# Numerical Study of Natural Convection Flow of Nanofluid Past a Circular Cone with Cattaneo–Christov Heat and Mass Flux Models

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

_{∞}and C

_{∞}are used to represent the constant ambient temperature and concentration far away from the surface. The flow is developed in an upward direction. The x-axis is taken along the surface of the cone, and the y-axis is taken normal to it as depicted in Figure 1.

_{E}and λ

_{C}are the relaxation time parameter for heat flux and mass flux, k represents the thermal conductivity, and D

_{m}is used to represent the mass diffusion coefficient. The above-generalized flux models become to Fourie and Fick’s laws, if ${\lambda}_{E}=0$, and ${\lambda}_{C}=0$, that is:

${D}_{T}$: | Thermophoretic diffusion coefficient; | $\alpha :$ | Thermal diffusivity; |

${D}_{B}$: | Brownian diffusion coefficient; | $\mathrm{g}:$ | Gravitational acceleration; |

$\tau :$ | The ratio of heat capacity of a nanoparticle to the base fluid; | $C$ | Concentration; |

$T$ | Temperature; |

## 3. Method of Solution

- (i)
- The higher-order differential equations are transformed into the first-order. For this purpose, lets us consider$${f}^{\prime}=U,{f}^{\u2033}=V,{\theta}^{\prime}=P,{\varphi}^{\prime}=Q,$$$${V}^{\prime}+\frac{7}{4}fV-\frac{1}{2}{U}^{2}+\theta =0,$$$${V}^{\prime}+\frac{7}{4}fV-\frac{1}{2}{U}^{2}+\theta =0,$$$$\frac{1}{Pr}{P}^{\prime}+\frac{7}{4}fP-{\delta}_{1}\left(\frac{35}{8}fUP+\frac{49}{16}{f}^{2}{P}^{\prime}\right)+{N}_{b}PQ+{N}_{t}{Q}^{2}=0,$$$$\frac{1}{Le}\left({Q}^{\prime}+\frac{Nt}{Nb}{P}^{\prime}\right)+\frac{7}{4}fQ-{\delta}_{2}\left(\left(\frac{35}{8}\right)fUQ+\left(\frac{49}{16}\right){f}^{2}{Q}^{\prime}\right)=0,$$$$f=0,\text{\hspace{1em}\hspace{1em}}U=0,\text{\hspace{1em}\hspace{1em}}\theta =1,\text{\hspace{1em}\hspace{1em}}\varphi =1\text{\hspace{1em}\hspace{1em}}at\text{\hspace{1em}\hspace{1em}}\eta =0,$$$$f=0,\text{\hspace{1em}\hspace{1em}}U=0,\text{\hspace{1em}\hspace{1em}}\theta =1,\text{\hspace{1em}\hspace{1em}}\varphi =1\text{\hspace{1em}\hspace{1em}}at\text{\hspace{1em}\hspace{1em}}\eta =0,$$$$U=0,\text{\hspace{1em}\hspace{1em}}\theta =0,\text{\hspace{1em}\hspace{1em}}\varphi =0\text{\hspace{1em}\hspace{1em}}as\text{\hspace{1em}\hspace{1em}}\eta \to \infty .$$
- (ii)
- The derivatives are discretized by the central difference formula$${\left(\right)}_{j-\frac{1}{2}}=\frac{1}{h}\left({\left(\right)}_{j}-{\left(\right)}_{j-1}\right),$$$${\left(\right)}_{j-\frac{1}{2}}=\frac{1}{2}\left({\left(\right)}_{j}+{\left(\right)}_{j-1}\right).$$
- (iii)
- The discretized nonlinear algebraic equations are linearized with the help of Newton’s technique. For this purpose, the functions at $\left(i+1\right)th$ iteration are written as$${\left(\right)}_{i+1}={\left(\right)}_{i}+\delta {\left(\right)}_{i},$$
- (iv)
- Linearized algebraic equations are finally solved through block-tridiagonal elimination method.

