Interplay of shear and bulk viscosity in generating flow in heavy-ion collisions

Abstract

We perform viscous hydrodynamic calculations in 2+1 dimensions to investigate the influence of bulk viscosity on the viscous suppression of elliptic flow in non-central heavy-ion collisions at RHIC energies. Bulk and shear viscous effects on the evolution of radial and elliptic flow are studied with different model assumptions for the transport coefficients. We find that the temperature dependence of the relaxation time for the bulk viscous pressure, especially its critical slowing down near the quark-hadron phase transition at T_c, partially offsets effects from the strong growth of the bulk viscosity itself near T_c, and that even small values of the specific shear viscosity eta/s of the fireball matter can be extracted without large uncertainties from poorly controlled bulk viscous effects. Comment: 13 pages, 7 figures, 1 table. Submitted to Physical Review C. v2: corrected typos in several entries in Table I

Full text

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arXiv:0909.1549v2 [nucl-th] 8 Sep 2009
Interplay of shear and bulk viscosity in generating flow in heavy-ion collisions
Huichao Song∗and Ulrich Heinz
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
(Dated: September 8, 2009)
We perform viscous hydrodynamic calculations in 2+1 dimensions to investigate the influence of
bulk viscosity on the viscous suppression of elliptic flow in non-central heavy-ion collisions at RHIC
energies. Bulk and shear viscous effects on the evolution of radial and elliptic flow are studied with
different model assumptions for the transport coefficients. We find that the temperature dependence
of the relaxation time for the bulk viscous pressure, especially its critical slowing down near the
quark-hadron phase transition at Tc, partially offsets effects from the strong growth of the bulk
viscosity itself near Tc, and that even small values of the specific shear viscosity η/s of the fireball
matter can be extracted without large uncertainties from poorly controlled bulk viscous effects.
PACS numbers: 25.75.-q, 12.38.Mh, 25.75.Ld, 24.10.Nz
I. INTRODUCTION
A question of widespread interest is that of the spe-
cific shear viscosity (shear viscosity per entropy density
η/s) of the quark-gluon plasma (QGP) created in nu-
clear collisions at the Relativistic Heavy Ion Collider
(RHIC). Ideal (i.e.
inviscid) fluid dynamics has been
quite successful in describing the transverse momentum
(pT) spectra and the elliptic flow coefficient v2(pT) of
the bulk of the thousands of particles created in each,
say, Au+Au collision [1]. The agreement between theory
and experiments improves further when one interfaces
a (3+1)-dimensional ideal fluid description of the QGP
phase with a hadron cascade during the late expansion
stage, in order to properly account for the highly viscous
evolution after hadronization of the QGP [3]. This suc-
cess strongly suggests that the QGP fireball created at
RHIC thermalizes very quickly and behaves like an al-
most perfect liquid [4], which implies that it must be a
strongly coupled plasma [5–7].
On the other hand, the quantum mechanical uncer-
tainty relation places a fundamental lower bound on
the specific shear viscosity of any medium [8], and ex-
plicit computation in a large class of very strongly cou-
pled quantum field theories (unfortunately not including
QCD) suggests that this limit is close to the so-called KSS
bound
s
??
than that of any other known (real) fluid [10, 11], with the
possible exception of strongly interacting systems of ul-
tracold fermionic atoms near the unitarity limit [12, 13]),
it is known that the anisotropic elliptic flow generated in
non-central relativistic heavy-ion collisions is very sensi-
tive to shear viscosity [14, 15]. The roots of this sensitiv-
ity lie in the exceedingly rapid expansion of the heavy-
ion collision fireballs, especially during the early expan-
sion stage which is characterized by very large compo-
nents of the velocity shear tensor [8]. Recent progress
η
KSS=
1
4π≈ 0.08 [9, 10]. While this is a very
small number (almost two orders of magnitude smaller
∗Correspond to song@mps.ohio-state.edu
in performing causal relativistic hydrodynamical simula-
tions of viscous fluids in 2+1 dimensions [16–28] revealed
that even very small specific shear viscosities, close to the
KSS bound, should leave easily identifiable experimental
signatures, in particular through a suppression of ellip-
tic flow. That in the experimental data this suppression
in not large enough to lead to immediate failure of the
ideal fluid approach suggests that the QGP viscosity, in
the temperature region probed by RHIC collisions, must
also be close to the KSS bound [29–33].
Viscous hydrodynamics, in comparison with experi-
mental data, allows in principle for an accurate deter-
mination of η/s. In practice, this requires excellent con-
trol of several other inputs that either are presently not
known with sufficient accuracy or have not yet been cor-
rectly implemented in the numerical simulations [34].
The largest prevailing uncertainty is related to the initial
source deformation that drives the elliptic flow which is
presently not known to better than 20-30% [2, 22, 35, 36]
(see, however, recent suggestions to eliminate this error
source [37, 38]). As shown in [22], this leads at present to
an O(100%) uncertainty in the extracted η/s value. Two
other effects of similar magnitude which, however, work
against each other and may largely cancel, are strong
viscous effects [2] and the non-equilibrium chemical com-
position [39–43] in the late hadronic phase. Finally (and
this is the point of the present paper) bulk viscous ef-
fects must be included in any study that aims to extract
the specific shear viscosity [34, 44]. However, even when
making generous allowances for all these uncertainties, it
appears clear that the effective shear viscosity to entropy
ratio of the QGP, averaged over the expansion history
of the fireballs created in RHIC collisions, cannot exceed
the conservative upper limit
η
s
???
QGP< 5 ×
1
4π.
This makes the QGP the most perfect liquid ever ob-
served in the laboratory.
In the present paper we use the (2+1)-dimensional vis-
cous relativistic fluid dynamic code VISH2+1 [18, 20, 21]
to study the effects of bulk viscosity and their interplay

