Solutions of conformal Israel-Stewart relativistic viscous fluid dynamics

We use symmetry arguments developed by Gubser to construct the first radially expanding explicit solutions of the Israel-Stewart formulation of hydrodynamics. Along with a general semi-analytical solution, an exact analytical solution is given which is valid in the cold plasma limit where viscous effects from shear viscosity and the relaxation time coefficient are important. The radially expanding solutions presented in this paper can be used as nontrivial checks of numerical algorithms employed in hydrodynamic simulations of the quark-gluon plasma formed in ultrarelativistic heavy ion collisions. We show this explicitly by comparing such analytic and semi-analytic solutions with the corresponding numerical solutions obtained using the music viscous hydrodynamics simulation code.

Figure 1

Comparison between the solutions for $\widehat{T}$ (left panel) and ${\overline{\pi }}_{\xi }^{\xi }$ (right panel) for $\eta /s=0.2,c=5$, and $\widehat{T}\left(0\right)=1.2$ found using different versions of the relativistic fluid equations. The solid black lines denote solutions of Israel-Stewart theory, results from relativistic Navier-Stokes theory are in dashed blue, while the dashed-dotted red curves correspond to the ideal fluid case.

Figure 2

Temperature and ${\tau }^2{\pi }^{\xi \xi }$ profiles in Israel-Stewart theory for $\tau =1.2$ fm (solid black curves), $\tau =1.5$ fm (dashed blue curves), and $\tau =2$ fm (dashed-dotted red curves) with $q=1{\mathrm{fm}}^{-1},\eta /s=0.2,c=5$, and $\widehat{T}\left(0\right)=1.2$.

Figure 3

$\tau r{u}^{\tau }{s}_{\mathrm{neq}}$, entropy production up to second order, $Q$, and total entropy production, $Q_{\mathrm{total}}$, profiles as a function of the radius in Israel-Stewart theory for $\tau =1$ fm (solid black curves), $\tau =1.5$ fm (dashed blue curves), and $\tau =2$ fm (dashed-dotted red curves).

Figure 4

Comparison between the solutions for temperature (left panel) and the $x$ component of the four-velocity (right panel) from Gubser flow and music (numerical), as a function of $x$. In this plot $\eta /s=0.2$ and ${\tau }_{R}T=5\eta /s$. The solid lines denote the semi-analytic solution while the points denote solutions obtained from music.

Figure 5

Comparison between the solutions for the $\xi \xi$ (left panel), $yy$ (right panel), and $xy$ (lower panel) components of the shear-stress tensor from Gubser flow and music (numerical), as a function of $x$. In this plot $\eta /s=0.2$ and ${\tau }_{R}T=5\eta /s$. The solid lines denote the semi-analytic solution while the points denote solutions obtained from music.

Figure 6

Numerical solutions of music obtained with $\chi =1.1$ (open circles) for the $xx$ (left panel) and $yy$ (right panel) components of the shear-stress tensor. The full circles correspond to the solutions obtained with $\chi =1.8$ and the solid lines correspond to the semi-analytic solution.

Figure 7

Comparison between the solutions for the total equilibrium entropy (left panel) and total nonequilibrium entropy (right panel) from Gubser flow and music (numerical), as a function of $\tau$. In both plots the total entropy is obtained by integrating corresponding entropy density from $r=0$ to $r=5$ fm. The solid lines denote the semi-analytic solution while the points denote solutions obtained from music. On the right panel, we also show the total nonequilibrium entropy obtained by integrating from $r=0$ to $r=7.5$ fm (dashed line) and $r=0$ to $r=12$ fm (dashed-dotted line).

Figure 8

Comparison between the solutions for temperature (left panel) and the $x$ component of the four-velocity (right panel) from Gubser flow and music (numerical), as a function of $x$. In this plot $\eta /s=0.2$ and ${\tau }_{R}T=5\eta /s$. The solid lines denote the analytic solution while the points denote solutions obtained from music.

Figure 9

Comparison between the solutions for the $\xi \xi$ (left panel), $yy$ (right panel), and $xy$ (lower panel) components of the shear-stress tensor from Gubser flow and music (numerical), as a function of $x$. In this plot $\eta /s=0.2$ and ${\tau }_{R}T=5\eta /s$. The solid lines denote the analytic solution while the points denote solutions obtained from music.