# Finite Difference Method to Evaluate the Characteristics of Optically Dense Gray Nanofluid Heat Transfer around the Surface of a Sphere and in the Plume Region

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## Abstract

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## 1. Introduction

## 2. Flow Analysis for Region-I

## 3. Solution Process

#### 3.1. Dimensionless Variables

#### 3.2. Primitive Variable Formulation

#### 3.3. Computational Technique

## 4. Flow Analysis for Plume Region-III

## 5. Solution Process

#### 5.1. Dimensionless Variables

#### 5.2. Primitive Variable Formulation

#### 5.3. Computational Technique

## 6. Results and Discussion

#### 6.1. Analysis of Heat and Fluid Flow Characteristics around the Sphere

_{,}the Nusselt number $NuG{r}^{-1/4}$, and the Sherwood number $ShG{r}^{-1/4}$ for diverse values of the governing parameters. These results are evaluated at various locations on the surface of the sphere. Table 1 presents the results of ${C}_{f}G{r}^{1/4}$, $NuG{r}^{-1/4}$, and $ShG{r}^{-1/4}$ for various values of the Schmidt number $Sc$ at different points of the surface of the sphere. It is noticed that $Sc$, ${C}_{f}G{r}^{1/4}$, $NuG{r}^{-1/4}$, and $ShG{r}^{-1/4}$ are higher at $X=\mathrm{0.1,3.1}$ and lower at $X=1.0,2.0$. On the other hand, $NuG{r}^{-1/4}$ and $ShG{r}^{-1/4}$ risefor $X=\mathrm{0.1,2.0,3.1}$ but reduceat $X=1.0$. Similarly, Table 2 depicts the effect of the Prandtl number on the evaluated physical properties. It has to be mentioned that the increase in $Pr$ causes the increase in ${C}_{f}G{r}^{1/4}$, but reductions occur in $NuG{r}^{-1/4}$ and $ShG{r}^{-1/4}$. In Table 3, the effects of $N$ on ${C}_{f}G{r}^{1/4}$, $NuG{r}^{-1/4}$, and $ShG{r}^{-1/4}$ are illustrated. Augmenting $N$ causes a decline in ${C}_{f}G{r}^{1/4}$ and an increase in $NuG{r}^{-1/4}$ and $ShG{r}^{-1/4}$. Table 4 portrays the trend of the same properties for different values of ${N}_{t}$. It can be seen that all the properties go down by increasing the values of ${N}_{t}$. Table 5 presents the behavior of the skin of friction coefficient, Nusselt number, and Sherwood number for various values of ${N}_{b}$. It is observed that each variable of interest increases against increasing values of ${N}_{b}$.

#### 6.2. Physical Behavior of Material Properties in the Plume Region-III

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\mathit{u}$ | Dimensionless velocity component in x direction |

$\mathit{v}$ | Dimensionless velocity component in $\mathit{y}$ direction |

$\mathit{w}$ | Dimensionless velocity component in $\mathit{z}$ direction in plume region |

$\mathit{x},\mathit{y}$ | Dimensionless axes along and normal to the surface of a sphere |

$\mathit{x},\mathit{z}$ | Dimensionless axes along and normal in the plume region |

$\mathit{U}$ | Primitive variable for velocity component in $\mathit{X}$ direction |

$\mathit{V}$ | Primitive variable for velocity component in $\mathit{Y}$ direction |

$\mathit{W}$ | Primitive variable for velocity component in $Z$ direction in plume region |

$\mathit{g}\left(\mathit{m}{\mathit{s}}^{-2}\right)$ | Gravitational acceleration |

$\mathit{T}\left(\mathit{K}\right)$ | Fluid temperature in boundary layer |

$\mathit{C}\left(\mathit{k}\mathit{g}{\mathit{m}}^{-3}\right)$ | Mass concentration in boundary layer |

${\mathit{C}}_{\mathit{P}}\left(\mathit{J}\mathit{k}{\mathit{g}}^{-1}.{\mathit{K}}^{-1}\right)$ | Specific heat at constant pressure |

