Theory of cylindrical dense packings of disks

We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analysing the rules for phyllotactic indices of related structures and the variation of the density for line-slip structures, close to the symmetric ones. We show that rhombic structures, which are of a lower density, are always unstable i.e. can be increased in density by small perturbation

Phyllotaxis, disk packing, and Fibonacci numbers

We consider the evolution of the packing of disks (representing the position of buds) that are introduced at the top of a surface which has the form of a growing stem. They migrate downwards, while conforming to three principles, applied locally: dense packing, homogeneity and continuity. We show that spiral structures characterised by the widely observed Fibonacci sequence (1,1,2,3,5,8,13...), as well as related structures, occur naturally under such rules. Typical results are presented in a animation.Comment: Accompanying animation is located here: https://youtu.be/gFKeOZTKpZ

Metallic foam processing from the liquid state

A model is developed to describe the formation of metallic foams in which liquid drainage acts to collapse the foam before it can freeze. Numerical solution of the foam drainage equation, combined with the equations of heat conduction, provides new insight into the competition between these two processes. It also stimulates and confirms a theoretical analysis which leads to criteria for creating uniform samples of frozen metal foam. The analysis suggests new experiments to clarify the role of the various processes leading to foam formation

Topological changes in a two-dimensional foam cluster

Experiments on a small cluster of bubbles in a nominally two-dimensional foam show an instability in which a topological change forces one of the bubbles to be ejected to the outside of the cluster at a point where this is not predicted by a two-dimensional model of a foam. This is interpreted in terms of the energy of the initial and ejected states and of the finite liquid content of the experimental system. A description of the distribution of liquid in various experimental set-ups suggests that the exact response may depend critically upon the type of system used. This is demonstrated experimentally with reference to small clusters of bubbles undergoing a single topological change

Rocking Newton’s cradle

In textbook descriptions of Newton’s cradle, it is generally claimed that displacing one ball will result in a collision that leads to another ball being ejected from the line, with all others remaining motionless. Hermann and Schmälzle, Hinch and Saint-Jean, and others have shown that a realistic description is more subtle. We present a simulation of Newton’s cradle that reproduces the break-up of the line of balls at the first collision, the eventual movement of all the balls in phase, and is in good agreement with our experimentally obtained data. The first effect is due to the finite elastic response of the balls, and the second is a result of viscoelastic dissipation in the impacts. We also analyze a dissipation-free ideal Newton’s cradle which displays complex dynamics.This work was funded by Enterprise Ireland (Basic Research Grant No. SC/2000/239/Y) for one of the authors (S. H.) and a Trinity College Dublin Research Studentship for another (G. D.