Thermal Effects on the Dynamics of Singularity Formation in Viscous Threads

这个项目研究了由非线性偏微分方程组控制的自由边界问题，其中局部加热控制了表面张力驱动流动中奇点形成的动力学。热效应可以导致有趣的动力学，这与等温情况下的动力学非常不同。该项目将研究温度对粘度的影响和表面张力在成型粘性丝线中所起的作用。正在研究的问题涉及粘度和/或表面张力随温度迅速变化的非均匀流体圆柱体和圆柱管。为了更好地理解流体圆柱体或管子的坍塌机理，该项目研究了描述以下问题的模型问题：1)圆柱形螺纹的夹断；2)无限介质中圆柱孔的坍塌；3)管子的坍塌；4)管子的退火；以及5)这些问题在流动中的延伸。本项目分析了温度对粘性材料螺纹的影响，这些材料在许多制造工艺中都被使用。由于表面张力的影响，粘性细线(如印刷中的喷墨)会被掐断成液滴，这往往会使液体的表面积最小化。夹断现象也出现在其他应用中，例如，在涂布流动中出现的薄膜中。加热可以显著改变表面张力，从而导致热毛细效应，其中温度梯度导致表面张力梯度，从而驱动流动。制造过程要求玻璃和聚合物材料在高温下变形，这会导致热粘性效应，因为由此产生的粘度变化很大。例如，拉动光纤电缆和形成用于电生理学的玻璃微电极。近年来，半导体纳米团簇和高纯度纳米线已经通过各种技术被生产出来，通常需要在退火阶段进行加热。在所有这些过程中，加热在控制最终产品的形状方面起着至关重要的作用。该项目将有助于详细了解加热对粘度和表面张力的影响，潜在地改进这些材料的制造工艺。

This project investigates free boundary problems governed by nonlinear partial differential equations where localized heating controls the dynamics of the formation of singularities in surface-tension-driven flows. Thermal effects can lead to interesting dynamics that are very different from the dynamics in isothermal cases. The project will investigate the role temperature effects on viscosity and surface tension play in shaping viscous threads. The problems under study involve nonuniform fluid cylinders and cylindrical tubes with viscosity and/or surface tension that change rapidly with temperature. To better understand the collapse mechanism of fluid cylinders or tubes, the project studies model problems that describe: 1) pinch off of cylindrical threads; 2) collapse of a cylindrical hole in an infinite medium; 3) collapse of tubes; 4) annealing of tubes, and 5) extensions of these problems with flow.This project analyzes the effects of temperature on threads of viscous material, which are used in many manufacturing processes. Thin viscous threads (such ink jets in printing) pinch off into droplets due to the effect of surface tension, which tends to minimize the surface area of the liquid. Pinch-off phenomena also occur in other applications, e.g., in thin films that arise in coating flows. Heating can significantly modify surface tension, which leads to thermocapillary effects, in which temperature gradients cause surface tension gradients that drive flows. Manufacturing processes require glass and polymeric materials to be at high temperatures to deform them, which leads to thermoviscous effects due to due to the resultant large changes in viscosity. Examples include the pulling of fiber optic cables and formation of glass microelectrodes used in electrophysiology. In recent years, semiconductor nanoclusters and high-purity nanowires have been produced using various techniques, and heating is normally required in the annealing stage. In all these processes, heating plays a crucial role in controlling the shape of final products. This project will contribute to detailed knowledge of the effects of heating on viscosity and surface tension, potentially leading to improved manufacturing processes for these materials.