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2
with shear viscosity in the buildup of radial and elliptic
flow. Some preliminary results were reported in [34, 44]
(see also [45] for related work). Our work is preceded
by three (0+1)-dimensional studies for systems undergo-
ing boost-invariant longitudinal expansion without trans-
verse flow [46–48] which explored the suggestion by Tor-
rieri, Tom´ aˇ sik and Mishustin [49] that in rapidly expand-
ing fireballs bulk viscosity can lead to such large negative
bulk pressures that the fluid becomes mechanically un-
stable against clustering and cavitation. Since bulk vis-
cosity is expected to be maximal near the quark-hadron
phase transition (see discussion in Section II), those stud-
ies predicted that bulk viscous effects become important
mostly during the second half of the fireball expansion
when the QGP undergoes hadronization. At that time
the scalar expansion rate θ≡∂µuµ(where uµ(x) the flow
4-velocity), which for 1-dimensional boost-invariant ex-
pansion equals θ=1/τ (where τ =√t2−z2is the longitu-
dinal proper time, with z indicating the longitudinal or
beam direction), is already small enough to significantly
temper the growth (in magnitude) of the (negative) bulk
pressure, leading to instability problems only for rela-
tively large peak values for the specific bulk viscosity ζ/s
[46–48].
Our work improves on these analysis by including a re-
alistic initial transverse density profile and the resulting
transverse flow in the fireball. This has two important
consequences: (i) The transverse flow increases the ex-
pansion rate θ, leading to larger bulk pressures |Π| for
given ζ/s. (ii) Some of the matter near the dilute trans-
verse edge of the fireball experiences large bulk viscosi-
ties already at very early times where the expansion rate
θ ∼ 1/τ is big. This leads to much more severe problems
with mechanical instability in VISH2+1 than for simple
1-dimensional expansion, and to correspondingly smaller
values for the upper limit for ζ/s that allows for stable
hydrodynamic evolution. Even more restrictive than the
condition for mechanical stability is the self-consistency
constraint for the validity of viscous hydrodynamics it-
self: the entire framework, which is based on a near-
equilibrium expansion, breaks down when viscous cor-
rections to the local equilibrium distribution function be-
come comparable to the thermal equilibrium terms. We
will show that this happens, for particles with typical
momenta p∼3T, even before the effective total pressure
becomes negative and mechanical instability sets in [50].
While the formalism may be able to qualitatively indicate
where and when cavitation sets in [46–48], we doubt that
the phenomenon itself can be self-consistently described
within the existing viscous hydrodynamic frameworks.
Obviously, viscous hydrodynamics can predict the vis-
cous suppression of elliptic flow reliably only within its
domain of validity. We therefore restrict our attention
to the parameter range where the bulk viscous pressure
stays everywhere sufficiently small that stable hydrody-
namic evolution is ensured. Within that range (which
we determine), we study the effects of bulk viscosity and
of the microscopic relaxation time for the bulk viscous
pressure on radial and elliptic flow, with and without ad-
ditional shear viscosity. For a fluid with constant specific
shear viscosity
s=
conditions and details of the temperature dependence of
the relaxation time, bulk viscosity increases the viscous
suppression of v2(pT) by 5 – 50%. This large range indi-
cates not only that bulk viscosity is a potentially serious
contaminant in the extraction of the specific shear vis-
cosity η/s from elliptic flow data, but also that a robust
theoretical effort is needed to better constrain the range
of reasonable values for the bulk viscosity and its associ-
ated relaxation time. We find that the uncertainty range
is drastically reduced, to the 10 − 20% level, if we im-
pose proportionality between the specific bulk viscosity
and its associated relaxation time, as indicated by kinetic
theory. The critical growth of the specific bulk viscosity
near the quark-hadron phase transition is then accom-
panied by critical slowing down of the dynamics of the
viscous bulk pressure. This diminishes the bulk viscous
contribution to the viscous suppression of elliptic flow.
The paper is organized as follows: In Section II we
review the present state of knowledge of the tempera-
ture dependence of the specific bulk and shear viscosi-
ties ζ/s and η/s and their associated microscopic relax-
ation times. Based on this analysis we introduce specific
parametrizations for (ζ/s)(T) and the bulk pressure re-
laxation time τΠ(T) which we use later in the numerical
simulations. Section III gives a brief summary of specific
features of the viscous hydrodynamic equations solved in
this work, referring to earlier work for a more general de-
scription. In Section IV we discuss generic effects of bulk
and shear viscosity on the hydrodynamic evolution of fire-
ball eccentricity and flow and their implications for the
final hadron spectra and elliptic flow. Section V discusses
the sensitivity of bulk viscous effects to the initialization
of the bulk viscous pressure and to its relaxation time. In
Section VI we explore the range of bulk viscosities that
allows for stable viscous hydrodynamic evolution. Con-
sequences of bulk viscous effects for the extraction of the
specific shear viscosity η/s from experimental elliptic flow
data are discussed in Section VII before summarizing our
findings in Section VIII.
η
1
4πwe find that, depending on initial
II.VISCOSITIES AND RELAXATION TIMES
The present state of knowledge of the viscous proper-
ties of strongly interacting matter at high temperatures
is nicely reviewed in [51] to which we refer for details.
Kinetic theory [52] and experiment [10, 11] show that
for non-relativistic fluids the specific shear viscosity η/s
typically reaches a minimum near the liquid-gas phase
transition, rising both towards lower temperatures in the
liquid phase and towards higher temperatures in the gas
phase. Lattice QCD [53], perturbative QCD [54], and
hadron cascade simulations [55] indicate that relativis-
tic QCD matter behaves analogously, but with the liquid
and gas phases interchanged (the liquid QGP phase ex-

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3
ists at higher temperature than the hadronic gas phase).
According to perturbative [54] and lattice [53] QCD, the
increase of η/s with temperature in the QGP phase is
weak over the temperature range explored in RHIC col-
lisions, suggestion the use of a constant η/s for the QGP
in hydrodynamic simulations. We here use the smallest
value for this constant permitted by the KSS conjecture
[10],
s
??
viscous effects into the extraction of such a small value
from experimental data. [We note that the assumption of
a constant η/s is unacceptable for quantitative attempts
to extract it from heavy-ion collision data; at least one
must account for a significant increase of this ratio during
hadronization and in the late hadronic phase [55].]
The relaxation time τπ for the shear viscous pressure
tensor πµνhas been computed for a relativistic Boltz-
mann gas [56, 57], in weakly coupled QCD [58], in lat-
tice QCD [59], and in N =4 SYM theory at infinite cou-
pling [60–62]. The results can be presented in the form
τπ=λ
T
τπ=
T
Theoretical knowledge of the specific bulk viscosity ζ/s
is more murky. For a non-interacting system of massless
quanta it vanishes exactly, due to conformal invariance.
Interactions lead to deviations from zero that usually re-
main small, except near phase transitions where the sys-
tem may develop large correlation lengths [63–65]. Ki-
netic theory gives
ation time approximation [67] and κ=15 for a system of
photons radiated by massive particles in thermal equilib-
rium [68]. The complete leading order result for weakly
coupled QCD [69] is roughly consistent with the latter
of these two, but adds a weak (decreasing) temperature
dependence. At high temperatures c2
ζ
ηis small of second order in the deviation. For strongly
coupled N =4 SYM theory one obtains a lower bound
for this ratio which is only linear in this deviation and
thus much larger:
η
KSS=
1
4π≈ 0.08, in order to extract reasonable
upper bounds for the uncertainties introduced by bulk
η
s, with λ bracketed by 2<
η
s=
∼λ<
∼6. We here use
3
1.5
2πT.
ζ
η=κ?1
3−c2
s
?2where κ=5/3 in relax-
s≈1
3, so the ratio
ζ
η≥2?1
3−c2
s
?[70]. For the hadron gas
ζ
η≪1 just below Tc
different authors [71–73] agree that
and that this ratio decreases towards lower temperatures.
There is no agreement on the sign of the temperature
dependence of the specific bulk viscosity ζ/s itself which
according to [71] increases towards lower temperature for
massive pions, but decreaes for massless pions [73]. How-
ever, there are general arguments [64, 74] that support
the idea that ζ/s should peak near the quark-hadron
phase transition, due to long-range correlations related
to the restoration of chiral symmetry; at a second-order
critical point ζ/s is predicted to diverge [65, 66].
We assume here that ζ/s quickly approaches zero once
T decreases below Tc; above Tc, we parametrize it as
ζ
s=
2π
?1
[70] for η/s = 1/(4π)), using c2
same lattice QCD data [75] that we used for our equa-
tion of state (EOS L, see [21] for details). The factor
?1
1
3− c2
s
?(which corresponds to the Buchel bound
s(T) extracted from the
3− c2
s
?
increases as we approach Tc from above; the
0 0.10.2 0.3
T (GeV)
0
0.02
0.04
0.06
ζ/s
{
ζ/s=
0 (HRG)
2*1/4π*(1/3-cs
2) (QGP)
C=1
FIG. 1: (Color online) Parametrization of the specific bulk
viscosity ζ/s as a function of temperature. (C is a multiplica-
tive scaling factor for the entire function, see text.)
resulting increase of ζ/s is qualitatively, but not quanti-
tatively consistent with a direct extraction of ζ/s from
lattice QCD [76] (for a critical discussion of this extrac-
tion see [65]) and with recent work in “holographic QCD”
[77]. We connect our parametrization above Tcto the as-
sumed zero value for ζ/s well below Tcby interpolating
with a Gaussian function (see Fig. 1). This results in a
peak value (ζ/s)(Tc) ≃ 0.04 – about half as big as our
choice for the (temperature independent) specific shear
viscosity η/s and consistent with strong coupling esti-
mates for strongly coupled conformal field theories using
the AdS/CFT correspondence [78] (which was also used
by Buchel when deriving his bound) and with holographic
QCD [77]. It is, however, more than 10 times smaller
than both the lattice QCD value extracted by Meyer [76]
and a recent AdS/CFT-based estimate by Buchel for a
non-conformal plasma [66]. We will see that this factor
10 has crucial implications for the applicability of viscous
hydrodynamics. To simulate larger bulk viscosities, we
scale the function shown in Fig. 1 (to which we will refer
as “minimal bulk viscosity” for brevity) by a constant
factor C >1.
Finally, we must specify the relaxation time τΠfor the
bulk viscous pressure Π about which even less is known
theoretically. In Israel-Stewart theory (both in its macro-
scopic and microscopic kinetic formulation [56]) one has
τΠ=ζβ0where β0is some combination of thermal equi-
librium integrals. This suggests that, if ζ/s peaks near
Tc due to growing correlation lengths, so does the re-
laxation time τΠfor the bulk pressure (“critical slowing
down” [79]). Buchel [66] found that in theories where
the specific heat diverges at Tc, cV∼1/?|1−Tc/T|, the
relaxation time can actually diverge at Tc even if ζ/s
remains finite, i.e. as T →Tc, τΠgrows more strongly
than ζ/s (see also the discussion in [65]). We use the
parametrization
τΠ(T) = max
?
˜ τ·ζ
s(T), 0.1fm
?
, with ˜ τ = 120fm/c. (1)

