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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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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
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
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
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).
Boltzmann equation and its basic properties
Velocity distribution function position time molecular velocity Molecular mass in at time Mass density in phase space Boltzmann equation (1872)
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
Nonlinear integro-differential equation Boltzmann equation collision integral [ : omitted ] Post-collisional velocities depending on molecular models Hard-sphere molecules
Entropy inequality ( H-theorem) Basic properties of Maxwellian (local, absolute) Conservation 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. Fluids9, 1658 Entropy inequalityAndries et al. (2000), Eur. J. Mech. B19, 813 revival
Initial and boundary conditions Initial condition Boundary condition [ : omitted ] No net mass flux across the boundary
(#) No net mass flux across the boundary arbitrary satisfies (#)
Conventional boundary condition [ : omitted ] Specular reflection [ does not satisfy (iii) ] Diffuse reflection No net mass flux across the boundary
Maxwell type Accommodation coefficient Cercignani-Lampis model Cercignani & Lampis (1971), Transp. Theor. Stat. Phys.1, 101
H-theorem H-function (Entropy inequality) Maxwellian Thermodynamic entropy per unit mass
spatially uniform never increases never increases Boltzmann’s H theorem Direction for evolution
Darrozes-Guiraud inequality Darrozes & Guiraud (1966) C. R. Acad. Sci., Paris A262, 1368 Equality: Cercignani (1975)
Free-molecular gas (collisionless gas; Knudsen gas) Time-independent case parameter
(Infinite domain) Initial-value problem Initial condition: Solution: Boundary-value problem Convex body given from BC Solved! BC :
Example Slit Mass flow rate: No flow
General boundary BC Integral equation for Diffuse reflection: Integral equation for
Conventional boundary condition [ : omitted ] 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 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
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
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