$\mathit{a}\left(\mathit{m}\right)$ | The radius of a sphere |

$\widehat{\mathit{r}}\left(\mathit{m}\right)$ | Dimensioned radial distance from the symmetric axis to the surface of a sphere |

${\mathit{D}}_{\mathit{m}}\left({\mathit{m}}^{2}{\mathit{s}}^{-1}\right)$ | Mass diffusion coefficient |

$\mathit{G}\mathit{r}$ | Grashof number |

$\mathit{N}$ | Radiation parameter |

${\mathit{N}}_{\mathit{t}}$ | Thermophoresis parameter |

${\mathit{N}}_{\mathit{b}}$ | Brownian motion parameter |

${\mathit{q}}_{\mathit{r}}$ | Radiative heat flux |

${\mathit{C}}_{\mathit{f}}$ | Skin friction coefficient |

$\mathit{N}\mathit{u}$ | Nusselt number |

$\mathit{S}\mathit{h}$ | Sherwood number |

Greek Symbols | |

${\beta}_{T}\left({K}^{-1}\right)$ | Volumetric coefficient thermal expansion |

${\beta}_{C}\left({K}^{-1}\right)$ | Volumetric coefficient concentration expansion |

$\alpha \left(m{s}^{-1}\right)$ | Thermal diffusivity |

$\theta $ | Dimensionless temperature |

$\varphi $ | Dimensionless mass concentration |

$\mu \left(Pa.s\right)$ | Dynamic viscosity |

$\nu \left({m}^{2}{s}^{-1}\right)$ | Kinematic viscosity |

$\rho \left(kg{m}^{-3}\right)$ | Fluid density |

$\kappa \left(W{m}^{-1}.{K}^{-1}\right)$ | Thermal conductivity |

$\sigma $ | Stefan–Boltzmann constant |

${k}^{*}$ | Mean absorption coefficient |

Subscripts | |

$\infty $ | Ambient conditions |

$w$ | Wall conditions |

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**Figure 2.**Outcomes of radiation parameter $N$ on velocity distribution $U$ when $Nb=1.5,Nt=1.5,Pr=7.0,$ and $Sc=10.0.$

**Figure 3.**Outcomes of radiation parameter $N$ on temperature distribution $\theta $ when $Nb=1.5,Nt=1.5,Pr=7.0,$ and $Sc=10.0$.

**Figure 4.**Outcomes of radiation parameter $N$ on mass concentration $\varphi $ when $Nb=1.5,Nt=1.5,Pr=7.0,$ and $Sc=10.0$.

**Figure 5.**Outcomes of thermophoresis parameter ${N}_{t}$ on velocity distribution $U$ when N = 0.5, Nb = 1.5, Pr = 7.0, and Sc = 10.0.

**Figure 6.**Outcomes of thermophoresis parameter ${N}_{t}$ on temperature distribution $\theta $ when N = 0.5, Nb = 1.5, Pr = 7.0, and Sc = 10.0.

**Figure 7.**Outcomes of thermophoresis parameter ${N}_{t}$ on mass concentration $\varphi $ when N = 0.5, Nb = 1.5, Pr = 7.0, and Sc = 10.0.

**Figure 8.**Outcomes of Brownian motion parameter ${N}_{b}$ on velocity distribution $U$ when $N=0.5,Pr=7.0,Nt=1.5$, and $Sc=10.0.$

**Figure 9.**Outcomes of Brownian motion parameter ${N}_{b}$ on temperature distribution $\theta $ when $N=0.5,Pr=7.0,Nt=1.5$, and $Sc=10.0.$

**Figure 10.**Outcomes of Brownian motion parameter ${N}_{b}$ on mass concentration $\varphi $ when $N=0.5,Pr=7.0,Nt=1.5$, and $Sc=10.0.$

**Figure 11.**Outcomes of Schmidt number $Sc$ on mass concentration $\varphi $ when $N=0.5,Nb=1.5,Nt$ = 1.5, and $Pr=7.0.$

**Figure 12.**Outcomes of radiation parameter $N$ on velocity distribution $U$, when $Nb=1.5,Nt=1.5,Pr=7.0$, and $Sc=10.0$.

