Instabilities, Waves, and Growth in the Dynamics of Filaments

该提案分为三个相互关联的研究主线：细丝不稳定性的非线性分析，弹性棒中波传播和奇点形成的研究，以及弹性生长的分析和建模。灯丝不稳定性的分析将通过使用和扩展由主要研究人员开发的非线性分析技术来进行。这些产生具有相当数学意义的非线性振幅方程系统的技术，将广泛应用于各种具有实际重要性的情况，包括可伸缩和可剪切杆、输送流体的管道和生长的丝状生物。在各种条件下，波在细丝中传播的研究将通过各种分析稳定性技术进行研究，而奇点形成的研究将借鉴用于研究非均匀介质中波传播的技术。与生物过程相关的弹性生长建模将根据连续体力学原理发展，并应用于丝状和三维结构。从微观到宏观，丝状结构在自然界和物理世界中都无处不在。在自然界中，我们看到藤蔓和植物卷须的蜿蜒运动；在光学显微镜下我们可以看到丝状细菌；通过电子显微镜我们可以看到DNA链的结构。如果说发展纤维结构的合理连续力学的最初动机是由索、弹簧和支柱的机械世界提供的，那么当代研究往往是由生物学背景推动的。这些问题，包括基本的自然增长现象，对经典公式提出了新的挑战。虽然弹性结构的静力理论有着悠久而杰出的历史，但动力效应理论在理论上和计算上仍然面临许多重要的挑战。在过去几年中，主要研究人员的工作表明，弹性细丝的控制方程可以简化为更易于处理的非线性演化方程，能够描述许多动态分岔，包括基本的扭扭转换。如果这些早期的工作集中在最简单的情况下，那么将这一理论扩展到大量具有实际意义的重要情况还有待发展。除了结构不对称和非均匀性的影响外，管道和软管中产生的内部流体流动的影响，以及生物建模所需的生长效应的结合，将在本奖项支持的工作中得到解决。一旦实现，这些公式将提供必要的数学工具来研究像鞭子、软管和细菌细丝这样不同的丝状结构中的构象变化、不稳定性和波传播。主要研究人员将把这些研究应用于各种机械和生物环境下的丝状和其他弹性结构的分析，例如爬藤植物和其他丝状生物表现出的手性的自发变化，细菌丝状物中的扭波传播，以及空中生长的细菌微生物和真菌的形态变化。在所有这些例子中，自然生长是形态动力学的一个基本组成部分，将进行各种调查，以提高我们对如何将这种基本的自然效应纳入自一致的数学模型的理解。这些将包括在杆的本构方程中包含生长项，到发展三维弹性结构生长的连续力学模型，例如在生物组织中出现的体生长描述。

The proposal is divided into three main lines of interconnected research: the nonlinear analysis of filament instabilities, the study of wave propagation and singularity formation in elastic rods, and the analysis and modeling of elastic growth. The analysis of filament instabilities will be carried out through the use and extension of nonlinear analysis techniques developed by the principal investigators. These techniques, which yield systems of nonlinear amplitude equations of considerable mathematical interest, will be applied to a wide variety of cases of practical importance including extensible and shearable rods, tubes conveying fluids, and growing filamentary organisms. The studies of wave propagation in filaments under various conditions will be investigated by a variety of analytical stability techniques, and the investigations of singularity formation will draw on adaptations of techniques used to study wave propagation in inhomogeneous media. The modeling of elastic growth pertinent to biological processes will be developed according to continuum mechanical principles and applied to both filamentary and three dimensional structures.Filamentary structures are ubiquitous in both the natural and physical worlds at scales ranging from the microscopic to the macroscopic. In nature we see the sinuous motion of vines and plant tendrils; under the optical microscope we can see filamentary bacteria; and through electron microscopy we can see the structure of DNA strands. If the original motivations for developing a rational continuum mechanics of filamentary structures was provided by the mechanical world of cables, springs and struts, contemporary research has often been motivated by the biological context. These problems, including the fundamental natural phenomenon of growth, raise new challenges to the classical formulations. While the static theory of elastic structures has a long and distinguished history, the theory of dynamical effects still presents many important challenges, both theoretically and computationally. Work of the principal investigators over the past few years has shown that the governing equations of elastic filaments, can be reduced to more tractable nonlinear evolution equations capable of describing many dynamical bifurcations including the fundamental twist-to-writhe conversion. If this earlier work concentrated on simplest cases, the extension of this theory to a whole host of important cases of practical importance has yet to be developed. In addition to the effects of structural asymmetry, and inhomogeneity, the effects of internal fluid flow that arise in pipes and hoses, and the incorporation of growth effects required for biological modeling, will be addressed in work supported by this award. Once achieved, these formulations will provide the necessary mathematical tools to study conformation changes, instability, and wave propagation in filamentary structures as varied as whips, hoses and bacterial filaments. The principal investigators will apply these studies to the analysis of filamentary and other elastic structures in a variety of mechanical and biological settings such as the spontaneous change in handedness exhibited by climbing plants and other filamentary organisms, twist wave propagation in bacterial filaments, and morphological changes in aerially growing bacterial microorganisms and fungi. In all these examples, natural growth is a fundamental component in the morphological dynamics, and a variety of investigations will be carried out to improve our understanding of how this fundamental natural effect can be incorporated into self-consistent mathematical models. These will range from the inclusion of growth terms in the constitutive equations for rods, to the development of continuum mechanical models of growing three dimensional elastic structures such as those arising in the description of bulk growth in biological tissues.