The most efficient way to pack equally sized spheres isotropically in three dimensions is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution based upon a mapping onto a one-dimensional problem. To test this theory we performed extensive simulations for mixtures of elastic spheres with hydrodynamic friction. The simulations show a general (but weak) dependence of the final (essentially hard sphere) packing density on fluid viscosity and on particle size but this can be eliminated by choosing a specific relation between mass and particle size, making the random close packed volume fraction well defined. Our theory agrees well with the simulations for bidisperse, tridisperse, and log-normal distributions and correctly reproduces the exact limits for large size ratios.
Flows of droplets through networks of microchannels differ significantly from the flow of simple fluids. Our report focuses on the paths of individual droplets through the simplest possible network: a channel that splits into two arms that subsequently recombine. This simple system exhibits complex patterns of flow: both periodic and irregular, depending on the frequency at which the drops are fed into the "loop." A numerical model explains these results and shows regions of regular patterns separated by regions of high complexity. Our results elicit new questions regarding the dynamics of flow of discrete elements of fluids through networks, and point to potential opportunities and difficulties in the design of integrated mini-laboratories operating on droplets.
clusters of olivine are present at the roof, forming a foreshortened mirror image of the coarsening-upwards component of the floor accumulation. The coarsening-upwards sequence records the growth of olivine crystals while in suspension in a convecting magma, and their aggregation into clusters, followed by settling over a prolonged period (with limited trapping at the roof). As olivine was progressively lost from the convecting magma, crystal accumulation on the (contemporaneous) floor of the PCU was increasingly dominated by plagioclase, most likely forming clusters and aggregates with augite and olivine, both of which form large poikilitic grains in the crinanite. While the PCU is unusual in being underlain by an earlier, still hot, intrusion that would have enhanced any driving force for convection, we conclude from comparison with microstructures in other sills that convection is likely in tabular bodies >100 m thickness.
A fractal design is shown to be highly efficient both as a load bearing structure and as a general metamaterial. Through changing the hierarchical order of the structure, the scaling of material required for stability against loading can be manipulated. We show that the transition from solid to hollow beams changes the scaling in a manner analogous to increasing the hierarchical order by one. An example second order solid beam frame is constructed using rapid prototyping techniques. The optimal hierarchical order of the structure is found for different values of loading. Possible fabrication methods and applications are then discussed.
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