Bulk viscosity-driven suppression of shear viscosity effects on the flow harmonics at energies available at the BNL Relativistic Heavy Ion Collider

The interplay between shear and bulk viscosities on the flow harmonics, $v_n$'s, at the BNL Relativistic Heavy Ion Collider is investigated using the newly developed relativistic $2+1$ hydrodynamical code v-usphydro that includes bulk and shear viscosity effects both in the hydrodynamic evolution and at freeze-out. While shear viscosity is known to attenuate the flow harmonics, we find that the inclusion of bulk viscosity decreases the shear viscosity-induced suppression of the flow harmonics, bringing them closer to their values in ideal hydrodynamical calculations. Depending on the value of the bulk viscosity to entropy density ratio, $\zeta /s$, in the quark-gluon plasma, the bulk viscosity-driven suppression of shear viscosity effects on the flow harmonics may require a reevaluation of the previous estimates of the shear viscosity to entropy density ratio, $\eta /s$, of the quark-gluon plasma previously extracted by comparing hydrodynamic calculations to heavy-ion data.

Figure 1

(Top) Temperature dependence of $\eta /s$ from Eq. (7) (dashed blue line) and $\zeta /s$ (multiplied by a factor of 10 for clarity) obtained from Eq. (9) (solid black line). (Bottom) The relaxation time coefficients ${\tau }_{\pi }$ from Eq. (8) for shear (dashed blue line) and ${\tau }_{\Pi }$ for bulk from Eq. (10) (solid black line).

Figure 2

Energy density distribution for a random event in peripheral ($20\%–30\%$) collisions at RHIC for different times. (a) Initial time ${\tau }_0=1$ fm. Energy distribution at $\tau =6$ fm for (b) ideal hydrodynamics, (c) viscous hydrodynamics with only bulk viscosity, (d) viscous hydrodynamics with only shear viscosity, and (e) viscous hydrodynamics with both bulk and shear viscosity effects. The transport coefficients are the ones shown in Fig. 1.

Figure 3

Ratio between the integrated $v_n$'s of viscous and ideal hydrodynamics of direct pions over all centralities at RHIC computed using the MOM for the bulk viscosity contribution at freeze-out. The circles correspond to the case where only shear viscosity is taken into account, the squares represent the case with both shear and bulk viscosity, while the diamonds correspond to the case where shear and bulk are included but $\zeta /s$ is multiplied by a factor of 10.

Figure 4

Ratio between the integrated $v_n$'s of viscous and ideal hydrodynamics of direct pions over all centralities at RHIC computed using the MOM and the MH formulas for the bulk viscosity contribution at freeze-out. The solid triangles correspond to the case with only bulk viscosity with $\delta f^{\text{bulk}}$ from MOM while the open triangles correspond to the analogous case computed with $\delta f^{\text{bulk}}$ from MH. The solid squares correspond to the case with shear and bulk viscosities with $\delta f^{\text{bulk}}$ from MOM, while open squares correspond to the analogous case computed with $\delta f^{\text{bulk}}$ from MH.

Figure 5

Comparison of $v_2$ of direct pions across the centrality classes $0\%–10\%$, $10\%–20\%$, $20\%–30\%$ [in column (a)] and $30\%–40\%$, $40\%–50\%$, and $50\%–60\%$ [in column (b)] for RHIC. The solid black lines denote the ideal hydrodynamical results, the short-dashed blue lines were computed taking into account only shear viscosity, the long-dashed green curves were computed using $\mathrm{shear}+\mathrm{bulk}$ with the MOM expression for the freeze-out, while the dark red dot-dashed curves correspond to $\mathrm{shear}+\mathrm{bulk}$ with the MH formula.

Figure 6

Comparison of $v_3$ of direct pions across the centrality classes $0\%–10\%$, $10\%–20\%$, $20\%–30\%$ [in column (a)] and $30\%–40\%$, $40\%–50\%$, and $50\%–60\%$ [in column (b)] for RHIC. The solid black lines denote the ideal hydrodynamical results, the short-dashed blue lines were computed taking into account only shear viscosity, the long-dashed green curves were computed using $\mathrm{shear}+\mathrm{bulk}$ with the MOM expression for the freeze-out, while the dark red dot-dashed curves correspond to $\mathrm{shear}+\mathrm{bulk}$ with the MH formula.

Figure 7

Comparison of $v_4$ of direct pions across the centrality classes $0\%–10\%$, $10\%–20\%$, $20\%–30\%$ [in column (a)] and $30\%–40\%$, $40\%–50\%$, and $50\%–60\%$ [in column (b)] for RHIC. The solid black lines denote the ideal hydrodynamical result, the short-dashed blue lines were computed taking into account only shear viscosity, the long-dashed green curves were computed using $\mathrm{shear}+\mathrm{bulk}$ with the MOM expression for the freeze-out, while the dark red dot-dashed curves correspond to $\mathrm{shear}+\mathrm{bulk}$ with the MH formula.

Figure 8

Comparison of $v_5$ of direct pions across the centrality classes $0\%–10\%$, $10\%–20\%$, $20\%–30\%$ [in column (a)] and $30\%–40\%$, $40\%–50\%$, and $50\%–60\%$ [in column (b)] for RHIC. The solid black lines denote the ideal hydrodynamical result, the short-dashed blue lines were computed taking into account only shear viscosity, the long-dashed green curves were computed using $\mathrm{shear}+\mathrm{bulk}$ with the MOM expression for the freeze-out, while the dark red dot-dashed curves correspond to $\mathrm{shear}+\mathrm{bulk}$ with the MH formula.

Figure 9

Ratio between the integrated $v_n$'s of viscous and ideal hydrodynamics of direct pions in the $20\%–30\%$ centrality class at RHIC computed using the MOM for the pure shear case $\eta /s=0.007$ (black dots) and the $\mathrm{bulk}+\mathrm{shear}$ calculation where $\zeta /s=\eta /s=0.007$ (red squares).

Figure 10

Comparison between the $v_n\left(p_T\right)$'s of direct pions for the centrality class $20\%–30\%$ at RHIC when bulk and shear viscosities have equal magnitude. The solid black curves denote the ideal hydrodynamics results, the short-dashed blue curves show the result in the case where there is only shear viscosity $\eta /s=0.007$, while the long-dashed green curves correspond to the case where $\zeta /s=\eta /s=0.007$ computed using the MOM approach.

Figure 11

Comparison between the TECHQM results (lines) and our results (dots) for the energy density $\varepsilon$, the shear stress components ${\pi }^{11}$ and ${\pi }^{33}$, as well as the $x$ component of the flow velocity. The comparison is made at the times $\tau =0.6$ (solid black lines), $1.6$ (blue short-dashed lines), and $2.6$ fm (red long-dashed lines).

Figure 12

Comparison between the semianalytical solutions of Ref. [36] (lines) and v-usphydro (dots). Here we compare for the time steps $\tau =1.0$ fm (first time step, which sets the initial condition for the fields) (solid black lines), $1.2$ fm (blue short-dashed lines), and $1.5$ fm (red long-dashed lines).