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4
This increases linearly with ζ/s as T →Tc, but imposes
a non-zero lower bound on τΠ, for reasons of numeri-
cal stability of VISH2+1. For comparison we also study
two constant relaxation time values, τΠ=0.5 and 5fm/c,
roughly corresponding to the smallest and largest values
of Eq. (1) for temperatures 1≤T/Tc≤2 if we set C =1.
III. VISCOUS HYDRODYNAMICS
We solve the following second order viscous hydrody-
namic equations (“Israel-Stewart equations” [56, 80–82]),
dµTµν= 0, Tµν= euµuν− (p+Π)∆µν+ πµν,
∆µα∆νβDπαβ= −1
−1
DΠ = −1
(2)
τπ(πµν−2ησµν)
2πµνηT
2ΠζT
τΠ
τπdλ
?τπ
?τΠ
ηTuλ
?
, (3)
τΠ(Π + ζθ) −1
dλ
ζTuλ
?
, (4)
in the two transverse spatial directions and time ((2+1)-
d), implementing boost-invariant longitudinal expansion
along the beam direction. We assume zero net baryon
density and thus vanishing heat conductivity. Here, Tµν
is the energy momentum tensor, πµνis the shear pressure
tensor, and Π is bulk pressure. dµdenotes the covariant
derivative components (see [16, 24] for details) in the
curvilear coordinates (τ,x,y,ηs) where τ =
the longitudinal proper time and ηs =
space-time rapidity.∆µν= gµν−uµuνprojects onto
the spatial components in the local rest frame (here
gµν= diag(1, −1, −1, −1/τ2) is the metric tensor in
(τ,x,y,ηs) coordinates); ∇µ=∆µνdν is the spatial gra-
dient and D=uµdµis the time derivative in that frame.
The driving forces for the shear and bulk viscous pres-
sures are the (symmetric and traceless) velocity stress
tensor σµν=∇?µuν?≡1
scalar expansion rate θ=dνuν=∇νuν, respectively. The
shear and bulk viscosities η and ζ and their associated re-
laxation times τπand τΠwere discussed in the preceding
Section.
The last terms in Eqs. (3) and (4) are of second order
in deviations from local equilibrium. For conformal sys-
tems they can be written in various equivalent forms, up
to higher order corrections [21]. Even for non-conformal
systems, such as QCD with the equation of state EOS L
used here (see below), the difference between the terms
as written down here and their various conformal approx-
imations are numerically insignificant [21] unless incon-
sistently large relaxation times are used. Other second
order terms that should be allowed for on the right hand
sides of Eqs. (3) and (4) were identified in [60, 83], and
some of their coefficients were derived in the weak cou-
pling limit in [58]. Recent code verification efforts by
the TECHQM Collaboration [84] indicate that these ad-
ditional terms have very little numerical influence. We
therefore ignore them in the present study.
√t2−z2is
2lnt+z
1
t−zis the
2(∇µuν+∇νuµ)−1
3∆µνθ and the
The explicit form of Eqs. (2-4) for longitudinally boost
invariant (i.e. ηs-independent) systems is given in [16].
The equations are closed by providing an equation of
state for which we use EOS L as described in Ref. [21].
We study Au+Au collisions with the same initial con-
ditions for the starting time τ0=0.6fm/c and Glauber
model initial energy density profiles as in Ref. [21],
with peak density e0≡e(r=0,τ0;b=0)=30GeV/fm3in
central (b=0) collisions.
we use either zero (Π(x,y,τ0;b)=πµν(x,y,τ0;b)=0) or
Navier-Stokes initial conditions (Π(x,y,τ0;b)=−ζθ(τ0)
and πµν(x,y,τ0;b)=2ησµν(τ0)), calculated from the ini-
tial velocity profile (which does not depend on x, y and b,
due to the absence of initial transverse flow). The actual
choice will be noted when discussing the results. As in
[21] we end the hydrodynamic evolution and compute the
final hadron spectra on a freeze-out surface of constant
temperature Tdec=130MeV.
For the viscous pressures
IV. VISCOUS EVOLUTION AND SPECTRA:
GENERIC FEATURES
In this section we compare generic effects on the hy-
drodynamic evolution and final particle spectra caused
by shear and bulk viscous effects separately. (Their com-
bined effects will be explored in Sects. V-VII.) To this
end we perform hydrodynamic comparison runs for cen-
tral (b=0) and non-central (b=7fm) Au+Au collisions,
using identical initial and freeze-out conditions, for (i) an
ideal fluid, (ii) a viscous fluid with only minimal shear
viscosityη
s=
imal bulk viscosity” (C =1) as defined in Sec. II.
the viscous runs we choose Navier-Stokes initial condi-
tions for the viscous pressures and equal relaxation times
τπ=τΠ=3η
sT. (As a caveat we note that in the bulk vis-
cous case results can depend sensitively on the initial
conditions, depending on the characteristics of the relax-
ation time for the bulk viscous pressure – see discussion
in Sec. V.) The results for case (ii) supplement those for
the smaller Cu+Cu collision system studied in [18, 20]
(although for a more realistic equation of state and using
Eq. (3) instead of the “simplified Israel-Stewart equa-
tion” employed in those earlier papers).
1
4π, and (iii) a viscous fluid with only “min-
In
A. Hydrodynamic evolution
Figure 2a shows the time evolution of the local tem-
perature in central Au+Au collisions for the three cases.
(We plot the temperature at a radius r=3fm from the
fireball center since at r=0 the curves for cases (i) and
(iii) are almost indistinguishable.) Compared with the
ideal fluid, shear viscosity reduces the work done by lon-
gitudinal pressure and thus slows down the cooling pro-
cess during the early stage; during the middle and late
stages, shear viscosity accelerates the cooling since the
positive transverse shear pressure leads to stronger radial

