ARTICLE IN PRESS. J. Non-Newtonian Fluid Mech. 154 (2008) Contents lists available at ScienceDirect. Journal of Non-Newtonian Fluid Mechanics

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1 J. Non-Newtonian Fluid Mech. 154 (2008) Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: Numerical simulation of calendering viscoplastic fluids Evan Mitsoulis School of Mining Engineering & Metallurgy, National Technical University of Athens, Zografou, Athens, Greece article info abstract Article history: Received 7 January 2008 Received in revised form 18 February 2008 Accepted 4 March 2008 Available online xxx Keywords: Calendering Viscoplasticity Yield stress Yielded/unyielded regions Bingham plastics Papanastasiou model Sheet thickness Numerical simulations have been undertaken for the process of calendering viscoplastic sheets with a finite thickness. The finite element method (FEM) is used to provide numerical results for a fixed entry thickness (known attachment point) under two-dimensional steady-state conditions. The Herschel Bulkley Papanastasiou model of viscoplasticity is used, which is valid for all ranges of deformation rates. Part of the solution is finding the shape of the free surfaces of the entering and exiting sheet. Yielded/unyielded regions are found a posteriori for a range of the dimensionless yield stress or Bingham number (Bn) from the Newtonian viscous fluid (Bn = 0) to a highly viscoplastic one (Bn = 1000). The 2D FEM results show limited unyielded regions between the rolls, in disagreement with the lubrication approximation theory (LAT), which predicts erroneous extended unyielded regions. However, LAT is good at predicting the excess sheet thickness over the thickness at the nip (hence the detachment point), the pressure distribution and all engineering quantities of interest in calendering. For thick entering sheets, viscoplasticity (and also shear-thinning) leads to excess sheet thickness as the dimensionless Bingham number increases; it reduces the vortex size in the fluid bank, and gives virtually no swelling at the exit from the rolls. All engineering quantities, given in a dimensionless form, increase substantially with the departure from the Newtonian values Elsevier B.V. All rights reserved. 1. Introduction Calendering is a process used in many industries, such as the paper, plastics, rubber, and steel industries, for the production of rolled sheets of specific thickness and final appearance. The process is shown schematically in Fig. 1, where two variants of the operation are shown: (a) a material is taken up from an infinite reservoir by two co-rotating rolls (calenders) to form a sheet of thickness 2H [1], (b) a material enters as a sheet of finite thickness 2H f and exits as a sheet of reduced thickness 2H [1]. The process has been extensively studied by many researchers over the last 50 years. Starting with Ardichvili [2] in 1938, the work was extended to Newtonian and Bingham plastics by Gaskell [3],to power-law fluids by McKelvey [4], Pearson [5], Chong [6], Brazinsky et al. [7], Agassant and Avenas [8] and others. The textbook by Middleman [1] summarizes the findings up to Non-isothermal effects have been studied by Kiparissides and Vlachopoulos [9,10]. All these analyses are based on the lubrication approximation theory (LAT) of Reynolds. Lifting this assumption leads to a twodimensional analysis (2D), as was done by Mitsoulis et al. [11] and Agassant and Espy [12]. These two works have shown very interesting results with intricate patterns dominated by large vortices in the fluid bank before the rolls, found both experimentally [12] and computationally [11,12]. Another important work offered new insights about the detachment of the sheet and its acquired free surface for Newtonian, pseudoplastic (power-law) and viscoelastic fluids [13]. Again, all works prior to 1990 have been summarized in the textbook by Agassant et al. [14]. More recent work has been done on the simultaneous calculation of roll deformation and polymer flow [15] and compared with experiments, while work on the detachment of the sheet from the rolls has been carried out both experimentally and analytically by Kalyon et al. [16]. 2.5D simulations have also appeared for the spreading of the sheet in the 3rd dimension [17], as well as a fully 3D simulation for Newtonian and power-law fluids [18]. On the other hand, the rheology of materials used in calendering is non-newtonian, exhibiting either pseudoplastic (shear-thinning or -thickening) or viscoplastic (presence of a yield stress) behaviour (see, e.g., Bird et al. [19]). Models describing pseudoplastic behaviour include the power-law, Carreau, Cross, etc. Models describing viscoplastic behaviour include the Bingham, Herschel Bulkley and Casson. The Herschel Bulkley model has the advantage of reducing with an appropriate choice of parameters to the Bingham, power-law or Newtonian model. In simple shear flow it takes the form [19,20]: = K n 1 ± y, for > y, (1a) address: mitsouli@metal.ntua.gr. = 0, for y, (1b) /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.jnnfm JNNFM-2827; No. of Pages 12

