# Dissolution-Driven Convection in a Porous Medium Due to Vertical Axis of Rotation and Magnetic Field

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. Basic Equations

#### 2.2. Basic Flow

#### 2.3. Linear Stability Analysis

#### 2.4. Stationary Mode

#### 2.5. Oscillatory Mode

## 3. Artificial Neural Network Modeling

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Schematic representation of a multilayer feed-forward network consisting of two inputs, one hidden layer, and two outputs.

**Figure 4.**Regression plots for training, validation, and testing; the targets are simulated data and the outputs are ANN-predicted data.

**Figure 5.**Neutral curves (solid lines represent the stationary convection and dotted lines represent the oscillatory convection) for $Dm=20$, $Ta=3.1$, $Va=20$: (

**a**) $Ha=0.5$, (

**b**) $Ha=0.6$, (

**c**) $Ha=0.7$.

**Figure 6.**Neutral curves (solid lines represent the stationary convection and dotted lines represent the oscillatory convection) for $Dm=20$, $Ha=0.6$, $Va=20$: (

**a**) $Ta=2.5$, (

**b**) $Ta=3.1$, (

**c**) $Ta=5$.

**Figure 7.**Neutral curves (solid lines represent the stationary convection and dotted lines represent the oscillatory convection) for $Dm=20$, $Ha=0.5$, $Ta=10$.

**Figure 8.**Plots of the critical $Ra$ as the function of $Ta$ for $Va=1,\phantom{\rule{3.33333pt}{0ex}}5,\phantom{\rule{3.33333pt}{0ex}}10,\phantom{\rule{3.33333pt}{0ex}}15$.

**Figure 9.**Plots of critical $Ra$ as the function of $Ta$ for $Dm=5,\phantom{\rule{3.33333pt}{0ex}}Va=5$, $Ha=0.2,\phantom{\rule{3.33333pt}{0ex}}0.3,\phantom{\rule{3.33333pt}{0ex}}0.4,\phantom{\rule{3.33333pt}{0ex}}0.5$.

**Figure 10.**Plots of critical $Ra$ as the function of $Ta$ for $Va=15,\phantom{\rule{3.33333pt}{0ex}}Ha=0.5$, $Dm=1,\phantom{\rule{3.33333pt}{0ex}}5,\phantom{\rule{3.33333pt}{0ex}}10,\phantom{\rule{3.33333pt}{0ex}}15$.

**Figure 11.**Comparison of the simulated and ANN-predicted critical Rayleigh number values for (

**a**) $Dm$, (

**b**) $Ha$, and (

**c**) $Ta$.

**Figure 12.**Comparison of the simulated and ANN-predicted critical Rayleigh number values for (

**a**) $Dm$, (

**b**) $Ha$, and (

**c**) $Ta$.

**Table 1.**Calculated stationary values of ${R}^{2}$, $RMSE$, and $RMRE$ at various values of $Ta$, $Dm$, and $Ha$.

Values | Stationary | |||
---|---|---|---|---|

${\mathit{R}}^{2}$ | $\mathbf{RMSE}$ | $\mathbf{RMRE}$ | ||

$Ta=0,5,10,\dots ,50$ | $Va=0.5,Ha=0.5,Dm=2$ | 0.999992 | 0.301510 | 0.549099 |

$Dm=0.5,1,1.5,\dots ,5$ | $Va=0.5,Ha=0.5,Ta=20$ | 0.999991 | 0.316226 | 0.562340 |

$Ha=0.1,0.2,0.3,\dots ,0.9$ | $Va=0.5,Dm=2,Ta=20$ | 0.999996 | 0.333332 | 0.5773497 |

**Table 2.**Calculated oscillatory values of ${R}^{2}$, $RMSE$, and $RMRE$ at various values of $Ta$, $Dm$, and $Ha$.

Values | Oscillatory | |||
---|---|---|---|---|

${\mathit{R}}^{2}$ | $\mathbf{RMSE}$ | $\mathbf{RMRE}$ | ||

$Ta=0,5,10,\dots ,50$ | $Va=0.5,Ha=0.5,Dm=2$ | 0.999999 | 0.447213 | 0.668740 |

$Dm=0.5,1,1.5,\dots ,5$ | $Va=0.5,Ha=0.5,Ta=20$ | 0.999994 | 0.316226 | 0.562340 |

$Ha=0.1,0.2,0.3,\dots ,0.9$ | $Va=0.5,Dm=2,Ta=20$ | 0.999966 | 0.333327 | 0.577345 |

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**MDPI and ACS Style**

Reddy, G.S.K.; Koteswararao, N.V.; Ravi, R.; Paidipati, K.K.; Chesneau, C.
Dissolution-Driven Convection in a Porous Medium Due to Vertical Axis of Rotation and Magnetic Field. *Math. Comput. Appl.* **2022**, *27*, 53.
https://doi.org/10.3390/mca27030053

**AMA Style**

Reddy GSK, Koteswararao NV, Ravi R, Paidipati KK, Chesneau C.
Dissolution-Driven Convection in a Porous Medium Due to Vertical Axis of Rotation and Magnetic Field. *Mathematical and Computational Applications*. 2022; 27(3):53.
https://doi.org/10.3390/mca27030053

**Chicago/Turabian Style**

Reddy, Gundlapally Shiva Kumar, Nilam Venkata Koteswararao, Ragoju Ravi, Kiran Kumar Paidipati, and Christophe Chesneau.
2022. "Dissolution-Driven Convection in a Porous Medium Due to Vertical Axis of Rotation and Magnetic Field" *Mathematical and Computational Applications* 27, no. 3: 53.
https://doi.org/10.3390/mca27030053