# A Self-Similar Approach to Study Nanofluid Flow Driven by a Stretching Curved Sheet

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## Abstract

**:**

## 1. Introduction

_{2}water with nano-particles in the presence of thermal radiation. Khan et al. [25] described heterogeneous-homogenous reactions with characteristics of heat transfer and electrically conducting boundary layer fluid flow over stretched sheets. Nasir et al. [26] discussed the computations and modeling of rotating three-dimensional MHD nano-fluid flow through an extending surface.

## 2. Mathematical Formulation

## 3. Computational Procedure

#### 3.1. Quasi-Linearization Method

#### Procedure Steps

- (1)
- ${f}^{(0)},{\theta}^{(0)}$ and ${\chi}^{(0)}$ are the initial guesses to assure the boundary conditions, which are specified in equation.
- (2)
- Set ${f}^{(1)}$ in Equation (28) to present the solution of the linear system.
- (3)
- We are solving a linear system by means of ${f}^{(1)}$ forgetting ${\theta}^{(1)}$ and ${\chi}^{(1)}$.
- (4)
- By using new initial guesses that are ${f}^{(1)},{\theta}^{(1)}$ and ${\chi}^{(1)}$ which converges to $f,\theta $ and $\chi $, repeating this process to create sequences ${f}^{(k)},{\theta}^{(k)}$ and ${\chi}^{(\mathrm{k})}.$
- (5)
- We are creating four sequences until$$\mathrm{max}\left\{\right||{f}^{(k+1)}-{f}^{(k)}|{|}_{{L}_{\infty}},\left|\right|{\theta}^{(k+1)}-{\theta}^{(k)}|{|}_{{L}_{\infty}},||{\chi}^{(k+1)}-{\chi}^{(k)}|{|}_{{L}_{\infty}}\}<{10}^{-6}$$

## 4. Results and Discussion

## 5. Conclusions

- Velocity profiles have shown enhancing behavior for higher values of ${\kappa}_{f}$ while decreasing for magnetic parameter $M$.
- Temperature profile $\theta $ enhances greater values of $Rd$.
- Skin friction reduces for larger values of chemical reaction parameter $\gamma $.
- The Prandtl number tends to reduce the rate of heat transfer.
- The Schmidt number causes an increase in concentration.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$l:m$ | Cartesian coordinates, [m] |

$u,v$ | Velocity components, [ms^{−1}] |

${c}_{p}$ | Specific heat, [${\mathrm{m}}^{2}{\mathrm{s}}^{-2}$] |

p | Pressure, [${\mathrm{kgm}}^{-1}{\mathrm{s}}^{-2}$] |

${k}_{nf}$ | Thermal conductivity of the nano-fluid, [${\mathrm{kgms}}^{-3}{\mathrm{K}}^{-1}$] |

T | Temperature, [K] |

Greek Symbols | |

${\mu}_{nf}$ | Dynamic viscosity, [Nsm^{−2}] |

${\rho}_{nf}$ | Density, [kgm^{−3}] |

${\upsilon}_{nf}$ | Kinematics viscosity, [${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$] |

${(\rho {c}_{p})}_{nf}$ | Heat capacitance of the nano-fluid, [${\mathrm{kgm}}^{-1}{\mathrm{s}}^{-2}{\mathrm{K}}^{-1}$] |

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**Figure 2.**Comparison of the present work by Yasmin et al. [44].

**Figure 3.**$f\left(\eta \right)$ for $M=0.1$, $\mathrm{Pr}=7$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$ and different ${\kappa}_{f}$.

**Figure 4.**${f}^{\prime}\left(\eta \right)$ for $M=0.1$, $\mathrm{Pr}=7$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$ and different ${\kappa}_{f}$.

**Figure 5.**$\chi \left(\eta \right)$ for $M=0.1$, $\mathrm{Pr}=7$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$ and different ${\kappa}_{f}$.

**Figure 6.**$f\left(\eta \right)$ for ${k}_{f}=10,\mathrm{Pr}=7$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$.

**Figure 7.**${f}^{\prime}\left(\eta \right)$ for ${k}_{f}=10,\mathrm{Pr}=7$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$.

**Figure 8.**$\chi \left(\eta \right)$ for ${k}_{f}=10,\mathrm{Pr}=7$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$.

**Figure 9.**$\theta \left(\eta \right)$ for ${k}_{f}=10$, $M=0.1$, $\varphi =0.1$, $Rd=0.5$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$.

**Figure 10.**$\theta \left(\eta \right)$ for ${k}_{f}=10$, $M=0.1$, $\varphi =0.1$, $\mathrm{Pr}=7$, $Sc=0.1$, $\gamma =0.1,{\lambda}_{1}=0.2$.

