A Stabilized Finite Element Method to Pseudop Flow Governed by the Sisko Relation

7/30/2019 A Stabilized Finite Element Method to Pseudop Flow Governed by the Sisko Relation

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Volume 31, N. 1, pp. 1935, 2012

Copyright 2012 SBMAC

ISSN 0101-8205 / ISSN 1807-0302 (Online)

www.scielo.br/cam

A stabilized nite element method to pseudoplastic

fow governed by the Sisko relation

MARCIO ANTNIO BORTOLOTI1 and JOS KARAM FILHO2

1Departamento de Cincias Exatas, Universidade Estadual do Sudoeste da Bahia, UESB,

Estrada do Bem-Querer, Km 4, 45083-900 Vitria da Conquista, BA, Brazil2Coordenao de Mecnica Computacional, Laboratrio Nacional de Computao Cientca,

LNCC, Av. Getlio Vargas, 333, 25651-070 Petrpolis, RJ, Brazil

E-mails: [email protected] / [email protected]

Abstract. In this work, a consistent stabilized mixed nite element ormulation or incom-

pressible pseudoplastic fuid fows governed by the Sisko constitutive equation is mathematically

analysed. This ormulation is constructed by adding least-squares o the governing equations and

o the incompressibility constraint, with discontinuous pressure approximations, allowing the use

o same order interpolations or the velocity and the pressure. Numerical results are presented to

conrm the mathematical stability analysis.

Mathematical subject classication: Primary: 65M60; Secondary: 65N30.

Key words: nite element method, numerical analysis, Sisko relation, non-Newtonian fow.

1 Introduction

In modeling some kinds o fuids, the Sisko constitutive relation comes out

rom the Cross model [6], when the apparent viscosity lies in a range between

the pseudoplastic region and the lower Newtonian plateau. A good alterna-

tive constitutive equation o the Sisko type or blood fow has been proposed

by [13], or example.

#CAM-240/10. Received: 10/VIII/10. Accepted: 10/XI/11.*Corresponding author


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20 STABILIZED FEM TO PSEUDOPLASTIC FLOW

The nonlinearity o this relation together with the incompressibility constraint

may generate numerical instabilities when some classical numerical methods

are used. For classical methods, in case o velocity and pressure ormulations,

it is well known that, even or the linear case, dierent interpolation orders or

these variables have to be used in order to satisy the Babuka-Brezzi stability

condition [2, 4].

In this work, to avoid the use o penalization methods or reduced integration

and to recover the stability and accuracy o the solution o same interpolation

orders in primitive variables, a consistent mixed stabilized nite element ormu-

lation is presented. It is constructed by adding the least-squares o the governing

equations and the incompressibility constraint, with continuous velocity anddiscontinuous pressure interpolations. The present ormulation is here mathem-

atically analysed based on Scheurers theorem, [12]. Stability conditions and

error estimates are established when the Sisko relation is considered. Numerical

examples are presented to conrm the stability analysis.

2 Denition o the problem

Let be a bounded domain in Rn where the positive integer n, denotes the

space dimension. We consider the stationary incompressible creep fow problem

governed by div = in , where : Rn Rn denotes the Cauchystress tensor or the fuid and denotes the body orces.

The governing equation, written above, is subjected to the incompressibility

constraint div u = 0 in , where u denotes the velocity eld.The Sisko model is characterized by a linear supersposition between the

Newtonian and the pseudoplastic eects, presenting a dependence o the viscos-ity with the shear-strain rate tensor, dening an apparent viscosity (|(u)|),leading to the stress tensor o the orm

= pI + (|(u)|)(u) with (|(u)|) = 1 + 2(|(u)|)and (s) = s2, where 1, 2 are two positive constitutive constants, ]1, 2[is the power index, p is the hydrostatic pressure, I Rn Rn is the identity

tensor, | | denotes the Euclidean tensor norm and(u) = 1

2

u + TuComp. Appl. Math., Vol. 31, N. 1, 2012


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MARCIO ANTNIO BORTOLOTI and JOS KARAM FILHO 21

is the symmetric part o the gradient ou.

