标题: Bouncing water drops 作者: Richard D, Quere D 来源出版物: EUROPHYSICS LETTERS 卷: 50 期: 6 页: 769-775 出版年: JUN 2000 被引频次: 91
实验结论：液滴和超疏水表面的碰撞近似是弹性碰撞（恢复系数约为0.9） 问题：如何用数量级分析估算液滴和基地的碰撞接触时间τ ？
Rayleigh-Bénard convection is a type of natural convection, occurring in a plane of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Rayleigh-Bénard convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of selforganizing nonlinear systems. ----From wiki
步骤一：Identifying Navier-Stokes equation
p x
2U z 2
0
步骤二：Approximating differential equations
2U ~ U , p ~ p ~ , ~ h
z 2
h2 x l l
l2
步骤三：Balancing leading terms
R
~ R2
步骤三：Balancing leading terms
R 2
~
R2
~ R3
事实上，Lord Rayleigh, Lamb & Chandrasekhar等 大师级学者都计算过空中自由液滴的谐振周期， 均为该毛细特征时间。说明液滴在基地的弹跳使 液滴作受迫振动。
毛细特征时间，毫米液滴 在10 ms 量级
U h2
~
h
h / 3/ 2
h
~
1
U
2/
3
1
Ca2 / 3
第37讲：弹性稳定性问题
软基地上薄膜的失稳
Elastic substrate
Evolution of Wrinkle Patterns
• Symmetry breaking in isotropic system:
– from spherical caps to elongated ridges – from labyrinth to herringbone.
• Symmetry breaking due to anisotropic strain
– from labyrinth to parallel stripes
• Controlling the wrinkle patterns
– On patterned substrates – By introducing initial defects
连续介质力学（固体） Continuum Mechanics Mechanics of Continua Mechanics of Continuous Media
第36-43讲
(Zhao Ya-Pu)
非线性力学国家重点实验室
ห้องสมุดไป่ตู้2010
第36讲：数量级分析的进一步讨论
陆明万，罗学富. 弹性理论基础，清华大学出版社，1990
T2 > T1 液体
Q
均匀加热
T1
T2
Convection cell
Convection Patterns
Cloud streets over Ithaca (photo by J. McCoy)
冯端、金国钧. 凝聚态物理学，上卷，高等教育出版社，2003
变化多端的液晶相变
John William Strutt, 3rd Baron Rayleigh
Capillary number
h ~ Ca2/ 3
η-viscosity, γ-surface tension Ca—capillary number
用数量级分析的方法确定LDD (Landau-Levich-Derjaguin) 标度率(scaling law)
Lord Rayleigh found that the period of a free droplet in free oscillation is
R3 4
Lord Rayleigh. The Theory of Sound, 1st edn. (London: Macmillan), 1877
对称破缺经典的力学例子：Rayleigh(理论)-Bénard(实验, 1900) 对流
The features of Rayleigh-Bénard convection can be obtained by a simple experiment first conducted by Henri Bénard, a French physicist, in 1900.
对称破缺：symmetry breaking 自发性对称破缺：spontaneous symmetry breaking
• Prof. Rui Huang
• Department of Aerospace Engineering and Engineering Mechanics
• The University of Texas at Austin
步骤一：Identifying equations of motion
ij
x j
2 xi t 2
步骤二：Approximating differential equations (γ为表面张力，R为液滴半径)
2 xi t 2
~ R /2,
ij
~
Laplace
pressure~
R
,
ij
x j
~ ij