Fluid Dynamics & Viscosity

1. Fluid Dynamics & Viscosity EU Hadron Physics I3 Report Frascati, May ’07, J. Aichelin & L.P. Csernai

2. In superstring theory, „based on analogy between black hole physics and equilibrium thermodynamics, ... there exist solutions called black branes, which are black holes with translationally invariant horizons. ... these solutions can be extended to hydrodynamics, ... and black branes possess hydrodynamic characteristics of ... fluids: viscosity, diffusion constants, etc.” In this model the authors concluded that η / s = 1 / 4π And then they „speculate” that in general η / s >1 / 4πorη / s > 1. They argue that this is a lower limit especially for such strongly interacting systems where up to now there is no reliable estimate for viscosity, like the QGP. According to the authors the viscosity of QGP must be lower than that of classical fluids.

3. <=1 [Kovtun, et al., PRL 2005]

4. Why viscosity ? • Viscosity leads to dissipation, it slows down, weakens, averages out flow phenomena, - helps to approach global equilibrium in a system. (No problem, the flow is present and strong in HI physics  viscosity is not too strong to eliminate the flow !!! ) • Viscosity prevents random, irregular, turbulent motion and instabilities, makes the flow LAMINAR, i.e. well defined and coherent, while too low viscosity leads to instabilities and turbulence. (Yes, flow seems to be regular and systematic in HI reactions  viscosity is large enough !!!) • Strictly speaking: Perfect fluid is absolutely unstable ! [LL]

5. RT – instabilities in Tokamak • The figure above shows three-dimensional isosurfaces of the pressure as the instability develops along ridges dominantly aligned along the ambient magnetic field.

6. Stability, Reynolds number - kinematic viscosity In an ideal fluid any small perturbation increases and leads to turbulentflow.For stability sufficiently largeviscosity and/orheat conductivity are needed! Re  1000 - 2000 (Calculations are also stabilized by numerical viscosity!) - density - viscosity - length - velocity Measured [Lijuan Ruan / STAR] scaling [A, E] of dimensionless v2 fluctuations , can be compared to constant Re contours. If the two are similar viscous effects are dominant in these fluctuations, and viscosity (or Re) can be extracted. (If not, Initial State (IS) fluctuations and FD fluctuations should be separated: complex theroretical task.)

7. Re – studies in HICs Theoretical [D. Molnar, U. Heinz, et al., ] η = 50 – 500 MeV/fm2c Re  10 – 100 Exp.: 50 – 800 MeV/nucleon energies 80’s[Bonasera, Schurmann, Csernai] scaling analysis of flow parameters. Re  7 – 8 !(more dilute, more viscous matter) In both cases η/s  1 (0.5 – 5) ,This is a value large enough to keep the flow laminar in Heavy Ion Collisions !!! NS – Star-quakes / Spin-up of rotation is observed  ηfinite

8. Stability, Reynolds number Interesting and important: in RFD detonationfronts are stabilized by radiation and heat conductivity. E.g. :- Rocket propulsion- Implosion,fission- and fusion reactions- Heavy Ion reactions

9. Preventing turbulence The instability of deflagration- (flame-) front is not desirable at supersonic fronts. With increasing temperature the radiation becomes dominant and stabilizes the flame front.

10. (Kovtun, et al., PRL 2005) • Csernai, Kapustaand McLerran have studied these results and assumptions and found that : • η vs. T has a typical decreasing and then increasing behaviour, due to classical reasons (Enskog’21) • η/s has a minimum exactly at the critical point in systems, which have a liquid-gas type of transition • η vs. Tshows a characteristic behaviour in all systems near the critical point (not only in the case of He).

11. Viscosity vs. T has a minimum at the 1st order phase transition. This might signal the phase transition if viscosity is measured. At lower energies this was done.

12. Viscosity – Momentum transfer Via VOIDS Via PARTICLES Liquid Gas

13. [Prakash, Venugopalan, .] Helium (NIST) QGP (Arnold, Moore, Yaffe) This phenomenon can help us to detect experimentally the critical point: η can be determined from (i) fluctuation of flow parameters and from (ii) scaling properties of flow parameters. Water (NIST)

14. Viscosity at the Frankfurt meeting • Formulation of dissipative processes in RFD, the physical processes governing these processes and their stabilityL. Turko, A.K. Chaduri, T. Koide, P. Van, A. Muronga, .. • Fluid dynamical models and the role of viscosity, numerical and parametric, and lack of viscosity in really perfect fluid, ANALYTIC solutionsI. Mishustin, T. Csorgo, M. Nagy, M. Chojnacki, U. Ornik, D. Strottman, I. Arsene, M. Gyulassy, B. Betz, E. Molnar, G. Denicol, P. Mota, … • Other subjects, important but not strongly related to viscosity, like freeze-out, initial states, role of EoS, … • All these are needed for experimental study of Vicsosity.

15. 1st measurement of viscosity • Roy Lacey et al. : • C. Nonaka – adiabatic expansion goes close to the CEP. • Low viscosity point can be reached in expansion • Using – Cs. & Cs.’s perfect FD model results, Tc, m.f.p, Cs, - η was estimated as η/s=0.09

16. The prediction is strongly based on model FD estimates. This is unavoidable both in AE scaling in flow analyses and in v2 value estimates. Thus, reliable 3d CRFD models are vital with known physical (parametric) and numerical (computational) viscosity.

17. Numerical viscosity • Shock test: asymptotic states are exactly known from an EoS. • Stationary shock profile should develop in a stable CRFD model Ls M. Chojnacki – Mathematica [WR]

18. Other questions related to numerical viscosity • Dissipation and entropy production may be eliminated by brute force in CFD. However, • Coarse graining, (finite cell size) eliminates SHORT wave-length flow patterns by averaging these out. The energy of these should be dissipated and transformed to heat. • If entropy increase is prevented (or forced to a given value), due to energy conservation, this extra energy is returned to the fluid in the form of LONG wavelength fluid motion! (see talk of Gabriel Denicol or instabilities in T. Mota’s talk.) • This can only be realistic if the physical process is such, that - short wavelength fluctuations die out due to some other reason - this other process is well represented by the method..