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5
0
5
10
15
τ−τ0(fm/c)
0.1
0.15
0.2
0.25
0.3
0.35
(a)
0.4
T (GeV)
5
10
τ−τ0(fm/c)
15
0.15
0.2
T (GeV)
r=3 fm
r=3 fm
0
5
10
15
τ-τ0(fm/c)
0
0.1
0.2
0.3
0.4
0.5
0.6
〈VT 〉
ideal hydro
viscous hydro (shear only)
viscous hydro (bulk only)
(b)
Au+Au, b=0 fm, EOSL
e0 = 30 GeV/fm3, τ0 = 0.6 fm/c
Tdec = 130 MeV
FIG. 2: (Color online) Time evolution of the local tempera-
ture (a) and average radial flow (b), from ideal hydrodynam-
ics (dashed blue), viscous hydrodynamics with only minimal
shear (solid red) or bulk (solid brown) viscosity. In (b) the
average radial flow is calculated with the Lorentz contracted
energy density γ⊥e in the transverse plane as weight func-
tion. The inset in (a) shows the late evolution with increased
resolution.
flow than for the ideal fluid (see Fig. 2b and Ref. [20] for
a full discussion). At late times, the shear viscous fireball
thus cools more rapidly than the ideal fluid [18, 20].
Bulk viscosity, on the other hand, reduces the work
done in all three directions, due to the isotropic negative
bulk pressure Π∼−ζθ driven by the positive expansion
rate θ >0. As a result, radial flow develops less rapidly
than for the ideal fluid (Fig. 2b), and the bulk viscous
fluid cools (slightly) more slowly than the ideal one dur-
ing all expansion stages (Fig. 2a). While the expansion
rate is largest at very early times, the bulk viscosity then
is very small throughout the fireball, except for a thin
region near the transverse edge of the fireball where the
matter is close to Tc; bulk viscous effects are therefore
almost negligible until most of the matter cools down to
Tc. At this time the longitudinal expansion rate has sig-
nificantly decreased, but transverse expansion picks up
some of the the slack, and we see significant bulk viscous
effects on radial flow evolution between 5 and 10fm/c.
Surprisingly, the consequences for the cooling rate are
significantly smaller than in the shear viscous case: with
the parameters studied here, the cooling rates for the
ideal and bulk viscous fluids almost agree.
We now turn to non-central collisions.
the fireball deformations in configuration and momen-
tum space, we use the spatial eccentricity εx=??y2−x2??
(where ??...?? denotes an energy density weighted av-
erage over the transverse plane [85]) and the momen-
tum anisotropies εp=?Txx
?Txx
0
+Tyy
weighted averages over the transverse plane of compo-
nents of the ideal fluid part of the energy-momentum
tensor, and thus measuring only the collective flow
anisotropy [20]) and ε′
?Txx+Tyy?(defined in terms
of the total energy-momentum tensor that contains the
viscous pressure components and thus additionally in-
cludes microscopic momentum anisotropies in the local
rest frame of the fluid [20]).
Figure 3a shows the time evolution of the spatial ec-
centricity εxfor non-central Au+Au collisions at b=7fm.
Compared with the ideal fluid, bulk viscosity decelerates
whereas shear viscosity initially accelerates the decrease
with time of the spatial eccentricity εx. This is a direct
consequence of the weaker radial flow in the bulk viscous
case and the stronger radial flow in the shear viscous case.
It is easy to see that isotropic radial expansion is enough
to decreases the spatial eccentricity εx [85]; anisotropic
flow, with larger flow velocities in the reaction plane than
perpendicular to it, only accelerates the rate with which
it decreases. At late times, the eccentricity for the shear
viscous fluid decreases more slowly than for the ideal one,
since the ideal fluid develops stronger elliptic flow (see
Fig. 3b and following discussion). In contrast, in the bulk
viscous case the slower rate of decrease of the eccentricity
is caused by weaker radial flow.
As the spatial eccentricity of the fireball decreases,
its momentum anisotropy increases. This is shown in
Fig. 3b.The dash-dotted lines for εp, which takes
only the ideal fluid part Tµν
0
energy-momentum tensor into account, show how hy-
drodynamic forces convert the spatial anisotropy into a
flow anisotropy. We observe that at early times the flow
anisotropy εp rides on the developing radial flow: com-
pared to the ideal fluid, εpdevelops a little faster in the
shear viscous fluid but a little more slowly in the bulk
viscous case, following the evolution of radial flow.
The difference between εp (dash-dotted lines) and ε′
(solid lines) stems from the viscous pressure components
in the energy-momentum tensor. It reflects a contribu-
tion to the momentum anisotropy that does not arise
from anisotropic collective flow but from viscous devia-
tions from isotropy of the microscopic momentum dis-
tribution f =feq+δf in the local fluid rest frame [20],
accounted for by the non-ideal terms in Tµν. Figure 3b
shows that for the shear viscous fluid these viscous correc-
To describe
??y2+x2??
0
−Tyy
0?
0?(defined in terms of un-
p=?Txx−Tyy?
=(e+p)uµuν− pgµνof the
p

Page 6

6
0
0.05
0.1
0.15
0.2
0.25
εx
ideal hydro
viscous hydro (shear only)
viscous hydro (bulk only)
024
6
8 10 12
τ−τ0(fm/c)
0
0.02
0.04
0.06
0.08
0.1
Au+Au, b=7 fm
EOS L
Tdec = 185 MeV
Tdec = 200 MeV
*
Tdec = 175 MeV
Tdec = 130 MeV
Tdec = 165 MeV
ε’p
εp
εp
X
XXX
X
X
*
*
*
*
+
+
+
++
+
(a)
(b)
FIG. 3: (Color online) Time evolution of spatial eccentricity
εx (a) and momentum anisotropy εp, ε′
viscous hydrodynamics (see text for details). In (b), the differ-
ent symbols along the bulk viscous fluid lines indicate central
(r=0) freeze-out times for different freeze-out temperatures
as described in the legend.
p(b), from ideal and
tions are large and negative (reflecting local momentum
anisotropies pointing out of the reaction plane [20], i.e.
opposite to the flow anisotropy), especially at early times
when the expansion rate and shear velocity components
σµνare large. In contrast, the bulk viscous fluid shows
significant viscous corrections only after about 2.5fm/c,
lasting until about 8fm/c, which is when in these non-
central (b=7fm) Au+Au collisions the bulk of the mat-
ter passes through the phase transition where ζ/s is large
(c.f. the temperature markers on the curves shown in
Fig. 3b). The bulk viscous pressure contribution to ε′
is positive, i.e. pointing into the reaction plane, parallel
to the collective flow anisotropy. (This was also recently
pointed out by Monnai and Hirano [86].)
For both shear and bulk viscosity, the sign of the vis-
cous pressure contributions to ε′
they act against the radial flow driven effects on the mo-
mentum anisotropy. At late times these contributions be-
come small in both cases, in the bulk viscous case driven
by the rapid disappearance of ζ below Tc (as modeled
p
pobeys a “Lenz rule”:
by us), in the shear viscous case by the more gradual
vanishing of the shear velocity tensor σµν[20]. We see
that in the bulk viscous case the radial flow effect on ε′
eventually wins over that from the local deviation from
equilibrium δf, whereas the opposite is true for the shear
viscous fluid. In both cases, the net effect at freeze-out is
thus a viscous suppression of the momentum anisotropy
below the ideal fluid limit (see purple stars in Fig. 3b).
The viscous suppression of ε′
cosity is 4-5 times stronger that that arising from bulk
viscosity.
p
presulting from shear vis-
B. Spectra and elliptic flow
From the hydrodynamic output at decoupling temper-
ature Tdec the spectra and their azimuthal anisotropy,
in particular the elliptic flow coefficient v2(pT), are com-
puted with a modified Cooper-Frye algorithm that takes
into account that in viscous hydrodynamics the local
phase-space distribution f(x,p) on the freeze-out surface
slightly deviates from thermal equilibrium, f =feq+δf,
due to small but non-zero viscous pressure components
[15, 20, 87, 88]. Figure 4a shows the pion pT-spectra for
central Au+Au collisions from ideal and viscous hydro-
dynamics. Compared to the spectrum from ideal fluid
dynamics, shear viscosity leads to flatter spectra while
bulk viscosity generates steeper ones. This is a direct
reflection of the stronger radial flow caused by the posi-
tive transverse shear pressure and the weaker radial flow
resulting from the negative bulk viscous pressure. The
slightly larger normalization of the viscous spectra is a
consequence of viscous entropy production which leads
to larger final multiplicities [21, 87].
Figure 4a shows that shear and bulk viscosity act
against each other in how they affect the slope of the pT-
spectra. When both viscosities are included together in
the viscous calculations, this reduces the amount of read-
justment needed in the initial conditions when fitting the
measured transverse momentum spectra with viscous in-
stead of ideal fluid dynamics [87]. The differential elliptic
flow v2(pT) for soft pions, on the other hand, is affected
by both bulk and shear viscosity in the same way: Fig-
ure 4b shows that the viscous reduction of ε′
translates directly into reduced elliptic flow v2of the fi-
nal hadron spectra. This is true in particular in the low-
pT region pT<1GeV/c (see inset in Fig. 4b). At larger
pT, the shear viscous v2 suppression further increases,
due to a negative contribution from δf along the freeze-
out surface. In contrast, bulk viscosity increases v2(pT)
above 1GeV/c because it results in steeper pT spectra.
At pT=0.5GeV/c (approximately the mean transverse
momentum for pions) we find that minimal bulk viscos-
ity suppresses v2by ∼5% while minimal shear viscosity
leads to a ∼20% suppression.
If this were the complete story, the additional ∼5%
bulk viscous v2 suppression would lead to a ∼25% re-
duction of the value for η/s that one might extract from
pin Fig. 3b

