Nov 1, 2009 - Wei Wang ab; Xikui Li a; Xianhong Han c a State Key Laboratory ..... similar scheme, one may refer to Li and Duan[24]. The iterative procedure ...
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Equal Low-Order Finite Element Simulation of the Planar Contraction Flow for Branched Polymer Melts Wei Wang ab; Xikui Li a; Xianhong Han c a State Key Laboratory of Structural Analysis of Industry Equipment, Dalian University of Technology, Dalian, P.R. China b Key Laboratory of Rubber-Plastics, Ministry of Education, Qingdao University of Science & Technology, Qingdao, P.R. China c Department of Plasticity Technology, Shanghai Jiaotong University, Shanghai, P.R. China Online Publication Date: 01 November 2009
To cite this Article Wang, Wei, Li, Xikui and Han, Xianhong(2009)'Equal Low-Order Finite Element Simulation of the Planar Contraction
Flow for Branched Polymer Melts',Polymer-Plastics Technology and Engineering,48:11,1158 — 1170 To link to this Article: DOI: 10.1080/03602550903147312 URL: http://dx.doi.org/10.1080/03602550903147312
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Polymer-Plastics Technology and Engineering, 48: 1158–1170, 2009 Copyright # Taylor & Francis Group, LLC ISSN: 0360-2559 print=1525-6111 online DOI: 10.1080/03602550903147312
Equal Low-Order Finite Element Simulation of the Planar Contraction Flow for Branched Polymer Melts Wei Wang1,2, Xikui Li1, and Xianhong Han3 1
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State Key Laboratory of Structural Analysis of Industry Equipment, Dalian University of Technology, Dalian, P.R. China 2 Key Laboratory of Rubber-Plastics, Ministry of Education, Qingdao University of Science & Technology, Qingdao, P.R. China 3 Department of Plasticity Technology, Shanghai Jiaotong University, Shanghai, P.R. China
The rheological characteristics of branched polymer melts is described by the modified XPP model, which is discretized by inconsistent streamline-upwind method. A finite increment calculus procedure is introduced to reformulate the mass equation and to overcome oscillations of the pressure field. Moreover, the governing equations are discretized and solved by the iterative fractional step algorithm. Thus the equal low-order finite elements for velocitypressure-stress variables are adopted to calculate the planar contraction viscoelastic flows. The influences of the Weissenberg number and the amount of branched-arms on the rheological behavior of the Pom-Pom molecule are discussed. Results demonstrate good agreement with those given in the literatures. Keywords Equal low-order finite element; Fractional step algorithm; Planar contraction; Rheological behavior; Viscoelastic flow; XPP model
INTRODUCTION Significant progress has been achieved in the development of viscoelastic constitutive models during the last decade. McLeish and Larson[1] first presented the PomPom model based on the tube theory and a simplified topology of branched polymer architecture. The constitutive equations of the model consist of two decoupled equations: one for the orientation and another for the stretch. A distinctive feature is the separation of relaxation times for the two timescales. The introduction of the Pom-Pom model gives great impetus to the extensive development of constitutive models for viscoelastic flows. The model provides an insight into the complex flow behavior of polymer melt processing that also impelled intensive investigations on this subject. Blackwell et al.[2] suggested a modification of the model and introduced local Address correspondence to Xikui Li, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, P.R. China. E-mail: xikuili@dlut. edu.cn
branch-point withdrawal before the molecules are fully ¨ ttinger[3] investigated the thermodynamic stretched. O admissibility of the Pom–Pom model. Although the model is found to be thermodynamically admissible, he proposed a modification for the orientation equation and reformulated the model written in a double-equation or singleequation form, known as DIPP, SIPP model respectively. As for the drawbacks in the Pom-Pom model, Verbeeten et al.[4] presented the extended Pom-Pom (XPP) model, which included the second normal stress difference and eliminated the finite extensibility condition from the original model. The XPP model gives excellent agreement with the rheological experiment data for elongation and shear flows. Moreover, by using the XPP model and the DEVSS= DG (discontinuous Galerkin) method, Verbeeten et al.[5,6] investigated complex viscoelastic flows and achieved good prediction with the experiment results. After such modifications, Clemeur et al.[7] developed the Double Convected Pom–Pom (DCPP) model along with an approach similar to that suggested by Verbeeten et al., in order to eliminate the defects of the model in the mathematical aspect. Furthermore, Inkson and Phillips[8] have also found that the XPP model possesses some disconcerting attributes, i.e., multiple solutions found in simple steady shear and uniaxial extensional flows. However, they indicated that the numerical integration solution of the transient problem with the model would not seem to exhibit such phenomenon of the ‘‘multiple solutions.’’ On the other hand, with the development of numerical algorithms and constitutive equations for viscoelastic polymer melts and solutions, viscoelastic flow problems are widely investigated. In particular, streamline upwind schemes are applied to deal with the convective terms in the constitutive equations, such as streamline-upwind= Petrov-Galerkin (SUPG) method[9], streamline-upwind (SU) formulation[10]. More importantly, the stable numerical schemes, such as the elastic-viscous split stress
