Mathematical Analysis of Some Fundamental Problems in Solid-Liquid Interaction

在过去的几十年里，固体在粘性液体中的运动已经成为应用研究的主要焦点之一。本计画将探讨固液交互作用的一些基本方面，以达成下列目标。第一个问题涉及粘性液体中刚体的振动诱导运动。一个特别重要的情况是当振动是由身体内的时间周期性位移的质量引起的。第二个是研究可变形固体在粘性液体中的运动。本研究将包括通过时间周期性形状变化来推进物体，以及确定和可能的分叉，受恒定力的弹性体的平衡构型。粘性液体与固体（刚性或可变形）的相互作用构成了理论和应用研究的复杂和有趣的主题。事实上，它不仅是基本的理论兴趣，但它也代表了许多工程，生物和医学研究在不同规模的焦点。重要的应用包括由于液体的冲击而导致的结构损坏，如水坝或船舶，动物运动，如鸟类的飞行和鱼类的游泳，短纤维复合材料的制造，其中向聚合物基质中添加短纤维状颗粒将增强材料的机械性能，通过电泳分离大分子，以及DNA测序。 固液相互作用的另一个同样重要的应用是在血液流动中，研究血液与动脉壁的相互作用，以及血细胞在血浆中的运动。

Over the past decades, the study of the motion of solids in a viscous liquid has become one of the main focuses of applied research. This project will address some fundamental aspects of solid-liquid interaction directed toward the accomplishment of the following objectives. The first one concerns the vibration-induced motion of a rigid body in a viscous liquid. A particularly significant case is when the vibration is induced by a time-periodically displaced mass inside the body. The second one is devoted to the motion of a deformable solid in a viscous liquid. This study will include propulsion of bodies by time-periodic shape-changes, as well as the determination, and possible bifurcation, of equilibrium configurations of elastic bodies subject to a constant force.The interaction of a viscous liquid with a solid body (either rigid or deformable) constitutes a complex and intriguing topic of theoretical and applied research. In fact, not only is it of fundamental theoretical interest, but it also represents the focus of many engineering, biological and medical studies at different scales. Important applications include damage of structures due to the impinging of the liquid, like in dams or ships, animal locomotion, like the flying of birds and the swimming of fish, manufacturing of short-fiber composites, where the addition of short fiber-like particles to a polymer matrix will enhance the mechanical properties of the material, separation of macromolecules by electrophoresis, and DNA sequencing. Another, but no less important, application of solid-liquid interaction occurs in blood flow, in the study of the interaction of blood with arterial walls, and of the motion of blood cells within the plasma.