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Decay of an oscillating disk in a gas:Case of a collision-less gas anda special Lorentz gasKazuo AokiDept.of Mech.Eng.and Sci.Kyoto University(in collaboration with Tetsuro Tsuji)Conference on Kinetic Theory and Related Fields(Department of Mathematics,POSTECHJune 22-24,2019)Decay of an oscillating diskIf ,thenEquation of motion of the disk:Exponential decayCollisionless gas (Free-molecular gas,Knudsen gas)Other types of gasExternal force Drag(Hookes law)GasDecay rate?Decay rateMathematical studyCaprino,Cavallaro,&Marchioro,M3AS(07)Monotonic decayBC:specular reflectionCollisionless gasCollisionless gasTime-independent caseparameterCollisionless gasBoltzmannequationHighly rarefied gasEffect of collisions:NeglectedMolecularvelocityMean free pathVelocity distribution functiontimepositionmolecular velocityMacroscopic quantitiesMolecular mass in at timegas const.Equation for :Boltzmann equationcollisionintegralBoltzmann equationNonlinear integro-differentialequation :omitted Dimensionless form:Knudsen numberTime-independent caseparameterCollisionless gasBoltzmannequationHighly rarefied gasEffect of collisions:NeglectedMolecularvelocityMean free pathInitial-value problem(Infinite domain)Initial condition:Solution:(Steady)boundary-valueproblemSingle convex bodygivenfrom BCBC:Solved!General boundaryBCIntegral equation forDiffuse 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 (Birkhuser,2019):omitted Conventional boundary conditionSpecular reflectionDiffuse reflectionNo net mass flux across the boundaryMaxwell typeAccommodation coefficientCercignani-Lampis modelCercignani,Lampis,TTSP(72)Initial and boundary-value problemDecay rateMathematical studyCaprino,Cavallaro,&Marchioro,M3AS(07)Monotonic decayBC:specular reflectionGuessBC:diffuse reflection,oscillatory caseNumerical studyCollisionless gasGas:EQ:IC:BC:Diffuse reflection on body surfaceBody:EQ:IC:Gas:EQ:IC:BC:Diffuse reflection on platePlate:EQ:IC:gas(unit area)left surfaceright surface1D case:Decay of oscillating plateNumerical results(decay rate)ParametersDouble logarithmic plotParametersNumerical results(decay rate)Double logarithmic plotPower-law decayDiffuse ref.Specular ref.LONG MEMORY effect(recollision)Single logarithmic plotIf the effect of recollision is neglectedParametersExponential decayno oscillationaround originImpingingmoleculesReflectedmolecules(diffuse reflection)ImpingingmoleculesInitial distributionLEFT SIDERIGHT SIDETRAJECTORY OF THE PLATEReflectedmolecules(diffuse reflection)Velocity of the plateVelocity of the platerecollisionenlarged for a large time(Marginal)VDF on the platePower-lawdecayenlarged figureLong memory effect(Marginal)VDF on the platePower-law decay Decay rate of kinetic energy is faster than potential energy No possibility of infinitely many oscillations around origin Decay of the plate velocityPower-law decayDensity2D&3D casesDisk(diameter ,without thickness)AxisymmetricNumerical evidence for(BC:diffuse reflection,non small )Special Lorentz gas(Toy model for gas)Gas molecules:Interaction with backgroundDestruction of long-memory effectEQ:IC:(Dimensionless)BC:Diffuse reflectionEQ for the disk,Knudsen numbermean free pathcharacteristic lengthRandomly distributed obstacles at restRe-emittedAbsorbedEvaporating dropletsNo collision betweengas moleculesGas moleculeMean free pathNumber densitySaturated stateCollisionless gasToy modelIndependent ofAlgebraic decay!Collisionless gasToy modelIndependent ofAlgebraic decay!Special Lorentz gas(Toy model for gas)Gas molecules:Interaction with backgroundDestruction of long-memory effectEQ:IC:(Dimensionless)BC:Diffuse reflectionEQ for the disk,Knudsen numbermean free pathcharacteristic lengthlong-memory effectVery special Lorentz gas(Very toy model for gas)EQ:IC:(Dimensionless)BC:Diffuse reflectionEQ for the disk,Knudsen numbermean free pathcharacteristic lengthPreviousmodelRandomly distributed moving obstaclesRe-emittedAbsorbedEvaporating dropletsNo collision betweengas moleculesGas molecule(velocity )Obstacles:MaxwellianCollisionless gasToy model 1Toy model 2Exponential decay!Collisionless gasToy model 1Toy model 2Exponential decay!Collisionless gasToy model 1Toy model 2Exponential decay!谢谢你的阅读v知识就是财富v丰富你的人生谢谢！
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