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SOLUTION OF PLANAR FLOW FOR BRANCHED POLYMER MELTS
(EVSS) formulation[11] and the discrete elastic-viscous split stress (DEVSS) method[12], are adopted to introduce an elliptic operator contribution. Some efficient and robust numerical algorithms for viscoelastic flow have been developed[13], such as EVSS-G, DEVSS=DG, DG, DEVSS=SUPG. Furthermore, fractional step algorithm or pressure projection algorithm was initially presented for incompressible flow problems in the finite difference method and then introduced to finite element context[14,15]. Today, it is extensively employed to calculate non-Newtonian flow in the context of finite element. Utilizing hybrid finite-element=volume algorithm, Aguayo et al.[16] simulated a 4:1 planar rounded-corner contraction flow governed by the XPP model and demonstrated the influence of the number of dangling arms at each end of the pom-pom molecule on the flow characteristics. Han and Li[17] calculated the incompressible non-isothermal nonNewtonian fluid flow using an iterative stabilized fractional step scheme and the characteristic-Galerkin method for energy conservation equation. Similar schemes for introducing the iterative procedure into the fractional step algorithms have been employed for fluid mechanics[18]. Recently, Li et al.[19] presented an adaptive method that coupled the finite element method and the meshfree method in the arbitrary Lagrangian–Eulerian (ALE) description for numerical simulation of injection molding processes. Nowadays, numerical simulation of viscoelastic flow problems is an effective way of investigating the complex viscoelastic flow mechanism related to practical engineering problems[20,21], such as plastics injection molding, extrusion, film blowing, fiber spinning and microscale polymer processing, so as to predict appropriate processing conditions, optimize the mold structure and improve the quality of finished products. The present methodology for the simulation of viscoelastic flow problems is based on the iterative stabilized fractional step scheme in combination with the FIC process and will be shown to be equally competitive with other approaches in numerical accuracy and stability. The objective of this study is to investigate the influence of parameters in the modified XPP model on the flow characteristics generated in a 4:1 planar contraction transient viscoelastic flow. The finite elements with equal low-order interpolation approximation for velocity-pressure-stress variables are adopted to calculate the 4:1 planar contraction viscoelastic flows for branched polymer melts. The influences of Weissenberg number and the amount of branched-arms at the end of the Pom-Pom molecule on the flow characteristics, such as streamline pattern, profiles of stress components, generated in this complex flow are discussed.
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GOVERNING EQUATIONS Conservation Laws For incompressible, isothermal flows the conservation equations for momentum and mass may be expressed as, respectively @u q þ u ru ¼ qg þ r T rp ð1Þ @t ru¼0
ð2Þ
where q is the fluid density, p the pressure, T the extra stress tensor and g gravity acceleration. Frequently, the extra stress tensor is defined in terms of an addition of viscous and viscoelastic contributions: T ¼ 2gv D þ s
ð3Þ
where gv is the viscosity of the purely viscous contribution, s the viscoelastic contribution, D ¼ 1=2(L þ LT) the rate of deformation tensor, in which L ¼ (ru)T is the velocity gradient tensor and ()T denotes the transpose of a tensor. Constitutive Equations A general way to describe the constitutive equation of a single mode is written as r
s þ f c ðs; DÞ þ k1 0b f d ¼ 2G0 D
ð4Þ
where fc and fd depend upon the chosen constitutive equation, k0b is the relaxation time of the backbone tube orientation, G0 the plateau modulus. In Eq. (4) the overhead symbol r denotes the upperconvected time derivative defined as r
s
@s þ u rs ðruÞT s s ru @t
ð5Þ
For the extended Pom-Pom(mXPP) model with modified stretch dynamics[6], fc and fd are respectively defined as fc ¼ 0 fd ¼
a 2 s þ F s þ G0 ðF 1ÞI G0
ð6Þ ð7Þ
where a is a material parameter (a > 0), defining the amount of anisotropy, and 1 1 atrðs2 Þ vðK1Þ F ðsÞ ¼ 2re 1 þ 2 1 ð8Þ K 3G0 K qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j k0b 2 with K ¼ 1 þ jtrðsÞ 3G0 , r ¼ ks , v ¼ q. Here, ks is the stretch relaxation time, q the amount of arms at the end of a backbone.
