Unsteady circular tube flow of compressible polymeric liquids subject to pressure-dependent wall slip

19 Pages·2008·0.71 MB·English

Unsteady circular tube flow of compressible polymeric
liquids subject to pressure-dependent wall slip
H. S. Tanga) and Dilhan M. Kalyonb)
Stevens Institute of Technology, Castle Point St., Hoboken, New Jersey 07030
(Received28May2007;finalrevisionreceived28December2007(cid:1)
Synopsis
A mathematical model is developed for the time-dependent circular tube flow of compressible
polymeric liquids subject to pressure-dependent slip at the wall and applied to a poly (cid:2)dimethyl
siloxane(cid:1)(cid:2)PDMS(cid:1).Theparametersofpressure-dependentwallslipvelocityandshearviscosityof
the PDMS were determined using combinations of small-amplitude oscillatory shear, steady
torsional and squeeze flows and were employed in the prediction of the time-dependent circular
tube flow behavior of the PDMS. The numerical solutions suggest that a steady tube flow is
generatedwhentheflowboundaryconditionatthewallisstable,thatis,eitheracontiguousstick
(cid:2)orweakslip(cid:1)oracontiguousstrongslipconditionalongtheentirelengthofthewall.Ontheother
hand, when the flow boundary condition changes from stick (cid:2)or weak slip(cid:1) to strong slip at any
locationalongthelengthofthewall,undampedperiodicoscillationsinpressureandmeanvelocity
are observed. The experimentally characterized and simulated tube flow curves of PDMS are
similar and the simulation findings for flow stability are in general consistent with the
experimentallyobservedflowinstabilitybehaviorofPDMS. ©2008TheSocietyofRheology.
(cid:3)DOI:10.1122/1.2837104(cid:4)
I. INTRODUCTION
Experimentaldatacollectedfromsimpleshearflowsofpolymericliquidssuggestthat
there are intimate links between the flow boundary condition at the wall and the time-
dependent formation of flow instabilities and extrudate distortions for polymeric liquids
(cid:3)Benbow and Lamb (cid:2)1963(cid:1); Ramamurthy (cid:2)1986(cid:1); Piau et al. (cid:2)1995(cid:1); Denn (cid:2)2001(cid:1); Lar-
razabal et al. (cid:2)2006(cid:1)(cid:4). Three distinct types of extrudate irregularities are considered to
have different initiating mechanisms, i.e., the shark skin region arising from slip and the
uniaxialstretchingthatoccursattheexitofthedie(cid:3)HatzikiriakosandMigler(cid:2)2005(cid:1)(cid:4),the
slip stick associated with wall slip (cid:3)Denn (cid:2)2001(cid:1); Georgiou (cid:2)2003(cid:1)(cid:4) and the gross bulk
extrudatedistortions,linkedtothedynamicsoftheconvergingflowattheentrancetothe
die (cid:3)Kim and Dealy (cid:2)2002(cid:1)(cid:4).Although the wall slip is assumed to be principally respon-
sibleforslip-sticktypeflowinstabilitiesitisanticipatedthatslipwillalsoplaysignificant
role in the development of flow instabilities in converging flow (cid:3)Joshi and Denn (cid:2)2003(cid:1)(cid:4)
and uniaxial stretching at the exit of the die. Empirical experimental evidence indicates
that flow instabilities are not encountered during pressure-driven flows for polymeric
a(cid:1)CurrentlywithCityCollege,CityUniversityofNewYork,NewYork.
b(cid:1)Authortowhomcorrespondenceshouldbeaddressed;electronicmail:[email protected]
©2008byTheSocietyofRheology,Inc.
J.Rheol.52(cid:2)2(cid:1),507-525March/April(cid:2)2008(cid:1) 0148-6055/2008/52(cid:2)2(cid:1)/507/19/$27.00 507
508 H.S.TANGANDD.M.KALYON
liquids or polymeric liquids incorporated with various additives, which generate either a
stable wall stick condition or a stable wall slip condition (cid:3)Ramamurthy (cid:2)1986(cid:1); Kalyon
and Gevgilili (cid:2)2003(cid:1); Fujiyama and Inata (cid:2)2002(cid:1)(cid:4).
The slip-based theoretical treatment of the development of polymer flow instabilities
in simple channels has generally relied on the considerations of the compressibility and
various empirical non-monotonic wall slip velocity versus wall shear stress expressions
(cid:3)Hatzikiriakos and Dealy (cid:2)1992a, 1992b(cid:1); Den Doelder et al. (cid:2)1998(cid:1); Georgiou (cid:2)2003(cid:1)(cid:4).
