Density-contrast instabilities in finite element modelling of slow viscous flow

Finite element methods are often used to model Earth processes involving slow viscous or viscoelastic flow. Inertial terms of the Navier-Stokes equations are neglected in very slow flows, so timestep size is not limited by the Courant instability. However, where there is advection of density contrasts in a gravitational field, over-advection can lead to numerically induced flow oscillations. We derive analytic results for the maximum stable timestep size in two cases: a free surface over a fluid of uniform density, and a free surface kept level by sedimentation/erosion, but with a density gradient in the underlying medium. Using parameters appropriate to the Earth's crust we show that the density-contrast instability occurs for timesteps larger than 3000 years for the constant-density case. For a fluid with a density gradient of 10 kg/m³ per km the solution is stable for timesteps up to about 200,000 years if full erosion/sedimentation is implemented.     

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Stereochemical Definition of the Natural Product (6R,10R,13R, 14R,16R,17R,19S,20S,21R,24S,25S,28S,30S,32R,33R,34R,36S,37S,39R)‐Azaspiracid‐3 by Total Synthesis and Comparative Analyses

The previously accepted structure of the marine toxin azaspiracid‐3 is revised based upon an original convergent and stereoselective total synthesis of the natural product. The development of a structural revision hypothesis, its testing, and corroboration are reported. Synthetic (6R,10R,13R,14R,16R,17R,19S,20S,21R,24S,25S,28S,30S,32R, 33R,34R,36S,37S,39R)‐azaspiracid‐3 chromatographically and spectroscopically matched naturally occurring azaspiracid‐3, whereas the previously assigned 20R epimer did not.     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

Potentiometric determination of equivalent weight of ortho-substituted benzonitriles

Two potentiometric procedures are described for the titrimetric determination of the equivalent weight of ortho-substituted benzonitriles. The methods are simple, accurate, and applicable to compounds which otherwise cannot be determined by titration. The procedures are not restricted to ortho-substituted benzonitriles and are generally applicable to nitriles in general, all of which readily undergo the amidoximation reaction. In addition, these methods can be performed using readily available, inexpensive equipment found in all laboratories.     

The use of in-situ test method EN 1793-6 for measuring the airborne sound insulation of noise barriers

The in situ measurement of the airborne sound insulation, as outlined in EN 1793-6:2012, is becoming a common means of quantifying the performance of road traffic noise reducing devices. Newly installed products can be tested to reveal any construction defects and periodic testing can help to identify long term weaknesses in a design. The method permits measurements to be conducted in the presence of background noise from traffic, through the use of impulse response measurement techniques, and is sensitive to sound leakage. Factors influencing the measured airborne sound insulation are discussed, with reference to measurements conducted on a range of traffic noise barriers located around Auckland, New Zealand. These include the influence of sound leakage in the form of hidden defects and visible air gaps, signal-to-noise ratio, and noise barrier height. The measurement results are found to be influenced by the presence of hidden defects and small air gaps, with larger air gaps making the choice of measurement position critical. A signal-to-noise ratio calculation method is proposed, and is used to show how the calculated airborne sound insulation varies with signal-to-noise ratio. It is shown that the measurement results are influenced by barrier height, through the need for reduced length Adrienne temporal windows to remove the diffraction components, prohibiting the direct comparison of results from noise barriers with differing heights.