16 Asymptotic analysis of flow in wavy tubes and simulation of the extrusion process.pdf

MATHEMATICAL METHODS IN THE APPLIED SCIENCES

Math. Meth. Appl. Sci. 2007; 30 :889–909

Published online 17 January 2007 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.814

MOS subject classiﬁcation: 35 B 27; 35 Q 30; 76M45

Asymptotic analysis of ﬂow in wavy tubes and simulation

of the extrusion process

A. Ainser 1 , D. Dupuy 2 , ∗ , † , G. P. Panasenko 3 and I. Sirakov 1

1 Laboratoire de Rheologie des Matieres Plastiques

Universite Jean Monnet

rue du Dr Paul Michelon

France

2 Ecole Nationale Superieure des Mines de Saint-Etienne (ENSM-SE)

Saint-Etienne 42 023

Departement Methodes et Modeles

Mathematiques pour l’Industrie (3MI)

158

cours Fauriel

Saint-Etienne 42 023

Cedex 2

France

3 Laboratoire de Mathematiques de l’Universite de Saint-Etienne (LaMUSE

EA3989)

rue du Dr Paul

Michelon , Saint-Etienne 42 023 , France

Communicated by E. Sanchez-Palencia

SUMMARY

The paper is devoted to the mathematical modelling of an extrusion process. Usually, an extruder has

a very complicated geometry. This generates a lot of difﬁculties for computations of three-dimensional

ﬂows. In the present paper, we develop and justify the asymptotic domain decomposition strategy in order

to parallelize the computational process and reduce the memory. The error estimates are proved for the

Stokes steady-state equation in the two-dimensional and three-dimensional cases. Then, the asymptotic

domain decomposition procedure is applied for numerical testing and computations of the non-Newtonian

ﬂuid simulating a real process of the polymer extrusion. Copyright

2007 John Wiley & Sons, Ltd.

KEY WORDS :

asymptotic expansion; partial asymptotic decomposition of domain; extrusion process

1. INTRODUCTION

We are interested in studying the ﬂow of a polymer through an extruder. The extrusion is also

used in food-processing industry or in industrial polymer processing. The main advantage of this

process is its capacity to form and blend polymer. The objective of this paper is to propose a

numerical asymptotic method to simulate the ﬂow in a twin screw extruder. The problem is quite

difﬁcult for computation because of the complexity of the domain and of the rheological behaviour

∗ Correspondence to: D. Dupuy, Ecole Nationale Superieure des Mines de Saint-Etienne (ENSM-SE), Departement

Methodes et Modeles Mathematiques pour l’Industrie (3MI), 158, cours Fauriel, Saint-Etienne 42 023, Cedex 2,

France.

† E-mail: dupuy@emse.fr

Contract / grant sponsor: Region Rhone-Alpes; contract / grant numbers: 02 020611 01, 02 020613 01

2007 John Wiley & Sons, Ltd.

Received 9 October 2006

890

A. AINSER ET AL.

of the polymer. In fact, the polymers are described by complex differential or integral viscoelastic

constitutive models (e.g. power law, Carreau model, etc.).

The main idea of this paper is to develop the method of asymptotic partial decomposition

of domain (M.A.P.D.D.) for the ﬂow of an incompressible ﬂuid through a wavy tube structure.

The principle of this method is described in

. The study of a ﬂow governed by the Stokes

equations in a tube structure (i.e. a domain which is a ﬁnite union of tubes with a constant section)

is developed in

[

1, 2

]

. There the tubes had a cylindrical form and were not ‘wavy’.

In this paper, we consider a small parameter

[

]

1, which represents the thickness and at the

same time the period of the domain, and we study the behaviour of the solution when

is very

small with respect to the length of the channel taken equal to one (see

). The goal of the

M.A.P.D.D. is to consider the periodic solution far from a neighbourhood of the boundary layer

effects and to keep the initial model in the neighbourhoods of the ends.

The present paper consists of two parts. The ﬁrst part is theoretical. We consider a simpliﬁed

(Newtonian) model for the extrusion process and develop an asymptotic analysis of the problemwith

appropriate estimates of the error in the two-dimensional (Section 2) and three-dimensional (Section

3) cases. This asymptotic analysis is applied then for the asymptotic domain decomposition strategy,

that is to cut off the boundary layer zones and to reduce the problem to an ε -periodic ﬂow out of

these boundary layer zones. In the second part (Section 4), we provide the numerical implementation

of the M.A.P.D.D. for some more realistic complex non-Newtonian ﬂows, simulating extrusion or

blood circulation invessels.

[

]

2. FLOW IN A TWO-DIMENSIONAL WAVY TUBE STRUCTURE

In this section, we study the stationary ﬂow of an incompressible ﬂuid through a two-dimensional

wavy tube. First, we introduce the domain and the equations which describe the motion of the ﬂuid.

