Journal of Non-Newtonian Fluid Mechanics

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1 J. Non-Newtonian Fluid Mech. 157 (2009) Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: Steady flow simulations of compressible PTFE paste extrusion under severe wall slip Evan Mitsoulis a,, Savvas G. Hatzikiriakos b a School of Mining Engineering and Metallurgy, National Technical University of Athens, Athens, Greece b Department of Chemical and Biological Engineering, The University of British Columbia, Vancouver, BC, Canada article info abstract Article history: Received 16 July 2008 Received in revised form 7 September 2008 Accepted 16 September 2008 Keywords: Polytetrafluoroethylene Axisymmetric contraction Paste extrusion Slip law Compressibility Structural parameter Flow type parameter In a recent article [P.D. Patil, J.J. Feng, S.G. Hatzikiriakos, Constitutive modelling and flow simulation of polytetrafluoroethylene paste extrusion, J. Non-Newtonian Fluid Mech., 139 (2006) 44 53], a rheological constitutive equation was proposed to model the flow of polytetrafluoroethylene (PTFE) paste. Steadystate flow simulations were carried out, governed by the conservation equations of mass and momentum under isothermal, incompressible conditions, and taking into account a linear slip law at the wall due to the presence of lubricant in the paste. The constitutive model was a combination of shear-thinning and shearthickening viscosity terms, depending on a structural parameter,, which obeyed a convective-transport equation. The present work re-examines the flow problem, taking into account the significant compressibility of the paste and implementing the slip law based on the consistent normal-to-the-surface unit vector. New results are shown as contours of the -parameter in the flow field as well as contours of the flow type parameter,, used in the model. They offer a better understanding of the flow behaviour of PTFE in flow through extrusion dies Elsevier B.V. All rights reserved. 1. Introduction In polytetrafluoroethylene (PTFE) paste extrusion, fine powder resin of individual particle diameter of approximately 0.2 m is first mixed with a lubricating liquid to form a paste at concentrations ranging from 16 to 25 wt% depending on the application [1]. The paste is then compacted at a typical pressure of 2 MPa to produce a cylindrical preform that can withstand its own gravity and that is free of air void [2]. The next step involves the extrusion of the preform using a ram extruder at a temperature slightly higher than 30 C [1]. This is usually followed by the evaporation of the lubricant through an oven and then sintering, for processes such as wire coating and tube fabrication. During flow in the extrusion step, continuous structural changes occur in the material, which make PTFE flow a complex process and difficult to describe and study rheologically [3]. As the particles squeeze against each other due to high normal forces, crystallites across their interface mechanically interlock. As the particles separate due to mixed shear and elongational flow in the contraction, these crystallites unwind producing fibrils that interconnect the individual particles [1,3]. Thus, the paste continuously changes from a viscous state (at the entrance) into a Corresponding author. address: mitsouli@metal.ntua.gr (E. Mitsoulis). mixed viscous and semi-solid state. Due to these changes the study of the rheology of PTFE paste becomes a difficult and complicated task. In a recent article, Patil et al. [4] proposed a rheological constitutive equation to model the flow of PTFE paste in tapered axisymmetric dies. The authors performed steady-state flow simulations, which were governed by the conservation equations of mass and momentum under isothermal, incompressible conditions, and took into account a linear slip law at the wall due to the presence of the lubricant. The constitutive model was a combination of shear-thinning and shear-thickening viscosity terms, linearly connected by a structural parameter,, defined as the percentage of fibrillated domains in the paste. This parameter obeyed a convective-transport equation (CTE) (see below for details of the set of equations). The paste flow was assumed to be incompressible. However, significant amounts of air voids exist in the paste due to low lubricant concentrations. Thus, the isothermal compressibility of the paste is expected to be significant. Experiments were undertaken in the present work to determine this in order to examine numerically the effect of compressibility. In summary, the present work re-examines the PTFE paste flow problem by taking into account the significant compressibility of the paste and by implementing a slip law based on the consistent normal-to-the-surface unit vector. New results are shown as contours of the -parameter in the flow field used in the model in order /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.jnnfm

