Tensors

Tensors and tensor algebra are presented. The concept of a tensor is defined in two ways: as something which yields a scalar from a set of vectors, and as something whose components transform a given way. The meaning and use of these definitions is expounded carefully, along with examples. The action of the metric and its inverse (index lowering and raising) is derived. The relation between geodesic coordinates and Christoffel symbols is obtained. The difference between partial differentiation and covariant differentiation is explained at length. The tensor density and Hodge dual are briefly introduced.

Partial differentiation of air density in mass metrology

There are four major uncertainty components to be considered when performing mass comparisons. They are uncertainties of weighing process, reference weight used, air buoyancy, and mass comparator. The systematic effect of air buoyancy can be greatly reduced if the air density and the densities of the test and reference weights are known. This paper will emphasis on the uncertainty due to air buoyancy correction only. To calculate the uncertainty of air density correction, partial derivatives of temperature, barometric pressure and humidity must be performed. In this paper, two methods for partial differentiation of air density components are discussed.

A New Kinematic Error Modelling and Identification Method of the Parallel Manipulator

Parallel manipulators have broad application prospects on hybrid machine tools. Kinematic error modelling and identification are two key processes to improve the accuracy of parallel manipulators. The traditional kinematic error modelling method adopts the partial differentiation of the ideal kinematic model. However, the partial differentiation method is pure mathematical calculation, which ignores physical meaning of error terms corresponding to each link. In the process of error identification, the Jacobian matrix obtained from the partial differentiation method is usually ill-conditioned, which leads to non-convergence of the identification process. In order to solve the above problems, this paper proposes a new kinematic error modelling method and an error identification model. Firstly, the basic error terms for single link are analyzed. Based on basic error terms, the kinematic error model is established by using the practical connection point of two adjacent links. Then, a new error identification model is derived from the kinematic error model. Finally, as a study case, a 3-DOF parallel tool head is used to verify the correctness of the proposed method. The numerical results show that the proposed method is effective and the accuracy of the 3-DOF parallel tool head improves significantly after compensation of error terms.

Meteorite evidence for partial differentiation and protracted accretion of planetesimals

Modern meteorite classification schemes assume that no single planetary body could be source of both unmelted (chondritic) and melted (achondritic) meteorites. This dichotomy is a natural outcome of formation models assuming that planetesimal accretion occurred nearly instantaneously. However, it has recently been proposed that the accretion of many planetesimals lasted over ≳1 million years (Ma). This could have resulted in partially differentiated internal structures, with individual bodies containing iron cores, achondritic silicate mantles, and chondritic crusts. This proposal can be tested by searching for a meteorite group containing evidence for these three layers. We combine synchrotron paleomagnetic analyses with thermal, impact, and collisional evolution models to show that the parent body of the enigmatic IIE iron meteorites was such a partially differentiated planetesimal. This implies that some chondrites and achondrites simultaneously coexisted on the same planetesimal, indicating that accretion was protracted and that apparently undifferentiated asteroids may contain melted interiors.

Numerical Simulation of Crude Oil Dispersion in Water

Dispersion is one of the fate processes of oil spill. This research has been carried out on the numerical simulation of the dispersion of crude oil using the model obtained from the work of Hamam (1987). The model was solved with the explicit, implicit and Crank-Nicolson methods of solution of partial differentiation equations with the aid of MATLAB, and the concentration of the crude oil dispersed in water was obtained. The results obtained revealed that the three methods could be used to study the process because the profiles given by all of them were very similar. Also discovered from the investigations carried out was that the concentration of crude oil was decreasing with time for a particular spatial point while, for a particular time, it was increasing along the length of the water body. It was also discovered that experiments would be very necessary in order for the validation of the results obtained from the simulations.

Thermal convection in the crust of the dwarf planet – I. Ceres

ABSTRACT Ceres is the largest body in the Main Belt, and it is characterized by a large abundance of water ice in its interior. This feature is suggested by its relatively low bulk density (2162 kg m−3), while its partial differentiation into a rocky core and icy crust is suggested by several geological and geochemical features: minerals and salts produced by aqueous alteration, icy patches on the surface, and lobate morphology interpreted as surface flows. In this work, we explore how the composition can influence the characteristics of thermal convection in the crust of Ceres. Our results suggest that the onset of thermal convection is difficult and when it occurs, it is short lived, which could imply that Ceres preserved deep liquid until present, as recently suggested by the work of Castillo-Rogez et al. Moreover, cryovolcanism could be driven by diapirism (chemical convection) rather than thermal convection.