Perfect Fluids |
In General > s.a. gas.
* Idea: A fluid with
isotropic pressure, no viscosity and (in general relativity) no heat conduction;
It is described by a four-velocity ua,
number density ρ, energy density μ, pressure p,
specific entropy s, temperature T, relativistic enthalpy h;
Of the scalar quantities, only two are independent, to define the rest use the equation
of state and the first law of thermodynamics; If μ = μ(ρ,
s), then
h = ∂μ / ∂ρ = (μ + p)/ρ , T = (∂μ / ∂s) / ρ , and p = ρ h − μ .
* Stress-energy tensor: Of the form
Tab = (μ + p) ua ub + p gab ,
derivable from the Lagrangian L = − μ, where the
variations are subject to constraints; The field equations are stress-energy
conservation, ∇a
Tab = 0, together with either
the continuity equation ∇a(ρ
ua) = 0,
or ua
∂a s = 0
[@ Coll et al a1902, hydrodynamic approach].
* 3+1 Equations: The conservation of stress-energy,
∇a Tab
= 0, gives, projected parallel and perpendicular to ua,
respectively,
∇a(μ ua) = − p ∇a ua and (μ + p) ua ∇a vb = − qab ∇a p
(conservation of mass-energy and momentum, respectively); The Newtonian limit of the second equation is the Euler equation.
Types, Couplings and Applications > s.a. relativistic cosmological
models and matter.
* Isentropic: One with
an equation of state of the form p = p(μ)
only; In this case, one can introduce the elastic potential (or internal
energy) ε and the density ρ, with μ
= ρ (1 + ε); For example, the ideal gas.
@ Null dust: Kijowski et al PRD(90);
Bičák & Kuchař PRD(97)gq [canonical general relativity].
@ Other types: Quevedo & Sussman CQG(95)gq/94 [non-isentropic, thermodynamics];
Jackiw et al JPA(04)hp [with internal symmetry and supersymmetry];
Majd & Momeni IJMPE(11)-a0903 [p = −ρ, statistical derivation of density of states];
Jacak & Steinberg PT(10)may [quarks and gluons in heavy-ion collisions];
> s.a. Chaplygin Gas; Logotropic Fluid;
Phantom Field.
> In general relativity:
see canonical general relativity with matter; relativistic
cosmology and cosmological models, in particular FLRW
spacetimes; 3-dimensional and numerical relativity;
solutions, solutions with symmetries, spherically
symmetric, and generation methods.
References
@ General: Thorne in(67);
Misner in(68);
in Hawking & Ellis 73, 69ff;
in Misner et al 73;
Smarr et al in(80);
Anile 89.
@ Variational formulation:
Schutz & Sorkin AP(77);
Kopczyński AP(90);
Dimitrov gq/99,
gq/99;
Elze et al JPG(99)hp;
Marsden et al JGP(01) [general continuum mechanics];
Ootsuka et al CQG(16)-a1605 [relativistic].
@ Lagrangian, Hamiltonian: Bao et al CMP(85);
Bombelli & Torrence CQG(90);
Brown CQG(93),
AP(96)gq/94;
Langlois in(94);
Peitz & Appl MNRAS(98)gq/97 [3+1];
Morrison RMP(98);
Antoniou & Pronko TMP(04)ht/01 [and Clebsch variables];
Manoff in(01)gq/02,
gq/03 [Lagrangian];
Bhat mp/04 [compressible];
Kolev DCDSA(07)-a0711 [Poisson brackets];
Roberts a0910 [Clebsch potential approach];
Faraoni PRD(09)-a0912 [inequivalence of Lagrangians with non-minimal gravity coupling];
Minazzoli & Harko PRD(12)-a1209 [barotropic fluid, Lagrangian];
Frisch & Villone EPJH(14)-a1402 [3D ideal incompressible flow, Cauchy's formulation];
Wongjun PRD(17)-a1602 [Lagrangian];
Minguzzi a1606 [action];
Datta a1807 [and analog spacetime];
Mendoza & Silva IJGMP-a2011.
@ Covariant: Bruni & Sopuerta CQG(03)gq [long-wavelength].
@ Numerical implementation: Papadopoulos & Font PRD(00)gq/99 [spherical];
Baumgarte et al AIP(99)gq [relativistic, 3D].
@ Relativistic:
Schutz PRD(71) [Hamiltonian];
Calzetta CQG(98)gq/97 [fluctuating];
Walton ap/05 [symmetric hyperbolic Euler equations];
Gourgoulhon EAS(06)gq-ln;
Borshch & Zhdanov Sigma(07)-a0709-proc [exact non-stationary non-homogeneous flows];
Banerjee et al EPJC(15)-a1409 [Hamiltonian, in equal-time and light-cone coordinate systems];
> s.a. Geometrization.
@ Quantization: Roberts MPAG(98)gq;
Endlich et al JHEP(11)-a1011 [canonical];
Gripaios & Sutherland PRL(15)-a1406 [as a low-energy, effective field theory].
@ Related topics: Zloshchastiev APPB(99)gq/98 [models for Lorentzian geometry];
Ghosh ht/01 [as gauge theory],
ht/01 [conserved quantities];
Verozub IJMPD(08)gq/07 [particles as moving along geodesics];
Mannheim et al GRG(10) [on perfect fluids as general matter models].
> Related topics: see viscosity;
Weyssenhoff Fluid [particles with intrinsic spin].
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