# New, Spherical Solutions of Non-Relativistic, Dissipative Hydrodynamics

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## Abstract

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## 1. Introduction

## 2. Navier–Stokes Equations of Non-Relativistic, Viscous Hydrodynamics

## 3. Temperature Dependence of the Speed of Sound

## 4. Scale and the Continuity Equations for Spherically Symmetric Hubble Flow

## 5. New, Exact Solutions for a Generic, Temperature-Dependent Speed of Sound

## 6. Solutions for a Temperature-Independent Pressure to Energy Density Ratio

#### 6.1. Analytic Solutions for a Spatially Homogeneous Pressure Distribution

#### 6.2. Analytic Solutions for a Spatially Inhomogeneous Pressure Profile

#### 6.3. Discussion: Attractor Behaviour in Other Hydrodynamical Solutions

## 7. Asymptotically Perfect Fluid Behaviour

#### 7.1. Asymptotic Analysis for the Case of a Homogeneous Pressure and Linear Bulk Viscosity

#### 7.2. Asymptotic Analysis for an Inhomogeneous Pressure and Linear Bulk Viscosity

## 8. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The time evolution of the temperature in the centre of the fireball ($s=0$). The solid black line corresponds to a perfect fluid solution, while the coloured lines correspond to our new, viscous solution of non-relativistic Navier–Stokes equation with a homogeneous pressure field for different values of initial kinematic bulk viscosities. In the left panel, we set the same initial temperatures, but in the right panel, the curves start from different initial conditions, and each of them approach the solid black line, the perfect fluid asymptote.

**Figure 2.**For a fixed asymptotic solution with fixed ${T}_{0}^{A}$, the initial value of the kinematic bulk viscosity is a function of ${T}_{0}$, and in this figure, the initial time is scaled out. This non-monotonic behaviour is described by Equation (48), and the maximum of the curve is given by Equation (51).

**Figure 3.**The evolution of the $R\left(t\right)$ scale of the fireball (

**left**), its time derivative $\dot{R}\left(t\right)$ (

**center**), and the temperature (

**right**) as a function of time for an exact solution of the non-relativistic Navier-Stokes equations for fixed ${T}_{0}$ = 250 MeV, ${R}_{0}$ = 5 fm and ${\dot{R}}_{0}$ = 0 initial parameters. We assume a nuclear fluid here with m = 940 MeV particle mass and a constant, temperature-independent κ parameter: κ = 3. The solid black line stands for a perfect fluid solution, while the dashed blue, the dotted--dashed green, the dashed yellow, and the dotted--dashed red lines correspond to our new viscous solution of non-relativistic Navier-Stokes equations for different values of ${\zeta}_{0}/{p}_{0}$ but for the same initial conditions.

**Figure 4.**The evolution of the $R\left(t\right)$ scale of the fireball (

**left**), the $\dot{R}\left(t\right)$ scale velocity (

**center**), and the temperature (

**right**) as a function of time for the solution of the non-relativistic Navier-Stokes equations for ${T}_{0}^{A}$ = 250 MeV, ${R}_{0}^{A}$ = 5 fm, and ${\dot{R}}_{0}^{A}$ = 0 initial parameters, utilising an m = 940 MeV for the particle mass and a constant, temperature-independent κ = 3. The solid black line stands for a perfect fluid solution, and this perfect fluid curve labelled by zero bulk viscosity is approached by each of the shown exact viscous solutions asymptotically, $T\left(t\right)$ ~ ${T}_{A}\left(t\right)$. The dashed blue, the dotted--dashed green, the dashed yellow, and the dotted--dashed red lines correspond to our new viscous solution of non-relativistic Navier-Stokes equations for different values of ${\zeta}_{0}/{p}_{0}$, but for the same asymptotic solutions.

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**MDPI and ACS Style**

Kasza, G.; Csernai, L.P.; Csörgő, T.
New, Spherical Solutions of Non-Relativistic, Dissipative Hydrodynamics. *Entropy* **2022**, *24*, 514.
https://doi.org/10.3390/e24040514

**AMA Style**

Kasza G, Csernai LP, Csörgő T.
New, Spherical Solutions of Non-Relativistic, Dissipative Hydrodynamics. *Entropy*. 2022; 24(4):514.
https://doi.org/10.3390/e24040514

**Chicago/Turabian Style**

Kasza, Gábor, László P. Csernai, and Tamás Csörgő.
2022. "New, Spherical Solutions of Non-Relativistic, Dissipative Hydrodynamics" *Entropy* 24, no. 4: 514.
https://doi.org/10.3390/e24040514