stokes fluid dynamics for a vapor-gas mixture derived from kinetic theory kazuo aoki department of...
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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Hamel model Onishi & Sone (84 unpublished)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Hamel model Onishi & Sone (84 unpublished)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)
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