Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

The formulation, testing and stability of 16% fat cream liqueurs

Cream liqueurs manufactured by a one-step process, where alcohol was added before homogenisation, were more stable than those processed by a two -step process which involved addition of alcohol after homogenisation. Using the one-step process, it was possible to produce creaming-stable liqueurs by using one pass through a homogeniser (27.6 MPa) equipped with "liquid whirl" valves. Test procedures to characterise cream liqueurs and to predict shelf life were studied in detail. A turbidity test proved simple, rapid and sensitive for characterising particle size and homogenisation efficiency. Prediction of age thickening/gelation in cream liqueurs during incubation at 45 °C depended on the age of the sample when incubated. Samples that gelled at 45 °C may not do so at ambient temperature. Commercial cream liqueurs were similar in gross chemical composition, and unlike experimentally produced liqueurs, these did not exhibit either age-gelation at ambient or elevated temperatures. Solutions of commercial sodium caseinates from different sources varied in their calcium sensitivity. When incorporated into cream liqueurs, caseinates influenced the rate of viscosity increase, coalescence and, possibly, gelation during incubated storage. Mild heat and alcohol treatment modified the properties of caseinate used to stabilise non-alcoholic emulsions, while the presence of alcohol in emulsions was important in preventing clustering of globules. The response to added trisodium citrate varied. In many cases, addition of the recommended level (0.18%) did not prevent gelation. Addition of small amounts of NaOH with 0.18 % trisodium citrate before homogenisation was beneficial. The stage at which citrate was added during processing was critical to the degree of viscosity increase (as opposed to gelation) in the product during 45 °C incubation. The component responsible for age-gelation was present in the milk-solids non fat portion of the cream and variations in the creams used were important in the age-gelation phenomenon Results indicated that, in addition to possibly Ca++, the micellar casein portion of serum may play a role in gelation. The role of the low molecular weight surfactants, sodium stearoyl lactylate and monodiglycerides in preventing gelation, was influenced by the presence of trisodium citrate. Clustering of fat globules and age-gelation were inhibited when 0.18 % citrate was included. Inclusion of sodium stearoyl lactylate, but not monodiglycerides, reduced the extent of viscosity increase at 45 °C in citrate containing liqueurs.