# Thermal Convection of an Ellis Fluid Saturating a Porous Layer with Constant Heat Flux Boundary Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Rheological Model

#### 2.2. Modified Darcy’s Law

#### 2.3. Governing Equations

#### 2.4. Basic State

#### 2.5. Linear Stability Analysis

## 3. Asymptotic Analysis for Vanishing Wavenumber

## 4. Results and Discussion

## 5. Conclusions

- There exists a suitable variable transformation that yields a compact representation of the stability eigenvalue problem;
- The critical conditions hold always for $k=0$. The threshold values can be obtained entirely analytically due to an asymptotic analysis performed for $k\to 0$;
- The non-Newtonian character of the fluid plays a destabilizing effect on the convective flow, namely an increasing value of the Ellis number yields a destabilization of the basic flow;
- For $\mathrm{El}\to 0$, the Ellis index a does not affect the stability conditions and the results coincide with those for the limit of Newtonian fluid already available in the literature (${k}_{c}=0$ and ${\mathrm{R}}_{c}=12$);
- For large values of the Ellis number, the power-law behavior is recovered. This means that the critical Rayleigh number tends to zero.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the porous layer heated from below and cooled from above by equal constant heat fluxes with horizontal throughflow.

**Figure 3.**Comparison between asymptotic solution for $k\to 0$ (red dashed curves) and the neutral stability condition obtained numerically for $k={10}^{-3}$ (black continuous curves); $\mathrm{R}$ versus a.

**Figure 4.**Comparison between asymptotic solution for $k\to 0$ (red dashed curves) and the neutral stability condition obtained numerically for $k={10}^{-3}$ (black solid curves); $\mathrm{R}$ versus $\tilde{\mathrm{El}}$.

**Figure 5.**Critical $\mathrm{R}$ as a function of the basic pressure gradient for different values of $\mathrm{El}$ for $a=0.2$ and $a=0.8$.

**Table 1.**Neutral stability values of $\mathrm{R}$ for small values of k for the Newtonian limit $\mathrm{El}\to 0$.

k | $\mathbf{R}$ |
---|---|

$0.1$ | 12.0114 |

$0.01$ | 12.0001 |

$0.001$ | 12.0000 |

0 | 12 |

$\tilde{\mathbf{El}}$ | $\mathit{a}=0.8$ | $\mathit{a}=0.6$ | $\mathit{a}=0.4$ | $\mathit{a}=0.2$ |
---|---|---|---|---|

0 | 12 | 12 | 12 | 12 |

$0.1$ | 10.666667 | 10.285714 | 9.6 | 8 |

$0.5$ | 7.3846154 | 6.5454545 | 5.3333333 | 3.4285714 |

1 | 5.3333333 | 4.5 | 3.4285714 | 2 |

10 | 0.88888889 | 0.67924528 | 0.46153846 | 0.23529412 |

100 | 0.095238095 | 0.071570577 | 0.047808765 | 0.023952096 |

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**MDPI and ACS Style**

Brandão, P.V.; Celli, M.; Barletta, A.; Lazzari, S.
Thermal Convection of an Ellis Fluid Saturating a Porous Layer with Constant Heat Flux Boundary Conditions. *Fluids* **2023**, *8*, 54.
https://doi.org/10.3390/fluids8020054

**AMA Style**

Brandão PV, Celli M, Barletta A, Lazzari S.
Thermal Convection of an Ellis Fluid Saturating a Porous Layer with Constant Heat Flux Boundary Conditions. *Fluids*. 2023; 8(2):54.
https://doi.org/10.3390/fluids8020054

**Chicago/Turabian Style**

Brandão, Pedro Vayssière, Michele Celli, Antonio Barletta, and Stefano Lazzari.
2023. "Thermal Convection of an Ellis Fluid Saturating a Porous Layer with Constant Heat Flux Boundary Conditions" *Fluids* 8, no. 2: 54.
https://doi.org/10.3390/fluids8020054