Mathematical Thermodynamics of Complex Fluids Cetraro, Italy 2015

• Main Author: Ball, John M
• Format: eBook Book
• Language: English
• Published: Cham Springer Nature 2017; Springer; Springer International Publishing AG; Springer International Publishing
• Edition: 1
• Series: Lecture Notes in Mathematics
• Subjects: Engineering Fluid Dynamics; Fluid mechanics-Congresses; Mathematical Physics; Mathematics; Mathematics and Statistics; Mathematics. Analysis; Partial Differential Equations; Soft and Granular Matter, Complex Fluids and Microfluidics; Solid Mechanics
• ISBN: 3319676008; 9783319676005; 9783319675992; 3319675990
• ISSN: 0075-8434; 1617-9692
• DOI: 10.1007/978-3-319-67600-5

The main goal of this book is to provide an overview of the state of the art in the mathematical modeling of complex fluids, with particular emphasis on its thermodynamical aspects. The central topics of the text, the modeling, analysis and numerical simulation of complex fluids, are of great intere...

Table of Contents:

• Intro -- Preface to CIME Lecture Notes -- Cetraro, June 2015 -- Contents -- Liquid Crystals and Their Defects -- 1 Introduction -- 2 What Are Liquid Crystals? -- 3 Models and Order Parameters -- 3.1 Molecular Dynamics -- 3.2 Order Parameters -- 4 The Isotropic to Nematic Phase Transition -- 4.1 Description Using the Probability Density Function -- 4.2 Description Using a Q-Tensor Model -- 4.3 Satisfaction of the Eigenvalue Constraints -- 5 The Landau-de Gennes Theory -- 5.1 Frame-Indifference and Material Symmetry -- 5.2 Bulk and Elastic Energies -- 6 The Constrained Landau-de Gennes and Oseen-Frank Theories -- 7 Boundary Conditions -- 7.1 Constrained Landau-de Gennes and Oseen-Frank -- 7.2 Landau-de Gennes -- 8 Orientability -- 9 Existence of Minimizers in the General Landau-de Gennes Theory -- 10 Description of Defects -- 10.1 Summary of Liquid Crystal Models -- 10.2 Function Spaces -- 10.3 Point Defects -- 10.4 Line Defects -- 10.5 Planar Defects -- 10.5.1 Nematic Elastomers -- 10.5.2 Order Reconstruction -- 10.5.3 Smectic Thin Films -- 10.5.4 Recovering Orientability -- References -- Mathematical Thermodynamics of Viscous Fluids -- 1 Fluids in Continuum Mechanics -- 1.1 Fluids in Equilibrium -- 1.2 Fluids in Motion -- 1.3 Field Equations -- 1.3.1 Mass Conservation -- 1.3.2 Momentum Balance -- 1.3.3 Energy and Entropy -- 1.4 Boundary Behavior -- 2 Mathematics of Viscous Compressible Fluids -- 2.1 Equation of Continuity -- 2.2 Momentum Equation -- 2.3 Energy-Entropy -- 2.3.1 Entropy Based Weak Formulation -- 2.3.2 Thermal Energy Weak Formulation -- 2.4 Constitutive Relations, Navier-Stokes-Fourier System -- 3 Well-Posedness, Approximation Scheme -- 3.1 An Approximation Scheme for the Navier-Stokes-Fourier System -- 3.1.1 Hypotheses -- 3.2 Time Discretization -- 3.3 Space Discretization -- 3.3.1 Mesh, Triangulation -- 3.3.2 FEM Structure
• 3.3.2 Rigorous Argument [à la Choffrut-Nobili-O '14] -- 3.4 Our New Bound -- 3.4.1 Ignoring Logarithms -- 3.4.2 Rigorous Argument -- References
• 3.3.3 FVM Structure -- 3.4 Approximation Scheme -- 3.5 Existence of Weak Solutions via the Numerical Scheme -- 3.6 Existence for the Entropy Formulation -- 3.6.1 Constitutive Relations -- 3.6.2 Global Existence -- 4 A Priori Estimates for the Entropy Formulation -- 4.1 Total Mass Conservation -- 4.2 Energy Estimates -- 4.3 A Priori Estimates Based on Energy Dissipation -- 4.4 Pressure Estimates -- 5 Weak Sequential Stability of the Solution Set of the Navier-Stokes-Fourier System -- 5.1 Div-Curl Lemma -- 5.2 Strong Convergence of the Temperature -- 5.3 Strong Convergence of the Density -- 6 Relative Energy, Dissipative Solutions, Stability -- 6.1 Relative (Modulated) Energy -- 6.2 Dissipative Solutions -- 6.3 Weak-Strong Uniqueness -- 6.4 Weak Solutions Based on the Thermal Energy Formulation -- 6.5 Synergy Analysis-Numerics -- 7 Viscosity Solutions, Inviscid Limits -- 7.1 Euler-Fourier System -- 7.1.1 Infinitely Many Weak Solutions -- 7.1.2 Infinitely Many Admissible Weak Solutions -- 7.2 Riemann Problem -- 7.2.1 Riemann Problem for the Full Euler System -- 7.3 Viscosity Solutions -- 7.4 Vanishing Dissipation Limit for the Navier-Stokes-Fourier System -- 7.4.1 Navier-Stokes-Fourier System -- 7.4.2 Relative Energy Inequality -- 7.4.3 Vanishing Dissipation Limit -- References -- Rigorous Bounds on Scaling Laws in Fluid Dynamics -- 1 Main Result on Thermal Convection -- 2 Derivation of the Model -- 2.1 A Stability Criterion -- 2.2 Dynamics and Boussinesq Approximation -- 2.3 Conservation Laws and Dissipation -- 2.4 Concavity of the Entropy Function -- 2.5 Boussinesq and Dissipation -- 2.6 Nondimensional Parameters and Nondimensionalization -- 3 Sketch of the Proof of Theorem 1 -- 3.1 Identities and Inequalities Involving Nu and Ra -- 3.2 The Constantin-Doering '96 Bound -- 3.3 The Constantin-Doering '99 Bound (Pr=∞) -- 3.3.1 Ignoring Logarithms