Double Bruhat Cells in Kac–Moody Groups and Integrable Systems

We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter elements in the affine Weyl group. In the process we extend various results on double Bruhat cells in simple algebraic groups to the setting of Kac–Moody groups. We also generalize some fundamental results in Poisson–Lie theory to the setting of ind-algebraic groups, which is of interest beyond our immediate applications to integrable systems.     

Mode beating in spot-size converter lasers

An anomalous modulation in the wavelength spectrum has been observed in lasers with spot-size converters. This intensity modulation is shown to be caused by beating between the fundamental lasing mode and radiation modes in the taper. This results in a periodic modulation in the net gain spectrum, which causes wavelength jumps between adjacent net gain maxima, and a drive current dependent spectral width that is expected to affect system performance. The amplitude of this spectral modulation is reduced significantly by either using an angled rear-facet which reflects the beating radiation modes away from the laser axis, or by using a nonlinear, adiabatic taper.     

Diffusion from sessile droplets through membranes

We analyze diffusion from a periodic array of hemispherical droplets through a membrane. We find that the multiple sources do not interact strongly, even when the droplets are closely spaced, so that the flux through the membrane appears nearly additive.     

Electron paramagnetic resonance investigation of molecular bond rupture due to ozone in deformed rubber

Electron paramagnetic resonance (EPR) provides a sensitive tool by which microscopic bond rupture can be monitored simultaneously with observations of macroscopic deformation and failure. Past techniques for studying fracture in semicrystalline polymers have been extended to investigate degradation of unfilled ruber in the presence of ozone. It was found that the rate of free radical production was linearly proportional to stretch ratio and ozone concentration and that stress relaxation and creep were not directly proportional to this production rate. The latter behavior was attributed to the particular dependence of crack density and growth on stress. It was concluded that at low strains, comparatively few surface cracks form; however, at higher strains, many more crack centers become active. Although many more surface cracks are present, they do not progress into the material as rapidly. Therefore, although more bonds were broken at higher strains and stresses, the stress relaxation rate and creep rates were not significantly increased.     

Effect of pressure on structure and extinction of near-limit hydrogen counterflow diffusion flames

Results of measurements of critical conditions for extinction and of temperature profiles in counterflow diffusion flames are reported. The fuel was a hydrogen–nitrogen mixture with 14 mole percent hydrogen, and the oxidizer was air. Pressures ranged from 0.1 MPa to 1.5 MPa; measurements were made in a facility especially constructed for carrying out counterflow combustion experiments at high pressures. With increasing pressure, the strain rate at extinction first increases and then decreases, in qualitative agreement with predictions, but there are observable quantitative differences. Temperature profiles, obtained employing an R-type thermocouple at a fixed strain rate of 100/s, agree well with predictions, within experimental uncertainty. The results may help to improve knowledge of underlying chemical-kinetic and transport parameters at elevated pressures.     

Effect of addition of a non-equidiffusional reactant to an equidiffusional diffusion flame

A Burke–Schumann (flame-sheet) formulation is developed for diffusion flames between a fuel and oxidiser with Lewis numbers of unity, subject to addition to the fuel and/or oxidiser stream of a different reactant for which the Lewis number differs from unity. This formulation is applied to laminar counterflow diffusion-flame experiments, reported here, in which hydrogen was added to either methane–nitrogen mixtures or oxygen–nitrogen mixtures at normal atmospheric pressure, with both feed streams at normal room temperature. Experimental conditions were adjusted to fix selected values of the stoichiometric mixture fraction and the adiabatic flame temperature, and the strain rate was increased gradually, maintaining the momentum balance of the two streams, until extinction occurred. At the selected sets of values, the strain rate at extinction was measured as a function of the hydrogen concentration in the fuel or oxidiser stream. The ratio of the fraction of the oxidiser flux that consumes hydrogen to the fraction that consumes fuel was calculated from the new Burke–Schumann formulation, and it was found that, within experimental uncertainty, the ratio of the extinction strain rate with hydrogen addition to that without was the same at any given value of this oxygen flux ratio, irrespective of whether the hydrogen was added on the fuel or oxidiser side. This experimental result was also in close agreement with computational predictions employing detailed chemistry. These results imply that differences in detailed hydrogen concentration profiles within the reaction zone have little or no influence on the chemical kinetics of extinction when the stoichiometric mixture fraction, the adiabatic flame temperature, and the proportion of oxygen that consumes the added fuel are fixed. This same correspondence may be expected to apply for other fuels and additives.     

