1. 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 Séminaire du Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie (Paris VI) (February 4, 2011)

2. 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 Subject

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

4. Systematic asymptotic analysis (for small Kn) based on kinetic theory Steady flows Fluid-dynamic description equations ?? BC’s ?? not obvious Pure vapor Sone & Onishi (78, 79), A & Sone (91), … Fluid-dynamic equations + BC’s in 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 Small deviation from saturated equilibrium state at rest Hamel model Onishi & Sone (84 unpublished)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 Small deviation from saturated equilibrium state at rest Hamel model Onishi & Sone (84 unpublished) Boltzmann eq.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 Large temperature and density variations Fluid limit Takata & A, TTSP (01) Corresponding to Stokes limit Rigorous result: Golse & Levermore, CPAM (02) (single component) Bardos, Golse, Saint-Raymond, … Fluid limit

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

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

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

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

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

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

12. Macroscopic quantities (perturbations)

14. Linearized Boltzmann eqs. BC (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) Saturation number density

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

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

17. Solvability conditions

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

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

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

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

22. 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) Bardos, Caflisch, & Nicolaenko (86): CPAM Maslova (82), Cercignani (86), Golse & Poupaud (89) Knudsen-layer problem Single-component gas Half-space problem for linearized Boltzmann eqs. Decomposition Grad (69) Conjecture Present study Numerical

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

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

25. Thermal creep Shear slip Diffusion slip

26. Evaporation or condensation Concentration gradient Temperature gradient Normal stress

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)