2 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) Table 1 Compressibility of PTFE paste as a function of the lubricant Isopar concentration at 35 C. Isopar (wt%) ˇ ( 10 8 Pa 1 ) to study in detail the effect of compressibility on the structure formation and flow kinematics of paste. These results offer a better understanding of the flow behaviour of PTFE in flow through extrusion dies. It has been shown in the past that compressibility plays a significant role in the flow of polymer melts even though these fluids are slightly compressible [5,6]. On the contrary, PTFE paste is highly compressible, and significant differences are expected by removing the assumption of incompressibility in PTFE paste flow. 2. Experimental The pastes used in this work were similar to those used in the study by Ochoa and Hatzikiriakos [7] and Patil et al. [4]. Thus, the same rheological and physical parameters are used in the simulation. The pastes were prepared using a high molecular weight PTFE resin (F-104) mixed with concentrations of Isopar M as the lubricant ranging from vol.% (17 wt%) to vol.% (20 wt%). The isothermal compressibility is defined as: ( ) ˇ = 1 V, (1) V p T where V is the volume, p is the pressure and T is the temperature. It is essentially a measure of the material to change volume under applied pressure at constant temperature. This compressibility was measured by means of a capillary rheometer as follows: the reservoir of the capillary rheometer was blanked off by a blind die. Consequently, it was loaded with a known amount of paste. After waiting approximately 20 min, the paste was compressed by moving the piston at a velocity of mm/s, while the change of pressure was continuously recorded. The pressure volume relationship thus obtained was a straight line for values of pressure higher than a certain value. This line was extrapolated to obtain the density of paste at low pressures ( 0 = 1.6 g/cm 3 ). Using a linear regression method, the isothermal compressibility was determined to change almost linearly from Pa 1 (17 wt% lubricant) to Pa 1 (20 wt% lubricant) as listed in Table 1. These values are at least one order of magnitude greater than those reported for molten polymers, typically 10 9 Pa 1 [5,6,8]. Therefore, PTFE paste is significantly more compressible compared to molten polymers, mainly due to the existence of air voids. 3. Governing equations and rheological modelling We consider the conservation equations of mass and momentum for compressible fluids under isothermal, creeping, steady flow conditions. These are written as [9 12]: ū + ( ū) = 0, (2) 0 = p +, (3) where is the density, ū is the velocity vector, p is the pressure and is the extra stress tensor. For a compressible fluid, pressure and density are connected as a first approximation through a simple linear thermodynamic equation of state [10 12]: = 0 (1 + ˇp), (4) where ˇ is the isothermal compressibility (defined by Eq. (1)) with the density to be 0 at reference pressure p 0 (=0). For weakly compressible flows, the values of ˇ range between 0 (incompressible fluids) and 0.02 MPa 1 (slightly to moderately compressible materials such as the pastes examined in this work, see also Table 1). The viscous stresses are given for inelastic non-newtonian compressible fluids by the relation [10 12]: ( = ( ) 2 ) ( ū)ī, (5) 3 where ( ) is the apparent non-newtonian viscosity, which is a function of the magnitude of the rate-of-strain tensor = ū + ū T,whichisgivenby ( 1 1 ) 1/2 = 2 II = 2 ( : ), (6) where II is the second invariant of II = ( : ) = ij ij, (7) i j The tensor Ī in Eq. (5) is the unit tensor. In the previous work [4], a simple linear combination of shearthinning and shear-thickening viscosities was assumed for PTFE paste, depending on a structural parameter, : = (1 ) (8) In the above, 1 and 2 are the shear-thinning and shear-thickening viscosities, respectively, that are expressed by a Carreau model [12]: i = 0,i [1 + ( i ) 2 ] (n i 1)/2, (9) where i = 1 refers to shear-thinning (n 1 < 1) and i = 2 refers to shearthickening (n 2 > 1). The model parameters are the zero-shear-rate viscosity, 0i, the power-law index, n i, and the time constant, i. Their values are listed in Table 2. For the structural parameter,, a convective-transport equation was introduced in Ref. [4] based on arguments of creation and destruction of fibrils (see [4] for details). The CTE is written as: ( ) ū =, (10) where is a flow type parameter, originally introduced by Giesekus [13] and later used by Fuller and Leal [14] and Astarita [15], and given by = ω + ω, (11) where and ω are the magnitudes of the rate-of-strain tensor and the vorticity tensor ω = ū ū T. Eq. (11) is the definition of Table 2 Constants used in the simulations for PTFE paste extrusion at 35 C. Parameter PTFE (35 C) (model, Eq. (9)) 0, MPa s n s 0, MPa s n s ˇ 0.02 MPa kg/m 3 ˇsl 1.92 m/(mpa s)

