The well known transport laws of Navier-Stokes and Fourier fail for the simulation of processes on lengthscales in the order of the mean free path of a particle that is when the Knudsen number is not small enough. Thus, the proper simulation of flows in rarefied gases requires a more detailed description. This book discusses classical and modern methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equation, for small and moderate Knudsen numbers, i.e. at and above the Navier-Stokes-Fourier level. The main methods discussed are the classical Chapman-Enskog and Grad approaches, as well as the new order of magnitude method, which avoids the short-comings of the classical methods, but retains their benefits. The relations between the various methods are carefully examined, and the resulting equations are compared and tested for a variety of standard problems. The book develops the topic starting from the basic description of an ideal gas, over the derivation of the Boltzmann equation, towards the various methods for deriving macroscopic transport equations, and the test problems which include stability of the equations, shock waves, and Couette flow.

Back Cover Text: This book addresses the study of the gaseous state of granular matter in the conditions of rapid flow caused by a violent and sustained excitation. In this regime, grains only touch each other during collisions and hence, kinetic theory is a very useful tool to study granular flows. The main difference with respect to ordinary or molecular fluids is that grains are macroscopic and so, their collisions are inelastic. Given the interest in the effects of collisional dissipation on granular media under rapid flow conditions, the emphasis of this book is on an idealized model (smooth inelastic hard spheres) that isolates this effect from other important properties of granular systems. In this simple model, the inelasticity of collisions is only accounted for by a (positive) constant coefficient of normal restitution. The author of this monograph uses a kinetic theory description (which can be considered as a mesoscopic description between statistical mechanics and hydrodynamics) to study granular flows from a microscopic point of view. In particular, the inelastic version of the Boltzmann and Enskog kinetic equations is the starting point of the analysis. Conventional methods such as Chapman-Enskog expansion, Grad’s moment method and/or kinetic models are generalized to dissipative systems to get the forms of the transport coefficients and hydrodynamics. The knowledge of granular hydrodynamics opens up the possibility of understanding interesting problems such as the spontaneous formation of density clusters and velocity vortices in freely cooling flows and/or the lack of energy equipartition in granular mixtures. Some of the topics covered in this monograph include: Navier-Stokes transport coefficients for granular gases at moderate densities Long-wavelength instability in freely cooling flows Non-Newtonian transport properties in granular shear flows Energy nonequipartition in freely cooling granular mixtures Diffusion in strongly sheared granular mixtures Exact solutions to the Boltzmann equation for inelastic Maxwell models

This book highlights a comprehensive description of the numerical methods in rarefied gas dynamics, which has strong applications ranging from space vehicle re-entry, micro-electromechanical systems, to shale gas extraction. The book consists of five major parts: The fast spectral method to solve the Boltzmann collision operator for dilute monatomic gas and the Enskog collision operator for dense granular gas; The general synthetic iterative scheme to solve the kinetic equations with the properties of fast convergence and asymptotic preserving; The kinetic modeling of monatomic and molecular gases, and the extraction of critical gas parameters from the experiment of Rayleigh-Brillouin scattering; The assessment of the fluid-dynamics equations derived from the Boltzmann equation and typical kinetic gas-surface boundary conditions; The applications of the fast spectral method and general synthetic iterative scheme to reveal the dynamics in some canonical rarefied gas flows. The book is suitable for postgraduates and researchers interested in rarefied gas dynamics and provides many numerical codes for them to begin with.

Computational fluid dynamics (CFD) studies the flow motion in a discretized space. Its basic scale resolved is the mesh size and time step. The CFD algorithm can be constructed through a direct modeling of flow motion in such a space. This book presents the principle of direct modeling for the CFD algorithm development, and the construction unified gas-kinetic scheme (UGKS). The UGKS accurately captures the gas evolution from rarefied to continuum flows. Numerically it provides a continuous spectrum of governing equation in the whole flow regimes.