**Figure 13.**Outcomes of radiation parameter $N$ on temperature distribution $\theta $ when $Nb=1.5,Nt=1.5,Pr=7.0$, and $Sc=10.0$.

**Figure 14.**Outcomes of radiation parameter $N$ on mass concentration $\varphi $ when $Nb=1.5,Nt=1.5,Pr=7.0$, and $Sc=10.0$.

**Table 1.**Outcomes of the Schmidt number $Sc$ on (a) the skin friction coefficient ${C}_{f}G{r}^{1/4}$, (b) Nusselt $NuG{r}^{-1/4}$, and (c) the Sherwood number $ShG{r}^{-1/4}$ when $Nt=0.2,Nb=10.0,N=1.1,$ and $Pr=7.0.$

X | ${\mathit{C}}_{\mathit{f}}\mathit{G}{\mathit{r}}^{1/4}$ | $\mathit{N}\mathit{u}\mathit{G}{\mathit{r}}^{-1/4}$ | $\mathit{S}\mathit{h}\mathit{G}{\mathit{r}}^{-1/4}$ | |||
---|---|---|---|---|---|---|

$\mathit{S}\mathit{c}=0.1$ | $\mathit{S}\mathit{c}=2.0$ | $\mathit{S}\mathit{c}=0.1$ | $\mathit{S}\mathit{c}=2.0$ | $\mathit{S}\mathit{c}=0.1$ | $\mathit{S}\mathit{c}=2.0$ | |

0.1 | 0.00981 | 0.01145 | 0.06308 | 0.06349 | 0.07619 | 0.07801 |

1.0 | 0.06199 | 0.04599 | 0.06578 | 0.08904 | 0.07682 | 0.10350 |

2.0 | 0.10721 | 0.04818 | 0.19978 | 0.09024 | 0.20579 | 0.10476 |

3.1 | 0.00534 | 0.00578 | 0.05831 | 0.05809 | 0.07276 | 0.07327 |

**Table 2.**Outcomes of the Prandtl number $Pr$ on (a) the skin friction coefficient ${C}_{f}G{r}^{1/4}$, (b) Nusselt $NuG{r}^{-1/4}$, and (c) the Sherwood number $ShG{r}^{-1/4}$ when $Nt=0.2,Nb=10.0,N=1.1,$ and $Sc=10.0.$

X | ${\mathit{C}}_{\mathit{f}}\mathit{G}{\mathit{r}}^{1/4}$ | $\mathit{N}\mathit{u}\mathit{G}{\mathit{r}}^{-1/4}$ | $\mathit{S}\mathit{h}\mathit{G}{\mathit{r}}^{-1/4}$ | |||
---|---|---|---|---|---|---|

$\mathit{P}\mathit{r}=7.0$ | $\mathit{P}\mathit{r}=10.0$ | $\mathit{P}\mathit{r}=7.0$ | $\mathit{P}\mathit{r}=10.0$ | $\mathit{P}\mathit{r}=7.0$ | $\mathit{P}\mathit{r}=10.0$ | |

0.1 | 0.01271 | 0.01483 | 0.06050 | 0.05501 | 0.07456 | 0.07243 |

1.0 | 0.08087 | 0.08239 | 0.10321 | 0.09503 | 0.12380 | 0.12105 |

2.0 | 0.08619 | 0.08762 | 0.10548 | 0.09702 | 0.12654 | 0.12361 |

3.1 | 0.00611 | 0.00742 | 0.05663 | 0.04794 | 0.07183 | 0.06604 |

**Table 3.**Outcomes of the radiation parameter $N$ on (a) the skin friction coefficient ${C}_{f}G{r}^{1/4}$, (b) Nusselt $NuG{r}^{-1/4}$, and (c) the Sherwood number $ShG{r}^{-1/4}$ when $Nt=0.2,Nb=0.3,Sc=10.0,$ and $Pr=7.0.$

