Implementation of a Slip Boundary Condition in a Finite Volume Code ...

II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008

Implementation of a Slip Boundary Condition in a Finite Volume Code Aimed to Predict Fluid Flows Ferrás, L.L.1, Nóbrega, J.M. 1, Pinho, F.T.2,3 e Carneiro, O.S.1 1

University of Minho, IPC – Institute for Polymers and Composites, Campus de Azurém, 4800-058 Guimarães, Portugal email: [email protected] http://www.dep.uminho.pt 2 University of Minho, Largo do Paço, 4704-553 Braga, Portugal 3 University of Porto, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal

Abstract This work describes the implementation of the slip boundary condition on solid walls in a numerical code, based on the finite volume method, for the solution of the Navier-Stokes system of equations. Specific numerical issues related to this implementation upon the flow and stress fields are discussed and three different slip models are compared. The numerical predictions are validated against simple problems with analytical solution, and other more complex flows are studied to test the accuracy and robustness of the slip model implementation. Keywords: Wall slip, finite volume method, Navier-Stokes equations

1

Introduction

In fluid dynamics, the continuity (Eq. 1) and momentum equations (Eq. 2) provide the tool to predict the behavior of complex flows, subject to appropriate boundary conditions, such as the impermeability and non-slip conditions of Eqs. (3) and (4), respectively. ∂ui =0 in Ω (1) ∂xi

∂ρ ui ∂ρ ui u j ∂p ∂τ ij + =− + + ρ gi ∂t ∂x j ∂xi ∂xi

in Ω

(2)

on ∂Ω (3) on ∂Ω (4) ui = 0 In Eqs. (1) to (4) ui is the ith velocity component, ρ is the fluid density, p is the pressure, τ is the deviatoric stress tensor, g is gravity, ni is the normal unit vector component and Ω is the physical domain. The deviatoric stress tensor τ must be given by an adequate rheological constitutive equation in order to obtain a closed set of equations. In this work the Newtonian fluid model of Eq. (5) is used ( µ (γɺ ) is the viscosity, γɺ is ui .ni = 0

the shear rate, i, j = 1, 2,3 ),

 ∂ui ∂u j  (5) +  .  ∂x  j ∂xi  The specification of boundary conditions is a necessity to guarantee the wellposedness of the problem. Often, as at an outlet, it is not clear what boundary condition should be imposed and there are strategies to reduce the impact of an ill posed condition upon the results [1]. Another quite important situation regards the interaction between fluids and solid walls. A survey through the literature of the twentieth century shows that in the vast majority of flow cases the boundary condition to be used is the Dirichlet boundary condition (Eq. 6), (6) ui = 0 on ∂Ω . which imposes that the fluid adheres to the wall, together with the impermeability condition. However, the former cannot be derived from first principles [2, 3, 4]. When this topic is discussed (Lamb [5], Batchelor [6] and Goldestein [7]) it is often to mention that slip may be wrong or that the use of no-slip stems from agreement of predictions with experiments (some of these experiments were not carried out carefully and consequently their results are contradictory). In any case it is now an established fact that for macro geometries the interaction

τ ij = µ ( γɺ ) 

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II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008 between small fluid molecules and walls is equivalent at the macro scale to a no-slip condition. As the ratio between mean free path and characteristic flow size increases slip effects become more important. There are many papers related to existence and measurements of the slip velocity (see [2, 8, 9] and references cited therein). In what regards long molecules, such as polymer melts, slip effects can actually be found at the macro scales leading to some interesting phenomena reviewed by Denn [10], such as shark skin, slip-stick and melt fracture. Other references on slip at the liquid-solid interface for polymers are by Potente et al [11] and Mitsoulis et al [12]. Slip of fluid at solid walls has a profound influence on velocity profiles (Fig.1) and affects the relationship between integral quantities such has pressure drop and flow rate. Therefore, the implementation of slip models in computational codes used to simulate such relevant industrial flows as extrusion of polymer melts is essential. no-slip

slip

x2 wall

x1 (a)

(b)

