# Boundary Layer Flow through Darcy–Brinkman Porous Medium in the Presence of Slip Effects and Porous Dissipation

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

Dimensionless Physical Parameters | Notations | Definitions |

Brinkmann parameter | $\gamma $ | ${\epsilon}^{2}\frac{{\mu}_{e}}{\mu}$ |

Porosity parameter | ${P}_{m}$ | $\frac{\mu {\epsilon}^{2}}{\rho \alpha {K}^{*}}$ |

Suction/injection parameter | $S$ | $\frac{{V}_{0}}{\sqrt{\alpha \upsilon}}$ |

Prandtl number | $\mathrm{Pr}$ | $\frac{\mu {C}_{p}}{\kappa}$ |

Eckert number | $Ec$ | $\frac{{\alpha}^{2}}{c{C}_{p}}$ |

Velocity slip parameter | $\beta $ | ${\beta}_{1}\sqrt{\frac{\alpha}{\upsilon}}$ |

Thermal slip parameter | $\delta $ | ${\delta}_{1}\sqrt{\frac{\alpha}{\upsilon}}$ |

## 3. Solution Methodologies

## 4. Results and Discussion

## 5. Conclusions

- (i)
- Both velocity and temperature decrease with the increase of suction parameter.
- (ii)
- The slip parameter has high impact on skin friction coefficient as compared with no-slip condition.
- (iii)
- Heat transfer rate is reduced due to increase in Eckert number and thermal slip parameter.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Comparison between analytical solution [22] and shooting method for different values of $\beta $ when $\gamma =1$ and ${P}_{m}=0.0,S=0.0$.

Velocity Slip Parameter $\mathit{\beta}$ | ${\mathit{g}}^{\prime}(0)$ | $-{\mathit{g}}^{\u2033}\left(0\right)$ | ||
---|---|---|---|---|

Andersson [22] | Present | Andersson [22] | Present | |

0.0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

0.1 | 0.9128 | 0.91278 | 0.8721 | 0.87215 |

0.2 | 0.8447 | 0.84471 | 0.7764 | 0.77645 |

0.5 | 0.7044 | 0.70436 | 0.5912 | 0.59127 |

1.0 | 0.5698 | 0.56974 | 0.4302 | 0.43025 |

2.0 | 0.4320 | 0.43183 | 0.2840 | 0.28408 |

5.0 | 0.2758 | 0.27530 | 0.1448 | 0.14493 |

10.0 | 0.1876 | 0.18670 | 0.0812 | 0.08132 |

20.0 | 0.1242 | 0.12285 | 0.0438 | 0.04385 |

50.0 | 0.0702 | 0.06801 | 0.0186 | 0.01863 |

100.0 | 0.0450 | 0.04225 | 0.0095 | 0.00957 |

**Table 2.**Skin friction coefficient ${S}_{fx}{\mathrm{Re}}_{x}^{1/2}=-{g}^{\u2033}(0)$ for no slip case $\beta =0$. Comparison between exact and numerical solution.

Physical Parameters | ${\mathit{S}}_{\mathit{f}\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{1/2}=-{\mathit{g}}^{\u2033}(0)$ | |||
---|---|---|---|---|

$\mathit{\gamma}$ | $\mathit{S}$ | ${\mathit{P}}_{\mathit{m}}$ | Exact (See Equation (11)) | Numerical (Shooting Method) |

1.0 | 1.0 | 0.5 | 1.82287 | 1.82287 |

2.0 | 1.0 | 0.5 | 1.15138 | 1.15140 |

3.0 | 1.0 | 0.5 | 0.89314 | 0.89324 |

0.5 | 0.0 | 0.3 | 1.61245 | 1.61245 |

0.5 | 1.0 | 0.3 | 2.89736 | 2.89736 |

0.5 | 2.0 | 0.3 | 4.56904 | 4.56905 |

2.0 | 0.5 | 0.0 | 0.84307 | 0.84336 |

2.0 | 0.5 | 0.4 | 0.97094 | 0.97098 |

2.0 | 0.5 | 0.8 | 1.08188 | 1.08188 |

**Table 3.**Skin friction coefficient ${S}_{fx}{\mathrm{Re}}_{x}^{1/2}=-{g}^{\u2033}(0)$ for slip case $\beta =1.0$. Comparison between Shooting method and MATLAB bvp4c.

