# Vadasz Number Effects on Convection in a Horizontal Porous Layer Subjected to Internal Heat Generation and G-Jitter

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

**V**, T and ${\mathrm{p}}_{\mathrm{r}}$ represent the dimensionless filtration velocity vector, temperature and reduced pressure, respectively, whilst ${\widehat{\mathbf{e}}}_{\mathbf{z}}$, is the unit vector in the z- direction. In the previous work [9], the author has put forward a motivation for specific cases when the Vadasz number is small and can be retained. In the instance of liquid metals this could be the case. In the current study when considering a high temperature reactor for the proof of concept geometry, an average porosity of ${\mathrm{\varphi}}^{*}\approx 0.4$, a Prandtl number $\mathrm{Pr}=0.7$, characteristic permeability ${\mathrm{k}}_{0}^{*}\approx 0.704$ and calculated Darcy number $\mathrm{D}\mathrm{a}\approx 0.155$ yields a Vadasz number $\mathrm{V}\mathrm{a}=1.81$. This implies that for systems involving gas reactors and porous media, there are instances in which the Vadasz number is close to unity and can be retained in the momentum equation. The solutions for the basic temperature and flow field is given as ${\mathrm{T}}_{\mathrm{B}}=\mathrm{B}+\mathrm{A}\mathrm{z}-1/2{\mathrm{z}}^{2}$ and ${\mathrm{V}}_{\mathrm{B}}=0$, where A and B are constants that may be determined, based on the imposed boundary for the various cases shown in Table 1. Since we will consider the derivative of the basic temperature in the energy equation, the value for the constant B is not required, and therefore will not be presented in Table 1.

## 3. Linear Stability Analysis

## 4. Results and Discussion

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of vibrating porous layer with thermal boundary conditions shown in Table 1.

**Figure 2.**Case 3: Critical Rayleigh number versus vibration frequency for selected values of Vadasz number (A = −0.5).

**Figure 3.**Critical Rayleigh number versus scaled Vadasz number at selected values of vibration frequency (Case 3: A = −0.5).

**Figure 4.**Critical Rayleigh number versus scaled Vadasz number at selected values of vibration frequency (Case 4: A = 1.5).

**Figure 5.**Benard Convection with g-jitter: Critical Rayleigh number versus vibration frequency at selected values of Vadasz number (no internal heat generation).

Case | Boundary Condition | A | Description |
---|---|---|---|

1 | $z=0:\frac{d{T}_{B}}{dz}=0$$z=1:{T}_{B}=0$ | 0 | Adiabatic bottom wall/Perfectly conducting top wall |

2 | $z=0:{T}_{B}=0$$z=1:{T}_{B}=0$ | $\frac{1}{2}$ | Perfectly conducting top and bottom walls |

3 | $z=0:{T}_{B}=1$$z=1:{T}_{B}=0$ | $-\frac{1}{2}$ | Perfectly conducting top and bottom walls |

4 | $z=0:{T}_{B}=0$$z=1:{T}_{B}=1$ | $\frac{3}{2}$ | Perfectly conducting top and bottom walls |

5 | $z=0:{T}_{B}=0$$z=1:\frac{d{T}_{B}}{dz}=0$ | 1 | Adiabatic top wall/Perfectly conducting bottom wall |

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**MDPI and ACS Style**

Govender, S.
Vadasz Number Effects on Convection in a Horizontal Porous Layer Subjected to Internal Heat Generation and G-Jitter. *Fluids* **2020**, *5*, 124.
https://doi.org/10.3390/fluids5030124

**AMA Style**

Govender S.
Vadasz Number Effects on Convection in a Horizontal Porous Layer Subjected to Internal Heat Generation and G-Jitter. *Fluids*. 2020; 5(3):124.
https://doi.org/10.3390/fluids5030124

**Chicago/Turabian Style**

Govender, Saneshan.
2020. "Vadasz Number Effects on Convection in a Horizontal Porous Layer Subjected to Internal Heat Generation and G-Jitter" *Fluids* 5, no. 3: 124.
https://doi.org/10.3390/fluids5030124