# Homotopy Semi-Numerical Modeling of Non-Newtonian Nanofluid Transport External to Multiple Geometries Using a Revised Buongiorno Model

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}-water based nanofluid. Vasu et al. [14] used the finite difference method to simulate the influence of thermophoresis and heat sink/source on double-diffusive convection of a short memory viscoelastic fluid with thermal radiative flux effects. Very recently, Vasu et al. [15] reported on numerical solutions for transient mixed convection flow of a nanofluid in the forward stagnation region of a spinning sphere under the nonlinear Boussinesq approximation. Ray et al. [16] studied electrically-conducting viscoplastic nanofluid bioconvection thin film transport phenomena from a time-dependent extending sheet. Sreenivasulu et al. [17] used Lie algebra and computational solvers to investigate the radiative heat transfer and slip effects on oblique hydromagnetic flow of a tangent hyperbolic (shear-thinning) fluid containing carbon nanotubes.

## 2. Mathematical Modeling

- (a)
- $m=1$ and $\phi \ne 0$: cone
- (b)
- $m=0$ and $\phi \ne 0$: wedge
- (c)
- $m=0$ and $\phi =0$: plate

## 3. Solution Using Homotopy Analysis Method

#### Convergence of Homotopy Series Solution

^{−7}. With the same set of parameters, the order of approximation as well as the computational time for the plate is generally less. The computations have been performed in system processor: Intel(R) Core (TM) i5–5200U [email protected] 2.20 GHz and system type: 64-bit MS Windows 10 operating system. The symbolic software Mathematica has been used to compute the results.

## 4. Results and Discussion

## 5. Conclusions

- Momentum boundary layer thickness is more significantly modified for the plate compared to the cone and wedge whereas thermal and concentration boundary layer is more significantly altered for the wedge geometry.
- Due to the boundary condition defined by modified Buongiorno model, the mass transfer rate (Sherwood number) decreases with increase in thermophoresis parameter and increases with increase in Brownian motion parameter for all the geometries.
- Brownian motion exerts a stronger influence on mass transfer rates (Sherwood numbers) for all the geometries when compared to heat transfer rates (Nusselt numbers).
- Increasing thermal Grashof number and solutal (nanoparticle) Grashof number reduce temperature and enhance wall heat transfer rates (Nusselt numbers).
- Increasing thermal Grashof number and solutal (nanoparticle) Grashof number both elevate the skin friction factor for all geometries considered (cone, wedge and plate).
- The Nusselt number for the cone is in excess of that for either a plate or wedge.
- The convective boundary condition parameter, i.e., Biot number, controls the thermal and concentration boundary layer significantly and improves the heat transfer rates (Nusselt numbers) and in particular achieves high magnitudes for the wedge (Falkner–Skan case).
- Increasing non-isothermal behaviour (rising value of wall temperature parameter) and non-iso-solutal effect (greater wall concentration parameter) magnifies the heat transfer and mass transfer rates (i.e., Nusselt and Sherwood numbers) for all geometries.
- Thermal and solute Grashof number enhances the momentum boundary layer and suppresses the thermal and concentration boundary layer for all geometries. Both thermal and species buoyancy force therefore increase the rate of heat and nanoparticle mass transfer to the wall.
- The dominant effect of increasing Schmidt number is to lessen the heat transfer rate and enhance skin friction and nanoparticle mass transfer rate for all geometries.
- Increasing Schmidt number, Brownian motion and non-iso-solutal wall parameter deplete the Nusselt numbers, i.e., reduce heat transfer rates at the walls of all the geometries studied.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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m (Order) | f″(0) | θ′(0) | ϕ′(0) |
---|---|---|---|

4 | −1.03007 | −0.89245 | 0.89245 |

8 | −1.03696 | −0.97244 | 0.97244 |

10 | −1.03724 | −0.98955 | 0.98955 |

12 | −1.03712 | −0.99906 | 0.99906 |

16 | −1.03673 | −1.00664 | −1.00665 |

20 | −1.03651 | −1.00822 | 1.00822 |

22 | −1.03658 | −1.00829 | 1.00829 |

**Table 2.**Comparison of the present study for $-{f}^{\u2033}(0)$ at $n=1,\text{}Gr=Gc=0$ for different S.

S | Exact [53] | Hassanien et al. [53] | Present |
---|---|---|---|

−1.5 | 0.500005 | 0.500005 | 0.500093 |

−1 | 0.618042 | 0.618042 | 0.6179949 |

−0.5 | 0.780781 | 0.780781 | 0.7807835 |

0 | 1.00000 | 1.00000 | 1.000000 |

0.5 | 1.280777 | 1.280777 | 1.280275 |

1 | 1.618034 | 1.618034 | 1.616967 |

1.5 | 2.00000 | 2.00000 | −1.99857 |

**Table 3.**Computational time and order of variation with respect to different residual error in temperature for the various geometries.