3 28 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) Table 3 Dimensionless parameters for the range of simulations for PTFE paste extrusion at 35 C. Apparent shear rate, A (s 1 ) Slip parameter B sl (Eq. (14)) Compressibility parameter B (Eq. (15)) Fig. 1. Flow domain and boundary conditions. The shaded part is the solution domain due to symmetry. the Giesekus function. It is interesting to note here that the quantity ω (vorticity) is not objective [15]. Deriving an objective quantity is clearly a challenge, as it involves the evaluation of angular velocity of eigenvectors. Some authors [16,17] have worked out such modifications, but it is certainly not a trivial matter. In this study and as a first approximation, we only consider Eq. (11). As explained in Refs. [13,14], = 0 in pure shear flow, = 1 in pure rotational flow, and = 1 in pure elongational flow. In a mixed flow, such as flow through an extrusion die considered here, 1 1. Since the CTE involves the square root of, only non-negative values are taken into the CTE. Also, 0 1, as it is explained in detail in Ref. [4]. The above rheological model is introduced into the conservation of momentum (Eq. (3)) and closes the system of equations. Boundary conditions are necessary for the solution of the above system of equations. Fig. 1 shows the solution domain and boundary conditions for the contraction geometry. Because of symmetry only one half of the flow domain is considered, as was done in our previous work [18]. Also, severe slip occurs at the die wall due to the lubricant, as found out and measured experimentally by Patil et al. [4]. Due to slip at the wall (boundary EDSC in Fig. 1), the experimental data have been fitted with a slip law, which takes the usual form [8]: u sl = ˇsl w b 1 w, (12) where u sl is the slip velocity, w is the shear stress at the die wall, ˇsl is the slip coefficient, and b is the slip exponent (b =1foralinear slip law applied to the PTFE paste). The value of ˇsl was found to be 1.92 m/(mpa s) (see [4] for details of the experimental results). For the 2D axisymmetric problem, the above law means that the tangential velocity, u t, is given by the slip law, while the normal velocity, u n, is set to zero, i.e.: ˇsl t n : b 1 ( t n : ) = t ū = u t, u n = n ū = 0, (13) where n is the unit outward normal vector to a surface and t is the tangential unit vector in the direction of flow. All lengths are scaled by the die radius R, all velocities by the average velocity U at the die exit, all pressures and stresses by 0,1 n 1 1 (U/R) n 1, and the density by 1 0. Then the dimensionless numbers which control the flow phenomena are [12]: (a) for slip at the wall, the slip parameter, B sl,givenby ( B sl = ˇsl 0,1 U ) n. (14) U R 1 n 1 1 When the slip parameter B sl = 0, we have no slip, while B sl 1 corresponds to macroscopically obvious slip (approach to perfectly plug velocity profile). For the present conditions (see below and Table 3), 1.2 < B sl < 2.6, which corresponds to severe slip; (b) for compressibility, the compressibility parameter, B, givenby B = ˇ 0,1 1 n 1 1 ( U ) n. (15) R The compressibility parameter B ranges from 0 (incompressible fluids) to 0.01 (weakly to moderately compressible fluids). For the present conditions (see below and Table 3), < B < 0.007, which is substantial for the material at hand. Because of compressibility effects, the domain length is also of importance. In the experiments, the die radius R = mm, the reservoir radius R res = mm, thus the contraction ratio D res /D = 18.75:1, and the area reduction ratio RR =(D res /D) 2 = 352. The barrel length (L res + L c ) was typically around 120 mm, thus giving for a die entrance angle of 2 =30,(L res + L c )/R around 470. The barrel length is important to give the correct pressure values only in transient experiments [5,6]. The steady-state values are very weakly dependent on the barrel length (see below) and there is almost no pressure drop until the contraction is reached. In the present simulations initial runs were made with various lengths (L res + L c )/R, but the final runs were made with L res /R around 40 50, to save space and obtain a more detailed solution in the contraction and die land area, where it matters. 