I N N F I N N F MM

Computational Rheology Isaac Newton Institute

Dynamics of Complex Fluids -10 Years on

Institute of non-Newtonian Fluid MechanicsInstitute of non-Newtonian Fluid MechanicsEPSRC Portfolio PartnershipEPSRC Portfolio Partnership

Juan P. AguayoHamid Tamaddon

Mike Webster

Schlumberger, UNAM-(Mexico), INNFM

Mike Webster

I N N F I N N F MM

Computational Rheology – Some Outstanding Challenges

To achieve highly elastic, high strain-rate/deformation rate solutions (polymer melts & polymer solutions)

To quantitatively predict pressure-drop, as well as flow field structures (vortices, stress distributions)

• To accurately represent transient flow evolution in complex flows

• To quantitatively predict multiple-scale response (multi-mode)

• To achieve compressible viscoelastic representations

I N N F I N N F MM

TRANSIENT & STEADYContraction Flows

EPTT Oldroyd

AxisymmetricPlanar Planar Axisymmetric

Fluid viscosity = 1.75Pa.s – 8:1 contraction, exit length

7.4mm

0

10

20

30

40

50

60

0 100 2 104 4 104 6 104 8 104 1 105 1.2 105 1.4 105

8:1 Axisymmetric contraction - Fluid viscosity 1.7Pa.s

Newtonian syrupBoger Fluid

Q (g

/s)

DP (Pa)

Pressure-drop vs flow-rate in contractions

Newtonian syrup

Boger fluid

0

2

4

6

8

10

12

0 100 5 104 1 105 1.5 105 2 105

20:1 Planar contraction - Fluid viscosity 1.7Pa.s

Newtonian SyrupBoger Fluid

Q (g

/s)

DP (Pa)

Fluid viscosity = 1.75Pa.s – 20:1 contraction, exit length

40mm

axisymmetric planar

Pressure drop (epd) vs. We, 4:1:4 axisymmetric

Szabo et al. with FENE-CRJ. Non-Newt. Fluid Mech. 72:73-86, 1997

epd

We

Schematic diagram for a) 4:1:4 contraction/expansion, b) 4:1 contraction

a)

1st 4th

2nd 3rd

Front-face Back-faceMid-plane

Quadrants

RuRu

Ru4

Symmetry line

Ru4

Ru4

b)

1st

2nd

End ofrounded-corner

Quadrants

RuRu

Rc

Symmetry line

RuRc =

4

3 Rc4

Fro

nt-f

ace

( P P ) ( P )

( P P ) ( P )fd en

fd en

Boger Boger

Newt Newt

P

Szabo et al.J. Non-Newt. Fluid Mech. 72:73-86, 1997

Rothstein and McKinleyJ. Non-Newt. Fluid Mech. 86:61-88, 1999J. Non-Newt. Fluid Mech. 98:33-63, 2001

Wapperom and KeuningsJ. Non-Newt. Fluid Mech. 97:267-281, 2001

downstreamupstreamfd PPP : Total pressure dropP

Excess pressure drop (epd -

P )

P

Pressure-drop (epd) vs. We, Oldroyd-B, a, c) axisymmetric, b, d) planar

c)4:1a)

d)b)

Axisymmetric

Planar

4:1:4 a)

b)

Oldroyd-B, =0.9

Pressure profile around constriction zone, 4:1:4 axisymmetric and planar case

++

++

+

+

+

+

++

++

+ +

P

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

0

5

10

15 [1] Axi, We = 1[2] Axi, We = 2[3] Axi, We = 3[4] Planar, We = 1[5] Planar, We = 2[6] Planar, We = 3

+

[1]

[2]

[3]

[4]

[5]

[6]

Oldroyd-B, =0.9

N1p 3D view – 4:1:4contraction/expansion

Axisymmetric

Planar

(P - PNewt) and stress profiles along wall, 4:1 and 4:1:4 axisymmetric case

Oldroyd-B, =0.9

4:1 4:1:4

11

-2 -1 0 1 2 3 4-10

0

10

20

30

40

50[1] We = 1[2] We = 2[3] We = 3

Fro

nt-f

ace

Mid

-pla

ne

Bac

k-fa

ce

4:1:4 Contraction/expansion, 11 at boundary wall

11

-2 -1 0 1 2 3 4-10

0

10

20

30

40

50

[1] We = 1[2] We = 2[3] We = 3

4:1 Contraction, 11 at boundary wall

Fro

nt-f

ace

End of rounded-corner

P-

PN

ewt

-2 -1 0 1 2 3 4-7

-6

-5

-4

-3

-2

-1

0

1

[1] We = 1[2] We = 2[3] We = 3Newtonian

4:1 Contraction, pressure at boundary wallF

ront

-fac

e

End of rounded-corner

[1]

