Theory of Multicomponent Fluids

311 Pages·1999·2.538 MB·English

by  Donald A. Drew, Stephen L. Passman (auth.)

Applied Mathematical Sciences
Volume 135
Editors
J.E.Marsden L.Sirovich
Advisors
S.Antman J.K.Hale P.Holmes
T.Kambe J.Keller K.Kirchga¨ssner
B.J.Matkowsky C.S.Peskin
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Donald A. Drew Stephen L. Passman
Theory of
Multicomponent Fluids
With40Illustrations
1 3
DonaldA.Drew StephenL.Passman
RickettsProfessorofAppliedMathematics SandiaNationalLaboratories
DepartmentofMathematicalScience P.O.Box5800
RensselaerPolytechnicInstitute Albuquerque,NM87185
Troy,NY12180-3590 USA
USA
Editors
J.E.Marsden L.Sirovich
ControlandDynamicalSystems,107-81 DivisionofAppliedMathematics
CaliforniaInstituteofTechnology BrownUniversity
Pasadena,CA91125 Providence,RI02912
USA USA
MathematicsSubjectClassification(1991):76T05,35B20
LibraryofCongressCataloging-in-PublicationData
Drew,DonaldA.(DonaldAllen),1945–
Theoryofmulticomponentfluids/DonaldA.Drew,StephenL.
Passman
p. cm.—(Appliedmathematicalsciences;135)
Includesbibliographicalreferencesandindex.
ISBN0-387-98380-5(hardcover:alk.paper)
1.Multiphaseflow. 2.Continuummechanics. I.Passman,Stephen
L. II.Title. III.Series:Appliedmathematicalsciences(Springer-
VerlagNewYork,Inc.);v.135.
TA357.5.M84D74 1998
620.1’06—dc21 98-18392
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ISBN0-387-98380-5 Springer-Verlag NewYork Berlin Heidelberg SPIN10657516
Contents
ListofFigures ix
Introduction 1
I Preliminaries 11
1 PhysicalReality,CorpuscularModels,ContinuumModels 13
2 ClassicalContinuumTheory 20
2.1. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2. BalanceEquations . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3. JumpConditions . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4. Frames,FrameIndifference,Objectivity . . . . . . . . . . . . 31
2.5. ConstitutiveEquations,Well-PosednessofBoundary-Value
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6. RepresentationTheorems . . . . . . . . . . . . . . . . . . . . 35
2.7. ThermodynamicProcesses . . . . . . . . . . . . . . . . . . . 36
3 ViscousandInviscidFluidsandElasticSolids 41
3.1. Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2. Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 KineticTheory 48
4.1. CollisionOperator . . . . . . . . . . . . . . . . . . . . . . . . 49
vi Contents
4.2. TheH-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3. MaxwellianDistribution . . . . . . . . . . . . . . . . . . . . . 53
4.4. SlightDisequilibrium . . . . . . . . . . . . . . . . . . . . . . 53
4.5. DenseGases . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 ClassicalTheoryofSolutions 59
II ContinuumTheory 63
6 ContinuumBalanceEquationsforMulticomponentFluids 65
6.1. MulticomponentMixtures . . . . . . . . . . . . . . . . . . . . 65
6.2. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3. BalanceEquations . . . . . . . . . . . . . . . . . . . . . . . . 68
6.4. MulticomponentEntropyInequalities . . . . . . . . . . . . . . 74
7 MixtureEquations 81
III AveragingTheory 85
8 Introduction 87
8.1. LocalConservationEquations . . . . . . . . . . . . . . . . . . 88
8.2. JumpConditions . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.3. SummaryoftheExactEquations . . . . . . . . . . . . . . . . 91
9 EnsembleAveraging 92
9.1. ResultsforEnsembleAveraging. . . . . . . . . . . . . . . . . 99
10 OtherAverages 105
10.1. MultiparticleDistributionFunctions . . . . . . . . . . . . . . 106
10.2. TimeAveraging . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3. VolumeAveraging . . . . . . . . . . . . . . . . . . . . . . . . 116
10.4. Desiderata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
11 AveragedEquations 121
11.1. AveragingBalanceEquations . . . . . . . . . . . . . . . . . . 121
11.2. DefinitionofAverageVariables . . . . . . . . . . . . . . . . . 122
11.3. AveragedBalanceEquations . . . . . . . . . . . . . . . . . . 126
12 PostulationalandAveragingApproaches 131
IV ModelingMulticomponentFlows 135
13 Introduction 137
Contents vii
14 ClosureFramework 140
14.1. CompletenessoftheFormulation . . . . . . . . . . . . . . . . 140
14.2. ConstitutiveEquations . . . . . . . . . . . . . . . . . . . . . . 141
14.3. FormsforConstitutiveEquations . . . . . . . . . . . . . . . . 144
14.4. EntropyRestrictions . . . . . . . . . . . . . . . . . . . . . . . 147
15 RelationofMicrostructuretoConstitutiveEquations 153
15.1. AveragingTechniques . . . . . . . . . . . . . . . . . . . . . . 154
15.2. CellModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
15.3. InviscidFluidFlowingAroundaSphere . . . . . . . . . . . . 162
15.4. ViscousFlowAroundaSphere . . . . . . . . . . . . . . . . . 177
16 Maxwell–BoltzmannDynamics 193