The role of cool-flame chemistry in quasi-steady combustion and extinction of n-heptane droplets

Experiments on the combustion of large n-heptane droplets, performed by the National Aeronautics and Space Administration in the International Space Station, revealed a second stage of continued quasi-steady burning, supported by low-temperature chemistry, that follows radiative extinction of the first stage of burning, which is supported by normal hot-flame chemistry. The second stage of combustion experienced diffusive extinction, after which a large vapour cloud was observed to form around the droplet. In the present work, a 770-step reduced chemical-kinetic mechanism and a new 62-step skeletal chemical-kinetic mechanism, developed as an extension of an earlier 56-step mechanism, are employed to calculate the droplet burning rates, flame structures, and extinction diameters for this cool-flame regime. The calculations are performed for quasi-steady burning with the mixture fraction as the independent variable, which is then related to the physical variables of droplet combustion. The predictions with the new mechanism, which agree well with measured autoignition times, reveal that, in decreasing order of abundance, H₂O, CO, H₂O₂, CH₂O, and C₂H₄ are the principal reaction products during the low-temperature stage and that, during this stage, there is substantial leakage of n-heptane and O₂ through the flame, and very little production of CO₂ with no soot in the mechanism. The fuel leakage has been suggested to be the source of the observed vapour cloud that forms after flame extinction. While the new skeletal chemical-kinetic mechanism facilitates understanding of the chemical kinetics and predicts ignition times well, its predicted droplet diameters at extinction are appreciably larger than observed experimentally, but predictions with the 770-step reduced chemical-kinetic mechanism are in reasonably good agreement with experiment. The computations show how the key ketohydroperoxide compounds control the diffusion-flame structure and its extinction.     

Residual stresses in continuous-tungsten-fibre-reinforced Kanthal-matrix composites

Residual stresses were measured in four Kanthal matrix-continuous-tungsten-fibre composites (with different tungsten fibre volume fractions V f = 10, 20, 30 and 70 vol.%) using neutron diffraction. Parallel to the fibres the stress in the Kanthal ranged from 40 MPa ( V f = 10 vol.%) to 1100 MPa ( V f = 70 vol.%) compared with m1877 MPa ( V f = 10 vol.%) to m400 MPa ( V f = 70 vol.%) for the tungsten. Perpendicular to the fibres the stress ranged from m52 MPa ( V f = 10 vol.%) to 620 MPa ( V f = 70 vol.%) in the Kanthal compared with m778 MPa ( V f = 10vol.%) to m195 MPa ( V f = 70 vol.%) in the tungsten. Assuming that the measured residual stresses were solely thermal in origin, predictions were made using concentric cylinder and finite-element models. In the absence of hardening data the assumed material behaviour was elastic-perfectly plastic and the predictions underestimated the measured stresses for all volume fractions. Nevertheless the model results were consistent with the experimental measurements. The transverse stress in the fibres is discussed in the context of the interface normal stress, which is significant to the global mechanical response.     

Self-avoiding random walk: A Brownian motion model with local time drift

Summary A natural model for a self-avoiding Brownian motion inR ^d, when specialised and simplified tod=1, becomes the stochastic differential equation , where {L(t, x):t0,xR} is the local time process ofX. ThoughX is not Markovian, an analogue of the Ray-Knight theorem holds for {L(,x):xR}, which allows one to prove in many cases of interest that exists almost surely, and to identify the limit.