Mathematical Sciences: Computational Two-Phase Viscous Drop Spreading

小行星9623092 由于与扩散过程相关的复杂运输， 随着焊料滴的扩散和涂覆过程， 提出了一个程序上的液滴是由一个薄膜覆盖的蔓延。 通常在焊接和其他涉及铺展的工艺中，液滴可能 变得非常薄，这表明存在多个长度尺度。 基于作者以前的研究，提出的第一种方法是一种 多尺度渐近分析（润滑理论）推导非线性 自由表面的扩散方程;那些非线性扩散方程 将使用以下方法的光谱精确实现来解决： 线 所有感兴趣的量都可以用这种近似计算 一旦找到所需的自由表面。第二种方法也是 提出，通过边界积分方程，近似下， Stokes流在液滴和液膜中流动。 这种方法放宽了要求 液滴和膜是薄的，但仍然要求流动是缓慢的。 我们 建议直接比较这两种方法， 评估每一个在不同情况下的效用。 拟议的工作将 发展模型只为流体动力学，并列入反应 将在随后的项目中提出影响。 制造业中的重要过程涉及流体的扩散 液滴在另一种流体的存在;一个例子是焊接，其中，在 在许多情况下，第二流体（焊剂）薄层的存在， 这个过程所需要的。 因为铅基焊料的使用 不再被允许，理解这种液滴扩散过程， 其他类型的焊接材料将非常有助于理解 替代焊接材料。项目中提出的工作正在发展中 开发一个完整的 扩散焊料滴的模型。该计划旨在发展 液滴扩散理论：液滴在覆盖有薄 不同流体的薄膜。计算机代码将被开发出来， 这一次，文龙的鸡巴又硬了起来，顶在了文龙的胸膛上。 下降总之，复杂的数学技术将被应用于 一个重要的技术问题。 计算机代码产生于 该项目预计将适用于相关的润湿问题， 不同的技术领域（如涂层工艺） 完成这个项目。

9623092 Braun Motivated by the complex transport associated with spreading process such as the spreading of solder droplets and coating processes, a research program is proposed on the spreading of a droplet that is covered by a film. Often in soldering and other processes involving spreading, the drop may become very thin which indicates the presence of multiple length scales. Based on the author's previous research, the first approach proposed is a multi-scale asymptotic analysis (lubrication theory) to derive nonlinear diffusion equations for the free surfaces; those nonlinear diffusion equations will be solved using a spectrally-accurate implementation of the method of lines. All quantities of interest may be computed in this approximation once the required free surfaces have been found. A second approach is also proposed, via boundary integral equations, under the approximation of Stokes flow in the drop and film. This approach relaxes the requirement that the drop and film be thin, but still requires that the flow be slow. We propose making a direct comparison between the two approaches in order to evaluate the utility of each in various situations. The work proposed will develop models only for the fluid dynamics, and inclusion of reactive effects will be proposed in a subsequent project. Important processes in manufacturing involve the spreading of a fluid droplet in the presence of another fluid; one example is soldering, where, in many instances, the presence of a thin layer of a second fluid, the flux, is required for the process to work at all. Because the use of lead-based solder is no longer allowed, understanding of this droplet spreading process for other kind of solder materials would be very helpful in understanding alternative solder materials. The work proposed in the project is developing some preliminary models necessary for the development of a complete model for a spreading solder drop. The proposed program is to develop theory for the spreading of a droplet on a flat plate that is covered by a thin film of a different fluid. Computer codes will be developed that predict how fast the drop spreads, and the shapes of the drop and the film that covers the drop. In summary, sophisticated mathematical techniques will be applied to a problem of technological importance. The computer codes resulting from this project are expected to be applicable to related wetting problems in different technological areas (e.g. coating processes) subsequent to the completion of this project.