## 4. Results and Discussion

## 5. Concluding Remarks

- With the increase of Brownian motion parameters, Sherwood number increases, whereas it gains reverse behavior against thermophoresis parameter.
- With the increase of thermophoresis and Brownian motion parameters, Nusselt number decreases.
- Nusselt number decreases by increasing Prandtl and Lewis numbers.
- Sherwood number increases by increasing Lewis numbers and decreasing Prandtl numbers.
- Nusselt number decreases by increasing thermal and concentration relaxation parameters $({\delta}_{1}\text{}\mathrm{and}\text{}{\delta}_{2}$).
- Sherwood number increases by increasing concentration relaxation parameter (${\delta}_{2})$.
- Sherwood number decreases by increasing thermal relaxation parameter $({\delta}_{1})$.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

D_{T} | thermophoretic diffusion coefficient; |

D_{B} | Brownian diffusion coefficient; |

T | Temperature; |

α | thermal diffusivity; |

g | Gravitational acceleration; |

C | Concentration; |

N_{b} | Brownian motion parameter |

N_{t} | thermophoresis parameter |

T | Temperature of the fluid |

T_{∞} | Ambient fluid temperature |

T_{w} | Surface temperature |

C | Solutal concentration |

C_{∞} | Ambient solutal concentration |

C_{f} | Skin friction coefficient |

C_{w} | Solutal concentration at the wall |

Nu | Nusselt number |

Pr | Prandtl number |

Sc | Schmidt number |

Sh | Sherwood number |

$\overline{u},\overline{v}$ | Dimensional velocity components in $\overline{x}$ and $\overline{y}$ directions |

u, v | Dimensionless velocity components in x and y directions |

$\overline{x},\overline{y}$ | Coordinates along and normal to the surface in dimensional form |

x, y | Coordinates along and normal to the surface in dimensionless form |

r | Radius of the base of cone |

Gr | Grashof number |

Le | Lewis number |

Greek symbols | |

$\gamma $ | Internal half angle of the cone |

τ | The ratio of heat capacity of a nanoparticle to the base fluid |

$\varphi $ | Dimensionless concentration |

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

$\psi $ | Stream function |

v | Kinematic viscosity |

$\mu $ | Dynamic viscosity |

ρ | Fluid density |

$c$ | Relaxation time of the mass flux |

${\lambda}_{E}$ | Relaxation time of the heat flux |

${\delta}_{1}$ | Dimensionless relaxation time of the heat flux |

${\delta}_{2}$ | Dimension relaxation time of the mass flux |

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**Figure 2.**Effect of ${\delta}_{1}\&{\delta}_{2}$ on velocity, temperature, and concentration profiles.

**Figure 6.**Variation in Nusselt number against $Nt$ for different values of ${\delta}_{1}\mathrm{and}{\delta}_{2}$.

**Figure 7.**Sherwood number graph against $Nt$ for variation of ${\delta}_{1}\mathrm{and}{\delta}_{2}$.

$\mathit{P}\mathit{r}$ | ${\mathit{f}}^{\u2033}\left(0,0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0,0\right)$ | ||
---|---|---|---|---|

Present | Yih et al. [8] | Present | Yih et al. [8] | |

0.0001 | 1.452616 | 1.6006 | 0.033845 | 0.0079 |

0.001 | 1.440436 | 1.5135 | 0.038402 | 0.0246 |

0.01 | 1.348483 | 1.3551 | 0.075460 | 0.0749 |

0.1 | 1.095916 | 1.0960 | 0.211345 | 0.2116 |

0.7 | 0.819591 | - | 0.451095 | - |

1 | 0.769428 | 0.7699 | 0.510399 | 0.5109 |

10 | 0.487697 | 0.4877 | 1.033989 | 1.0339 |

100 | 0.289635 | 0.2896 | 1.922854 | 1.9226 |

1000 | 0.166145 | 0.1661 | 3.470171 | 3.4696 |

10000 | 0.094042 | 0.0940 | 6.200679 | 6.1984 |

**Table 2.**The $Nu\mathrm{and}\text{}Sh$ values for different $NtandNb$ when $Le=Pr=1.0,{\delta}_{1}={\delta}_{2}=0.1$.

$\mathit{N}\mathit{t}/\mathit{N}\mathit{b}$ | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | |

