# Presentation: 282490

## Non-Newtonian nature of Causal Hydrodynamics

- T. Koide
- (Universidade Federal do Rio de Janeiro)

**G.S. Denicol (UFRJ),T. Kodama (UFRJ),Ph. Mota (UFRJ)**

- Because of causality,  the relativistic dissipative fluid will be
- a non-Newtonian fluid. Thus
- The GKN formula should be modified.
- 1/4can be a lower bound of the shear of Newtonian fluids.
- The fluid expands to vacuum by forming a stationary wave.
- The additional viscosity is still necessary stabilize solutions.

## Infinite speed in diffusion process

- Random walk

## Infinite speed in diffusion process

- Dispersion relation

## Relativistic NR equation(1+1)

- At             , the dispersion relation is reduced to the diffusion type.

- >0.  stable

- Linear perturbation analysis

- Hiscock&Lindblom(‘85)

## Introduction of memory effect

- Equation of continuity

- Fick’s law

- Acausal (Cattaneo)
- Breaking sum rules (Kadanoff&Martin, T.K.)

## Landau-Lifshitz theory 

- Landau-Lifshitz theory

- Equation of Continuity

- heat current

- bulk viscosity

- shear viscosity

## Introduction of Memory Effect

- T.K.,Denicol,Mota&Kodama(2007)

- Entropy four current

**Landau-Lifshitz**

**Memory effect**

**Causal Hydrodynamics**

## Dispersion relation(1+1)

- Stable in the sense of the linear analysis.

- blue and red

- Speed of light

## Linear analysis of stability

- Hiscock& Lindblam (‘85), Kouno,Maruyama,Takagi&Saito (‘90)

- The Eckert theory is more unstable than the LL theory.

## Stability around scaling solution

## Newtonian fluid

**Newtonian fluid**

**Memory effect**

**Causal Hydrodynamics**

**Landau-Lifshitz**

## Anomalous viscosity

**Newtonian**

**Dilatant**

**Thixotropic**
**Rheopectic **

**Pseudoplastic**

**Bingham flow**

**Velocity gradient**

**Viscosity**

- sludge, paint,
- blood, ketchup

- latex, paper pulp, clay solns.

- quicksand,
- candy compounds

- tars, inks,
- glue

- Navier-Stokes

- QGP ?

## Common sense?

- The transport coefficients are given by the Green-Kubo-Nakano formula.
- The lower bound of the shear viscosity of the CD hydro. is given by             .
- The viscosity is small correction.
- We do not need any additional viscosity in the CD hydro.

- Not trivial !!

## Linear response theory

- system

- Hamiltonian

- External perturbation

## Green-Kubo-Nakano formula

- Exact!

- In Newtonian fluid, transport coefficients are defined by

- We ignore the memory effect

- GKN formula

## D=?

- Newtonian

- Non-Newtonian

## Markov app.

## Generalization of GKN formula

-      In the generalized formula, we need        and
- 2.       characterizes the deviation from the GKN formula.
- When      vanishes in the low momentum limit,
-     the new formula reproduces the GKN formula.
- 4. The result obtained in the new formula is consistent with
-     sum rules.

- T.K.&Maruyama (2004), T.K.(2005), T.K.(2007), T.K.&Kodama (2008)

- GKN formula

- New formula

## Lower limit of shear viscosity?

- 1) GKN formula

- 2) Absorption cross section

- G: Newton’s constant

- Kovtun,Son&Starinets(‘05)

## Lower limit of shear viscosity?

- 1) GKN formula

- 2) Absorption cross section

- G: Newton’s constant

**Only for Newtonian!!**

- Kovtun,Son&Starinets(‘05)

## Common sense?

- The transport coefficients are given by the Green-Kubo-Nakano formula.
- The lower bound of the shear viscosity of the CD hydro. is given by             .
- The viscosity is small correction.

## In the following calculations, 

- we consider only the 1+1 dimensional fluid.

## Expansion of fluid

- It seems to be
- a stationary wave.

## Universal relation between pressure and viscosity 

- We assume that the fluid forms a stationary wave at the
- boundary to vacuum. Then at the boundary,

- pressure

- (-1) * viscosity

- The causal fluid expands as a stationary wave.
- Viscosity can be the same order as energy density and pressure.

## Common sense?

- The transport coefficients are given by the Green-Kubo-Nakano formula.
- The lower bound of the shear viscosity of the CD hydro. is given by             .
- The viscosity is small correction.
- We do not need any additional viscosity in the CD hydro.

## Numerical oscillation

- The frequency is the size of the grid.

- unstable

## Usual “artificial” viscosity

- Navier-Stokes type:

- Burnett type:

- Neumann-Richtmyer type:

- Generalization to relativistic cases ?

- Causal?

## Additional causal viscosity

- We need to introduce the artificial viscosity consistent with
- causality.

- :the size of the grid   (0.01 fm)

## Slide 30

## Slide 31

## Effect to entropy production

- The effect of
- the additional viscosity is

- This is the same order of
- the breaking of
- the energy conservation
- in our code.

## Summary

- The transport coefficients are given by the Green-Kubo-Nakano formula.
- The lower bound of the shear viscosity of the CD hydro. is given by             .
- The viscosity is small correction.
- We do not need any additional viscosity in the CD hydro.

- No. We need a new formula
- for non-Newtonian fluids

- Not yet clear for
- causal dissipative hydro.

- We need.

- No. We have a relation
- and forms a stationary wave.

## We have to check

- Is the new formula really useful or not.
- What is the lower bound of the causal shear viscosity?
- CD hydro is a stable theory?
- Are numerical oscillations we encountered  due to the problem of numerical calculation or the problem of the theory (turbulence)?

## That is, the causal hydrodynamics 

- will be one of non-Newtonian fluids!!

- Newtonian

- Non-Newtonian

## Stability and Causality(1+1)

- Linear perturbation analysis of the causal hydro.

- Sound velocity is faster than ideal.

- Stable, Independent of momentum

- T.K. Denicol,Kodama&Mota (‘07)

## It is difficult to cut high momentum modes to avoid unstable modes.

- We need to find a new theory to satisfy causality.

- Model equation of hydro

- assumption

- Non-linear effect

## Additional viscosity