Table Of ContentSPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY
THERMAL ENGINEERING AND APPLIED SCIENCE
Dennis A. Siginer
Stability of Non-
Linear Constitutive
Formulations for
Viscoelastic Fluids
123
SpringerBriefs in Applied Sciences and Technology
Thermal Engineering and Applied Sciences
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FrancisA.Kulacki,UniversityofMinnesota,USA
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Dennis A. Siginer
Stability of Non-Linear
Constitutive Formulations
for Viscoelastic Fluids
DennisA.Siginer
CentrodeInvestigacio´nenCreatividad
yEducacio´nSuperior
UniversidaddeSantiagodeChile
Santiago,Chile
ISSN2191-530X ISSN2191-5318(electronic)
ISBN978-3-319-02416-5 ISBN978-3-319-02417-2(eBook)
DOI10.1007/978-3-319-02417-2
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thismonographtogetherwithitscomplimentaryvolume,DynamicsofTubeFlow
ofViscoelasticFluidsandNon-colloidalSuspensionsinthisseriesisanattemptto
give an overall comprehensive view of a complex field, which has developed
rapidly in the second half of the twentieth century, still far from being settled on
firm grounds, that of nonlinear constitutive equation formulations for viscoelastic
fluid media and their impact on the dynamics of viscoelastic fluid flow in tubes.
This monograph covers the development of constitutive equation formulations in
their historical context together with the latest progress made, and Dynamics of
Tube Flow of Viscoelastic Fluids and Suspensions covers the state-of-the-art
knowledge in predicting the flow of viscoelastic fluids and suspensions in tubes
highlighting the historical as well as the most recent findings. Most if not all
viscoelastic fluids in industrial manufacturing processes flow through tubes,
whicharenotnecessarilycircular,atonetimeoranotherduringtheprocessingof
thematerial.Itiscriticallyimportantthattheflowofnonlinearviscoelasticfluidsin
tubes can be predicted on a sound basis, thus the raison d0e^tre of the volume on
DynamicsofTubeFlowofViscoelasticFluidsandNon-colloidalSuspensions.As
flowbehaviorpredictionsaredirectlyrelatedtotheconstitutiveformulationsused,
DynamicsofTubeFlowofViscoelasticFluidsandNon-colloidalSuspensionsrelies
heavilyonthisvolumewhenviscoelasticfluidsaretheworkingfluids.
The science of rheology defined as the study of the deformation and flow of
matter wasvirtuallysinglehandedlyfoundedandthenameinventedbyProfessor
BinghamofLafayetteCollegeinthelate1920s.Rheologyisawideencompassing
science which covers the study of the deformation and flow of diverse materials
such as polymers, suspensions, asphalt, lubricants, paints, plastics, rubber, and
biofluids,allofwhichdisplaynon-Newtonianbehaviorwhensubjectedtoexternal
stimuliandasaresultdeformandflowinamannernotpredictablebyNewtonian
mechanics.
v
vi Preface
The development of rheology, which had gotten to a slow start, took a boost
during WWII as materials used in various applications, in flame throwers for
instance, were found to be viscoelastic. As Truesdell and Noll famously wrote
(Truesdell C, Noll W (1992) The nonlinear field theories of mechanics, 2nd edn.
Springer, Berlin) “By 1949 all work on the foundations of rheology done before
1945hadbeenrenderedobsolete.”IntheyearsfollowingWWIItheemergenceand
rapid growth of the synthetic fiber and polymer processing industries, the appear-
ance of liquid detergents, multigrade oils, non-drip paints, and contact adhesives,
anddevelopmentsinpharmaceuticalandfoodindustriesandbiotechnologyspurred
thedevelopmentofrheology.Alltheseexamplesclearlyillustratetherelevanceof
rheologicalstudiestolifeandindustry.Therelianceofallthesefieldsonrheolog-
icalstudiesisattheverybasisofmanyifnotalloftheamazingdevelopmentsand
successstoriesendingupwithmanyoftheproductsusedbythepublicatlargein
everydaylife.
