13 Time-dependent tube flow of.pdf

Time-dependent tube ﬂow of compressible suspensions

subject to pressure dependent wall slip: Ramiﬁcations

on development of ﬂow instabilities

H. S. Tang a) and Dilhan M. Kalyon b)

Stevens Institute of Technology, Castle Point St., Hoboken, New Jersey 07030

(Received 1 August 2007; ﬁnal revision received 19 May 2008

Synopsis

A mathematical model developed earlier for the time-dependent circular tube ﬂow of compressible

polymer melts subject to pressure-dependent wall slip

Tang and Kalyon, J. Rheol 52 , 507–525

was applied to the tube ﬂow of polymeric suspensions with rigid particles. The model

relies on the apparent slip mechanism for suspension ﬂow with the additional caveat that the

polymeric binder slips at the wall according to a pressure-dependent wall slip condition. The

numerical simulations of the tube ﬂow of concentrated suspensions suggest that steady ﬂow is

generated when the ﬂow boundary condition at the wall is a contiguous strong slip condition along

the entire length of the tube wall. The ﬁndings of the simulations are consistent with the

experimental ﬂow curves and ﬂow instability data collected on suspensions of a poly dimethyl

siloxane , which itself exhibits wall slip, compounded with rigid and hollow spherical particles in

the 10–40% by volume range. Increasing the concentration of rigid particles gives rise to the

expansion of the range of ﬂow rates over which the ﬂow remains stable, as consistent with the

2008

DOI: 10.1122/1.2955508

I. INTRODUCTION

The development of ﬂow instabilities during the ﬂow of concentrated suspensions of

rigid particles incorporated into polymeric binders in simple capillary dies

cylindrical

reservoir connected to a converging entry region into a straight circular tube

can involve

time-dependent changes in the concentration of particles or changes in the shape of the

extrudates emerging from tube ﬂow. Time-dependent changes in the concentration of

particles generally occur on the basis of time-periodic oscillations in extrusion pressure,

upon the formation and break-up of mats of solids at the capillary entrance and the

ﬁltration of the binder phase

. The time-

dependent changes in the concentration of rigid particles of extrudates of concentrated

suspensions become especially signiﬁcant with decreasing binder shear viscosity

Yaras et al.

1994

; Rough et al.

2000

ﬁltra-

tion based ﬂow instabilities are especially prevalent with Newtonian binders

, increasing

particle size, increasing capillary convergence ratio

reservoir diameter over the tube

diameter

. On the other hand, with

increasing shear viscosity of the binder phase, concentrated suspensions exhibit time and

apparent shear rate-dependent changes in the shape of extrudates emerging from tube

, and decreasing ﬂow rate

Yilmazer et al.

1989

Currently with City College, City University of New York.

Author to whom correspondence should be addressed; electronic mail: dkalyon@stevens.edu

J. Rheol. 52 5 , 1069-1090 September/October

2008

0148-6055/2008/52 5 /1069/22/$27.00

1069

1070

H. S. TANG AND D. M. KALYON

ﬂow

akin to ﬂow instabilities associated with various types

of extrudate shape distortions observed in the tube ﬂow of polymeric melts

Birinci and Kalyon

2006

Benbow and

Lamb

1963

; Kalika and Denn

1987

; Denn

2001

; Robert et al.

2004

; Hatzikiriakos

and Migler

The effects of the presence and the concentration of rigid particles on the development

of shape distortions of the extrudates of suspensions are pronounced and contradictory.

For an elastomer that does not exhibit ﬂow instabilities over a wide range of shear rates

in capillary or rectangular slit ﬂows, the incorporation of particles induced extrudate

shape distortions over the same apparent shear rate range

2005

.On

the other hand, for other polymer melts that exhibit instabilities upon ﬂow through cy-

lindrical dies, the incorporation of particles reduces the range of apparent shear rates over

which ﬂow instabilities are observed at relatively low particle concentrations

Tang and Kalyon

2004a

Rosen-

baum et al.

2000

; Birinci and Kalyon

2006

and eliminates them altogether at rela-

tively high particle concentrations

. Currently, there is no

explanation for this type of contradictory behavior involving the induction or elimination

of ﬂow instabilities upon the incorporation of particles into polymeric melts.

The slip-based theoretical treatment of the development of ﬂow instabilities of poly-

mer melts in simple channels has generally relied on the considerations of the compress-

ibility and various empirical nonmonotonic wall slip velocity versus wall shear stress

expressions

Birinci and Kalyon

2006

Hatzikiriakos and Dealy

1992a

; Den Doelder et al.

