Multicomponent Flow Modeling

The goal of this is book to give a detailed presentation of multicomponent flow models and to investigate the mathematical structure and properties of the resulting system of partial differential equations. These developments are also illustrated by simulating numerically a typical laminar flame. Our aim in the chapters is to treat the general situation of multicomponent flows, taking into account complex chemistry and detailed transport phe­ nomena. In this book, we have adopted an interdisciplinary approach that en­ compasses a physical, mathematical, and numerical point of view. In par­ ticular, the links between molecular models, macroscopic models, mathe­ matical structure, and mathematical properties are emphasized. We also often mention flame models since combustion is an excellent prototype of multicomponent flow. This book still does not pretend to be a complete survey of existing models and related mathematical results. In particular, many subjects like multi phase-flows , turbulence modeling, specific applications, porous me­ dia, biological models, or magneto-hydrodynamics are not covered. We rather emphasize the fundamental modeling of multicomponent gaseous flows and the qualitative properties of the resulting systems of partial dif­ ferential equations. Part of this book was taught at the post-graduate level at the Uni­ versity of Paris, the University of Versailles, and at Ecole Poly technique in 1998-1999 to students of applied mathematics.

1. Introduction. - 2. Fundamental Equations. - 2. 1. Introduction. - 2. 2. Conservation equations. - 2. 3. Thermodynamics. - 2. 4. Chemistry. - 2. 5. Transport fluxes. - 2. 6. Entropy. - 2. 7. Boundary conditions. - 2. 8. Notes. - 2. 9. References. - 3. Approximate and Simplified Models. - 3. 1. Introduction. - 3. 2. One-reaction chemistry. - 3. 3. Small Mach number flows. - 3. 4. Coupling. - 3. 5. Notes. - 3. 6. References. - 4. Derivation from the Kinetic Theory. - 4. 1. Introduction. - 4. 2. Kinetic framework. - 4. 3. Kinetic entropy. - 4. 4. Enskog expansion. - 4. 5. Zero-order approximation. - 4. 6. First-order approximation. - 4. 7. Transport linear systems. - 4. 8. Notes. - 4. 9. References. - 5. Transport Coefficients. - 5. 1. Introduction. - 5. 2. Transport algorithms. - 5. 3. Molecular parameters. - 5. 4. Shear viscosity. - 5. 5. Volume viscosity. - 5. 6. Diffusion matrix. - 5. 7. Thermal conductivity. - 5. 8. Thermal diffusion ratios. - 5. 9. Partial thermal conductivity. - 5. 10. Thermal diffusion coefficients. - 5. 11. Notes. - 5. 12. References. - 6. Mathematics of Thermochemistry. - 6. 1. Introduction. - 6. 2. Thermodynamics with volume densities. - 6. 3. Thermodynamics with mass densities. - 6. 4. Chemistry sources. - 6. 5. Positive equilibrium points. - 6. 6. Boundary equilibrium points. - 6. 7. Inequalities near equilibrium. - 6. 8. A global stability inequality. - 6. 9. Notes. - 6. 10. References. - 7. Mathematics of Transport Coefficients. - 7. 1. Introduction. - 7. 2. Assumptions on transport coefficients. - 7. 3. Properties of diffusion matrices. - 7. 4. Properties of other coefficients. - 7. 5. Diagonal diffusion. - 7. 6. Diffusion inequalities. - 7. 7. Stefan-Maxwell equations. - 7. 8. Notes. - 7. 9. References. - 8. Symmetrization. - 8. 1. Introduction. - 8. 2. Vector notation. - 8. 3. Quasilinear form. - 8. 4. Symmetrization and entropic variables. - 8. 5. Normal forms. - 8. 6. Symmetrization for multicomponent flows. - 8. 7. Normal forms for multicomponent flows. - 8. 8. Notes. - 8. 9. References. - 9. Asymptotic Stability. - 9. 1. Introduction. - 9. 2. Governing equations. - 9. 3. Local dissipative structure. - 9. 4. Global existence theorem. - 9. 5. Decay estimates. - 9. 6. Local dissipativity for multicomponent flows. - 9. 7. Global existence for multicomponent flows. - 9. 8. Notes. - 9. 9. References. - 10. Chemical Equilibrium Flows. - 10. 1. Introduction. - 10. 2. Governing equations. - 10. 3. Entropy and symmetrization. - 10. 4. Normal forms. - 10. 5. Global existence. - 10. 6. Notes. - 10. 7. References. - 11. Anchored Waves. - 11. 1. Introduction. - 11. 2. Governing equations. - 11. 3. First properties. - 11. 4. Existence on a bounded domain. - 11. 5. Existence of solutions. - 11. 6. Notes. - 11. 7. References. - 12. Numerical Simulations. - 12. 1. Introduction. - 12. 2. Laminar flame model. - 12. 3. Computational considerations. - 12. 4. Hydrogen-Air Bunsen flame. - 12. 5. References.