2010-4-18 the effects of viscosity on hydrodynamical evolution of qgp 苏中乾 大连理工大学...
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2010-4-18 DaLian university and technology
The effects of viscosity on hydrodynamical evolution of QGP
苏中乾大连理工大学
Dalian University of Technology
Contents
• I. Formalization of viscous hydrodynamics
• II. Equation of state (EOS) and initial conditions
• III. Numerical results
• IV. Summery
The effects of viscosity on hydrodynamical evolution of QGP
Viscous hydrodynamics
• Why viscous?
• The ideal fluid description works well in almost central Au+Au collisions near midrapidity at top RHIC energy, but gradually breaks down in more peripheral collisions, at forward rapidity, or at lower collision energies, indicating the onset of dissipative effects. And viscous fluid is a real fluid.
Viscous hydrodynamics
Energy-momentum tensor decomposition:
( ) 13
( ) 12
( )
( )
u T u
p
T
a a
a a a
Energy density in fluid rest frame
Pressure in fluid rest frame,
Bulk viscous pressure
Shear stress tensor,
Symmetrized tensor
Symmetrized, traceless spatial projection
13p T
0, 0u u
( )T u u p
0S
0N Charge conservation
0T Energy-momentum conservation
The second law of thermodynamics
Starting point : conservation laws
Viscous hydrodynamics
( )T u u p 0T
Energy-momentum tensor conservation: Thinking about zero baryon system, we can ignore the effects of vorticity ( ) and heat conductivity of flow ( )0 0q
Relaxation equations for shear stress tensor and bulk pressure
122 ( )u
Tu T
12 ( )u
Tu T
Second-order Israel-Stewart equation
Navier-Stokes (NS) equations: simple, but: unstable and acausal equations of motion
Viscous hydrodynamics
With , viscous hydrodynamics reduces to ideal hydrodynamics
• Use Bjorken picture and thinking about azimuthally symmetric systems:
, 0
/zv z t (Boost invariant)
( , , , )r
Notice: once we know and , velocity and the temperature at will be Obtained by Lorentz boost:
viscous hydrodynamics equations
Close equations with EOS ( )p p
( , , , )r
Boost-invariant and azimuthally symmetric systems:
Use coordinates and solve:
hydrodynamic equations for
with
kinetic relaxation equations for shear viscous tensor and bulk viscous pressure
shear
bulk
Energy-momentum
Equation of state and initial condition
with
for
the MIT bag equation of state with bag constant
First: QCD lattice results (Cross over EOS: entropy density of the system is function of temperature)
•Input: EOS (influence on the dynamic of evolution)( )p p
Second: Equation of state for massless quarks and gluons gas
where (the number of quark flovers)
A. Muronga & D. Rischke; nucl-th/0407114v2
The result equation of state
EOS results
0.1T Tc
/ 3.6r dQ dH
The Solid line:
The dashed line: quark and gluon gas equation of state.
The initial conditions
The components of the shear tensor and bulk pressure
0 maximum energy density for central collisions at initial time
initial transverse radius of source.0 6R fm
the initial temperature at the center of source and initial proper time
Initial conditions: Initial energy density distribution in the transverse plane use Woods-saxon parameterization
Viscous vs. ideal hydrodynamics——Temperature and velocity
For ideal hydrodynamic equation the temperature cool faster thanthat the shear viscosity are included, this means viscous effect lead to slow down of expansion of system.
1/ 2
4s
Viscous vs. ideal hydrodynamics——shear and bulk viscous
The profiles of shear tensor and bulk pressure components in unit of at different times as function of
0p
0/r R
Our results VS A.Muronga & D. Rischke
A. Muronga & D. Rischkenucl-th/0407114v2
Ideal quark &gluon gasEOS
Cross over EOSand woods-saxondistribution
Viscous vs. ideal hydrodynamics——Space-time diagrams of freeze-out
At a fixed temperature, a cross over transition equation of state lead to a larger space-time of freeze-out and a little larger while we take the effect of viscosity into account.
The source with considering of viscosity, reaches to Tf have a larger space until a certain time and then it will be a littlesmaller.
The reason : at binging due to effect of viscosity, source will expand slowly, however after a while, this effect will be dissipated.R.Baier & P.Romatschke: Eur. Phys. J. C 51, 677 (2007).D. A. Teaney: J.Phys. G 30, S1247 (2004).A. K. Chaudhuri: Phys. Rev. C 74, 044904 (2006).
• I. For ideal fluid the temperature cool faster than that the viscosity are included, this means viscous effect lead to slow down of the temperature decrease and the expansion of system.
II. A cross over transition equation of state lead to a larger space-time of freeze-out than no phase transition and a little larger while we take the effect of viscosity into account.
Summery
Thank you!