2 78 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) Fig. 2. Schematic representation of the yielded/unyielded zones separated by the yield line y 0 in calendering of viscoplastic materials exhibiting a yield stress according to Gaskell [3] using the lubrication approximation theory (LAT). The shaded regions are unyielded, representing a plastic solid. Fig. 1. Schematic representation of the calendering process and definition of variables: (a) the calendered sheet is considered as coming from an infinite reservoir far away from the nip, (b) calendering with a finite sheet, whose thickness is reduced from H f to H. Note the attachment point x and detachment point. f where is the shear stress, the shear rate (=du/dy), y the yield stress, K the consistency index, and n is the power-law index. Note that if n = 1 and K = (a constant), the Herschel Bulkley model reduces to the Bingham model. If y = 0, the power-law model is recovered, and if y = 0 and n = 1, the Newtonian model is obtained. It should be noted that in viscoplastic models, when the shear stress falls below y, a solid structure is formed (unyielded). Also in viscoplasticity, a dimensionless Bingham number, Bn, is defined by: Bn = y K ( H0, (2) U where H 0 is a characteristic length (half the thickness at the nip) and U is a characteristic velocity (the roll speed). The Bingham number ranges from 0 (purely viscous fluid) to (unyielded solid). Another definition of a dimensionless yield stress, y, involves the pressure gradient at the nip and ranges, respectively, from 0 to 1 [21 24]. The pioneering work by Gaskell [3] on calendering of viscoplastic materials was based on LAT and showed schematically the yielded/unyielded zones for a Bingham plastic (see Fig. 2). Chung [21] gave some numerical examples of the analysis and verified Gaskell s presumed unyielded zones. Recently, Sofou and Mitsoulis [22 24] undertook an analysis of calendering based on LAT for both pseudoplastic and viscoplastic materials for finite sheets (including the case of an infinite reservoir) and provided results from full parametric studies of the power-law index n and the dimensionless yield stress, y. They showed the development of the yielded/unyielded zones in the value range for y from 0 (purely viscous fluid) to 1 (rigid plastic). These zones, although extended and interesting in shape and in qualitative agreement with the original work of Gaskell [3], were found to be a direct consequence of using LAT. It was argued along the lines of the work by Lipscomb and Denn [25] that such zones are erroneous and cannot be so, due to rapid velocity rearrangements under the rolls. Therefore, a fully 2D anal- ysis became necessary to find out the correct extent and shape of the unyielded zones and how these affect the other variables of the process. It is the purpose of this work to undertake a 2D analysis of calendering for viscoplastic materials using the continuous regularized Herschel Bulkley Papanastasiou model, which has shown good predicting capabilities of yielded/unyielded regions in other flows of viscoplastic materials [20,26 28]. The present work will also examine the previous study by Zheng and Tanner [13] for setting a criterion to find the detachment point of the exiting sheets. It will also include the free surfaces present before and after leaving the rolls. Two different cases of calendering will be studied with high aspect ratios R/H 0 of 100 and Finally, conclusions will be drawn about the capabilities of LAT as compared with the 2D FEM analysis, and its usefulness in predicting engineering quantities in calendering of viscoplastic materials. Fig. 3. Dimensionless sheet thickness H/H 0 as a function of entering sheet thickness H f /H 0 for different values of the Bingham number Bn according to LAT [22]. Note that for a certain value of H f /H 0 = 15.85, the results are independent of the amount of yield stress. The vertical lines, denoted by 1 and 2, correspond to the cases of calendering-1 and calendering-2, respectively, studied here.

