Theory and numerical approach in kinetic theory of gases (Part 1)

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

2. Pre-introduction

3. 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)

4. 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 ontime, position Area: number density (density ) (Macroscopic) fluid mechanics

5. Reference state Fast flow Cold flow Low-density flow

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

7. 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 dynamicsincluding ordinary gas dynamics as a limit)

8. Introduction

9. 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

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

11. Free-molecular flow Fluid-dynamic (continuum) limit (necessary cond.) Fluid dynamics arbitrary Molecular gas dynamics (Kinetic theory of gases) 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).

12. Boltzmann equation and its basic properties

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

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

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

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

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

18. Mass conservation Boltzmann equation for collisionlessgas

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

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

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

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

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

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

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

30. 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

31. ES model Holway (1966), Phys. Fluids9, 1658 Entropy inequalityAndries, Le Tallec, Perlat, & Perthame (2000), (H-theorem) Eur. J. Mech. B19, 813 revival

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

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

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

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

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

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

38. Darrozes-Guiraud inequality Darrozes & Guiraud (1966) C. R. Acad. Sci., Paris A262, 1368 Equality: Cercignani (1975)

39. Non-dimensionalization

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

41. Dimensionless form (hat omitted) Strouhal number Knudsen number

42. Dimensionless form (Macroscopic variables)

43. Dimensionless form (Macroscopic variables) (hat omitted)