Download - Decay of an oscillating disk in a gas: Case of a collision-less gas and a special Lorentz gas
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!!
