Transcript 幻灯片 1

Flow Vorticity and Rotation
in Peripheral HIC
Dujuan Wang
University of Bergen, Norway
2014 CBCOS, Wuhan, 11/05/2014
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Outline
• Introduction
• Vorticity for LHC, FAIR & NICA
• Rotation in an exact hydro model
• Summary
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1. Introduction

Pre-equilibrium stage 
Initial state

Quark Gluon Plasma 
FD/hydrodynamics 
Particle In Cell (PIC) code

Freeze out, and simultaneously
“hadronization” 
Phase transition on hyper-surface
 Partons/hadrons
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Relativistic Fluid dynamics model
Relativistic fluid dynamics (FD) is based on the conservation laws
and the assumption of local equilibrium ( EoS)
N  ,  0
[ N  dˆ  ]  0
T  ,   0
[T  dˆ  ]  0
4-flow:

N  (n, j )

energy-momentum tensor: T 
For perfect fluid:
d3p  
  0 p p f ( x, p )
p
T   (e  P )u  u  Pg 
In Local Rest (LR) frame = (e, P, P, P);
g   g  diag(1,1,1,1)
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tilted initial state, big initial angular momentum
Flow velocity
Structure and
asymmetries of I.S. are
maintained in nearly
perfect expansion.
Pressure gradient
[L.P.Csernai, V.K.Magas,H.Stoecker,D.D.Strottman, PRC 84,024914(2011)]
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The rotation and Kelvin Helmholtz Instability (KHI)
More details in
Laszlo’ talk
Straight line 
Sinusoidal wave
for peripheral collisions
[L.P.Csernai, D.D.Strottman, Cs.Anderlik,
PRC 85, 054901(2012)]
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2. Vorticity
Definitions:
Classical flow:
[L.P. Csernai, V.K. Magas, D.J. Wang,
PRC 87, 034906(2013)]
Relativistic flow:
The vorticity in [x,z]
plane is considered.
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More details:
Weights:
+0
+-
++
0+
Etot: total energy in a y layer
Ncell: total num. ptcls. In this y layer
In [x,z] plane:
Corner cells
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Vorticity @ LHC energy:
In Reaction Plane t=0.17 fm/c
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In Reaction Plane t=3.56 fm/c
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In Reaction Plane t=6.94 fm/c
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b5
All y layer added up
at t=0.17 fm/c
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b5
All y layer added up
at t=3.56 fm/c
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Average Vorticity in summary
Decrease with time
Bigger for more peripheral collision
Viscosity damps the vorticity
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Circulation:
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Vorticity @ NICA , 9.3GeV:
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Vorticity @ FAIR, 8 GeV
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3, Rotation in an exact hydro model
Hydrodynamic basic equations
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The variables:
Scaling variable:
Csorgo, arxiv: 1309.4390[nucl.-th]
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More details:
rhs:
y
cylindrical coordinates:
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lhs:
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Kinetic energy:
Expansion energy at the surface
Rotational energy at the surface
Expansion energy at the longitudinal direction
sρM & syM:
Boundary of spatial integral
(α and β are independent of time)
For infinity case:
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Internal energy:
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Solutions:
The solution:
Runge-Kutta method: Solve first order DE
initial condition for R and Y is needed, and the constants Q and W
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Table 1 : data extracted from
L.P. Csernai, D.D Strottman and Cs Anderlik, PRC 85, 054901 (2012)
R : average transverse
radius
Y: the length of the system
in the direction of the
rotation axis
θ : polar angle of rotation
ω : anglar velocity
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Energy time dependence:
Energy conserved !
decreasing internal energy
and rotational energy
leads the increasing of
kinetic energy .
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Spatial expanding:
Smaller initial radius
parameter
overestimates the radial
expansion velocity
due to the lack of dissipation
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Expansion Velocity:
In both cases the
expansion in the radial
direction is large.
Radial expansion
increases faster,
due to the centrifugal
force from the rotation.
It increases by near to 10
percent due to the
rotation.
the expansion in the
direction of the axis of
rotation is less.
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Summary
• High initial angular momentum exist for
peripheral collisions and the presence of KHI is
essential to generate rotation.
• Vorticity is significant even for NICA and FAIR
energy.
• The exact model can be well realized with
parameters extracted from our PICR FD model
Thank you for your attention!
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Table 2 : Time dependence of
characteristic parameters of
the exact fuid dynamical model.
Large extension in the beam
direction is neglected.
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α and β
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