[2]

[3]

Newt

P-

PN

ewt

-2 -1 0 1 2 3 4-1

0

1

2

3

4

5

6

7

[1] We = 1[2] We = 2[3] We = 3NewtonianF

ront

-fac

e

Mid

-pla

ne

Bac

k-fa

ce

4:1:4 Contraction/expansion,pressure at boundary wall

(P - PNewt) and stress profiles along wall – 4:1:4 planar and axisymmetric case

Oldroyd-B, =0.9

P-

PN

ewt

-2 -1 0 1 2 3 4-1

0

1

2

3

4

5

6

7

[1] We = 1[2] We = 2[3] We = 3Newtonian

Fro

nt-f

ace

Mid

-pla

ne

Bac

k-fa

ce

4:1:4 Axisymmetric contraction/expansion,pressure at boundary wall

[1][2][3]

Newt

11

-2 -1 0 1 2 3 4-10

0

10

20

30

40

50[1] We = 1[2] We = 2[3] We = 3

Fro

nt-f

ace

Mid

-pla

ne

Bac

k-fa

ce

4:1:4 Axisymmetric contraction/expansion,11 at boundary wall

P-

PN

ewt

-2 -1 0 1 2 3 4-1

0

1

2

3

4

5

6

7

[1] We = 1[2] We = 2[3] We = 3NewtonianF

ront

-fac

e

Mid

-pla

ne

Bac

k-fa

ce

4:1:4 Planar contraction/expansion,pressure at boundary wall

11

-2 -1 0 1 2 3 4-10

0

10

20

30

40

50

[1] We = 1[2] We = 2[3] We = 3

Fro

nt-f

ace

Mid

-pla

ne

Bac

k-fa

ce

4:1:4 Planar contraction/expansion,11 at boundary wall

Planar Axisymmetric

epd

epd

We

Pressure-drop (epd) vs. We, 4:1:4 axisymmetric, alternative models

We

(P - PNewt) profiles along wall – 4:1:4 axisymmetric, increasing

upturn epd

monotonic decrease epd

upturn & enhanced epd

P-

PN

ewt

-19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2-5

0

5

10

15

We = 0.5We = 1.5Newtonian

4:1:4 Contraction/expansion, pressure at boundary wall, = 1/9

P-

PN

ewt

-19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2-5

0

5

10

15

We = 1We = 2We = 3Newtonian

4:1:4 Contraction/expansion, pressure at boundary wall, = 0.9

P-

PN

ewt

-19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2-5

0

5

10

15

We = 1We = 2We = 5Newtonian

4:1:4 Contraction/expansion, pressure at boundary wall, = 0.99

Alternative differential pressure-drop measure

Boger Newt Boger Newt

Newt Newt

( P ) ( P ) [P P ]1

( P ) ( P )en en en

en en

P

Boger Newt[P P ] 0fd

0)P( Newt en

0]P[Pspup

entryNewt

Since & by calibration

Boger Newt Boger Newt

Newt Newt

[P P ] [P P ]1

( P ) ( P )en

en en

P

0]PP[ exitNewtBoger

1P

Rate of dissipation & pressure-drop, 4:1:4

D QP

inlet exit Boger[P P ] Q BogerD inle exit[P P ] Qt Newt NewtD

0

NewtNewt inlet Newt exit[P P ] [P P ]

Q

D D

rate of dissipationinlet exitP P P

Boger NewtP P 0fd

Newt 0entryup sp

P P

definition

NewtNewt entry Newt

constrictionup spone

[P P ] ( 1)( P )Q en

z

D D

P

Seeking {P – 1} > 0 Newt 0constrictionzone

D D

,

epd

epd

Pressure-drop (epd) vs. a) We, b) upstream sampling distance, 4:1:4 axisymmetric

We

= 0.99

= 1/9

We = 1

sal = 0.553sal = 0.552

We = 2

sal = 0.556 sal = 0.524

We = 5

sal = 0.562 sal = 0.656

We = 1

sal = 2.06 sal = 0.091

We = 0.1

sal = 0.707 sal = 0.507

We = 1.5

sal = 3.08 sal = 0.041

4:1:4 axisymmetric vortex cell size, Oldroyd-B, change

upturn & enhanced epd

mono-dec epd

=0.99

=1/9

a) Oldroyd-B extensional viscosity,

b) Shear and extensional viscosity,

c) Shear and extensional viscosity,

Rheological properties: Oldroyd-B, LPTT, EPTT, SXPP

NEW BOGER fluid modelling

& Pressure DropAxisymmetric contraction

Planar contraction

Centreline pressure gradient4:1:4 axisymmetric, Oldroyd-B

=0.99

=0.9

=1/9