Thermocapillary migration of a droplet with a thermal source at large Reynolds and Marangoni numbers

The {\it unsteady} process for thermocapillary droplet migration at large Reynolds and Marangoni numbers has been previously reported by identifying a nonconservative integral thermal flux across the surface in the {\it steady} thermocapillary droplet migration, [Wu and Hu, J. Math. Phys. {\bf 54} 023102, (2013)]. Here we add a thermal source in the droplet to preserve the integral thermal flux across the surface as conservative, so that thermocapillary droplet migration at large Reynolds and Marangoni numbers can reach a {\it quasi-steady} process. Under assumptions of {\it quasi-steady} state and non-deformation of the droplet, we make an analytical result for the {\it steady} thermocapillary migration of droplet with the thermal source at large Reynolds and Marangoni numbers. The result shows that the thermocapillary droplet migration speed slowly increases with the increase of Marangoni number.Comment: 3 figure

Hysteresis in three-dimensional multi-layer molecularly thin-film lubrication

For three-dimensional multi-layer molecularly thin-film lubrication system with elastic substrates, roles of hysteresis on tribological properties are investigated by using the multiscale simulation method. It is found that multiple stick-slip transitions with/without hysteresis loops appear in a sliding process and form a quasi-periodic progress with lattice distance. For the few-/multi-layer thin-film lubrication system, as the load increases, the hysteresis length monotonously increases/tends to keep constant. The hysteresis is mainly caused by the relaxation of metastable states of solid atoms in the elastic substrates, which delays the system back to its equilibrium states. In the quasi-periodic shearing progress, the effective elastic coefficients and the hysteresis lengths approximately remain unchanged, which reveals that although the hysteresis loops with the same lengths appear in the sliding process, the total systematic energy is still conserved. These findings not only provide a profound understanding of roles of hysteresis in the thin-film lubrication system but also show the effects of film layers and loads on the systematic tribological properties, which are of great significance for practical applications.</p

Streamline topology and dilute particle dynamics in a Karman vortex street flow

Three types of streamline topology in a Karman vortex street flow are shown under the variation of spatial parameters. For the motion of dilute particles in the K\'arm\'an vortex street flow, there exist a route of bifurcation to a chaotic orbit and more attractors in a bifurcation diagram for the proportion of particle density to fluid density. Along with the increase of spatial parameters in the flow filed, the bifurcation process is suspended, as well as more and more attractors emerge. In the motion of dilute particles, a drag term and gravity term dominate and result in the bifurcation phenomenon.Comment: 16 pages, 9 figure

Rotation numbers of invariant manifolds around unstable periodic orbits for the diamagnetic Kepler problem

In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincar\'e section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincar\'e section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.Comment: 8 figures, 1 tabl

Thermocapillary migration of a droplet with a thermal source at large Reynolds and Marangoni numbers

The unsteady process for thermocapillary droplet migration at large Reynolds and Marangoni numbers has been previously reported by identifying a nonconservative integral thermal flux across the surface in the steady thermocapillary droplet migration (Wu and Hu, 2013) [15]. Here we add a thermal source in the droplet to preserve the integral thermal flux across the surface as conservative, so that thermocapillary droplet migration at large Reynolds and Marangoni numbers can reach a quasi-steady process. Under assumptions of quasi-steady state and non-deformation of the droplet, we make an analytical result for the steady thermocapillary migration of droplet with the thermal source at large Reynolds and Marangoni numbers. The result shows that the thermocapillary droplet migration speed slowly increases with the increase of Marangoni number. (C) 2014 Elsevier Ltd. All rights reserved

薄膜润滑系统的多尺度模拟研究

组合粗粒化方法和原子模拟处理含有弹性固体基底的薄膜润滑系统。在固体基底远区的有限元粗粒化过程中采用局域和非局域单元覆盖。研究表明,采用局域单元覆盖的自由能修正粗粒化和原子模拟混合方法能够有效地预测出系统全原子模拟的系统热力学参量,即剪应力,平均分离距离和横向应力曲线。但是,非局域单元覆盖的自由能修正粗粒化方法在连续和原子描述混合区可导致系统非物理滑移,这主要是基底近壁区非各项均匀同性

Thermocapillary migration of a deformed droplet in the combined vertical temperature gradient and thermal radiation

Thermocapillary migration of a deformed droplet in the combined vertical temperature gradient and thermal radiations with uniform and non uniform fluxes is first analyzed. The creeping flow solutions show that the deformed droplet has a slender or a cardioid shape, which depends on the form of the radiation flux. The deviation from a sphere depends not only on the viscosity and the conductivity ratios of two phase fluids but also on capillary and thermal radiation numbers. Moreover, in the roles of interfacial rheology on thermocapillary migration of a deformed droplet, only the surface dilatational viscosity and the surface internal energy can reduce the steady migration velocity, but the surface shear viscosity has not any effects on the steady migration velocity. The surface shear and dilatational viscosities affect the deformation of the droplet by increasing the viscosity ratio of two phase fluids. The surface internal energy directly reduces the deformation of the droplet. However, the deformed droplet still keeps its original shape without the influence of interfacial rheology. Furthermore, it is found that, based on the net force balance condition of the droplet, the normal stress balance at the interface can be used to determine the steady migration velocity, which is not affected by the surface deformation in the creeping flow. From the expressions of the normal/the tangential stress balance, it can be proved that the surface shear viscosity does not affect the steady migration velocity. The results could not only provide a valuable understanding of thermocapillary migration of a deformed droplet with/without the interfacial rheology in a vertical temperature gradient controlled by thermal radiation but also inspire its potential practical applications in microgravity and microfluidic fields

大Marangoni数下液滴热毛细迁移的物理机制

液滴(气泡)在流体介质中的输运过程是很多自然现象和工程应用中的关键基础问题。在重力的作用下,当两相流体介质密度不同时,液滴基于浮力和重力运动。在微重力环境下,浸含在流体介质中的液滴(气泡)的浮力效应基本消失,而其界面的表面张力变为控制液滴(气泡)运动的主要驱动力。由于表面张力反比于界面的温度,因此外加不均匀温度场将导致液滴(气泡)出现移动

Thermocapillary droplet migration in a vertical temperature gradient controlled by thermal radiations

Thermocapillary migration of a droplet in a vertical temperature gradient controlled by uniform and non-uniform thermal radiations is theoretically analyzed and numerically investigated. A non-dimensionlized thermal radiation number is proposed to quantitatively depict the intensity ratio of the thermal radiation flux to the uniform temperature gradient. From the momentum and energy equations at zero limits of Reynolds and Marangoni numbers, analytical results for the uniform and non-uniform thermal radiations are determined. The steady migration velocity raises with the increasing of the thermal radiation number. By using the front-tracking method, it is observed that thermocapillary droplet migration under the uniform thermal radiation at moderate Marangoni and moderate thermal radiation numbers reaches a steady process. The steady migration velocity decreases with the increasing of Marangoni number and increases with the increasing of thermal radiation number. Moreover, the intensity of thermal energy transferred from the interface to both fluids depends on the volume heat capacity ratio. For the larger/smaller volume heat capacity ratio, more heat is transferred into the continuous phase fluid/the droplet. Furthermore, when the uniform thermal radiation is replaced by the non-uniform ones, the time evolutions, the structures of temperature fields, and parameter dependencies of thermocapillary droplet migration at moderate Marangoni and moderate thermal radiation numbers remain qualitatively unchanged. This study provides a profound understanding of thermocapillary droplet migration in a vertical temperature gradient controlled by thermal radiations, which is of great significance for practical applications in microgravity and microfluidic fields