**Figure 11.**$\chi \left(\eta \right)$ for ${k}_{f}=10$, $M=0.1$, $\varphi =0.1$, $\mathrm{Pr}=7$, $Rd=0.5$, $\gamma =0.1,{\lambda}_{1}=0.2$.

**Figure 12.**$\chi \left(\eta \right)$ for ${k}_{f}=10$, $M=0.1$, $\varphi =0.1$, $\mathrm{Pr}=7$, $Rd=0.5$, $Sc=0.1,{\lambda}_{1}=0.2$.

c_{p} (J/kgK) | ρ (kg/m^{3}) | k (W/mK) | β × 10^{5} (K^{−1}) | |
---|---|---|---|---|

Pure water | 4179 | 997.1 | 0.613 | 21 |

Cu | 385 | 89.33 | 401 | 1.67 |

**Table 2.**Numerical values of the radius of curvature and Hartmann number for fixed ${\kappa}_{f}=10$, $M=0.1$, $\mathrm{Pr}=7$, $Sc=0.1$, $\varphi =0.1$, $\gamma =0.1$, $Rd=0.5$, $\lambda =0.2$.

κ_{f} | M | $-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\chi}}^{\prime}(0)$ |
---|---|---|---|---|

0.1 | 9.93356 | 2.6934 | 3.280395 | |

0.3 | 3.88908 | 1.9148 | 1.651969 | |

0.5 | 2.73516 | 1.8277 | 1.260395 | |

0.7 | 2.27919 | 1.8065 | 1.077154 | |

0.9 | 2.04465 | 1.7993 | 0.96929 | |

1 | 1.67012 | 1.7318 | 0.571286 | |

1.5 | 1.89177 | 1.6689 | 0.570398 | |

2 | 2.17357 | 1.5891 | 0.569365 | |

2.5 | 2.49816 | 1.4987 | 0.5683 | |

3 | 2.85166 | 1.4034 | 0.567278 |

**Table 3.**Effect of the Prandlt number and Radiation parameter for fixed ${\kappa}_{f}=10$, $M=0.1$, $\mathrm{Pr}=7$, $Sc=0.1$, $\varphi =0.1$, $\gamma =0.1$, $Rd=0.5$, $\lambda =0.2$.

Pr | Rd | $-{\mathit{\theta}}^{\prime}(0)$ |
---|---|---|

05 | 1.461766 | |

10 | 2.222069 | |

15 | 2.839614 | |

20 | 3.359658 | |

25 | 3.815000 | |

0.10 | 2.076521 | |

0.30 | 1.916277 | |

0.60 | 1.728912 | |

0.90 | 1.585597 | |

0.99 | 1.548987 |

**Table 4.**Effect of the Schmidt number and chemical reaction parameter for fixed ${\kappa}_{f}=10$, $M=0.1$, $\mathrm{Pr}=7$, $Sc=0.1$, $\varphi =0.1$, $\gamma =0.1$, $Rd=0.5$, $\lambda =0.2$.

$\mathit{S}\mathit{c}$ | $\mathit{\gamma}$ | $-{\mathit{\chi}}^{\prime}(0)$ |
---|---|---|

0.1 | 0.572105 | |

0.2 | 0.595779 | |

0.3 | 0.619490 | |

0.4 | 0.643226 | |

0.5 | 0.666975 | |

0.1 | 0.629969 | |

0.3 | 0.661002 | |

0.6 | 0.691290 | |

0.9 | 0.720873 | |

0.99 | 0.749786 |

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## Share and Cite

**MDPI and ACS Style**

Ali, K.; Jamshed, W.; Ahmad, S.; Bashir, H.; Ahmad, S.; Tag El Din, E.S.M.
A Self-Similar Approach to Study Nanofluid Flow Driven by a Stretching Curved Sheet. *Symmetry* **2022**, *14*, 1991.
https://doi.org/10.3390/sym14101991

**AMA Style**

Ali K, Jamshed W, Ahmad S, Bashir H, Ahmad S, Tag El Din ESM.
A Self-Similar Approach to Study Nanofluid Flow Driven by a Stretching Curved Sheet. *Symmetry*. 2022; 14(10):1991.
https://doi.org/10.3390/sym14101991

**Chicago/Turabian Style**

Ali, Kashif, Wasim Jamshed, Sohail Ahmad, Hina Bashir, Shahzad Ahmad, and El Sayed M. Tag El Din.
2022. "A Self-Similar Approach to Study Nanofluid Flow Driven by a Stretching Curved Sheet" *Symmetry* 14, no. 10: 1991.
https://doi.org/10.3390/sym14101991