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摘要: 本文综述剪切流中的弥散问题. 经典Taylor–Aris理论源于截面平均浓度的长时间渐近展开, 并不适用于刻画早期阶段所呈现的多尺度动力学结构. 针对剪切弥散全过程, 本文在Lagrange坐标系下揭示其多尺度扩散叠加的物理本质, 构建统一时空尺度的弥散模型. 通过引入兼具空间各向异性和时间非线性特征的等效扩散度张量, 实现层流与湍流弥散全过程的对流与扩散动力学解耦. 在此基础上, 进一步阐述解析模型向主动物质领域的跨尺度拓展, 涵盖浮游微生物和人工粒子的弥散机理与研究进展, 重点考虑旋转动力学和剪切弥散的耦合效应及其引发的反常标度律. 结合复杂系统从个体行为到集体涌现的演化规律, 探讨发展剪切弥散解析理论的基础价值, 展望水体环境中污染物迁移转化、微藻水动力学聚集以及微纳机器人协同调控等应用前景.Abstract: This review revisits Taylor dispersion in shear flows. Since the classical theory relies on long-time asymptotic expansions of mean concentration, it inevitably overlooks the multi-scale dynamics inherent in early regimes. To address the entire evolution of shear dispersion, we unveil its physical nature as a set of superposable diffusive processes from a Lagrangian perspective—an approach naturally yields an analytic streamwise dispersion model. By formulating an equivalent diffusivity tensor characterized by spatial anisotropy and temporal nonlinearity, we successfully decouple the dynamics of advection and diffusion in laminar and turbulent flows across timescales. Further, we examine the dispersion of active-matter systems, including suspensions of motile micro-organisms and synthetic particles. Special emphasis is placed on the interplay between rotational dynamics and shear dispersion, particularly regarding the anomalous scaling. Finally, we discuss the fundamental significance of analytic theory for shear dispersion in complex systems, with applications spanning from pollutant transport and the hydrodynamic focusing of micro-algae, to the coordinated control of micro-robots.
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Key words:
- diffusion /
- shear dispersion /
- multi-scale modelling /
- analytic theory /
- fluid mechanics
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图 1 多尺度弥散过程. (a) 剪切弥散初始段、过渡段和渐近段的示意图, 其演化可通过在流动路径的不同位置测量峰值宽度得到 (Moser & Baker 2021). (b) 微通道中纳米粒子的多尺度弥散演化阶段 (Vilquin et al 2021)
图 2 湍流场中的弥散 (Xia et al 2013). (a) 弱电磁驱动湍流. (b) 强电磁驱动湍流. (c)法拉第波驱动的二维湍流
图 3 复杂流动中剪切弥散过程的理论与实验对比. (a) 微通道中带电纳米粒子在开放吸附边界下的Taylor 弥散过程, 实验结果(左列)和Taylor–Aris弥散理论(右列)的对比 (Vilquin et al 2023). (b) 植被水流中弥散系数随均匀植被有效直径的变化 (Tanino & Nepf 2008). (c) 不同横截面管道中溶质弥散的空间分布 (Aminian et al 2016). (d) 圆管泊肃叶流中点源弥散的全过程演化, Guan & Chen (2024) 的理论预测与Aminian et al (2016) 的实验观测定性一致
图 4 主动物质的水动力学聚集和致旋性捕获及非定常等效捕获机理. (a) 水动力学聚集现象 (Kessler 1985): 极地雪藻在下沉流(左侧)和上涌流(右侧)中分别呈现中心和壁面聚集. 由于细胞比水沉, 高浓度细胞在上涌流中下沉, 形成致旋型羽流和失稳. (b) 致旋性捕获现象 (Durham et al 2009): 在水平剪切流中, 致旋型微生物在弱剪切区向上迁移, 而在强剪切区则快速旋转并被捕获, 形成浮游植物薄层, 这可能是赤潮等自然灾害的潜在机理. (c) 在上涌流中, 朝向中轴线内侧的致旋型主动粒子的运动($ -\text{π} \lt \phi \lt 0 $)是不稳定的, 因为重力矩$ {T}_{G} $和黏性力矩$ {T}_{\nu } $的共同作用使粒子发生向外、向上的运动. 因此, 主动粒子在管壁和中轴线之间被短暂捕获, 最终随致旋性效应的时间积累而聚集在管壁. 在下沉流中, 致旋型主动粒子稳定地向内侧迁移($ 0 \lt \phi \lt \text{π} $). Guan et al (2023)揭示了非定常等效捕获与水动力学聚集竞争下的丰富现象