3 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) Mathematical modelling 2.1. Governing equations We follow here the analysis of calendering as given by Middleman [1]. The flow is governed by the usual conservation equations of mass and momentum under isothermal creeping flow conditions for an incompressible viscous fluid in a 2D Cartesian coordinate system. The constitutive equation that relates the stresses to the velocity gradients is the generalized Newtonian fluid. In full tensorial form the constitutive equation is written as: =, (3) where is the apparent viscosity given by the Herschel Bulkley Papanastasiou model [20,29]: = K n 1 + y [1 exp( m )]. (4) In the above, m is the regularization parameter (with units of time), which controls the exponential growth of stress, and is the magnitude of the rate-of-strain tensor, = v + v T, which is given by: [ 1 1 ] 1/2 = 2 II = 2 { : }, (5) where II is the second invariant of. Similarly for the second invariant of the stress tensor II. The criterion to track down yielded/unyielded regions is for the material to flow (yield) when the magnitude of the extra stress tensor exceeds the yield stress y, i.e., [ 1 1 ] 1/2 yielded : = 2 II = 2 { : } > y, (6a) [ 1 1 ] 1/2 unyielded : = 2 II = 2 { : } y. (6b) Contours of = y separate the yielded from the unyielded regions and are drawn a posteriori from the numerical solution. In calendering the following dimensionless parameters are introduced [1]: x x =, y = y, h = h = 1 + x2, 2RH H 0 0 H 0 2RH 0 P = P ( H0, 2 = Q 1, (7) K U 2UH 0 where is a dimensionless flow rate (or leave-off distance), Q the flow rate, and the rest of the symbols are defined in Fig. 1. For Herschel Bulkley fluids and making use of LAT, Sofou and Mitsoulis [22,23] have shown that the following dimensionless pressure gradient is obtained: ( 2n + 1 dp dx = n 2R H 0 ( 2 x 2 ) ( 2 x 2 ) n 1 [(1 + x 2 ) + n(n + 1)y 0 ]n 1, (8) (1 + x 2 y )n+1 0 where the dimensionless yield line y is related to the Bn number 0 according to: y 0 = Bn 2R dp /dx. (9) H 0 Eq. (8) incorporates both cases: for < x < the pressure gradient is positive, and for x the pressure gradient is negative. Integration of Eq. (8) requires boundary conditions for the pressure gradient dp /dx and the pressure P as well. In the case of an infinite reservoir, the standard conditions are [1]: P = dp dx = 0, at x =, (10a) P = 0, at x =. (10b) Note that Eq. (10a) is referred to as the Swift condition [13] and is instrumental in the solution of the LAT problem. It assumes that the sheet detaches with the velocity of the rolls U, and at detachment the shear stress is also zero [1]. Then the pressure is given by the integral: ( 2n + 1 P = n 2R H 0 (2 x 2 ) ( 2 x 2 ) n 1 x [(1 + x 2 ) + n(n + 1)y 0 ]n 1 dx. (11) (1 + x 2 y 0 )n+1 The extrema of the pressure distribution occur at x =, ±. The dimensionless leave-off distance corresponding to an infinite reservoir can be found from the above equation knowing that P (x = ) = 0. Therefore can be found from the relation: ( 2 0 = x 2 ) ( 2 x 2 ) n 1 1 [(1 + x 2 ) + n(n + 1)y dx 0 ]n (1 + x 2 y )n+1 0 = I(n, Bn)dx. (12) The above integral has no analytical solution for the general case of Herschel Bulkley fluids. Therefore, a numerical solution must be found, based on some numerical algorithm for solving non-linear equations, as was done in [22,23] Sheet thickness Once is found as a function of n and Bn, then all other quantities of interest are readily available. The exiting sheet thickness H from an infinite reservoir is given by: H = H. (13) 0 For the case of calendering sheets with a finite thickness, the leave-off distance is substituted by and by x. The exiting f sheet thickness H is given by: H = 1 + 2, (14) H 0 while the thickness of the entering sheet H f is entering the analysis according to the definition: ( ) 1/2 Hf x = 1. (15) f H 0 The results for the sheet thickness of viscoplastic Bingham fluids (n = 1), based on LAT, have been obtained by Sofou and Mitsoulis [22] and are reproduced here in Fig. 3. For a given entering sheet thickness H f /H 0 and Bn number, the analysis gives the exiting sheet thickness H/H 0, and vice versa. It is worth noting that for H f /H 0 < 15.85, the Newtonian fluid gives the biggest exit thickness, while the reverse is true for H f /H 0 > At H f /H 0 = 15.85, all fluids have the same exit thickness. In Fig. 3, the vertical lines numbered 1 and 2 refer to the 2D cases studied in the present work.

4 80 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) (ii) the roll-separating force per unit width, F/W(n, Bn), defined by: F W (n, Bn) = x f ( U P(x)dx = K RF(n, Bn), H 0 (17a) ( 2n + 1 [ ] with F(n, Bn) = 2 I(n, Bn)dx dx,(17b) n x x f Fig. 4. Flow domain and boundary conditions for the 2D FEM analysis Operating variables The operating variables used in engineering calculations are also of interest [1]: (i) the maximum pressure, P(n, Bn), defined by: P(n, Bn) = P max 2R/H 0 = ( 2n + 1 I(n, Bn)dx, (16) n (iii) the torque for both rolls, T(n, Bn), defined by: ( U T(n, Bn) = 2WR xy y=h(x) dx = WK R RH 0 E(n, Bn), H 0 x f (18a) with E(n, Bn) = 2 ( 2n I(n, Bn)(1 + x 2 )dx. n x f (18b) 3. Method of solution The constitutive equation for the viscoplastic fluids must be solved together with the conservation equations and appropriate Fig. 5. Finite elements meshes used in the computations. Upper half shows a mesh with 8232 elements (M2), lower half mesh has 2058 elements (M1): (a) calendering-2, (b) calendering-1 (deformed mesh from Newtonian solution), (c) calendering-1, blow-up near the separation point at exit (deformed mesh from Newtonian solution). Note the small swelling of the free surface after detachment.