1998

; Georgiou

. For polymeric liquids the pressure dependence of the wall slip velocity can be

signiﬁcant

2003

Vinogradov and Ivanova

1967

; Hatzikiriakos and Dealy

1992b

; Person

and Denn

with the slip velocity decreasing with in-

creasing pressure. Since the ﬂow curves characterized over relatively high apparent shear

rates are subject to wall slip, and since such slip is a function of wall shear stress, as well

as pressure, new methodologies, including the use of inverse problem solutions and

squeeze ﬂow, are necessary for the characterization of the parameters of pressure-

dependent wall slip and shear viscosity material functions

1997

; Tang and Kalyon

2008

With suspensions the primary cause of wall slip is the apparent slip mechanism asso-

ciated with the formation of a pure binder or binder-rich apparent slip layer at the wall

Tang and Kalyon

2008

. In the following, ﬁrst the analysis of the apparent slip mechanism of

concentrated suspensions is expanded to include the hitherto-neglected slip of the poly-

meric binder at the wall. Second, a mathematical model and corresponding numerical

simulation results of the time-dependent isothermal tube ﬂow of compressible suspen-

sions with rigid spherical particles, subject to pressure-dependent wall slip, are provided

and applied to a model suspension at differing concentrations of rigid particles. Third,

procedures for the experimental determination of the parameters of pressure-dependent

wall slip and shear viscosity material function for suspensions

Reiner

1960

based principally on the

inverse-problem solution methodologies

are presented. Finally, the capillary ﬂow data of

suspensions of a poly

, PDMS, incorporated with rigid spherical glass

particles, collected with relatively long capillary dies, are compared directly with the

time-dependent numerical simulation results of tube ﬂow to elucidate the underlying

mechanisms for the development of ﬂow instabilities for concentrated suspensions.

dimethyl siloxane

II. APPARENT WALL SLIP BEHAVIOR OF CONCENTRATED SUSPENSIONS

WITH RIGID PARTICLES

The wall slip behavior of the concentrated suspensions of rigid particles with rela-

tively low aspect ratios, incorporated into a non-Newtonian binder, is considered to occur

on the basis of the apparent slip mechanism subject to the wall slip of the polymeric

TIME-DEPENDENT TUBE FLOW OF SUSPENSIONS

1071

FIG. 1. Schematics of the apparent slip mechanism of suspensions with the binder liquid exhibiting slip at the

wall with slip velocity u sb .

binder

Fig. 1

. The apparent slip layer

Vand layer, i.e., a zone which is free of particles

and thus consists solely of the binder

has a thickness,

, at the wall. Since the thickness

of the slip layer,

, would be signiﬁcantly smaller than the channel gap, the formation of

the slip layer gives the appearance of wall slip; hence the “apparent slip” at the wall

Reiner

1960

; Cohen and Metzner

1985

; Yilmazer and Kalyon

1989

; Kok Hartman

et al.

. This mechanism is depicted in an

exaggerated manner in Fig. 1 , which shows the apparent slip layer formed next to the

wall of the tubular die. Such apparent wall slip may or may not be signiﬁcant in com-

parison to the mean velocity in the channel, i.e., constituting the weak versus strong wall

slip of the suspension. The rigidity of the particles can alter the dynamics of the forma-

tion of the apparent slip layer

2002

; Tabuteau et al.

2004

; Kalyon

2005

Unlike the earlier analyses in the treatment of apparent slip, here the binder itself is

also considered to be subject to a slip condition with the wall slip velocity of the poly-

meric binder, u sb , following a hyperbolic tangent-type dependence on the wall shear

stress,

Meeker et al.

2004

; Adams et al.

2004

Tang and Kalyon

2004a

; Tang and Kalyon

2008

b s 1

u sb =

0.5 + 0.5 tanh

w −

where

b and s 1 are the slip coefﬁcient

referred to as Navier’s slip coefﬁcient for s 1

and slip exponent of the polymeric binder, respectively, and

is a positive constant

describing the sharpness of the weak-to-strong slip transition in the slip

velocity of the polymeric binder at the critical wall shear stress,

typically 1–20

c .

, for viscoelastic polymer melts the slip

velocity should be a function of the pressure and the ﬁrst and second normal stress

differences at the wall. For polymeric suspensions with rigid particles the development of

normal stress differences would also be affected by the formation of anisotropic particle

clusters

As noted by Hatzikiriakos and Dealy

1992b

. For the materials considered in

our experimental study the normal stress effects were determined to be small and only the

pressure effect is considered in our analysis. The slip coefﬁcient for the binder,

Nott and Brady

1994

; Zarraga et al.

2000

b ,is

assumed to vary inversely with pressure, akin to the slip of compressible gases during

ﬂow through simple conduits

Knudsen

1950

1072

H. S. TANG AND D. M. KALYON

p a

b =

where p is pressure at any location in the die, p a is the atmospheric pressure,

0 is the slip

coefﬁcient of the binder at atmospheric pressure. The exponent

becomes equal to one

for Knudsen ﬂow but needs to be determined experimentally for a polymeric melt

Tang

and Kalyon

2008

and its suspensions with rigid particles with

dependent on the

concentration of particles, i.e.,

. The available experimental data for polymer melts

Vinogradov and Ivanova

1967

; Hatzikiriakos and Dealy

1992b

indicate that the slip

coefﬁcient decreases with increasing pressure

. The principal under-

lying mechanisms for pressure dependence of wall slip are considered to be the entrain-

ment of air into the binder phase of the suspension and the residence time dependence of

the establishment of the apparent slip condition as discussed later.

