Fluid solution

Résumé

In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics, fluid solutions are often employed as stellar models. (It might help to think of a perfect gas as a special case of a perfect fluid.) In cosmology, fluid solutions are often used as cosmological models. Mathematical definition The stress–energy tensor of a relativistic fluid can be written in the form :T^{ab} = \mu , u^a , u^b + p , h^{ab} + \left( u^a , q^b + q^a , u^b \right) + \pi^{ab} Here

the world lines of the fluid elements are the integral curves of the velocity vector u^a,
the projection tensor h_{ab} = g_{ab} + u_a , u_b projects other tensors onto hyperplane elements orthogonal to u^a,
the matter density is given by the scalar function \mu,
the pressure is given by the

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https://en.wikipedia.org/wiki/Fluid_solution

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Two-fluid solutions for Langmuir probes in collisionless and isothermal plasma, over all space and bias potential

Ivo Furno, Philippe Frédérique Bruno Guittienne, Alan Howling

This paper presents solutions for the classical one-dimensional (1D radial and Cartesian) problem of Langmuir probes in a collisionless, isothermal plasma. The method is based on two-fluid equations derived from the first two moments of Vlasov's equation. In contrast to commonly used approximations, electron inertia and ion temperature are not neglected so that the fluid equations are symmetric in terms of electrons and ions. The fluid equations are reduced analytically so that the electric potential is the only remaining spatial function, which is numerically determined using Poisson's equation. The single radial solution applies continuously over the whole region from the probe up to the unperturbed plasma, in contrast to theories which separate the probe boundary region into a charged sheath and a quasi-neutral pre-sheath, and is valid for all values of probe bias potential. Current-voltage characteristics are computed for cylindrical and spherical probes, which exhibit non-saturation of the ion and electron currents. The 1D Cartesian case is also analysed, and the Bohm criterion is recovered only in the limit of large radius probes. Published by AIP Publishing.

2018

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Using the Lambert function, Guittienne et al. [Phys. Plasmas 25, 093519 (2018)] derived two-fluid solutions for radial Langmuir probes in collisionless and isothermal plasma. In this Brief Communication, we point out the close analogy with classical compressible fluid dynamics, where the simultaneous flows of the ion and electron fluids experience opposite electrostatic body forces in the inward radial flow of the plasma, which behaves as a converging nozzle. Hence, the assumed boundary condition of sonic flow of the repelled species at the probe is explained as choked flow. The sonic passage from subsonic to supersonic flow of the attracted species at the sonic radius is also interpreted using classical fluid dynamics. Moreover, the Lambert function can provide a general solution for one-dimensional, isothermal compressible fluids, with several applications.

2019

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