4. Method of solution The numerical solution is obtained with the finite element method (FEM), employing as primary variables the two velocities, pressure, and structural parameter (u v p formulation). Noting the similarity of the CTE with the energy equation, we have substituted in our previous formulation the temperature, T, with the structural parameter, [18 20]. We use Lagrangian quadrilateral elements with biquadratic interpolation for the velocities and the structural parameter, and bilinear interpolation for the pressures. The densities, appearing in the compressible conservation equations, are substituted by the equation of state (Eq. (4)) through the pressures and their linear interpolation per element. An important extra feature of the governing equations is the handling of the purely convective-transport equation (Eq. (10)), which has no diffusion (conduction) terms. The CTE is thus the energy equation of heat transfer with c p = 1 (where is the density and c p is the heat capacity), k = 0 (where k is the thermal diffusivity), and the RHS term corresponding to the viscous dissipation term. This equation can be solved with the standard Streamline- Upwind/Petrov-Galerkin (SU/PG) numerical scheme of Brooks and Hughes [21] on bilinear quadrilateral Lagrangian elements. A problem though arises, as the SU/PG scheme holds for the presence of diffusion (conduction) terms, which give rise to the Peclet number (Pe = c p UH/k). Obviously, for k =0,wegetPe =, and there is no solution for the CTE.

4 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) One way to handle this is to add a pseudo-diffusive term in the CTE, made however small is desired. Namely, the equation is written as: ū }{{} = k 2 }{{} convection diffusion }{{} source +. (16) }{{} RHS-source The above equation is then solved for with SU/PG in a finite element grid, which results from the original by subdivision of each element into 2 2 bilinear quadrilaterals, with convection diffusion and source terms (all involving ), and with the RHS-source term having the contribution derived from the kinematics. Our FEM formulation for Eq. (16) leads to a system [A]{} = {b}, where [A] is the stiffness matrix, containing the convection, diffusion, and source terms, {} is the vector of unknowns, and {b} is the load vector, containing the RHS-source term [19]. Obviously, the system is nonlinear, since is present both in the CTE and the momentum equations through the viscosity, and Picard iterations are performed. The solution depends on the diffusion term added (hence on the Peclet number). Numerical tests showed that the results depended somewhat on k (hence on the Pe) for k >10 3 (Pe <10 +3 ), but values of k =10 5 and k =10 9 gave indistinguishable results. The examples given in Ref. [21] were faithfully reproduced with our code as a check. Also, as further checks we reproduced the analytical solutions for the o.d.e. + d/dz = const. with boundary conditions (0)=0. The finite element meshes used in the simulations are shown in Fig. 2 for the 2 =30 geometry (both the full domain and blowup sections of it). The entry has been set at 120R and the exit at +40R (L/D = 20) as was the case in the experiments. Mesh M1 has 6272 elements, 26,013 nodes and 82,030 unknown degrees of freedom (DOF) with 29 points in the radial direction at entry. A more dense mesh M2 (not shown) with 25,088 elements (resulting by subdividing M1 into four sub-elements) is used for the -solution of the convective-transport equation according to the SUPG formulation. These meshes are a progression in grid density from 2400 elements to 3600 elements to finally 6272 elements to guarantee results, which are mesh-independent. Another feature of the solution is the proper imposition of a slip boundary condition along the solid die walls. There are two points of sharp change in the geometry at the beginning and the end of the conical section (points D and S, respectively, in Fig. 1). The consistent method of Engelman et al. [22] and Page et al. [23] for the incompressible case was modified to take into account the density changes due to compressibility assumption. We found that the correct implementation of the slip boundary condition was essential for obeying the mass balance (no loss of mass) and for obtaining good and convergent results either with sharp corners or with rounded ones. However, for the cases of no slip and/or compressibility (especially at the highest apparent shear rates) the solution at the sharp corners showed discontinuities in the velocity gradients, which degenerated with each iteration and eventually led to divergence. Therefore, it was found necessary to round the corners, as done by Patil et al. [4], by using a local radius of a circle of 20R. An extra feature of the solution is the imposition of the flow rate at entry and based on that flow rate, the inlet velocity profile is found by one-dimensional FEM taking into account the model under study, i.e. [12]: ṁ = 2 Rres u z (r)rdr, (17) 0 where ṁ is the mass flow rate and R res is the reservoir radius. For incompressible fluids, ṁ may be substituted by Q, the volumetric flow rate, which is equal at entry and exit. For compressible fluids, it is ṁ, which is constant at entry and exit, but not Q, because of density changes according to the equation of state. The imposition of a numerically found velocity profile at entry results in a smooth development of the flow field, since the same order of interpolation is used for the velocities both for the 1D and the 2D problems. The CTE (10) contains the square root of, with the specific treatment that when < 0, then = 0. The square-root function exhibits an infinite slope at the origin. This means that a quantity is suddenly introduced along trajectories. This behaviour was noticed at entry but it quickly died out within the first column of elements. Therefore, no special treatment was made for this, except for adding part of the reservoir in the full flow domain, something which was not done in Ref. [4]. 