Pandelitsa Panaseti, Maria Philippou, Zacharias Kountouriotis,

Georgios Georgiou

Department of Mathematics and Statistics, University of Cyprus

VPF2015: Viscoplastic Fluids: From Theory to ApplicationBanff, 25-30 October, 2015

CONFINED VISCOPLASTIC FLOWS WITH

WALL SLIP

Wall slip

b

wu

( )u y

y

Navier slip

w wu Slip coefficient

wu

w

b

Slip length

Slip lengths of simple fluids are in the nanometer scale but they are much larger for complex flluids such as

wormlike micellar solutions, concentrated emulsions, foams or colloidal suspensions.

Wall slip occurs in many flows leading to interesting phenomena and instabilities. Viscoplastic materials, such as polymeric solutions, suspensions, and gels, are known to exhibit wall slip (Yilmazer and Kalyon, 1989; Barnes, 1995; Ballesta et al., 2012).

Slip velocity

Navier slip equation and its power-law generalization have been used in many studies of flows of yield stress fluids (Aktas et al. , JOR, 2014)

sw wu

Kalyon (2005) analyzed the apparent slip flows of Herschel-Bulkley fluids in various geometries assuming that the apparent slip layer consists solely of the binder and its thickness is independent of the flow rate and the nature of the flow mechanism.

With non-Newtonian binders, s1.

No slip when β=∞

Motivation

A.L. Vayssade et al., Dynamical role of slip continuities in confined flows, Phys. Rev. E 89, 052309 (2014).

Vayssade and co-workers (2014) imaged the motion of well characterized softy glassy suspensions in microchannels whose inner walls impose different slip velocities. They showed that flows in confined geometries are controlled by slip heterogeneities.

As the channel height decreases the flow ceases to be symmetric and slip heterogeneities effects become important.

Herschel-Bulkley fluids with n=1/2 Poiseuille flow with asymmetric Navier slip

FLOW

x

y

H

1 1 1w wu

2 2 2w wu

1G G2G

I. SLIDING II. LOWER YIELDING III. LOWER & UPPER YIELDING

1wu

2wu

1wu

2wu

1y

1wu

2wu

1y

2y

0

3 flow regimes

Increasing pressure gradient

Solutions for Bingham flow (n=1)

1 2

( )Ix

GHu y

1GG

2G

I. SLIDING II. LOWER YIELDING III. LOWER & UPPER YIELDING

1wu

2wu

1wu

2wu

1y

1wu

2wu

1y

2y

0

2 21 1 1 1

21 1 1

( ) , 02

( )

,2

wIIx

w

Gu y y y y y

u yG

u y y y H

2 21 1 1 1

21 1 1 2

2 22 2 2 2

( ) , 02

( ) ,2

( ) ( ) ,2

w

IIIx w

w

Gu y y y y y

Gu y u y y y y

Gu H y y y y y H

2 01 2 2 1 2 1 2 1 2

1

2 0 02

1 2 0

( ) 2 2( ) ,

1 2 2 /,

2(1 2 / )

B B B B B B B G G Gy GHH B GH

G GB B GH GH

2 0 022

1 2 0

1 2 2 /,

2(1 2 / )

B GHyG G

H B B GH GH

02

21 22 1 1 2

2 /

1 1 1 4( ) / ( )2

HG

B BB B B B

21

1 2

21 0 01 2 2 1 2 1 2 1 2

2 02

1 2 0

, 0

( ) 2 2( ) ,

1 2 2 /,

2(1 2 / )

w

BG G

B B

B B B B B B B G G GGH GH GH

B GHG G

B B GH

11

1

wwu

2 12

2 2

w ww

GHu

021

1

1GH

Critical pressure gradients

Lower wall shear stress

Slip velocities

Yield points

Velocity

, 1, 2ii

B iH

No slip when Βi=0

Asymmetric velocity profiles (B1=0.1, B2=1)

n=4/3

n=1 n=1/2

For n<1, velocity tends to become more symmetric as the pressure gradient is increased approaching the power-law-fluid limit.

1/

1/ 10

, 1, 2n

i ni

kB i

H

Lengths are scaled by H, stresses by τ0, the pressure gradient by τ0/H , and the velocity by H(τ0/k)1/n

0

n 100

,

k ,

γ 0

τ γ

* 2

1 2

By

B B * 1

2y

* 2

1 2

1 2

2(1 )

By

B B

Experimental data

A.L. Vayssade et al., Dynamical role of slip continuities in confined flows, Phys. Rev. E 89, 052309 (2014).

Asymmetry parameter

Slip velocity parameter

General result (independent of the slip law) for n<1 and high pressure gradients.

1/2

1 1/

n

SS Un

1 2

11

2S y y

2 1

2w w

S

u uU

Normalized velocity profiles

Can we reproduce the experimental results assuming that Navier slip occurs along both walls?

