Relativistic Hydrodynamics 2025, 14 - Non-Selfgravitating Fluids

Non-selfgravitating fluids are fluids whose total mass and energy is sufficiently small that the right side of the Einstein equation can be evaluated to 0. This reduces the system equations to rest mass, energy and momentum conservation. For 1D flows in a geometry independent form, obtaining the conservation equations, it becomes convenient to introduce a similarity variable ξ with tξ = ±x¹, whose absolute value is at most 1. The dimensionless similarity variable ξ can be viewed either as position of a point in the solution at time t, or as the velocity some part of the solution moves at (usually discontinuity features). With this in mind, write the eqs. of conservation in terms of self-similar variables for each of the speeds and variables

which are invariant under ξ→-ξ, v→-v.

A nontrivial class of self-similar solutions for expanding spherically symmetric flows are boiling bubbles. The bubble surface is treated as a reaction front. Self-similar motion can appear only when the radius is small enough that no interaction between neighboring bubbles occurs, but large enough that the corrections from surface tension are negligible. If the surface is modeled by a weak deflagration front, ξ = 0 is the center at any given time, and ξ_s = c_s is the expanding sonic radius, determining possible weak discontinuities, and ξ = 1 is the edge of the future light cone. At least 2 discontinuities are introduced in ξ ∈ [0, 1].

Another self-similar solution can be found for evaporating drops. The system contracts, so ξ < 0. The fluid behind the deflagration front is under a compression wave, because a fluid element crossing the front is strongly decompressed from its null-state, then compressed up to the value in the unperturbed fluid as it moves away. At the sonic radius, it joins via weak discontinuity onto another solution with zero velocity and constant energy density. As the fluid behind the front is at rest and supersonic, the fluid behind the front moves at supersonic speed relative to the front rest frame. Solutions for these systems acting as detonations are impossible.

Spherical flows around compact gravitating objects are derived from a stationary spherical flow with postive-definite radial velocity u on a gravitating compact object of mass M, with the fluid obeying a polytropic eq. of state, reducing to the Bernoulli equation

The flow exists in both the Bernoulli and continuity eqs. for all fixed radial positions. Their intersections can be modeled geometrically, and in these cases, the mass accretion rate is either too large or too small to form a smooth solution. In the case that it exists, either at two points at either side of the line u = c_s bisecting their plane and also at that point, those curves meet tangentially. This radial position is the "sonic radius". Otherwise, the intersection is on either side of said line, but never at a single point. Through the gravitation equation and the "Eddington Luminosity" L_Edd, the Eddington limit of the luminosity produced through accretion is defined.

These are the basics that can be find for spherical accretion onto a black hole as well, though filtered through the Schwarzschild metric restricting the accretion solution. One can assume for this case, that the fluid is isentropic, and so for the fluid velocity,

with the latter for the critical radius