Mathematical Sciences: Computational Two-Phase Viscous Drop Spreading

9623092布劳恩受到与铺展过程相关的复杂传输的推动，例如焊滴的铺展和涂层过程，提出了一项关于被薄膜覆盖的液滴铺展的研究计划。通常在焊接和其他涉及铺展的过程中，液滴可能会变得非常薄，这表明存在多个长度的鳞片。在作者以前研究的基础上，第一种方法是用多尺度渐近分析(润滑理论)来推导自由表面的非线性扩散方程；这些非线性扩散方程将使用谱精确的直线法来求解。一旦找到了所需的自由面，就可以用这个近似值计算出所有感兴趣的量。在液滴和液膜中的Stokes流近似下，通过边界积分方程组提出了第二种方法。这种方法放宽了液滴和薄膜必须很薄的要求，但仍然要求流动缓慢。我们建议对这两种方法进行直接比较，以评估每种方法在不同情况下的效用。拟议的工作将只开发流体动力学的模型，并将在后续的项目中提出包含反应影响的模型。制造中的重要过程涉及在存在另一种流体的情况下铺开液滴；一个例子是焊接，在许多情况下，存在第二种流体的薄层，即焊剂，这是工艺运行所必需的。由于不再允许使用铅基焊料，因此了解其他类型焊料的这种液滴扩散过程将非常有助于理解替代焊料材料。该项目中提出的工作是开发一些初步模型，以开发一个完整的铺展焊点模型。拟议的计划是开发液滴在平板上传播的理论，平板上覆盖着一层不同流体的薄膜。将开发计算机代码来预测液滴传播的速度，以及液滴的形状和覆盖液滴的薄膜。总而言之，复杂的数学技巧将被应用到一个具有技术重要性的问题上。预计该项目完成后，该项目产生的计算机代码将适用于不同技术领域(如涂层工艺)的相关润湿问题。

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.