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NUMERICAL METHODS Pressure-Stabilized Method Based on the FIC Process It is known that in numerical solutions for incompressible Navier–Stokes equations, the velocity–pressure interpolation satisfies the usual compatible condition (known as the Ladyzhenskaya–Babuska–Brezzi (LBB) or inf–sup condition) between the function spaces for velocity and pressure that prevents locking and spurious oscillation phenomena. Although some elements satisfying the LBB condition, i.e., compatible u p interpolations may be used, one prefers to use the equal low-order u p interpolations owing to its efficiency in computational cost and convenience for using an adaptive strategy. Hence, the equal low-order elements for velocities, pressure, discrete rate-of-strains and stresses, e.g., T3P3 will be applied to simulate viscoelastic flows in this investigation. To restrain even eliminate spurious oscillations of pressure field due to the incompressibility of fluids and avoid the calculation of high order spatial derivatives required in the original finite increment calculus (FIC) process of O~ nate[22], a modified FIC process for pressure stabilized mass conservation equation is given in the literature[19,23]. Based on the modified FIC process, the stabilized mass conservation equation can be written as r u kd r ðu þ rp=qÞ ¼ 0
ð9Þ
DEVSS/SU Method for Constitutive Equation The onset of numerical instability in the simulation of viscoelastic flows mainly stems from the two sources, one is the loss of characteristics of the governing equations symbolizing elliptic behavior and the other is due to the highly hyperbolic nature of viscoelastic flow equations, particularly in the convective dominated case. To restrain and overcome the instability from the first source, the EVSS method may be employed to retain an elliptic term and consequently the viscous contribution in the momentum equation. Unfortunately, the constitutive equation needs to be modified. Moreover, the convected derivative of the rate of strain tensor emerges, which requires a second-order derivative of the velocity field. So, it is difficult to apply this approach to complex constitutive equation, such as the XPP and DCPP models. To circumvent these difficulties, we employ a modification of the EVSS formulation, known as the discrete EVSS method (DEVSS)[12]. In this method a stabilizing elliptic term is introduced in the discrete version of the momentum equation in the absence of a purely viscous contribution or as the viscous contribution is negligible in comparison with the viscoelastic contribution and can be conveniently numerically implemented. To construct the DEVSS method, an auxiliary variable D as a discrete counterpart of the rate of deformation tensor D is introduced and determined in terms of satisfaction of DD¼0
with /
@u þ u ru r T=q g @t
ð12Þ
in the following weighted average form, i.e., ð10Þ ðW; D DÞ ¼ 0
ð13Þ
d
where k is termed ‘‘intrinsic time’’ per unit volume, u is an auxiliary variable. The terms underlined in Eq. (9) are the stabilization terms introduced by the FIC process. Note that if kd ¼ 0 in Eq. (9), the conventional mass conservation equation is recovered. The value of kd to be taken in the present numerical simulation will be given in the following section. With the use of Eq. (10) the momentum Eq. (1) can be re-written as / ¼ rp=q
ð11Þ
i.e., actually the introduced auxiliary variable u is defined as the first-order derivative of pressure multiplied with the value of 1=q. Hence, with the use of Eq. (11) the calculation of high (second)-order spatial derivatives of pressure required in the original finite increment calculus (FIC) process of O~ nate[22] can be avoided.