Fig.1. Velocity profile (a) with no-slip (b) with slip. This research team has developed a finite volume three-dimensional code for the numerical solution of flow and heat transfer equations of non-Newtonian fluids [13], but always assuming the validity of the no-slip condition. In the present contribution this procedure is extended to deal with slip at solid walls. At this stage Newtonian fluids are considered, to be extended in the future to viscoelastic models. 1.1 Slip models In 1827, Navier [14] (and later Maxwell) proposed a slip model described next. The idea is inspired by solid mechanics as depicted in Fig. 2, where two blocks A and B are represented. Between the two bodies a normal      force Fn and a friction force Ft act. According to the Amontons-Coulomb law Ft = η Fn , the friction force Ft , is  proportional to the normal load Fn (independent of the area of contact of the two solids), and η is a constant of proportionality called the coefficient of friction.  Fn

A

 Ft

B Fig.2. Illustration of the Amontons-Coulomb law for two solids. Transposing this idea to fluid mechanics (Fig. 3), assume that “B” is the solid wall and “A” is a fluid layer near the wall. Navier, argued that in the presence of slip, the liquid motion must be opposed by a force proportional to the relative velocity between the first liquid layer and the solid wall (see also [9, 15, 16]). x2  ut

x1



τt

Fluid Layer



τt

 ut

Solid Wall wall

Fig.3. friction type - boundary slip condition (fluid-solid interaction) For a two dimensional flow, and assuming Couette flow near the wall, the Navier’s slip condition is given by (Eq. 7, 8),

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II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008   ∂u  µ  2  τ x1 x1 τ x1 x2    ∂x1  τ = = τ x2 x1 τ x2 x2   µ  ∂v + ∂u      ∂x1 ∂x2 



τ t = n.τ .(1 − nn) = µ

 ∂u ∂v   +   ∂x2 ∂x1     ∂v   µ 2    ∂x2 

µ

∂u ∂x2

 ∂u  1 − us =  µ  k  ∂x2  ∂u ⇔ us = − ( k µ ) ∂x2

(7)

where k −1 is the friction coefficient, and k µ is called the slip length ( Ls ) assumed independent of the shear rate. The term (1 − nn) is a tensor that projects a vector into its tangent component. A generalization of Eq. 7 gives what is called here the first model of slip (model 1).   (8) ut = − k τ t (model 1) . Several authors [2, 15] try to explain the existence of slip and its dependence on parameters like surface roughness, dissolved gas and bubbles, wetting characteristics, shear rate, electrical properties and pressure [2], and this list keeps increasing with time. As in solid mechanics, the friction law works as a model that can be used for a large number of materials and surfaces (in a continuum framework), fitting experimental data. However, this is not a fundamental law and the smaller the size of the flow system the less valid the continuum approach is and new molecular models need to be used to properly account for wall-fluid interactions. However, our system of equations is for the continuum and so there is the need to bridge the gap between the molecular scales and the fluid particle concept of continuum mechanics (for works related to molecular and continuum models/theory see [15, 19]). To fit experimental observations different slip models were created, such as those stating the dependence of the friction coefficient on wall shear rate or stress [20] (Eq. 9),  (9) k = f (τ t ) . Therefore, since the main objective here is the numerical implementation of slip models for a variety of situations two other models, given by Eqs. (10) and (11) were implemented.  τ   (10) us = −k2 τ t n  t (model 2) k2 ∈ ℝ + , n ∈ ℝ + \ {1} . n τt    τ + k4  τ t  (11) us = − k3 ln  t (model 3) k4 , k3 ∈ ℝ + .    k4  τ t These models consider non-linear relationships between the stress vector tangent to the wall and the slip velocity vector. In “model 2” a simple nonlinearity law is applied, and “model 3” is an asymptotic relationship. With these models, new boundary conditions are created, but there are other problems related to mathematical issues, because we need to know if the selected boundary guarantees the wellposedness of the problem. Several and important notes related to this subject can be found on [16, 17, 18] and references cited therein. In the next Section the numerical issues related to the slip boundary conditions are described in detail and discussed. Case studies are analyzed in Section 3 to assess the accuracy and robustness of the code and the work ends in Section 4 with the main conclusions.