FLOW

x

y

H

1 1 1w wu

2 2 2w wu

Inability to reproduce the data

All combinations of B1 and B2 we tried could not reproduce both the asymmetry parameter and the velocity profile patterns.Possible explanations: (1) The Navier-slip assumption is not valid.(2) Flow in the experiments not fully developed.

Velocity overshoots similar to those observed in flow development.

Objectives

Systematic numerical investigation of the development of viscoplastic flow in both pipes and channels at low Reynolds numbersUse of alternative definitions of the development length Study the effect of wall slip using Navier’s slip lawInvestigation of viscoplastic flow in a channel with asymmetric slip – Comparisons with experiments

Outline1. Literature review2. Viscoplastic flow development in a pipe.

Different definitions of the development length. Effect of slip.

3. Preliminary results for flow in a channel with asymmetric slip

Development of Newtonian flow with no slip

Contradictory data in early literature and many different empirical correlations L/D=f(Re) with more emphasis on high Re Accurate results by Durst et al. (2005) for Re in [0,10000]

Development length L: most commonly defined as the length required for the cross-sectional maximum velocity to attain 99% of its fully developed value.

1/1.61.6 1.6/ (0.619) (0.0567 )L D Re

1/1.61.6 1.6/ (0.631) (0.0442 )L H Re

Pipes

Channels

Durst et al., ASME J. Fluids Eng. 127, 1154-1160 (2005); Ferrás et al., ASME J. Fluids Eng. 134, 104503 (2012);

UDRe

These empirical equations are not accurate for low Re.

0.6023 for Re=0

0.6286 for Re=0

1.21/1.81.8 1.8

1.5

1 3.15 / 0.28 // (0.631) (0.047 )

1 3.82 / 0.018 /

B Re BL H Re

B Re B

Correlation for channels (0<Re<100)

Ferrás et al., ASME J. Fluids Eng. 134, 104503 (2012); Kountouriotis et al. , submitted (2015)

Development of Newtonian flow with wall slip

Inclusion of slip is important in Newtonian flows in microchannels (industrial micro- and nanotechnologies) Ferrás et al. (2012) studied the development of plane Poiseuille flow with Navier slip

* *1

2w wBu

L varies nonmonotonically with B exhibiting a maximum. Slip supresses the velocity overshoots near the entrance.

Re=0, No slip Re=0, Slip (B=10)

HB

A wall development length is relevant

Centerline and wall development lengths

Lc: the length required for the centerline velocity to attain 99% of its fully developed value (continuous)Lw: the length required for the slip velocity to attain 101% of its fully developed value (dashed)

AXISYMMETRIC FLOW CHANNEL FLOW

In the presence of finite slip, the flow development in the channel is slower near the wall than at the midplane, i.e., Lw>Lc.

Kountouriotis et al. , submitted (2015)

Development of viscoplastic flow with no slip

Wilson and Taylor (1996) : to permit the flow development of a yield stress fluid the ideal models must be relaxed to permit some deformation of the unyielded material (they used the biviscosity model). Al Khatib and Wilson (2001) showed that as the biviscosity model approaches the ideal Bingham model the approach to parallel flows becomes infinitely delayed.

Wilson and Taylor , JNNFM 65, 165-176 (1996). Al Khatib and Wilson, JNNFM 100, 1-8 (2001).

In the case of a n ideal yield-stress fluid the flow is kinematically impossible (Wilson and Taylor, 1996).

Development of viscoplastic flow with no slip

Early works ignored the low Re case. It was incorrectly predicted that L=0 for Re=0. Based on the centerline velocity development, Vradis et al. (1993) reported that the velocity profiles develop faster with higher values of the yield stress, which “is to be expected given the increase of the core radius”.

Vradis et al., Int. J. Heat Mass Transfer 36, 543-522 (1993). Ookawara et al., J. Chem. Eng. Japan 33, 675-678 (2000)

The flow develops faster at the center of the plug region and slower at its boundary (Ookawara et al. (2001) .

Development of viscoplastic flow with no slip

Ookawara et al. (2000) provided a correlation in which L is independent of Bn at low Re.

Poole and Chhabra (2010) employed FLUENT (biviscosity model and Papanastasiou’s with a rather low M) to calculate L95 and showed that it varies nonmonotonically with Bn exhibiting a minimum.

Ookawara et al., J. Chem. Eng. Japan 33, 675-678 (2000); Poole and Chhabra, ASME J. Fluids Eng. 132, 034501 (2010).

L95 : the axial distance required for the velocity to reach 99% of the calculated maximum value at a radial location corresponding to 95% of the plug radius.

“Expanded” definition of the development length by Ookawara et al. (2000)

Governing equations

Continuity equation

Momentum equation

Papanastasiou equation

0 up u u τ

0

1 exp( )m

τ γ

, 0w w ru u

0

Bingham fluid

Papanastasiou model

Increasing m

For sufficiently large values of m, the Papanastasiou model provides a satisfactory approximation of the Bingham model, while the need of determining the yielded and unyielded regions in the flow field is eliminated.