where W is admissible weighting function. Taking the divergence of the Eq. (12) and adding it into the momentum conservation Equation (1) gives qðut þ u ruÞ ¼ rp þ r ð2gv D þ sÞ þ qg 2ar ðD DÞ
ð14Þ
where ut denotes @u=@t, a is a positive parameter. In general, a is chosen to ge. Hence, using g ¼ ge þ gv, we obtain qðut þ u ruÞ ¼ rp þ r ð2gD þ sÞ þ qg r ð2ge DÞ
ð15Þ
After applying the DEVSS method, Eq. (15) is the stabilized form of the momentum equation. With the aid of the modified FIC process and the DEVSS scheme, Eqs. (4), (9),
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SOLUTION OF PLANAR FLOW FOR BRANCHED POLYMER MELTS
(11), (12) and (15) construct the stabilized governing equations, which include three unknown variables u, p, s and two auxiliary variables u; D. As the second source of numerical instability in the simulation of viscoelastic flows, i.e., the highly hyperbolic nature of viscoelastic constitutive equation, is concerned, inconsistent SU method is employed since its merits in circumventing oscillations of the resulting stress fields at steep stress boundary layers or near singularities as compared with the SUPG method and performing well in convergence up to high values of the Weissenberg number for the sudden contraction flow problems. In present method, as the constitutive equation is spatially discretized the upwind weighting functions is only applied to the convective term of the equation, while other terms in the equation are still pre-multiplied with standard weighting functions used in the standard Galerkin approximation, i.e., equal to the shape functions. The weak form of the Eq. (4) is written as: D E r Ns ; s þ f c ðs; DÞ þ k1 f 2G D þ hWu rNs ; u rsi ¼ 0 0 0b d ð16Þ where h&,&i denotes the standard inner product, Ns the weight function of standard Galerkin formulation, Wu r Ns the upwind term of SU method, W ¼ hs=2U where hs characteristic length along the streamline, U characteristic velocity. Pressure-Stabilized Iterative Fractional Step Algorithm Let An and Anþ1 be the variable at time tn and tnþ1, respectively, (e.g., A ¼ u; p; s; u; D), with Dt ¼ tnþ1 tn, Dp ¼ pnþ1 pn. At time tnþh, each variable is defined as Anþh ¼ (1 h)An þ hAnþ1, where, h 2 [0, 1] with h ¼ 0, 0.5, 1 for the explicit, Crank–Nicolson implicit and the backward Euler implicit forms, respectively. Within a typical time sub-interval [tn, tnþ1], the temporal discretization forms of Eqs. (15), (9), (11), (12) and (4) are respectively written as q nþ1 ðu un Þ ¼ qg ðqu ruÞnþh Dt þ ST ðgD0 ÞSunþh rpnþh þ r snþh r 2ge Dnþh ð17Þ r unþ1 kd ðr unþ1 þ q1 r2 pnþ1 Þ ¼ 0
ð18Þ
1 unþ1 ¼ rpnþ1 q
ð19Þ
Dnþ1 ¼ Dnþ1
ð20Þ
1 nþ1 ðs sn Þ ¼ ½u rs þ ðruÞT s þ s ru Dt nþh f c ðs; DÞ k1 0b f d þ 2G0 D
ð21Þ
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where, for two dimensional problems D0 ¼ diag(2 2 1), and S is the strain matrix (operator) linking the strain rates to the velocities and is defined by @ @x1 S ¼ 0 T
0 @ @x2
@x2 @x1 @ @x1
ð22Þ
Adopting the Crank-Nicolson implicit difference scheme, i.e., h ¼ 0.5, and introducing an intermediate velocity u , we can split Eq. (17) as q
I3 hST gD0 S ðu un Þ ¼ qg ðqu ruÞnþh
Dt þ ST ðgD0 ÞSun þ r snþh crpn r 2ge Dnþh q Dt
I3 hST gD0 S ðunþ1 u Þ ¼ rðpnþh cpn Þ
ð23Þ ð24Þ
where c ¼ 0, 1 corresponds to non-incremental and incremental versions of the split algorithm respectively. In the present work, c ¼ 1 is specified. Note that the u terms in Eqs. (23) and (24) are removed from the right-hand sides to the left-hand sides, so that it is beneficial to enhancing the numerical stability. Taking the divergence of the vectorial Eq. (24) results q r ðunþ1 u Þ Dt þ hr ½ST ðgD0 ÞSÞðunþ1 u Þ
r2 ðpnþh cpn Þ ¼
ð25Þ
Since the term underlined on the right-hand side of Eq. (25) is the third order spatial derivatives of the velocity u, it is simply omitted for convenience. Then substituting Eq. (18) into the above equation and rearranging, we obtain ! kd q hþ r2 ðpnþ1 pn Þ ¼ ðr u kd r /nþ1 Þ Dt Dt ! kd þ 1 c r2 pn Dt
ð26Þ
Note that if kd ¼ 0 Eq. (26) is degenerated to the Poisson equation resulting from standard fractional step method. In fact, kd is a stabilization parameter introduced by the FIC process and can be chosen as[23] kd ¼
4t 2U 1 þ h2 h
ð27Þ
where U is the characteristic velocity, t ¼ l=q the kinematic viscosity and h the typical element size.
