1. Flow Vorticity and Rotation in Peripheral HIC Dujuan Wang 1 2014 CBCOS, Wuhan, 11/05/2014 University of Bergen, Norway

2. Introduction Vorticity for LHC, FAIR & NICA Rotation in an exact hydro model SummaryOutline 2

3. 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 3

4. 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: In Local Rest (LR) frame = (e, P, P, P); For perfect fluid: 4

5. 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 5

6. The rotation and Kelvin Helmholtz Instability (KHI) [L.P.Csernai, D.D.Strottman, Cs.Anderlik, PRC 85, 054901(2012)] 6 More details in Laszlo’ talk Straight line  Sinusoidal wave for peripheral collisions

7. 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)] 7

8. 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: 8

9. In Reaction Plane t=0.17 fm/c Vorticity @ LHC energy: 9

10. In Reaction Plane t=3.56 fm/c 10

11. In Reaction Plane t=6.94 fm/c 11

12. All y layer added up at t=0.17 fm/c b5 12

13. All y layer added up at t=3.56 fm/c b5 13

14. Average Vorticity in summary Decrease with time Bigger for more peripheral collision Viscosity damps the vorticity 14

15. Circulation: 15

16. Vorticity @ NICA, 9.3GeV: 16

17. Vorticity @ FAIR, 8 GeV 17

18. 3, Rotation in an exact hydro model Hydrodynamic basic equations 18

19. The variables: Csorgo, arxiv: 1309.4390[nucl.-th] Scaling variable: 19

20. cylindrical coordinates: rhs: More details: y 20

21. lhs: 21

22. 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) s ρM & s yM : Boundary of spatial integral 22

23. Internal energy: 23

24. The solution: Runge-Kutta method: Solve first order DE initial condition for R and Y is needed, and the constants Q and W Solutions: 24

25. 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 25

26. Energy time dependence: Energy conserved ! decreasing internal energy and rotational energy leads the increasing of kinetic energy. 26

27. Smaller initial radius parameter overestimates the radial expansion velocity due to the lack of dissipation Spatial expanding: 27

28. 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: 28

29. 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 29

30. 30

31. Table 2 : Time dependence of characteristic parameters of the exact fuid dynamical model. Large extension in the beam direction is neglected. 31

32. α and β 32

33. 33