0.1 | 0.47256 | 0.29844 | 0.45902 | 0.10786 | 0.44588 | −0.06091 | 0.43312 | −0.2091 | 0.42074 | −0.33788 |

0.2 | 0.45094 | 0.41891 | 0.43785 | 0.33380 | 0.42515 | 0.25910 | 0.41284 | 0.19421 | 0.40089 | 0.13854 |

0.3 | 0.42988 | 0.46077 | 0.41724 | 0.4106 | 0.40499 | 0.36703 | 0.39311 | 0.32968 | 0.38161 | 0.29816 |

0.4 | 0.40938 | 0.48295 | 0.39720 | 0.45007 | 0.38540 | 0.42190 | 0.37396 | 0.39814 | 0.36290 | 0.37853 |

0.5 | 0.38947 | 0.49724 | 0.37774 | 0.47459 | 0.36639 | 0.45551 | 0.3554 | 0.43978 | 0.34476 | 0.42718 |

**Table 3.**$Nu\mathrm{and}\text{}Sh$ data for different $Pr\mathrm{and}Le$ when $Nt=Nb={\delta}_{1}={\delta}_{2}=0.1$.

$\mathit{P}\mathit{r}/\mathit{L}\mathit{e}$ | 2 | 3 | 5 | |||
---|---|---|---|---|---|---|

$\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | |

2 | 0.54899 | 0.44100 | 0.59205 | 0.37685 | 0.63310 | 0.30480 |

3 | 0.53896 | 0.61442 | 0.57534 | 0.56004 | 0.6035 | 0.50459 |

5 | 0.52795 | 0.84874 | 0.55651 | 0.80533 | 0.5693 | 0.76937 |

**Table 4.**$Nu\mathrm{and}\text{}Sh$ values for different ${\gamma}_{1}\mathrm{and}{\gamma}_{2}$ when $Nt=Nb=0.1,Pr=Le=1.0$.

${\mathit{\delta}}_{1}\to /{\mathit{\delta}}_{2}\downarrow $ | 0.0 | 0.2 | 0.3 | 0.4 | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | $\mathit{N}\mathit{u}$ | $\mathit{S}\mathit{h}$ | |

0.0 | 0.47432 | 0.29986 | 0.47140 | 0.29258 | 0.47015 | 0.28898 | 0.46904 | 0.28545 |

0.2 | 0.47384 | 0.30440 | 0.47091 | 0.29714 | 0.46966 | 0.29351 | 0.46855 | 0.28992 |

0.3 | 0.47358 | 0.30685 | 0.47064 | 0.29961 | 0.46939 | 0.29597 | 0.46827 | 0.29235 |

0.4 | 0.47331 | 0.30945 | 0.47036 | 0.30221 | 0.46910 | 0.29857 | 0.46798 | 0.29493 |

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**MDPI and ACS Style**

Iqbal, M.S.; Khan, W.; Mustafa, I.; Ghaffari, A.
Numerical Study of Natural Convection Flow of Nanofluid Past a Circular Cone with Cattaneo–Christov Heat and Mass Flux Models. *Symmetry* **2019**, *11*, 1363.
https://doi.org/10.3390/sym11111363

**AMA Style**

Iqbal MS, Khan W, Mustafa I, Ghaffari A.
Numerical Study of Natural Convection Flow of Nanofluid Past a Circular Cone with Cattaneo–Christov Heat and Mass Flux Models. *Symmetry*. 2019; 11(11):1363.
https://doi.org/10.3390/sym11111363

**Chicago/Turabian Style**

Iqbal, Muhammad Saleem, Waqar Khan, Irfan Mustafa, and Abuzar Ghaffari.
2019. "Numerical Study of Natural Convection Flow of Nanofluid Past a Circular Cone with Cattaneo–Christov Heat and Mass Flux Models" *Symmetry* 11, no. 11: 1363.
https://doi.org/10.3390/sym11111363