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W. WANG ET AL.
Generally, Eqs. (23), (26) and (24) are known as prediction, projection and pressure correction steps, respectively. As for Eq. (21), it is noted that fc in the mXPP model vanishes, and fd(s)nþh can be written as f d ðsÞ
nþh
¼
f 0d ðsÞnþh
þ hðs
nþ1
where Z
M¼
X Z
C¼
n
s Þ
ð28Þ
f 0d (s)nþh ¼ fd(s)nþh h(snþ1 sn),
with h ¼ 0.5 in this where investigation. Substituting Eq. (28) into Eq. (21) and moving h(snþ1 n s ) to the left-hand side of Eq. (21), we obtain 1 h þ ðsnþ1 sn Þ Dt k0b
L¼
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ð30Þ
Note that for /, taking Nu ¼ Nu in this paper, so we obtain / ¼ Nu /. By using the standard Galerkin procedure, the weak forms of Eqs. (23), (26), (19), (20) and (24) along with the weak forms of the corresponding natural boundary conditions can be, respectively, written as q M þ hgKu ð un Þ u Dt ¼ qC unþh gKu un pn þ 2ge KD D Ks snþh þ cLT
nþh
þ fs
D¼
Z
MD ¼
¼
!
Z
1 nþ1 ¼ DT Mu pnþ1 q
Dt
ðSNu ÞT Ns dX
X
Z
ðSNu ÞT ND dX
X
ðrNp ÞT Nu dX Kp ¼
Z
ðrNp ÞT ðrNp ÞdX ð36Þ
X
NTD ND dX NTu qgdC þ
C
fp ¼ h
Z
HD ¼ 1=2 Z
Z
NTD D0 SNu dX
X nþh NTut dC
Ct
NTp ni
@Dp @pn þ ð1 cÞ dC @xi @xi
C
The inconsistent streamline upwind method (SU) is employed to spatially discretize the viscoelastic constitutive Eq. (29) that results in
1 h 0 nþh þ Ms Ds ¼ ½Cs s þ Qu þ G0 Eu k1 0b Fd Dt k0b ð37Þ
where Ms ¼
Z
NTs Ns dX
Cs ¼
X
Q¼
Z
¼
WTs ðunþh rÞNs dX
X
NTs D0 ðSs Þnþh Nu dX
X
Z
Z
E¼
Z
NTs D0 Nu dX
ð38Þ
X
NTs f 0d dX
with
kd pn þ f s þ 1 c Kp Dt
2
ð32Þ
q
NTp ðr Nu ÞdX KD ¼
Z
X
q qkd nþ1 L u Du Dt Dt
MD D
rÞNu dX Ks ¼
X
F0d
kd hþ pnþ1 pn Þ Kp ð Dt
X
NTu ðunþh
X
ð31Þ
!