Furthermore, it is determined that wall slip of polymers occurs as a function of normal
stress/pressure at the wall with wall slip velocities decreasing with increasing normal
stress/pressure (cid:3)Vinogradov and Ivanova (cid:2)1967(cid:1); Kalika and Denn (cid:2)1987(cid:1); Hatzikiriakos
and Dealy (cid:2)1992a, 1992b(cid:1); Person and Denn (cid:2)1997(cid:1)(cid:4). If the wall slip behavior is consid-
ered to be a function of pressure then obtaining the parameters of the shear viscosity
material function in the relatively high shear rate range using the conventional pressure-
driven capillary and rectangular slit rheometers becomes a significant challenge. For
example,HatzikiriakosandDealy(cid:2)1992a(cid:1)havedemonstratedthedifficultiesencountered
in the application of the Bagley correction procedures for capillary rheometry (cid:2)due to a
lack of linearity of the pressure drop versus capillary length over the diameter curves(cid:1).
Currently there are no general methodologies available for the determination of the pa-
rametersofpressure-dependentwallslipandtheshearviscositymaterialfunctions(cid:2)under
conditions in which wall slip is significant(cid:1) simultaneously.
Inthisstudy,amathematicalmodelisfirstprovidedforthetime-dependentisothermal
laminar flow of compressible generalized Newtonian fluids through long circular tubes
subject to pressure-dependent wall slip. Second, experimental procedures for the deter-
mination of the parameters of pressure-dependent wall slip and shear viscosity material
function of polymeric liquids (cid:2)based principally on the inverse-problem solution meth-
odologies(cid:1) are presented. These procedures rely on the collection of small-amplitude
oscillatoryshear,steadytorsionalandsqueezeflowdata.Finally,capillaryflowdataofa
poly (cid:2)dimethyl siloxane(cid:1), PDMS, collected with relatively long capillaries with system-
aticallyvarieddiametersatconstantlength/diameterratio,arecompareddirectlywiththe
numerical simulation results of unsteady circular tube flow to probe the underlying
mechanisms for the development of flow instabilities for polymeric liquids.
II. WALL SLIP BEHAVIOR OF POLYMERIC LIQUIDS
Various polymer melts, especially linear polymers, exhibit wall slip (cid:3)Ramamurthy
(cid:2)1986(cid:1); Kalika and Denn (cid:2)1987(cid:1); Hatzikiriakos and Dealy (cid:2)1991, 1992a(cid:1); Migler et al.
(cid:2)1993(cid:1); Chen et al. (cid:2)1993(cid:1); Awati et al. (cid:2)2000(cid:1); Münstedt et al. (cid:2)2000(cid:1); Gevgilili and
Kalyon (cid:2)2001(cid:1)(cid:4).Wall slip is considered to be one of the major factors that can affect the
formation of extrudate distortions (cid:3)Benbow and Lamb (cid:2)1963(cid:1); Kissi and Piau (cid:2)1990(cid:1)(cid:4).
The literature suggests that there may be multiple mechanisms for the wall slip of poly-
mer melts. For example, with PDMS Migler et al. (cid:2)1993(cid:1) have measured the velocity
distribution directly within a distance of 100 nm of the wall and have observed a sharp
transitionbetweenaregimeofweakslipandoneofstrongslip.Thetransitionfromweak
to strong slip in simple shear is considered to occur at a material-dependent wall shear
stress,valueofwhichdependsonthesurfacedensityofsurface-anchoredchainsandthe
molecular weight of the polymer melt (cid:3)Brochard and de Gennes (cid:2)1992(cid:1); Léger et al.
(cid:2)1997(cid:1)(cid:4).Lowmolecularweightpolymerfractions,aswellasvariousprocessingadditives
with relatively low shear viscosity, are generally present in commercial polymers. Such
species can also generate wall slip on the basis of an apparent slip (cid:3)Kalyon (cid:2)2005(cid:1)(cid:4)
mechanism.
UNSTEADYCIRCULARTUBEFLOW 509
Ramamurthy (cid:2)1986(cid:1) reported a critical wall shear stress of 0.14 MPa for the onset of
wall slip for linear low density polyethylene. Hatzikiriakos and Dealy (cid:2)1991(cid:1) reported a
critical wall shear stress of 0.09 MPa for the onset of wall slip for high density polyeth-
ylene (cid:2)HDPE(cid:1) using a sliding plate rheometer. In their study of high-density polyethyl-
enes, Wang and Drda (cid:2)1997(cid:1) determined a critical stress of about 0.3 MPa (cid:2)based on the
discontinuity of the flow curve(cid:1) for high density polyethylene at 180 °C. Kissi and Piau
(cid:2)1990(cid:1)carriedoutflowvisualizationexperimentswithPDMSusingtransparentdieswith
tracer particles and were able to document wall slip in the 0.05–0.07 MPa wall shear
stress range. On the other hand, Benbow and Lamb (cid:2)1963(cid:1) determined a critical shear
stressof0.07 MPafortheonsetofwallslipforPDMS.Chenetal.(cid:2)1993(cid:1)usedMooney’s
method for capillary flow and determined that wall slip of linear-low density polyethyl-
ene,priortotheonsetofextrudatedistortionsuponflowthroughacapillarydie,isrelated
to the materials of construction and the roughness of the surface of the capillary.
