# Mixed Convection Flow over an Elastic, Porous Surface with Viscous Dissipation: A Robust Spectral Computational Approach

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Modeling

#### Similarity Analysis

## 3. Implementation of Numerical Method

#### 3.1. Basic Steps to Apply Spectral Local Linearization Method

#### 3.2. Implementation of the Spectral Local Linearization Method

#### 3.3. Convergence Analysis

## 4. Numerical Results and Discussion

## 5. Summary and Conclusions

- (i)
- The magnetic and permeability parameters augment the temperature profile, while decreasing the velocity profile.
- (ii)
- The mixed convection parameter increases the velocity profile, while reducing the temperature profile.
- (iii)
- The Eckert number and the heat generation parameter have a significant effect to increase the temperature profile and the thickness of the thermal boundary layer.
- (iv)
- Increasing the Prandtl number reduces the temperature profile, because the momentum diffusivity is more dominant and the increased flow dampens the development of the temperature profile.
- (v)
- The numerical results obtained with the Spectral local linearization method are consistent with the results obtained with other related methodologies such as bvp4c.
- (vi)
- When the skin friction profile is compared to previously published data, it is found that the current numerical results show a good agreement and this validates the proposed method.
- (vii)
- The SLLM does not lose accuracy as the number of collocation points increases.
- (viii)
- For all of the model parameters investigated in this work, the method has been found to rapidly converge to the respective solutions.
- (ix)
- When compared to other approaches, the suggested methodology is computationally more efficient and shows better performance with fewer collocation points $\left(N\right)$ and four iterations. Because this method is more reliable, simple, and efficient, the SLLM methodology is considered more suitable for the solution of nonlinear boundary value problems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Ceramic engineering [7] | Simultaneous mass and heat transfer in disordered media occurs during the burnout/drying of the binder system from green compacts during the colloidal process of ceramics. |

Chemical engineering [8] | During the interim eventual and storage of nuclear waste, as well as in packet bed reactors. |

Ground water hyrology [9] | The investigation of seepage water through river beds and underground water resources. |

Industrial engineering [10] | The primary goal of filtration analysis is to examine the movement of fluid through a porous medium, leaving behind unwanted material. As a result of the mass deposition, the porous medium is constantly changing and altering the system’s pressure drop properties. |

Mechanical engineering [11] | To achieve effective insulation, solid conduction must be minimized, porosity must be maximized to reduce effective thermal conductivity, and free convection must be suppressed. The same concept is useful when producing high-performance insulators for cryogenic containers. |

Petroleum engineering [12] | For oil recovery mechanisms. |

Geophysics [13] | In the analysis of geo-pressurized reservoirs, and extraction of geothermal energy. |

Function Name | Calls | Total Time (s) | Self Time (s) |
---|---|---|---|

SLLM code | 1 | 0.284 | 0.202 |

Newplot | 2 | 0.033 | 0.013 |

xlabel | 2 | 0.029 | 0.023 |

newplot > ObserveAxesNextPlot | 2 | 0.017 | 0.004 |

cla | 2 | 0.013 | 0.007 |

hold | 2 | 0.011 | 0.009 |

ylabel | 2 | 0.010 | 0.008 |

convertStringToCharArgs | 4 | 0.007 | 0.005 |

graphics/private/clo | 2 | 0.005 | 0.005 |

gobjects | 4 | 0.003 | 0.003 |

convertStringsToChars | 4 | 0.002 | 0.001 |

markfigure | 2 | 0.001 | 0.001 |

graphics/private/claNotify | 2 | 0.001 | 0.001 |

graph2d/private/labelcheck | 4 | 0.001 | 0.001 |

convertStringsToChars > convertStrings | 16 | 0.001 | 0.001 |

axescheck | 2 | 0.000 | 0.000 |

newplot > ObserveFigureNextPlo | 2 | 0.000 | 0.000 |

Iterations | N | Skin Friction Coefficient | Nusselt Number |
---|---|---|---|

4 | 5 | $-1.15627$ | $1.17653$ |

4 | 10 | $-1.29290$ | $1.43602$ |

4 | 100 | $-1.30162$ | $1.44072$ |

− | bvp4c | $-1.30162$ | $1.44072$ |

**Table 4.**Numerical comparison between the present numerical method and bvp4c for Skin friction against different values of emerging parameters.

