Thermodynamics of Flowing Systems with Internal Microstructure

, by Beris, Antony N.; Edwards, Brian J.

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ISBN: 9780195076943 | 019507694X
Cover: Hardcover
Copyright: 5/26/1994

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This much-needed monograph presents a systematic, step-by-step approach to the continuum modeling of flow phenomena exhibited within materials endowed with a complex internal microstructure, such as polymers and liquid crystals. By combining the principles of Hamiltonian mechanics with those of irreversible thermodynamics, Antony N. Beris and Brian J. Edwards, renowned authorities on the subject, expertly describe the complex interplay between conservative and dissipative processes. Throughout the book, the authors emphasize the evaluation of the free energy--largely based on ideas from statistical mechanics--and how to fit the values of the phenomenological parameters against those of microscopic models. With Thermodynamics of Flowing Systems in hand, mathematicians, engineers, and physicists involved with the theoretical study of flow behavior in structurally complex media now have a superb, self-contained theoretical framework on which to base their modeling efforts.

Theory	p. 1
Introduction	p. 3
Overview	p. 3
The challenge of multiple time and length scales	p. 7
The energy as the fundamental quantity	p. 13
The generalized bracket approach	p. 15
A simple application: The damped oscillator	p. 18
Symplectic geometry in optics	p. 23
Introduction	p. 23
Theories of optics	p. 25
Symplectic structure	p. 26
The symplectic vector space	p. 26
The symplectic transformation	p. 28
Gaussian and linear optics	p. 29
Gaussian optics	p. 29
Linear optics	p. 36
Geometrical optics	p. 37
The symplectic structure of geometrical optics	p. 37
Fermat's principle	p. 40
An overview of wave optics and electromagnetism	p. 41
Hamiltonian mechanics of discrete particle systems	p. 45
The calculus of variations	p. 45
Hamilton's principle of least action	p. 47
The Poisson bracket description of Hamilton's equations of motion	p. 52
Properties of the Poisson bracket	p. 53
The Liouville equation	p. 58
The optical/mechanical analogy	p. 59
A historical aside on the principle of least action	p. 59
Fermat's reception	p. 61
Maupertuis' contribution	p. 62
Hamilton on the principle of least action	p. 64
Equilibrium thermodynamics	p. 67
The fundamental equation of thermodynamics	p. 67
Other fundamental relationships of thermodynamics	p. 71
The fundamental equation for a multicomponent system	p. 76
Extensive variable formulation	p. 76
Specific variable formulation	p. 78
Density variable formulation	p. 80
Equilibrium thermodynamics of a material with internal microstructure	p. 81
Additivity in compound systems	p. 83
Poisson brackets in continuous media	p. 87
The material description of ideal fluid flow	p. 88
The canonical Poisson bracket for ideal fluid flow	p. 94
The calculus of functionals	p. 94
The continuum Poisson bracket	p. 96
The spatial description of ideal fluid flow	p. 97
Ideal fluid flow with constraints: The incompressible fluid	p. 108
Nonlinear elasticity	p. 113
The material description of nonlinear elasticity	p. 113
Spatial bracket derivation	p. 114
Spatial bracket in terms of a tensor with unit trace	p. 118
The relation between thermodynamics and hydrodynamics	p. 122
The global system	p. 122
The compound system	p. 124
Non-equilibrium thermodynamics	p. 125
Irreversibility and stability	p. 126
Systems with internal variables	p. 128
The Clausius inequality	p. 132
Non-equilibrium thermodynamics of flowing systems	p. 134
The entropy balance equation	p. 134
The linear phenomenological relations	p. 137
Frame indifference and material invariance	p. 138
The principle of microscopic reversibility	p. 143
The nature of affinities and fluxes	p. 144
The Onsager/Casimir reciprocal relations	p. 149
Einstein's fluctuation theory	p. 150
The Onsager/Casimir reciprocal relations	p. 153
Transformation properties of the reciprocal relations	p. 157
Affinities and fluxes for continua	p. 160
Density description of the entropy production rate	p. 160
Description of the entropy production rate in terms of fluxes	p. 162
The dissipation bracket	p. 165
The general dissipation bracket	p. 167
Definition and properties of the dissipation bracket	p. 169
Equivalence of the energetic and entropic formalisms	p. 178
The hydrodynamic equations for a single-component system	p. 182
The general isotropic fluid	p. 183
The incompressible Navier/Stokes equations	p. 189
The hydrodynamic equations for a polar fluid	p. 195
The hydrodynamic equations for a multicomponent fluid	p. 198
Applications	p. 207
Incompressible viscoelastic fluids	p. 209