With the above considerations, together with boundary condition o Dirichlet

type, the resulting problem is: nd (u, p)

C2()

C1() such that

div ((|(u)|)(u)) + p = in

div u = 0 in u = u on

(1)

where denotes the boundary o.

Physically, pseudoplastic fows are characterized by a viscosity decreasing

continuously and smoothly with increasing o shear rate, and this behaviour

occurs in a limited range o shear rate, generating the viscosity plateaus, wherewe can see that the apparent viscosity (s) is a bounded continuous unction

such that

(s) 0 (2)or 0 < g0 s g with g0 and g being the shear rate nite limits and0 and corresponding to the nite limiting Newtonian plateaus or low and

high shear rate, respectively, [3].

It can be seen that rom continuous classical Galerkin ormulation associatedwith problem (1), we can obtain

(u)0 C

0 (3)

or all u W1,20 (), where C is the constant in Poincars inequality.

3 Petrov-Galerkin-Like ormulation

To generate the stabilized nite element method proposed here, the ollowing

denitions will be used.

Let Lp() = {u|u is measurable,

|u(x)|pd < } be the class o allmeasurable unctions u, such that, u is p-integrable in and let Lp0 () = {u Lp(),

u d = 0} be the class o unctions in Lp() such that u has null

mean. Let Wm,p() be the Sobolev space Wm,p() = {u Lp()|D u Lp(), 0

||

m}

with

D u = ||u

1x1 mxmComp. Appl. Math., Vol. 31, N. 1, 2012


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where i is a natural integer and || = 1 + + m . The Wm,p0 () is denedas the space o unctions u Wm,p() such that D u = 0 on , or all with

|

| m

1, [1].

The norm in the space Wm,p() is dened as

um,p =

||mD up

1/p

, 1 p < ,

where up is the Lp() norm dened as

up = |u(x)|pd x1/p

, x .In this paper, we will denote u1 = u1,2 and u0 = u2.

We assume or simplicity Rn, a polygonal domain discretized by aclassical uniorm mesh o nite elements with Ne elements, such that

=Nee=1

e

, ei ej = or all i = j

where e denotes the interior o the et h element and e is its closure.Let Skh () be the nite element space o the Lagrangean continuous poly-

nomials in o degree k and Qlh () the nite element space o the Lagrangean

discontinuous polynomials in o degree l. Thus we can dene the approxima-

tion spaces Vh = (Skh () W1,20 ())n and Wh = Qlh() L2() to velocityand pressure respectively, that can be generated by triangles or quadrilaterals.

Remark 3.1. For the Galerkin method, k and l must be dierent orders even ork 2 and one can ollow [7, 8] and [9], or example, to see the limitations orthe combinations ok and l.

Remark 3.2. With the present consistent stabilized ormulation, all the combi-

nations o dierent orders are possible, optimal and suboptimal, but it is possible

to use same orders ork and l, with complete polynomials, providing k 2, asollows.

In this work, to obtain velocity and pressure approximations to problem (1),

we dene the ollowing variational orm, constructed by adding the least squares

Comp. Appl. Math., Vol. 31, N. 1, 2012


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o the linear momentum and o the continuity equations to the Galekin ormula-

tion, generating the ollowing problem with homogeneous boundary condition

considered without lost o generalities:

Problem PGhd. Given W1,2(),thedualoW1,2(),nd Uh VhWh,such that

(Ah(Uh ), Vh) + Bh(ph , vh) = Fh(Vh ) Vh Vh WhBh(qh , uh) = 0 qh Wh

where

(Ah (Uh ), Vh ) = ((|(uh)|)(uh),(vh)) + 2(div uh, div vh)

+ 1h2

(uh + ph, vh + qh)h ,

(4)

Bh(ph , vh) = (ph , div vh), (5)

Fh(Vh )=

(vh)+

1h2

(,

vh

+ qh )h, (6)

with h denoting the mesh parameter, (|(uh)|) the apparent viscosity,

(u, v) =

u v d x, (u, v)h =Nee=1

e

u v d x,

1 and 2 being positive constants denoted as stability parameters, uh =div ((|(uh)|)(uh)), Uh = {uh, ph}, Vh = {vh, qh} and being a dimensional

parameter. We can note that when 1 = 2 = 0, Problem PGhd reduces tothe Galerkin ormulation which, or interpolations o same order, is unstable

exhibiting spurious pressure modes or presenting the locking o the velocity

eld, [9]. The nonlinear Problem PGhd preserves the good properties o the

linear analogous o [11] in the sense that it accommodates, or k 2 equal-order interpolations or velocity and pressure as will be shown in the ollowing

analysis and conrmed later by the obtained numerical results.