2

Implementation of the slip models

The code used for this work is explained in detail elsewhere [13]. The equations are transformed to a general coordinate system and integrated over hexahedral control volumes. For the momentum Eq. 2 this leads to Eq. (12) written in the new coordinates (ξ1 , ξ 2 , ξ3 ) (Einstein notation is used thoughtout this paper to write down the governing equations, and is assumed that i, j , k = 1, 2,3 )

1 ∂ ( J ρ ui ) 1 ∂ ( ρβ lj u j ui ) 1 ∂ ( βli P ) 1 ∂ ( βljτ ij ) + =− + + ρ gi J J J ∂ξ l J ∂ξ l ∂t ∂ξ l

(12)

βlj are the metrics of the coordinates change and J is the Jacobian of the coordinate transformation. The diffusive and convective terms are the only ones that have contributions from the boundary. However, for a non-porous wall the convective term vanishes and only the diffusive term (Eq. 13) remains and needs to be modified to implement the slip condition.

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II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008

for i = 1

∂ ( β ljτ 1 j ) ∂ξl

=

∂ ( β11τ 11 ) ∂ ( β12τ 12 ) ∂ ( β13τ 13 ) ∂ ( β 21τ 11 ) ∂ ( β 22τ 12 ) ∂ ( β 22τ 13 ) + + + + + + ∂ξ1 ∂ξ1 ∂ξ1 ∂ξ 2 ∂ξ 2 ∂ξ 2 (13)

∂ ( β31τ 11 ) ∂ ( β32τ 12 ) ∂ ( β33τ 13 ) + + ∂ξ 3 ∂ξ3 ∂ξ 3 Following the finite volume method, integrating the diffusive term over the control volume V p , gives Eq. (14) after application of Gauss' theorem 1 ∫∫∫Vp J div ( β1 jτ 1 j , β 2 jτ1 j , β3 jτ1 j ) Jd ξ1d ξ2 dξ3 = Gauss Theorem = ∫∫S ( β1 jτ 1 j , β 2 jτ1 j , β3 jτ1 j ) .n dS =

∫∫ ( β τ , β τ , β τ ) .n dS + ∫∫ ( β τ + ∫∫ ( β τ , β τ , β τ ) .n dS + ∫∫ ( β τ Se

Ss

1j 1j

1j 1j

2 j 1j

3 j 1j

2 j 1j

e

3j 1j

So

s

St

1j 1j

, β 2 jτ 1 j , β 3 jτ 1 j ) .n dS w + ∫∫

1j 1j

, β 2 jτ 1 j , β 3 jτ 1 j ) .n dSt

Sn

(β τ + ∫∫ ( β τ

1j 1j

, β 2 jτ 1 j , β 3 jτ 1 j ) .n dS n

Sb

1j 1j

, β 2 jτ 1 j , β3 jτ 1 j ) .n dSb

(14)

where e, w, n, s, t, b, stand for the six faces of the control volume (the previous and following results were obtained bearing in mind just the case i = 1 ). Considering just the east face, it can be found that (Eq. 14) results in, ∫∫ ( β1 jτ 1 j , β 2 jτ1 j , β3 jτ1 j ) .n dSe = ∫∫ β1 jτ1 j dSe = ( β1 jτ1 j ) ∫∫ dSe = ( β1 jτ1 j ) δξ 2δξ3 = (b11τ11 + b12τ12 + b13τ13 )e (15) Se

where β1 j =

e

Se

b1 j

δξ 2δξ3

e

Se

, ( b11 , b12 , b13 ) is the vector area and δξ 2δξ3 are the new computational control volume

dimensions [21]. The discretization of the stress terms (Eq.16) in the new coordinate system (Eq.17),  ∂u ∂u  2 ∂u τ ij = µ  i + j  − µδ ij k  ∂x j ∂xi  3 ∂xk   ∂u  2 µ ∂u µ  ∂u τ ij =  βlj i + β li j  − δ ij βlk k J  ∂ξl ∂ξ l  3 J ∂ξ l is carried out assuming the following profile for velocity derivatives, φe − φo  ∂φ  .   ≃ ξ δξ ∂  p

(16) (17)