Page 7

7
0
0.5
1
1.5
2
pT(GeV)
10-1
100
101
102
(1/2π)dN/dypTdpT(GeV-2)
ideal hydro
viscous hydro (shear only)
viscous hydro (bulk only)
π−
Au+Au, b=0 fm,
e0=30GeV/fm3,
Tdec=130 MeV
EOS L
(a)
τ0=0.6fm/c
0
0.5
1
1.5
2
pT(GeV)
0
0.1
0.2
v2
ideal hydro
viscous hydro (shear only)
viscous hydro (bulk only)
0
0.1
0.2
pT(GeV)
0.30.4
0.5
0
0.02
0.04
0.06
v
(b)
2
Au+Au, b=7 fm
EOS L
FIG. 4: (Color online) pT spectra and elliptic flow v2(pT) for directly emitted pions (i.e. without resonance decay contributions).
experimental elliptic flow data, by comparing them with
an ideal fluid dynamical baseline as advertised in [22].
This is a non-negligible effect. Since the input for the
bulk viscosity used in our calculations is fraught with
large theoretical uncertainties, as discussed in Sec. II,
this points to a likelihood for correspondingly large un-
certainties in the empirical extraction of η/s from ex-
periment. In the following Section we follow this line of
thought further, by investigating the additional sensitiv-
ity of bulk viscous effects to the initial conditions for the
bulk viscous pressure and to its relaxation time.
V. SENSITIVITY OF BULK VISCOUS
DYNAMICS TO INITIAL CONDITIONS AND
RELAXATION TIMES
In this Section we now focus entirely on bulk viscosity
and investigate what happens when we change the initial
value for the bulk viscous pressure Π and its relaxation
time τΠ. We study only “minimal bulk viscosity” as de-
fined in Sec. II (i.e. C =1), leaving the discussion of
larger values to the following Section.
Figure 5 shows, for peripheral Au+Au collisions at
b=7fm, the differential elliptic flow for pions (a) and the
time evolution of the average value of the bulk viscous
pressure ?Π? (b), for the two initial conditions (zero and
Navier-Stokes) for Π and the three choices of relaxation
time scales τΠdiscussed in Sec. II. At pT=0.5GeV/c,
Fig. 5a indicates a bulk viscous v2 suppression that
ranges (for C =1) from ∼2% to ∼10%. For the short re-
laxation time τΠ=0.5fm/c (solid and dotted green lines)
the suppression is insensitive to the initialization of Π,
yielding about 8% suppression below the ideal fluid value
at pT=0.5GeV/c for both zero and Navier-Stokes initial
values. Figure 5b explains the underlying reason for this
observation: for this short relaxation time, ?Π? quickly
loses all memory of its initial value, relaxing for both ini-
tial conditions to the same trajectory after ∼1 − 2fm/c
(i.e. after a few relaxation times, similar to what we saw
earlier [20] for the shear pressure components). This also
demonstrates that most of the finally observed bulk vis-
cous v2suppression is generated during the middle part
of the expansion when most of the matter cools through
the phase transition. If it were dominated by large nega-
tive bulk viscous pressures in the outer shell of the fireball
at early times, we should see stronger sensitivity to the
initial value for Π.
This changes completely if we chose a 10 times longer
relaxation time, τΠ=5fm/c (solid and dotted magenta
lines in Fig. 5). Now the bulk viscous v2suppression be-
comes extremely sensitive to the initialization of Π. For
zero initialization, the average bulk pressure ?Π? always
remains small, leading to very small (O(2%)) final sup-
pression effects for v2. For Navier-Stokes intialization,
?Π? is initially very large and negative (due to the large
initial expansion rate) and, instead of relaxing to smaller
values as predicted by Navier-Stokes theory (and real-
ized by the solid green line corresponding to short τΠ), it
remains larger than the N-S value for about 4fm/c. As
seen in Fig. 5a, this leads to a much larger v2suppression
of about 10%.
The “critical slowing down” scenario which uses a tem-
perature dependent relaxation time that follows the be-
haviour of
?
black curves in Fig. 5. In this case the bulk viscous pres-
sure quickly relaxes to its Navier-Stokes value in the in-
terior of the fireball where the temperature is high and
the relaxation time is short; near the edge of the fire-
ball, however, where the temperature is near Tcand the
relaxation time is long, it remembers its initial value (ei-
ther zero or the large negative initial N-S value) for a
long period. With some reflection one convinces oneself
that this implies that for both zero and N-S initializa-
tions the magnitude of the average bulk viscous pressure
?Π? remains below the value observed for the short re-
laxation time. This is seen in Fig. 5b when comparing
the black and green curves. Correspondingly, the viscous
v2suppression seen in part (a) of the Figure is for both
ζ
s
?
(T) is shown by the solid and dash-dotted