X | ${\mathit{C}}_{\mathit{f}}\mathit{G}{\mathit{r}}^{1/4}$ | $\mathit{N}\mathit{u}\mathit{G}{\mathit{r}}^{-1/4}$ | $\mathit{S}\mathit{h}\mathit{G}{\mathit{r}}^{-1/4}$ | |||
---|---|---|---|---|---|---|

$\mathit{N}=0.7$ | $\mathit{N}=1.1$ | $\mathit{N}=0.7$ | $\mathit{N}=1.1$ | $\mathit{N}=0.7$ | $\mathit{N}=1.1$ | |

0.1 | 0.01812 | 0.01421 | 0.01271 | 0.06050 | 0.07342 | 0.07456 |

1.0 | 0.15241 | 0.08169 | 0.08087 | 0.10321 | 0.12246 | 0.12380 |

2.0 | 0.16466 | 0.08695 | 0.08619 | 0.10548 | 0.12508 | 0.12654 |

3.1 | 0.02562 | 0.00611 | 0.05072 | 0.05663 | 0.00679 | 0.07183 |

**Table 4.**Outcomes of the thermophoresis parameter ${N}_{t}$ on (a) the skin friction coefficient ${C}_{f}G{r}^{1/4}$, (b) Nusselt $NuG{r}^{-1/4}$, and (c) the Sherwood number $ShG{r}^{-1/4}$ when $Sc=10.0,Nb=0.3,N=1.0$, and Pr = 7.0.

X | ${\mathit{C}}_{\mathit{f}}\mathit{G}{\mathit{r}}^{1/4}$ | $\mathit{N}\mathit{u}\mathit{G}{\mathit{r}}^{-1/4}$ | $\mathit{S}\mathit{h}\mathit{G}{\mathit{r}}^{-1/4}$ | |||
---|---|---|---|---|---|---|

$\mathit{N}\mathit{t}=0.2$ | $\mathit{N}\mathit{t}=0.9$ | $\mathit{N}\mathit{t}=0.2$ | $\mathit{N}\mathit{t}=0.9$ | $\mathit{N}\mathit{t}=0.2$ | $\mathit{N}\mathit{t}=0.9$ | |

0.1 | 0.01271 | 0.01093 | 0.06050 | 0.00835 | 0.07456 | 0.02171 |

1.0 | 0.08087 | 0.25808 | 0.10321 | 2.87762 | 0.12380 | 0.18380 |

2.0 | 0.08619 | 0.26958 | 0.10548 | 0.09417 | 0.12654 | 0.19084 |

3.1 | 0.00611 | 0.01412 | 0.05663 | 0.06653 | 0.07183 | 0.05583 |

**Table 5.**Outcomes of the Brownian motion parameter ${N}_{b}$ on (a) the skin friction coefficient ${C}_{f}G{r}^{1/4}$, (b) Nusselt $NuG{r}^{-1/4}$, and (c) the Sherwood number $ShG{r}^{-1/4}$ when $Nt=0.2,Sc=10.0,N=1.1,$ and $Pr=7.0$.

X | ${\mathit{C}}_{\mathit{f}}\mathit{G}{\mathit{r}}^{1/4}$ | $\mathit{N}\mathit{u}\mathit{G}{\mathit{r}}^{-1/4}$ | $\mathit{S}\mathit{h}\mathit{G}{\mathit{r}}^{-1/4}$ | |||
---|---|---|---|---|---|---|

$\mathit{N}\mathit{b}=0.3$ | $\mathit{N}\mathit{b}=0.9$ | $\mathit{N}\mathit{b}=0.3$ | $\mathit{N}\mathit{b}=0.9$ | $\mathit{N}\mathit{b}=0.3$ | $\mathit{N}\mathit{b}=0.9$ | |