Then, we construct an asymptotic expansion of the solution and we calculate some estimates in

order to justify this approach. Finally, we apply the variational method of asymptotic decomposition

of domain to construct an approximated solution, called the partially decomposed solution.

2.1. Setting of the problem

For 0 i 2, let h i

2 ) be a function of class C

: R → ( 0 ,

( R ) such that there exists a constant m :

h i

Moreover, we assume that h 1 and h 2 are one-periodic functions.

From these functions, we deﬁne three cellular domains Y i

(

)

0 for all t

∈ R

2 ]

⊂[

]× (

h i

Y i ={ (

y 1 ,

y 2 ) ∈ R

y 1 <

1and

−

(

y 1 )<

y 2 <

(

y 1 ) }

(1)

h i

We denote by

={ (

y 1 , ±

(

y 1 )) :

y 1 <

}

the uppe r a nd the lo wer bo und ary of the domain

( 1

e 1 ) ∪ 0

∪ ( 2

Y i for 0

2, and we assume th at the curve given by

−

e 1 )

is Lipschitz

is the set of interior points.

2 ,

continuous. For any set

⊂ R

is the closure and

∈ N )

be a small parameter. We denote by G 0

Let

ε = (

)(

the domain obtained from Y 0

by a homothetic contraction:

G 0 ={ x = ε y : y ∈ Y 0 }

2007 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2007; 30 :889–909

DOI: 10.1002/mma

891

ASYMPTOTIC ANALYSIS OF FLOW IN WAVY TUBES

and we deﬁne two-periodic channels from the cellular domains Y 1 and Y 2 :

l = 1 ε( Y 1 − l e 1 )

G 1 =

l = 1 ε( Y 2 + l e 1 )

G 2 =

In this study, we consider the thin domain G ε depending on the small parameter ε :

G ε =[

G 2 ]

G 1 ∪

G 0 ∪

i =

ε ±

the lateral side of the domain G ε and

Denote by

G ε ∩{

x 1 =

} (

=±

)

be the upper and

G ε .

the lower part of the boundary

2 with vanishing divergence

and equal to zero on the boundaries ε ± . The space V ( G ε ) is the subspace of H div ( G ε ) of functions

vanishing on t he whole boundary G ε . We denote by W i the space of divergence free functions

Let H div (

G ε )

H 1

G ε ) ]

be the space of vector-valued functions from

[

(

v ∈ C per ( l ∈ Z ( Y i + l e 1 )) , where C per is the space of one periodic in y 1 functions of class C ∞ .

The closure of

H 1

W i with respect to the usual norm of

[

(

Y i ) ]

is denoted by W

(

Y i )

Let u ε be a function of H div (

G ε )

such that u ε (

) = u i (

/ε) (

)

. Moreover, we suppose

that there exists

∈ R

such that the following condition is satisﬁed:

h + ( 0 )

− h − ( 0 ) ( u 1 ) 1 (

h + ( 1 )

− h − ( 1 ) ( u 2 ) 1 (

y 2 )

d y 2 =

y 2 )

d y 2 =

(2)

where

(.) 1 stands for the ﬁrst component.

The steady-state ﬂow of an incompressible Newtonian ﬂuid through the two-dimensional domain

G ε

is governed by the following Stokes system:

⎨

⎩

v ε +∇

p ε =

in G ε

−

div v ε = 0

in G ε

(3)

v ε =

ε ± ,

v ε = ε

u ε

(

=±

)

Theorem 2.1

Assume that f ∈[ H − 1

( G ε ) ]

2 and u ε

is a function of H div ( G ε )

satisfying (2). Then, problem (3)

has a unique solution ( v ε , p ε ) in H div ( G ε ) × L 2

( G ε )/ R.

Proof

From assumption (2), the compatibility condition is veriﬁed and we use the Lax–Milgram theorem

to prove the existence and the uniqueness of the solution as in [ 5 ] .

2.2. Asymptotic solution

For simplicity, we suppose in the sequel that

(

x 1 ,

x 2 ) = (

f 1 (

x 1 ),

)

(4)

with f 1 a known function of L 2

( − 1 , 1 ) .

2007 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2007; 30 :889–909

DOI: 10.1002/mma

892

A. AINSER ET AL.

Let be a cut-off function ∈ C

( R ) and

⎨

⎩

0f t

( t ) =

(5)

1f t

and we denote

An asymptotic expansion is sought in the form

⎨

1 (

) = ( −

)

and

2 (

) = (

)

l = 1 v i x

i x 1

2 v bl x

v a ( x ) = ε

+ ε

(6)

q i ( x 1 ) + ε p i x

i x 1

+ ε p bl x

⎩

l = 1

p a ( x ) =

( v i , p i )

The terms

are periodic solutions of cell problems and q i

is solution of a differential

equation

represent the boundary layer effects. In this case, they

appear in three zones of the domain: near the extremities

( i =

)

. The terms

( v bl , p bl )

1 , and in the junction of

two-periodic tubes. Let us give some details of the construction of these functions.

