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Decay of an oscillating disk in a gas:Case of a collision-less gas and
a special Lorentz gas
Kazuo Aoki
Dept. of Mech. Eng. and Sci.
Kyoto University
(in collaboration with Tetsuro Tsuji)
Conference on Kinetic Theory and Related Fields(Department of Mathematics, POSTECHJune 22-24, 2011)
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Decay of an oscillating disk
If , then
Equation of motion of the disk :
Exponential decay
Collisionless gas (Free-molecular gas, Knudsen gas)Other types of gas
External force Drag(Hooke’s law)
Gas
Decay rate ???
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Decay rate Mathematical study
Caprino, Cavallaro, & Marchioro,M3AS (07)
Monotonic decay
BC: specular reflection
Collisionless gas Collisionless gas
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Time-independent case
parameter
Collisionless gas
Boltzmannequation
Highly rarefied gas
Effect of collisions: NeglectedMolecularvelocity
Mean free path
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Velocity distribution function
time position molecular velocity
Macroscopic quantities
Molecular mass in at time
gas const.
Equation for : Boltzmann equation
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collisionintegral
Boltzmann equation Nonlinear integro-differentialequation
[ : omitted ]
Dimensionless form:
: Knudsen number
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Time-independent case
parameter
Collisionless gas
Boltzmannequation
Highly rarefied gas
Effect of collisions: NeglectedMolecularvelocity
Mean free path
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Initial-value problem (Infinite domain)
Initial condition:
Solution:
(Steady) boundary-valueproblem
Single convex body
given
from BC
BC :
Solved!
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General boundary
BC
Integral equation for
Diffuse reflection:Maxwell type:
Integral equation forExact solution! Sone, J. Mec. Theor. Appl. (84,85)
General situation, effect of boundary temperature Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications (Birkhäuser, 2007)
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[ : omitted ]
Conventional boundary condition
Specular reflection
Diffuse reflection
No net mass flux across the boundary
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Maxwell type
Accommodation coefficient
Cercignani-Lampis model
Cercignani, Lampis, TTSP (72)
Initial and boundary-value problem
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Decay rate Mathematical study
Caprino, Cavallaro, & Marchioro,M3AS (07)
Monotonic decay
BC: specular reflection
Guess BC: diffuse reflection, oscillatory case
Numerical study
Collisionless gas
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Gas:
EQ:
IC:
BC: Diffuse reflection on body surface
Body:
EQ:
IC:
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Gas:
EQ:
IC:
BC: Diffuse reflection on plate
Plate:
EQ:
IC:
gas
(unit area)
left surface
right surface
1D case: Decay of oscillating plate
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Numerical results (decay rate)
Parameters
Double logarithmic plot
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Parameters
Numerical results (decay rate)
Double logarithmic plot Power-law decay
Diffuse ref.
Specular ref.
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LONG MEMORY effect(recollision)
Single logarithmic plot
If the effect of recollision is neglected…
Parameters
Exponential decay
no oscillationaround origin
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Impingingmolecules Reflected
molecules(diffuse reflection)
Impingingmolecules
Initial distribution
LEFT SIDE RIGHT SIDETRAJECTORY OF THE PLATE
Reflectedmolecules(diffuse reflection)
Velocity of the plate
Velocity of the plate
recollisionenlarged
for a large time
(Marginal) VDF on the plate
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Power-lawdecay
enlarged figure
Long memory effect
(Marginal) VDF on the plate
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Power-law decay
• Decay rate of kinetic energy is faster than potential energy• No possibility of infinitely many oscillations around origin
Decay of the plate velocity
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2D & 3D cases Disk (diameter , without thickness)
[Axisymmetric]
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Numerical evidence for
( BC: diffuse reflection, non small )
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Special Lorentz gas (Toy model for gas)
Gas molecules: Interaction with background
Destruction of long-memory effect
EQ:
IC:
(Dimensionless)
BC: Diffuse reflection
EQ for the disk, …
Knudsen number
mean free path
characteristic length
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Randomly distributed obstacles at rest
Re-emitted
Absorbed
Evaporating droplets
No collision betweengas molecules
Gas molecule
Mean free path
Number density Saturated state
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Collisionless gas
Toy model
Independent of Algebraic decay!
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Collisionless gas
Toy model
Independent of Algebraic decay!
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Special Lorentz gas (Toy model for gas)
Gas molecules: Interaction with background
Destruction of long-memory effect
EQ:
IC:
(Dimensionless)
BC: Diffuse reflection
EQ for the disk, …
Knudsen number
mean free path
characteristic length
long-memory effect
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Very special Lorentz gas (Very toy model for gas)
EQ:
IC:
(Dimensionless)
BC: Diffuse reflection
EQ for the disk, …
Knudsen number
mean free path
characteristic length
Previousmodel
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Randomly distributed moving obstacles
Re-emitted
Absorbed
Evaporating droplets
No collision betweengas molecules
Gas molecule
(velocity )
Obstacles:Maxwellian
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Collisionless gas
Toy model 1
Toy model 2 Exponential decay!!
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Collisionless gas
Toy model 1
Toy model 2 Exponential decay!!
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Collisionless gas
Toy model 1
Toy model 2 Exponential decay!!