Theory and numerical approach in kinetic theory of gases (Part 1) 2018 International Graduate Summer School on “Frontiers of Applied and Computational Mathematics” (Shanghai Jiao Tong University, July 9-21, 2018) Theory and numerical approach in kinetic theory of gases (Part 1) Kazuo Aoki Dept. of Math., National Cheng Kung University, Tainan and NCTS, National Taiwan University, Taipei

Pre-introduction

Average speed ~ Sound speed ~ 340 m/s Gas Ensamble of many molecules (Avogadro number　６.02×1023) Average speed ~ Sound speed ~ 340 m/s Ordinary gas (Air in this room) Frequent collisions Mean free path (Average distance between two successive collisions) ~ 10-6 cm Average Wind (5 m/s)

Velocity distribution of gas molecules Velocity of gas molecules Velocity distribution of gas molecules Ordinary gas (frequent collision) Gaussian (Maxwellian) distribution determines the shape 1D schematic figure Number of molecules Local equilibrium Temperature Gas const. Local: vary depending on time, position Area: number density (density ) (Macroscopic) fluid mechanics

Reference state Fast flow Cold flow Low-density flow

Low-density gas Gas in microscales Collision: not frequent 　Deviation from Gaussian (local equilibrium) Measure of deviation Knudsen number Ordinary gas Odinaty size Mean free path Characteristic length

The distribution itself Boltzmann equation (1872) Fluid dynamic limit (Continuum limit) Free-molecular flow Local equilibrium (Macroscopic) fluid mechanics General : The shape is not determined by The distribution itself Boltzmann equation (1872) Ludwig Boltzmann (1844 -1906) Molecular gas dynamics Gas dynamics for the distribution itself (More general gas dynamics including ordinary gas dynamics as a limit)

Introduction

Classical kinetic theory of gases Non-mathematical (Formal asymptotics & simulations) Monatomic ideal gas, No external force Diameter (or range of influence) We assume that we can take a small volume in the gas, containing many molecules (say molecules) Negligible volume fraction Finite mean free path Binary collision is dominant. Boltzmann-Grad limit

Ordinary gas flows Fluid dynamics Local thermodynamic equilibrium Low-density gas flows (high atmosphere, vacuum) Gas flows in microscales (MEMS, aerosols) Non equilibrium Deviation from local equilibrium Knudsen number mean free path characteristic length Fluid-dynamic (continuum) limit Free-molecular flow

Molecular gas dynamics (Kinetic theory of gases) arbitrary Fluid-dynamic (continuum) limit Free-molecular flow Fluid dynamics (necessary cond.) Molecular gas dynamics (Kinetic theory of gases) arbitrary Microscopic information Boltzmann equation Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhäuser, 2002). Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications (Birkhäuser, 2007). H. Grad, “Principles of the kinetic theory of gases” in Handbuch der Physik (Springer, 1958) Band XII, 205-294 C. Cercignani, The Boltzmann equation and Its Applications (Springer, 1987). C. Cercignani, R. Illner, & M. Pulvirenti, The Mathematical Theory of Dilute Gases (Springer, 1994).

Boltzmann equation and its basic properties

Velocity distribution function time position molecular velocity Molecular mass in at time Mass density in phase space Boltzmann equation (1872)

Velocity distribution function time position molecular velocity Molecular mass in at time Macroscopic quantities density flow velocity temperature gas const. ( Boltzmann const.) stress heat flow

Nonlinear integro-differential equation Boltzmann equation collision integral [ : omitted ] Post-collisional velocities depending on molecular models Hard-sphere molecules

Inverse power intermolecular force Singular at Maxwell molecule Hard sphere Angular cutoff Hard potential Soft potential

Remarks No collision (Liouville theorem) Mass in the box:

Mass conservation Boltzmann equation for collisionless gas

collision With collision Molecular number going out during Molecular number coming in during Boltzmann equation Expressions of in terms of

Basic properties of Maxwellian (local, absolute) Conservation Entropy inequality ( H-theorem) equality

Basic properties of Maxwellian (local, absolute) Conservation Entropy inequality ( H-theorem) equality

Uniform equilibrium state Absolute Maxwellian: Uniform equilibrium state Local Maxwellian: If are such that , is an exact solution. Example: rigid-body rotation angular velocity

Maxwellian (local, absolute) Conservation Entropy inequality ( H-theorem) equality

Conservation equations (mass) (momentum) (energy) Internal energy (per unit mass) Summation convention is used

Maxwellian (local, absolute) Conservation Entropy inequality ( H-theorem) equality

Satisfying three basic properties Corresponding to Maxwell molecule Model equations BGK model Bhatnagar, Gross, & Krook (1954), Phys. Rev. 94, 511 Welander (1954), Ark. Fys. 7, 507 Satisfying three basic properties Corresponding to Maxwell molecule Drawback

Entropy inequality Andries, Le Tallec, Perlat, & Perthame (2000), ES model Holway (1966), Phys. Fluids 9, 1658 Entropy inequality Andries, Le Tallec, Perlat, & Perthame (2000), (H-theorem) Eur. J. Mech. B 19, 813 revival

Initial and boundary conditions Initial condition Boundary condition [ : omitted ] No net mass flux across the boundary

No net mass flux across the boundary (#) No net mass flux across the boundary arbitrary satisfies (#)

Conventional boundary condition [ : omitted ] Specular reflection [ does not satisfy (iii) ] Diffuse reflection No net mass flux across the boundary

Accommodation coefficient Maxwell type Accommodation coefficient Cercignani-Lampis model Cercignani & Lampis (1971), Transp. Theor. Stat. Phys. 1, 101

H-theorem H-function (Entropy inequality) Maxwellian Thermodynamic entropy per unit mass

spatially uniform never increases Boltzmann’s H theorem Direction for evolution

Darrozes-Guiraud inequality C. R. Acad. Sci., Paris A 262, 1368 Equality: Cercignani (1975)

Non-dimensionalization

Dimensionless variables Subscript 0: Reference variables mean collision frequency mean free path [Dimensional form] at equilibrium at rest

Dimensionless form (hat omitted) Strouhal number Knudsen number

Dimensionless form (Macroscopic variables)

Dimensionless form (Macroscopic variables) (hat omitted)