Geometries | Residual Error | Order | Computational Time (in Seconds) |
---|---|---|---|

Cone | 10^{−3} | 4 | 4.1638 |

10^{−5} | 6 | 10.8407 | |

10^{−7} | 12 | 76.2564 | |

Wedge | 10^{−3} | 2 | 0.758698 |

10^{−5} | 6 | 8.5965 | |

10^{−7} | 12 | 67.8979 | |

Plate | 10^{−3} | 2 | 0.75515 |

10^{−5} | 4 | 3.2184 | |

10^{−7} | 10 | 41.0949 |

**Table 4.**Effect of different parameters on skin factor, Nusselt number and Sherwood number for Cone.

$\mathbf{Pr}$ | $\mathit{G}\mathit{r}$ | $\mathit{G}\mathit{c}$ | $\mathit{S}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | $\mathit{S}\mathit{c}$ | $\mathit{N}\mathit{t}$ | $\mathit{N}\mathit{b}$ | $\mathit{\gamma}$ | $-{\mathit{C}}_{\mathit{f}}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1.0483845 | 0.780282 | −0.78028 | |||||||||

2 | 1.1143002 | 1.00665 | −1.00665 | |||||||||

5 | 1.164296 | 1.26109 | −1.26109 | |||||||||

1 | 0 | 1.1701714 | 1.006335 | −1.00634 | ||||||||

0.5 | 1.1143002 | 1.00665 | −1.00665 | |||||||||

1 | 1.0579479 | 1.007142 | −1.00714 | |||||||||

0.5 | 0 | 1.188345 | 1.002249 | −1.00225 | ||||||||

0.5 | 1.1143002 | 1.00665 | −1.00665 | |||||||||

1 | 1.0463852 | 1.011312 | −1.01131 | |||||||||

0.5 | 0 | 0.9553758 | 0.942301 | −0.9423 | ||||||||

0.2 | 1.2824339 | 1.06812 | −1.06812 | |||||||||

0.4 | 1.6362823 | 1.178777 | −1.17878 | |||||||||

0.1 | 0 | 1.0942366 | 0.878849 | −0.87885 | ||||||||

0.5 | 1.1053063 | 0.949277 | −0.94928 | |||||||||

1 | 1.1143002 | 1.00665 | −1.00665 | |||||||||

0 | 1.096399 | 1.007789 | −1.00779 | |||||||||

0.5 | 1.1065814 | 1.007163 | −1.00716 | |||||||||

1 | 1.1143002 | 1.00665 | −1.00665 | |||||||||

1 | 1.1424273 | 1.007178 | −1.00718 | |||||||||

2 | 1.0713618 | 1.00337 | −1.00337 | |||||||||

3 | 1.0576191 | 1.000187 | −1.00019 | |||||||||

1.2 | 0.1 | 1.1143002 | 1.00665 | −1.00665 | ||||||||

0.3 | 0.9815406 | 1.007427 | −3.02228 | |||||||||

0.5 | 0.8671823 | 1.0101 | −5.0505 | |||||||||

0.1 | 0.1 | 1.1143002 | 1.00665 | −1.00665 | ||||||||

0.3 | 1.1629507 | 1.003694 | −0.33456 | |||||||||

0.5 | 1.1730216 | 1.003114 | −0.20062 | |||||||||

0.1 | 1 | 1.1577563 | 0.670264 | −0.67026 | ||||||||

2 | 1.1143002 | 1.00665 | −1.00665 | |||||||||

4 | 1.072742 | 1.341994 | −1.34199 |

**Table 5.**Effect of different parameters on skin factor, Nusselt number and Sherwood number for Wedge.

$\mathbf{Pr}$ | $\mathit{G}\mathit{r}$ | $\mathit{G}\mathit{c}$ | $\mathit{S}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | $\mathit{S}\mathit{c}$ | $\mathit{N}\mathit{t}$ | $\mathit{N}\mathit{b}$ | $\mathit{\gamma}$ | $-{\mathit{C}}_{\mathit{f}}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.673389 | 0.682206 | −0.682206 | |||||||||