5. Flow simulations 5.1. Flow field Fig. 2. View of the finite element meshes used in the simulations (mesh M1 with 6272 elements): (a) full view, (b) blow-up view near the tapered entry, (c) blow-up view near the die entry. Die design with 2 =30, RR = 352, L/D = 20. The flow simulations have been performed with the above conservation, constitutive, state, and convective-transport equations, their boundary conditions, and the parameters of Table 2 (a representative value of the isothermal compressibility is chosen to be Pa 1 ). The reservoir has been included for a better representation of entry into the conical die. Reference results are given for one apparent shear rate ( A = 4Q/R 3 = 5869 s 1 ) and one die design (2 =30 ). Both incompressible (B = 0) and compressible conditions (ˇ = 0.02 MPa 1, B = 0.006) have been assumed with no slip (B sl = 0) or slip (B sl = 1.44) at the walls. The purpose is to find out the effect of different parameters on the results, and the relative importance of the various forces at play in paste extrusion. Results are shown as contours of the two new variables, specific to the model, namely, the structural parameter,, and the flow type parameter,. The results are shown first for B = B sl =0inFig. 3. We observe that starts with 0 at entry and at the walls, and then most of the changes occur in an envelope in the conical section, where it reaches a maximum at the centreline. This envelope is representative of elongational flow in the vicinity of the centreline, as the

5 30 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) Fig. 3. Contours of (a) structural parameter (), (b) flow type parameter ( ), and (c) blow-up view of near the die entry in PTFE paste extrusion at 35 C using the parameters listed in Table 1. Incompressible flow with no slip at the wall for A = 5869 s 1,2 =30, RR = 352, L/D =20(B = B sl = 0). Fig. 4. Contours of (a) structural parameter (), (b) flow type parameter ( ), and (c) blow-up view of near the die entry in PTFE paste extrusion at 35 C using the parameters listed in Table 1. Incompressible flow with slip at the wall for A = 5869 s 1,2 =30, RR = 352, L/D =20(B =0,B sl = 1.44). material accelerates towards the die entry. The contours of in Fig. 3b show the envelope being much smaller along the centreline, but because of its definition, can take values between 1 and +1, the latter being the maximum value at the centreline just after the die entrance. For a better view near the die entry, blownup contours of are shown in Fig. 3c, and reveal a larger area where is close to 1, as expected. It should be noted that can also take negative values, which in the present case reach Fig. 4 shows the corresponding results for incompressible flow (B = 0) but with slip at the wall (B sl = 1.44). At this elevated apparent shear rate, the slip is substantial, yielding an almost flat velocity profile in the die. The results with slip show an increase of the - parameter in Fig. 4a, with now reaching a maximum of about 0.84 at the wall ( not being 0 at the wall when slip is allowed), and the envelope is now wider and closer to the wall compared with the no slip case. The results are also different and more interesting for the -parameter in Fig. 4b, where high values are reached close to the die walls at the entry to the conical section. Also negative values appear before the entry to the tapered region and after the entry to the die region, which correspond to a rotational motion of the material there. A blown-up region near the die entry is shown in Fig. 4c, and clearly shows the area of negative -values, due to rotation (minimum value of 0.98). Fig. 5 shows the corresponding results for compressible flow (B = 0.006) with slip at the wall (B sl = 1.44). The results are not very different from the incompressible case with slip of Fig. 4. Apparently, compressibility does not influence the - and -parameters as much as slip does. Any differences will be apparent in the axial distributions of variables, which now follow Axial distributions Fig. 6 shows the axial distribution of the axial velocity for the three cases studied. For the case of no compressibility and no slip Fig. 5. Contours of (a) structural parameter (), (b) flow type parameter ( ), and (c) blow-up view of near the die entry in PTFE paste extrusion at 35 C using the parameters listed in Table 1. Compressible flow with slip at the wall for A = 5869 s 1,2 =30, RR = 352, L/D =20(B = 0.006, B sl = 1.44).