KalyonandGevgilili(cid:2)2003(cid:1)havefocusedonthewallslipandflowinstabilitybehav-
iorofthreepolymersduringextrusionthroughcapillarydies.Twoofthesepolymers,i.e.,
a poly (cid:2)dimethyl siloxane(cid:1) and HDPE, exhibited easily detectable strong wall slip in
simpleshearflowwithcriticalwallshearstressvaluesof0.07and0.2 MPa.Ontheother
hand,thethirdmaterialoftheirstudy,i.e.,anoxetane-basedalternatingblockcopolymer,
BAMO/AMMO,withhardblocksconsistingof(cid:3)3,3-bis(cid:2)azidomethyl(cid:1)oxetane,BAMO(cid:4)
and with soft blocks of (cid:2)3-azidometyl-3-metyloxetane, AMMO(cid:1) did not exhibit strong
slip in steady torsional flow (cid:3)Kalyon and Gevgilili (cid:2)2003(cid:1)(cid:4) over a very wide range of
strains and strain rates. The extrudates of the HDPE and the PDMS emerging from
capillary flow were distortion free at wall shear stresses, which were lower than their
respective critical shear stress values. On the other hand, the BAMO/AMMO thermo-
plastic elastomer of their study, which did not exhibit strong slip over a broad range of
shearratesinsimpleshearflow,didnotexhibitextrudatedistortionsduringcapillaryflow
overthesamerangeofshearrates.Thisobservationagainemphasizedtheimportanceof
wall slip in affecting the development of flow instabilities and extrudate distortions.
The relationship between the onset of strong wall slip of HDPE and its processability
wasfurtherinvestigatedbyusingacontinuousshearrollextruder(cid:3)Kalyonetal.(cid:2)2004(cid:1)(cid:4).
The continuous shear roll extrusion is a continuous processing method that is based
solelyontheuseoftwoparallelandgroovedrolls.Sincethereisnobarreltoenclosethe
rolls, the conveying of the melt takes place upon its sticking to one roll and its slipping
at the surface of the second roll. Thus, the shear roll extruder can be considered to be a
differential slip tester for polymer liquids (cid:2)the stick or slip can be immediately observed
and affected by changes in roll speeds, temperatures, and surface roughnesses(cid:1). These
experiments have revealed that the complete detachment of a HDPE melt from both roll
surfaces occurs when the shear stress in the nip region between the two rolls surpasses
the critical shear stress, (cid:1), at which strong wall slip is onset, providing a direct link
c
between onset of strong wall slip and processability.
The weak to strong transition in slip velocity, u , at the critical wall shear stress, (cid:1),
s c
canbedescribedinconjunctionwithahyperbolictangentdependence(cid:3)TangandKalyon
(cid:2)2004a(cid:1)(cid:4):
u =(cid:2)(cid:1)s1(cid:3)0.5+0.5tanh(cid:2)(cid:3)(cid:2)(cid:1) −(cid:1)(cid:1)(cid:1)(cid:4). (cid:2)1(cid:1)
s w w c
Here, (cid:1) is the shear stress at the wall, (cid:2)and s are the Navier’s slip coefficient and slip
w 1
exponent, respectively, for the polymers and (cid:3)is a positive constant describing the
sharpness of the weak-to-strong slip transition in the slip velocity of the polymer at the
criticalwallshearstress,(cid:1).Thepowerlawdependenceoftheslipvelocityonwallshear
c
stress arises directly in the apparent slip mechanism when the fluid constituting the
510 H.S.TANGANDD.M.KALYON
apparent slip layer is non-Newtonian (cid:3)Kalyon (cid:2)2005(cid:1)(cid:4). This power law dependence was
assumed to hold also for the wall slip of the polymer melt. The hyperbolic tangent
function allows the incorporation of the concept of the critical wall shear (cid:2)at which the
transition from weak to strong slip at the wall occurs(cid:1) to describe the wall slip behavior
of the polymer melt.