Page 8

8
0 0.10.2 0.30.40.5
pT (GeV)
0
0.05
v2
Glauber initialization:
e0 = 30 GeV/fm3
τ0 = 0.6 fm/c
Tdec = 130 MeV
Au+Au, b=7 fm
EOS L
(a)
ideal hydro
viscous hydro
(min. bulk viscosity only)
0
5
10
τ−τ0(fm/c)
-0.1
0
<Π> (GeV/fm )
3
τΠ= 0.5 fm/c
τΠ= (ζ/s)(T) (120 fm/c)
Π = 0
Π = −ζ ( ∂ u)
Π = 0
Π = −ζ ( ∂ u)
{
{
{
η/s=0, ζ/s = minimal
-- min. bulk viscosity only
.
.
(b)
viscous hydrodynamics
τΠ= 5 fm/c
Π = −ζ ( ∂ u)
Π = 0
.
.
FIG. 5: (Color online) (a) Differential elliptic flow v2(pT) for
directly emitted pions (without resonance decays) from ideal
and viscous hydrodynamics, including only minimal (C =1)
bulk viscosity. (b) Time evolution of the bulk pressure ?Π?
averaged over the transverse plane (weighted by the energy
density) from viscous hydrodynamics. Different curves cor-
respond to different initializations and relaxation times, as
indicated (see text for discussion).
initializations smaller for the “critical slowing down” sce-
nario than for a short constant relaxation time. When
comparing the “critical slowing down” scenario with the
long constant relaxation time, the viscous v2suppression
is significantly smaller for N-S initialization (O(7%) vs.
O(10%)) and about equally small (O(2%)) for zero ini-
tialization.
Since these findings contradict at least our own naive
first expectations, we briefly reiterate the main point:
taking into account the critical slowing down of the bulk
viscous pressure dynamics near Tc where ζ/s becomes
large leads to weaker bulk viscous suppression effects on
the elliptic flow than seen for both short and long con-
stant (i.e. T-independent) relaxation times τΠ.
VI. LARGER ζ/s AND THE BREAKDOWN OF
VISCOUS FLUID DYNAMICS
As noted in Sec. II, the peak value of ζ/s in our
parametrization shown in Fig. 1 is about 10 times smaller
than some other estimates [66, 76]. When one tries to
simply multiply the function shown in Fig. 1 by C =10,
one finds that (except for special circumstances discussed
below) the viscous hydrodynamic code crashes. The rea-
son is that sufficiently large bulk viscosity can lead to
fireball regions where the effective total isotropic pressure
p+Π (thermal + bulk viscous pressure) becomes negative
and the medium becomes mechanically unstable and will
tend to break up [46–49].
ponents of the shear viscous pressure (in particular its
longitudinal component πηη) are usually also negative,
instability can set in even somewhat earlier [47, 48]. In
numerical simulations this manifests itself through the
exponential amplification of local numerical errors which
will eventually stop the code from running.
We point out that even before the fluid becomes me-
chanically unstable one has left the region of applica-
bility of viscous hydrodynamics. The viscous hydrody-
namic formalism is based on a near-equilibrium expan-
sion; its validity assumes that the viscous corrections to
the energy-momentum tensor are small compared with
the ideal fluid terms. In other words, if the condi-
tion (|Π|+|πµν|)/(e+p)≪1 is violated for any compo-
nent (µν), the evolution based on equations (2-4) can no
longer be trusted. Ignoring the shear pressure and set-
ting e+p=sT ≈4p for a QGP, the instability threshold
p+Π=0 translates into |Π|/(e+p)≈1
ciently small to trust the continued validity of the equa-
tions. The following alternate consideration leads to the
same conclusion: If the fluid can be described by quasi-
particles, the viscous terms in the energy-momentum ten-
sor correspond to deviations of the local phase-space dis-
tribution f(x,p)=feq+δf from local equilibrium. Using
Grad’s 14-moment method, the deviation δf is expanded
up to quadratic order in momentum [15, 56, 57, 83, 86]
and (for a fluid with only bulk viscosity and massless
particles at midrapidity y =0) can thus be written in the
form
In fact, since certain com-
4which is not suffi-
δf
feq
= ap2
T
T2
Π
e+p,
(5)
where a is a slowly varying function of temperature with
magnitude of order unity [86]. When p+Π=0 such that
Π
e+p=−1
typical thermal momenta pT≃3T the deviation δf/feq
is negative with magnitude 1 or larger, rendering the to-
tal distribution function f negative, which is unphysi-
cal. Clearly the deviations from local equilibrium are
too large and the formalism breaks down.
In this Section we explore the range of bulk viscosities
that are allowed without leaving the region of validity of
second-order (Israel-Stewart) viscous hydrodynamics. As
4, this means that for midrapidity particles with

Page 9

9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(ζ/s)max(Tc)
0
5
10
15
20
Cmax
0.11 10
τΠ(fm/c)
0
0.1
0.2
0.3
0.4
0.5
∆S/S0
(a)
N-S initialization Zero initialization
η/s, τ0 (fm/c)
0.0, 0.6
0.08, 0.6
0.08, 0.6
0.16, 0.6
0.08, 1.0
0.08, 1.0
0.08, 2.0
η/s, τ0 (fm/c)
0.0, 0.6
(b)
Au+Au, b=7 fm, EOS L
*
*
FIG. 6: (Color online) Upper limits for ζ/s (a) and viscous
entropy production (b) as a function of bulk viscous relax-
ation time τΠ, for zero and Navier-Stokes initialization. The
stars indicate the results for the temperature dependent re-
laxation time (1) with Navier-Stokes initial conditions, for
τ0=0.6fm/c. For τΠgiven by Eq. (1) and zero initial con-
ditions, there is no upper limit for ζ/s, i.e. the flud remains
stable for all values of C.
in the preceding Section, we study both zero and Navier-
Stokes initial conditions and the same three choices for
the bulk viscous relaxation time τΠ, but we now also in-
clude runs where the fluid has an additional shear viscos-
ity η/s=(1÷2)/(4π), with shear viscous relaxation time
τπ=3η/(sT), and we vary the time τ0when we start the
hydrodynamic evolution. (For later starting times, we
downscale the initial peak entropy density s0 such that
the total initial entropy ∼ s0τ0 is held constant.) For
the specific bulk viscosity
?
form shown in Fig. 1, but multiplied by an arbitrary con-
stant C >1. For each set of initial conditions, τΠ, and η/s
we determine the largest value Cmaxthat still allows for
stable running of the code, i.e. where the effective to-
tal isotropic pressure p+Π does not violate the stability
criterium p+Π>0 anywhere inside the freeze-out surface.
Figure 6 shows the upper limit (ζ/s)max(Tc) (on the
left vertical axis) and the corresponding maximal C-value
Cmax (on the right vertical axis) as a function of the
bulk viscous relaxation time τΠ. We see that it depends
strongly on the initialization.
For Navier-Stokes (N-S) initial conditions (ζ/s)max(Tc)
is insensitive to the relaxation time τΠ. In this case the
ζ
s
?
(T) we take the functional
magnitude of the average bulk pressure Π decreases more
or less monotonically with time (see Fig. 5b). Violations
of the positivity condition p+Π>0 thus always happen
at the starting time τ0, at transverse positions where
the matter is close to the phase transition. This leads
to a (ζ/s)max(Tc) that is controlled by initial conditions
and independent of the relaxation time. (This includes
the temperature dependent relaxation time (1) – see the
star in Fig. 6a.) Correspondingly, (ζ/s)max(Tc) does not
depend on the value of η/s when shear viscosity is in-
cluded. When one starts the hydrodynamic evolution
later, (ζ/s)max(Tc) increases with τ0. The dependence
on τ0 arises from the strong dependence of the initial
bulk pressure Π=−ζθ =−ζ∂·u on τ0, through the ex-
pansion rate θ(τ0)=1/τ0. This is illustrated by the solid
red, dashed magenta and dotted orange lines in Fig. 6a:
as one increases τ0from 0.6 to 1 and 2fm/c, the maximal
(ζ/s)max(Tc) increases from 0.05 to 0.09 and 0.18.
For zero initialization Π(τ0)=0 one finds a qualita-
tively similar dependence of (ζ/s)max(Tc) on the start-
ing time τ0: The curves Cmax(τΠ) move up as one in-
creases τ0 from 0.6 to 1.0fm/c (solid and dashed green
lines). However, in contrast to the N-S initialization, the
(ζ/s)max(Tc) curves now show a strong dependence on
relaxation time τΠ, rising monotonically with τΠ. The
reason is that it takes some time for the bulk pressure
Π to develop large enough magnitudes to violate the
positivity condition p+Π>0; again this happens typi-
cally in regions where the matter is close to the phase
transition.For larger relaxation times Π moves away
from its zero initial value more slowly, rendering the
fluid more stable and resulting in a monotonic increase
of (ζ/s)max(Tc) with τΠ. For τΠ<1fm/c we find ”univer-
sal” (ζ/s)max(Tc)−τΠcurves that do not depend on the
shear viscosity η/s (solid black, green and blue curves),
but move upwards as we increase the starting time τ0.
This is because the violation of the positivity condition
p+Π>0 then generally happens at early times τ <3fm/c
when the flow profiles are not yet significantly affected by
shear viscous effects. For the two viscous fluid lines with
η/s=0.08 and 0.16 (solid blue and green lines) one sees
that they continue to overlap even for τΠ>1fm/c after
they have broken away from the η/s=0 line.
ideal fluid (η/s=0) the phase transition generates large
velocity gradients near the phase transition that generate
locally large expansion rates, causing instability at lower
values of ζ/s. Shear viscosity smoothes out these large
gradients, as discussed in Ref. [20], allowing the fluid to
evolve stably up to larger values of ζ/s. Bulk viscosity ζ
alone has no smoothing influence on sharp structures gen-
erated by a phase transition. For a zero initial value for
Π, shear viscosity thus helps crucially in stabilizing the
evolution of the viscous fluid against mechanical instabil-
ities caused by strongly negative bulk viscous pressure,
especially for large relaxation times τΠ.
Very interesting is our finding that, for zero initial con-
ditions, there is no limit on Cmaxif one accounts for crit-
ical slowing down of the bulk pressure dynamics near Tc
In the

