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研究生:和正平
研究生(外文):C. P. He
論文名稱:二維非彈性顆粒子之簇集現象
論文名稱(外文):Clustering Phenomena of Inelastic Granular Particles in Two Dimensions
指導教授:黎璧賢黎璧賢引用關係
指導教授(外文):P. Y. Lai
學位類別:碩士
校院名稱:國立中央大學
系所名稱:物理研究所
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2000
畢業學年度:88
語文別:英文
中文關鍵詞:顆粒子簇集現象
外文關鍵詞:GranularClustering
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本文以MD方法模擬二維非彈性顆粒子系統的冷卻過程﹙freely cooling system﹚,探討不同彈性係數下簇集現象形成的情況,並觀查不同的邊界條件對此現象的影響。當系統處於齊次狀態﹙homogeneous state﹚時,系統能量衰減正比於t -2,此結果與Haff’s cooling law相符合,而系統在非齊次態時簇集現象﹙clustering﹚將會展現。在非齊次狀態下,能量衰減與彈性係數成並非單純的線性關係。這是因為空間分佈的不均勻造成速率較小的顆粒子有較高的碰撞機率,並使彈性系數小於臨界彈性系數的系統反而擁有較高的系統總能量。本文末並觀察具有切線方向磨擦之系統的冷卻過程。
We consider the dynamics of an ensemble of identical, inelastic, hard disks in a square
domain, with three kinds of dierent boundary conditions, (i) double periodic bound-
aries, (ii) a pair of smooth, elastic walls in the x-direction and periodic boundaries in
the y-direction, (iii) four smooth and elastic walls. Starting with the almost elastic
case, in which the coeÆcient of restitution is just a little less than 1, the homoge-
neous regime resembles a classical non-dissipative gas and there is no large structure.
When decreases, the system becomes inhomogeneous and spatial non-uniformity
occurs. Clusters appear when is even smaller, large clusters of disks form, break,
and reform. As time goes by, the cluster stays in a status of hydrodynamic shear
state, or collapse. Inelastic collapse, which is a dynamic singularity of the binary
collision model, is caused by the many-body dynamics. Numerical simulations show
that the energy decay in the homogeneous regime is proportion to t
Contents
1 Introduction 7
2 Theory 10
2.1 Ha''s Cooling Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 The Momentum Conservation Equation . . . . . . . . . . . . . 11
2.1.3 The Energy Conservation Equation . . . . . . . . . . . . . . . 11
2.1.4 Main Assumptions and The Result of Ha''s Cooling Law . . . 12
2.2 Clustering and Collapse in Simulations . . . . . . . . . . . . . . . . . 14
2.2.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Inelastic Collapse . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Molecular dynamics Method and Simulation details 17
3.1 Molecular dynamics of Hard Systems . . . . . . . . . . . . . . . . . . 17
3.1.1 Collision Times between Two Particles . . . . . . . . . . . . . 18
3.1.2 Collision Times between a Particle and the Wall . . . . . . . . 19
3.2 The Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Particle-Particle Collision . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Particle-Wall Collision . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 23
4 Results and Discussions 25
4.1 Inelastic Particles with Smooth surface . . . . . . . . . . . . . . . . . 25
4.1.1 Probability of Non-Collapse . . . . . . . . . . . . . . . . . . . 25
4.1.2 Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.3 Energy Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.4 Number of Collisions and Time . . . . . . . . . . . . . . . . . 48
4.2 Inelastic Rough Particles with Periodic Boundary Conditions . . . . . 59
5 Conclusion and Outlook 66
6 Appendix 68
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