1,721,052 research outputs found
Liouvillian integrability of gravitating static isothermal fluid spheres
No full textWe examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = -1 and n = -3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = -5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka-Volterra quadratic polynomial differential system in \documentclass[12pt]{minimal}\begin{document}\end{document}R2, and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka-Volterra system possesses Liouvillian first integrals for n = -1, -3, -5, which descend from the existence of invariant algebraic curves of degree one, and for n = -6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka-Volterra system, and consequently of the second order ODE for the mass function must be non-Liouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely
Effects of surface tension on the collapse time of an empty bubble
No full textThe collapse of an empty spherical bubble in an ideal liquid, in the absence of viscosity and surface tension, was studied by Lord Rayleigh. Using energy conservation, he derived an exact expression for the total collapse time as a function of the initial radius of the bubble, the density of the liquid, and the far-field pressure. In the present work, we extend Rayleigh's expression to include surface tension effects. Results are found to depend on a dimensionless parameter ε that measures the ratio between the work done by surface tension and that done by pressure during the collapse. This parameter is small for large bubbles but can be of order unity or larger for bubbles of small radius and, eventually, small pressure. We show that the ratio between the collapse time in the presence of surface tension and Rayleigh's collapse time is proportional to a definite integral that is a smooth, monotonically decreasing function of ε. This function can be easily bounded analytically for any value of ε, yielding a simple and accurate approximation for the collapse time that, for all practical purposes, provides a complete analytical solution to the problem at hand. We finally extend results to the case of a hyperspherical collapsing empty bubble
Simple analytic approximations for the Blasius problem
No full textThe classical boundary layer problem formulated by Heinrich Blasius more than a century ago is revisited, with the purpose of deriving simple and accurate analytical approximations to its solution. This is achieved through the combined use of a generalized Padé approach and of an integral iteration scheme devised by Hermann Weyl. The iteration scheme is also used to derive very accurate bounds for the value of the second derivative of the Blasius function at the origin, which plays a crucial role in this problem. © 2015 Elsevier B.V. All rights reserved
Le porte di Palestrina dai Colonna ai Barberini, in Entrare in città: di archi e di porte
No full textThe Basilicata Wealth Fund
No full textThis paper introduces a novel resource policy for the resource-rich Southern Italian region of Basilicata. This policy consists of establishing a regional wealth fund, the Basilicata Wealth Fund, according to which royalty revenues from natural resources in Basilicata can be stored as low-risk financial assets in order to sustain future regional expenditures. The paper pins down and simulates the dynamics of a set of fiscal expenditure rules. The Basilicata Wealth Fund fosters transparency accountability as regards fiscal spending, hence avoiding misuse of resource revenues
A high-resolution modelling study of the circulation along the Campania coastal system, with a special focus on the Gulf of Naples
No full textA high-resolution modelling study of the circulation along the Campania coastal system (CCS) in the southern Tyrrhenian Sea is presented. The area includes three adjacent, semi-enclosed shallow basins (the gulfs of Gaeta, Naples and Salerno) facing the open deep sea, and represents an ideal site for investigating relevant dynamical processes of general interest in coastal oceanography. A Campania Regional sigma-coordinate Ocean Model (CROM) has been implemented in this region, with a 1/144° resolution; nesting with an operational circulation model covering the whole Tyrrhenian Sea with a 1/48° resolution is performed. Forcing is provided by surface momentum, heat, and freshwater fluxes computed from the non-hydrostatic SKIRON/Eta atmospheric modelling system outputs. A winter and a summer period of the reference year 2009 are analyzed in detail. The relative importance of the flow induced by remote large-scale currents through topographic interactions and of that induced locally by the wind are found to vary, even over a weekly time scale, in all of the three gulfs; the Gulf of Salerno appears to be the location where remote and local forcing are more often competing. An analysis of the high frequency variability shows that, besides the current changes induced by the typical mid-latitude atmospheric synoptic variability, inertial currents in the open sea and sea breeze-induced currents in the gulfs are present. Model-data comparison is performed in the CCS with altimeter data and satellite-derived turbidity distributions, and in the Gulf of Naples with the latter and with measurements performed with a high-frequency radar system. Significant model-data agreement is generally found. © 2016 Elsevier Ltd
Aspects of the summer circulation in the eastern Ligurian Sea
No full textWater exchanges between the Tyrrhenian Sea and the Liguro-Provençal basin occur through the Corsica Channel, a narrow passage between Corsica and the Elba Island, near the Italian coast. It has long been known that the Channel hosts a robust Tyrrhenian outflow in winter, which penetrates far into the Ligurian Sea, reaching the site where deep-water formation takes place. The summer circulation in the area, and, more generally, in the eastern part of the Ligurian Sea, is less well known. Several experimental and modelling works in the last decade have pointed out the presence of an anticyclone in the Channel area, sometimes referred to as Ligurian anticyclone, which could modulate the exchanges between the two basins in summer, and could strongly affect the dispersion properties in the south-eastern Ligurian Sea, with implications on the local, delicate marine environment. In this work, we focus on this structure, and on the role it plays in the local circulation, seeking information from long datasets of satellite observations (sea level, sea surface temperature and turbidity) that are now available. We find that the Ligurian anticyclone is a recurrent feature of the local circulation in summer. Its core is located between Cap Corse (the peninsula at the northern end of Corsica) and the Elba Island, but the anticyclone may extend till the Italian coast, and it may sometimes be more elongated in the latitudinal direction, and/or more displaced towards north. In some years, a companion anticyclonic vortex is found more to the north, in the north-eastern portion of the Ligurian Sea, but the robustness of this structure is unclear. Analysis of the ERA5 winds indicates that the Ligurian anticyclone is likely driven by an area of negative wind stress curl usually present near Cap Corse in summer, even though contributions from the surrounding circulation cannot be a priori excluded
Approximate analytic solutions to the isothermal Lane-Emden equation
No full textWe derive accurate analytic approximations to the solution of the isothermal Lane-Emden equation, a basic equation in Astrophysics that describes the Newtonian equilibrium structure of self-gravitating, isothermal fluid spheres. The solutions we obtain, using analytic arguments and rational approximations, have simple forms, and are accurate over a radial extent that is much larger than that covered by conventional series expansions around the origin. Our best approximation has a maximum error on density of 0.04 % at 10 core radii, and is still within 1 % from an accurate numerical solution at a radius three times larger. © 2014 Springer Science+Business Media Dordrecht
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