66
43. C. D. Han, Rheology in Polymer Processing, Ch. 5, Acad. Press, New York (1976).
44. Joseph, D. D. Fluid Dynamics of Viscoelastic Liquids, Springer, New York (1990).
45. Delvaux V., Crochet M. J. “Numerical simulations of delayed die swell” Rheologica Acta (1990), 29:1-10.
46. Yang, B.; Lee, L. J. “Effect of die temperature on the flow of polymer melts. Part I: Flow inside the die” Polymer Engineering and Science (1987), 27(14), 1079-87
47. Orbey, N.; Dealy, J.M. “Isothermal swell of extrudate from annular dies: effects of die geometry, flow rate, and resin characteristics” Polymer Engineering and Science (1984), 24(7), 511-18
48. Ghijsels, A.; De Clippeleir, J. “Melt strength behavior of polypropylenes” International Polymer Processing (1994), 9(3), 252-7
49. Huneault, M. A.; Lafleur, P. G. “Extrudate swell and draw down effects on extruded profile dimensions and shape” Polymer Engineering and Science (1990), 30(23),1544-50
50. Agarwal, P.K; Bagley, E. B. ”Viscosity, modulus, and die swell of glass bead filled polystyrene-acrylonitrile copolymer” Polymer Engineering and Science (1978), 18(4), 286-91
51. Eggen, S.; Hinrichsen, E. L. “Swell and distortions of high-density polyethylene extruded through capillary dies” Polymer Eng. and Science (1996), 36(3), 410-24
52. Pecora, L.; DiRaddo, R.W. “An experimental study on the effect of die geometry on swell and sag in the parison extrusion stage” CNRC, Canada, 1993.
53. Huang, D.; White, J. L. “Experimental and theoretical investigation of extrudate swell of polymer melts from small (length)/(cross-section) ratio slit and capillary dies” Polymer Engineering and Science (1980), 20(3), 182-9
133
54. Han, C. D., Lamonte, R. R.; Drexler, L. H “Melt spinning. IV. Spinning through a ribbon die” Journal of Applied Polymer Science (1973), 17(4), 1165-72
55. Kurtz, S. J. “Visualization of Exit Fracture in the Sharkskin Process” PPS 10th Ann. Meeting, Akron, Ohio 8-9 (1994).
56. Wang, S., Drda, P. A.; Inn, Y. “Exploring molecular origins of sharkskin, partial slip, and slope change in flow curves of linear low-density polyethylene” Journal of Rheology (New York) (1996), 40(5), 875-898.
57. Vinogradov, G. V.; Protasov, V. P.; Dreval, V. E. “The rheology of flexible-chain polymers at high shear rates and stresses, the critical process of spurting, and supercritical conditions of their movement below the glass temperature” Rheologica Acta (1984), 23(1), 46-61.
58. Piau, J. M.; El K., Toussaint, F; “Distortions of polymer melt extrudates and their elimination using slippery surfaces” Rheologica Acta (1995), 34(1), 40-57
59. Leonov A. I. and Prokunin A. N., Non-linear Phenomena in Flows of Viscoelastic (Ch.11) Polymer Fluids, Chapman & Hall, New York (1994).
60. Larson R. G. “Constitutive Equations for Polymer Melts and Solutions, Butterworth, Boston (1988).
61. Leonov A. I. ‘Analysis of simple constitutive equations for viscoelastic liquids’, J. Non-Newtonian Fluid Mech., 42, 323–350 (1992).
62. Kwon, Y. Leonov, A.I. “Stability constraints in the formulation of viscoelastic constitutive equations” J. Non-Newtonian Fluid Mech. (1995), 58, 25-46.
63. Simhambhatla, M, Leonov, A. I. “On the rheological modeling of filled polymers with particle-matrix interactions” Rheologica Acta (1995), 34(4), 329-38
64. Cable, P. J. Boger, D. V. “A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids: Part I. Vortex characteristics in stable flow” AIChE Journal (1978), 24(5), 869-79
65. Black, J. R.; Denn, M. M. “Converging flow of a viscoelastic liquid” Journal of Non- Newtonian Fluid Mechanics (1976), 1(1), 83-92.
66. Rothenberger, R., McCoy, D. H. Denn, M. M. “Flow instability in polymer melt extrusion” Transactions of the Society of Rheology (1973), 17(2), 259-69.
134
67. Cogswell, F. N. “Converging flow of polymer melts in extrusion dies” Polymer Engineering and Science (1972), 12(1), 64-73.
68. Cogswell, F. N. “Stretching flow instabilities at the exits of extrusion dies” Journal of Non-Newtonian Fluid Mechanics (1977), 2(1), 37-47.
