# Stagnation Point Flow and Heat Transfer over an Exponentially Stretching/Shrinking Sheet in CNT with Homogeneous–Heterogeneous Reaction: Stability Analysis

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

_{o}and there is no autocatalyst B in the external flow. Then, the homogeneous reaction (l) assures that the reaction rate in the external flow and at the outer edge of the boundary layer will be zero. In this case there is a simple relation between the concentrations $a$ and $b$, and this leads to the standard forward stagnation point boundary layer flow together with a single equation for the concentration of reactant $A$.

## 3. Stability of Solutions

## 4. Results and Discussion

## 5. Conclusions

- The skin friction and heat transfer rates increase linearly with nanoparticle volume fraction.
- Kerosene-based CNTs have higher heat transfer rates and skin friction compared to water-based CNTs.
- Single-wall CNTs show greater impact on skin friction and heat transfer rate than multi-wall CNTs in both water and kerosene.
- The concentration of species A at the surface increases with an increase in heterogeneous reaction parameter ${K}_{s}$ and Schmidt number $Sc$, while it decreases when homogeneous reaction parameter $K$ is increased.
- The range of solutions were widely expanded for exponentially shrinking cases compared with linear cases.
- The existence of unique solutions occurs for an exponentially stretching surface $\left(\epsilon >0\right)$, whereas dual solutions occur for an exponentially shrinking surface $\left({\epsilon}_{c}<\epsilon <0\right)$.
- The first solution is stable and physically realizable, while the second solution is unstable.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$a,b$ | concentrations of the chemical species |

$A,B$ | chemical species |

${a}_{o}$ | constant concentration |

$c,d$ | constants |

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

${C}_{p}$ | specific heat at constant pressure |

${D}_{A},{D}_{B}$ | diffusion coefficients |

$f$ | dimensionless stream function |

$g\left(\eta \right)$ | concentration of species A |

$h\left(\eta \right)$ | concentration of species B |

$k$ | thermal conductivity |

$K$ | strength of the homogeneous reaction |

${K}_{s}$ | strength of the heterogeneous reaction |

${k}_{1},{k}_{s}$ | constants |

${k}_{hm},{k}_{ht}$ | constants |

$L$ | characteristic length of a sheet |

$N{u}_{x}$ | local Nusselt number |

$\mathrm{Pr}$ | Prandtl number |

${q}_{w}$ | surface heat flux |

${\mathrm{Re}}_{x}$ | local Reynolds numbers |

$Sc$ | Schmidt number |

$t$ | time |

$T$ | temperature |

${T}_{o}$ | temperature constant |

$u,v$ | velocity components along the x- and y- directions, respectively |

${U}_{w}$ | stretching/shrinking velocity |

${U}_{\infty}$ | velocity of inviscid flow |

$x,y$ | cartesian coordinates along the surface and normal to it, respectively |

Greek symbols | |

$\alpha $ | thermal diffusivity |

$\delta $ | ratio of the diffusion coefficient |

$\phi $ | nanoparticle volume fraction |

$\theta $ | dimensionless temperature |

$\gamma $ | unknown eigenvalues |

$\epsilon $ | stretching/shrinking parameter |

$\nu $ | kinematic viscosity |

$\mu $ | dynamic viscosity |

$\rho $ | fluid density |

$\rho {C}_{p}$ | heat capacity of the fluid |

$\tau $ | dimensionless time variable |

${\tau}_{w}$ | surface shear stress |

$\psi $ | stream function |

$\eta $ | similarity variable |

Subscripts | |

$w$ | condition at the surface of the plate |

$\infty $ | ambient condition |

$CNT$ | carbon nanotubes |

$f$ | fluid |

$nf$ | nanofluid |

Superscript | |

$\u2019$ | differentiation with respect to $\eta $ |

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**Figure 10.**Effect of different nanoparticles and base fluids on $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ with $\phi $.

**Table 1.**Thermophysical properties of carbon nanotubes (CNTs)–nanofluid (Khan et al. [41]).

Physical Properties | Base Fluids | Nanoparticle | ||
---|---|---|---|---|

Water (Pr = 6.2) | Kerosene (Pr = 21) | SWCNT | MWCNT | |

$\rho \left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\right)$ | 997 | 783 | 2600 | 1600 |

${c}_{p}\left(\mathrm{J}/\mathrm{k}\mathrm{g}\text{}\mathrm{K}\right)$ | 4179 | 2090 | 425 | 796 |

$k\left(\mathrm{W}/\mathrm{m}\text{}\mathrm{K}\right)$ | 0.613 | 0.145 | 6600 | 3000 |

**Table 2.**Smallest eigenvalues of $\gamma $ for different values of $\epsilon $ when $\phi =0.1,K=0.2,{K}_{s}=1,Sc=1.2$, and $\mathrm{Pr}=6.2$ (Water–SWCNT).

$\mathit{\epsilon}$ | First Solution | Second Solution |
---|---|---|

−1.48706 | 0.0147 | −0.0147 |

−1.487 | 0.0413 | −0.0413 |

−1.485 | 0.2271 | −0.2261 |

−1.482 | 0.3557 | −0.3533 |

−1.48 | 0.4202 | −0.4168 |

−1.42 | 1.2926 | −1.2599 |

−1.4 | 1.4705 | −1.4278 |

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**MDPI and ACS Style**

Anuar, N.S.; Bachok, N.; Arifin, N.M.; Rosali, H.
Stagnation Point Flow and Heat Transfer over an Exponentially Stretching/Shrinking Sheet in CNT with Homogeneous–Heterogeneous Reaction: Stability Analysis. *Symmetry* **2019**, *11*, 522.
https://doi.org/10.3390/sym11040522

**AMA Style**

Anuar NS, Bachok N, Arifin NM, Rosali H.
Stagnation Point Flow and Heat Transfer over an Exponentially Stretching/Shrinking Sheet in CNT with Homogeneous–Heterogeneous Reaction: Stability Analysis. *Symmetry*. 2019; 11(4):522.
https://doi.org/10.3390/sym11040522

**Chicago/Turabian Style**

Anuar, Nur Syazana, Norfifah Bachok, Norihan Md Arifin, and Haliza Rosali.
2019. "Stagnation Point Flow and Heat Transfer over an Exponentially Stretching/Shrinking Sheet in CNT with Homogeneous–Heterogeneous Reaction: Stability Analysis" *Symmetry* 11, no. 4: 522.
https://doi.org/10.3390/sym11040522