$\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathbf{\Lambda}$ | $\mathbf{\Gamma}$ | $\mathit{\delta}$ | $\mathit{\lambda}$ | $\mathit{\varsigma}$ | $\mathit{\varphi}$ | SLLM | bvp4c |
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.2 | 0.5 | 1 | 0.1 | 0.2 | 0 | 1 | $-1.30162$ | $-1.30162$ |

1 | $-1.65671$ | $-1.65671$ | |||||||

1.5 | $-1.85064$ | $-1.85064$ | |||||||

0.5 | $-1.44264$ | $-1.44264$ | |||||||

0.8 | $-1.57383$ | $-1.57383$ | |||||||

1.2 | $-1.73640$ | $-1.73640$ | |||||||

0.4 | $-1.34991$ | $-1.34991$ | |||||||

1.0 | $-1.07167$ | $-1.07167$ | |||||||

1.5 | $-0.85505$ | $-0.85505$ | |||||||

0.71 | $-1.27273$ | $-1.27273$ | |||||||

2 | $-1.40238$ | $-1.40238$ | |||||||

3 | $-1.44594$ | $-1.44594$ | |||||||

$-0.1$ | $-1.31463$ | $-1.31463$ | |||||||

0 | $-1.30860$ | $-1.30860$ | |||||||

0.12 | $-1.30008$ | $-1.30008$ | |||||||

0.3 | $-1.29782$ | $-1.29782$ | |||||||

0.4 | $-1.29404$ | $-1.29404$ | |||||||

0.5 | $-1.29029$ | $-1.29029$ | |||||||

$-0.5$ | $-1.00143$ | $-1.00143$ | |||||||

0 | $-1.301625$ | $-1.301625$ | |||||||

0.5 | $-1.70488$ | $-1.70488$ | |||||||

$-0.6$ | $0.942750$ | $0.942750$ | |||||||

0 | $0.45307$ | $0.45307$ | |||||||

0.6 | $-0.48097$ | $-0.48097$ |

**Table 5.**Numerical comparison between present method and bvp4c for Nusselt number against different values of the other parameters.

$\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathbf{\Lambda}$ | $\mathbf{\Gamma}$ | $\mathit{\delta}$ | $\mathit{\lambda}$ | SLLM | bvp4c |
---|---|---|---|---|---|---|---|

0.2 | 0.2 | 0.5 | 1 | 0.1 | 0.2 | $1.44072$ | $1.44072$ |

1 | $1.25957$ | $1.25957$ | |||||

1.5 | $1.16052$ | $1.16052$ | |||||

0.5 | $1.36888$ | $1.36888$ | |||||

0.8 | $1.30192$ | $1.30192$ | |||||

1.2 | $1.21886$ | $1.21886$ | |||||

0.4 | $1.42396$ | $1.42396$ | |||||

1 | $1.50962$ | $1.50962$ | |||||

1.5 | $1.56319$ | $1.56319$ | |||||

0.71 | $1.18551$ | $1.18551$ | |||||

2 | $2.15499$ | $2.15499$ | |||||

3 | $2.69120$ | $2.69120$ | |||||

$-0.1$ | $1.55523$ | $1.55523$ | |||||

0 | $1.49965$ | $1.49965$ | |||||

0.12 | $1.42846$ | $1.42846$ | |||||

0.3 | $1.38320$ | $1.38320$ | |||||

0.4 | $1.32612$ | $1.32612$ | |||||

0.5 | $1.26948$ | $1.26948$ |

**Table 6.**Numerical comparison between present results with previously report results [45] for skin friction profile.

$\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathbf{\Lambda}$ | $\mathit{\varsigma}$ | $\mathit{\varphi}$ | Present Results | Fang et al. [45] |
---|---|---|---|---|---|---|

0 | 0 | 0 | $-{\textstyle \frac{1}{6}}$ | 1 | $-1.0234$ | $-1.0234$ |

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**MDPI and ACS Style**

Zhang, L.; Tariq, N.; Bhatti, M.M.; Michaelides, E.E.
Mixed Convection Flow over an Elastic, Porous Surface with Viscous Dissipation: A Robust Spectral Computational Approach. *Fractal Fract.* **2022**, *6*, 263.
https://doi.org/10.3390/fractalfract6050263

**AMA Style**

Zhang L, Tariq N, Bhatti MM, Michaelides EE.
Mixed Convection Flow over an Elastic, Porous Surface with Viscous Dissipation: A Robust Spectral Computational Approach. *Fractal and Fractional*. 2022; 6(5):263.
https://doi.org/10.3390/fractalfract6050263

**Chicago/Turabian Style**

Zhang, Lijun, Nafisa Tariq, Muhammad Mubashir Bhatti, and Efstathios E. Michaelides.
2022. "Mixed Convection Flow over an Elastic, Porous Surface with Viscous Dissipation: A Robust Spectral Computational Approach" *Fractal and Fractional* 6, no. 5: 263.
https://doi.org/10.3390/fractalfract6050263