This ormulation is consistent, being easy to veriy that the exact solution oproblem dened in (1), U = {u, p}, satises the PGhd problem.



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4 Finite element analysis

The nite element analysis is developed here by considering solutions in Hilbert

Spaces, as in [10]. We start the analysis by rewriting the discontinuous pres-sure approximation ph as

ph = ph + ph , (7)with ph Wh and ph Wh where Wh = {ph Wh L20(e), peh =ph} is the subspace o the pressure with zero mean at the element level andWh = {ph Wh; peh = 0, peh =

e

peh de/

ede} is the subspace o

the piecewise constant pressure, where peh represents the constraint o ph in

element e.The discontinuous pressure allows the satisaction o the incompressibility

constraint at element level in contrast to the continuous approximations, which

satises the constraint only in global sense. Considering this segregation, the

Problem PGhd can be rewritten as the ollowing variational orm, which consid-

ers the pressure variable ph written as unctions o ph Wh() and ph Wh ,as was described above:

Problem PGhd. Given W1,2(), nd {uh, ph , ph} Vh Wh Wh,such that

(Ah(Uh ), V

h ) + Bh (ph, vh) = Fh (Vh ) Vh Vh Wh

Bh (q h , uh) = 0 qh Wh

where

(Ah(Uh ), V

h ) = ((|(uh)|)(uh),(vh)) + Bh (ph , vh) + Bh(qh , uh)

+ 2(div uh, div vh) +1h

2

(uh + ph , vh + qh )h ,

(8)

Bh (ph , vh) = (ph , div vh), (9)

Bh(ph , vh) = (ph , div vh), (10)

Fh (Vh ) = (vh) +1h

2

(, vh + qh )h, (11)Uh = {uh , ph} and Vh = {vh , qh }.



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Lemma 4.1. For(s) bounded, continuous and smooth real unction such that

|d(s)/ds| M, there exists a positive constantC, independent oh, such thath

uh

0h

C

(uh)

0 where

u

20h

=(u, u)h .

Proo. From the inverse estimate

hdiv (uh)0h Ch(uh)0, (12)

typical o nite element methods, [5], 0 < (s) 0 and |d(s)/ds| M, we have the inverse estimate proposed, with C = 0Ch + M.

Lemma 4.2. Assuming the same considerations o Lemma 4.1, there is apositive constant Cl, independent oh, such that, h uh + vh0h Cl(uh) (vh)0 or alluh, vh Vh with h > 0.

Proo. From the triangular inequality, the Lemma 4.1 and (12) we have

h2 uh + vh20h 4[(20C2h + M2)(uh vh)20

+(1

+C2h )

(vh)

20

(uh)

(vh)

20h

].

The mean value theorem yields

h2 uh + vh20h 4

(20C2h + M2)(uh) (vh)20

+ (1 + C2h )(vh)20 sup01

(|((1 )uh + vh)|)20huh vh20h

.

Since |d(s)/ds| M or all uh Vh and rom Korns inequality, we canconclude the result, with the constant Cl given by

Cl = 4

(20C2h + M2) + (1 + C2h )M2C2K sup

vhVh(vh)20

where CK is the constant o the Korns inequality. The lemma is obtained as a

consequence o

(uh )0 C(1, 2, , C, 1, )0 or all uh Vh, (13)

that is, naturally, obtained rom Problem PGhd and consequently yields Cl as a

nite constant, where C is the constant o the Poincar inequality.



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Denition 4.3. Let|Uh| = Uh + h(div (uh)0h + (uh)0h + ph0h)be a mesh-dependent norm on the product space H10 () L2(), where hdenotes the mesh parameter and

Uh

2

= uh

21+

ph

20 is the norm dened in

Vh Wh.

