Fluid Mechanics of Viscoplasticity

405 Pages·2022·6.943 MB·English

by  R. R. Huilgol, Georgios C. Georgiou

Raja R. Huilgol
Georgios C. Georgiou
Fluid Mechanics
of Viscoplasticity
Second Edition
Fluid Mechanics of Viscoplasticity
·
Raja R. Huilgol Georgios C. Georgiou
Fluid Mechanics
of Viscoplasticity
Second Edition
RajaR.Huilgol GeorgiosC.Georgiou
CollegeofScienceandEngineering DepartmentofMathematicsandStatistics
FlindersUniversity UniversityofCyprus
Adelaide,SA,Australia Nicosia,Cyprus
ISBN 978-3-030-98502-8 ISBN 978-3-030-98503-5 (eBook)
https://doi.org/10.1007/978-3-030-98503-5
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Preface
In the seven years since the first edition appeared, several significant additions to
model the flows of viscoplastic fluids have occurred. In this volume, these novel
aspectsareincorporated.
Specifically,theparticlevelocitybasedlatticeBoltzmannmethoddemonstrates
thatfifteenfictitiousparticlesatapoint,movinginspecifieddirectionswithallocated
speeds and satisfying a set of evolution equations, produce Cauchy’s equations of
continuummechanicsforallfluidstowithinanapproximation.Anothersetofseven
particles,withcorrespondingevolutionequations,delivertheenergyequation.Since
thenumericalmodellingoftheseevolutionequationscanbeachievedthroughfinite
differences, the computational time is reduced significantly when compared with
finiteelementmethods.Importantly,thereisnolossofaccuracyeither.Therelevant
equationsarederivedinChap.3and,inChap.10,theseequationsareusedtomodel
the flow of a Bingham fluid in a pipe of square cross-section, and the thermally
inducedlid-drivenflowinacavity.
NumerousanalyticalsolutionstotheflowsofHerschel-Bulkleyfluidsareincluded
inthismonograph,alongwiththeuseoftheLambertWfunctiontostudysuchflows,
usingthePapanastasioumodel.Inaddition,theantiplaneshearflow,thehodograph
transformation,effectsofwallslip,andflowsofmaterialswithpressuredependent
rheologicalparametersarealsoexaminedindetailinChap.5.
In Chap. 7, the lubrication paradox is re-examined and an integral equation
approachishighlightedtocircumventthisparadoxintheflowsofaHerschel-Bulkley
fluid.Variousslumptestsarealsodiscussed.
It has been found recently that the Cheeger set of the cross-section of a pipe
determinestheminimumpressuredropperunitlengthtoinitiatethesteadyflowofa
Binghamfluid.Equally,theCheegersetofthecross-sectiondeterminesthesettling
ofrigidbodiesinBinghamfluids.ThesenewdevelopmentsappearinChap.9.
Finally,inChap.10,apartfrommodellingtheequationsintwoseparateflowsof
BinghamfluidsderivedfromtheparticlevelocitybasedlatticeBoltzmannmethod,
ashootingmethodisemployedtomodeltheflowofBinghamandHerschel-Bulkley
fluidinanannulus.Thisapproach,whichisfasterthanthatutilisingthefiniteelement
method,canbeextendedtohelicalflowsaswell.
vii
viii Preface
Itistobeexpectedthatthematerialfromtheearliereditionisretained,although
manyimportantadditionsandtheirexpositioncanbefoundhere.
Wehopethatourjointventureisofsufficientbreadthanddepthtoconvincethe
readertoembarkonavoyageofscientificdiscovery.
Adelaide,Australia RajaR.Huilgol
Nicosia,Cyprus GeorgiosC.Georgiou
Acknowledgements
MuchofthematerialinChaps.7–10ofthismonographisderivedfromtheresearch
funded by the Australian Research Council and Moldflow Pty. Ltd. under the
LinkageGrantsduring2005–2009awardedtoRajaHuilgol.TheworkofGholam-
reza Kefayati reproduced in Chaps. 3 and 10 was supported by an International
PostgraduateResearchFellowshipfromFlindersUniversity,Adelaide,from2013to
2016.
Figures 6.1–6.4, 7.1, 7.2, 9.2–9.6, 10.1–10.8, as well as Tables 3.1, 10.2–10.8,
