Boundary-value problems ofthe Boltzmann equation:

Asymptotic and numerical analyses(Part 1)

Kazuo Aoki

Dept. of Mech. Eng. and Sci.

Kyoto University

Intensive Lecture Series(Postech, June 20-21, 2011)

Introduction

We assume that we can take a smallvolume in the gas, containing manymolecules (say molecules)

Monatomic ideal gas, No external force

Classical kinetic theory of gasesNon-mathematical (Formal asymptotics & simulations)

Diameter (or range of influence)

Negligible volume fraction

Finite mean free path

Binary collision is dominant.

Boltzmann-Grad limit

mean free path characteristic length

Ordinary gas flows Fluid dynamicsLocal thermodynamic equilibrium

Low-density gas flows (high atmosphere, vacuum)Gas flows in microscales (MEMS, aerosols)

Non equilibrium

Deviation from local equilibrium Knudsen number

Fluid-dynamic(continuum) limit

Free-molecularflow

Fluid-dynamic(continuum) limit

Free-molecularflow

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, 205-294C. 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 andits basic properties

Velocity distribution function

time position molecular velocity

Molecular mass in at time

Mass density inphase space

Boltzmann equation (1872)

Velocity distribution function

time position molecular velocity

Macroscopic quantities

Molecular mass in at time

gas const. ( Boltzmann const.)

density

flow velocity

temperature

stress

heat flow

collisionintegral

Post-collisionalvelocities

Boltzmann equation Nonlinear integro-differentialequation

depending onmolecular models

[ : omitted ]

Hard-spheremolecules

Conservation

Entropy inequality( H-theorem)

Basic properties of

Maxwellian (local, absolute)

equality

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

ES model Holway (1966), Phys. Fluids 9, 1658

Entropy inequality Andries et al. (2000), Eur. J. Mech. B 19, 813 revival

[ : omitted ]

Initial condition

Boundary condition

No net mass flux across the boundary

Initial and boundary conditions

No net mass flux across the boundary

(#)

satisfies (#)

arbitrary

[ : omitted ]

Conventional boundary condition

Specular reflection

Diffuse reflection

No net mass flux across the boundary

[ does not satisfy (iii) ]

Maxwell type

Accommodation coefficient

Cercignani-Lampis model

Cercignani & Lampis (1971), Transp. Theor. Stat. Phys. 1, 101

H-function

(Entropy inequality)

Maxwellian

Thermodynamic entropy per unit mass

H-theorem

spatially uniform never increases

never increases

Boltzmann’s H theorem Direction for evolution

Darrozes & Guiraud (1966)C. R. Acad. Sci., Paris A 262, 1368

Darrozes-Guiraud inequality

Equality:

Cercignani (1975)

Highly rarefied gas

Free-molecular gas (collisionless gas; Knudsen gas)

Time-independent case

parameter

Initial-value problem (Infinite domain)

Initial condition:

Solution:

Boundary-value problem

Convex body

given

from BC

BC :

Solved!

Example

Slit

Mass flow rate:

No flow

General boundary

BC

Integral equation for

Diffuse reflection:

Integral equation for

[ : omitted ]

Conventional boundary condition

Specular reflection

Diffuse reflection

No net mass flux across the boundary

Maxwell type

Accommodation coefficient

Cercignani-Lampis model

Cercignani & Lampis (1971) TTSP

Statics: Effect of boundary temperature

Sone (1984), J. Mec. Theor. Appl. 3, 315; (1985) ibid 4, 1

Maxwell-type (diffuse-specular) condition

Closed or open domain, boundary at restarbitrary shape and arrangement

Arbitrary distribution of boundary temperature,accommodation coefficient

Path of a specularly reflected molecule

Exact solution

Condition Molecules starting from infinity :

Converges uniformly with respect to for

Reduces to for diffuse reflection

No flow !

Temperature field does not cause a flowin a free-molecular gas.

A, Bardos, Golse, Kogan, & Sone,Eur. J. Mech. B-Fluids (1993) Functional analytic approach

Example 1

Similarly,

No flow Same as slit-case!

Sone (1985)

Example 2 Sone & Tanaka (1986), RGD15

Example 3 A, Sone, & Ohwada (1986), RGD15

Numerical