5 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) Table 1 Geometry data used in the simulations of calendering Geometry Radius, R (cm) Minimum gap, 2H 0 (cm) Aspect ratio, R/H 0 ( ) Entry thickness, H f /H 0 ( ) Calendering Calendering Fig. 6. Pressure distribution along the flow field in calendering-1. Results based on LAT and FEM (for the centreline and the wall/free surfaces). boundary conditions. Fig. 4 shows the solution domain and boundary conditions for the symmetric problem. Because of symmetry, only one half of the flow domain is considered, as was done previously [13,22 24]. The boundary conditions are (for flow from left to right): Fig. 8. Shear stress distribution along the flow field in calendering-1. Results based on LAT and FEM (for the wall/free surfaces). (a) symmetry along the centreline AB (v y =0, xy = 0); (bo slip along the roll walls ED (tangential velocity v t = U, normal velocity v n = 0); (c) along the exit free surface DC, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0) and no flow through the surface ( v n = 0), where n and t are the normal and tangential vectors to the surface, and = pī + is the total stress; Fig. 7. Blow-up of pressure distribution near the separation point region in calendering-1. Results based on LAT and FEM (for the centreline and the wall/free surface). Fig. 9. Blow-up of shear stress distribution near the separation point region in calendering-1. Results based on LAT and FEM (for the wall/free surface).

6 82 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) (d) along the outflow boundary BC, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0); (e) along the inflow boundary AF, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0); (f) along the entry free surface FE, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0) and no flow through the surface ( v n = 0). The reference pressure is also set to zero at point B. All lengths are scaled with the minimum gap H 0, all velocities with the roll speed U, and all pressures and stresses with K(U/H 0. Also the stress-growth exponent m gives rise to the dimensionless stress-growth exponent M=mU/H 0. The numerical solution is obtained with the finite element method (FEM), using the program UVPTH, originally developed for multilayer flows [30,31], which employs as primary variables the two velocities, pressure, temperature and free surface location (u v p T h formulation). It uses 9-node Lagrangian quadrilateral elements with biquadratic interpolation for the velocities, temperatures and free surface location, and bilinear interpolation for the pressures. The free surface is found in a coupled way as part of the solution for the primary variables. In the present work we have selected two geometries: (a) calendering-1 with aspect ratio R/H 0 = 100, and (b) calendering-2 with aspect ratio R/H 0 = The first case refers to the work by Zheng and Tanner [13], with the purpose of setting the detachment point from the rolls and comparing the corresponding results for Newtonian and power-law fluids. The second case refers to previous Newtonian simulations by Mitsoulis et al. [11], and corresponds to strong calendering due to the large aspect ratio. Data for the two geometries are given in Table 1, while in Fig. 3 they are represented by the vertical lines numbered 1 and 2, based on the H f /H 0 ratio. For each geometry we have used two meshes, which are shown (put together for brevity) in Fig. 5 for the case of calendering-2 (Fig. 5a) and calendering-1 (Fig. 5b and c). The meshes of Fig. 5a are shown at the beginning of the solution process undeformed, while those of Fig. 5b and c after the Newtonian solution has been achieved, hence the deformed domain due to the free surfaces at entry (Fig. 5b) and near the exit (Fig. 5c). In each geometry mesh M1 (lower half) has 2058 elements, while mesh M2 (upper half) has 8232 elements, and is produced by subdividing each element of M1 into 4 sub-elements. Mesh M2 gives nodes and unknown degrees of freedom (DOF) with 57 points in the transverse direction. The less dense mesh M1 with 2058 elements was used primarily for preliminary runs to gain experience with two-dimensional calendering flows. More elements have been concentrated near the attachment and detachment points and near the rolls, where most of the changes are expected. The entering sheet is usually set with a length of 3H f and the exiting sheet with a length of 4H to guarantee a fully developed profile at entry and exit. Knowing that viscoplastic fluids as opposed to viscoelastic fluids do not swell much [26,32], such lengths are adequate for a full rearrangement of the velocity profile. The adequacy of the entry and exit lengths was also checked for each run by plotting the centreline and surface velocity profiles and observing their levelling off and matching in these regions. 4. Method of finding the separation point In calendering a difficulty arises from the indeterminate position of the separation point. According to LAT, this is resolved by the Swift condition, which requires that P =dp/dx =0 at. This additional assumption has to be used for the evaluation of (see, e.g., [1] and [22 24]). As noted in [13], the Swift condition does not apply to a two-dimensional analysis. Instead, Zheng and Tanner [13] use the condition that a zero tangential traction should signify the point of separation. Thus, in their boundary element method (BEM) calcula- Fig. 10. Streamline contours in calendering-1: (a) Newtonian fluid, (b) power-law fluid (n = 0.4). Fig. 11. Pressure distribution along the rolls for power-law fluids using FEM.