16.1. CollisionEffects . . . . . . . . . . . . . . . . . . . . . . . . . 194
16.2. FluidVelocityEffects . . . . . . . . . . . . . . . . . . . . . . 194
17 InterfacialArea 199
17.1. GeometryModels . . . . . . . . . . . . . . . . . . . . . . . . 200
17.2. EvolutionofGeometricStatistics . . . . . . . . . . . . . . . . 205
17.3. AverageGeometricalProperties . . . . . . . . . . . . . . . . . 210
17.4. ACoalescenceandBreakupModelandGeometry . . . . . . . 217
17.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
18 EquationsofMotionforDiluteFlow 221
18.1. ConstitutiveEquations . . . . . . . . . . . . . . . . . . . . . . 221
18.2. DispersedFlowEquationsofMotion . . . . . . . . . . . . . . 231
V Consequences 235
19 NatureoftheEquations 237
19.1. SpecialCasesoftheEquations . . . . . . . . . . . . . . . . . 238
20 Well-Posedness 243
20.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
20.2. CharacteristicValues . . . . . . . . . . . . . . . . . . . . . . 244
20.3. TheSimplestModel . . . . . . . . . . . . . . . . . . . . . . . 248
20.4. EffectofViscosity . . . . . . . . . . . . . . . . . . . . . . . . 249
20.5. InertialEffects . . . . . . . . . . . . . . . . . . . . . . . . . . 251
20.6. SummaryObservations . . . . . . . . . . . . . . . . . . . . . 252
21 SolutionsforShearingFlows 254
21.1. FieldEquations . . . . . . . . . . . . . . . . . . . . . . . . . 255
21.2. KinematicsandDynamicsofShearingFlow . . . . . . . . . . 258
viii Contents
22 WaveDynamics 273
22.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
22.2. AcousticPropagation . . . . . . . . . . . . . . . . . . . . . . 274
22.3. VolumetricWaves . . . . . . . . . . . . . . . . . . . . . . . . 282
22.4. CharacteristicsandLinearStability . . . . . . . . . . . . . . . 286
22.5. InletStepResponse . . . . . . . . . . . . . . . . . . . . . . . 288
22.6. NonlinearWaves . . . . . . . . . . . . . . . . . . . . . . . . . 294
References 297
Index 303
List of Figures
1.1 Mass-to-volumeratiosforvarioussizedballs . . . . . . . . . . 15
2.1 A“flake”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Atetrahedron. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Atypicalconfiguration. . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Atypicalconfigurationwithacut. . . . . . . . . . . . . . . . . 27
2.5 Anotherconfigurationwithacut. . . . . . . . . . . . . . . . . . 27
2.6 Cylindricalvolumeillustratingbalancesacrossinterfaces. . . . . 31
4.1 Collisiongeometry. . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Cylindricalvolumeforpotentialcollisions. . . . . . . . . . . . 51
4.3 ExcludedVolume . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Shieldingeffect . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.1 Configurationsandmappings. . . . . . . . . . . . . . . . . . . 68
9.1 Realizations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.1 Excludedvolume . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.2 Timehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
15.1 Nearest-neighborand“tophat”distributions . . . . . . . . . . . 160
15.2 A“cell” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
15.3 Cellaveragingoftheinterfacialforce. . . . . . . . . . . . . . . 166
15.4 Volumeaverage.. . . . . . . . . . . . . . . . . . . . . . . . . . 176
15.5 Cellapproximationtothevolumeaverage. . . . . . . . . . . . . 176
x ListofFigures
15.6 Effectiveviscosity. . . . . . . . . . . . . . . . . . . . . . . . . 190
17.1 Steady-statevaluesofs versusα . . . . . . . . . . . . . . . . . 205
d
17.2 Principalcoordinatesandradiiofcurvature. . . . . . . . . . . . 207
17.3 Surfacecoalescence. . . . . . . . . . . . . . . . . . . . . . . . 216
19.1 Momentumbalanceformixture . . . . . . . . . . . . . . . . . 239
20.1 Normalandtangentialcoordinates. . . . . . . . . . . . . . . . . 244
20.2 Characteristicvalues . . . . . . . . . . . . . . . . . . . . . . . 252
21.1 Simpleshearingflow. . . . . . . . . . . . . . . . . . . . . . . . 262
21.2 Volumefractionprofile. . . . . . . . . . . . . . . . . . . . . . . 271
21.3 Shearstress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
22.1 Speedofsoundversusvolumefraction. . . . . . . . . . . . . . 279
22.2 Speedofsoundversusfrequency. . . . . . . . . . . . . . . . . . 282
22.3 Attenuationversusfrequency.. . . . . . . . . . . . . . . . . . . 283
22.4 Spreadingcharacteristics. . . . . . . . . . . . . . . . . . . . . . 285
22.5 Convergingcharacteristics,leadingtoashock. . . . . . . . . . . 285
22.6 “Density”oftheconservedmomentum. . . . . . . . . . . . . . 290
22.7 Evolutionofaninitialdiscontinuity . . . . . . . . . . . . . . . 291
22.8 Shockandrarefactionloci. . . . . . . . . . . . . . . . . . . . . 292
22.9 Travelingwaveprofile. . . . . . . . . . . . . . . . . . . . . . . 295