Stokes fluid dynamics for a vapor-gas mixture derived from kinetic theory

1. Séminaire du Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie (Paris VI) (February 4, 2011) Stokes fluid dynamics for a vapor-gas mixture derived from kinetic theory Kazuo Aoki Department of Mechanical Engineering and Science Kyoto University, Japan in collaboration with Shigeru Takata & Takuya Okamura

2. Subject Fluid-dynamic treatment of slow flows of a mixture of - a vapor and a noncondensable gas - with surface evaporation/condensation - near-continuum regime (small Knudsen number) - based on kinetic theory

3. Vapor flows with evaporation/ condensation on interfaces Introduction Important subject in RGD (Boltzmann equation) Vapor is not in local equilibrium near the interfaces, even for small Knudsen numbers (near continuum regime). (Continuum limit ) mean free path characteristic length

4. equations ?? BC’s ?? not obvious Fluid-dynamic description Systematicasymptotic analysis (for small Kn) based on kinetic theory Steady flows Pure vapor Sone & Onishi (78, 79), A & Sone (91), … Fluid-dynamic equations + BC’sin various situations Vapor+ Noncondensable (NC) gas Vapor (A) + NC gas (B) Fluid-dynamic equations ?? BC’s ?? Small deviation from saturated equilibrium state at rest Hamel model Onishi & Sone (84 unpublished)

5. Vapor+ Noncondensable (NC) gas Vapor (A) + NC gas (B) Fluid-dynamic equations ?? BC’s ?? Small deviation from saturated equilibrium state at rest Hamel model Onishi & Sone (84 unpublished) Boltzmann eq. Present study Corresponding to Stokes limit Rigorous result: Golse & Levermore, CPAM (02) (single component) Bardos, Golse, Saint-Raymond, … Fluid limit Largetemperature and density variations Fluid limit Takata & A, TTSP (01)

6. Problem Steady flows of vaporand NC gas at smallKn for arbitrary geometry and for small deviation from saturated equilibrium state at rest Vapor (A) + NC gas (B) Linearized Boltzmann equation for a binary mixture hard-sphere gases B.C. Vapor - Conventional condition NC gas - Diffuse reflection

7. (before linearization) Preliminaries Dimensionless variables (normalized by ) Velocity distribution functions VaporNC gas position molecular velocity Molecular number of component in Boltzmann equations

8. Collision integrals (hard-sphere molecules) Macroscopic quantities

9. Boundary condition Vapor (number density) (pressure) of vapor in saturated equilibrium state at evap. cond. New approach:Frezzotti, Yano, …. Diffuse reflection (no net mass flux) NC gas

10. Analysis • Linearization(around saturated equilibrium state at rest) • SmallKnudsen number concentration of ref. state reference mfp of vapor reference length

11. Linearized Boltzmann eqs. Linearized collision operator (hard-sphere molecules)

12. Macroscopic quantities (perturbations)

14. Linearized Boltzmann eqs. BC Saturation number density (Formal) asymptotic analysis for Sone (69, 71, … 91, … 02, …07, …) • Kinetic Theory and Fluid Dynamics (Birkhäuser, 02) • Molecular Gas Dynamics: Theory, Techniques, and Applications (B, 07)

15. Fluid-dynamic equations Hilbert solution (expansion) Macroscopic quantities Linearized Boltzmann eqs. Sequence of integral equations

16. Sequence of integral equations Solutions Linearized local Maxwellians (common flow velocity and temperature) Solvability conditions Constraints for F-D quantities Stokes set of equations (to any order of )

17. Solvability conditions

18. Stokes equations function of ** Any ! Auxiliary relations eq. of state

19. thermal diffusion diffusion functions of ** Takata, Yasuda, A, Shibata, RGD23 (03)

20. Knudsen layer and slip boundary conditions Hilbert solution does not satisfy kinetic B.C. Solution: Knudsen-layer correction Hilbert solution Stretched normal coordinate Half-space problem for linearized Boltzmann eqs. Eqs. and BC for

21. Knudsen-layer problem Half-space problem for linearized Boltzmann eqs. Undeterminedconsts. Solution exists uniquely iff take special values A, Bardos, & Takata, J. Stat. Phys. (03) Boundary values of BC for Stokes equations

22. Knudsen-layer problem Single-component gas Grad (69) Conjecture Bardos, Caflisch, & Nicolaenko (86): CPAM Maslova (82), Cercignani (86), Golse & Poupaud (89) Half-space problem for linearized Boltzmann eqs. Decomposition Numerical • Shear slip Yasuda, Takata, A , Phys. Fluids (03) • Thermal slip (creep) Takata, Yasuda, Kosuge, & A, PF (03) • Diffusion slip Takata, RGD22 (01) • Temperature jump Takata, Yasuda, A, & Kosuge, PF (06) • Partial pressure jump • Jump due to evaporation/condensation • Yasuda, Takata, & A (05): PF • Jump due to deformation of • boundary (in its surface) Present study

23. Stokes eqs. function of BC No-slip condition (No evaporation/condensation) Vapor no. density Saturation no. density

24. Slip condition slip coefficients function of : Present study Others : Previous study Numerical sol. of LBE Database Takata, RGD22 (01); Takata, Yasuda, A, & Kosuge, Phys. Fluids (03, 06); Yasuda, Takata, & A, Phys. Fluids (04, 05)

25. Thermal creep Shear slip Diffusion slip

26. Temperature gradient Evaporation or condensation Normal stress Concentration gradient

27. Slip coefficients Reference concentration : Vapor : NC gas

28. Summary We have derived - Stokes equations - Slip boundary conditions - Knudsen-layer corrections describing slow flows of a mixture of a vapor and a noncondensable gas with surface evaporation/ condensation in the near-continuum regime (small Knudsen number) from Boltzmann equations and kinetic boundary conditions. Possible applications evaporation from droplet, thermophoresis, diffusiophoresis, …… (work in progress)