HatzikiriakosandDealy(cid:2)1992a(cid:1)haveindicatedthatforviscoelasticpolymericliquids
the slip velocity should be a function of the pressure and the first and second normal
stress differences at the wall but only the pressure effect is considered here, following
PersonandDenn(cid:2)1997(cid:1).Kalyonetal.(cid:2)1995(cid:1)haveinvestigatedthepressure-drivenflow
of a concentrated suspension through a slit die, using high-speed cinematography and a
microscope attached to a transparent die surface (cid:2)at a location which is close to the exit
of the die(cid:1). The high-speed cinematography has revealed for the first time that the die
surface is covered in a time-dependent fashion with air films. The bubbles of air were
observed to approach the die surface, attach and flatten, and then slide, roll and detach
uponcontactwiththewallfor30(cid:4)10 msundertheexperimentalconditionsapplied.The
ramificationsofthelubricationofthediewallbythevaporphasehavebeendiscussedin
conjunction with the processing behavior of concentrated suspensions (cid:3)Aral and Kalyon
(cid:2)1995(cid:1)(cid:4). Imaging experiments with various other fluids have also revealed the formation
of a vapor layer (cid:2)some as nanobubbles(cid:1) at the wall (cid:3)Tyrrell andAttard (cid:2)2001(cid:1)(cid:4).
Theformationofsuchanapparentsliplayer,entrainingsomeofthevaporphaseatthe
wall, would be affected by the bulk pressure. Thus, the slip velocity at the wall should
diminish with increasing pressure.This provides an important mechanism to explain the
experimentallyobservedpressuredependenceofwallslipofpolymermelts(cid:3)Vinogradov
and Ivanova (cid:2)1967(cid:1); Kalika and Denn (cid:2)1987(cid:1); Hatzikiriakos and Dealy (cid:2)1992a, 1992b(cid:1);
Person and Denn (cid:2)1997(cid:1)(cid:4). Here, considering that the presence of a gas phase at the wall
is the important factor to give rise to the observed pressure dependence of wall slip on
pressure, the Navier’s slip coefficient for the polymer, (cid:2), is assumed to vary inversely
with pressure, akin to the experimentally observed wall slip behavior of compressible
gases during flow through simple conduits (cid:3)Knudsen (cid:2)1950(cid:1)(cid:4):
(cid:5) (cid:6)
(cid:5)
p
(cid:2)=(cid:2) a , (cid:2)2(cid:1)
0 p
wherepisthelocalpressure,p istheatmosphericpressure,and(cid:2) istheslipcoefficient
a 0
ofthepolymeratatmosphericpressure.Theexponent(cid:5)isexperimentallyobservedtobe
equal to one for slip flow of gases (cid:3)Knudsen (cid:2)1950(cid:1)(cid:4). It is an empirical factor that needs
to be determined experimentally for polymeric liquids.
III. UNSTEADY LAMINAR FLOW OF COMPRESSIBLE GENERALIZED
NEWTONIAN FLUIDS THROUGH LONG CIRCULAR TUBES SUBJECT TO
PRESSURE-DEPENDENT WALL SLIP
The unsteady isothermal laminar flow of a compressible fluid contained in a long
circular tube of length L and radius R resulting from the sudden imposition of a flow
condition at one end of the tube is considered. Consistent with the earlier literature
investigating the dynamics of the flow of polymeric liquids in circular tubes, the com-
pressible fluid is assumed to be purely viscous, i.e., generalized Newtonian fluid (cid:3)Hatz-
ikiriakos and Dealy (cid:2)1992a(cid:1); Den Doelder et al. (cid:2)1998(cid:1); Georgiou (cid:2)2003(cid:1)(cid:4). It is assumed
that Ostwald-de Waele or “power law” behavior represents the shear viscosity, (cid:1) =
rz
−m(cid:7)dV /dr(cid:7)n−1(cid:2)dV /dr(cid:1), where m and n are the consistency index and the power law
z z
index parameters of the power law equation and V (cid:2)r(cid:1) is the velocity component in the
z
UNSTEADYCIRCULARTUBEFLOW 511
flow direction, z.The inertial and gravitational effects are assumed to be negligible.The
integrations of the governing equations of the continuity and momentum equations over
the cross-sectional area,A, provide
(cid:8)(cid:5) (cid:6) (cid:9)
(cid:1)p (cid:1) (cid:6)
+ p−p + a V =0, (cid:2)3(cid:1)
(cid:1)t (cid:1)z a (cid:7)
(cid:1)V 1(cid:1)(cid:8)V2 2(cid:1) 1 (cid:1)p