ðSNu ÞT D0 SNu dX
X
Z
fs ¼
A ¼ Na A
Ku ¼
Z
X
Z
nþh 0 ð29Þ ¼ ½u rs þ ðruÞT s þ s ru k1 0b f d þ 2G0 D
Space Discretization Using the Finite Element Method In the context of finite elements, let Na (a ¼ u, p, s, u, D, respectively) denote the shape function of each variable A, i.e., A ¼ u; p; s; u; D, respectively, and A their nodal values within each element. So, the unknown variables may be spatially approximated as
NTu Nu dX
nþ1
¼ HD unþ1
M þ hgKu ð u Þ ¼ LT ð pnþh c pn Þ unþ1
3 0 0 s2k @=@xk 0 s3k @=@xk s1k @=@xk 5 STs ¼ 4 0 s2k @=@xk 0 s1k @=@xk s3k @=@xk 0 0 s2k @=@xk s1k @=@xk 0 0 s3k @=@xk
ð39Þ
ð33Þ ð34Þ
ð35Þ
The iterative procedure is particularly introduced into the pressure stabilized fractional step algorithm based on the FIC process in combination with the DEVSS scheme and the temporal discretization using the Crank-Nicolson implicit difference scheme in this investigation, abbreviated as the I PS DEVSS CNBS scheme. Regarding the pressure stability analysis of the
SOLUTION OF PLANAR FLOW FOR BRANCHED POLYMER MELTS
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similar scheme, one may refer to Li and Duan[24]. The iterative procedure of the scheme can be found in the literature[25]. For calculations presented in the current study, the iterative convergence criterion is set to be e ¼ 106, and time step Dt ¼ 103 s is employed. COMPLEX VISCOELASTIC FLOW SIMULATION Planar Contraction Flow Problem The 4:1 contraction flow is a benchmark test problem and has been extensively studied[16,17,26]. The numerical results for the 4:1 contraction viscoelastic flow problem display that the viscoelastic fluid at the zone far from the contraction region is subjected to a simple shear and that in the contraction channel is subjected to a pure extensional deformation along the centerline (elongation flow), while a mixture of shear and elongation exists near the re-entrant corners. In fact, such type of contraction flows also widely exist in industrial polymer processes, such as injection molding, extrusion flow etc. A schematic diagram of the upper half of the 4:1 planar contraction geometry is shown in Figure 1. The lengths of upstream and downstream channel are 30 m and 40 m, respectively. The triangular mesh in contraction region is displayed in Figure 2, and the data to characterize the computational mesh for the simulation are given in Table 1, where the smallest size of mesh is listed. The parameters of the XPP model for fitting the IUPAC A LDPE melt at T ¼ 150 C (see EPAPS Document No. E-JORHD2–45-013104[4]) are given in Table 2. The density of 918 kg=m3 for IUPAC A LDPE melt at T ¼ 150 C is used. Boundary Conditions At the inlet we prescribe a Poiseuille velocity profile that ensures the specified flow rate and, due to memory effects of viscoelastic fluids, all components of the viscoelastic extra-stress prescribed by the same method described in ref.[16], along with the discrete approximation of the rate-of-strain tensor D are prescribed as initial conditions. Moreover, we assume that the downstream exit length is
FIG. 2. Computational mesh around the re-entrant corner.
chosen long enough so that at the outlet a fully developed Poiseuille flow condition is also imposed. No-slip conditions are imposed on wall and symmetry conditions are specified on the centerline y ¼ 0. At the exit, the pressure of the symmetric point is set to zero. By taking a half of the downstream channel height H and the downstream two-dimensional mean velocity U as characteristic values, the flow averaged Weissenberg number We and Reynolds number Re can be evaluated respectively by We ¼
k0b U H
ð40Þ
where g ¼ ge þ gv, ge ¼ G0k0b, (gv=ge ¼ 1=8 as given in ref.[16]). To specify different values of the We number, various inflow rates of Q ¼ 0.1, 0.25, 0.5, 0.75, 1.0, 2.5 m2=s are adopted corresponding to different downstream mean velocity U, respectively and k0b is fixed so as to ensure the same rheological characteristics of Pom-Pom molecule.
TABLE 1 The data to characterize the computational mesh
Elements FIG. 1. Schematic of 4:1 abrupt planar contraction.
