This article deals with a survey of the Newtonian fluid dynamics equations with some historical notes and a discussion related to the solvability of fluid flows problems. On the way, we discuss briefly various questions related with the behavior of fluid flows governed by these fluid dynamics equations. Some of these questions are, in fact, closely connected with the well-posedness of fluid dynamics problems. We touch on the problem of the fluid dynamics limit of the Boltzmann equation of the kinetic theory of gases. An overview of some rigorous recent results on the existence and uniqueness of solutions of the fluid flows problems concludes this paper. At the end, after some concluding remarks, there are 306 references.

A wide variety of high-performance applications require materials for which shape control is maintained under substantial stress, and that have minimal density. Bio-inspired hexagonal and square honeycomb structures and lattice materials based on repeating unit cells composed of webs or trusses, when made from materials of high elastic stiffness and low density, represent some of the lightest, stiffest and strongest materials available today. Recent advances in 3D printing and automated assembly have enabled such complicated material geometries to be fabricated at low (and declining) cost. These mechanical metamaterials have properties that are a function of their mesoscale geometry as well as their constituents, leading to combinations of properties that are unobtainable in solid materials; however, a material geometry that achieves the theoretical upper bounds for isotropic elasticity and strain energy storage (the Hashin-Shtrikman upper bounds) has yet to be identified. Here we evaluate the manner in which strain energy distributes under load in a representative selection of material geometries, to identify the morphological features associated with high elastic performance. Using finite-element models, supported by analytical methods, and a heuristic optimization scheme, we identify a material geometry that achieves the Hashin-Shtrikman upper bounds on isotropic elastic stiffness. Previous work has focused on truss networks and anisotropic honeycombs, neither of which can achieve this theoretical limit. We find that stiff but well distributed networks of plates are required to transfer loads efficiently between neighbouring members. The resulting low-density mechanical metamaterials have many advantageous properties: their mesoscale geometry can facilitate large crushing strains with high energy absorption, optical bandgaps and mechanically tunable acoustic bandgaps, high thermal insulation, buoyancy, and fluid storage and transport. Our relatively simple design can be manufactured using origami-like sheet folding and bonding methods.

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Theoretical studies have indicated that truss core panels with a tetragonal topology support bending and compression loads at lower weight than competing concepts. The goal of this study is to validate this prediction by implementing an experimental protocol that probes the key mechanical characteristics while addressing node eccentricity and structural robustness. For this purpose, panels have been fabricated from a beryllium-copper alloy using a rapid prototyping approach and investment casting. Measurements were performed on these panels in flexure, shear and compression. Numerical simulations were conducted for these same configurations. The measurements reveal complete consistency with the stiffness and limit load predictions, as well as providing a vivid illustration of asymmetric structural responses that arises because the bending behavior of optimized panels is dependent on truss orientation. Ó

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