Lemma 4.4 (Equivalence o the norms). There exists a positive constant

such thatUh |Uh| Uh or allUh Vh.

Proo. The inequality Uh |Uh| is immediate. In other hand, rom thedenition o|Uh|, rom the inverse estimate (12) and rom the inverse estimate

or the pressure, see [5],

hph0h Cpph0,

we have |Uh| Uh + (Ch + 1)(uh)0 + Cpph0. Using the classicalinequality,

1ndiv u0 (u)0 u1, (14)

we complete the proo o Lemma 4.4, with =

1+

max{

Ch +

1, Cp}

.

With the above results we can establish the ollowing result that will be needed

later to generate the estimates in Theorem 4.9.

Theorem 4.5. There exists a positive constant c such that

|(Ah(U) Ah(Vh), Uh Vh )| c|U Vh | Uh Vh

or allU W1,2

0 () L2

() andUh , Vh Vh Wh.Proo. By the consistency o the problem PGhd and rom (4) we have

|(Ah (U) A(Vh ), Uh Vh )| 1|((u vh),(uh vh))|

+ 2|((|(u)|)(u) (|(vh)|)(vh),(uh vh))| + |Bh (ph qh , u vh)|

+ 2 |(div u div vh, div uh div vh)| + |Bh (p qh , uh vh)|

+ 1h2

u + vh + (p qh ), uh + vh + (ph qh )h



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From the continuity o the two rst terms in the right hand side above, [12], and

using the inequality (14) we have

|(Ah(U) Ah(Vh ), Uh Vh )| [(1 + 2 + n2)u vh1

+np qh 0]uh vh1 +1h

(|(u)|)div (u) (|(vh)|)div (vh)0h+p qh 0h + (u)(u) (vh)(vh)0h

uh vh1 + ph qh 0.By using Lemma 4.2 and identiying the | | norm, we can conclude

|(Ah (U) Ah(Vh ), Uh Vh )| c|U Vh | Uh Vh ,

where c = max{1 + 2 + n2,

n, 1

}.

Theorem 4.6. Let Kh = {vh Vh, Bh (q h , vh) = 0, or all qh Wh}. Then,

there exists a positive constant e such that (Ah (Uh ) Ah(Vh ), Uh Vh ) eUh Vh 2 or allUh , Vh Kh Wh .

Proo. From (8) we obtain

(Ah(Uh ) Ah(Vh ), Uh Vh ) 1(uh vh)20

+ 2div uh div vh20h + 2Bh (ph qh , uh vh)

+ 1h2

uh + vh + (ph qh )20h

+ 2

(|(uh)|)(uh) (|(vh)|)(vh),(uh vh)

.

From the ellipticity o the last term in the right hand side above, presented in

[12], and using the Young inequality with = 12

we have

(Ah(Uh ) Ah (Vh ), Uh Vh ) ( 2 + 1)(uh) (vh)20

+ 1h2

uh + vh + (ph qh )20h

1

2ph qh 20.



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Applying again the Young inequality it yields

(Ah(Uh ) Ah (Vh ), Uh Vh ) ( 2 + 1)(uh) (vh)20

12

ph qh 20 +1h

2

(1 1

) uh + vh20h

+ 1h2

(1 )ph qh 20h,

with > 0. From Lemma 4.2 and considering r1 and r2 as two positive con-

stants, such that 1r1

+ 1r2

= 1, we can rewrite the previous inequality as

(Ah (Uh ) Ah(Vh ), Uh Vh ) 2 + 1r1

(uh) (vh)20

+ h2

2 + 1r2Cl

+ 1

1 1

uh + vh20h

+ 1h2

(1 )ph qh 20h

1

2ph qh 20.

Choosing =

1r2Cl

( 2+1)+1r2Cl), we obtain

(Ah (Uh ) Ah(Vh ), Uh Vh )

2 + 1r1

(uh) (vh)20

12

ph qh 20 +1( 2 + 1)h2

(2 + 1) + 1r2Cl)(ph qh )20h .

