R. J. Poole - web.fe.up.ptfpinho/pdfs/Aafonso_ SoR_2007.pdf · R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007 Numerical method - brief description

M. A. Alves Departamento de Engenharia Química, CEFT, Faculdade de Engenharia da

Universidade do Porto, Portugal, [email protected]

A. Afonso Departamento de Engenharia Química, CEFT, Faculdade de Engenharia da

Universidade do Porto, Portugal, [email protected]

F. T. Pinho Escola de Engenharia, Universidade do Minho, Portugal, [email protected]

CEFT, Faculdade de Engenharia Universidade do Porto, Portugal, [email protected]

The Society of Rheology 79th annual meeting, 7th to 11th October 2007 Salt Lake City, USA

P. J. Oliveira Departamento de Eng. Electromecânica, Universidade da Beira Interior,

Covilhã, Portugal, [email protected]

R. J. Poole Dept. Engineering, Mechanical Engineering, University of Liverpool

Liverpool L69 3GH, UK, [email protected]

Outline

R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007

•! Motivation and Previous work (UCM)

•! Governing equations/numerical method

•! Effect of solvent viscosity (Oldroyd-B) and inertia

•! Effect of extensional viscosity (PTT)

•! Conclusions

Motivation


Arratia et al, Physical Review Letters 96, 144502 (2006)

Microfluidic flow in a “cross channel” geometry

650 !m

500 !m

Newtonian:

Re < 10-2

PAA Boger fluid:

Re < 10-2 (De=4.5)

Motivation and Previous work


Poole, Alves and Oliveira, accepted in Physical Review Letters (2007)

Successfully used a numerical technique with a simple

viscoelastic constitutive equation (UCM) to model this steady

asymmetry under creeping flow conditions.

Newtonian De=0.4




Purely-elastic: inertia decreases the degree of asymmetry and

stabilizes the flow

De

DQ

0.25 0.3 0.35 0.4 0.45-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Re = 0

Re = 1

Re = 2

Re = 5

DQ = A (De - DeCR)0.5

Q q1

q2 q1

q2 Q

Q

Q

Q

qq

qq

qqDQ 12

12

12!

=+

!=

DQ=0 ! symmetric

DQ=±1 ! completely asymmetric

Flow asymmetry:




Purely-elastic: inertia decreases the degree of asymmetry and

stabilizes the flow

De

Re

0.3 0.4 0.50

1

2

3

4

Re-De Map !

Symmetric

Steady asymmetric

Unsteady asymmetric

Governing equations


( ) ( ) ( )TDf

Dt! " #$ % $ % =& '( )

Au A A u A

(Mass conservation)

(Momentum conservation)

(Constitutive equation, based on the conformation tensor, A)

•!Incompressible Viscoelastic fluid

( ),

( ) ( ),

(tr )( ),

f

Y

!"#

= !$#! !%

A I

A A I

A A I

•! Examples: Upper Convected Maxwell - UCM (!=0)

Oldroyd-B (0<!<1)

Phan-Thien and Tanner (0<!<1),

with

( )

( )

1 tr 3(tr )

exp tr 3Y

!

!

+ "#$= %

"& '$ ( )*

AA

A

(linear)

(exponential)

s s

o s P

! !"

! ! !# =

+

0=!" u

( ) ( ) Auuu

!"#+"+"!"+#"= $%

&$&' 1

oT

opDt

D


Numerical method - brief description

•! Finite Volume Method

Oliveira, Pinho and Pinto (1998)

Oliveira and Pinho (1999)

•! Structured, collocated and non-orthogonal meshes.