4800
Nodes
Degrees of freedoms ðu; p; s; DÞ
Dxmin
Dymin
2537
22833
0.0141
0.0304
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TABLE 2 Some parameters of the IUPAC A LDPE melt at T ¼ 150 C Maxwell parameters G0 (Pa) 859.6
XPP model
k0b (s)
q
k0b=ks
a
20
4
2
0.025
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RESULTS AND DISCUSSION Numerical Solutions for Different Weissenberg Numbers For viscoelastic flow, the Weissenberg number, We, is an important dimensionless parameter that denotes the ratio of the relaxation time of the polymer to the macroscale flow time (see Eq. (40)). The elasticity of viscoelastic
fluid increases with increasing value of Weissenberg number. We will investigate the features of the viscoelastic flow using the mXPP model with different values of We in the contractive geometry. In this subsection, the material parameters of the XPP model depicted in Table 2 are adopted. Different values of We can be taken with different downstream mean velocities. The streamline patterns of the contraction flow with different We are illustrated in Figure 3. With increasing We, the salient-corner vortex cell size is clearly enlarged, which is in agreement with the results about 4:1 planar rounded-corner contraction flow of Aguayo et al.[16]. The sizes of the salient-corner vortex are described with the lengths of its horizontal and vertical sides along the cavity walls, i.e., Lx and Ly as shown in Figure 3 in which the geometry and sizes of the salient-corner vortex and their evolution can be observed. It is illustrated in Figure 3 that
FIG. 3. Streamline patterns for the 4:1 contraction flow for the XPP model. (a) We ¼ 2.0 Lx ¼ 1.70 m Ly ¼ 1.98 m; (b) We ¼ 5.0 Lx ¼ 1.75 m Ly ¼ 2.14 m; (c) We ¼ 10.0 Lx ¼ 1.92 m Ly ¼ 2.63 m; (d) We ¼ 15.0 Lx ¼ 2.01 m Ly ¼ 2.92 m; (e) We ¼ 20.0 Lx ¼ 2.15 m Ly ¼ 2.94 m; (f) We ¼ 50.0 Lx ¼ 2.37 m Ly ¼ 2.95 m.
SOLUTION OF PLANAR FLOW FOR BRANCHED POLYMER MELTS
Variations of salient-corner vortex cell size with increasing We.
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FIG. 4.
FIG. 5.
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Lx augments with increasing We, while Ly increases with increasing We only as We 15 and then keeps a steady value as We increases. The values of Lx and Ly and their variations against We are plotted in Figure 4. The stress components and the first normal stress difference N1 of the 4:1 planar viscoelastic contraction flow near the re-entrant corner along the downstream wall, i.e., y ¼ 1, are shown in Figure 5, respectively. It is observed from the figure that whatever values of We are used the peak values of these mechanical quantities occur in the vicinity of the sharp corner. In the contraction channel, the stress syy becomes negative and N1 > 0 that is in agreement with the behavior of realistic polymer melts. The comparison of the numerical results obtained by using the proposed scheme with experimental results[4–6] further validates good performance of the XPP model in the prediction of the rheological behaviors of branched polymer melts. Moreover, as one can see in Fig. 5 that, the peak of each stress component amplifies with increasing We, and keeps higher steady value for larger We.
Profiles along downstream wall (y ¼ 1) for different We.
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W. WANG ET AL.
FIG. 6.
Profiles along symmetry line (y ¼ 0) for different We.
The profiles of stress, horizontal velocity and the first normal stress difference along the centerline y ¼ 0 are depicted in Figure 6. It is observed that in the downstream channel and near the sharp corner, the Weissenberg number has great effect on the physical quantities in the flow domain. For larger We, the stress syy shows higher peak value around the sharp corner, while far from the entrance of the contraction zone (x < 20), the magnitude of syy is much less and approaches zero. With increasing We, the peak of each stress component augments. Furthermore, the value of We presents notable influence on the dimensionless horizontal velocity along
FIG. 7. Contour of pressure, We ¼ 20, material parameters listed in Table 2. pmin ¼ 89 Pa, pmax ¼ 344105 Pa.
FIG. 8. Convergence history for We ¼ 20 using equal low-order elements (T3P3).