+ + w + =0, (cid:2)4(cid:1)
(cid:1)t 2 (cid:1)z (cid:2)(cid:7)(cid:2)p−p (cid:1)+(cid:6)(cid:1)R (cid:2)(cid:7)(cid:2)p−p (cid:1)+(cid:6)(cid:1)(cid:1)z
a a a a
where t is the time, V is the cross-section-averaged z velocity, the density is given by
(cid:6)=(cid:7)(cid:2)p−p (cid:1)+(cid:6), i.e., (cid:6)and (cid:6), are the density values of the melt at pressures p and p ,
a a a a
respectively, (cid:7) is the compressibility coefficient, and (cid:8)=(cid:2)1/(cid:2)AV2(cid:1)(cid:10)(cid:10) V2dA. 1(cid:1)(cid:8)V2 was
A z 2 (cid:1)z
assumed to be negligible for creeping flow conditions. Under steady conditions the wall
stress, (cid:1), can be determined from the prevailing steady local pressure gradient,
w
(cid:2)−(cid:1)p/(cid:1)z(cid:1), i.e.,
(cid:5) (cid:6)
R (cid:1)p
(cid:1) =− , (cid:2)5(cid:1)
w 2 (cid:1)z
and the local averaged velocity, V, can be obtained as
(cid:5) (cid:6)
R(cid:2)1+1/n(cid:1) (cid:1)p 1/n
V= − +u . (cid:2)6(cid:1)
(cid:2)2m(cid:1)1/n(cid:2)3+1/n(cid:1) (cid:1)z s
The boundary conditions at the entry and exit planes of the tube are
V=V at z=0, (cid:2)7(cid:1)
o
p=p at z=L. (cid:2)8(cid:1)
a
Thedetailsofnumericalmethodsusedforthesolutionoftheseequationsarediscussedin
Appendix.
IV. DETERMINATION OF THE PARAMETERS OF SHEAR VISCOSITY
AND PRESSURE-DEPENDENT WALL SLIP VELOCITY
A. Materials
Apoly (cid:2)dimethyl siloxane(cid:1), PDMS, manufactured by GE Silicones (cid:2)SE-30(cid:1) was used
intheexperimentalstudies.Ithasadensityof958 kg/m3 underambientpressure(cid:2)Table
I(cid:1). This PDMS was used in our earlier experimental investigations focusing on the flow
instabilities of polymeric liquids on one hand and as the binder of concentrated suspen-
sions used in studies concerned with the development of flow instabilities during capil-
lary and rectangular slit die flows (cid:3)Kalyon and Gevgilili (cid:2)2003(cid:1); Birinci and Kalyon
(cid:2)2006(cid:1)(cid:4).
B. Experimental apparati and procedures
An Advanced Rheometric Expansion System rheometer, originally from Rheometric
Scientific, Inc., Piscataway, NJ (cid:2)currently TA Instruments(cid:1), was utilized in conjunction
with steady torsional flow, and small-amplitude oscillatory shear flows using cone-and-
plateandparallel-diskconfigurations.Theenvironmentalchamberwasequippedwithan
imagingwindowandauxiliaryopticsforcontinuousmonitoringofthefreesurfaceofthe
specimen (cid:3)Aral and Kalyon (cid:2)1994(cid:1); Gevgilili and Kalyon (cid:2)2001(cid:1)(cid:4).Ahigh-speed camera,
512 H.S.TANGANDD.M.KALYON
TABLE I. Parameters of PDMS associated with density, wall slip and
K-BKZequation
Densityatambient,(cid:6): 958kg/m3
a
Compressibility,(cid:7): 1.4(cid:9)10−6s2/m2
Criticalshearstress,(cid:1): 70,000Pa
c
Pressurecoefficientofslip,(cid:5): 0.7
Navier’sslipcoefficient,(cid:2): 5(cid:9)10−16m/(cid:2)s-Pas1(cid:1)
0
Slipexponent,s : 3.1
1
(cid:3): 20
Dampingfunctionparameters:
f: 1.0
n : 0.283
1
n : -
2
Relaxationtime,(cid:10),s: Relaxationstrength,G,Pa
i i
0.0003 61,040
0.006 47,140
0.034 40,730
0.16 29,210
0.72 11,970
3.27 2,435
21.1 290
80 38
Powerlawparameters:
For(cid:11)˙(cid:12)40s−1: m=19,000Pasnandn=0.37
For(cid:11)˙(cid:13)40s−1: m=31,000Pasnandn=0.21
capable of recording at filming speeds as high as 2000 frames per second, was a part of
the setup to allow the monitoring of the free surface of the specimen. During steady
torsional flow a straight-line marker was placed on the edges of the cone/plate and the
freesurfaceofthepolymermelt(cid:3)Kalyonetal.(cid:2)1993(cid:1);AralandKalyon(cid:2)1994(cid:1);Gevgilili
and Kalyon (cid:2)2001(cid:1)(cid:4). The discontinuities in the marker line that develop between the
surface of the plates of the rheometer and the polymeric liquid suggest the initiation of
strongwallslip,fromwhichthecriticalshearstressfortheonsetofstrongwallslipcould
be determined. The torque data of the steady torsional flow of the PDMS at various
nominalshearrateswereusedforthedeterminationoftheslipparameters(cid:2) ands upon
0 1
the solution of the inverse problem.