•! Discretization (formally 2nd order)

•! Diffusive terms: central differences (CDS)

•! Advective terms, high resolution scheme: CUBISTA

Alves, Pinho and Oliveira (2003)

•! Dependent variables evaluated at cell centers;

•! Special formulations for cell-face velocities and stresses;


Computational domain, boundary conditions

Q q1

q2 q1

q2

Q

Q

Q

20D

D

20D

20D

20D

Q=UD

De=!U/D

Inlet Boundary Conditions:

Fully-developed u(y) and "(y)

Outlet Boundary Conditions:

0=!! y"


Effect of mesh refinement

NC DOF (!xMIN)/D

M1 12 801 76806 0.02

M2 50 601 303606 0.01

x/D

y/D

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1


Effect of mesh refinement (Oldroyd-B, !=1/9, De=0.35 and Re=0)

* * * * * * * * * * ************************* * * * * * * * * * *

* * * * * ****

*

*

*

*

*

*

*

*** ******* ***

*

*

*

*

*

*

*

**** ** * * *

x/D

u/U

Txx

-1 -0.5 0 0.5 1

-1.5

-1

-0.5

0

0.5

1

1.5

-2.5

-2

-1.5

-1

-0.5

0

NC DOF (!xMIN)/D

M1 12 801 76806 0.02

M2 50 601 303606 0.01


Oldroyd-B Results – ! effect

•! Effect of increasing the solvent viscosity (creeping flow)

De

DQ

0.3 0.4 0.5 0.6 0.7-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

!=0

!=1/9

!=2/9

!=3/9

Increases the critical DeCR

For ">3/9 flow became asymmetric

unsteady;

Q q1

q2 q1

q2 Q

Q

Q

Q

qq

qq

qqDQ 12

12

12!

=+

!=

DQ=0 ! symmetric

DQ=±1 ! completely asymmetric

Flow asymmetry:


Oldroyd-B Results – ! effect

•! Streamlines (creeping flow and !=1/9)


Oldroyd-B Results – Inertia effect

•! Effect of increasing the Reynolds number (!=1/9)


Decreases the degree of

asymmetry;

For Re>2, asymmetric

unsteady flow.

De

DQ

0.3 0.4 0.5 0.6 0.7-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Re=0

Re=1

Re=2

Re=3

Re=4

DQ

0.71

0.72


Oldroyd-B Results – Inertia effect

•!Re.vs.De map (!=1/9)

De

Re

0.3 0.4 0.5 0.60

1

2

3

4

Symmetric

Steady asymmetric

Unsteady asymmetric

De

DQ

0.5 1 1.5 2 2.5 3-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1


PTT Results

#=0

#=0.02

#=0.04

#=0.06

#=0.08


Decreases the degree of

asymmetry (#<0.04);

Increases the degree of

asymmetry and in De extension (#>0.04);

For #>0.08, asymmetric

stable flow desappears .

•!Effect of varying # parameter in PTT model (creeping flow and !=1/9)

De

!

0.5 1 1.5 2 2.5 3

0

0.05

0.1


PTT Results

•! # .vs.De stability map (Creeping flow and !=1/9)

Symmetric

Steady asymmetric

Unsteady asymmetric


PTT Results

•! Streamlines (creeping flow, #=0.02 and !=1/9)

Conclusions


•! Increasing the solvent viscosity (increasing !) increases the critical De. For !>2/9 the first steady instability disappears and the flow becomes unsteady (but still asymmetric);

•! Increasing the level of inertia (increasing Re) shifts the onset of the instability to higher De and decreases the degree of asymmetry. Essentially inertia appears to stabilize the flow.

•!For the PTT model, decreasing the extensional viscosity (increasing #) increases the critical De.

•!We propose that this flow would make a good “benchmark” case for purely-elastic instabilities

Acknowledgments

•! Fundação para a Ciência e Tecnologia:

•!PROJECT: BD/28288/2007

•!PROJECT POCI/EME/59338

•! Fundação Luso-Americana:

•!PROJECT: 544/2007

•! The Society of Rheology:

•!Student Travel Grant


Documents

R. J. Poole - web.fe.up.ptfpinho/pdfs/Aafonso_ SoR_2007.pdf · R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007 Numerical method - brief description