Viscous Gubser flow with conserved charges to benchmark fluid simulations

We present semi-analytical solutions for the evolution of both the temperature and chemical potentials for viscous Gubser flow with conserved charges. Such a solution can be especially useful in testing numerical codes intended to simulate relativistic fluids with large chemical potentials. The freeze-out hypersurface profiles for constant energy density are calculated, along with the corresponding normal vectors and presented as a new unit test for numerical codes. We also compare the influence of the equation of state on the semi-analytical solutions. We benchmark the newly developed Smoothed Particle Hydrodynamics (SPH) code CCAKE that includes both shear viscosity and three conserved charges. The numerical solutions are in excellent agreement with the semi-analytical solution and also are able to accurately reproduce the hypersurface at freeze-out.

💡 Research Summary

In this work the authors develop semi‑analytical solutions for a viscous conformal fluid undergoing Gubser flow while carrying three conserved charges – baryon number (B), strangeness (S) and electric charge (Q). The presence of conserved currents introduces non‑trivial chemical potentials μB, μS and μQ, which couple to the temperature evolution only through the equation of state (EoS). By exploiting the high degree of symmetry of Gubser flow (SO(3)q × SU(1,1) × Z2) the authors map the Milne coordinates (τ, r, φ, η) onto de Sitter coordinates (ρ, θ, φ, η) via a Weyl rescaling and a specific coordinate transformation. In de Sitter space the fluid four‑velocity becomes static (ûμ = (−1,0,0,0)), allowing the full set of relativistic hydrodynamic equations (energy‑momentum conservation, Israel‑Stewart shear‑stress relaxation, and charge‑conservation without diffusion) to reduce to a system of ordinary differential equations in the single variable ρ.

Two conformal equations of state are considered. EoS1 corresponds to a free massless quark‑gluon plasma with pressure (P = \frac{g_{\rm QGP}\pi^{2}}{90}T^{4} + \sum_{Y}\frac{g_{q}}{216}\mu_{Y}^{2}T^{2} + \frac{\mu_{Y}^{4}}{3888\pi^{2}}), where Y∈{B,S,Q}. EoS2 is a simplified linear combination (P = \frac{g_{\rm QGP}\pi^{2}}{90}T^{4}\bigl