69. Denson, C. D. “Implications of extensional flows in polymer fabrication processes” Polymer Engineering and Science (1973), 13(2), 125-30
70. Halmos, A. L.; Boger, D. V. “Flow of viscoelastic polymer solutions through an abrupt 2-to-1 expansion” Transactions of the Society of Rheology (1976), 20(2), 253-64
71. Kaye, S. E.; Rosen, S. L. “Influence of conical entry regions on laminar entrance losses for Newtonian fluids” AIChE Journal (1973), 19(5), 1033-5
72. Debbaut, B.; Crochet, M. J. “Further results on the flow of a viscoelastic fluid through an abrupt contraction” Journal of Non-Newtonian Fluid Mechanics (1986), 20, 173-85.
73. Abdul-Karem, T. Binding, D.M.; Sindelar, M. “Contraction and expansion flows of non-Newtonian fluids” Composites Manufacturing (1993), 4(2), 109-16
74. Sato, Toru, Richardson, Stephen M. “Explicit numerical simulation of time- dependent viscoelastic flow problems by a finite element/finite volume method” Journal of Non-Newtonian Fluid Mechanics (1994), 51(3), 249-75
75. Baloch, A.; Townsend, P. Webster, M. F. “Extensional effects in flows through contractions with abrupt or rounded corners” Journal of Non-Newtonian Fluid Mechanics (1994), 54(1-3), 285-302
76. Baloch, A.; Townsend, P. Webster, M. F. “On vortex development in viscoelastic expansion and contraction flows” Journal of Non-Newtonian Fluid Mechanics (1996), 65(2/3), 133-149
77. Bulters, M. J. H.; Meijer, H. E. H. “Modeling of pullout in solution spinning; analogy with the vortex size in contraction flows” Theor. Appl. Rheol. Proc. Int. Congr. Rheol, 11th (1992), 1, 171-3
78. Alves, Ml A. Oliveira, P. J. “On the effect of contraction ratio in viscoelastic flow through abrupt contractions.” Journal of Non-Newtonian Fluid Mechanics (2004), 122(1-3), 117-130
135
79. Yesilata, B. “Unsteady vortex dynamics of a viscoelastic flow in a circular sudden contraction geometry” Int. Journal of Transport Phenomena (2002), 4, 193-207
80. Rothstein, J. P. McKinley, G. H. “The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop” Journal of Non-Newtonian Fluid Mechanics (2001), 98(1), 33-63
81. Mitsoulis, E. “Numerical simulation of entry flow of the IUPAC-LDPE melt” Journal of Non-Newtonian Fluid Mechanics (2001), 97(1), 13-30
82. Naka, Y; Chiba, K; Nakamura, K. “Unstable entry flow of polymer solutions through a tubular contraction” Nihon Reoroji Gakkaishi (1998), 26(1), 41-47
83. Paskhin, E. D. “Motion of polymer liquids under unstable conditions and in channel terminals” Rheologica Acta (1978), 17(6), 663-75
84. Muller, S. J.; Shaqfeh, E. S. G.; Larson, R. G. “Experimental studies of the onset of oscillatory instability in viscoelastic Taylor-Couette flow” Journal of Non-Newtonian Fluid Mechanics (1993), 46(2-3), 315-30
85. White, S. A.; Baird, D. G. “Flow visualization and birefringence studies on planar entry flow behavior of polymer melts” Journal of Non-Newtonian Fluid Mechanics (1988), 29 245-67
86. Tremblay, B. “Visualization of the flow of linear low-density polyethylene/low density polyethylene blends through sudden contractions” Journal of Non-Newtonian Fluid Mechanics (1992), 43(1), 1-29
87. Guido, S.; Frallicciardi, P.; Grizzuti, N.; Marrucci, G. “Rheo-optics of hydroxypropyl cellulose solutions in Poiseuille flow” Rheologica Acta (1994), 33(1), 22-8
88. Binnington, R. J.; Boger, D. V.; Troup, G. “Point velocity measurement in slow flows using laser speckle photography” Journal of Non-Newtonian Fluid Mechanics, 12 (1983) 255-267.