7 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) tions, where the velocities and tractions are the primary variables, they monitor the tangential traction on the roll with which the fluid is in contact, and when this becomes zero (or by interpolation) they locate the position of separation. Their condition is equivalent to having a zero shear stress at the point of separation, i.e., xy =0 at x =. However, when FEM is used, the primary variables are the velocities and pressures, and all stresses near this point behave in a singular way, because of the change of the velocity boundary condition from a fixed no-slip value to a non-fixed finite value dictated by the free surface boundary condition. Therefore, no condition of the type xy =0 at x = can be used. In coating flows, which are similar to calendering, the position of the separation point is found from an extra equation assuming a dynamic contact angle, which enters the system of equations, as explained in the works by Coyle et al. [33 35]. However, according to Kistler and Scriven [36] for very viscous materials with negligible surface tension effects (capillary number set to ), such as viscoplastic materials considered here, the dynamic contact angle should not be set, but it is part of the solution and is often close to 180 (or more precisely between 160 and 180 ). Although this line of work is probably the best in finding the separation point, it is rather involved, requires an initial guess close to the solution, and cannot be used effectively for all parameter ranges of pseudoplastic and viscoplastic fluids. In the course of the present work, different criteria and methods of approach were tried. One approach akin to that of Zheng and Tanner [13] was instead of using the tangential traction to use the roll speed as the desired criterion. Namely, for a given roll speed U and entry sheet thickness 2H f (where the sheet first bites the rolls) we set the separation point (point D in Fig. 4) as that given by LAT. This constitutes the initial guess. Usually this is a very good guess for the 2D problem, but not accurate enough. The lack of accuracy shows in the velocity profile at outflow, which is not equal to the roll speed set, i.e., u x U at x = x ex (point C in Fig. 4). A new position is then found according to Newton s method so that u x = U at x = x ex. This requires remeshing of the new flow domain, but it is not very difficult for the problem at hand, since most changes occur near the separation point. Newton s method converges quadratically to a desired accuracy (<10 6 ) within 3 4 iterations on (i.e., the flow domain). The same method can be used for the problem of a fixed exit thickness (fixed ) while finding the desired entry thickness 2H f of the incoming sheet (point E in Fig. 4). However, experimental evidence [37] has shown that the sheet does not leave with the roll speed but with a little lesser speed, so this method does not hold. Furthermore, the results so obtained for were in the opposite Fig. 12. The effect of the exponent m and different meshes M1 and M2 used in the determination of the unyielded zone (shaded) in calendering-2 of viscoplastic fluids obeying the Bingham Papanastasiou model (Bn = 10). The shaded regions are unyielded. Full flow field shown in (a) (c). Blow-ups of the exit region are shown in (d) (f). The vertical lines near the exit denote the detachment point as given by LAT.

8 84 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) Table 2 Power-law fluid results in calendering-1 Power-law index, n ( ) Leave-off distance, ( ) Maximum vortex intensity, v,max ( ) Maximum pressure, P ( ) Roll-separating force, F ( ) Roll torque, E ( ) direction than the results from LAT of Fig. 3, so this method was abandoned. Another approach was to employ no criterion for a set given by LAT and to move the separation point whenever the free surface nodes were attached or penetrated the roll. This however never occurred, and the free surface nodes kept always clear of the rolls for any. The inclusion of slip near and at the separation point did not help either, as it influenced the solution of the pressure and shear stress distributions, altering the results. To overcome this apparent problem of indeterminate solution for the position of the separation point, the approach was adopted to accept as given by LAT. A comparison was then made with the results of Zheng and Tanner [13] (calendering-1) to find out and quantify any discrepancies between the two works. Thus, we present here details of the solution for Newtonian fluids for the conditions given in [13], in which a Newtonian value of = is found, while our value, as found by LAT and set as the separation point, is = (difference <0.1%). Fig. 6 shows the pressure distribution along the flow field, as found by LAT and FEM (both at the centreline and the roll/free surfaces). There is good overall agreement between the two methods. There are differences at the separation point,, and at the attachment point of the sheet, x. These differences are more obvious f in Fig. 7, where a blow-up section of the pressure distributions is shown in the separation point region. Note that the pressure value at the separation point is zero according to LAT, due to the Swift condition. Using FEM for the calculation of the pressure distribution on the roll, the separation point presents a singularity. The pressure Fig. 13. Progressive growth of the unyielded zones as the dimensionless Bingham number, Bn, increases in calendering-2 of viscoplastic fluids obeying the Bingham Papanastasiou model. The shaded regions are unyielded.