6 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) Fig. 6. Axial velocity distribution for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip. (B = B sl = 0), the velocity rises smoothly in the conical section to the maximum value of about 630 mm/s in the die centreline, having zero values at the die walls. For no compressibility but with slip (B =0, B sl = 1.44), the velocity rises to values of about 390 mm/s at the wall and 400 mm/s at the centreline in the die region. Thus, slip is substantial, as there is a small but discernible difference between centreline and wall. For the case of both compressibility and slip, the velocity rises significantly higher in the die region, reaching at the exit 620 mm/s at the wall and 640 mm/s at the centreline. Thus, compressibility raises the velocity profiles as the material proceeds towards the exit, considerably. This is due to the decompression of the material as it approaches the exit where the pressure decreases substantially. This additional acceleration causes extension of the paste in the die that influences the -parameter. Fig. 7 shows the axial distribution of pressure for the three cases studied, plus the base case of no fibrillation ( = 0) for checking purposes (purely shear-thinning Carreau model). The latter gives an entry pressure of 52.4 MPa. For the case of incompressibility and no slip (B = B sl = 0), the pressure falls smoothly in the conical section starting from a maximum value of about 57.5 MPa at entry and going to zero at the die exit. Thus, the inclusion of the CTE with its -parameter serves to increase the pressure, as expected due to the shear-thickening contribution. For no compressibility but with slip (B =0,B sl = 1.44), the pressure falls to values of about 25.5 MPa at the entry, due to the effect of substantial slip. For the case of both compressibility and slip, the pressure is now higher, reaching about 31 MPa at the entry. Thus, compressibility raises the pressure by about 20% compared with the incompressible case, a significant contribution. Fig. 8 depicts the axial distribution of the structural parameter,, at the centreline and the wall for the three cases studied. There are now considerable differences in the behaviour of for all cases. At the centreline the structural parameter rises continuously from the entry to reach a maximum of 1 in the conical region before entry to the die, and then continues at that level towards the exit. This behaviour is the same for all the cases. It can be traced to the exponential increase of the elongational rate dv z /dz, which according to the CTE give values of = 1. These values are then carried to the exit due to the convective nature of CTE. At the wall the structural Fig. 7. Axial pressure distribution along the wall for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip. For the case of B = B sl = 0, the distributions are shown both along the centreline (lower bend) and the wall. parameter shows more pronounced differences for the three cases. It is 0 for incompressibility and no slip (even if it was not set to 0, it gave very small values close to 0 there). When slip is included it rises in the conical region from 0 to almost 1, and then as the material enters the die it decreases exponentially, to below 0.1 for incompressible conditions and to about 0.3 for compressible conditions (due to additional acceleration in the die). Thus, compressibility serves to raise the structural parameter at the exit of the die. As the pressure decreases towards the exit, the material decompresses significantly. This causes significant acceleration, which contributes to Fig. 8. Axial distribution for the structural parameter,, along the centreline and the wall for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip.

