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In recent years there has been vigorous debate whether asymptotic transverse macrodispersion exists in steady three-dimensional (3D) groundwater flows in the purely advective limit. This question is tied to the topology of 3D flow paths (termed the Lagrangian kinematics), specifically whether streamlines can undergo braiding motions or can wander freely in the transverse direction. In this study we determine which Darcy flows do admit asymptotic transverse macrodispersion for purely advective transport on the basis of the conductivity structure. We prove that porous media with smooth, locally isotropic hydraulic conductivity exhibit zero transverse macrodispersion under pure advection due to constraints on the Lagrangian kinematics of these flows, whereas either non-smooth or locally anisotropic conductivity fields can generate transverse macrodispersion. This has implications for upscaling locally isotropic porous media to the block scale as this can result in a locally anisotropic conductivity, leading to non-zero macrodispersion at the block scale that is spurious in that it does not arise for the fully resolved Darcy scale flow. We also show that conventional numerical methods for computation of particle trajectories do not explicitly preserve the kinematic constraints associated with locally isotropic Darcy flow, and propose a novel psuedo-symplectic method that preserves these constraints. These results provide insights into the mechanisms that govern transverse macrodispersion in groundwater flow, and unify seemingly contradictory results in the literature in a consistent framework. These insights call into question the ability of smooth, locally isotropic conductivity fields to represent flow and transport in real heterogeneous porous media., The authors thank the reviewers for their constructive feedback which has improved the manuscript. M.D. acknowledges the support of the Spanish Research Agency (https://doi.org/10.13039/501100011033), Spanish Ministry of Science and Innovation and European Regional Development Fund “A way of making Europe” through Grants CEX2018-000794-S and HydroPore PID2019-106887GB-C31. Open access publishing facilitated by RMIT University, as part of the Wiley - RMIT University agreement via the Council of Australian University Librarians.

A crane hook or a lifting hook is used for picking up the load with the help of devices such as a hoist, chain or wire ropes. It is subjected to bending stresses which makes it highly prone to failure. To prevent structural failure of a crane hook, we must study the stresses induced due to loading and unloading as well as stress concentration pattern. This review paper looks at the findings established in previous publications to determine the optimum cross section and material combination.

Stretching of fluid elements by a heterogeneous flow field, such as the flow through a porous media or geophysical flows such as atmospheric or oceanic vortices, is known to enhance mixing rates of scalar fields[1]. While the mechanisms leading to the elongation of material lines are well understood, predicting mixing rates still remains a challenge particularly when there is a reconnection (or aggregation) between several parts of the mixing interface, leading, at large mixing time, to a so-called coalescence regime[1][2]. In this presentation, we numerically study this coalescence dynamics through scalar transport in two different flow fields, the Rankine vortex and Stokes flow through a periodic bead pack[3]. The former is typical of large-scale turbulent flows [4] whereas the second is generic of small-scale laminar flows in porous media [5]. Both flows show a net elongation of the mixing interfaces, although at very different rates. To solve the transport problem in these flows, we use a Lagrangian method (the diffusive strip method[6]). This method allows us to reconstruct, at high resolution, the scalar concentration fields and to compute the evolution of the distribution of concentrations levels, scalar dissipation rate and scalar power spectrum in time. The signature of coalescence is clearly observed in both flows and we assess the influence of coalescence on the difference in mixing behaviour for the two flows. We finally discuss how coalescence may affect the reaction kinetics of mixing-limited reactive flows. The analysis proposed sheds light on fundamental aspects of transport and mixing in earth surface and subsurface flows.[1] Emmanuel Villermaux. Mixing versus stirring. Annual Review of Fluid Mechanics, 51:245–273, 2019.[2] Tanguy Le Borgne, Marco Dentz, and Emmanuel Villermaux. The lamellar description of mixing in porous media. Journal of Fluid Mechanics, 770:458–498, 2015.[3] Régis Turuban, David R Lester, Tanguy Le Borgne, and Yves Méheust. Space-group symmetries generate chaotic fluid advection in crystalline granular media. Physical review letters, 120(2):024501, 2018.[4] RT Pierrehumbert. Large-scale horizontal mixing in planetary atmospheres. Physics of Fluids A: Fluid Dynamics, 3(5):1250–1260, 1991.[5] Brian Berkowitz, Andrea Cortis, Marco Dentz, and Harvey Scher. Modeling non-fickian transport in geological formations as a continuous time random walk. Reviews of Geophysics, 44(2), 2006.[6] Patrice Meunier and Emmanuel Villermaux. The diffusive strip method for scalar mixing in two dimensions. Journal of fluid mechanics, 662:134–172, 2010.

International audience; We analyze the dynamics of solute mixing and reaction in a mixing-limited reactive flow by considering the transport of a tracer in a linear shear flow and in a Rankine vortex. The action of a shear flow, in general, achieves stretching of fluid elements due to the heterogeneous nature of the flow. A vortex flow exhibits not only stretching but also folding of fluid elements in a way that brings adjacent fluid elements closer at every turn. A strong stretching along the tangential direction is accompanied by a concomitant thinning in the radial direction leading to a strong diffusive flux, which may cause the material from neighboring regions of the mixing interface to aggregate. Through a Lagrangian concentration evolution technique, the diffusive strip method, we obtain the concentration field and pinpoint the signature of coalescence of two neighboring concentration regions by analyzing the concentration distribution profiles. The role of substrate deformation on the reaction kinetics of a classical heterogeneous chemical reaction is also studied where we derive analytical expressions for the coupling between the rate of product formation and the Peclet number in different time limits. Finally, the impact of coalescence on reaction rates is studied for a Rankine vortex, a result that holds important implications for simple bimolecular reactions. This analysis is useful to understand scalar dispersion in vortical flow structures and the consequences of stretching-enhanced diffusion in mixing-limited reactive flows.