Tidal friction in close-in satellites and exoplanets. The Darwin theory re-visited

This report is a review of Darwin's classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) the capture into a 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.Comment: 30 pages, 7 figures, corrected typo

Stability of aluminium reduction cells with mean flow

We report results of the linear stability analysis undertaken to investigate the effect of the mean flow of liquid metal on the stability of aluminum reduction cells. A simplified model of the cell is considered that consists of thin layers of aluminum and cryolite superimposed in an infinite horizontal channel with electrically non-conducting walls. A vertical uniform magnetic field and an electric current are applied in the opposite directions. In the basic steady state, a uniform flow of aluminum is assumed, while cryolite is at rest. The onset of the instability is caused by the action of two different mechanisms. The first is the Kelvin-Helmholtz instability of the mean flow. The second, essentially the MHD mechanism, is a consequence of destabilizing electromagnetic (Lorentz) forces produced by nonuniformities of the electric current due to interface deflections. We use the shallow water approximation and solve the problem for the cases of pure Kelvin-Helmholtz (zero magnetic field) and pure MHD (zero mean flow) instabilities and for the general case. We compute the stability chart and derive the parameters that determine the stability threshold. It is found that, while both playing a destabilizing role, the instability mechanisms do not affect each other. In particular, a uniform mean flow changes the direction of propagation of interfacial waves but leaves the MHD stability threshold unaltered. Figs 4, Refs 12

Cell Voltage Noise in Aluminium Smelting

Aluminium reduction process is a nonlinear, time varying electrochemical process, in which alumina (Al2O3) dissolved in cryolite (Na3AlF6) is electrolysed to produce molten aluminium. In industrial practice the term noise refers to unsteady voltage signal in the reduction cell or random and/or uncontrolled fluctuations in the cell voltage; it can be used to identify problems in the cell operation. The main objectives of the reduction process is to achieve as high a current efficiency as possible with minimum energy consumption. Noises of certain frequencies are generally considered unfavorable to cell current efficiency and energy consumption. Therefore, noise control is an important issue in commercial cell automatic control system. The present paper analyses noise from different sources during cell operation and their possible causes with help of Fast Fourier Transformation techniques in terms of frequency and amplitudes of the fluctuations. An advanced data acquisition system was used for collecting the data required for the noise studies. The characterization of the three types of noise generated is discussed. Real-time algorithms have been implemented to allow on-line estimation of the noise sources and intensities.by A. Agnihotri, S. U. Pathak, and Jyoti Mukhopadhya