Simulating elliptic flow with viscous hydrodynamics

TL;DR

This work develops a 2+1D boost-invariant viscous hydrodynamics framework based on the Ottinger–Grmela second-order theory to study non-central Au–Au collisions at RHIC. It compares the impact of shear viscosity with and without off-equilibrium corrections to the distribution function on the differential elliptic flow $v_2(p_T)$ and on transverse spectra, using a constant $\eta/s$ and a Bjorken-like initial state. The main findings are that viscous modifications to the flow field are modest, while the off-equilibrium correction $\delta f$ can substantially suppress $v_2(p_T)$ at larger $p_T$, and that the hydrodynamic description remains reliable up to roughly $\eta/s \approx 0.3$ before the stress deviates from Navier–Stokes expectations. The work discusses freezeout criteria, the dependence on initial conditions and the equation of state, and compares with other viscous-hydrodynamics studies to delineate the regime of applicability for RHIC phenomenology. Collectively, these results highlight the crucial role of nonequilibrium spectral corrections in interpreting elliptic flow and establish a practical framework for evaluating when second-order viscous hydrodynamics provides a trustworthy description of heavy-ion collision dynamics.

Abstract

In this work we simulate a viscous hydrodynamical model of non-central Au-Au collisions in 2+1 dimensions, assuming longitudinal boost invariance. The model fluid equations were proposed by Öttinger and Grmela \cite{OG}. Freezeout is signaled when the viscous corrections become large relative to the ideal terms. Then viscous corrections to the transverse momentum and differential elliptic flow spectra are calculated. When viscous corrections to the thermal distribution function are not included, the effects of viscosity on elliptic flow are modest. However, when these corrections are included, the elliptic flow is strongly modified at large $p_T$. We also investigate the stability of the viscous results by comparing the non-ideal components of the stress tensor ($π^{ij}$) and their influence on the $v_2$ spectrum to the expectation of the Navier-Stokes equations ($π^{ij} = -η\llangle \partial_i u_j \rrangle$). We argue that when the stress tensor deviates from the Navier-Stokes form the dissipative corrections to spectra are too large for a hydrodynamic description to be reliable. For typical RHIC initial conditions this happens for $η/s \gsim 0.3$.

Figures (15)

• Figure 1: (Color online) Plot of the energy density per unit rapidity (left) and of the transverse velocity (right) at times of $\tau=1,3,6,9$ fm/c, for $\eta/s=0.2$ (solid red line) and for ideal hydrodynamics (dotted blue line).
• Figure 2: (Color online) Contour plot of energy density per unit rapidity in the transverse plane. The contour values working outward are for $\tau=1$ fm/c: 15, 10, 5, 1, 0.1, for $\tau=3$ fm/c: 10, 5, 1, 0.1, for $\tau=6$ fm/c: 3, 2, 1, 0.5 and for $\tau=9$ fm/c: 0.5, 0.375, 0.25, in units of GeV/fm$^2$.
• Figure 3: (Color online) Contour plot of transverse velocity, $v_\perp=\sqrt{v_x^2+v_y^2}$. The inner most contour is for $v_\perp=0.1$ and increases in steps of $\Delta v_\perp = 0.15$.
• Figure 4: (Color online) Time evolution of the spatial ellipticity $\epsilon_x$, the momentum anisotropy $\epsilon_p$, and the energy density weighted transverse flow $\langle\langle v_\perp \rangle\rangle$, see Eq. \ref{['eq:anis']}.
• Figure 5: (Color online) Contour plot of various freezeout surfaces for central Au-Au collisions. Left: Surfaces from ideal hydrodynamics where the freezeout condition is set by the parameter $\chi$=1.5, 3 and 4.5. Right: Corresponding viscous solution where $\eta/s$ was fixed by the condition $\frac{\eta}{p}\partial_\mu u^\mu=0.6$. The thin solid black curve shows the contour set by $\frac{\eta}{p}\partial_\mu u^\mu=0.225$ for comparison.