9 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) Table 3 Bingham Papanastasiou fluid results in calendering-2 Bingham number, Bn ( ) Leave-off distance, ( ) Maximum vortex intensity, v,max ( ) Maximum pressure, P ( ) Roll-separating force, F ( ) Roll torque, E ( ) there becomes discontinuous showing the high peaks associated with the change in boundary conditions, as is the case in extrudate swell [32]. On the other hand on the symmetry line, the pressure value is not zero, but just before the separation point it becomes negative, levels off and shows a minimum, which is tantamount to having a zero pressure gradient at this point, just as the Swift condition stipulates (dp/dx = 0). Thus, the LAT assumption for the separation point is not a bad assumption, and as mentioned above it differs from the 2D calculation by Zheng and Tanner [13] byavery small percentage. Fig. 8 shows the shear stress distribution along the flow field, as found by LAT and FEM (on the roll/free surfaces), while Fig. 9 shows a blow-up in the separation point region. Again good agreement is observed between LAT and FEM, while the discrepancies for x < are due to using a parabola in LAT and a circle in FEM for the definition of the roll curvature. As shown in Fig. 9, the FEM shear stress distribution near the separation point gives the tell-tale spikes due to the singularity, before becoming zero on the free surface, while the value predicted by LAT is zero. These graphs clearly show the difficulty of the problem at the separation, which is avoided using the boundary element method, since the pressure is not a primary variable. 5. Results and discussion 5.1. Power-law fluids First the calculations are pursued for pseudoplastic power-law fluids showing shear-thinning behaviour (0 < n 1), as was done by Zheng and Tanner [13]. The results for shear-thinning fluids based on LAT are well known and can be found in Middleman [1], while a more thorough investigation has been given by Sofou and Mitsoulis [22,23]. Here we pursue sample calculations to show the influence of the upstream free surface on the vortex pattern and examine the influence of the power-law index on the vortex size and intensity. Again this is the case of calendering-1, with a fixed attachment of entry sheet thickness H f /H 0 = 20, and the rest of the values given in Table 1. The novelty is the simultaneous calculation of the upstream and downstream free surfaces, while the value of is taken from LAT for a given power-law index. According to LAT for an infinite reservoir, the Newtonian values (for n =1) of = and H/H 0 = are a starting point, after which it is noted that shear-thinning increases the values, reaching at the limit for n =0, = and H/H 0 = (found analytically) [22,23]. For the problem at hand with a finite sheet entering Fig. 14. Blow-up of Fig. 13 in the nip region and near the exit. The vertical lines near the exit denote the detachment point as given by LAT.