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FIG. 9. Profiles along downstream wall (y ¼ 1) for various q: We ¼ 20.
the centerline, as shown in the lower left plot of Figure 6. This observation is consistent with the results reported by Alves et al.[27] for Oldroyd-B and PTT fluids. The smooth pressure contours shown in Figure 7 confirm that the proposed scheme with introduction of the FIC process into the mass conservation equation can effectively restrain even eliminate spurious oscillations of resulting pressure spatial distribution due to incompressibility of fluids. Similar method has already been successfully utilized to calculate the incompressible non-isothermal non-Newtonian fluid flows and the mold filling process [23]. In addition, the convergence history of iterative procedure is demonstrated in Figure 8. Some temporal oscillations of pressure are observed, while the velocity and stress are stable convergence. From Figure 8, we may observe that for viscoelastic flow simulation, the velocity and pressure fields have strongly coupling effect, while the stress fields have weak dependence on the pressure field. Hence, as the I PS DEVSS CNBS scheme is applied in this
study, the FIC process and DEVSS method have different effects on pressure and stress fields, respectively. Influence of the Amount of Arms q In this subsection, the rheological behavior of branched polymer melts with increasing value of the amount of arms at the end of a backbone, i.e., q, is to be discussed. For the convenience of comparison, other material parameters used in the mXPP model and the We of 20 are fixed. Figure 9 shows the profiles of each extra stress component and N1 along the downstream wall i.e., y ¼ 1. The variation of q has great influence on sxx, sxy and N1 around the sharp corner and along the contraction channel. Moreover, note that with increasing q, magnitudes of sxx remarkably increase near the entrance of the contraction channel. Hence, the polymer melts with many branched arms may undergo larger elongation in the downstream region. However, the effect of q on syy is negligible.
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FIG. 10. Profiles along symmetry line (y ¼ 0) for various q: We ¼ 20.
The distributions of syy, sxx, p and horizontal velocities for various q along the centerline are illustrated in Figure 10. It is observed that various q have distinct influence on sxx and pressure rather than syy and u. These pattern are in excellent agreement with the results reported by Aguayo et al.[16] for the planar rounded-corner contraction flow. As an example, we only present the resulting contours of each extra stress component for We ¼ 20 shown in Figure 11. In view of the stress singularity at the sharp corner and a thin layer at the upper wall after the re-entrant corner displayed in Figure 11 where high stress gradients exist, the refined meshes at and near those areas are required. The minimal sizes of mesh are listed in Table 1. Finally, it should be reported that we achieve the convergent solutions for We ¼ {2, 5, 10, 15, 20 and 50}. Moreover, the velocities in x-direction at different cross-sections (x ¼ 20, 50) for various q are shown in Figure 12. We observe that q has very little effect on the horizontal velocity away from the centerline. However,
FIG. 11. Contours of stress components for We ¼ 20, q ¼ 4, a ¼ 0.025. (a) Tyy; (b) Txx; (c) Txy.
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The influence of We and q on flow characteristics, such as streamline pattern, contours of stresses and profiles of stress components, generated in a 4:1 planar contraction transient flow are discussed. Numerical results show that the responses of the mXPP model around the re-entrant corner is significantly influenced by such parametric variation, some of which are in good agreement with the results of Aguayo et al.[16]. These are beneficial to extending this numerical study to non-isotherm viscoelastic fluid injection molding and complex polymer processing.
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ACKNOWLEDGMENTS The authors are pleased to acknowledge the support of this work by the National Natural Science Foundation of China through Contract=Grant numbers 10590354, 10672033 and 10272027. The first author would also like to acknowledge Prof. Q.Y. Wu for many stimulating discussions. REFERENCES
FIG. 12. Horizontal Velocities at different cross-section of channel for various q. (a) At cross-section of upstream channel x ¼ 20; (b) At crosssection of downstream channel x ¼ 50.
with increasing q, horizontal velocity slightly increases in the neighborhood of the symmetry axis. CONCLUSIONS The viscoelastic flow in a 4:1 planar contraction geometry is investigated using the iterative stabilized fractional step scheme in combination with the introduction of the FIC process into the scheme to reformulate the mass conservation equations, along with DEVSS=SU technique for the constitutive equation of the mXPP model. The equal low-order triangular elements are utilized to discretize the computational domain and solve for the velocities, pressure, discrete rate-of-strains and stresses. The spurious oscillations of pressure field resulting from the incompressibility of fluids are restrained. The stable numerical solutions for more complex viscoelastic flows can be expected by the I PS DEVSS CNBS scheme adopted in the present study.
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