# Maximum Entropy Production Is Not a Steady State Attractor for 2D Fluid Convection

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## Abstract

**:**

## 1. Introduction

## 2. Model System

## 3. Results

## 4. Discussion

## 5. Further Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BC | Boundary condition |

LBM | Lattice Boltzmann Model |

MEP | Maximum Entropy Production |

## References

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**Figure 1.**Model system diagram showing the various heat fluxes that comprise the boundary energy balances. Boundaries are periodic in the horizontal direction and enforce the no-slip velocity condition.

**Figure 2.**Macroscopic transport properties of the model system. The boundary temperatures ${T}_{a}$ (red dot–dash line) and ${T}_{b}$ (blue dashed line) and dimensionless entropy production $\overline{\sigma}$ (solid black line) are plotted as a function of the normalized heat flux $\overline{{J}_{ab}}$.

**Figure 3.**Normalized steady state heat flux $\overline{{J}_{ab}}$ as a function of fluid viscosity ν and thermal diffusivity χ. The grey plane represents the heat flux value corresponding to maximum entropy production (MEP).

**Table 1.**Mean Rayleigh numbers for each Lattice Boltzmann Model parameter combination (velocity distribution relaxation parameter ${\tau}_{\nu}$ and internal energy distribution relaxation parameter ${\tau}_{c}$).

${\mathit{\tau}}_{\mathit{\nu}}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 | 1.00 | 1.05 | ||

${\tau}_{c}$ | 0.55 | 4.7 $\times {10}^{4}$ | 3.9 $\times {10}^{4}$ | 3.2 $\times {10}^{4}$ | 2.8 $\times {10}^{4}$ | 2.5 $\times {10}^{4}$ | 2.2 $\times {10}^{4}$ | 2.0 $\times {10}^{4}$ | 1.8 $\times {10}^{4}$ |

0.60 | 2.1 $\times {10}^{4}$ | 1.7 $\times {10}^{4}$ | 1.4 $\times {10}^{4}$ | 1.2 $\times {10}^{4}$ | 1.1 $\times {10}^{4}$ | 9.7 $\times {10}^{3}$ | 8.8 $\times {10}^{3}$ | 8.1 $\times {10}^{3}$ | |

0.65 | 1.2 $\times {10}^{4}$ | 1.0 $\times {10}^{4}$ | 8.5 $\times {10}^{3}$ | 7.4 $\times {10}^{3}$ | 6.6 $\times {10}^{3}$ | 6.0 $\times {10}^{3}$ | 5.4 $\times {10}^{3}$ | 5.0 $\times {10}^{3}$ | |

0.70 | 8.5 $\times {10}^{3}$ | 7.0 $\times {10}^{3}$ | 6.0 $\times {10}^{3}$ | 5.2 $\times {10}^{3}$ | 4.7 $\times {10}^{3}$ | 4.2 $\times {10}^{3}$ | 3.8 $\times {10}^{3}$ | 3.5 $\times {10}^{3}$ | |

0.75 | 7.0 $\times {10}^{3}$ | 5.3 $\times {10}^{3}$ | 4.5 $\times {10}^{3}$ | 4.0 $\times {10}^{3}$ | 3.6 $\times {10}^{3}$ | 3.2 $\times {10}^{3}$ | 3.0 $\times {10}^{3}$ | 2.8 $\times {10}^{3}$ | |

0.80 | 5.1 $\times {10}^{3}$ | 4.2 $\times {10}^{3}$ | 3.6 $\times {10}^{3}$ | 3.2 $\times {10}^{3}$ | 2.9 $\times {10}^{3}$ | 2.6 $\times {10}^{3}$ | 2.4 $\times {10}^{3}$ | 2.3 $\times {10}^{3}$ |

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**MDPI and ACS Style**

Bartlett, S.; Virgo, N.
Maximum Entropy Production Is Not a Steady State Attractor for 2D Fluid Convection. *Entropy* **2016**, *18*, 431.
https://doi.org/10.3390/e18120431

**AMA Style**

Bartlett S, Virgo N.
Maximum Entropy Production Is Not a Steady State Attractor for 2D Fluid Convection. *Entropy*. 2016; 18(12):431.
https://doi.org/10.3390/e18120431

**Chicago/Turabian Style**

Bartlett, Stuart, and Nathaniel Virgo.
2016. "Maximum Entropy Production Is Not a Steady State Attractor for 2D Fluid Convection" *Entropy* 18, no. 12: 431.
https://doi.org/10.3390/e18120431