Pandelitsa - PowerPoint Presentation

Slide1

Pandelitsa

Panaseti, Maria Philippou, Zacharias Kountouriotis, Georgios GeorgiouDepartment of Mathematics and Statistics, University of Cyprus

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

CONFINED

VISCOPLASTIC

FLOWS WITH WALL SLIPSlide2

Wall slip

Navier slip

Slip

coefficient

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)

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, s1.

No

slip when

β

=∞Slide3

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

3 flow regimes

Increasing pressure gradientSlide4

Solutions for Bingham flow (

n=1)

Critical pressure gradients

Lower wall shear stress

Slip velocities

Yield points

Velocity

No

slip

when

Β

i

=

0Slide5

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.

Lengths are scaled by

H

, stresses by

τ

0

, the pressure gradient by

τ

0/H , and the velocity by H(τ0/k)1/n Slide6

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.

Normalized velocity profiles

Can we reproduce the experimental results assuming that

Navier

slip occurs along both walls? Slide7

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.Slide8

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

Outline

Literature review

Viscoplastic

flow development in a pipe.

Different definitions of the development length.

Effect of slip.

Preliminary results for flow in a channel with asymmetric slipSlide9

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.

Pipes

Channels

Durst et al.,

ASME

J. Fluids Eng. 127, 1154-1160 (2005);

Ferrás

et al.,

ASME

J. Fluids Eng. 134, 104503 (2012);

These empirical equations are not accurate for low

Re

.

0.6023 for

Re

=00.6286 for Re=0Slide10

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

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

Re=0, No slipRe

=0, Slip (B=10)

A wall development length is relevantSlide11

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.,

L

w

>

L

c

.

Kountouriotis

et al. , submitted (2015)Slide12

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). Slide13

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) . Slide14

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

L

95

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) Slide15

Governing equations

Continuity equationMomentum equationPapanastasiou equation

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

condi

tions

NAVIER

SLIP

SYMMETRYSlide16

Dimensionless governing equations

Continuity equationMomentum equation

Papanastasiou equation

Boundary

condi

tions

NAVIER

SLIP

SYMMETRY

Reynolds number

Bingham number

Growth number

Slip number

Dimensionless numbersSlide17

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 elementsNumber of unknowns

Size of smallest element

Mesh 1

4844

44605

0.019

Mesh2

18102

165287

0.005

Mesh 3

42581

387823

0.002

B

Re

Mesh 1

Mesh 2

Mesh 3

L

L

L



0

5

0.62857

0.69250

0.62855

0.69252

0.62854

0.69252

10000

0

5

0.62866

0.69262

0.62864

0.69264

0.62863

0.69265

1000

0

5

0.62949

0.69375

0.62947

0.69377

0.62947

0.69378

100

0

5

0.63784

0.70537

0.63782

0.70538

0.63783

0.70537

10

0

5

0.68185

0.78019

0.68185

0.78021

0.68185

0.78021

1

0

5

0.55650

0.67971

0.55649

0.67971

0.55649

0.67971

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. Slide18

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

maxSlide19

Development length - Definitions

Bn=2, B=20Fully-developed profile

PLUGYIELDED

L

w

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

L

95

:

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.

L

g

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

L

m

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

c: the length required for the centerline velocity to attain 99% of its fully developed value.Slide20

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)

L

95

and

Lc

are well below the other three development lengths.Slide21

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

L

95

and

L

c

are not good choices for measuring

viscoplastic

flow development.Slide22

Comparison with Poole and

Chhabra (2010)Poole and

Chhabra, ASME J. Fluids Eng. 132, 034501 (2010).Slide23

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)

L

w

is similar to and eventually coincides with

L

g. Slide24

Effect of slip on the development length (

Bn=2)Non-monotonic SLIDING REGIMENO SLIP

Axial velocity, 0.1:0.1:2

Monotonic

L

95

and

L

c

are well below

L

g

,

L

m

, and Lw

.Slide25

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

Slide26

Conclusions

Future work

Extend caclulations

to Herschel-

Bulkley

fluids (

n

=1/2).

Study the effect of

Re

(in the low

Re

regime) for both the

axisymmetric

and planar geometries

Complete 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 slip

Alternative 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 stress

Preliminary results of viscoplastic flow in a channel with slip heterogeneities have been presented