Non-Newtonian Fluid Mechanics, which is an integral part of rheology, really
made big strides only after WWII, and has been developing at a rapid rate ever
since.Thedevelopmentofreliableconstitutiveformulationstopredictthebehavior
of flowing substances with nonlinear strain–stress relationships is quite a difficult
proposition by comparison with Newtonian fluid mechanics with linear
stress–strainrelationship.Thelatterdoesenjoyahead-startoftwocenturiestracing
back its inception to Newton and luminaries like Euler and Bernoulli. With the
former the nonlinear structure does not allow the merging of the constitutive
equations for the stress components with the linear momentum equation as it is
thecasewithNewtonianfluidsendingupwiththeNavier–Stokesequations.Thus
the practitioner ends up with six additional scalar equations to be solved in three
dimensionsforthesixindependentcomponentsofthesymmetricstresstensor.The
difficultiesinsolvingintandemthissetofnonlinearequations,whichmayinvolve
both inertial and constitutive nonlinearities, cannot be underestimated. Perhaps
equally importantlyatthispoint intimeinthe developmentofthe science we are
not fortunate enough to have developed a single constitutive formulation, which
may lend itself to most applications and yield reasonably accurate predictions
together with the field balance equations. The field is littered with a plethora of
equations, some of which may yield reasonable predictions in some flows and
utterlyunacceptablepredictionsinothers.Thusweendupwithclassesofequations
that would apply to classes of flows and fluids, an ad hoc concept at best that
hopefullywillgiveway1daytoauniversalequation,whichmayapplytoallfluids
inallmotions.Inadditionthestabilityoftheseequationsisaveryimportantissue.
Any givenconstitutiveequationshouldbestableintheHadamardanddissipative
senseandshouldnotviolatethebasicprinciplesofThermodynamics.
Efforts have not been spared to be thorough in the presentation with commen-
tariesaboutthesuccessesandfailuresofeachtheoryandthereasonsbehindthem.
The link between different theories and the naturally unfolding succession of
theories over time borne out of the necessity of better predictions as well as the
Preface vii
challenges in the field at this time are given much emphasis at the expanse of a
detailed in-depth development of various theories. This monograph provides a
snapshot of a fast developing topic and a bridge connecting new research results
withatimelyandcomprehensiveliteraturereview.Foradetailedin-depthexposi-
tionofanyonesubjectincludedinthisbookthereaderisreferredtotheextensive
reference list given at the very end of this monograph. The responsibility for any
mistakes and misquotes that may have crept up into the text in spite of extensive
checkingremainssolelywiththeauthor.
Santiago,Chile DennisA.Siginer
Contents
1 Preamble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Viscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 ComputationalApproachtoSolutions. . . . . . . . . . . . . . . . . . . . . . 4
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 ConstitutiveFormulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Non-localConstitutiveFormulations. . . . . . . . . . . . . . . . . . . . . . . 12
2.2 LocalConstitutiveFormulations. . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 DifferentApproachestoLocalConstitutive
Formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 LinearViscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 Non-linearViscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 ConstitutiveEquationFormulationsBasedonRational
ContinuumMechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 ConstantStretchHistoryFlows. . . . . . . . . . . . . . . . . . . . . 31
2.4.2 SimpleFluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 ConstitutiveEquationFormulationConsistent
withThermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 MaximizationoftheRateofDissipation. . . . . . . . . . . . . . . 49
2.5.2 BurgersEquation. .. . . . . . . .. . . . . . . . .. . . . . . . . .. . . . 56
2.5.3 MinimumFreeEnergyandMaximum
RecoverableWork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5.4 ImplicitConstitutiveTheories. . . . . . . . . . . . . . . . . . . . . . 68
2.6 Maxwell-LikeDifferentialEquations. . . . . . . . . . . . . . . . . . . . . . . 74
2.7 SingleIntegralConstitutiveEquations. . . . . . . . . . . . . . . . . . . . . . 76
ix