It is assumed that the Ostwald-de Waele or “power law” behavior represents the

behavior of

thus

is positive

the shear viscosity of

the binder phase in tube ﬂow,

i.e.,

rz =

n b −1

− m b

is the axial z velocity and m b and n b are the con-

sistency index and the power law index parameters of the Ostwald-de-Waele “power-

law” equation for the pure binder, respectively. The slip velocity, u s , for the apparent slip

mechanism

, where u

consists of the contributions of the slip of the binder, u sb , and the contribution of the

apparent slip mechanism, u sa :

Kalyon,

2005

subject to the slip of the binder at the tube wall

Fig. 1

u s = u sa + u sb = s b

s b +1

1− R

1−

+ u sb .

m s b

s b +1

Considering that the reciprocal power-law index of the binder, i.e., s b =1

n b , is positive

and assuming an integer value—the use of Binomial Theorem provides

s b +1

1−

R , the slip velocity at the interface between the apparent slip layer and the

bulk of the suspension, u s , i.e., at r = R −

1−

s b +1

, becomes:

u s = u sa + u sb = s b

a s b + u sb ,

m s b

+ u sb =

where

a is the Navier’s slip coefﬁcient resulting from the apparent slip mechanism, i.e.,

m s b

a =

Kalyon

2005

. A correlation for the apparent slip layer thickness,

, and the

m sb

Navier’s slip coefﬁcient

a =

D p /

1−

/ m

can be used

Kalyon

2005

, where D p

is the particle diameter,

m is the maxi-

mum packing fraction of solids. This correlation for the mean value of the slip layer

thickness

is the volume percent of rigid particles, and

over the length of the die

applies for concentrated suspensions with rigid

particles exhibiting low aspect ratios at

m and suggests that the ﬂow of the concen-

trated suspension approaches plug ﬂow as

→ m . This is seen below from the simpliﬁed

analysis of the apparent wall slip of a Newtonian suspension with viscosity,

s and with

a Newtonian binder with viscosity

b and on the basis of a no-slip condition at the wall

for the Newtonian binder, which constitutes the apparent slip layer. The total ﬂow rate, Q ,

versus the wall shear stress,

w , relationship for the apparent slip ﬂow of a Newtonian

suspension in a tube with radius, R , becomes:

TIME-DEPENDENT TUBE FLOW OF SUSPENSIONS

1073

R 3

Q = Q s +

R 2 u s +

R 2

b w +

R 2

where Q s =

/ b w . The ratio of the ﬂow rate due to slip Q s over the total ﬂow rate,

Q , i.e., Q s /

Q , indicates the relative importance of wall slip at any wall shear stress,

w ,

and is given by:

Q s

Q =

R +

Upon replacing the relative viscosity of

the suspension,

r =

s / b , with

r =

−2.0

−

/ m

Krieger and Dougherty

1959

and the apparent slip layer thickness,

with

D p

=1−

/ m

Kalyon

2005

, Q s /

Q becomes

Q s

Q =

1+ R

4 D p

1−

Therefore, the application of the apparent slip mechanism to a Newtonian suspension

suggests that as

→ m , Q s /

1, i.e., the ﬂow approaches plug ﬂow with increasing

concentration of the rigid particles and that apparent slip has a negligible effect for

relatively small concentration of rigid particles, i.e.,

→

D p . Such

plug ﬂow behavior is indeed observed during the Poiseuille ﬂow of Newtonian suspen-

sions with increasing

→

0, Q s /

0 since R

→

Thus, with the contributions of the slip of the binder and the apparent slip mechanism

the apparent wall slip velocity, u s , of the suspension becomes:

, irrespective of the ﬂow rate

Karnis and Mason

1967

s b .

p a

+ D p

m s b

1−

s 1

u s =

0.5 + 0.5 tanh

w −

The rheological characterization of the polymeric binder provides

c .

Upon the characterization of the shear viscosity and the wall slip behavior of the binder

of the suspension and the physical properties of the rigid phase

0 , s 1 , s b , m b , and

the harmonic mean

particle diameter, D p , volume fraction,

, and maximum packing fraction of the particles,

the only unknown in Eq.

that describes the wall slip behavior of the suspensions

with the same binder is,

The typical slip velocity, u s , versus the wall shear stress and pressure behavior gener-

ated by Eq.

=0.2 is shown in Fig. 2 . Various properties

of this model suspension are listed in Table I . The wall slip velocity is affected by both

the pressure and the wall shear stress and undergoes signiﬁcant changes at various critical

wall shear stress and pressure combinations representing the weak to strong slip transition

that Eq.

for a model suspension with

represents. Figure 2 suggests that if the critical pressure and the wall shear

stress conditions

were

to be encountered at any location along the length of the ﬂow channel, signiﬁcant

changes in the ﬂow boundary condition at the wall will develop.

at which the slip behavior is converted from weak to strong slip

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