# Viscous anisotropic hydrodynamics for the Gubser flow

## Abstract

In this work we describe the dynamics of a highly anisotropic system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion (Gubser flow) for arbitrary shear viscosity to entropy density ratio. We derive the equations of motion of dissipative anisotropic hydrodynamics by applying to this situation the moments method recently derived by Moln´ar et al. (MNR) [1, 2], based on an expansion around an arbitrary anisotropic one-particle distribution function. One requires an additional evolution equation in order to close the conservation laws. This is achieved by selecting the relaxation equation for the longitudinal pressure with a suitable Landau matching condition. As a result one obtains two coupled differential equations for the energy density and the longitudinal pressure which respect the *SO*(3)q Ⓧ *SO*(1, 1) Ⓧ *Z* _{2} symmetry of the Gubser flow in the deSitter space. These equations are solved numerically and compared with the predictions of the recently found exact solution of the relaxation-time-approximation Boltzmann equation subject to the same flow. We also compare our numerical results with other fluid dynamical models. As a result we observe that the MNR description of anisotropic fluid dynamics reproduces the space-time evolution of the system than all other currently known hydrodynamical approaches.

- Authors:

- North Carolina State Univ., Raleigh, NC (United States); The Ohio State Univ., Columbus, OH (United States)
- The Ohio State Univ., Columbus, OH (United States)

- Publication Date:

- Research Org.:
- North Carolina State Univ., Raleigh, NC (United States); The Ohio State Univ., Columbus, OH (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC) Office of Nuclear Physics

- OSTI Identifier:
- 1502397

- Grant/Contract Number:
- [FG02-03ER41260; SC0004286]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Nuclear Physics. A

- Additional Journal Information:
- [ Journal Volume: 967; Journal Issue: C]; Journal ID: ISSN 0375-9474

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; relativistic heavy-ion collisions; quark-gluon plasma; anisotropic hydrodynamics; Boltzmann equation; viscous fluid dynamics

### Citation Formats

```
Martinez, M., McNelis, M., and Heinz, U. Viscous anisotropic hydrodynamics for the Gubser flow. United States: N. p., 2017.
Web. doi:10.1016/j.nuclphysa.2017.04.012.
```

```
Martinez, M., McNelis, M., & Heinz, U. Viscous anisotropic hydrodynamics for the Gubser flow. United States. doi:10.1016/j.nuclphysa.2017.04.012.
```

```
Martinez, M., McNelis, M., and Heinz, U. Mon .
"Viscous anisotropic hydrodynamics for the Gubser flow". United States. doi:10.1016/j.nuclphysa.2017.04.012. https://www.osti.gov/servlets/purl/1502397.
```

```
@article{osti_1502397,
```

title = {Viscous anisotropic hydrodynamics for the Gubser flow},

author = {Martinez, M. and McNelis, M. and Heinz, U.},

abstractNote = {In this work we describe the dynamics of a highly anisotropic system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion (Gubser flow) for arbitrary shear viscosity to entropy density ratio. We derive the equations of motion of dissipative anisotropic hydrodynamics by applying to this situation the moments method recently derived by Moln´ar et al. (MNR) [1, 2], based on an expansion around an arbitrary anisotropic one-particle distribution function. One requires an additional evolution equation in order to close the conservation laws. This is achieved by selecting the relaxation equation for the longitudinal pressure with a suitable Landau matching condition. As a result one obtains two coupled differential equations for the energy density and the longitudinal pressure which respect the SO(3)q Ⓧ SO(1, 1) Ⓧ Z2 symmetry of the Gubser flow in the deSitter space. These equations are solved numerically and compared with the predictions of the recently found exact solution of the relaxation-time-approximation Boltzmann equation subject to the same flow. We also compare our numerical results with other fluid dynamical models. As a result we observe that the MNR description of anisotropic fluid dynamics reproduces the space-time evolution of the system than all other currently known hydrodynamical approaches.},

doi = {10.1016/j.nuclphysa.2017.04.012},

journal = {Nuclear Physics. A},

number = [C],

volume = [967],

place = {United States},

year = {2017},

month = {9}

}

#### Figures / Tables:

_{$L$}-matching [1, 2,more »

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*