Geometric methods for fluid-structure interactions

流固耦合的几何方法

• 批准号: RGPIN-2018-05751
• 负责人: Putkaradze, Vakhtang
• 金额: $ 1.46万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677169
• 关键词: Geometric; methods; fluid; structure; interactions

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.

当草或树叶在风中弯曲时，运动会变形狭窄的通道的内部结构，并将其移动到其中。寻找油的地质学家会用声音波探测地球，从而将多孔介质和液体移动以复杂的方式移动。血液通过我们的动脉脉动和静脉脉动会以每一个心跳变形弹性墙。流体和结构的相互作用无处不在，可以很容易地找到日常生活中的更多例子。描述流体和结构的相互作用总是具有挑战性的。该项目将使用几何力学方法创建一个统一的框架，用于考虑流体和结构之间的相互作用。 *****该项目的重点将放在流体在弹性材料内流动的情况，例如窄管和多孔材料介质。该项目的统一主题是使用一般的几何思想，例如空间的对称性和分析力学方法（变化程序），从而得出了第一原理的派生和分析。几何力学的方法允许治疗多种问题。第一组问题涉及输送流体的管的机制，这是与工程（例如化学）和生物医学应用（血流）相关的问题。该项目中开发的方法还将导致变异计算方法的发展，这些方法可以保留线性和角动量，以及由于离散化而导致的质量，力和扭矩的人工源和下沉。 第二组问题涉及柔性多孔介质的动态，例如用水的海绵，由这种媒体制成的床单和棒。该项目中开发的理论将允许分析流体填充的板和杆的运动，尤其是在运动下计算内部耗散，这在没有几何方法的情况下很难计算。最后，我们将Hamel的力学理论应用于我们的问题。此方法基于为当前问题选择最方便的速度，将进一步简化分析，并允许在此处考虑的复杂问题找到最方便的速度坐标。 ***我们还合并了描述大脑神经胶质瘤生长的几何弹性模型，鉴于大脑的体积受到限制，并且大脑周围的大脑围绕胶质瘤变化，因此重点是其机械效应。了解力学和其他应力对神经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的，并且基于对称性的基本原理，因此可以将它们应用于来自不同物理学的各种实际问题，构成了此类研究的背景，而不是此处描述的5年项目的范围。