Bondi mass with a cosmological constant

The mass loss of an isolated gravitating system due to energy carried away by gravitational waves with a cosmological constant $\Lambda\in\R$ was recently worked out, using the Newman-Penrose-Unti approach. In that same article, an expression for the Bondi mass of the isolated system, $M_\Lambda$, for the $\Lambda>0$ case was proposed. The stipulated mass $M_\Lambda$ would ensure that in the absence of any incoming gravitational radiation from elsewhere, the emitted gravitational waves must carry away a positive-definite energy. That suggested quantity however, introduced a $\Lambda$-correction term to the Bondi mass $M_B$ (where $M_B$ is the usual Bondi mass for asymptotically flat spacetimes) which would involve not just information on the state of the system at that moment, but ostensibly also its past history. In this paper, we derive the identical mass-loss equation using an integral formula on a hypersurface formulated by Frauendiener based on the Nester-Witten identity, and argue that one may adopt a generalisation of the Bondi mass with $\Lambda\in\R$ \emph{without any correction}, viz. $M_\Lambda=M_B$ for any $\Lambda\in\R$. Furthermore with $M_\Lambda=M_B$, we show that for \emph{purely quadrupole gravitational waves} given off by the isolated system (i.e. when the "Bondi news" $\sigma^o$ comprises only the $l=2$ components of the "spherical harmonics with spin-weight 2"), the energy carried away is \emph{manifestly positive-definite} for the $\Lambda>0$ case. For a general $\sigma^o$ having higher multipole moments, this perspicuous property in the $\Lambda>0$ case still holds if those $l>2$ contributions are weak --- more precisely, if they satisfy any of the inequalities given in this paper.Comment: 29 pages, accepted for publication by Physical Review

Helicalised fractals

We formulate the helicaliser, which replaces a given smooth curve by another curve that winds around it. In our analysis, we relate this formulation to the geometrical properties of the self-similar circular fractal (the discrete version of the curved helical fractal). Iterative applications of the helicaliser to a given curve yields a set of helicalisations, with the infinitely helicalised object being a fractal. We derive the Hausdorff dimension for the infinitely helicalised straight line and circle, showing that it takes the form of the self-similar dimension for a self-similar fractal, with lower bound of 1. Upper bounds to the Hausdorff dimension as functions of $\omega$ have been determined for the linear helical fractal, curved helical fractal and circular fractal, based on the no-self-intersection constraint. For large number of windings $\omega\rightarrow\infty$, the upper bounds all have the limit of 2. This would suggest that carrying out a topological analysis on the structure of chromosomes by modelling it as a two-dimensional surface may be beneficial towards further understanding on the dynamics of DNA packaging.Comment: 25 pages, 10 figures. v3: Detailed derivation of the Hausdorff dimension included. Accepted by Chaos, Solitons & Fractal

No-boarding buses: Synchronisation for efficiency

We investigate a no-boarding policy in a system of $N$ buses serving $M$ bus stops in a loop, which is an entrainment mechanism to keep buses synchronised in a reasonably staggered configuration. Buses always allow alighting, but would disallow boarding if certain criteria are met. For an analytically tractable theory, buses move with the same natural speed (applicable to programmable self-driving buses), where the average waiting time experienced by passengers waiting at the bus stop for a bus to arrive can be calculated. The analytical results show that a no-boarding policy can dramatically reduce the average waiting time, as compared to the usual situation without the no-boarding policy. Subsequently, we carry out simulations to verify these theoretical analyses, also extending the simulations to typical human-driven buses with different natural speeds based on real data. Finally, a simple general adaptive algorithm is implemented to dynamically determine when to implement no-boarding in a simulation for a real university shuttle bus service.Comment: 49 pages, 9 figures. Video available here: https://www.youtube.com/watch?v=SBNqvTr1Aj

Unsteady flow, clusters and bands in a model shear-thickening fluid

We analyse the flow curves of a two-dimensional assembly of granular particles which are interacting via frictional contact forces. For packing fractions slightly below jamming, the fluid undergoes a large scale instability, implying a range of stress and strainrates where no stationary flow can exist. Whereas small systems were shown previously to exhibit hysteretic jumps between the low and high stress branches, large systems exhibit continuous shear thickening arising from averaging unsteady, spatially heterogeneous flows. The observed large scale patterns as well as their dynamics are found to depend on strainrate: At the lower end of the unstable region, force chains merge to form giant bands that span the system in compressional direction and propagate in dilational direction. At the upper end, we observe large scale clusters which extend along the dilational direction and propagate along the compressional direction. Both patterns, bands and clusters, come in with infinite correlation length similar to the sudden onset of system-spanning plugs in impact experiments

Characterization and flow of food and mineral powders : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering at Massey University, Manawatū, New Zealand

Powders are important commodities across different industries, such as the food and pharmaceutical industries. In these industries, powders are usually made, mixed, milled, packaged, and stored; these operations require the powders to move and flow under desired conditions and different stress levels. Failure to flow will cause hindrances to production; therefore knowledge of powder flow or flowability is important. There is a constant demand for accurate, reliable, and robust measurement and characterization methods for powder flowability. Powders behave differently under varying conditions; the behaviour of a powder is influenced by particle size distribution, and powder handling and processing conditions. There is to date no one “standard” method to characterize powder flowability; it is common to use a variety of methods and devices to measure flow properties and provide insight into the behaviour and flow characteristics of powders under different conditions. The flow properties of model food and mineral powders were measured and assessed by shear testing, compression via tapping, fluidization, and powder tumbling. Shear testing was done with an annular shear cell following Jenike (1964) and Berry, Bradley and McGregor (2014). Compression via tapping was performed according to a procedure in the dairy industry (Niro, 1978) and the European Pharmacopoeia (Schüssele & Bauer-Brandl, 2003). Fluidization was used to measure powder bed expansion and bed collapse following the powder classification framework provided by Geldart and co-workers (Geldart, 1973; Geldart, Harnby, & Wong, 1984; Geldart & Wong, 1984, 1985). Powder tumbling was performed in a novel Gravitational Displacement Rheometer, GDR, which measured the motion and avalanche activity of powders that moved under their own weight when rotated in a cylinder at different drum speed levels. The flow data from each characterization method were evaluated individually with regards to particle size distribution and then assessed collectively. The findings presented and discussed include the i) demonstration of the dominant influence of surface-volume mean particle diameter on powder flow properties, ii) characterization of flowability based on Jenike’s arbitrary flow divisions, iii) development of new correlations for the estimation of powder cohesion and bulk density at low preconsolidation stresses, iv) demonstration of hopper outlet diameter as a measure of flowability, v) demonstration of the limited utility of Hausner ratio as a flowability index, vi) substantiation of von Neumann ratio as a sensitive and useful indicator for identifying the onset of bubbling in fluidized beds using bed pressure fluctuation data, and vii) demonstration of the utility of standard deviation of the GDR load cell signal as an indicator of powder avalanche activity. These findings provide improved understanding and knowledge of powder flowability; they can be used to assist and facilitate the development of new techniques and solutions relevant to the handling and processing of powders especially in the food and pharmaceutical industries