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Flow Vorticity and Rotation in Peripheral HIC

Dujuan Wang

2014 CBCOS, Wuhan, 11/05/2014

University of Bergen, Norway

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• Introduction

• Vorticity for LHC, FAIR & NICA

• Rotation in an exact hydro model

• Summary

Outline

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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)

4-flow:

energy-momentum tensor: ),(0

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pxfppp

pdT

),( jnN

PguuPeT )(

In Local Rest (LR) frame = (e, P, P, P);

For perfect fluid:

)1,1,1,1( diaggg

0]ˆ[

0]ˆ[

dT

dN

0,

0,

T

N

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tilted initial state, big initial angular momentum

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

[L.P.Csernai, V.K.Magas,H.Stoecker,D.D.Strottman, PRC 84,024914(2011)]

Flow velocity

Pressure gradient

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The rotation and Kelvin Helmholtz Instability (KHI)

[L.P.Csernai, D.D.Strottman, Cs.Anderlik, PRC 85, 054901(2012)]

More details in Laszlo’ talk

Straight line Sinusoidal wave for peripheral collisions

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Classical flow:

Relativistic flow:

2. Vorticity

The vorticity in [x,z]plane is considered.

Definitions:

[L.P. Csernai, V.K. Magas, D.J. Wang, PRC 87, 034906(2013)]

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Weights:

+0

0+

+++-

In [x,z] plane:

Etot: total energy in a y layerNcell: total num. ptcls. In this y layer

Corner cells

More details:

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In Reaction Plane t=0.17 fm/cVorticity @ LHC energy:

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

b5

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

b5

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Average Vorticity in summary

Decrease with timeBigger for more peripheral collisionViscosity 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 modelHydrodynamic basic equations

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The variables:

Csorgo, arxiv: 1309.4390[nucl.-th]Scaling variable:

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cylindrical coordinates:

rhs:

More details:

y

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lhs:

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Expansion energy at the surface

Expansion energy at the longitudinal direction

Rotational energy at the surface

For infinity case:

Kinetic energy:

(α and β are independent of time)

sρM & syM:Boundary of spatial integral

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Internal energy:

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The solution:

Runge-Kutta method: Solve first order DE initial condition for R and Y is needed, and the constants Q and W

Solutions:

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Table 1 : data extracted fromL.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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Smaller initial radius parameter

overestimates the radial expansion velocitydue to the lack of dissipation

Spatial expanding:

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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 ofrotation is less.

Expansion Velocity:

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Summary

Thank you for your attention!

• 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

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Table 2 : Time dependence of characteristic parameters ofthe exact fuid dynamical model. Large extension in the beam direction is neglected.

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α and β

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