Let

ε − 1

and

w i

, i

(

v i ,

p i ) = (

)

be the unique solution of the cell problem:

⎨

⎩

Find ( w i

, i

) ∈ W ( Y i ) × L 2

( Y i )/ R such that

= i e 1 in l ∈ Z ( Y i + l e 1 )

− w i

+∇ i

for i

2and

∈ R

ﬁxed by the condition

h i

( 0 )

w i

( 0 ) (

) 1 (

y 2 )

d y 2 =

− h i

with the rate of ﬂow deﬁned from (2).

We suppose that the terms q 1 and q 2 satisfy the following Darcy equations:

⎨

⎩

d q 1

d x 1 (

f 1 (

x 1 ) −

x 1 ) = 1 ,

x 1 ∈ ( −

)

d q 2

d x 1 ( x 1 ) = 2 ,

f 1 ( x 1 ) −

x 1 ∈ (ε, 1 )

The boundary layer term, denoted as ( v bl , p bl ) , has the following form:

⎨

v bl ( y ) = v bl − 1 y +

e 1

1 ( y 1 ) + v bl0 ( y ) + v bl1 y −

e 1

2 ( y 1 )

p bl − 1 y

e 1

p bl1 y

e 1

⎩

p bl (

) =

1 (

y 1 ) +

p bl0 (

) +

−

2 (

y 1 )

where y is the microscopic variable deﬁned by y = x /ε .

2007 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2007; 30 :889–909

DOI: 10.1002/mma

893

ASYMPTOTIC ANALYSIS OF FLOW IN WAVY TUBES

The boundary layer terms ( v bl − 1 , p bl − 1 ) and ( v bl1 , p bl1 ) are the solutions of the Stokes problem

set in a semi-inﬁnite wavy channel and the boundary layer in x 1 =

( v bl0 , p bl0 )

, satisﬁes the

Stokes problem in an inﬁnite wa vy channel.

Let us denote

1 =[ l = 1 ( Y 1 + l e 1 ) ] and

2 =[ l = 1 ( Y 2 − l e 1 ) ] and for i =

2, we deﬁne

={ y 1 > 0and y 2 =± h i

( y 1 )) } and i ( 0 ) = i ∩{ y 1 = 0 } . We consider the following problems:

⎨

− v bl − 1 +∇ p bl − 1 = 0in 1

div v bl − 1 = 0

in 1

(7)

⎩

1 ,

v bl − 1 =

v bl − 1 + v 1 = u on

1 (

)

and

⎧

⎨

− v bl1 +∇ p bl1 = 0n 2

div v bl1 = 0

in 2

(8)

⎩

2 ,

v bl1 =

v bl1 +

v 2 = u on

2 (

)

Theorem 2.2

Problems (7) and (8) admit a unique solution

in H div ( 1 ) × L loc ( 1 )/ R and

( v bl − 1 , p bl − 1 )

in H div ( 2 ) ×

L loc ( 2 )/ R

(

v bl1 ,

Moreover, these solutions are exponentially decaying: there exists c

p bl1 )

0and

0 such that for

all R

v bl − 1 [ H 1

c exp ( − R )

( 1 ∩{| y 1 | > R } ) ]

(9)

v bl1 [ H 1 ( 2 ∩{| y 1 | < R } ) ] 2

c exp

( − R )

Proof

The existence and uniqueness result can be found in [ 5, Theorem 2.1 p. 23 ] . The decay estimate

is a generalization of the properties proved in [ 6 ] .

The boundary layer in x 1 =

0 is a little bit more complicated. Indeed, this co nst ruction im pli es

the stu dy of the Stokes equation set in the wavy inﬁnite channel

={[ l = 1 ( Y 1 − l e 1 ) ]∪ Y 0 ∪

[ l = 1 ( Y 2 + l e 1 ) ]} . Hence, we consider the problem

⎧

⎨

⎩

−

v bl0 +∇

p bl0 =

F ε

div v bl0 =− ε

(10)

v bl0 =

lim

| y 1 |→∞

v bl0 (

y 1 ,

y 2 ) =

Functions F ε

ε are deﬁned via the periodic functions v i , p i , via q i and via the boundary

layer terms solution of (7) and (8) in such a way

and

F ε ( y ) =

f 1 (ε y 1 ) e 1 { 1 −[ ( − y 1 ) + ( y 1 ) ]} − f ε (ε y )

2007 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2007; 30 :889–909

DOI: 10.1002/mma

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