# Three-Dimensional Non-Linearly Thermally Radiated Flow of Jeffrey Nanoliquid towards a Stretchy Surface with Convective Boundary and Cattaneo–Christov Flux

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

**Contunity Equation**

**Momentum Equation**

**Temperature Equation**

**Nano-Particle Volume Fraction Equation**

## 3. Convergence of the Solution

## 4. Computational Results and Discussion

## 5. Conclusions

- The thickening of the thermal boundary occurs while raising the thermal radiation.
- On increasing the thermal radiation, the local heat transfer diminishes and the local heat transfer raises with a raise in the Deborah number.
- The thickness of the momentum boundary layer reduces by boosting the ratio of the relaxation to retardation time; however, the skin friction rises by raising the ratio of the relaxation to retardation time.
- While boosting the thermal Biot number, the thermal boundary layer thickness rises, which results in a rise in the heat transfer rate.
- The local heat (mass) transfer rate diminishes (rises) when the Brownian motion parameter is raised.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CCHF | Cattaneo–Christov heat flux |

HAM | Homotopy Analysis Method |

Nomenclature | |

$\mathit{c}$ | ratio of stretching rates |

${c}_{p}$ | specific heat |

h | heat transfer coefficient |

k | thermal conductivity |

${k}^{\ast}$ | mean absorption coefficient |

${v}_{1},{v}_{2},{v}_{3}$ | velocity components taken along the x-, y- and z-axes |

$Cn$ | concentration |

${D}_{B}$ | Brownian motion |

${D}_{T}$ | thermophoresis coefficient |

$Nb$ | Brownian motion parameter |

$Nt$ | thermophoresis parameter |

$Pr$ | Prandtl number |

q | heat flux |

$Rd$ | radiation parameter |

$Sc$ | Schmidt number |

$Te$ | temperature |

Greek Symbols | |

$\alpha $ | thermal Biot number |

$\beta $ | Deborah number |

$\gamma $ | thermal relaxation time parameter |

${\lambda}_{1}$ | ratio of relaxation to retardation time |

${\lambda}_{2}$ | retardation time |

${\lambda}_{3}$ | thermal relaxation |

$\nu $ | kinematic viscosity |

$\rho $ | density |

${\sigma}^{\ast}$ | Stefan–Boltzmann constant |

$\tau $ | ratio between the effective nanoparticle materials and fluid heat capacity |

${\theta}_{w}$ | temperature ratio parameter |

## References

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**Figure 2.**$\mathit{h}$-curve for ${\varphi}_{1}^{\u2033}\left(0\right)$, ${\varphi}_{2}^{\u2033}\left(0\right)$, ${\varphi}_{3}^{\prime}\left(0\right)$, and ${\varphi}_{4}^{\prime}\left(0\right)$.

**Figure 3.**Influence of ${\lambda}_{1}$ on (

**a**) ${\varphi}_{1}^{\prime}\left(\zeta \right)$, (

**b**) ${\varphi}_{2}^{\prime}\left(\zeta \right)$, (

**c**) ${\varphi}_{3}\left(\zeta \right)$, and (

**d**) ${\varphi}_{4}\left(\zeta \right)$.

**Figure 4.**Influence of $\beta $ on (

**a**) ${\varphi}_{1}^{\prime}\left(\zeta \right)$, (

**b**) ${\varphi}_{2}^{\prime}\left(\zeta \right)$, (

**c**) ${\varphi}_{3}\left(\zeta \right)$, and (

**d**) ${\varphi}_{4}\left(\zeta \right)$.

**Figure 5.**Influence of c on (

**a**) ${\varphi}_{1}^{\prime}\left(\zeta \right)$, (

**b**) ${\varphi}_{3}\left(\zeta \right)$, and (

**c**) ${\varphi}_{4}\left(\zeta \right)$.

**Figure 6.**Influence of (

**a**) $Rd$ and (

**b**) $\gamma $ on ${\varphi}_{3}\left(\zeta \right)$, and (

**c**) $\alpha $ on ${\varphi}_{4}\left(\zeta \right)$.

**Figure 7.**Influence of ${\lambda}_{1}$ on (

**a**) $R{e}_{x}^{\frac{1}{2}}{C}_{{f}_{x}}$ and (

**b**) $R{e}_{y}^{\frac{1}{2}}{C}_{{f}_{y}}$.

**Table 1.**Convergence of the series $-{\varphi}_{1}^{\u2033}\left(0\right)$, $-{\varphi}_{2}^{\u2033}\left(0\right)$, $-{\varphi}_{3}^{\prime}\left(0\right)$, and $-{\varphi}_{4}^{\prime}\left(0\right)$ for $\alpha =0.3$, $\beta =0.2$, $c=0.3$, $\gamma =0.2$, ${\lambda}_{1}=0.3$, $Nb=0.2$, $Nt=0.2$, $Pr=1.0$, $Rd=0.3$, $Sc=0.6$, and ${\theta}_{w}=0.3$.