Page 10

10
via Eq. (1). In this case the bulk pressure, starting at
zero, never grows sufficiently large to threaten mechani-
cal stability of the fluid, irrespective of how large the bulk
viscosity becomes at Tc! As the peak value (ζ/s)(Tc) is
increased, so is the time it takes Π to evolve towards its
Navier-Stokes value, and this never happens fast enough
to violate the stability condition p+Π>0.
Figure 6b shows the viscous entropy production for the
maximally allowed bulk viscosities shown in Fig. 6a. Not
surprisingly, viscous entropy production increases with
shear viscosity η/s and decreases when hydrodynamics
is started later, with correspondingly smaller initial ex-
pansion rates [21]. The dependence on (ζ/s)max(Tc) is
non-monotonic, however. The reason is that the bulk
viscous entropy production rate ∼Π2/(2ζ) depends not
only on how large ζ is but also on how close Π is to its
Navier-Stokes limit, and this in turn depends on τΠ.
VII.
EXPERIMENTAL DATA: UNCERTAINTIES
INTRODUCED BY BULK VISCOSITY
TOWARDS EXTRACTING η/s FROM
Given the fact that bulk viscosity contributes to the
viscous suppression of elliptic flow (see Fig. 5a), and as-
suming that bulk and shear viscous effects cannot be sep-
arated by studying other experimental observables, the
question arises naturally how much of an irreducible un-
certainty this will introduce into the extraction of the
specific shear viscosity η/s from experimental elliptic flow
measurements. More precisely, if the QGP should turn
out to be a “most perfect liquid” with “minimal” shear
viscosity η/s=1/4π, with what kind of accuracy can we
hope to verify this experimentally if bulk viscosity is the
only quantity beyond our theoretical and experimental
control?
To answer this question, we used VISH2+1 to com-
pute the differential elliptic flow of directly emit-
ted pions (without resonance decay contributions) for
200AGeV Au+Au collisions at b=7fm, assuming the
fireball medium to have constant specific shear viscosity
η/s=1/4π but allowing the bulk viscosity ζ/s to vary
over the entire range allowed by the mechanical stabil-
ity criterium p+Π>0. In doing so we assumed a fixed
shape of the temperature dependence of ζ/s as shown
in Fig. 1 but let its normalization vary between C =1
and Cmax(τΠ) where the latter is the largest value within
the range of applicability of Israel-Stewart viscous fluid
dynamics, shown in Fig. 6a. We allowed for two fixed
values of 0.5 and 5fm/c for the bulk viscous relaxation
time τΠas well as for “critical slowing down” according
to Eq. (1), and we studied both zero and Navier-Stokes
initial values for the viscous pressure components. All
calculations assume τ0=0.6fm/c as starting time. The
results are presented in Fig. 7 and Table I.
Generically one observes that, even for minimal shear
viscosity near the KSS bound, the shear viscous contri-
bution to the elliptic flow suppression far exceeds the
0
0.05
v2
ideal hydro
visc hydro (only min shear visc)
visc hydro (min shear visc + bulk visc)
0 0.1 0.20.30.4
0.5
pT(GeV)
0
0.05
v2
Au+Au, b=7 fm,
{
{
C=1.0
τΠ = 5.0 fm/c{
τΠ = 0.5 fm/c
C=1.0
C=1.3
τΠ = 0.5 fm/c
τΠ = 5.0 fm/c{
η/s=1/4π
Π (τ 0) = −ζ(∂
EOS L
πmn (τ 0) = 2ησmn
u)
.
{
C=1.3
C=1.0
C=100
C=1.0
C=3.5
η/s=1/4π
πmn (τ 0) = 0
Π (τ 0) = 0
(a)
(b)
τΠ = ζ/s 120 fm/c
.
{
C=1.0
C=1.3
C=10.3
C=1.0
τΠ = ζ/s 120 fm/c
.
FIG. 7: (Color online) v2(pT) for directly emitted pions from
ideal and viscous hydrodynamics with Navier-Stokes (a) or
zero (b) initial conditions for the viscous pressures. Shown
are results for minimal shear viscosity η/s=1/4π and bulk
viscosities ranging from “minimal” (C =1) to the maximal
values from Fig. 6a that still allow for stable viscous evolution,
for three choices of the bulk viscous relaxation time τΠ.
bulk viscous contribution. This is good news since it
means that the uncertainty introduced into the extrac-
tion of η/s by theoretically poorly controlled bulk viscous
effects remains limited and is, in fact, quite small, espe-
cially if the real fireballs created in heavy-ion collisions
do not completely saturate the KSS bound. On a more
quantitative level, one finds that for pions with typical
transverse momentum pT=0.5GeV/c the elliptic flow is
suppressed by just over 16% below the ideal fluid value if
the expanding matter has only shear, but no bulk viscos-
ity, and that this suppression increases to values between
17% and 25% if bulk viscosity is added. The largest bulk
viscous suppression is found for fixed relaxation times τΠ
and zero initialization if the bulk viscosity is increased
all the way up to its upper allowed limit. In these cases
the additional suppression can be as large as 50% of the
suppression found for the fluid with only (minimal) shear
viscosity. If one takes into account that the evolution of
the bulk viscous pressure slows down near Tcwhere ζ/s
is largest, the additional bulk viscous suppression never
exceeds 20% of the shear viscous elliptic flow suppression,

