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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
Motivation and Previous work
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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.
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 →
Motivation and Previous work
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
Poole, Alves and Oliveira, accepted in Physical Review Letters (2007)
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
1
q2
q1
q2 Q
Q
Q
Q
qqDQ 12
12
12−
=+−
=
DQ=0 → symmetric
DQ=±1 → completely asymmetric
Flow asymmetry:
Motivation and Previous work
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
( ) ( ) ( )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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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;
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
Computational domain, boundary conditions
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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;
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
Oldroyd-B Results – ββββ effect
• Streamlines (creeping flow and ββββ=1/9)
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
Oldroyd-B Results – Inertia effect
• Effect of increasing Reynolds number (ββββ=1/9)
Increase of critical DeCR
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
Increase of critical DeCR
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.
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
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
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007
Acknowledgments
• Fundação para a Ciência e Tecnologia:
•PROJECT: BD/28288/2007
• Fundação Luso-Americana:
•PROJECT: 544/2007
• The Society of Rheology:
•Student Travel Grant
R. J. Poole, M.A. Alves, A. Afonso, F.T. Pinho and P.J. Oliveira, 79th SoR Meeting, 2007