After applying this rule for the discretization, the stress terms can be written as,

)

(

f

3 3 f 2 f f   µ   3 f f   b u b u blkf [ ∆uk ]l  ∆ + ∆ − τ ijf =  δ [ ] ∑ ∑∑ lj i li j ij     l l 3 l =1 k =1  δ V   l =1 

(18)

where f = e, n, t , w, s, b indicates the face of the control volume and, [ ∆uk ]l means the variation of the uk f

velocity component along the l direction and δ ij is the Kronecker delta. In the code, the stress term (Eq. 18) is divided in two parts (Eq.19), one that is explicitly (Eq.20) calculated and other that is treated implicitly (Eq.21). For more details see [13]. (19) τ ijf = (τ ijf )im + (τ ijf )ex

)

(

3 3 3 f 2 f f   µ   3 f f f (τ ijf )ex =    ∑ blj [ ∆ui ]l + ∑ bli  ∆u j  l − δ ij ∑∑ blk [ ∆uk ]l  3 l =1 k =1  δV   l≠ f l =1  f

(

)

(20)

f

f  µ  f (21) (τ ijf )im =   b fj [ ∆ui ] f .  δV  As an example, it has been seen that for the first velocity component and for an east face we obtain (Eq. 15). In general, the contribution of the stress term to the boundary, for the “ i ”momentum equation is (Eq.22). 3

source(ui )f = ∑ ( bffj (τ ijf ) ) = bff 1 (τ if1 ) + bff 2 (τ if2 ) + bff 3 (τ if3 ) j =1

3 3 3 3 f 2 f f  f   µ   2  3 f f f f f source(ui )f = D f ( uiF − uiP ) +    ∑ b fj blj [ ∆ui ]l  + ∑ bli ∑ b fj  ∆u j  l − b fiδ ij ∑∑ blk [ ∆uk ]l  (22)     δ V   ∑ 3 j =1 l =1 k =1  l =1   l ≠ f  j =1 implícit   f

explicit

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II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008 1

f

2  2 2  3  µ  Df =   ( b f ) , b f = b f =  ∑ ( b fi )  (f area )  δV   i =1  In order to implement the no-slip boundary condition, we need to impose that the velocity variations along the tangent directions to the wall are null, that can achieved by (Eq.23), 2  3 f f f  (23)  ∑ b fj blj [ ∆ui ]l  = 0 ∑ l ≠ f  j =1 

2.1 Numerical implementation of the slip boundary condition For the numerical implementation of the slip boundary condition the existence of a Couette flow near the wall is assumed, as was previously the case for the no-slip boundary condition. To simplify the implementation, it is better to use the stress vector form (Eq. 24),  T = (T1 , T2 , T3 )

T1 = τ 11n1 + τ 12 n2 + τ 13 n3

 n = ( n1 , n2 , n3 ) normal unit vector

T2 = τ 21n1 + τ 22 n2 + τ 23 n3

(24)

T3 = τ 31n1 + τ 32 n2 + τ 33 n3 The contribution of the stress term to the boundary, for the “ i ” momentum equation is (Eq.25) 3

source(ui )f = ∑ ( bffj (τ ijf ) ) = b f (τ if1 )n1 + (τ if2 )n2 + (τ if3 )n3  = b f Ti

(25)

j =1

Wall

Tt

Tn

F

δp

T P'

P

Fig.4. Stress term at the wall.

Bearing in mind Fig.4 and assuming that a zero normal stress for the fluid layer near the wall, for the Couette flow the tangent stress contribution is (Eq.26) (the zero normal stress is a consequence of considering Couette flow for Newtonian fluid. The normal stress will be non-zero for a viscoelastic fluid)  ∂uti , (26) Tt = µ f ∂n where uti is the tangent velocity for the near wall fluid (for ease of understanding we assume that the wall is fixed). Fig.4 hints the following discretization (Eq. 27) for Eq. 26 ,  ∂uti utf ( slip ) − utpi ' Tt = µ f ≈ µf i δp ∂n

(27)

where utif ( slip ) is the slip velocity and the following approximation is made utpi ' ≃ utpi . In this way, the contribution of the stress term for the “ i ”momentum equation is (Eq. 28), utf ( slip ) − utpi bf f f source(ui )f = b f Ti = b f Ttif = b f µ f i = µ uti ( slip ) − ( uip − u jp n j ni )  δ δ p

bf

p

(28)

µ  −u p + utfi ( slip ) + u jp n j ni  = δp  i f

The system of algebraic equations formed by the discretized transport equations and the boundary conditions is solved by an iterative procedure based on SIMPLE method [22]. For more details see [23].