By using the inequality

h2

qh

20h

qh

20, (15)

as in [10], and the Korns inequality, we have

(Ah(Uh ) Ah (Vh ), Uh Vh ) e(uh vh21 + ph qh 20)

with

e = min

( 2 + 1)Cr1

,1( 2 + 1)

(2 + 1) + 1r2Cl 1

2

,

since1( 2

+1)

(2 + 1) + 1r2Cl 1

2> 0. (16)



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The inequality (16) gives a sucient condition, providing a useul relation to

be used to choose the stabilizing parameters.

Theorem 4.7. There exists a positive constant B such that

Bh (p ph , uh vh) Bp ph0Uh Vh

or all p L2(), Uh , Vh Vh Wh.

Proo. This result comes rom the application o the Hlder-Schwarz inequal-

ity and by the use o (14).

Theorem 4.8. For k 2 and since Vh Vh Wh, there exists a positiveconstanth, independent oh, such that

supvhVh

|Bh (qh , vh)|vh1

hq h0 or all qh Wh.

Proo. This result may be seen in [7].

With the above results, we can establish the ollowing error approximation

estimates.

Theorem 4.9. There exists a positive constant h, independent oh, such that

the ollowing estimate holds |U Uh| h|U Vh|.

Proo. From the denition o (Ah (), ) and the consistency o the ormula-tion we can write

Ah (Uh ) Ah (Vh ), Uh Vh = Ah (U) Ah (Vh ), Uh Vh + Ah (Uh ) Ah (U), Uh Vh (17)

with Ah(U

h ) Ah (U), Uh Vh = Bhp q h , uh vh. (18)

Replacing (18) in (17) we have

Ah(Uh ) Ah(Vh ), Uh Vh = Ah (U) Ah (Vh ), Uh Vh

+ Bhp q h, Uh Vh

.



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From Theorem 4.5, Theorem 4.6, Theorem 4.7 and considering the norm equiv-

alence between | | and established in Lemma 4.4, we have

|U Uh | 1 + ce |U Vh | + Be p q h0. (19)In order to obtain an estimate to p ph0, we note that rom PGhd problem,we have

Bhph , uh vh = Ah(U) Ah (Uh ), Uh Vh + Bhp, uh vh.

Since vh Kh , then

Bhph qh , uh vh = Ah (U) Ah(Uh ), Uh Vh

+ Bhp qh , uh vh

.

Using Theorem 4.5 and Theorem 4.7 we have

supwhVh

Bh (ph qh , wh)wh1 + ph qh 0

|U Vh | +p qh0.

By the use o Theorem 4.8, we can see that

p ph0 1

hU Vh h +

1 + 1h

p q h0. (20)

Combining (19) and (20) we have

|U Uh | h|U Vh |,

where

h = 1 +1

h +1

e max

c, B

,

since ph20 = ph20 + ph20. From Theorem 4.9, applying inverse estimates and the very classical interpo-

lation results presented in, or example, Chapter 3 o [5], we obtain the ollowing

error estimate

U Uh h(2 + Ch )c1hk|u|k+1 + h (1 + Cp)c2hl+1|p|l+1 (21)

with c1, c2 R, uh Skh () and ph Qlh () and | |m+1 the semi-norm othe Wm,2().



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5 Numerical results

In order to obtain numerical results or the nite element method presented here

to nonlinear problem, the ollowing numerical algorithm will be used. There aremany methods to solve nonlinear equations. In this case, we lag nonlinear terms

in the system o equations and start with an initial guess generating a sequence o

unctions that is expected to converge or the solution. In this sense, our scheme

is constructed by: given u0h Vh, lets nd (unh, pnh ) Vh Wh, n = 1, 2, 3, . . .such that

((

|(unh)

|)(un+1h ),(vh))

+Bh (p

n+1h , vh)

+Bh (qh, u

n+1h )

+1h2

(div ((unh)(un+1h )) + pn+1h , vh + qh)h

+2(div un+1h , div vh) = Fh (Vh ), Vh Vh Wh.

