# Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Application of Homotopy Analysis Method

#### 3.1. A Short Description of HAM

- ${y}_{0}\left(t\right)$ is a solution of equation$${\left.\mathcal{H}[\Phi (t;q),q]\right|}_{q=0}=0,$$

#### 3.2. Application of HAM to System (12)–(14)

**Theorem**

**1.**

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PDE(s) | Partial Differential Equation(s) |

MHD | Magnetohydrodynamic |

HAM | Homotopy Analysis Method |

DE(s) | Differential Equation(s) |

ODE(s) | Ordinary Differential Equation(s) |

APG | Adverse Pressure Gradient |

FPG | Favorable Pressure Gradient |

PG | Pressure Gradient |

ADV | Adverse Pressure Effect |

RAD | Radiation |

e | Edge |

$Oxy$ | coordinate system |

x, y | coordinates |

${U}_{\infty}$ | free stream velocity |

L | length of the plate |

$\delta $ | dimensional boundary layer thickness |

u, v | dimensional velocity components of the fluid |

$\rho $ | density of the fluid |

p | pressure of the fluid |

$\nu $ | kinematic viscosity of the fluid |

$\mu $ | dynamic viscosity of the fluid |

${C}_{p}$ | specific temperature of the fluid under constant pressure |

T | temperature of the fluid |

k | coefficient of thermal conductivity |

$\frac{\partial {q}_{r}}{\partial y}$ | local radiant absorption |

${q}_{r}$ | radiative energy flux |

${T}_{w}$ | temperature of the flat plate |

${T}_{e}$ | temperature of the fluid at the edge of the boundary layer |

${u}_{e}$ | velocity of the fluid at the edge of the boundary layer |

$\alpha $ | absorption coefficient |

$\sigma $ | Stefan–Boltzmann constant |

$\eta (x,y)$ | dimensionless coordinate variable |

$\Psi $ | stream function |

f | dimensionless stream function |

$\theta $ | dimensionless temperature of the fluid |

$Pr$ | Prandtl number |

$\overline{x}$ | dimensionless coordinate variable |

${\overline{u}}_{e}$ | dimensionless velocity of the fluid at the edge of the boundary layer |

$\overline{\delta}$ | dimensionless boundary layer thickness |

$\mathcal{H}$ | homotopy |

q | embedding parameter |

$\mathcal{L}$ | auxiliary linear operator |

$\mathcal{N}$ | auxiliary nonlinear operator |

$H\left(t\right)$, ${H}_{f}\left(\eta \right)$, ${H}_{\theta}\left(\eta \right)$ | auxiliary functions |

ℏ, ${\hslash}_{f}$, ${\hslash}_{\theta}$ | convergence control parameters |

${S}_{B}$ | set of base functions |

${E}_{K}$, ${E}_{K}^{f}$, ${E}_{K}^{\theta}$ | “discrete square residual” errors |

R | radiation parameter |

${{C}_{f}}_{x}$ | local skin friction coefficient |

$S{t}_{x}$ | local Stanton number |

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**Figure 2.**(

**a**) $\hslash -$ curve for the dimensionless velocity, $f{\phantom{\rule{0.166667em}{0ex}}}^{\prime}\left(\eta \right)$ for 15 terms. (

**b**) $\hslash -$ curve for the dimensionless temperature, $\theta \phantom{\rule{0.166667em}{0ex}}\left(\eta \right)$ for 15 terms.

**Figure 3.**(

**a**) The dimensionless velocity, $f{\phantom{\rule{0.166667em}{0ex}}}^{\prime}\left(\eta \right)$ with respect to $\eta $. (

**b**) The dimensionless temperature, $\theta \phantom{\rule{0.166667em}{0ex}}\left(\eta \right)$ with respect to $\eta $, for the case where $L=8$, for adverse pressure gradient (APG) and favorable pressure gradient (FPG) cases.

**Figure 4.**Comparison of the dimensionless velocity, ${f}^{\prime}\left(\eta \right)$ with respect to the dimensionless distance $\eta $ for the analytical (HAM, 20 terms), and the numerical solution obtained with the numerical approach introduced in [5].

**Figure 5.**Skin friction coefficient, ${f}_{w}^{\u2033}$ with respect to the dimensionless x-direction.

**Figure 6.**Heat transfer coefficient, ${\theta}_{w}^{\prime}$ with respect to the dimensionless x-direction.

**Figure 7.**Effect of radiation on the dimensionless temperature, $\theta \phantom{\rule{0.166667em}{0ex}}\left(\eta \right)$, depending on the parameter, R, and the adverse pressure gradient, with respect to $\eta $.

Order | $\mathit{f}{\phantom{\rule{0.166667em}{0ex}}}^{\prime}$-error | ${\mathit{\hslash}}_{\mathit{f}}$ | $\mathit{\theta}$-error | ${\mathit{\hslash}}_{\mathit{\theta}}$ |
---|---|---|---|---|

2 | 0.00239722 | −0.485458 | 0.0021427 | −0.637823 |

4 | 0.000427796 | −0.803283 | 0.000710738 | −0.596724 |

6 | 0.0000874967 | −0.824688 | 0.000489771 | −0.63417 |

8 | 0.0000507706 | −0.892126 | 0.00045183 | −0.653759 |

10 | 0.0000637655 | −0.87467 | 0.000441161 | −0.681927 |

12 | 0.0000661033 | −0.651043 | 0.000436139 | −0.68495 |

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## Share and Cite

**MDPI and ACS Style**

Xenos, M.A.; Petropoulou, E.N.; Siokis, A.; Mahabaleshwar, U.S.
Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation. *Symmetry* **2020**, *12*, 710.
https://doi.org/10.3390/sym12050710

**AMA Style**

Xenos MA, Petropoulou EN, Siokis A, Mahabaleshwar US.
Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation. *Symmetry*. 2020; 12(5):710.
https://doi.org/10.3390/sym12050710

**Chicago/Turabian Style**

Xenos, Michalis A., Eugenia N. Petropoulou, Anastasios Siokis, and U. S. Mahabaleshwar.
2020. "Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation" *Symmetry* 12, no. 5: 710.
https://doi.org/10.3390/sym12050710