Geometric methods for fluid-structure interactions

当草或树叶在风中弯曲时，运动会使狭窄通道的内部结构变形，并将流体移动到内部。一位寻找石油的地质学家用声波探测地球，声波以一种复杂的方式移动着地球内部的多孔介质和流体。血液在我们的动脉和静脉中搏动，每次心跳都会使弹性壁变形。流体和结构的相互作用无处不在，日常生活中更多的例子可以很容易地找到。描述流体和结构的相互作用总是具有挑战性的。这个项目将创建一个统一的框架来考虑流体和结构之间的相互作用，使用几何力学的方法。*该项目的重点将放在流体在弹性材料内流动的情况上，例如窄管和多孔材料介质。该项目的统一主题是使用一般几何思想，如空间的对称性，以及分析力学的方法(变分程序)，从第一原理推导和分析方程。几何力学的方法可以处理各种各样的问题。第一组问题涉及输送液体的管子的力学，这是一个与工程(例如化学)和生物医学应用(血液流动)相关的问题。本项目中开发的方法还将导致变分计算方法的发展，这些方法保证线动量和角动量守恒，并且由于离散化而不存在质量、力和力矩的人为源和汇。第二组问题涉及柔性多孔介质的动力学，例如充满水的海绵，以及由这种介质制成的薄片和棒。这个项目中发展的理论将可以分析充满流体的薄板和杆的运动，特别是在运动下的内部耗散计算，这是很难计算的，如果没有几何方法。最后，我们将哈默尔的力学理论应用到我们的问题中。这种方法基于对手头的问题选择最方便的速度，将进一步简化分析，并允许为这里所考虑的复杂问题找到最方便的速度坐标。*我们还纳入了几何弹性模型来描述脑内胶质瘤的生长，重点放在它的机械效应上，因为大脑的体积受到限制，而且胶质瘤周围的脑物质成分发生了变化。了解力学和附加压力对胶质瘤生长的影响最终可能有助于治疗建议。此外，由于我们在这里开发的方法是一般性的，并且基于基本的对称原理，它们可以应用于来自不同物理的各种实际问题，构成了这里描述的五年项目范围之外的研究的背景。

When a grass or tree leaf bends in the wind, the motion deforms the internal structure of narrow channels and moves the fluid inside. A geologist looking for oil probes the Earth with sounds waves, which move the porous media and fluid inside it in a complex manner. Blood pulsing through our arteries and veins deforms the elastic walls with every heartbeat. Interactions of fluids and structures are everywhere, and many more examples from everyday life can be easily found. Describing interaction of fluids and structures is always challenging. This project will create a unified framework for considering interactions between fluids and structures, using the approach of geometric mechanics. ******The focus of the project will be on the cases when fluids are flowing inside the elastic materials, such as narrow tubes and porous material media. The unifying theme of the project is the use of general geometric ideas, such as the symmetry of space, and methods of analytical mechanics (variational procedure), yielding the derivation and analysis of equations from the first principles. The methods of geometric mechanics allow to treat a wide variety of problems. The first set of problems concerns the mechanics of tubes conveying fluid, a problem which is relevant for engineering (e.g. chemical), and biomedical applications (blood flow). The methods developed in this project will also lead to the development of variational computational methods which guarantee conservation of linear and angular momenta and absence of artificial sources and sinks of mass, forces and torques due to discretization. The second set of problems concerns the dynamics of flexible porous media, such as sponge filled with water, and sheets and rods made of such media. The theory developed in this project will allow to analyze the motion of fluid-filled sheets and rods, and in particular, compute internal dissipation under the motion, which is difficult to compute without geometric methods. Finally, we apply Hamel's theory of mechanics to our problems. This method, based on choosing the most convenient velocities for the problem at hand, will further simplify the analysis and allow to find the most convenient velocity coordinates for the complex problems considered here. ***We are also incorporating geometric elasticity models for describing the growth of glioma in the brain, with the focus on its mechanical effects, given that the volume of the brain is constrained and the composition the brain matter around glioma changes. Understanding the effects of mechanics and additional stresses on glioma growth may eventually contribute to treatment recommendations. In addition, since the methods we develop here are general, and are based on fundamental principles of symmetry, they can be applied to a wide variety of practical problems coming from different physics, forming the background for studies beyond the scope of the 5-year project described here.