# An Eigenvalues Approach for a Two-Dimensional Porous Medium Based Upon Weak, Normal and Strong Thermal Conductivities

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Formulation of the Problem

## 4. Results and Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 2.**The variations in temperature $\mathsf{\Theta}$ along $y$ when x $=0.5$, for weak, normal and strong conductivities.

**Figure 3.**The changes in the volume fraction field of the voids distribution φ along y when x = 0.5, for weak, normal and strong conductivities.

**Figure 4.**The variation in horizontal displacement u along y when x = 0.5, for weak, normal and strong conductivities.

**Figure 5.**The variation in vertical displacement $v$ along $y$ when x $=0.5$, for weak, normal and strong conductivities.

**Figure 6.**The variation in stress ${\sigma}_{xx}$ along $y$ when x $=0.5$, for weak, normal and strong conductivities.

**Figure 7.**The variation in stress ${\sigma}_{xy}$ along $y$ when x $=0.5$, for weak, normal and strong conductivities.

**Figure 8.**The variation in temperature $\mathsf{\Theta}$ along $x$ when y $=0.5$, for weak, normal and strong conductivities.

**Figure 9.**The changes in the volume fraction field of the voids distribution $\phi $ along $x$ when y $=0.5$, for weak, normal and strong conductivities.

**Figure 10.**The variation in horizontal displacement $u$ along $x$ when y $=0.5$, for weak, normal and strong conductivities.

**Figure 11.**The variation in vertical displacement $v$ along $x$ when y $=0.5$, for weak, normal and strong conductivities.

**Figure 12.**The variation in stress ${\sigma}_{xx}$ along $x$ when y $=0.5$, for weak, normal and strong conductivities.

**Figure 13.**The variation in stress ${\sigma}_{xy}$ along $x$ when y $=0.5$, for weak, normal and strong conductivities.

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**MDPI and ACS Style**

Alzahrani, F.; Hobiny, A.; Abbas, I.; Marin, M.
An Eigenvalues Approach for a Two-Dimensional Porous Medium Based Upon Weak, Normal and Strong Thermal Conductivities. *Symmetry* **2020**, *12*, 848.
https://doi.org/10.3390/sym12050848

**AMA Style**

Alzahrani F, Hobiny A, Abbas I, Marin M.
An Eigenvalues Approach for a Two-Dimensional Porous Medium Based Upon Weak, Normal and Strong Thermal Conductivities. *Symmetry*. 2020; 12(5):848.
https://doi.org/10.3390/sym12050848

**Chicago/Turabian Style**

Alzahrani, Faris, Aatef Hobiny, Ibrahim Abbas, and Marin Marin.
2020. "An Eigenvalues Approach for a Two-Dimensional Porous Medium Based Upon Weak, Normal and Strong Thermal Conductivities" *Symmetry* 12, no. 5: 848.
https://doi.org/10.3390/sym12050848