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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, POSTECH June 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 (Hookes law) Gas Decay rate ???

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

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Time-independent case parameter Collisionless gas Boltzmann equation Highly rarefied gas Effect of collisions: Neglected Molecular velocity 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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collision integral Boltzmann equation Nonlinear integro-differential equation [ : omitted ] Dimensionless form: : Knudsen number

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

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

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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)

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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, M 3 AS (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 oscillation around origin

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

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Power-law decay 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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Power-law decay

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Density

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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 between gas 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 Previous model

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Randomly distributed moving obstacles Re-emitted Absorbed Evaporating droplets No collision between gas 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!!

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