89. Dudderar, T. D. Simpkins, P. G. “Laser speckle photography in a fluid medium” Nature (London, United Kingdom) (1977), 270(5632), 45-7
90. Isayev, A. I.; Upadhyay, R. K. “Two-dimensional viscoelastic flows: xperimentation and modeling” Journal of Non-Newtonian Fluid Mechanics (1985), 19(2), 135-60
91. Nguyen, T. H.; Boger, D. V. “The kinematics and stability of die entry flows. Journal of Non-Newtonian Fluid Mechanics (1979), 5, 353-68
136
92. White, S. A.; Gotsis, A. D.; Baird, D. G. “Review of the entry flow problem: experimental and numerical” Journal of Non-Newtonian Fluid Mechanics (1987), 24(2), 121-60
93. Ballenger, Timothy F. White, J. L. “Development of the velocity field in polymer melts in a reservoir approaching a capillary die” Journal of Applied Polymer Science (1971), 15(8), 1949-62
94. Ballenger, Timothy F. White, J. L “Experimental study of flow patterns in polymer fluids in the reservoir of a capillary rheometer” Chemical Engineering Science (1970), 25(7), 1191-5
95. Ramamurthy, A. V. “Velocity measurements in the die entry region of a capillary rheometer” Journal of Rheology (New York, United States) (1980), 24(2), 167-88
96. Birman, V.B.; Leonov, Arkady I.; “Study of high Deborah number flow of polyisobutylene in square die” Rheologica Acta (1999), 38(3), 235-240
97. M.B. Bush, R.I. Tanner and N. Phan-Thien, “A boundary element investigation of extrudate swell” J. Non-Newt. Fluid Mech., 18 (1985) 143.
98. Wesson, R. D.; Papanastasiou, T. C.; Wilkes, J. O. “Analysis of plane extrudate- swell of highly elastic liquids with molecular constitutive equations” Journal of Non- Newtonian Fluid Mechanics (1989), 32(2), 157-73.
99. Kwon, Y.; Leonov, A. I. “Stability constraints in the formulation of viscoelastic constitutive equations” Journal of Non-Newtonian Fluid Mechanics (1995), 58(1), 25-46
100. Upadhyay, R. K.; Isayev, A. I. “Elongational flow behavior of polymeric fluids according to the Leonov model” Rheologica Acta (1983), 22(6), 557-68
101. Hulsen, M. A. Van der Zanden, J. “Numerical simulation of contraction flows using a multi-mode Giesekus model” Journal of Non-Newtonian Fluid Mechanics (1991), 38(2-3), 183-221
102. Isayev, A. I. Huang, Y. H. “Two-dimensional planar flow of a viscoelastic plastic medium” Rheologica Acta (1993), 32(2), 181-191
103. Purnode, B.; Legat, V. “Hyperbolicity and change of type in flows of FENE-P fluids” Journal of Non-Newtonian Fluid Mechanics (1996), 65(1), 111-129
137
104. Beris, A. N. Armstrong, R. C.; Brown, R. A. “Spectral/finite-element calculations of the flow of a Maxwell fluid between eccentric rotating cylinders” Journal of Non-Newtonian Fluid Mechanics (1987), 22(2), 129-67
105. King, R C. Apelian, M R.; Armstrong, R C.; Brown, R A. “Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries” Journal of Non-Newtonian Fluid Mechanics (1988), 29, 147-216
106. Lawler, J. V.; Muller, S. J.; Brown, R. A.; Armstrong, R. C. “Laser Doppler velocimetry measurements of velocity fields and transitions in viscoelastic fluids” Journal of Non-Newtonian Fluid Mechanics (1986), 20, 51-92
107. Hinch, E.J. “The flow of an Oldroyd fluid around a sharp corner” Journal of Non- Newtonian Fluid Mechanics (1993), 50(2-3), 161-71
108. Renardy, M.”A matched solution for corner flow of the upper convected Maxwell fluid” Journal of Non-Newtonian Fluid Mechanics (1995), 58(1), 83-9
109. Renardy, M. “Re-entrant corner behavior of the PTT fluid” Journal of Non- Newtonian Fluid Mechanics (1997), 69(1), 99-104.
110. Binding, D. M. “An approximate analysis for contraction and converging flows” Journal of Non-Newtonian Fluid Mechanics (1988), 27(2), 173-89
111. Jeong, J., Leonov, A. I. “A quasi-1D model for fast contraction flows of polymer melts” Journal of Non-Newtonian Fluid Mechanics (2004), 118(2-3), 157-173.
112. Zahorski, S. “Instability of disturbed elongational flows of viscoelastic fluids” Archives of Mechanics (1995), 47(4), 647-59
113. McKinley, G H.; Hassager, O. “The Considered condition and rapid stretching of linear and branched polymer melts” Journal of Rheology (New York) (1999), 43(5), 1195-1212
114. Marchal J. M., Crochet, M. J. “A new mixed finite element for calculating viscoelastic flow”, Journal of Non-Newtonian Fluid Mechanics, (1987), 26, 77-114
115. Rao R. R., Finlayson, B. A. “Viscoelastic flow simulation using cubic stress finite elements”, Journal of Non-Newtonian Fluid Mechanics, 43, 61-82 (1992).
116. Mckinley G. H.,Raiford W.P., Brown R.A., “ Non-linear dynamics of viscoelastic flow in axisymmetric abrupt contractions”, Journal of Non-Newtonian Fluid Mechanics, 223, 411-56 (1991).
138