Spreading Problems in Fluid Mechanics

在这项提案中，计划研究液体沿固体衬底的扩散问题和液体沿液体界面的扩散问题。纳维-斯托克斯方程将支配流体的运动。由于接触线上可能出现奇点，所以研究液体在固体衬底上的扩散既有趣又困难。在可能存在多个自由边界的情况下，沿液体界面扩散是一个具有挑战性的问题。这里，除了液/液和液/气边界外，在液/液/气相交的线上还可以存在三重结点。预测这三条相线的动态特性是本方案的目的之一。我们将感兴趣的是界面干净的情况和存在表面活性剂的情况。从数学上讲，我们面临着具有耦合的运动界面的问题，奇点可能沿着这些界面存在。对于这些复杂的自由边界问题，将采用渐近方法和数值方法进行研究。例如，我们将考虑薄膜极限，其中完整的三维方程组可以简化为一个耦合的非线性发展方程组。我们还将应用Level-Set数值方法求解完全的非线性方程组。当液滴停留在固体表面时，液滴的自由边界由气液界面和液/气/固三相之间的接触线来表征。这种三相线路通常被称为接触线。每当三个相接触时，它就会发生。这是日常生活中常见的现象，当一个人把油倒进煎锅或部分装满的酒杯时，就可以观察到这种现象。它还出现在许多具有实际意义的领域。例如工业涂布工艺、管道中的气液混合物运输、肺部药滴的扩散和海上液体废物(如石油或化学品)的扩散。后一个问题是接触线(在这种情况下称为三重结)出现在液/液/气界面的交叉点处的情况的一个例子。尽管接触线出现在这么多的地区，但我们对它的了解非常有限。模型仍在开发中，由于与接触线附近的解的行为相关的数学奇异性，需要确定解析法和数值解技术。

0104935MiksisIn this proposal, it is planned to investigate both the problem of the spreading of a liquid along a solid substrate and the spreading of a liquid along a liquid interface. The Navier-Stokes equations will govern the motion of the fluids. The investigation of the spreading of a liquid along a solid substrate is interesting and difficult because of the singularities that can occur at the contact line. Spreading along a liquid interface is a challenging problem where multiple free boundaries can exist. Here, in addition to the liquid/liquid and liquid/gas boundaries, a triple junction can exist at the liquid/liquid/gas line of intersection. Predicting the dynamics of these three phase lines is one of the aims of this proposal. We will be interested in situations where the interfaces are clean and situations where surfactants are present. Mathematically we are faced with problems having coupled moving interfaces along which singularities can exits. Both asymptotic and numerical methods will be used to study these complex free boundary problems. For example, we will consider the thin film limit where the complete three-dimensional system of equations can be reduced to a coupled system of nonlinear evolution equations. We will also apply the level-set numerical method to solve the complete nonlinear system of equations.When a liquid drop is resting on a solid surface, the free boundary of the drop is characterized by the gas/liquid interface, and the line of contact between the liquid/gas/solid phases. This three-phase line is usually referred to as the contact line. It occurs whenever three phases come into contact. It is a familiar phenomena of everyday life, and can be observed when one pours oil into a frying pan or in a partially filled wine glass. It also occurs in many areas of practical interest. Examples are industrial coating processes, the transport of gas/liquid mixtures in pipes, the spreading of droplets of medication in the lungs and the spreading of liquid wastes (e.g. oil or chemicals) on the sea. The latter problem is an example of a situation where the contact line (refereed to as the triple junction in this case) occurs at the intersection of a liquid/liquid/gas interface. Even though the contact line occurs in these many areas, our understanding of it is very limited. Models are still being developed and because of the mathematical singularities associated with the behavior of the solutions in the neighborhood of the contact line, solution techniques, both analytical and numerical, need to be identified.