An Instron Floor Tester was employed in conjunction with a capillary rheometer to
study the development of extrudate distortions of PDMS upon exit from capillaries with
differing diameters at constant capillary length over diameter ratio. The shapes of the
extruded samples were captured immediately upon exit from the capillary using a high-
speedcamera.Thetemperaturedistributionsoftheextrudatesemergingfromthediewere
also monitored using a ThermaCam thermal imaging camera (cid:3)Birinci and Kalyon
(cid:2)2006(cid:1)(cid:4). Typically, the surface temperature of the PDMS extrudate upon exit from the
capillarydieisobservedtoincreaseupto4 °Criseinthe2–40 s−1range(cid:2)overrelatively
long runs and with capillary length/diameter ratio of 60(cid:1).The viscous energy dissipation
effect is thus important in affecting the flow curve but was ignored in our current iso-
thermal analysis.
An Instron capillary rheometer was used in conjunction with a plug to seal the reser-
voir kept under isothermal conditions for the pressure-volume-temperature experiments.
The compressibility coefficient, (cid:7), of the PDMS, was determined to be 1.4(cid:9)10−6
UNSTEADYCIRCULARTUBEFLOW 513
FIG.1. SmallamplitudeoscillatorysheardataofPDMSandthebestfitoftherelaxationspectra.
(cid:2)s2/m2(cid:1), which agreed with the available pressure-volume-temperature data of PDMS
fitted with the Sanchez and Lacombe Lattice Fluid Model (cid:3)Brandrup et al. (cid:2)1999(cid:1)(cid:4).
C. Characterization of the shear viscosity of the PDMS
The pressure dependence of the slip velocity suggests that the shear viscosity of the
PDMS cannot be characterized in the relevant high shear rate range using conventional
pressure-driven viscometers.This was also recognized in the investigation of Smillo and
Dealy (cid:2)2006(cid:1), in which the magnitude of complex viscosity values was utilized to rep-
resent slip-free shear viscosity values on the basis of the Cox–Merz rule. Here the shear
viscosity of the PDMS at the relevant high shear rate range was predicted using a fac-
torized K-BKZ integral constitutive equation in conjunction with an exponential type
damping function (cid:3)Bernstein et al. (cid:2)1963(cid:1); Wagner (cid:2)1976(cid:1); Gevgilili (cid:2)2003(cid:1)(cid:4)
(cid:11)
t
(cid:1)(cid:2)t(cid:1)= M(cid:2)t−t(cid:1)(cid:1)h(cid:2)I ,I (cid:1)B(cid:2)t(cid:1)(cid:1)dt(cid:1), (cid:2)9(cid:1)
1 2
−(cid:14)
whereh(cid:2)I ,I (cid:1)isthedampingfunctionandI andI arethefirstandsecondinvariantsof
1 2 1 2
the relative Finger strain tensor, B. M(cid:2)t−t(cid:1)(cid:1) is the rubberlike-liquid memory function,
which can be determined from linear viscoelastic data. The determination of the param-
eters of the memory function and the damping function needs to be undertaken using
simpleshearflowconditionsatwhichthewallslipofthePDMSisnegligible(cid:2)i.e.,under
conditionsforwhichtheshearstressislessthanthecriticalshearstress,(cid:1),atwhichthe
c
weak to strong slip transition takes place(cid:1). Thus, for example, the relaxation modulus
versus time and strain data obtained upon the step strain experiment could not be used
due to the significant slip observed during the step strain experiment (cid:3)Gevgilili and
Kalyon (cid:2)2001(cid:1)(cid:4). The parameters of the constitutive equation were determined upon the
best fit of the relaxation spectra from the dynamic properties, collected using small-
amplitude oscillatory shear, and the shear stress growth and shear stress relaxation upon
cessation of steady shear material functions collected at the relatively low shear rates at
which wall slip of the PDMS was negligible.