10 86 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) Fig. 15. Streamline contours in calendering-2 of viscoplastic fluids obeying the Bingham Papanastasiou model for different Bingham numbers. with H f /H 0 = 20, the results are given in Table 2. Note that for n = 0.4, Zheng and Tanner [13] give a value of about 0.472, compared with = given by LAT (a difference <1.3%). Some important details of the present FEM solutions are given in Table 2. They concern the vortex intensity, v,max (made dimensionless and normalized by the stream function value at the final exit point C, C = H C U C ), and the three dimensionless operating variables, namely, the maximum pressure, P, the roll-separating force, F, and the torque, E (which is usually the measured quantity in calendering). These are within ±2% of the corresponding LAT values. The streamline patterns for the Newtonian and a power-law fluid with n = 0.4 are shown in Fig. 10. Because of the inclusion of the free surface upstream of the entering sheet, a big smooth vortex appears (with almost twice the flow rate recirculating in the fluid bank, v,max = 1.91), which becomes smaller as the shearthinning character of the fluid increases, and disappears for n = 0.2. This is not surprising, since a more shear-thinning fluid is less flexible to move (has a higher viscosity in regions of low shear rates such as the regions in the fluid bank). This is also the reason why the Newtonian fluid shows a big curvature and a lower entering sheet thickness to accommodate the traction-free boundary conditions of the upstream flow domain. On the other hand, the swelling of the exiting sheet, defined as the thickness ratio between points C and D (see Fig. 4), is negligible, being 1.6% for the Newtonian fluid (see also Fig. 5c) and getting smaller as the power-law index decreases, down to 0.5% for n = 0.2. Correspondingly, the final sheet speed is 0.98U for the Newtonian, going down to 0.97U for n = 0.2. Therefore, shear-thinning serves to reduce the admittedly small swelling in calendering, which is also the case for the benchmark extrudate swell problem from long extrusion dies [32]. The dimensionless pressure distributions along the rolls are shown in Fig. 11 for different values of the power-law index. When scaled with the maximum Newtonian pressure, they show a reduced value and a bigger spread of the flow domain, as also found out by LAT. The present results are in general agreement with those by Zheng and Tanner [13]. The same is valid for the roll-separating force, but it must be noted that in the present work the results are given in a dimensionless form according to Eqs. (16) (18). Under these circumstances, shear-thinning serves to increase these values. The results in graphical form are similar to the ones given by Sofou and Mitsoulis [22] and are not repeated here Viscoplastic fluids Calculations were carried out for a wide range of Bn values, i.e., 0 Bn Results were obtained for the calendering-2 geometry with the Bingham Papanastasiou model (n = 1) for a given ratio

11 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) of entering sheet thickness, H f /H 0, based on the attachment point E Yielded/unyielded regions Because of the yield stress, viscoplastic fluids have the characteristics of both viscous fluids and plastic solids. The yield line, y 0, separates the two regions, called yielded and unyielded, respectively. Several cases of viscoplastic flows have been analysed in the literature and have shown explicitly these regions, e.g., flow through contractions [20,26]. In calendering, such regions have been shown only schematically by Gaskell [3] (see Fig. 2), and calculated based on LAT by Chung [21], and more recently by Sofou and Mitsoulis [22,23]. As mentioned in the introduction, the interesting yielded/unyielded regions found by LAT are erroneous, so a 2D analysis is needed to find out where these regions might lie. With regard to Fig. 4, in calendering finite sheets of viscoplastic materials, we observe that the entering and exiting sheets, some distance from the attachment and detachment points, have a constant thickness and move with a plug velocity as a solid plastic sheet. Therefore, they constitute unyielded regions (shaded) before coming into contact with the rolls and after leaving them. But apart from these regions, it is not a priori clear where the unyielded regions between the rolls might be. For the present 2D quantitative analysis, we set the value of Bn (i.e., the value of y ) and the value of the regularization parameter m = 200 s (following which the dimensionless parameter M = mu/h 0 is calculated), and determine the yielded/unyielded zones based on this. For Bn = 0, we have purely viscous flow and the area is all yielded. At the other extreme, for Bn, we have a purely plastic solid, and the area is all unyielded. It is, however, difficult to see how this latter case can be achieved in calendering, since U 0, and there will always be yielded area due to the changing geometry, unless the material reverts to massive slip at the wall (not considered here). Representative results are given for the development of yielded/unyielded regions with the Bingham Papanastasiou continuous model for the calendering-2 case having an entering sheet thickness H f /H Due to the length of the domain, it is necessary to present each result in two, namely, firstly for the full domain and secondly for the nip region and near the exit. The points of detachment will be denoted by vertical lines. The results for the regularized viscoplastic model, as given by Eqs. (3) and (4), for the location of the yielded/unyielded regions employing the criterion of Eqs. (6a) and (6b), are known to depend on the value of the exponent m and the mesh used [38,39]. This is evidenced in Fig. 12, where we show for a value of Bn = 10, the location of the unyielded (shaded) regions for two values of the exponent m (100 and 200 s) and two meshes (M1 and M2). It is seen that the effect of m is bigger than the effect of the finite element mesh. Higher values of m lead to more oscillations in the yielded/unyielded contour and a lower rate of convergence, due to the exponential nature of the