High-Frequency Self-Excited Oscillations in Buckled Elastic-Walled Tubes

许多生物系统涉及液体或气体在具有柔性或弹性壁的管状导管内的流动。例子包括静脉和动脉中的血液流动，以及气道和肺中的空气流动。工业和医疗设备也可以包含具有弹性壁的流体填充管。(Even具有看似刚性壁的管道将具有一定程度的柔性，并且可以被认为是弹性的。因此，重要的是要了解这种流动的基本物理学，并有数学模型，使我们能够预测这种流动将如何表现。在这个项目中，复杂的数学技术将被用来研究这一领域的一个特殊问题。实验表明，即使流体沿着弹性壁管稳定地泵送，壁也会自发地开始振荡。在这种情况下，定常流被称为不稳定流。在其他情况下，最初施加的变形将导致振幅减小的振荡，并且管返回到稳定配置。在这些情况下，稳定流被认为是稳定的。对于给定的工业设置或生物系统，重要的是要知道流动是稳定的还是不稳定的，因为它会导致完全不同的行为。有时候，不稳定的情况所导致的振荡是有益的（例如，帮助混合或去除异物），有时不在这项研究中，数学建模和数值模拟将有助于提高我们对这些振荡不稳定性原因的理解，并帮助我们精确预测给定管道中的流动何时稳定和不稳定。在流动不稳定的情况下，我们也将能够预测所产生的振荡的形式、频率和增长率。这些结果将反过来建议人们在不同情况下可以防止或促进振荡的方法。

Many biological systems involve the flow of a liquid or gas within a tube-like conduit that has a flexible or elastic wall. Examples include blood flow in veins and arteries, and air flow in the airways and lungs. Industrial and medical devices may also contain fluid-filled tubes with elastic walls. (Even pipes with seemingly rigid walls will have some degree of flexibility, and can be regarded as being elastic.) It is therefore important to understand the underlying physics of such flows, and to have mathematical models that allow us to predict how such flows will behave. In this project, sophisticated mathematical techniques will be used to investigate a particular problem in this area.Experiments show that even with a steady pumping of fluid along an elastic-walled tube, the wall can spontaneously begin to oscillate in and out. In such cases, the steady flow is said to be unstable. In other cases, an initially imposed deformation will be seen to result in oscillations of decreasing amplitude, and the tube returns to a steady configuration. In these cases the steady flow is said to be stable. For a given industrial setup or biological system, it is important to know whether the flow will be stable or unstable, as it will lead to quite different behaviours. Sometimes the oscillations resulting from an unstable situation will be beneficial (e.g. to help with mixing or dislodging foreign objects) and sometimes not (e.g. leading to energy losses or unwanted vibrations).In this research, mathematical modelling and numerical simulations will help improve our understanding of the causes these oscillatory instabilities, and help us predict precisely when a flow in a given tube will be stable and unstable. In cases where the flow is unstable, we will also be able to predict the form, frequency and the growth rate of the resulting oscillations. These results will, in turn, suggest ways in which one might prevent or promote the oscillations in different situations.