Author | Cercignani, Carlo. author |
---|---|
Title | The Boltzmann Equation and Its Applications [electronic resource] / by Carlo Cercignani |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1039-9 |
Descript | XIII, 455 p. online resource |
I. Basic Principles of The Kinetic Theory of Gases -- 1. Introduction -- 2. Probability -- 3. Phase space and Liouvilleโs theorem -- 4. Hard spheres and rigid walls. Mean free path -- 5. Scattering of a volume element in phase space -- 6. Time averages, ergodic hypothesis and equilibrium states -- References -- II. The Boltzmann Equation -- 1. The problem of nonequilibrium states -- 2. Equations for the many particle distribution functions for a gas of rigid spheres -- 3. The Boltzmann equation for rigid spheres -- 4. Generalizations -- 5. Details of the collision term -- 6. Elementary properties of the collision operator. Collision invariants -- 7. Solution of the equation Q(f,f) = 0 -- 8. Connection between the microscopic description and the macroscopic description of gas dynamics -- 9. Non-cutoff potentials and grazing collisions. Fokker-Planck equation -- 10. Model equations -- References -- III. Gas-Surface Interaction and the H-Theorem -- 1. Boundary conditions and the gas-surface interaction -- 2. Computation of scattering kernels -- 3. Reciprocity -- 4. A remarkable inequality -- 5. Maxwellโs boundary conditions. Accommodation coefficients -- 6. Mathematical models for gas-surface interaction -- 7. Physical models for gas-surface interaction -- 8. Scattering of molecular beams -- 9. The H-theorem. Irreversibility -- 10. Equilibrium states and Maxwellian distributions -- References -- IV, Linear Transport -- 1. The linearized collision operator -- 2. The linearized Boltzmann equation -- 3. The linear Boltzmann equation. Neutron transport and radiative transfer -- 4. Uniqueness of the solution for initial and boundary value problems -- 5. Further investigation of the linearized collision term -- 6. The decay to equilibrium and the spectrum of the collision operator -- 7. Steady one-dimensional problems. Transport coefficients -- 8. The general case -- 9. Linearized kinetic models -- 10. The variational principle -- 11. Greenโs function -- 12. The integral equation approach -- References -- V. Small and Large Mean Free Paths -- 1. The Knudsen number -- 2. The Hilbert expansion -- 3. The Chapman-Enskog expansion -- 4. Criticism of the Chapman-Enskog method -- 5. Initial, boundary and shock layers -- 6. Further remarks on the Chapman-Enskog method and the computation of transport coefficients -- 7. Free molecule flow past a convex body -- 8. Free molecule flow in presence of nonconvex boundaries -- 9. Nearly free-molecule flows -- References -- VI. Analytical Solutions of Models -- 1. The method of elementary solutions -- 2. Splitting of a one-dimensional model equation -- 3. Elementary solutions of the simplest transport equation -- 4. Application of the general method to the Kramers and Milne problems -- 5. Application to the flow between parallel plates and the critical problem of a slab -- 6. Unsteady solutions of kinetic models with constant collision frequency -- 7. Analytical solutions of specific problems -- 8. More general models -- 9. Some special cases -- 10. Unsteady solutions of kinetic models with velocity dependent collision frequency -- 11. Analytic continuation -- 12. Sound propagation in monatomic gases -- 13. Two-dimensional and three-dimensional problems. Flow past solid bodies -- 14. Fluctuations and light scattering -- References -- VII. The Transition Regime -- 1. Introduction -- 2. Moment and discrete ordinate methods -- 3. The variational method -- 4. Monte Carlo methods -- 5. Problems of flow and heat transfer in regions bounded by planes or cylinders -- 6. Shock-wave structure -- 7. External flows -- 8. Expansion of a gas into a vacuum -- References -- VIII. Theorems on the Solutions of the Boltzmann Equation -- 1. Introduction -- 2. The space homogeneous case -- 3. Mollified and other modified versions of the Boltzmann equation -- 4. Nonstandard analysis approach to the Boltzmann equation -- 5. Local existence and validity of the Boltzmann equation -- 6. Global existence near equilibrium -- 7. Perturbations of vacuum -- 8. Homoenergetic solutions -- 9. Boundary value problems. The linearized and weakly nonlinear cases -- 10. Nonlinear boundary value problems -- 11. Concluding remarks -- References -- References -- Author Index