3

Validation and case studies

The analytical solution of the Navier-Stokes equations with slip at walls is possible for very simple domains, and with severe restrictions in the flow behavior (mathematical model). One such case is Poiseuille flow in a channel with slip. So, in order to validate the code (and to establish its accuracy) numerical predictions are carried out for

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II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008

this flow, but prior to this the corresponding analytical solution is derived next. Then, two more complex cases are numerically investigated to assess the code robustness. 3.1 Case Study 1 - Validation agains the analytical solution for Poiseuille flow in a channel with wall slip For fully-developed channel flow of a Newtonian fluid the Navier-Stokes equations become,   ∂u ∂v  ∂u  τ xx = µ  2    ∂x + ∂y = 0  ∂x        Du  ∂v  ∂P ∂τ xx ∂τ xy (29) =− + + τ yy = µ  2    ∂x ∂x ∂y    ∂y   Dt   Dv ∂P ∂τ yx ∂τ yy  ∂u ∂v   τ yx = τ yx = µ  +   =− + +    ∂y ∂x ∂y  Dt  ∂y ∂x    The flow is steady and unidirectional along the x direction, see Fig. 5, so the following applies. Du Dv ∂P ∂ = = 0 , v = 0, = 0, =0 (30) ∂y ∂x Dt  Dt       a

c

b

The final form of the x- momentum equation becomes, ∂  ∂u  ∂P = cte . µ  = ∂x  ∂y  ∂y

(31)

x2

wall

H x1

Fig.5. Flow profile with slip.

The solution of Eq. 31 can be obtained by imposing a mean velocity U (Eq. 32) and the slip boundary conditions for the three different slip models (Eqs. 33, 34, 35) (for more details see [23]), H

1 U= H

2

∫

−H

( )

u ( y ) dy ,

(32)

2

  us = k1τ t ⇒ us H = k1µ ∂u 2 ∂y y = H

k1 ∈ ℝ + ,

(33)

2

n

   τ    (34) us = −k2 τ t n  t ⇒ us H = k2  − µ ∂u k2 ∈ ℝ + , n ∈ ℝ + \ {1} , n  2 y ∂ H  y = τt  2       τ + k4  τ t  H = −k ln  µ ∂u 1 + 1 k , k ∈ ℝ + . us = −k3 ln  t u (35) ⇒   s 2 3  ∂y 4 3    k4  τ t y = H k4 2   The analytical solutions for the three models are given by Eqs. (36) - (38). For the last two slip models the solutions are actually “semi-analytic”, meaning that was necessary to use a numerical method to achieve the analytic solution since the parameters c1 * and c2 * must be numerically determined.

( )

( )

 H2  U  y 2 − k1 µ H −  4  u( y) =   H2  − k1 µ H  −  6 

6

k1 ∈ ℝ + ,

(36)

II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008

)

(

n 2 c * u ( y ) = 1  y 2 − H  + k2 − µ c1 * H 2  4  2 2 c *  − µ c2 * H  u ( y ) = 2  y 2 − H  + k3 ln  + 1 2  4   2 k4 

k2 ∈ ℝ + , c1* ∈ ℝ ,

(37)

k3 , k4 ,∈ ℝ + , c2 * ∈ ℝ .