Physical Parameters | ${\mathit{S}}_{\mathit{f}\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{1/2}=-{\mathit{g}}^{\u2033}(0)$ | |||
---|---|---|---|---|

$\mathit{\gamma}$ | $\mathit{S}$ | ${\mathit{P}}_{\mathit{m}}$ | Shooting Method | bvp4c |

1.0 | 1.0 | 0.5 | 0.610511 | 0.610497 |

2.0 | 1.0 | 0.5 | 0.500008 | 0.500006 |

3.0 | 1.0 | 0.5 | 0.439566 | 0.439507 |

0.5 | 0.0 | 0.3 | 0.550438 | 0.550437 |

0.5 | 1.0 | 0.3 | 0.712228 | 0.712227 |

0.5 | 2.0 | 0.3 | 0.808872 | 0.808872 |

2.0 | 0.5 | 0.0 | 0.406493 | 0.406209 |

2.0 | 0.5 | 0.4 | 0.452006 | 0.451987 |

2.0 | 0.5 | 0.8 | 0.485908 | 0.485905 |

**Table 4.**Local Nusselt number ${N}_{Rx}{\mathrm{Re}}_{x}^{-1/2}=-{\theta}^{\prime}\left(0\right)$ when $\beta =1.0\text{}\mathrm{and}\text{}S=0.5$ Comparison between Shooting method and MATLAB bvp4c.

Physical Parameters | ${\mathit{N}}_{\mathit{R}\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}=-{\mathit{\theta}}^{\prime}\left(0\right)$ | |||||
---|---|---|---|---|---|---|

$\mathbf{Pr}$ | $\mathit{E}\mathit{c}$ | $\mathit{\delta}$ | ${\mathit{P}}_{\mathit{m}}$ | $\mathit{\gamma}$ | Shooting Method | bvp4c |

0.7 | 0.5 | 1.0 | 0.4 | 2.0 | 0.456141 | 0.456203 |

1.2 | 0.5 | 1.0 | 0.4 | 2.0 | 0.538161 | 0.538197 |

6.8 | 0.5 | 1.0 | 0.4 | 2.0 | 0.738928 | 0.738983 |

3.0 | 0.0 | 1.0 | 0.4 | 2.0 | 0.738078 | 0.738124 |

3.0 | 0.6 | 1.0 | 0.4 | 2.0 | 0.642319 | 0.642382 |

3.0 | 1.2 | 1.0 | 0.4 | 2.0 | 0.546560 | 0.546591 |

3.0 | 1.0 | 0.0 | 0.4 | 2.0 | 2.208602 | 2.208638 |

3.0 | 1.0 | 0.6 | 0.4 | 2.0 | 0.820808 | 0.820821 |

3.0 | 1.0 | 1.2 | 0.4 | 2.0 | 0.504071 | 0.504105 |

3.0 | 1.0 | 1.0 | 0.0 | 2.0 | 0.640207 | 0.640288 |

3.0 | 1.0 | 1.0 | 0.5 | 2.0 | 0.566207 | 0.566224 |

3.0 | 1.0 | 1.0 | 1.0 | 2.0 | 0.517044 | 0.517133 |

3.0 | 1.0 | 1.0 | 0.4 | 1.0 | 0.619665 | 0.619690 |

3.0 | 1.0 | 1.0 | 0.4 | 2.0 | 0.578480 | 0.578501 |

3.0 | 1.0 | 1.0 | 0.4 | 3.0 | 0.546450 | 0.546487 |

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**MDPI and ACS Style**

Kausar, M.S.; Hussanan, A.; Mamat, M.; Ahmad, B.
Boundary Layer Flow through Darcy–Brinkman Porous Medium in the Presence of Slip Effects and Porous Dissipation. *Symmetry* **2019**, *11*, 659.
https://doi.org/10.3390/sym11050659

**AMA Style**

Kausar MS, Hussanan A, Mamat M, Ahmad B.
Boundary Layer Flow through Darcy–Brinkman Porous Medium in the Presence of Slip Effects and Porous Dissipation. *Symmetry*. 2019; 11(5):659.
https://doi.org/10.3390/sym11050659

**Chicago/Turabian Style**

Kausar, Muhammad Salman, Abid Hussanan, Mustafa Mamat, and Babar Ahmad.
2019. "Boundary Layer Flow through Darcy–Brinkman Porous Medium in the Presence of Slip Effects and Porous Dissipation" *Symmetry* 11, no. 5: 659.
https://doi.org/10.3390/sym11050659