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研究已经成为应用研究的主要焦点之一。这个项目将解决固体-液体相互作用的一些基本方面，旨在实现以下目标。第一个问题涉及刚体在粘性液体中的振动诱导运动。一个特别重要的情况是，振动是由体内时间周期性移位的质量引起的。第二门专门研究可变形固体在粘性液体中的运动。这项研究将包括物体的时间周期形状变化的推进，以及恒定力作用下弹性体平衡构型的确定和可能的分叉。粘性液体与固体(无论是刚性的还是可变形的)的相互作用构成了一个复杂而有趣的理论和应用研究课题。事实上，它不仅具有基本的理论意义，而且也代表了许多不同规模的工程、生物和医学研究的重点。重要的应用包括：由于液体撞击造成的结构破坏，如在大坝或船只中；动物的运动，如鸟类的飞行和鱼类的游动；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提高材料的机械性能；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，在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时。

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.