Boundary conditions

NAVIER SLIP

0, 0rz ru SYMMETRY

0

0zz

r

p

u

0

z

r

u U

u

( )T γ u u

Dimensionless governing equations

Continuity equation

Momentum equation

Papanastasiou equation

* * 0 u

* * * * * * *1

2pRe u u τ

** *

*

1 exp( )

21

MBn

τ γ

* * *, 01

2w w rBu u Boundary conditions

NAVIER SLIP

* *0, 0rz ru SYMMETRY

* *

*

0

0

zz

r

p

u

*

*

1

0

z

r

u

u

Reynolds number

Bingham number

Growth number

Slip number

2 URRe

02 RBn

U

mUM

R

2 RB

Dimensionless numbers

Numerical Method - Convergence

Finite elements (u-v-p formulation) with standard biquadratic basis functions for the two velocity components and bilinear ones for the pressure field.

Galerkin forms of the continuity and the momentum equations.

Newton-Raphson iterative scheme with a convergence tolerance equal to 10-4.

Lmesh=20 (sufficient for Re<20); M= 106

Number of

elements

Number of unknowns

Size of smallest element

Mesh 1 4844 44605 0.019Mesh2 18102 165287 0.005Mesh 3 42581 387823 0.002

B Re Mesh 1 Mesh 2 Mesh 3

L L L 0

50.628570.69250

0.628550.69252

0.628540.69252

10000 05

0.628660.69262

0.628640.69264

0.628630.69265

1000 05

0.629490.69375

0.629470.69377

0.629470.69378

100 05

0.637840.70537

0.637820.70538

0.637830.70537

10 05

0.681850.78019

0.681850.78021

0.681850.78021

1 05

0.556500.67971

0.556490.67971

0.556490.67971

Denser meshes are needed as Bn is increased.

The convergence of the numerical results has been studied using both uniform and non-uniform meshes of different refinement.

Development length

The development length is a function of r. L is the length required for the velocity to attain -- 99% of its fully developed value if u(r)>1 -- 101% its fully developed value if u(r)<1.

NEWTONIAN FLOW Pipe Channel

Flow development slower near the wall. Flow development slower near the axis.

max max

Development length - Definitions

Bn=2, B=20

Fully-developed profile

PLUG YIELDED

Lw: the length required for the slip velocity to attain 101% of its fully developed value.

L95 : the axial distance required for the velocity to reach 99% of the calculated maximum value at a radial location corresponding to 95% of the plug radius.

Lg: global development length (the length required for the velocity to attain 99% of its fully developed value if the latter is >1 (101% otherwise)

Lm: the length at the first local maximum near the yield point.

Lc: the length required for the centerline velocity to attain 99% of its fully developed value.

Effect of Bn (no slip)

Bn=0 (Newtonian) Bn=2 Bn=10 Bn=50

Velocity overshoots diminish with increasing Bingham number (note the y axis scale differences)

L95 and Lc are well below the other three development lengths.

Variation of L with the Bn (no slip)

“The flow develops faster as Bn is increased. “

Non-monotonic

Monotonic increase

Axial velocity, 0.1:0.2:2

L95 and Lc are not good choices for measuring viscoplastic flow development.

Comparison with Poole and Chhabra (2010)

Poole and Chhabra, ASME J. Fluids Eng. 132, 034501 (2010).

Effect of Slip (Bn=2)

B= (No slip) B=100 B=4 B=2.5

Sliding flow for B<Bn(note the y axis scale difference)

Lw is similar to and eventually coincides with Lg.

Effect of slip on the development length (Bn=2)

Non-monotonic

SLIDING REGIME NO SLIP

2 RB

Axial velocity, 0.1:0.1:2

Monotonic

L95 and Lc are well below Lg , Lm , and Lw .

Flow development with asymmetric slip

B=0 (No slip) B=1 B=0.1

Bn=1, n=1, Re=0, No slip at the lower plate

Axial velocity, 0.1:0.2:2

Full slip

Conclusions

Future workExtend caclulations to Herschel-Bulkley fluids (n=1/2). Study the effect of Re (in the low Re regime) for both the axisymmetric and planar geometriesComplete the analysis of channel flow with asymmetric slip Comparisons with experiments using different power-law slip models at the two walls.

We have investigated the creeping flow development of yield stress fluids in pipes with wall slipAlternative definitions of the development length have been proposed.L95 and Lc are not good choices for measuring viscoplastic flow development (with or without slip).The development length is monotonically increasing with the Bingham number The development length increases initially and then decreases with wall slipThe velocity overshoots are suppressed by both slip and yield stressPreliminary results of viscoplastic flow in a channel with slip heterogeneities have been presented