Transport processes such as the dispersion and mixing of solutes are governed by the interplay of advection and diffusion, where advection acts to organise fluid streamlines and diffusion acts to randomise solute molecules. Thus, the structure and organisation of streamlines, termed the Lagrangian kinematics of the flow, is central to the understanding and modelling of these transport processes. A key question is whether the streamlines in three-dimensional (3D) Darcy flows can wander freely through the fluid domain, or whether all streamlines of the flow are organised into a series of smooth, non-intersecting two-dimensional (2D) surfaces. The existence of such a foliation of surfaces constrains the Lagrangian kinematics in a manner similar to that of 2D flows, which in turn constrains the allowable transport processes. In a series of pioneering studies, Sposito (Water Resour. Res., 30(8):2395–2401, 1994; Adv. Water Resour., 24(7):793–801, 2001) argues that steady Darcy flow in locally isotropic media gives rise to Lamb surfaces, 2D material surfaces which are spanned by both the streamlines and vortex lines (field lines of the vorticity vector) of the flow. Hence, the existence of these surfaces renders the kinematics of such 3D steady Darcy flow as two dimensions. This topological constraint strongly affects transverse mixing and dispersion because 2D steady flow fields limit the rate of deformation of fluid elements and can only admit zero hydrodynamic transverse dispersion. In this study, however, we show that Lamb surfaces are not ubiquitous to all steady Darcy flows in locally isotropic media. We derive the conditions for when Lamb surfaces exist in such Darcy flows, and discuss the implications of these findings for the transport, mixing, and dispersion of solutes.

中文翻译：

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达西流中的流体动力色散和羔羊表面

诸如溶质的分散和混合之类的传输过程受平流和扩散的相互作用控制，其中平流作用是组织流体流线，而扩散作用是使溶质分子随机化。因此，流线的结构和组织，称为流动的拉格朗日运动学，是理解和建模这些传输过程的核心。一个关键问题是三维 (3D) 达西流中的流线是否可以在流体域中自由游荡，或者流的所有流线是否都组织成一系列平滑、不相交的二维 (2D) 表面。这种表面叶理的存在以类似于二维流动的方式约束拉格朗日运动学，这反过来又约束了允许的传输过程。在一系列开创性研究中，Sposito (Water Resour. Res., 30(8):2395–2401, 1994; Adv. Water Resour., 24(7):793–801, 2001) 认为稳定的达西流在局部各向同性介质产生 Lamb 表面，即由流线和涡流线（涡流矢量的场线）跨越的二维材料表面。因此，这些表面的存在将这种 3D 稳定达西流的运动学呈现为二维。这种拓扑约束强烈影响横向混合和分散，因为二维稳态流场限制了流体元素的变形率，并且只能允许零流体动力学横向分散。然而，在这项研究中，我们表明兰姆表面对于局部各向同性介质中的所有稳定达西流并不普遍。