regularized model. In all cases the essential features of the solution remain the same. Higher values of m slightly reduce the apparently unyielded regions (islands between the rolls), while the effect of the finite element mesh is minimal. Similar results were obtained in all the other cases studied and are not repeated here. The results for the development of yielded/unyielded regions are given in Fig. 13 for the full domain and in Fig. 14 for the nip region and near the exit. The progressive growth of the unyielded regions as Bn increases becomes evident. For small Bn numbers (Bn 0.1) the area between the rolls is basically all yielded, but an unyielded island appears upstream, due to the big region of the flow bank. This island grows with Bn, and for Bn 100 it joins the unyielded region of the incoming sheet. There are unyielded regions of the entering and exiting sheet, and these come closer to the rolls as the Fig. 16. Pressure distribution along the rolls in calendering-2 of viscoplastic fluids obeying the Bingham Papanastasiou model using FEM. Bn number increases, as expected. In contrast to what LAT predicts and for all Bn numbers, unyielded regions do not appear near the nip attached to the rolls (cf. Fig. 2 with Fig. 14). However, small islands appear at the centreline nip region, which grow in thickness with Bn. Again, LAT does not produce such regions as found by the 2D FEM analysis Flow field variables Fig. 15 shows the streamline patterns as Bn increases. As was the case in calendering-1, for the Newtonian fluid (Bn = 0) there is a strong vortex forming in the fluid bank upstream. Since the domain is now somewhat larger than in the previous case, the vortex is bigger in size and intensity, with more than 2.5 times the flow rate recirculating in the fluid bank ( v,max = 2.56). As Bn increases the vortex diminishes in size and intensity and has disappeared for Bn 1. This behaviour is equivalent to the shear-thinning behaviour of Fig. 10, but the phenomenon here is more prominent. For high Bn numbers the streamlines appear as straight lines with little or no curvature upstream, which is tantamount to a plug velocity profile and an unyielded region. Again, the swelling at exit is very small (max. of 0.5% for the Newtonian fluid, after which it decreases down to zero for Bn 10). Correspondingly, the final sheet speed is 0.995U for the Newtonian, going down to 0.977U for Bn = This tendency of viscoplasticity to reduce the swelling is in agreement with the results of extrudate swell from long extrusion dies [32]. The dimensionless pressure distributions along the rolls are shown in Fig. 16 for different values of the Bn number. Due to the very high maximum values when compared with the maximum Newtonian pressure, they have been scaled with their corresponding maxima for each Bn number, which are given in Table 3 as P. The curves show a bigger spread of the flow domain than the Newtonian fluids, as also found out by LAT, and a more triangular shape as the Bn number increases. They are in good agreement with those by Sofou and Mitsoulis [22]. Some other features of the 2D analysis are included in Table 3. These refer to the leave-off distance,, as found by LAT, the vortex intensity, v,max, and the three dimensionless operating variables, namely, the maximum pressure, P, the roll-separating force, F, and the torque, E. Regarding the operating variables, as in the previous

12 88 E. Mitsoulis / J. Non-Newtonian Fluid Mech. 154 (2008) case viscoplasticity serves to increase these values and does so more dramatically than in the case of power-law fluids. The results in graphical form are similar to the ones given by Sofou and Mitsoulis [22] and are not repeated here. 6. Conclusions The work presented above confirms the utility of lubrication analysis for inelastic pseudoplastic and viscoplastic fluids, and the validity of the Swift exit boundary conditions (P =dp/dx =0at x = ) in these cases. The lubrication approximation has been used to derive numerical solutions for the dimensionless leave-off distance,, and sheet thickness in calendering finite sheets of pseudoplastic and viscoplastic fluids. The two-dimensional FEM analysis was then used to solve for the flow field including upstream and downstream free surfaces. In the case of pseudoplastic power-law fluids, it was found that shear-thinning decreases the big vortices present in the fluid bank and eliminates them for extremely shear-thinning fluids (n 0.2). Swelling at the exit is almost non-existent, with values around 1% and decreasing as shear-thinning increases. The operating variables (maximum pressure, roll-separating force, and torque) are as predicted by LAT. In the case of viscoplastic materials, the sheet thickness is increased as the Bn number increases. New results show qualitatively as well as quantitatively the development and evolution of yielded/unyielded regions associated with viscoplasticity for a wide range of dimensionless yield stress values, 0 Bn <. Most of the flow field between the rolls is yielded due to the shearing motion of the rolls. Unyielded islands appear only in the large area of the fluid bank and they quickly connect with those of the incoming sheet as Bn increases. Near the nip there are also small unyielded regions around the centreline that remain small and limited as Bn increases. The present results are in contrast with previous ones obtained with LAT [3,21 24], which provides erroneous and much bigger yielded/unyielded regions. Operating variables, such as maximum pressure, roll-separating force, and power input to the rolls, all increase when given in a dimensionless form with increasing pseudoplasticity and viscoplasticity, in accordance with LAT. The problem of accurately finding the detachment point still remains unresolved for flows with unknown dynamic contact angles. Here the problem was circumvented by using LAT, but a thorough investigation must be made to find out possible differences, especially for small R/H 0 ratios (case of rolling) where LAT is no longer valid. However, the results found here for the yielded/unyielded regions are not expected to be altered. 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