Partially dissipative systems with applications to fluid-solid interaction problems

部分耗散系统在流固相互作用问题中的应用

• 批准号: RGPIN-2021-03129
• 负责人: Mazzone, Giusy
• 金额: $ 1.31万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742391
• 关键词: Partially; dissipative; systems; applications; fluid

The interactions of viscous incompressible fluids with (rigid or elastic) solids are ubiquitous in engineering, geophysical and biological applications. The long-term goal of our research program is to provide new analytical techniques for a systematic investigation of stability and long-time behaviour of fluid-solids interacting systems. In this proposal, we will focus on several multiphysics fluid-rigid body interaction problems that can be mathematically cast as "partially dissipative" systems of differential equations. A nonlinear evolution equation is said to be partially dissipative if the following two properties are satisfied: (P1) there exists an energy functional that is non-increasing along the trajectories; and (P2) there exists a vector field (depending on the solution) that is conserved at all times. The short-term objective is to characterize the long-time behaviour of a large class of solutions to three related fluid-rigid body interaction problems. The governing equations are partially dissipative and feature the coupling of the Navier-Stokes equations with the balance of momentums for the solids in presence of magnetic fields and/or heat transfer. The following topics will establish a stimulating research programme for the training of 3 PhD, 2 Master's and 2 NSERC USRA students: (1) Asymptotic stability of rigid body motions for solids with a fluid-filled gap. Consider a system of two solids, one contained inside the other, and with the gap between them completely filled by a viscous incompressible fluid. We will study the asymptotic stability of permanent rotations (i.e., steady states given by the system rotating as a whole rigid body with constant angular velocity). (2) Long-time behaviour of fluid-filled rigid bodies in a magnetic field. We consider a rigid body with a cavity completely filled by an electrically conducting fluid. We will investigate whether motions of the coupled fluid-solid system are able to generate and sustain a magnetic field within the system. (3) Long-time dynamics of rigid bodies with a fluid-filled gap subject to both temperature gradients and a gravitational field. The objective is to determine whether solid rotations inhibit the formation of fluid convecting motions at large times. Every problem proposed above is characterized by having different sources of dissipation while keeping a conservative component. From a mathematical point of view, these problems are examples of abstract nonlinear evolution equations satisfying (P1) and (P2). Existing theories concerning stability and long-time behaviour of solutions are applicable mainly to solutions in a strong functional setting. The far-reaching impact is to provide a novel theory for the long-time behaviour of weak solutions to partially dissipative systems. Besides their mathematical significance, problems (1)-(3) provide phenomenological models for applications in geophysics and engineering.

粘性不可压缩流体与（刚性或弹性）固体的相互作用在工程、地球物理和生物学应用中是普遍存在的。我们的研究计划的长期目标是提供新的分析技术的稳定性和流体-固体相互作用系统的长期行为的系统调查。在这个建议中，我们将集中在几个多物理场流体-刚体相互作用的问题，可以在数学上铸造为“部分耗散”微分方程系统。一个非线性演化方程被称为部分耗散的，如果满足以下两个性质：（P1）存在一个能量泛函，它沿轨迹沿着不增;（P2）存在一个向量场（取决于解），它在任何时候都是守恒的。 短期目标是表征三个相关的流体-刚体相互作用问题的一大类解决方案的长期行为。控制方程是部分耗散的，并具有耦合的Navier-Stokes方程与存在磁场和/或热传递的固体的动量平衡。以下课题将为培养3名博士、2名硕士和2名NSERC USRA学生建立一个有启发性的研究计划：（1）充满流体间隙的固体刚体运动的渐近稳定性。考虑两个固体的系统，一个包含在另一个里面，它们之间的差距完全由粘性不可压缩流体填充。我们将研究永久旋转的渐近稳定性（即，由作为具有恒定角速度的整个刚体旋转的系统给出的稳定状态）。 (2)磁场中充满流体的刚体的长期行为。我们考虑一个空腔完全被导电流体充满的刚体。我们将研究耦合流固系统的运动是否能够在系统内产生和维持磁场。 (3)具有充液间隙的刚体在温度梯度和重力场作用下的长时间动力学。我们的目标是确定是否固体旋转抑制流体对流运动的形成在大的时间。 上述每一个问题的特点是有不同的耗散源，同时保持保守的组成部分。从数学的角度来看，这些问题是满足（P1）和（P2）的抽象非线性发展方程的例子。现有的理论关于稳定性和长期行为的解决方案主要适用于解决方案在一个强功能的设置。其深远的影响是为部分耗散系统弱解的长时间行为提供了一种新的理论。问题（1）-（3）除了它们的数学意义外，还为物理学和工程学的应用提供了唯象模型。