havebeenreproducedfromthefollowingpublications:
1. Title:JournalofNon-NewtonianFluidMechanics.Vol.Number:123.Authors:
I.A.FrigaardandD.P.Ryan.Titleofarticle:Flowofavisco-plasticfluidina
channelofslowlyvaryingwidth.PageNos:67–83.CopyrightElsevier(2004).
2. Title:JournalofNon-NewtonianFluidMechanics.Vol.Number:136.Author:
R.R.Huilgol.Titleofarticle:Asystematicproceduretodeterminetheminimum
pressure gradient required for the flow of viscoplastic fluids in pipes of
symmetriccross-section.PageNos:140–146.CopyrightElsevier(2006).
3. Title: International Journal of Heat and Mass Transfer. Vol. Number: 103.
Authors:G.H.R.KefayatiandR.R.Huilgol.Titleofarticle:LatticeBoltzmann
MethodforsimulationofmixedconvectionofaBinghamfluidinalid-driven
cavity.PageNos:725–743.CopyrightElsevier(2016).
4. Title:EuropeanJournalofMechanicsB/Fluids.Vol.Number:65.Authors:G.
H.R.KefayatiandR.R.Huilgol.Titleofarticle:LatticeBoltzmannMethodfor
simulationofthesteadyflowofaBinghamfluidinapipeofsquarecross-section.
PageNos:412–422.CopyrightElsevier(2017).
5. Title:JournalofNon-NewtonianFluidMechanics.Vol.Number:251.Authors:
R. R. Huilgol and G. H. R. Kefayati. Title of article: A particle distribution
functionapproachtotheequationsofcontinuummechanicsinCartesian,cylin-
drical and spherical coordinates: Newtonian and non-Newtonian fluids. Page
Nos:119-131.CopyrightElsevier(2018).
ix
x Acknowledgements
6. Title:JournalofNon-NewtonianFluidMechanics.Vol.Number:265.Authors:
R. R. Huilgol, A. N. Alexandrou and G. C. Georgiou. Title of article: Start-
up plane Poiseuille flow of a Bingham fluid. Page Nos: 133–139. Copyright
Elsevier(2019).
Contents
1 TheBasicFeaturesofViscoplasticity ............................ 1
1.1 BinghamFluidatRestinaChannel ......................... 1
1.2 SignoftheShearStress ................................... 2
1.3 CriticalPressureDropandtheConstitutiveRelation ........... 3
1.4 TheSolution ............................................. 5
1.5 FlowRate ............................................... 6
1.6 InherentNonlinearity ..................................... 7
1.7 Non-dimensionalisation ................................... 7
1.8 TheBuckinghamEquation ................................ 10
1.9 FreeBoundaryProblems .................................. 12
1.10 TheMinimiserandtheVariationalInequality ................. 15
1.11 EffectsofWallSlip ....................................... 17
1.12 ExperimentalEvidenceandModelling ...................... 19
References .................................................... 23
2 KinematicsofFluidFlow ...................................... 25
2.1 KinematicalPreliminaries ................................. 26
2.2 RelationBetweentheVelocityandDeformationGradients ..... 28
2.3 RigidMotion ............................................ 30
2.4 PolarDecomposition,SpinandStretching ................... 31
2.5 SteadyVelocityFieldsandTheirRivlin-EricksenTensors ...... 33
2.6 AppendixA:Divergence,Curl,Rivlin-EricksenTensor
andSpinTensor .......................................... 35
References .................................................... 37
3 FundamentalEquations:ContinuumMechanicsandLattice
BoltzmannModels ............................................ 39
3.1 Introduction ............................................. 39
3.2 ConservationofMass ..................................... 41
3.3 Cauchy’sFirstLawofMotion .............................. 41
3.4 Cauchy’sSecondLawofMotion ........................... 43
3.5 ConservationofEnergy ................................... 45
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