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)

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 ???

Decay rate Mathematical study

Caprino, Cavallaro, & Marchioro,M3AS (07)

Monotonic decay

BC: specular reflection

Collisionless gas Collisionless gas

Time-independent case

parameter

Collisionless gas

Boltzmannequation

Highly rarefied gas

Effect of collisions: NeglectedMolecularvelocity

Mean free path

Velocity distribution function

time position molecular velocity

Macroscopic quantities

Molecular mass in at time

gas const.

Equation for : Boltzmann equation

collisionintegral

Boltzmann equation Nonlinear integro-differentialequation

[ : omitted ]

Dimensionless form:

: Knudsen number

Time-independent case

parameter

Collisionless gas

Boltzmannequation

Highly rarefied gas

Effect of collisions: NeglectedMolecularvelocity

Mean free path

Initial-value problem (Infinite domain)

Initial condition:

Solution:

(Steady) boundary-valueproblem

Single convex body

given

from BC

BC :

Solved!

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)

[ : omitted ]

Conventional boundary condition

Specular reflection

Diffuse reflection

No net mass flux across the boundary

Maxwell type

Accommodation coefficient

Cercignani-Lampis model

Cercignani, Lampis, TTSP (72)

Initial and boundary-value problem

Decay rate Mathematical study

Caprino, Cavallaro, & Marchioro,M3AS (07)

Monotonic decay

BC: specular reflection

Guess BC: diffuse reflection, oscillatory case

Numerical study

Collisionless gas

Gas:

EQ:

IC:

BC: Diffuse reflection on body surface

Body:

EQ:

IC:

Gas:

EQ:

IC:

BC: Diffuse reflection on plate

Plate:

EQ:

IC:

gas

(unit area)

left surface

right surface

1D case: Decay of oscillating plate

Numerical results (decay rate)

Parameters

Double logarithmic plot

Parameters

Numerical results (decay rate)

Double logarithmic plot Power-law decay

Diffuse ref.

Specular ref.

LONG MEMORY effect(recollision)

Single logarithmic plot

If the effect of recollision is neglected…

Parameters

Exponential decay

no oscillationaround origin

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

Power-lawdecay

enlarged figure

Long memory effect

(Marginal) VDF on the plate

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

Power-law decay

Density

2D & 3D cases Disk (diameter , without thickness)

[Axisymmetric]

Numerical evidence for

( BC: diffuse reflection, non small )

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

Randomly distributed obstacles at rest

Re-emitted

Absorbed

Evaporating droplets

No collision betweengas molecules

Gas molecule

Mean free path

Number density Saturated state

Collisionless gas

Toy model

Independent of Algebraic decay!

Collisionless gas

Toy model

Independent of Algebraic decay!

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

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

Randomly distributed moving obstacles

Re-emitted

Absorbed

Evaporating droplets

No collision betweengas molecules

Gas molecule

(velocity )

Obstacles:Maxwellian

Collisionless gas

Toy model 1

Toy model 2 Exponential decay!!

Collisionless gas

Toy model 1

Toy model 2 Exponential decay!!

Collisionless gas

Toy model 1

Toy model 2 Exponential decay!!