This volume is the fifth in a series of proceedings which started in 1999. The contributions include the latest results on the theory of wave propagation, extended thermodynamics, and the stability of the solutions to partial differential equations. Sample Chapter(s). Chapter 1: Reciprocal Transformations and Integrable Hamiltonian Hydrodynamic Type Systems (334 KB). Contents: Quantitative Estimates for the Large Time Behavior of a Reaction-Diffusion Equation with Rational Reaction Term (M Bisi et al.); Linearized Euler''s Variational Equations in Lagrangian Coordinates (G Boillat & Y J Peng); Restabilizing Forcing for a Diffusive Prey-Predator Model (B Buonomo & S Rionero); Fluid Dynamical Features of the Weak KAM Theory (F Cardin); Ricci Flow Deformation of Cosmological Initial Data Sets (M Carfora & T Buchert); Fuchsian Partial Differential Equations (Y Choquet-Bruhat); Analytic Structure of the Four-Wave Mixing Model in Photoreactive Material (R Conte & S Bugaychuk); A Note about Waves in Dissipative and Dispersive Solids (M Destrade & G Saccomandi); Exponential and Algebraic Relaxation in Kinetic Models for Wealth Distribution (B Dring et al.); Solitary Waves in Dispersive Materials (J Engelbrecht et al.); A GinzburgOCoLandau Model for the Ice-Water and Liquid-Vapor Phase Transitions (M Fabrizio); Stability Considerations for Reaction-Diffusion Systems (J N Flavin); A Mechanical Model for Liquid Nanolayers (H Gouin); A Particle Method for a Lotka-Volterra System with Nonlinear Cross and Self-Diffusion (M Groppi & M Sammartino); Transport Properties of Chemically Reacting Gas Mixtures (G M Kremer); Navier-Stokes in Aperture Domains: Existence with Bounded Flux and Qualitative Properties (P Maremonti); On Two-Pulse Interaction in a Class of Model Elastic Materials (A Mentrelli et al.); On a Particle-Size Segregation Equation (C Mineo & M Torrisi); Problems of Stability and Waves in Biological Systems (G Mulone); Multiple Cold and Hot Second Sound Shocks in HE II (A Muracchini & L Seccia); Differential Equations and Lie Symmetries (F Oliveri et al.); Bifurcation Analysis of Equilibria in Competitive Logistic Networks with Adaptation (A Raimondi & C Tebaldi); Poiseuille Flow of a Fluid Overlying a Porous Media (B Straughan); Analysis of Heat Conduction Phenomena in a One-Dimensional Hard-Point Gas by Extended Thermodynamics (S Tanigushi et al.); On Waves in Weakly Nonlinear Poroelastic Materials Modeling Impacts of Meteorites (K Wilmanski et al.); and other papers. Readership: Researchers in mathematics, physics, chemistry and engineering."

Model reduction and coarse-graining are important in many areas of science and engineering. How does a system with many degrees of freedom become one with fewer? How can a reversible micro-description be adapted to the dissipative macroscopic model? These crucial questions, as well as many other related problems, are discussed in this book. All contributions are by experts whose specialities span a wide range of fields within science and engineering.

This volume is the fifth in a series of proceedings which started in 1999. The contributions include the latest results on the theory of wave propagation, extended thermodynamics, and the stability of the solutions to partial differential equations.

This book presents generalized heat-conduction laws which, from a mesoscopic perspective, are relevant to new applications (especially in nanoscale heat transfer, nanoscale thermoelectric phenomena, and in diffusive-to-ballistic regime) and at the same time keep up with the pace of current microscopic research. The equations presented in the book are compatible with generalized formulations of nonequilibrium thermodynamics, going beyond the local-equilibrium. The book includes six main chapters, together with a preface and a final section devoted to the future perspectives, as well as an extensive bibliography.

This book covers concepts and the latest developments on microscale flow and heat transfer phenomena involving a gas. The book is organised in two parts: the first part focuses on the fluid flow and heat transfer characteristics of gaseous slip flows. The second part presents modelling of such flows using higher-order continuum transport equations. The Navier-Stokes equations based solution is provided to various problems in the slip regime. Several interesting characteristics of slip flows along with useful empirical correlations are documented in the first part of the book. The examples bring out the failure of the conventional equations to adequately describe various phenomena at the microscale. Thereby the readers are introduced to higher order continuum transport (Burnett and Grad) equations, which can potentially overcome these limitations. A clear and easy to follow step by step derivation of the Burnett and Grad equations (superset of the Navier-Stokes equations) is provided in the second part of the book. Analytical solution of these equations, the latest developments in the field, along with scope for future work in this area are also brought out. Presents characteristics of flow in the slip and transition regimes for a clear understanding of microscale flow problems; Provides a derivation of Navier-Stokes equations from microscopic viewpoint; Features a clear and easy to follow step-by-step approach to derive Burnett and Grad equations; Describes a complete compilation of few known exact solutions of the Burnett and Grad equations, along with a discussion of the solution aided with plots; Introduces the variants of the Navier-Stokes, Burnett and Grad equations, including the recently proposed Onsager-Burnett and O13 moment equations.