Page 11

11
TABLE I: Pion elliptic flow at pT =0.5GeV/c for b=7fm 200AGeV Au+Au collisions from ideal and viscous hydrodynamics,
with different choices of initial conditions, bulk viscous relaxation times τΠ, and bulk viscosities (parametrized by C). The
last of the 3 columns in each initialization block gives the viscous suppression of v2 at pT =0.5GeV/c in terms of the percent
deviation from the ideal fluid baseline (=100%).
zero initialization Navier-Stokes initialization
η/sτΠ(fm/c)Cv2(0.5GeV/c)(%)
v2
v2,ideal(%)
Cv2(0.5GeV/c)(%)
v2
v2,ideal(%)
0–0 5.7551000 5.755100
0.08–0 4.82183.80 4.81183.6
0.08
0.08
0.08
0.08
0.08
0.08
0.5
0.5
5.0
5.0
1 4.668
4.356
4.770
4.323
4.743
4.656
81.1
75.7
82.9
75.1
82.5
80.9
1 4.627
4.576
4.601
4.534
4.660
4.615
80.4
79.5
79.9
78.8
81.0
80.2
3.5
1
10.3
1
100
1.3
1
1.3
1
1.3
Eq.(1)
Eq.(1)
with 10-15% being a typical range (light blue and green
curves in Fig. 7).
An important caveat is, however, that for Navier-
Stokes initial conditions the allowed maximal bulk vis-
cosities are small, much below recent Lattice QCD esti-
mates [76]. If larger values are realized by Nature, they
invalidate the use of viscous hydrodynamics, at least at
early times [89–92]. The problems in this case arise from
the large bulk viscosity in a thin layer near the transverse
edge of the fireball where the matter is close to Tc. It is
only in this region that the viscous hydrodynamic de-
scription breaks down. Since the problematic factor, the
scalar expansion rate θ, decreases initially very rapidly,
these initially unstable fluid regions move quickly back
to mechancal stability. Since the momentum anisotropy
does not develop instantaneously, we find it hard to be-
lieve that the existence of this unstable external layer has
much influence on the evolution and final value of the el-
liptic flow, and one should get very similar results from
simulations in which the initial bulk viscous pressure Π
is restricted by hand to values below the threshold for vi-
olating the positivity condition p+Π>0. If this is indeed
the case, the results presented in this Section show that
bulk viscosity, even if theoretically not well controlled,
will not introduce large uncertainties into the extraction
of η/s from elliptic flow data.
VIII. CONCLUDING REMARKS
The present study shows that bulk viscosity, as long
as it is small enough that in expanding heavy-ion colli-
sion fireballs the negative bulk viscous pressure does not
become larger than the thermodynamic pressure, affects
the elliptic flow of the final hadrons much more weakly
than does shear viscosity. So, as long as the expanding
fireball can be described by viscous fluid dynamics, it is
possible to extract its shear viscosity (even if it is as small
as
s
??
η
KSS=
1
4π) with good accuracy from a comparison of
viscous hydrodynamic simulations with experimental el-
liptic flow data. Accounting for the critical slowing down
of viscous bulk pressure dynamics near Tc, we showed
that any contamination from bulk viscosity ζ/s is <20%
(for much of the parameter space it is even <10%), and
that its relative importance decreases further if η/s is
larger than the KSS bound.
However, we also saw that the stability condition
p+Π>0 is very restrictive and easily violated if the peak
value of ζ/s near Tc reaches values close to those esti-
mated from Lattice QCD [76] and from some strong cou-
pling approaches [66], and if the bulk viscous pressure
Π approaches its Navier-Stokes limit Π=−ζ∂·u. When
this occurs (typically at early times when the scalar ex-
pansion rate is largest, in a thin layer around Tc close
to the transverse edge of the fireball), the viscous fluid
dynamical description breaks down. Our analysis shows
that the phenomenon of “critical slowing down” can play
a crucial role in preventing this from happening. Kinetic
theory for weakly coupled systems [56, 93] and a recent
analysis by Buchel of strongly coupled systems [66] sug-
gest that the same microscopic physics (namely growing
correlation lengths due to critical fluctuations) that gen-
erate a peak of ζ/s at Tcalso causes the relaxation time
τΠfor the bulk viscous pressure to grow and possibly di-
verge at Tc even while ζ/s itself remains finite. When
using the model Eq. (1) for a temperature dependent τΠ
inspired by these ideas we saw that, unless Π is initialized
at its Navier-Stokes limit, it never reaches it during the
short time span of a heavy-ion collision in those fireball
region where ζ/s peaks and Π could thus become very
large. This reduces the problem of applicability of vis-
cous hydrodynamics at early times to a question of initial
conditions for Π, especially in that thin transverse layer
where (after local equilibrium is reached) the tempera-
ture happens to be close to Tc.
Determining these initial conditions (as opposed to
guessing them as we have done here) requires a theoreti-
cal description of the early pre-equilibrium evolution and
Landau-matching the corresponding energy-momentum
tensor to its viscous fluid dynamic form, Eq. (2) (in the

Page 12

12
spirit of Ref. [92] but generalized from 0+1 to 2+1 di-
mensions). At this point we lack the tools for doing this.
Let us, however, make a few comments in anticipation of
completion of that task. Consider a small fireball region
that is just reaching local thermal equilibrium at a tem-
perature close to Tc and undergoing self-similar boost-
invariant longitudinal expansion while transverse expan-
sion is negligible. Let us also assume that at this point
in time the bulk viscous pressure in the region is large
and negative, leading to negative effective total isotropic
pressure and causing the fluid to be mechanically unsta-
ble. What will happen? The fluid will begin to rupture,
forming little voids, and if the region were to remain in a
state of negative total pressure, it would eventually frag-
ment. However, since the considered region is undergoing
rapid expansion and cooling, it will quickly exit from its
state of mechanical instability. Furthermore, during the
short period of instability the hydrodynamic growth of
voids will be hampered by the large value of the relax-
ation time τΠ. By the time the considered region becomes
mechanically stable again, we expect it to be riddled with
small holes, but otherwise intact. The small voids formed
during the period of instability will re-collapse by cavita-
tion, and the region will quickly re-equilibrate due to the
now much shorter relaxation time below Tc. No whole-
sale breakup of the fluid will occur, due to lack of time.
Similar arguments hold later when the bulk of the matter
in the center of the fireball passes through Tc, only that
in this case the viscous bulk pressure may never grow
large enough to generate mechanical instability, due to
critical slowing down.
In summary, unlike the authors of Ref. [48], we do
not expect any dramatic macroscopic phenomena trig-
gered by the transient mechanical instability arising from
possibly large, but short-lived negative bulk pressures in
fireball regions passing through the hadronization phase
transition. For this reason we believe that a modified vis-
cous hydrodynamic treatment, where one limits by hand
the growth of the viscous bulk pressure so that it al-
ways remains below the instability threshold [94], will
not lead to impermissible distortions of the real (non-
equilibrium) dynamics in the (small) space-time regions
whose description lies outside the hydrodynamic domain.
This is important for future viscous hydrodynamic stud-
ies of heavy-ion collisions with fluctuating and granular
initial conditions [95] which are more realistic than the
smooth initial profiles presently used.
Acknowledgments
We gratefully acknowledge informative discussions
with K. Dusling, P. Petreczky, and J. Randrup, and
thank T. Hirano and G. Moore for constructive comments
on the manuscript. U.H. is indebted to K. Rajagopal for
a very fruitful exchange of ideas which clarified much of
the argument presented in the last Section of this paper.
This work was supported by U.S. Department of Energy
under contract DE-FG02-01ER41190.
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ζ
s(Tdec)=0