Purely elastic instabilities in a cross -slot flowwebx.ubi.pt/~pjpo/SOR2007b.pdf · M. A. Alves Departamento de Engenharia Química, CEFT, Faculdade de Engenharia da Universidade

M. A. AlvesDepartamento de Engenharia Química, CEFT, Faculdade de Engenharia da

Universidade do Porto, Portugal, [email protected]

Purely elastic instabilities in a crossPurely elastic instabilities in a cross--slot flowslot flow

A. AfonsoDepartamento de Engenharia Química, CEFT, Faculdade de Engenharia da

Universidade do Porto, Portugal, [email protected]

F. T. PinhoEscola 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 2007Salt Lake City, USA

P. J. OliveiraDepartamento de Eng. Electromecânica, Universidade da Beira Interior,

Covilhã, Portugal, [email protected]

R. J. PooleDept. Engineering, Mechanical Engineering, University of Liverpool

Liverpool L69 3GH, UK, [email protected], [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 µm

500 µm

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=0.45




Successfully used a numerical technique with a simple

viscoelastic constitutive equation (UCM) to model this steady

asymmetry under creeping-flow conditions.

De = 0De = 0 .1De = 0 .2De = 0 .3De = 0 .31De = 0 .315De = 0 .32De = 0 .33De = 0 .34De = 0 .35De = 0 .36De = 0 .37De = 0 .38De = 0 .39De = 0 .40De = 0 .41De = 0 .42De = 0 .43De = 0 .44De = 0 .45

Streamlines →




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

Qq

1

q2

q1

q2 Q

Q

Q

Q

qq

qq

qqDQ 12

12

12−

=+−

=

DQ=0 → symmetric

DQ=±1 → completely asymmetric

Flow asymmetry:



Afonso, Alves, Pinho and Oliveira, XV IWNMNNF, Rhodes (2007)

Secondary instability in which the flow becomes unsteady and

fluctuates non-periodically in time.

UCM model

De=1.0

→Re=1.0

Governing equations


( ) ( ) ( )TDf

Dtλ − ∇ − ∇ =

Au A A u A

( ) ( ) ( )TDf

D tλ − ∇ − ∇ =

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 (ββββ=0)

Oldroyd-B (0<ββββ<1)

Phan-Thien and Tanner (0<ββββ<1), with

( )( )

1 tr 3(tr )

exp tr 3Y

εε

+ −= −

AA

A

(linear)

(exponential)

s s

o s P

η ηβ

η η η≡ =

+0=⋅∇ u

g


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

Qq

1

q2

q1

q2

Q

Q

Q

20D

D

20D

20D

20D

Q=UD

De=λU/D

Inlet Boundary Conditions:

Fully-developed u(y) and ττττ(y)

Outlet Boundary Conditions:

0=∂∂ yφ


Effect of mesh refinement

0.01

0.02

(∆xMIN)/D

7680612 801M1

30360650 601M2

DOFNC

x/D

y/D

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1


Effect of mesh refinement (Oldroyd-B, ββββ=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

0.01

0.02

(∆xMIN)/D

7680612 801M1

30360650 601M2

DOFNC


Oldroyd-B Results – ββββ effect

• Effect of increasing 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

ββββ=0ββββ=1/9

ββββ=2/9

ββββ=3/9

Increase of critical DeCR

For ββββ>3/9 flow became asymmetricunsteady;


Oldroyd-B Results – ββββ effect

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


Oldroyd-B Results – Inertia effect

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


Decrease in degree of asymmetry;

For Re>2, asymmetricunsteady 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 (streamlines)-REMOVE

•Re.vs.De map (ββββ=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

•Effect of varying ε ε ε ε parameter in PTT model (creeping flow

and ββββ=1/9)εεεε=0

εεεε=0.02εεεε=0.04 εεεε=0.06

εεεε=0.08


Decrease in degree of

asymmetry (εεεε<0.04);

Increase in degree of asymmetry and in maximum De

(εεεε>0.04);

For εεεε>0.08, asymmetricstable flow disappears.


PTT Results

• εεεε .vs.De stability map (Creeping flow and ββββ=1/9)

De

ε

1 2 30

0.05

0.1

Symmetric

Steady Asymmetric

Unsteady Asymmetric


PTT Results

• Streamlines (creeping flow, εεεε=0.02 and ββββ=1/9)

Conclusions

• Increasing the solvent viscosity (increasing β) increases the critical De. For β>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 ε) 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

• Fundação Luso-Americana:

•PROJECT: 544/2007

• The Society of Rheology:

•Student Travel Grant


Documents

Purely elastic instabilities in a cross -slot flowwebx.ubi.pt/~pjpo/SOR2007b.pdf · M. A. Alves Departamento de Engenharia Química, CEFT, Faculdade de Engenharia da Universidade