0

0.2

0.4

0.6

0.8

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

Horizontal Velocity

1 =0.1, 2 = 1.0, = 1.31 =1.0, 2 = 0.1, = 1.31 =1.0, 2 = 1.0, = 1.3

Newtonian Flow

Figure 1 Horizontal velocity at x = 0.5

The algorithm above, was applied to obtain numerical results or the clas-

sical driven cavity fow problem with bounbary conditions: u(x) = (1, 0) onx [0, 1] {1} and u(x) = (0, 0) on the other boundaries. A nite element



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(a) Velocity eld (b) Pressure eld

Figure 2 Results or1 = 1.0, 2 = 0.1 and = 1.3 using 1 = 1.0 and 2 = 10.0.







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0

5

10

15

20

25

30

1.31.41.51.61.71.81.92.0

Iterations

1 = 1 and 2 = 11 = 1 and 2 = 10

1 = 1 and 2 = 100

1

= 1 and 2

= 1000

Figure 5 Number o iterations with T ol 106 or 17 17 nodes with biquadraticelements in the case o various combinations between 1 and 2.

mesh o 17 17 nodes and 8 8 biquadratic quadrilateral elements has beenused. The numerical results were perormed using the ollowing stabilizingparameters: 1 = 1.0 and 2 = 10.0. For the convergence o the algorithm,we imposed a tolerence o 106. Numerical results are shown or some com-

binations o the constitutive parameters 1 and 2. In Figure 1 we can note

the characteristic o the pseudoplastic behavior comparing the velocity proles

on x = 0.5 or the i combinations presented. Velocity and pressure eldsare shown in Figures 2-4 or the same i combinations o those in Figure 1.

We can see, rom these results, how 1 and 2 control the Newtonian and the

pseudoplastic contributions respectively. We note magnitude decreasing in the

velocity and in the pressure elds and also the fatteness o the pressure next to

the two corners (0, 1) and (1, 1) due to the pseudoplastic eect, as expected.

Formulations that use the continuous pressure interpolations may present lack

o accuracy in those regions, where critical boundary conditions exist. Unlike,

discontinuous pressure interpolations, as is the case here, recover the accuracy at

those regions, due to the satisaction o the incompressibility constraint locally.The convergence behaviour o the algorithm used, together with the Petrov-



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Galerkin-like ormulation PGhd, is shown in Figure 5, as a unction o the

power index or our Sisko fuids. It can be seen that the greater is the non-

Newtonian eect, the larger is the number o iteractions required to achieve

convergence, as expected or a xed mesh. Note that even or a higher nonlin-

earity (higher power index ) convergence and stability are achieved.

6 Summary and conclusions

In this work, a consistent stabilized mixed Petrov-Galerkin-like nite element

ormulation in primitive variables, with continuous velocity and discontinuous

pressure interpolations, has been mathematically analyzed or fows governedby the nonlinear Sisko relation. Stability, convergence and error estimates have

been proven or same order interpolations o the primitive variables or any

combinations when k 2.To generate the mathematical stability conditions, it was possible to split the

discontinuous pressure. Only the constant by part pressure resulted as respon-

sible to ulll the LBB condition. The other part, the null mean pressure part,

contributed to achieve the required ellipticity in the Scheurers theorem sense

together with the stabilizing terms. For this ormulation ellipticity was the key

or the stability, since the constant part o the pressure ullls in standard ways

the LBB. It was possible rom the ellipticity to provide a sucient condition to

choose the stabilizing parameters not only as a unction o the quasi-newtonian

constitutive parameter but considering both constitutive constants coming rom

the Sisko relation.

Numerical results have been presented or the benchmark driven cavity fow

problem to conrm the mathematical analysis.

From the results, stability and convergence have been reached or several com-

binations o the constitutive parameters o the Sisko relation, ranging rom lower

to highly pseudoplastic (low and/or high 2) eects, although with more in-

teractions in the last case.

The use o discontinuous pressure interpolations ensured accuracy or the

pressure eld even in the regions where discontinuous boundary conditions

are present, since, now, the weakened internal constraint is satised locally, incontrast with continuous approximations.



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Documents

A Stabilized Finite Element Method to Pseudop Flow Governed by the Sisko Relation