m-th-Order Approximation | $-{\mathit{\varphi}}_{1}^{\u2033}\left(0\right)$ | $-{\mathit{\varphi}}_{2}^{\u2033}\left(0\right)$ | $-{\mathit{\varphi}}_{3}^{\prime}\left(0\right)$ | $-{\mathit{\varphi}}_{4}^{\prime}\left(0\right)$ |
---|---|---|---|---|

1 | $1.0625$ | $0.2873$ | $0.2186$ | $0.7573$ |

5 | $1.1055$ | $0.2774$ | $0.2016$ | $0.4537$ |

10 | $1.1064$ | $0.2774$ | $0.1980$ | $0.3766$ |

15 | $1.1064$ | $0.2774$ | $0.1974$ | $0.3487$ |

20 | $1.1064$ | $0.2774$ | $0.1974$ | $0.3357$ |

25 | $1.1064$ | $0.2774$ | $0.1974$ | $0.3292$ |

30 | $1.1064$ | $0.2774$ | $0.1975$ | $0.3258$ |

35 | $1.1064$ | $0.2774$ | $0.1975$ | $0.3239$ |

**Table 2.**Numerical values of the skin-friction coefficient for the fixed values $\alpha =0.3$, $\gamma =0.2$, $Nb=0.2$, $Nt=0.2$, $Pr=1.0$, $Rd=0.3$, $Sc=0.6$, and ${\theta}_{w}=0.3$.

${\mathit{\lambda}}_{1}$ | $\mathit{\beta}$ | c | $-{\mathit{Re}}_{\mathit{x}}^{1/2}{\mathit{C}}_{{\mathit{f}}_{\mathit{x}}}$ | $-{\mathit{Re}}_{\mathit{y}}^{1/2}{\mathit{C}}_{{\mathit{f}}_{\mathit{y}}}$ |
---|---|---|---|---|

$0.1$ | $0.2$ | $0.3$ | $1.8505$ | $0.4639$ |

$0.2$ | $0.2$ | $0.3$ | $1.7717$ | $0.4441$ |

$0.3$ | $0.2$ | $0.3$ | $1.7022$ | $0.4267$ |

$0.4$ | $0.2$ | $0.3$ | $1.6403$ | $0.4112$ |

$0.5$ | $0.2$ | $0.3$ | $1.5847$ | $0.3973$ |

$0.3$ | $0.1$ | $0.3$ | $1.7739$ | $0.4268$ |

$0.3$ | $0.2$ | $0.3$ | $1.7022$ | $0.4267$ |

$0.3$ | $0.3$ | $0.3$ | $1.6387$ | $0.4265$ |

$0.3$ | $0.4$ | $0.3$ | $1.5819$ | $0.4263$ |

$0.3$ | $0.5$ | $0.3$ | $1.5306$ | $0.4259$ |

$0.3$ | $0.2$ | $0.1$ | $1.6361$ | $0.1204$ |

$0.3$ | $0.2$ | $0.2$ | $1.6696$ | $0.2641$ |

$0.3$ | $0.2$ | $0.3$ | $1.7022$ | $0.4267$ |

$0.3$ | $0.2$ | $0.4$ | $1.7341$ | $0.6053$ |

$0.3$ | $0.2$ | $0.5$ | $1.7656$ | $0.7976$ |

**Table 3.**Numerical values of the Nusselt number and Sherwood number for the fixed values $\alpha =0.3$, $c=0.3$, $Pr=1.0$, $Sc=0.6$, and ${\theta}_{w}=0.3$.