7 32 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) Fig. 9. Axial distribution for the flow type parameter,, along the centreline and the wall for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip. the extensional nature of the flow. Since extension increases fibrillation and structural formation, this is the reason why increases to higher values in the compressible flow. Fig. 9 shows the axial distribution of the flow type parameter,, at the centreline and the wall for the three cases studied. As was the case with, we observe differences in the -behaviour along the centreline and the wall for the three cases. At the centreline and for all cases, the -parameter starts from 0 at entry and then rises immediately to reach 1 and stays there up to the exit (pure extensional flow). At the wall and for the case of no compressibility and no slip (B = B sl = 0), the -parameter remains at 0. However, when slip is included it rises rapidly from 0 to 1 at entry to the die, after which it decreases quickly back to zero towards the die exit. For the case of both compressibility and slip, the behaviour is similar as with no compressibility, except that after entering the die, the -parameter increases towards the exit to reflect the acceleration of the paste in the die. Just after the die entry rapid changes appear at the die wall due to the singularity there, although this has been alleviated with the curved boundary Effect of flow and design parameters We continue with the effect of the die entrance angle on the structural parameter. We present results for the reference case of compressibility and slip at the apparent shear rate of A = 1875 s 1. According to Table 3, under these conditions B sl = 2.55 and B = The results are shown in Fig. 10a and b, along the centreline and the wall, respectively. At the centreline and in all cases of different angles ranging from 8 to 90, reaches 1 at the die entry and remains approximately constant (slightly decreasing towards the die exit). The shorter the conical die the sharper the increase in the value of, which grows according to the elongational component of the flow. At the wall, increases again up to the die entry, but its maximum never reaches 1 and is smaller for smaller angles (0.8 for 2 =8 ). Then after entry to the die, follows an exponential decrease and does not seem to be affected much by the preceding geometry, its final level being predominantly a function of L/D and, of course, the apparent shear rate as follows. Fig. 10. The effect of die entrance angle, 2, on the structural parameter,. Axial distribution of along (a) the centreline, and (b) the wall for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip included (B = 0.003, B sl = 2.55). The effect of the apparent shear rate for a given geometry (2 =30 ) is shown in Fig. 11. The increase is almost identical on the tapered wall for all flow rates, but the exponential decrease in the die is a strong function of apparent shear rate. As the latter increases so does the structural parameter,, of the material, with values from 0.1 to 0.5 in the range of the experimentally attained A -values. The small kinks around z/r = 66 and 0 are due to using the design with sharp corners in these simulations. The corresponding results in the centreline are not shown here as in all cases rises to 1 and remains approximately there at all apparent shear rates (as in Fig. 10a). Finally, the effect of L/D on the structural parameter for a given geometry (2 =90 ) and apparent flow rate ( A = 1875 s 1 )is shown in Fig. 12. Again, a longer die allows the material to relax and to reach lower values of, according to its exponential decrease in