2 | 0.731789 | 0.888087 | −0.888087 | |||||||||

5 | 0.784382 | 1.128502 | −1.128502 | |||||||||

1 | 0 | 0.821404 | 0.883647 | −0.883647 | ||||||||

0.5 | 0.731789 | 0.888087 | −0.888087 | |||||||||

1 | 0.647859 | 0.891888 | −0.891888 | |||||||||

0.5 | 0 | 0.796872 | 0.88223 | −0.88223 | ||||||||

0.5 | 0.731789 | 0.888087 | −0.888087 | |||||||||

1 | 0.670713 | 0.894365 | −0.894365 | |||||||||

0.5 | 0 | 0.662437 | 0.852889 | −0.852889 | ||||||||

0.2 | 0.805917 | 0.923339 | −0.923339 | |||||||||

0.4 | 0.96738 | 0.993094 | −0.993094 | |||||||||

0.1 | 0 | 0.692245 | 0.665117 | −0.665117 | ||||||||

0.5 | 0.715419 | 0.79364 | −0.79364 | |||||||||

1 | 0.731789 | 0.888087 | −0.888087 | |||||||||

0 | 0.727855 | 0.888483 | −0.888482 | |||||||||

0.5 | 0.72746 | 0.888416 | −0.888416 | |||||||||

1 | 0.731789 | 0.888087 | −0.888087 | |||||||||

1 | 0.769299 | 0.886881 | −0.886881 | |||||||||

2 | 0.66277 | 0.888964 | −0.888963 | |||||||||

3 | 0.631357 | 0.888297 | −0.88829 | |||||||||

1.2 | 0.1 | 0.731789 | 0.888087 | −0.888087 | ||||||||

0.3 | 0.611019 | 0.893689 | −2.681066 | |||||||||

0.5 | 0.5012 | 0.901164 | −4.505822 | |||||||||

0.1 | 0.1 | 0.731789 | 0.888087 | −0.888087 | ||||||||

0.3 | 0.774666 | 0.884138 | −0.294713 | |||||||||

0.5 | 0.783482 | 0.88337 | −0.176674 | |||||||||

0.1 | 1 | 0.777708 | 0.614742 | −0.614742 | ||||||||

2 | 0.731789 | 0.888087 | −0.888087 | |||||||||

4 | 0.691919 | 1.141768 | −1.141768 |

**Table 6.**Effect of different parameters on skin factor, Nusselt number and Sherwood number for Plate.

$\mathbf{Pr}$ | $\mathit{G}\mathit{r}$ | $\mathit{G}\mathit{c}$ | $\mathit{S}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | $\mathit{S}\mathit{c}$ | $\mathit{N}\mathit{t}$ | $\mathit{N}\mathit{b}$ | $\mathit{\gamma}$ | $-{\mathit{C}}_{\mathit{f}}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.5967038 | 0.689775 | −0.68978 | |||||||||

2 | 0.6719077 | 0.892232 | −0.89223 | |||||||||

5 | 0.7421422 | 1.130564 | −1.13056 | |||||||||

1 | 0 | 0.7925617 | 0.886282 | −0.88628 | ||||||||

0.5 | 0.6719077 | 0.892232 | −0.89223 | |||||||||

1 | 0.5608891 | 0.896999 | −0.897 | |||||||||

0.5 | 0 | 0.7587574 | 0.883975 | −0.88398 | ||||||||

0.5 | 0.6719077 | 0.892232 | −0.89223 | |||||||||

1 | 0.5926799 | 0.901358 | −0.90136 | |||||||||

0.5 | 0 | 0.6056181 | 0.857314 | −0.85731 | ||||||||

0.2 | 0.7435705 | 0.927157 | −0.92716 | |||||||||

0.4 | 0.9023529 | 0.996141 | −0.99614 | |||||||||

0.1 | 0 | 0.619437 | 0.669709 | −0.66971 | ||||||||

0.5 | 0.6656502 | 0.892576 | −0.89258 | |||||||||

1 | 0.6719077 | 0.892232 | −0.89223 | |||||||||

0 | 0.666216 | 0.892418 | −0.89242 | |||||||||

0.5 | 0.6656502 | 0.892576 | −0.89258 | |||||||||

1 | 0.6719077 | 0.892232 | −0.89223 | |||||||||

1 | 0.7220157 | 0.890116 | −0.89012 | |||||||||

2 | 0.5816218 | 0.894629 | −0.89463 | |||||||||

3 | 0.5412732 | 0.894543 | −0.89454 | |||||||||

1.2 | 0.1 | 0.6719077 | 0.892232 | −0.89223 | ||||||||

0.3 | 0.5166001 | 0.904096 | −2.71229 | |||||||||

0.5 | 0.3788474 | 0.920086 | −4.60043 | |||||||||

0.1 | 0.1 | 0.6719077 | 0.892232 | −0.89223 | ||||||||

0.3 | 0.7287665 | 0.886638 | −0.29555 | |||||||||

0.5 | 0.7406224 | 0.885563 | −0.17711 | |||||||||

0.1 | 1 | 0.7334558 | 0.616243 | −0.61624 | ||||||||

2 | 0.6719077 | 0.892232 | −0.89223 | |||||||||

4 | 0.6192847 | 1.149904 | −1.1499 |

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**MDPI and ACS Style**

Ray, A.K.; Vasu, B.; Bég, O.A.; Gorla, R.S.R.; Murthy, P.V.S.N. Homotopy Semi-Numerical Modeling of Non-Newtonian Nanofluid Transport External to Multiple Geometries Using a Revised Buongiorno Model. *Inventions* **2019**, *4*, 54.
https://doi.org/10.3390/inventions4040054

**AMA Style**

Ray AK, Vasu B, Bég OA, Gorla RSR, Murthy PVSN. Homotopy Semi-Numerical Modeling of Non-Newtonian Nanofluid Transport External to Multiple Geometries Using a Revised Buongiorno Model. *Inventions*. 2019; 4(4):54.
https://doi.org/10.3390/inventions4040054

**Chicago/Turabian Style**

Ray, Atul Kumar, Buddakkagari Vasu, O. Anwar Bég, Rama S.R. Gorla, and P.V.S.N. Murthy. 2019. "Homotopy Semi-Numerical Modeling of Non-Newtonian Nanofluid Transport External to Multiple Geometries Using a Revised Buongiorno Model" *Inventions* 4, no. 4: 54.
https://doi.org/10.3390/inventions4040054