Change in the nature and enhancement of Brownian motion in shear flows caused by the non-modal growth of thermal fluctuations

经典布朗运动代表静止流体中由于与周围热运动流体分子碰撞而产生的小被动粒子的随机运动。非均匀流的存在显著地改变了布朗运动的特性。理解和描述这种变化的本质是至关重要的，因为小的主动/被动颗粒在非均匀流体流动中的运动具有越来越重要的实际意义。这自然导致近年来不同研究界对非均匀流中活跃粒子的布朗运动的兴趣显著增加。然而，迄今为止所做的研究并没有涵盖流体流动对布朗运动对人造微/纳米机器甚至自然粒子设计的影响。在这种情况下，彻底了解非均匀流体流动对主动/被动布朗粒子运动的影响是至关重要的。在这个方向上迈出第一步——研究剪切流中被动粒子的布朗运动（特别是热波动的非模态增长的影响）是这个项目的目标。20世纪90年代，水动力稳定性群落揭示了非均质/剪切流动的非正态性。非正态性的实质是：正则/谱数学分析的算子对应的特征模态是非正交的，在研究剪切流动中扰动的线性动力学时被证明是非最优的。这种情况引发了这些流动中线性过程的数学方法范式的变化，将重点从长期渐近分析转移到短期行为的研究。因此，在理解和描述剪切流动中的线性现象方面取得了突破。研究发现，由于剪切流动的非正态性而引起的线性现象，会导致非常特定的-代数/瞬态-扰动增长。在层流剪切流动中，流动的非正态性显著影响流体分子的热运动，最终改变了静止流体的波动背景，使其各向异性增强。具体而言，形成的波动背景的速度场具有规律性（主要是在流方向上的空间相干性），并且其强度显著超过了波数空间特定区域内流体分子的热速度。自然，这种新的流体速度场的各向异性和增强的波动背景显著地影响了布朗运动的性质和强度。本文的目的是通过对剪切流非正态性修正波动背景下的朗之万方程的重构，揭示和研究层流平面剪切流中被动粒子布朗运动的新特征。

Classical Brownian motion represents the random motion of small passive particles in a quiescent fluid due to collisions with the surrounding fluid molecules in thermal motion. The presence of an inhomogeneous flow significantly changes the character of Brownian motion. It is of fundamental importance to comprehend and describe the essence of this change since the motion of small active/passive particles in inhomogeneous fluid flows acquires more and more practical importance. This, naturally, led to a significant increase in interest to Brownian motion of active particles in inhomogeneous flows from the different research communities in recent years. Nevertheless, the research done to date does not cover the impact of fluid flow subtleties on Brownian motion on design of man-made micro/nanomachines and even of natural particles. In such cases, a thorough understanding of the influence of inhomogeneous fluid flow on the motion of active/passive Brownian particles is critical. To take the first step in this direction – to investigate the Brownian motion of passive particles in shear flows (specifically, the influence of non-modal growth of thermal fluctuations) is the aim of this project.The hydrodynamic stability community revealed the non-normal nature of inhomogeneous/shear flows in the 1990s. The essence of the non-normality is as follows: the corresponding eigenmodes of the operators of canonical/spectral mathematical analysis are non-orthogonal and turned out to be non-optimal in the study of the linear dynamics of perturbations in shear flows. This circumstance initiated the change of paradigm of the mathematical approach of linear processes in these flows, moving focus from the long-time asymptotic analysis to the study of short-time behavior. As a result, a breakthrough in the understanding and description of linear phenomena in shear flows was achieved. It was found that the linear phenomena induced due to the shear flow non-normality, lead to the very specific - algebraic/transient - growth of perturbations. In laminar shear flows, the flow non-normality significantly affects the thermal motion of fluid molecules and eventually modifies the fluctuation background of quiescent fluid -- makes it anisotropic and enhanced. Specifically, the velocity field of the formed fluctuation background acquires regularity (spatial coherence mostly in streamwise direction) and moreover, its strength significantly exceeds the thermal velocity of fluid molecules in specific area of wave number space. Naturally, this new - anisotropic and enhanced - fluctuating background of fluid velocity field significantly affects the nature and intensity of Brownian motion. Our targeted aim is to reveal and study the new character of Brownian motion of passive particles in laminar plane shear flow, by the reconstruction of Langevin equation due to the shear flow non-normality modified fluctuation background.