图 5 角度空间中的弥散与基于趋向性的控制策略. (a) 振荡剪切流中的角度分布, 下方是通过保角变换得到的云图, 其第一行和第二行分别为随机游走模拟和实验结果 (Leahy et al 2013). (b) 基于趋向性控制主动粒子的集体行为: 对称破缺现象的全局机理和局部机理 (Théry et al 2024)
图 6 浮游微生物的多种游动模式与机理 (Koch & Subramanian 2011). (a) 细菌通过螺旋状鞭毛束的旋转推动前进. (b) 游动细菌产生的偶极子流场. (c) 跑动–翻转(run-and-tumble)运动. (d) 伸展的流体运动会使推动者(pusher, 如细菌)沿特定方向排列, 使其力偶极子(红色箭头)增强流动, 而拉动者(puller, 如微藻)则会阻碍流动
图 7 浮游微生物密度分布的实验与理论对比. (a) 左图为抛物型流动中极地雪藻(C. nivalis)与赤潮异弯藻(H. akashiwo)形成的浮游植物薄层. 中间图片为归一化密度分布的实验数据(实线)与数值模拟(虚线)对比, 在长时间尺度下吻合得较好. 右图中垂向游动速度的实验(实线)与数值(虚线)定量一致, 蓝色条带为观测标准差. 在$ S\approx 0 $处的峰值速度的衰减(灰线)与重力致旋性理论的预测相符, 这表明致旋性可能是导致有害藻类薄层形成的重要机理 (Durham et al 2009). (b) 二维圆柱点涡流动中重力致旋性微藻密度$ n $随径向位置$ r $的分布. (各类符号对应不同时刻)的密度分布实验数据与(实线)Gauss解在长时间后高度吻合. Gauss解中, $ \Omega $表示圆柱转动的恒定角速度, $ g $是重力加速度常数, $ {v}_{s} $表示游动速度大小, $ B $为重力致旋性对应的特征重定向时间, $ \lambda =1/(2B{D}_{r}) $是重力致旋性参数, $ {D}_{r} $是旋转扩散度, 而$ F(\lambda ) $是关于$ \lambda $的无量纲函数, 具体表达式详见文献 (Cencini et al 2019). 小图为渐近长时间后微藻群体的平均径向位置随旋转频率$ f $的变化, 红点是实验测量值, 实线为理论预测对应的解析解. (c) 三维球形点涡流动中枯草芽孢杆菌(B. subtilis)的概率密度随游动方向的分布, 左侧为Sokolov & Aranson (2016) 的实验结果, 右侧为Smoluchowski模型的理论预测结果, 实验数据经归一化后落在本文解析理论所预测的统一曲线上 (Guan, Ling et al 2026)
图 8 微生物群体输运、微流控混合与细菌湍流. (a) 两只草履虫因碰撞导致的角度变化, 红色符号表示实验结果, 蓝色符号为基于squirmer模型的数值模拟 (Ishikawa 2025). (b) 概率密度分布的实验测量与理论模拟结果对比, 包括新月柄杆菌 (▲, Aranson 2022)、大肠杆菌 (▼, Berke et al 2008) 及公牛精子 (■, Rothschild 1963), 虚线表示均匀分布. 插图则展示了对应于不同鞭毛长度和旋转扩散度的数值模拟分布 (Aranson 2022). (c) 被动微通道混合器, 加粗斜线表示通道底部的交错人字形沟槽, 下方的荧光显微图像展示了等量荧光溶液和非荧光溶液的混合搅拌过程 (Stroock et al 2002). 随着横截面上溶质的快速混合, 沿主流方向的剪切弥散效应显著减弱. (d) 两图分别为枯草芽孢杆菌Lagrange湍流的可视化实验观测和连续介质模型预测的二维云图 (Dunkel et al 2013)
表 1 剪切弥散理论发展史
文献 代表性贡献 文献 代表性贡献 Taylor (1953, 1954b) 开创剪切弥散理论研究 Yasuda (1984) 分析振荡流中的弥散特性 Townsend et al (1954) 揭示湍流剪切流中扩散机理 Chatwin & Allen (1985) 建立河流与河口弥散模型 Aris (1956) 建立严格的Taylor–Aris理论 Brenner (1993) 提出广义Taylor弥散理论 Batchelor (1957) 奠定湍流弥散的数学基础 Stone & Brenner (1999) 发展高维流动的弥散理论 Saffman (1960) 给出渐近短时间的近似源解 Mei & Vernescu (2010) 引入均质化方法分析弥散 Lighthill ( 1966) 寻找早期弥散阶段的理论解 Lauga (2011) 研究往复运动粒子的弥散 Gill (1967) 构建反应–对流–扩散模型 Ishikawa & Pedley (2014) 构建微生物群体输运模型 Fischer (1973, 1976) 系统综述剪切弥散理论 Jiang & Chen (2019) 解析主动粒子弥散理论 Smith (1982) 提出弥散过程的Gauss拟合 Guan & Chen (2024) 建立沿流线的弥散理论 注: “Taylor obtained his solution by a display of the sort of brilliance we can only admire.” — H. B. Fischer (1979) 
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