The dynamic properties of the PDMS are shown in Fig. 1. The collected storage
modulus,G(cid:1),andlossmodulus,G(cid:2),dataasafunctionoffrequency,(cid:15),wereemployedto
determine the relaxation strength, G, versus the relaxation time, (cid:10) as follows:
i i
(cid:12) G(cid:10)2(cid:15)2 (cid:12) G(cid:10)(cid:15)
G(cid:1)(cid:2)(cid:15)(cid:1)= i i , G(cid:2)(cid:2)(cid:15)(cid:1)= i i . (cid:2)10(cid:1)
1+(cid:10)2(cid:15)2 1+(cid:10)2(cid:15)2
i i i i
514 H.S.TANGANDD.M.KALYON
FIG.2. Shearstressgrowthupontheinceptionofflowandshearstressrelaxationuponthecessationofsteady
sheardataofPDMSandthebestfitofthesheardampingfunctionparameters.
Thecharacterizationofthediscreterelaxationspectra,i.e.,therelaxationstrength,G,
i
versustherelaxationtime,(cid:10),valueswascarriedoutinconjunctionwiththeGeneralized
i
Reduced Gradient type of nonlinear optimization by using a pattern search method.
Thedeterminationoftheparametersofthesheardampingfunctionh (cid:2)(cid:11)(cid:1),asafunction
s
of shear strain, (cid:11), i.e., f, n and n (cid:3)Osaki (cid:2)1976(cid:1); Laun (cid:2)1978(cid:1)(cid:4)
1 2
h (cid:2)(cid:11)(cid:1)=f exp(cid:2)−n (cid:11)(cid:1)+(cid:2)1−f(cid:1)exp(cid:2)−n (cid:11)(cid:1) (cid:2)11(cid:1)
s 1 2
required the use of two additional material functions, i.e., the shear stress growth upon
theinceptionofsimpleshearflow,(cid:16)+(cid:2)t,(cid:11)˙(cid:1),andtherelaxationoftheshearingstressupon
the cessation of steady shear flow, (cid:16)−(cid:2)t,(cid:11)˙(cid:1), data at shear rates under which wall slip is
negligible for PDMS (cid:3)Kalyon and Gevgilili (cid:2)2003(cid:1)(cid:4). The shear stress growth and relax-
ation material functions predicted on the basis of the double exponential damping func-
tion in conjunction with the Wagner postulate of the K-BKZ equation were available
from (cid:3)Kalyon et al. (cid:2)1988(cid:1)(cid:4). The parameters f, n and n of the damping function for
1 2
shear were sought upon the solution of the inverse problem to minimize the objective
functions evaluated by the forward difference method.
Theinverseproblemsolutionproceduresuggestedthattwoparametersaresufficientto
representtheshearstressgrowthandrelaxationbehaviorandprovided f andn valuesof
1
1 and 0.283, respectively. The typical comparisons between the time-dependent shear
stressgrowthupontheinceptionofsteadytorsionalflowandshearstressrelaxationupon
the cessation of steady shear data and the fit of the inverse problem solutions are shown
in Fig. 2. The agreement is acceptable.
The set of parameters G, (cid:10), f and n , determined using conditions under which wall
i i 1
slipisnegligible,allowedthedeterminationoftheshearviscositymaterialfunction,(cid:16)(cid:2)(cid:11)˙(cid:1)
which was further fitted with two sets of power law parameters for computational con-
venience (cid:2)Table I(cid:1). The accuracy of the parameters representing the shear viscosity ma-
terial function of PDMS are obviously limited by the capability of the Wagner postulate
of the K-BKZ equation to predict the shear viscosity over a broad range of shear rates.
Theremaybeothermethodstodeterminetheshearviscosityofthepolymermeltwithout
using pressure-driven flows, for example, from the extrapolation of the shear viscosity
datacollectedinthelowapparentshearraterange(cid:2)determinedatwallshearstressvalues
smaller than the critical wall shear stress at which slip effects are negligible(cid:1) to the high
UNSTEADYCIRCULARTUBEFLOW 515
FIG.3. Normalforce,F,versusgap,h,duringsqueezeflowofPDMSasafunctionoftheplatevelocity,U.
apparent shear rate range but such procedures would not have been more accurate than
the use of the K-BKZ equation to determine the shear viscosity behavior over a broad
range of deformation rates.
D. Characterization of the wall slip behavior of the PDMS
The steady torsional flow experiments of PDMS, carried out in conjunction with the
straight-linemarkermethodandimageanalysis(cid:3)KalyonandGevgilili(cid:2)2003(cid:1)(cid:4),wereused
directly for the determination of the critical wall shear stress, (cid:1) (cid:2)at which strong slip is
c
onset(cid:1)(cid:4). The additional slip parameters of the PDMS melt, (cid:2) and s , were determined
0 1
from the inverse problem solution using the steady torque, T, data collected at different
apparent shear rates in torsional flow
(cid:11) (cid:5) (cid:6)
R rW−2u n
T=2(cid:17) r2(cid:1)dr, (cid:1) =m s , (cid:2)12(cid:1)
w w H
0
where H is the gap, R is the disk radius, W is the angular velocity, u is the slip velocity
s
of the PDMS given from Eq. (cid:2)1(cid:1).The best fit of the slip parameters, (cid:2) and s , from the
0 1
inverse problem solution generated the values given in Table I.
