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)
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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
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(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
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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Hamel model Onishi & Sone (84 unpublished)
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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Hamel model Onishi & Sone (84 unpublished)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
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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
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Dimensionless variables (normalized by )
Velocity distribution functions
Vapor NC gas
Boltzmann equations
Molecular number of component in
position molecular velocity
Preliminaries (before linearization)
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Macroscopic quantities
Collision integrals (hard-sphere molecules)
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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, ….
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Linearization (around saturated equilibrium state at rest)
Small Knudsen number
concentration of ref. state
reference mfp of vapor reference length
Analysis
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Linearized collision operator (hard-sphere molecules)
Linearized Boltzmann eqs.
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Macroscopic quantities (perturbations)
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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
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LinearizedBoltzmann eqs.
Hilbert solution (expansion)
Macroscopic quantities
Sequence of integral equations
Fluid-dynamic equations
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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
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Solvability conditions
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Stokes equations
Auxiliary relationseq. of state
function of ** Any !
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diffusion thermaldiffusion
functions of **
Takata, Yasuda, A, Shibata, RGD23 (03)
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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
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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
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• 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
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Stokeseqs.
BC
Vapor no. densitySaturation no. density
No-slip condition(No evaporation/condensation)
function of
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: 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
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Thermalcreep
Shearslip
Diffusion slip
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Evaporation orcondensation
Concentrationgradient
Temperaturegradient
Normalstress
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Slip coefficients
Reference concentration
: Vapor : NC gas
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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)