Flow Vorticity and Rotation in Peripheral HIC

Flow Vorticity and Rotation in Peripheral HIC

Dujuan Wang

University of Bergen, Norway

• Introduction

• Vorticity for LHC, FAIR & NICA

• Rotation in an exact hydro model

• Summary

Outline

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

Relativistic Fluid dynamics model

Relativistic fluid dynamics (FD) is based on the conservation laws and the assumption of local equilibrium ( EoS)

4-flow:

energy-momentum tensor: ^T

^  ^d _p

^{p p}

^p

^{f ( x , p )}

) , ( n j N

 









e P u u Pg

T  (  ) 

For perfect fluid:

0 ˆ ]

[

0 ˆ ]

[





 

 



 d T

d N 0

,

0 ,





 

 

T

N

tilted initial state, big initial angular momentum

Structure and

asymmetries of I.S. are maintained in nearly perfect expansion.

Flow velocity

The rotation and Kelvin Helmholtz Instability (KHI)

More details in Laszlo’ talk

Straight line  Sinusoidal wave

for peripheral collisions

Classical flow:

Relativistic flow:

2. Vorticity

Definitions:

[L.P. Csernai, V.K. Magas, D.J. Wang,

PRC 87, 034906(2013)]

Weights:

+0

0+

+- ++

In [x,z] plane:

E^tot: total energy in a y layer

N^cell: total num. ptcls. In this y layer

Corner cells

More details:

In Reaction Plane t=0.17 fm/c

Vorticity @ LHC energy:

In Reaction Plane t=3.56 fm/c

In Reaction Plane t=6.94 fm/c

All y layer added up at t=0.17 fm/c

b5

All y layer added up at t=3.56 fm/c

b5

Average Vorticity in summary

Decrease with time

Bigger for more peripheral collision

Circulation:

Vorticity @ NICA , 9.3GeV:

Vorticity @ FAIR, 8 GeV

3, Rotation in an exact hydro model

Hydrodynamic basic equations

The variables:

Csorgo, arxiv: 1309.4390[nucl.-th]

Scaling variable:

cylindrical coordinates:

rhs:

More details:

y

lhs:

Expansion energy at the surface

Expansion energy at the longitudinal direction Rotational energy at the surface

Kinetic energy:

(α and β are independent of time)

s_ρM & s_yM:

Boundary of spatial integral

Internal energy:

The solution:

Solutions:

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

Energy time dependence:

Energy conserved !

decreasing internal energy and rotational energy

leads the increasing of

kinetic energy .

Smaller initial radius parameter

overestimates the radial expansion velocity

due to the lack of dissipation

Spatial expanding:

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.

Expansion Velocity:

Summary

• High initial angular momentum exist for periphe ral collisions and the presence of KHI is essential to generate rotation.

• Vorticity is significant even for NICA and FAIR en ergy.

• The exact model can be well realized with param

eters extracted from our PICR FD model

Table 2 : Time dependence of characteristic parameters of the exact fuid dynamical model.

Large extension in the beam direction is neglected.

α and β