0.1 | 0.01271 | 0.03029 | 0.06050 | 0.03671 | 0.07456 | 0.07692 |

1.0 | 0.08087 | 0.12463 | 0.10321 | 0.06132 | 0.12380 | 0.11358 |

2.0 | 0.08619 | 0.13128 | 0.10548 | 0.06242 | 0.12654 | 0.11540 |

3.1 | 0.00611 | 0.01716 | 0.05663 | 0.02969 | 0.07183 | 0.07301 |

**Table 6.**Outcomes of radiation parameter $N$ on (a) the skin friction coefficient ${C}_{f}G{r}^{1/4}$, (b) Nusselt $NuG{r}^{-1/4}$, and (c) the Sherwood number $ShG{r}^{-1/4}$ when $Nt=0.2$, $Nb=0.3,Sc=1.0$, and $Pr=7.0$.

X | ${\mathit{C}}_{\mathit{f}}\mathit{G}{\mathit{r}}^{1/4}$ | $\mathit{N}\mathit{u}\mathit{G}{\mathit{r}}^{-1/4}$ | $\mathit{S}\mathit{h}\mathit{G}{\mathit{r}}^{-1/4}$ | |||
---|---|---|---|---|---|---|

$\mathit{N}=0.5$ | $\mathit{N}=0.7$ | $\mathit{N}=0.5$ | $\mathit{N}=0.7$ | $\mathit{N}=0.5$ | $\mathit{N}=0.7$ | |

0.0 | 50.75594 | 46.76019 | 0.95139 | 0.23755 | 192.65468 | 179.60024 |

0.1 | 32.08902 | 34.08479 | 7.68669 | 7.03547 | 82.65567 | 75.70414 |

1.0 | 15.67088 | 13.73597 | 9.27360 | 11.81349 | 0.11835 | 0.12578 |

2.0 | 10.61131 | 8.79781 | 9.27483 | 11.81388 | 0.11340 | 0.11728 |

3.0 | 7.79274 | 5.95524 | 18.88779 | 17.98556 | 3.34746 | 4.44476 |

4.0 | 5.76776 | 3.84707 | 18.63854 | 18.20510 | 2.30285 | 4.74118 |

5.0 | 4.13138 | 2.18996 | 18.63854 | 17.75627 | 2.29610 | 6.89948 |

6.0 | 2.79540 | 1.27903 | 17.91728 | 16.67107 | 3.51993 | 9.09658 |

7.0 | 1.68713 | 0.95814 | 16.81319 | 15.59305 | 6.43594 | 12.18080 |

8.0 | 1.14654 | 0.78669 | 16.92500 | 14.97582 | 9.41521 | 13.28487 |

9.0 | 0.91312 | 0.68151 | 15.51050 | 14.72355 | 11.44516 | 13.84803 |

10.0 | 0.77717 | 0.60034 | 15.29638 | 14.52141 | 12.45936 | 14.1815 |

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**MDPI and ACS Style**

Ashraf, M.; Khan, A.; Abbas, A.; Hussanan, A.; Ghachem, K.; Maatki, C.; Kolsi, L.
Finite Difference Method to Evaluate the Characteristics of Optically Dense Gray Nanofluid Heat Transfer around the Surface of a Sphere and in the Plume Region. *Mathematics* **2023**, *11*, 908.
https://doi.org/10.3390/math11040908

**AMA Style**

Ashraf M, Khan A, Abbas A, Hussanan A, Ghachem K, Maatki C, Kolsi L.
Finite Difference Method to Evaluate the Characteristics of Optically Dense Gray Nanofluid Heat Transfer around the Surface of a Sphere and in the Plume Region. *Mathematics*. 2023; 11(4):908.
https://doi.org/10.3390/math11040908

**Chicago/Turabian Style**

Ashraf, Muhammad, Anwar Khan, Amir Abbas, Abid Hussanan, Kaouther Ghachem, Chemseddine Maatki, and Lioua Kolsi.
2023. "Finite Difference Method to Evaluate the Characteristics of Optically Dense Gray Nanofluid Heat Transfer around the Surface of a Sphere and in the Plume Region" *Mathematics* 11, no. 4: 908.
https://doi.org/10.3390/math11040908