Boundary-value problems of the 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 small volume in the gas, containing many molecules (say molecules) Monatomic ideal gas, No external force Classical kinetic theory of gases Non-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 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 Fluid-dynamic (continuum) limit Free-molecular flow

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).

Boltzmann equation and its basic properties

Velocity distribution function time position molecular velocity Molecular mass in at time Mass density in phase 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

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

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 rest arbitrary 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 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