# The Effect of the Vadasz Number on the Onset of Thermal Convection in Rotating Bidispersive Porous Media

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Linear Instability

#### 3.1. Steady Convection

#### 3.2. Oscillatory Convection

## 4. Numerical Results

- (1)
- analyze the asymptotic behaviour of ${R}_{O}$ with respect to ${\mathcal{T}}^{2}$ and J; and,
- (2)
- compare ${R}_{S}$ and ${R}_{O}$ to establish whether the convection arises through a steady state (stationary convection) or via an oscillatory state (oscillatory convection).

- (i)
- if ${\mathcal{T}}^{2}<{\mathcal{T}}_{c}^{2}$ or if $\{{\mathcal{T}}^{2}>{\mathcal{T}}_{c}^{2},\text{}J{J}_{c}\}$, then convection can only arise via a steady state;
- (ii)
- if $\{{\mathcal{T}}^{2}>{\mathcal{T}}_{c}^{2},\text{}J{J}_{c}\}$, convection can only arise via an oscillatory state.

## 5. Conclusions

- ${R}_{S}$ does not depend on the acceleration coefficient, i.e., inertial effects do not affect ${R}_{S}$;
- ${R}_{S}$ increases with the Taylor number, i.e., ${\mathcal{T}}^{2}$ has—as one is expected—a stabilizing effect on the onset of steady convection; and,

- ${R}_{O}$ is a decreasing function of J and there exists a threshold ${J}^{*}\in (0.31,0.32)$ for the inertia coefficient, such that ${R}_{O}$ exists and convection arises via an oscillatory state; and,
- ${R}_{O}$ is an increasing functions of ${\mathcal{T}}^{2}$ and there exists a threshold ${{\mathcal{T}}^{*}}^{2}$ for the Taylor number, such that, for ${\mathcal{T}}^{2}>{{\mathcal{T}}^{*}}^{2}$, the convection arises via an oscillatory state.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Asymptotic behaviour of ${R}_{O}$ with respect to J for $\{\gamma =0.8;\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}{\mathcal{T}}^{2}=10\}$.

**Figure 4.**Behaviour of ${R}_{S}$ and ${R}_{O}$ with respect to $\mathcal{T}$. The other parameters are set as $\{\gamma =0.8;\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}J=0.5\}$.

**Figure 5.**Behaviour of ${f}_{s}\left({a}^{2}\right)$ and ${f}_{o}\left({a}^{2}\right)$ for $\gamma =0.8;\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}J=1.5$ and ${\mathcal{T}}^{2}=30$.

**Figure 6.**Behaviour of ${f}_{s}\left({a}^{2}\right)$ and ${f}_{o}\left({a}^{2}\right)$ for $\gamma =0.8;\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}J=1.5$ and ${\mathcal{T}}^{2}=7.1$.

**Figure 7.**Plot of ${f}_{o}\left({a}^{2}\right)$ for $\gamma =0.8,\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}J=10$.

**Figure 8.**Plot of ${f}_{o}\left({a}^{2}\right)$ for $\gamma =0.8,\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}{\mathcal{T}}^{2}=10$.

**Table 1.**Critical threshold of J, from which ${R}_{O}$ exists and convection occurs via an oscillatory state in the case $\{\gamma =0.8,\text{}{k}_{r}=1.5,\text{}\eta =0.2,\text{}{\mathcal{T}}^{2}=10$}.

J | ${\mathit{a}}_{\mathit{o}}^{2}$ | ${\mathit{R}}_{\mathit{O}}$ |
---|---|---|

0 | ∄ | ∄ |

0.25 | ∄ | ∄ |

0.31 | ∄ | ∄ |

0.32 | 15.3410 | 51.9150 |

0.35 | 14.7812 | 59.7010 |

0.4 | 14.0459 | 49.0951 |

0.7 | 11.9923 | 44.4657 |

1 | 11.2881 | 42.7649 |

5 | 10.3196 | 40.0417 |

10 | 10.2433 | 39.7591 |

**Table 2.**Critical threshold of ${\mathcal{T}}^{2}$ from which ${R}_{O}$ exists and convection occurs via an oscillatory state in the case $\{\gamma =0.8,{k}_{r}=1.5,\eta =0.2,J=0.5$}.

${\mathcal{T}}^{2}$ | ${\mathit{a}}_{\mathit{s}}^{2}$ | ${\mathit{a}}_{\mathit{o}}^{2}$ | ${\mathit{R}}_{\mathit{S}}$ | ${\mathit{R}}_{\mathit{O}}$ |
---|---|---|---|---|

7 | 15.3031 | ∄ | 45.5293 | ∄ |

7.07 | 15.3234 | 12.5623 | 45.6906 | 46.0730 |

7.1 | 15.3320 | 12.5674 | 45.7596 | 46.0815 |

7.2 | 15.3603 | 12.5844 | 45.9885 | 46.1098 |

7.26 | 15.3769 | 12.5946 | 46.1253 | 46.1268 |

7.27 | 15.2796 | 12.5963 | 46.1480 | 46.1296 |

7.3 | 15.3878 | 12.6014 | 46.2162 | 46.1381 |

7.5 | 15.4410 | 12.6353 | 46.6677 | 46.1945 |

10 | 15.919 | 13.0503 | 51.9256 | 46.8876 |

20 | 16.1538 | 14.5717 | 67.9956 | 49.4695 |

50 | 15.2776 | 18.2109 | 95.5668 | 55.9909 |

$\mathit{\gamma}$ | ${\mathit{k}}_{\mathit{r}}$ | $\mathit{\eta}$ | ${\mathcal{T}}^{2}$ | J | ${\mathit{a}}_{\mathit{s}}^{2}$ | ${\mathit{a}}_{\mathit{o}}^{2}$ | ${\mathit{R}}_{\mathit{S}}$ | ${\mathit{R}}_{\mathit{O}}$ | CONVECTION |
---|---|---|---|---|---|---|---|---|---|

10 | 7 | 2 | 30 | 1.3 | 22.74 | 15.78 | 198.64 | 202.89 | STEADY |

10 | 7 | 2 | 100 | 1.3 | 37.07 | 21.08 | 437.66 | 289.54 | OSCILLATORY |

0.8 | 1.5 | 0.2 | 300 | 10 | 19.25 | 13.92 | 193.25 | 63.84 | OSCILLATORY |

0.8 | 0.5 | 0.2 | 100 | 1 | 21.27 | ∄ | 120.07 | ∄ | STEADY |

0.8 | 0.5 | 0.2 | 100 | 1.5 | 21.27 | 17.10 | 120.07 | 44.72 | OSCILLATORY |

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**MDPI and ACS Style**

Capone, F.; De Luca, R.
The Effect of the Vadasz Number on the Onset of Thermal Convection in Rotating Bidispersive Porous Media. *Fluids* **2020**, *5*, 173.
https://doi.org/10.3390/fluids5040173

**AMA Style**

Capone F, De Luca R.
The Effect of the Vadasz Number on the Onset of Thermal Convection in Rotating Bidispersive Porous Media. *Fluids*. 2020; 5(4):173.
https://doi.org/10.3390/fluids5040173

**Chicago/Turabian Style**

Capone, Florinda, and Roberta De Luca.
2020. "The Effect of the Vadasz Number on the Onset of Thermal Convection in Rotating Bidispersive Porous Media" *Fluids* 5, no. 4: 173.
https://doi.org/10.3390/fluids5040173