The pressure coefficient for wall slip, (cid:5), of the PDMS melt (cid:3)Eq. (cid:2)2(cid:1)(cid:4) was determined
from the squeeze flow experiments. The squeeze flow offers a combination of drag and
pressure-driven flow and conveniently covers a broad range of deformation rates. Our
finiteelementmethodbasedsourcecodesdevelopedfortheanalysisofthesqueezeflow
of viscoplastic fluids subject to wall slip (cid:3)Lawal and Kalyon (cid:2)2000(cid:1); Kalyon and Tang
(cid:2)2007(cid:1)(cid:4) were modified to allow the imposition of the pressure dependence of wall slip
(cid:3)Eqs.(cid:2)1(cid:1)and(cid:2)2(cid:1)(cid:4)andthedeterminationoftheslipcoefficient,(cid:5),upontheimpositionof
the inverse problem solution methodologies (cid:3)Tang and Kalyon (cid:2)2004b(cid:1)(cid:4). The agreement
betweenthenormalforce,F,versustimedatacollectedwithsqueezeflowandthosethat
were numerically predicted using the (cid:5)value of 0.7 (cid:2)obtained from the inverse problem
solution(cid:1) are shown in Fig. 3. All of the determined parameters of PDMS are listed in
Table I and the resulting behavior of wall slip velocity versus pressure and wall shear
stress is shown in Fig. 4.
Both the pressure and the wall shear stress play significant roles in affecting the
developmentofthewallslipbehaviorofPDMS(cid:2)Fig.4(cid:1).Thewallslipvelocityvaluesare
negligible at wall shear stress values smaller than the critical wall shear stress, (cid:1), of
c
516 H.S.TANGANDD.M.KALYON
FIG.4. WallslipvelocityversuspressureandwallshearstressbehaviorofPDMS.
70 kPa. Wall slip velocity values decrease with increasing pressure with the pressure
effect becoming more pronounced at the relatively higher wall shear stress values. The
wall slip behavior observed in Fig. 4 suggests that significant distributions in the flow
boundary condition could occur in circular tube flow as governed by the prevailing
density, pressure and wall shear stress conditions.
V. NUMERICAL SIMULATIONAND EXPERIMENTAL RESULTS
The governing Eqs. (cid:2)1(cid:1)–(cid:2)6(cid:1) were numerically solved to analyze the time-dependent
tube flow of PDMS for circular tubes with diameters of 0.83, 1.5, and 2.5 mm at a
constanttubeL/Dratioof40.Thesedimensionsarethesameasthoseofthestraightland
sections of the capillaries that were used to generate the flow curves as well as the
experimental data related to the development of extrudate shape distortions and flow
instabilities of the same PDMS and its suspensions with rigid spherical particles(cid:3)Birinci
and Kalyon (cid:2)2006(cid:1)(cid:4).
Thenumericalsimulationresultsforthesolutionofthetime-dependentflowbehavior
ofthePDMSinatubewithadiameterof1.5 mmatanapparentshearrateof10 s−1 are
shown in Figs. 5 and 6 (cid:2)determined at I(cid:18)21, CFL(cid:18)0.25, and Von(cid:18)0.25 where I is the
number of mesh nodes, CFLis the Courant-Friedrichs-Levy number, andVon is the von
Neumann number—seeAppendix for details(cid:1). Figure 5 shows the pressure, mean veloc-
ity, slip velocity, and shear stress as a function of time at a specific location next to the
exit(cid:2)z=0.9 L(cid:1)andFig.6showsthedistributionsofpressure,meanvelocity,slipvelocity,
andshearstressovertheentirelengthofthetubeattime=0.9 s.Thesenumericalresults
indicate that a steady state in flow could be reached at this apparent shear rate (cid:2)Fig. 5(cid:1).
Under the steady state conditions the wall shear stress increases from 52 to 52.2 kPa in
the axial, i.e., the z direction over the length of the tube (cid:2)Fig. 6(cid:1) and the mean velocity
increases only slightly with increasing axial distance due to the compressibility of the
polymer. The wall slip velocity is negligible over the entire length of the tube (cid:2)Fig. 6(cid:1).
The numerical solution was determined to be independent of the grid spacing and time
interval upon systematically varying the values of mesh density and CFLand von Neu-
mannnumbers.Overall,thesimulationresultsrevealednofluctuationsinpressure,shear