Incompressible and isothermal viscoelastic fluid models in terms of a single conformation tensor	p. 216
Simple phenomenological spring/dashpot-type models	p. 222
Elastic dumbbell models of dilute polymer solutions arising through kinetic theory	p. 236
Single internal variable models for concentrated polymer solutions and melts	p. 252
Dissipation and admissibility criteria for single-mode viscoelastic fluids	p. 264
The non-negative character of the conformation tensor for single-variable viscoelastic fluids	p. 272
Evolutionary character of the governing equations of single-variable viscoelastic fluids	p. 281
Incompressible viscoelastic fluid models in terms of multiple conformation tensors	p. 288
Guidelines on multiple-variable viscoelastic fluid models from the perspective of linear viscoelasticity	p. 289
Phenomenological multiple-variable models	p. 299
The bead/spring-chain model of kinetic theory	p. 320
Transport phenomena in viscoelastic fluids	p. 327
Compressible and non-isothermal viscoelastic fluid models	p. 328
Modeling of the rheology and flow-induced concentration changes in polymer solutions	p. 335
Single-fluid model of polymer-solution hydro-dynamics allowing for concentration variations	p. 336
Two-fluid model for polymer solutions	p. 341
Comparison of various theories for polymer solution hydrodynamics	p. 347
Surface effects on the microstructure and concentration in incompressible and isothermal viscoelastic fluid flows	p. 353
Introduction	p. 354
Derivation of the governing equations in terms of the Hamiltonian	p. 358
Determination of the Hamiltonian functional	p. 361
Calculations with the resulting model equations	p. 369
Comparison with available experimental data and previous theoretical work	p. 382
Effects of surface adsorption	p. 387
Non-conventional transport phenomena	p. 397
Relaxational phenomena in heat and mass transfer	p. 398
Flux-relaxation models	p. 400
The bracket formulation of relaxational phenomena in heat conduction	p. 403
The bracket description of relaxational phenomena in mass diffusion	p. 409
Heat and mass flux relaxation in the kinetic theory of gases	p. 414
Relaxational mass transfer in polymeric systems	p. 423
Phase transitions in inhomogeneous media	p. 441
The kinetics of phase transitions	p. 441
The bracket description of the kinetics of phase transitions	p. 444
The inertial description of incompressible viscoelastic fluids	p. 447
The dynamical theory of liquid crystals	p. 453
Introduction to liquid crystals	p. 456
Thermodynamics of liquid crystals under static conditions	p. 474
Equilibrium thermodynamics of homogeneous liquid crystals	p. 475
The Oseen/Frank description of non-homogeneous nematic liquid crystals	p. 487
The LE and Doi models for flowing liquidcrystalline systems	p. 499
The Leslie/Ericksen theory	p. 500
The scalar/vector theory	p. 506
The Doi theory	p. 507
The Leslie coefficients: Implications for flow behavior and molecular determination	p. 513
The bracket description of the LE theory	p. 526
The inertial form	p. 526
The non-inertial form	p. 532
The conformation tensor theory	p. 535
The inertial form	p. 537
The non-inertial form	p. 546
Comparison of the conformation tensor theory to previous theories	p. 550
Reduction to the Leslie/Ericksen theory	p. 551
Reduction to the scalar-vector theory	p. 552
Reduction to the original Doi theory	p. 556
Reduction to the extended Doi theory	p. 560
Equivalent Leslie coefficients for the general conformation tensor theory	p. 566
Concluding remarks	p. 569
Multi-fluid transport/reaction models with application in the modeling of weakly ionized plasma dynamics	p. 571
Introduction	p. 574
Elements from the modeling of weakly ionized plasma dynamics	p. 574
Multi-fluid descriptions of nonlinear kinetics	p. 576
The non-dissipative multi-fluid system	p. 577
The dissipative multi-fluid system	p. 583
Chemical reactions in a multicomponent single-fluid system	p. 591
Chemical reactions in multi-fluid systems	p. 594
Weakly ionized plasma model	p. 599
Conclusions	p. 601
Epilogue	p. 603
Introduction to differential manifolds	p. 605
Differential manifolds	p. 605
Curves	p. 606
Tangent spaces	p. 607
Differential forms	p. 608
Symplectic forms and symplectic transformations	p. 611
The Lagrangian manifold	p. 613
The Hamiltonian vector field	p. 614
The Poisson bracket	p. 617
The Legendre dual transformation	p. 621
Poisson brackets for arbitrary second-rank deformation tensors	p. 623
Calculations of the random-flight model	p. 627
Calculation of the end-to-end distribution function near a solid surface	p. 627
Calculation of the partition function	p. 628
Bibliography	p. 631
Author Index	p. 671
Subject Index	p. 679
Table of Contents provided by Syndetics. All Rights Reserved.

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