1. 2010-4-18 The effects of viscosity on hydrodynamical evolution of QGP 苏中乾 大连理工大学 Dalian University of Technology

2. 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

3. 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.

4. Viscous hydrodynamics Energy-momentum tensor decomposition: Energy density in fluid rest frame Pressure in fluid rest frame, Bulk viscous pressure Shear stress tensor, Symmetrized tensor Symmetrized, traceless spatial projection Charge conservation Energy-momentum conservation The second law of thermodynamics Starting point : conservation laws

5. Viscous hydrodynamics Energy-momentum tensor conservation: Thinking about zero baryon system, we can i gnore the effects of vorticity ( ) and heat conductivity of flow ( ) Relaxation equations for shear stress tensor and bulk pressure Second-order Israel-Stewart equation Navier-Stokes (NS) equations: simple, but: unstable and acausal equations of motion

6. Viscous hydrodynamics With, viscous hydrodynamics reduces to ideal hydrodynamics Use Bjorken picture and thinking about azimuthally symmetric systems: (Boost invariant) Notice: once we know and, velocity and the temperature at will be Obtained by Lorentz boost:

7. viscous hydrodynamics equations Close equations with EOS 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

8. 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) Second ： Equation of state for massless quarks and gluons gas where (the number of quark flovers) A. Muronga & D. Rischke; nucl-th/0407114v2

9. The result equation of state EOS results The Solid line: The dashed line: quark and gluon gas equation of state.

10. The initial conditions The components of the shear tensor and bulk pressure maximum energy density for central collisions at initial time initial transverse radius of source. 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

11. Viscous vs. ideal hydrodynamics——Temperature and velocity For ideal hydrodynamic equation the temperature cool faster than that the shear viscosity are included, this means viscous effect lead to slow down of expansion of system.

12. 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

13. Our results VS A.Muronga & D. Rischke A.Muronga & D. Rischke nucl-th/0407114v2 Ideal quark &gluon gas EOS Cross over EOS and woods-saxon distribution

14. 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 T f have a larger space until a certain time and then it will be a little smaller. 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).

15. 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

16. Thank you!