(38)

Fig.6 (a) compares numerical predictions by the code in various meshes with the analytical solution of Eq. (36) pertaining to slip model 1 (for the other two models see Fig. 6, (b), (c)). In this simulation (model 1) the amount of slip was k=1e-7 (7 mm/s) and it was imposed a constant flow of 60 mm/s. The number designating the mesh (MN) refers to the number (N) of computational cells across half of the channel ( x2 direction). As shown, there is very good agreement between predictions and the analytical solution and the numerical results, even for coarse meshes. The data were normalized with the bulk velocity ( u0 ) and the centreplane-to-wall distance ( x0 ). us u0

analytic solution

(a)  Mesh   M2  M4   M8  M16   M32

tt = us -Relative error (e r %)   9,842   2,646  0,679   0,174  0,047  x2 x0

us u0

us u0

(b)

analytic solution

analytic solution M2 M4 M8 M16 M32

(c)

M2 M4 M8 M16 M32

 Mesh   M2  M4   M8  M16   M32

us -(e r %) 3,786 0,835 0, 084 0,107 0,100

tt -(e r %)   2,540  0,557   0, 056  0, 071   0, 060 

 Mesh   M2  M4   M8  M16   M32 x2 x0

us -(e r %) 0, 009 0, 001 0, 001 0,001 0,001

tt -(e r %)   9, 492  1, 444   1, 360  1,359   1,350  x2 x0

Fig.6. Analytic and numerical solution for different number of control volumes along the “ x2 ”direction. (a) model 1, k1 =1e-7, (b) model 2, k2 =1e-8; n=1,5 (c) model 3, k3 =1e-5, k4 =1e-3.

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II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008 3.2 Case study 2 – Sudden contraction Flow To study the effect of the slip on the vortex dynamics appearing in a sudden contraction flow, simulations were carried out in a 2:1 contraction channel with the dimensions of Fig.7. The flow had an outlet Reynolds number of 0,00024, the flow was not fully developed and the slip model used was model 1. The number of computational cells used was 100 along the x1 direction and 20 along the x2 direction.

wall

recirculation zone

x2

x1

wall

outlet

inlet

wall

streamlines

symmetry plan

Fig.7. Contraction channel.

This case study was employed to evaluate the robustness of the code in calculations where the slip velocity changes abruptly its direction, as is the case inside the recirculation shown in Fig. 9.

(a)

(b)

Fig.8. Vortex dimensions (a) small slip velocity with k=1E-7 (b) large slip velocity with k=1E-6. vs v0

us u0

(b)

(a)

x1

Fig.9. Slip velocity along the wall (a) “u”component (b) “v” component.

8

x2

II Conferência Nacional de Métodos Numéricos em Mecânica de Fluidos e Termodinâmica Universidade de Aveiro, 8-9 de Maio de 2008

As shown (Fig 8), the increase of the slip intensity promotes a change in the vortex dimensions with the vortex diminishing as slip increases. In Fig. 9 we can see the slip velocity along the wall in the recirculation zone. In this zone there is an abrupt change in the velocity direction, but the code is robust and does not diverge. 3.3 Case study 3 – Flow through an extrusion Die In the third test case the code was used to simulate the flow through a complex geometry representing an extrusion die used in polymer processing. We used 10 computational cells along the die thickness and imposed a constant flow rate of 15 mm/s. The results for different slip velocities can be seen in Fig.10. k = 0.5 E − 7 u s = 3.8 mm / s

k = 1.0 E − 7 u s = 6.8 mm / s

(a)

(b)

k = 0.5 E − 6 u s = 14.5 mm / s

k = 0.25 E − 6 u s = 12.3 mm / s

(d)

(c)

u ( mm / s )

0

5.8

11.6

17.4

23.3

29.1

33.0

Fig.10. Extrusion die velocity profiles for different friction coefficients.

As expected, for a constant flow rate, the increase of the slip intensity, promotes an increase of the wall slip velocity through the entire channel wall. It also can be seen that the velocity distribution at the exit of the extrusion die gets more homogeneous if the material slips.

4

Conclusion

In this work, the main issues concerning the implementation of the three different slip models in a finite volume based code, were discussed. The tests perform evidenced a good agreement between analytic solutions and the numerical predictions. The studies done with more complex models showed that the code was able to perform the calculation, providing results that were expected from the physical point of view.

Acknowledgements The authors gratefully acknowledge funding by FEDER via FCT, Fundação para a Ciência e Tecnologia, under the POCI 2010 (project POCI/EME/58657/2004) and Plurianual programs.

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