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

Conference on Kinetic Theory and Related Fields (Department of Mathematics, POSTECH June 22-24, 2011). 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

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

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  1. Conference on Kinetic Theory and Related Fields (Department of Mathematics, POSTECH June 22-24, 2011) Decay of an oscillating disk in a gas:Case of a collision-less gas anda special Lorentz gas Kazuo Aoki Dept. of Mech. Eng. and Sci. Kyoto University (in collaboration with Tetsuro Tsuji)

  2. Decay of an oscillating disk Gas Equation of motion of the disk : External force Drag (Hooke’s law) If , then Exponential decay Collisionless gas (Free-molecular gas, Knudsen gas) Other types of gas Decay rate ???

  3. Collisionless gas Collisionless gas Mathematical study Decay rate Caprino, Cavallaro, & Marchioro, M3AS (07) BC:specular reflection Monotonic decay

  4. Highly rarefied gas Collisionless gas Effect of collisions: Neglected Molecular velocity Mean free path Boltzmann equation Time-independent case parameter

  5. Velocity distribution function position time molecular velocity Molecular mass in at time Macroscopic quantities gas const. Equation for : Boltzmann equation

  6. Nonlinear integro-differential equation Boltzmann equation collision integral [ : omitted ] Dimensionless form: : Knudsen number

  7. Highly rarefied gas Collisionless gas Effect of collisions: Neglected Molecular velocity Mean free path Boltzmann equation Time-independent case parameter

  8. (Infinite domain) Initial-value problem Initial condition: Solution: (Steady) boundary-value problem Single convex body given from BC Solved! BC :

  9. General boundary BC Integral equation for Diffuse reflection: Maxwell type: Integral equation for Exact 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)

  10. Conventional boundary condition [ : omitted ] Specular reflection Diffuse reflection No net mass flux across the boundary

  11. Maxwell type Accommodation coefficient Cercignani-Lampis model Cercignani, Lampis, TTSP (72) Initial and boundary-value problem

  12. Collisionless gas Mathematical study Decay rate Caprino, Cavallaro, & Marchioro, M3AS (07) BC:specular reflection Monotonic decay Guess BC:diffuse reflection, oscillatory case Numerical study

  13. Gas: EQ: IC: BC:Diffuse reflection on body surface Body: EQ: IC:

  14. 1D case: Decay of oscillating plate gas Gas: EQ: IC: BC:Diffuse reflection on plate Plate: (unit area) EQ: left surface IC: right surface

  15. Numerical results (decay rate) Parameters Double logarithmic plot

  16. Numerical results (decay rate) Parameters Diffuse ref. Specular ref. Double logarithmic plot Power-law decay

  17. Parameters If the effect of recollision is neglected… no oscillation around origin Exponential decay LONG MEMORY effect (recollision) Single logarithmic plot

  18. (Marginal) VDF on the plate LEFT SIDE TRAJECTORY OF THE PLATE RIGHT SIDE Velocity of the plate Velocity of the plate Reflected molecules (diffuse reflection) Impinging molecules Reflected molecules (diffuse reflection) Impinging molecules recollision enlarged for a large time Initial distribution

  19. (Marginal) VDF on the plate enlarged figure Power-law decay Long memory effect

  20. Decay of the plate velocity Power-law decay • Decay rate of kinetic energy • is faster than potential energy • No possibility of infinitely • manyoscillations around • origin

  21. Power-law decay

  22. Density

  23. 2D & 3D cases Disk (diameter , without thickness) [Axisymmetric]

  24. Numerical evidence for ( BC:diffuse reflection, non small )

  25. Special Lorentz gas (Toy model for gas) Gas molecules: Interaction with background Destruction of long-memory effect (Dimensionless) EQ: IC: Knudsen number BC: Diffuse reflection mean free path EQ for the disk, … characteristic length

  26. Randomly distributed obstacles at rest Evaporating droplets No collision between gas molecules Re-emitted Gas molecule Absorbed Mean free path Number density Saturated state

  27. Collisionless gas Toy model Independent of Algebraic decay!

  28. Collisionless gas Toy model Independent of Algebraic decay!

  29. Special Lorentz gas (Toy model for gas) Gas molecules: Interaction with background Destruction of long-memory effect (Dimensionless) long-memory effect EQ: IC: Knudsen number BC: Diffuse reflection mean free path EQ for the disk, … characteristic length

  30. Very special Lorentz gas (Very toy model for gas) (Dimensionless) EQ: Previous model IC: Knudsen number BC: Diffuse reflection mean free path EQ for the disk, … characteristic length

  31. (velocity ) Randomly distributed moving obstacles Evaporating droplets No collision between gas molecules Re-emitted Gas molecule Absorbed Obstacles: Maxwellian

  32. Collisionless gas Toy model 1 Toy model 2 Exponential decay!!

  33. Collisionless gas Toy model 1 Toy model 2 Exponential decay!!

  34. Collisionless gas Toy model 1 Toy model 2 Exponential decay!!

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