1. 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

2. Pre-introduction

3. 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)

4. 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

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

6. 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

7. 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
( )
Molecular gas dynamics
Gas dynamics for the distribution itself (More general
gas dynamics including 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. 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

11. 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,
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
time
position
molecular velocity
Molecular mass in
at time
Mass density in
phase space
Boltzmann equation (1872)

14. 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

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
collisionless gas

19. collision
With 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. Uniform equilibrium state
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. 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

31. 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

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
(#)
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. Accommodation coefficient
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
Boltzmann’s H theorem
Direction for evolution

38. Darrozes-Guiraud inequality
C. R. Acad. Sci., Paris A 262, 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)