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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