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 LionsUniversité Pierre et Marie Curie (Paris VI) (February 4, 2011)

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

(Continuum limit )

Introduction Vapor flows with evaporation/condensation on interfaces

Important subject in RGD (Boltzmann equation)

Vapor is not in local equilibriumnear the interfaces, even forsmall Knudsen numbers(near continuum regime).

mean free path characteristic length

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Hamel model Onishi & Sone (84 unpublished)

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Hamel model Onishi & Sone (84 unpublished)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 limitRigorous result: Golse & Levermore, CPAM (02)

(single component)Bardos, Golse, Saint-Raymond, … Fluid limit

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 ofvapor and NC gasat small Knfor arbitrary geometryand for small deviation from saturatedequilibrium state at rest

Problem

Dimensionless variables (normalized by )

Velocity distribution functions

Vapor NC gas

Boltzmann equations

Molecular number of component in

position molecular velocity

Preliminaries (before linearization)

Macroscopic quantities

Collision integrals (hard-sphere molecules)

Boundary condition

evap.

cond.

Vapor

(number density) (pressure)of vapor in saturatedequilibrium state at

NC gas Diffuse reflection (no net mass flux)

New approach: Frezzotti, Yano, ….

Linearization (around saturated equilibrium state at rest)

Small Knudsen number

concentration of ref. state

reference mfp of vapor reference length

Analysis

Linearized collision operator (hard-sphere molecules)

Linearized Boltzmann eqs.

Macroscopic quantities (perturbations)

Linearized Boltzmann eqs.

BC

(Formal) asymptotic analysis forSone (69, 71, … 91, … 02, …07, …)

• Kinetic Theory and Fluid Dynamics (Birkhäuser, 02)• Molecular Gas Dynamics: Theory, Techniques, and Applications (B, 07)

Saturation number density

LinearizedBoltzmann eqs.

Hilbert solution (expansion)

Macroscopic quantities

Sequence of integral equations

Fluid-dynamic equations

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

Solvability conditions

Stokes equations

Auxiliary relationseq. of state

function of ** Any !

diffusion thermaldiffusion

functions of **

Takata, Yasuda, A, Shibata, RGD23 (03)

Hilbert solution does not satisfy kinetic B.C.

Hilbert solution Knudsen-layercorrection

Stretched normal coordinate

Solution:

Eqs. and BC for Half-space problem forlinearized Boltzmann eqs.

Knudsen layer and slip boundary conditions

Knudsen-layer problem

Undetermined consts.

Half-space problem forlinearized Boltzmann eqs.

Solution exists uniquely iff take special values

Boundary values ofA, Bardos, & Takata, J. Stat. Phys. (03)

BC for Stokes equations

• 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): CPAMMaslova (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

Stokeseqs.

BC

Vapor no. densitySaturation no. density

No-slip condition(No evaporation/condensation)

function of

: Present studyOthers : Previous study

Slip conditionslip coefficientsfunction of

Takata, RGD22 (01); Takata, Yasuda, A, & Kosuge, Phys. Fluids (03, 06);Yasuda, Takata, & A, Phys. Fluids (04, 05)

Database Numericalsol. of LBE

Thermalcreep

Shearslip

Diffusion slip

Evaporation orcondensation

Concentrationgradient

Temperaturegradient

Normalstress

Slip coefficients

Reference concentration

: Vapor : NC gas

Summary

We have derived- Stokes equations- Slip boundary conditions- Knudsen-layer correctionsdescribing slow flows of a mixture of a vapor anda noncondensable gas with surface evaporation/condensation in the near-continuumregime (small Knudsen number) from Boltzmannequations and kinetic boundary conditions.

Possible applications

evaporation from droplet, thermophoresis,diffusiophoresis, …… (work in progress)