8 E. Mitsoulis, S.G. Hatzikiriakos / J. Non-Newtonian Fluid Mech. 157 (2009) through dies. The recently proposed model [4] was solved for different cases of extrusion rate and die geometry, with the emphasis being on the relative effects of compressibility and slip. Both of them are important for the paste at hand and alter significantly the results for velocity, pressure, - and -parameters. The -parameter depends on die length, contraction angle, and of course on the flow rate, with higher flow rates and longer dies offering more structure in the material in paste extrusion. These observations are experimentally consistent [1 3,7]. Acknowledgements Financial assistance was provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the National Technical University of Athens (NTUA), Greece, in the form of a basic research project, code-named KARATHEODORI. This assistance is gratefully acknowledged. References Fig. 11. The effect of apparent shear rate, A, on the structural parameter,. Axial distribution of along the wall for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip included. Fig. 12. The effect of die length, L/D, on the structural parameter,. Axial distribution of along the wall for PTFE paste extrusion at 35 C using the parameters listed in Table 2. Effects of compressibility and slip included. the die. The results are not affected in the tapered die for the same contraction angle. Again, the small kinks around z/r = 10and0are due to using the design with sharp corners in these simulations. 6. Conclusions The present results establish the behaviour of the structural parameter,, and the flow type parameter,, in the presence or absence of compressibility and slip, in paste extrusion of PTFE resins [1] S. Ebnesajjad, Fluoroplastics, vol. 1: Non-Melt Processible Fluoroplastics, Plastics Design Library, William Andrew Corp., New York, [2] A.B. Ariawan, S. Ebnesajjad, S.G. Hatzikiriakos, Preforming behavior of polytetrafluoroethylene paste, Powder Technol. 121 (2001) [3] A.B. Ariawan, S. Ebnesajjad, S.G. Hatzikiriakos, Paste extrusion of polytetrafluoroethylene (PTFE) fine powder resins, Can. J. Chem. Eng. 80 (2002) [4] P.D. Patil, J.J. Feng, S.G. Hatzikiriakos, Constitutive modelling and flow simulation of polytetrafluoroethylene paste extrusion, J. Non-Newtonian Fluid Mech. 139 (2006) [5] S.G. Hatzikiriakos, J.M. Dealy, Start-up pressure transients in a capillary rheometer, Polym. Eng. Sci. 34 (1994) [6] E. Mitsoulis, O. Delgadillo-Velazquez, S.G. Hatzikiriakos, Transient capillary rheometry: compressibility effects, J. Non-Newtonian Fluid Mech. 145 (2007) [7] I. Ochoa, S.G. Hatzikiriakos, Polytetrafluoroethylene paste preforming: viscosity and surface tension effects, Powder Technol. 146 (2004) [8] S.G. Hatzikiriakos, J.M. Dealy, Wall slip of molten high density polyethylenes. II. Capillary rheometer studies, J. Rheol. 36 (1992) [9] R.I. Tanner, Engineering Rheology, Oxford University Press, Oxford, [10] G.C. Georgiou, M.J. Crochet, Compressible viscous flow with slip at the wall, J. Rheol. 38 (1994) [11] G.C. Georgiou, M.J. Crochet, Time-dependent extrudate swell problem with slip at the wall, J. Rheol. 38 (1994) [12] G.C. Georgiou, The time-dependent, compressible Poiseuille and extrudateswell flows of a Carreau fluid with slip at the wall, J. Non-Newtonian Fluid Mech. 109 (2003) [13] H. Giesekus, Strömungen mit konstantem Geschwindigkeitsgradienten und die Bewegung von darin suspendierten Teilchen. Teil II. Ebene Strömungen und eine experimentelle Anordnung zu ihrer Realisierung, Rheol. Acta 2 (1962) [14] G.G. Fuller, L.G. Leal, Flow birefringence of dilute polymer solutions in twodimensional flows, Rheol. Acta 19 (1980) [15] G. Astarita, Objective and generally applicable criteria for flow classification, J. Non-Newtonian Fluid Mech. 6 (1979) [16] E. Ryssel, P.O. Brunn, Flow of a quasi-newtonian fluid through a planar contraction, J. Non-Newtonian Fluid Mech. 85 (1999) [17] G. Mompean, L. Thais, Assessment of a general equilibrium assumption for development of algebraic viscoelastic models, J. Non-Newtonian Fluid Mech. 145 (2007) [18] E. Mitsoulis, S.G. Hatzikiriakos, Bagley correction: the effect of contraction angle and its prediction, Rheol. Acta 42 (2003) [19] K.M. Huebner, E.A. Thornton, The Finite Element Method for Engineers, Wiley, New York, [20] E. Mitsoulis, I.B. Kazatchkov, S.G. Hatzikiriakos, The effect of slip in the flow of a branched PP polymer: experiments and simulations, Rheol. Acta 44 (2005) [21] A.N. Brooks, T.J.R. Hughes, Streamline-Upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comp. Meth. Appl. Mech. Eng. 32 (1982) [22] M.S. Engelman, R.L. Sani, P.M. Gresho, The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow, Int. J. Numer. Meth. Fluids 2 (1982) [23] Page, M.-I. Farinas, A. Garon, Imposition of slip boundary conditions without the explicit computation of consistent normals, Commun. Numer. Meth. Eng. 13 (1997)