# Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Linear Stability Analysis

#### 3.1. Stationary Convection:

#### 3.2. Oscillatory Convection

## 4. Discussion

## 5. Conclusions

- ${R}_{{T}_{sc}}^{c}$ and ${R}_{{T}_{oc}}^{c}$ increase as the Taylor number increases, indicating that $Ta$ has a stabilizing effect on the onset of convection.
- ${R}_{{T}_{sc}}^{c}$ and ${R}_{{T}_{oc}}^{c}$ are increasing functions of ${R}_{c}$ and decreasing functions of ${\kappa}_{r}$.
- S does not show any effect on ${R}_{{T}_{oc}}^{c}$, as ${R}_{{T}_{oc}}^{c}$ is independent of S.
- There exists a threshold ${R}_{c}^{*}\in (0.45,0.46)$ for the solute Rayleigh number such that, if ${R}_{c}>{R}_{c}^{*}$, then the convection arises via an oscillatory mode.
- The oscillatory convection sets in and, as soon as the value of S attains a critical value (∈(0.6, 0.7)), the convection ceases to be oscillatory, and stationary convection occurs as the first bifurcation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${C}_{a}$ | Acceleration coefficient |

${\kappa}_{f}$ | Permeability in macro pores |

${\kappa}_{p}$ | Permeability in micro pores |

$\zeta $ | Interaction coefficient |

$\mu $ | Fluid viscosity |

g | Gravity |

$\alpha $ | Coefficient of thermal expansion |

${\alpha}_{c}$ | Density coefficient for salinity |

$\sigma $ | Heat capacity ratio |

$\u03f5$ | Macro porosity |

$\delta $ | Micro porosity |

$\rho $ | Density |

${k}_{s}$ | Thermal conductivity of the solid |

${k}_{f}$ | Thermal conductivity of the fluid |

${\left(\rho c\right)}_{s}$ | Product of density and specific heat in the solid skeleton |

${\left(\rho c\right)}_{f}$ | Product of density and specific heat in the pores |

${\rho}_{0}$ | Reference density |

${k}_{m}$ | Thermal conductivity |

${P}^{f}$ | Pressure in macro pores |

${P}^{p}$ | Pressure in micro pores |

T | Temperature |

C | Salt concentration field |

R | Rayleigh number |

${R}_{C}$ | Solutal Rayleigh number |

$Ta$ | Taylor number |

$Le$ | Lewis number |

S | Soret number |

d | Length |

Superscripts | |

′ | Perturbated quantity |

c | Critical value |

Subscripts | |

b | Base state |

0 | Reference valve |

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**Figure 2.**Neutral curves for the different values of $Ta$ and for the fixed values of $\gamma =0.5,\eta =0.2,{\kappa}_{r}=1,$${R}_{C}=50$, and $S=0.5$ for the stationary mode.

**Figure 3.**Neutral curves for the different values of S and for the fixed values of $\gamma =0.5$, $\eta =0.2$, ${\kappa}_{r}=1$, ${R}_{C}=50$, and $Ta=20$ for the stationary mode.

**Figure 4.**Neutral curves for the different values of ${R}_{C}$ and for the fixed values of $\gamma =0.5,\eta =0.2,$${\kappa}_{r}=1,S=0.2$, and $Ta=50$ for the stationary mode.

**Figure 5.**Neutral curves for the different values of ${\kappa}_{r}$ and for the fixed values of $\gamma =0.5,\eta =0.2,$${R}_{C}=50,S=0.2$, and $Ta=50$ for the stationary mode.

**Figure 6.**Neutral curves for the different values of $Ta$ and for the fixed values of $\gamma =0.5,\eta =0.2,$${R}_{C}=50$, and ${\kappa}_{r}=1$ for the oscillatory mode.

**Figure 7.**Neutral curves for the different values of ${R}_{C}$ and for the fixed values of $\gamma =0.5,\eta =0.2,$$Ta=50$, and ${\kappa}_{r}=1$ for the oscillatory mode.

