Mathematical analysis of bubble rings in ideal fluids

理想流体中气泡环的数学分析

• 批准号: 531098047
• 负责人: Professor Dr. Christian Seis
• 金额: --
• 依托单位: Institut für Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/531098047?language=en
• 关键词: Mathematical; analysis; bubble; rings; ideal

In this project, the evolution of toroidal bubble vortices is to be investigated. Bubble vortices are special vortices that occur in two-phase fluids. A typical and fascinating example is an air bubble ring in water created by dolphins or beluga whales. The underlying mathematical model is given by the two-phase Euler equations with surface tension. One major goal is a thorough mathematical construction of steady rings that move without changing their shape, and of perturbations of these. Such traveling waves are known for the classical Euler equations, but their existence is unknown for surface tension dependent models. Of particular interest is the role of the surface tension for the shape of the ring, which will be investigated. A second objective of this project is to understand how the effect of surface tension can be exploited to rigorously justify certain nonlinear motion laws of one or more interacting bubble rings. The understanding of such motion laws for the classical Euler equations is poor, and it is expected that the regularising effect of surface tension helps to mathematically tame the problem.

在该项目中，将研究环形气泡涡流的演化。气泡涡流是两相流体中出现的特殊涡流。一个典型且有趣的例子是海豚或白鲸在水中形成的气泡环。基本数学模型由具有表面张力的两相欧拉方程给出。一个主要目标是对在不改变形状的情况下移动的稳定环及其扰动进行彻底的数学构造。这种行波在经典欧拉方程中是已知的，但它们的存在对于表面张力相关模型来说是未知的。特别令人感兴趣的是表面张力对环形状的影响，我们将对此进行研究。该项目的第二个目标是了解如何利用表面张力的影响来严格证明一个或多个相互作用的气泡环的某些非线性运动定律。对经典欧拉方程的此类运动定律的理解很差，预计表面张力的正则化效应有助于在数学上解决该问题。