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Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 1)

Intensive Lecture Series (Postech, June 20-21, 2011). Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 1). Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University. Introduction. Classical kinetic theory of gases

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Boundary-value problems of the Boltzmann equation: Asymptotic and numerical analyses (Part 1)

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  1. Intensive Lecture Series (Postech, June 20-21, 2011) Boundary-value problems ofthe Boltzmann equation:Asymptotic and numerical analyses(Part 1) Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University

  2. Introduction

  3. 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

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

  5. 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).

  6. Boltzmann equation and its basic properties

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

  8. 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

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

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

  11. 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

  12. ES model Holway (1966), Phys. Fluids9, 1658 Entropy inequalityAndries et al. (2000), Eur. J. Mech. B19, 813 revival

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

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

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

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

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

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

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

  20. Highly rarefied gas

  21. Free-molecular gas (collisionless gas; Knudsen gas) Time-independent case parameter

  22. (Infinite domain) Initial-value problem Initial condition: Solution: Boundary-value problem Convex body given from BC Solved! BC :

  23. Example Slit Mass flow rate: No flow

  24. General boundary BC Integral equation for Diffuse reflection: Integral equation for

  25. Conventional boundary condition [ : omitted ] Specular reflection Diffuse reflection No net mass flux across the boundary

  26. Maxwell type Accommodation coefficient Cercignani-Lampis model Cercignani & Lampis (1971) TTSP

  27. Statics: Effect of boundary temperature Sone (1984), J. Mec. Theor. Appl. 3, 315; (1985) ibid 4, 1 Closed or open domain, boundary at rest arbitrary shape and arrangement Maxwell-type(diffuse-specular)condition Arbitrary distribution of boundary temperature, accommodation coefficient Path of a specularly reflected molecule

  28. Exact solution

  29. Condition Molecules starting from infinity : Converges uniformly with respect to for Reduces to for diffuse reflection No flow ! Temperature field does not cause a flow in a free-molecular gas. A, Bardos, Golse, Kogan, & Sone, Eur. J. Mech. B-Fluids (1993) Functional analytic approach

  30. Example 1 Similarly, No flow Same as slit-case! Sone (1985)

  31. Example 2 Sone & Tanaka (1986), RGD15

  32. Example 3 A, Sone, & Ohwada (1986), RGD15 Numerical

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