**Figure 8.**Neutral curves for the different values of ${\kappa}_{r}$, and for the fixed values of $\gamma =0.5,\eta =0.2,Ta=50$ and ${R}_{C}=100$ for the oscillatory mode.

**Table 1.**Critical stationary and oscillatory Rayleigh numbers for different values of $Rc$ and the fixed values of ${\kappa}_{r}=1$, $Ta=5$, and $S=0.5$.

Rc | Stationary R | Stationary a | Oscillatory R | Oscillatory a | Instability |
---|---|---|---|---|---|

0 | 61.6464 | 3.9578 | 62.7612 | 3.9578 | Stationary |

1 | 62.1464 | 3.9578 | 62.7793 | 3.9578 | Stationary |

2 | 62.6464 | 3.9578 | 62.7973 | 3.9578 | Stationary |

3 | 63.1464 | 3.9578 | 62.8154 | 3.9578 | Oscillatory |

4 | 63.6464 | 3.9578 | 62.8335 | 3.9578 | Oscillatory |

5 | 64.1464 | 3.9578 | 62.8516 | 3.9578 | Oscillatory |

**Table 2.**Critical stationary and oscillatory Rayleigh numbers for the different values of S and the fixed values of ${\kappa}_{r}=1$, $Ta=50$, and $Rc=50$.

S | Stationary R | Stationary a | Oscillatory R | Oscillatory a | Instability |
---|---|---|---|---|---|

0.1 | 1018.7706 | 8.2527 | 992.2842 | 8.2527 | Oscillatory |

0.2 | 1013.7706 | 8.2527 | 992.2842 | 8.2527 | Oscillatory |

0.3 | 1008.7706 | 8.2527 | 992.2842 | 8.2527 | Oscillatory |

0.4 | 1003.7706 | 8.2527 | 992.2842 | 8.2527 | Oscillatory |

0.5 | 998.7706 | 8.2527 | 992.2842 | 8.2527 | Oscillatory |

0.6 | 993.7706 | 8.2527 | 992.2842 | 8.2527 | Oscillatory |

0.7 | 988.7706 | 8.2527 | 992.2842 | 8.2527 | Stationary |

0.8 | 983.7706 | 8.2527 | 992.2842 | 8.2527 | Stationary |

0.9 | 978.7706 | 8.2527 | 992.2842 | 8.2527 | Stationary |

**Table 3.**Critical stationary and oscillatory Rayleigh numbers for the different values of ${\kappa}_{r}$ and the fixed values of $S=0.8$, $Ta=50$, and $Rc=50$.

${\mathit{\kappa}}_{\mathit{r}}$ | Stationary R | Stationary a | Oscillatory R | Oscillatory a | Instability |
---|---|---|---|---|---|

1 | 983.7706 | 8.2540 | 992.2842 | 8.2540 | Stationary |

2 | 691.3454 | 6.4421 | 694.5709 | 6.4421 | Stationary |

3 | 588.8136 | 5.5171 | 590.1849 | 5.5171 | Stationary |

4 | 546.4455 | 4.9469 | 547.0506 | 4.9469 | Stationary |

5 | 531.2631 | 4.5668 | 531.5936 | 4.5668 | Stationary |

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**MDPI and ACS Style**

Ramchandraiah, C.; Kishan, N.; Reddy, G.S.K.; Paidipati, K.K.; Chesneau, C.
Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect. *Math. Comput. Appl.* **2022**, *27*, 56.
https://doi.org/10.3390/mca27040056

**AMA Style**

Ramchandraiah C, Kishan N, Reddy GSK, Paidipati KK, Chesneau C.
Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect. *Mathematical and Computational Applications*. 2022; 27(4):56.
https://doi.org/10.3390/mca27040056

**Chicago/Turabian Style**

Ramchandraiah, Chirnam, Naikoti Kishan, Gundlapally Shiva Kumar Reddy, Kiran Kumar Paidipati, and Christophe Chesneau.
2022. "Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect" *Mathematical and Computational Applications* 27, no. 4: 56.
https://doi.org/10.3390/mca27040056