$\mathit{Rd}$ | ${\mathit{\lambda}}_{1}$ | $\mathit{\beta}$ | $\mathit{Nt}$ | $\mathit{Nb}$ | $\mathit{\gamma}$ | ${\mathit{Re}}_{\mathit{x}}^{-1/2}\mathit{Nu}$ | ${\mathit{Re}}_{\mathit{x}}^{-1/2}\mathit{Sh}$ |
---|---|---|---|---|---|---|---|

$0.1$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.2019$ | $0.3453$ |

$0.2$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.2007$ | $0.3471$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1996$ | $0.3487$ |

$0.4$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1986$ | $0.3503$ |

$0.5$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1976$ | $0.3518$ |

$0.3$ | $0.1$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.2016$ | $0.3613$ |

$0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.2006$ | $0.3548$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1996$ | $0.3487$ |

$0.3$ | $0.4$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1986$ | $0.3431$ |

$0.3$ | $0.5$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1977$ | $0.3380$ |

$0.3$ | $0.3$ | $0.1$ | $0.2$ | $0.2$ | $0.2$ | $0.1985$ | $0.3424$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1996$ | $0.3487$ |

$0.3$ | $0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2005$ | $0.3546$ |

$0.3$ | $0.3$ | $0.4$ | $0.2$ | $0.2$ | $0.2$ | $0.2014$ | $0.3600$ |

$0.3$ | $0.3$ | $0.5$ | $0.2$ | $0.2$ | $0.2$ | $0.2022$ | $0.3651$ |

$0.3$ | $0.3$ | $0.2$ | $0.1$ | $0.2$ | $0.2$ | $0.2001$ | $0.4146$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1996$ | $0.3487$ |

$0.3$ | $0.3$ | $0.2$ | $0.3$ | $0.2$ | $0.2$ | $0.1990$ | $0.2835$ |

$0.3$ | $0.3$ | $0.2$ | $0.4$ | $0.2$ | $0.2$ | $0.1985$ | $0.2191$ |

$0.3$ | $0.3$ | $0.2$ | $0.5$ | $0.2$ | $0.2$ | $0.1979$ | $0.1553$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.1$ | $0.2$ | $0.2020$ | $0.2098$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1996$ | $0.3487$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.3$ | $0.2$ | $0.1971$ | $0.3951$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.4$ | $0.2$ | $0.1946$ | $0.4183$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.5$ | $0.2$ | $0.1920$ | $0.4323$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.1$ | $0.1985$ | $0.3496$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.2$ | $0.1996$ | $0.3487$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.3$ | $0.2007$ | $0.3478$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.4$ | $0.2019$ | $0.3467$ |

$0.3$ | $0.3$ | $0.2$ | $0.2$ | $0.2$ | $0.5$ | $0.2032$ | $0.3456$ |

**Table 4.**Comparision of current study with Hayat et al. [27] for $\beta =0.2$, ${\lambda}_{1}=0.3$, and $c=0.3$.

m-th-Order Approximation | Hayat et al. [27] $-{\mathit{\varphi}}_{1}^{\u2033}\left(0\right)$ | Present $-{\mathit{\varphi}}_{1}^{\u2033}\left(0\right)$ | Hayat et al. [27] $-{\mathit{\varphi}}_{2}^{\u2033}\left(0\right)$ | Present $-{\mathit{\varphi}}_{2}^{\u2033}\left(0\right)$ |
---|---|---|---|---|

1 | $1.10000$ | $1.0375$ | $0.27960$ | $0.29235$ |

5 | $1.10638$ | $1.0947$ | $0.27744$ | $0.27983$ |

10 | $1.10643$ | $1.1053$ | $0.27737$ | $0.27753$ |

15 | $1.10643$ | $1.10633$ | $0.27737$ | $0.27737$ |

20 | $1.10643$ | $1.10643$ | $0.27737$ | $0.27737$ |

25 | $1.10643$ | $1.10643$ | $0.27737$ | $0.27737$ |

35 | $1.10643$ | $1.10643$ | $0.27737$ | $0.27737$ |

50 | $1.10643$ | $1.10643$ | $0.27737$ | $0.27737$ |

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**MDPI and ACS Style**

Jagan, K.; Sivasankaran, S.
Three-Dimensional Non-Linearly Thermally Radiated Flow of Jeffrey Nanoliquid towards a Stretchy Surface with Convective Boundary and Cattaneo–Christov Flux. *Math. Comput. Appl.* **2022**, *27*, 98.
https://doi.org/10.3390/mca27060098

**AMA Style**

Jagan K, Sivasankaran S.
Three-Dimensional Non-Linearly Thermally Radiated Flow of Jeffrey Nanoliquid towards a Stretchy Surface with Convective Boundary and Cattaneo–Christov Flux. *Mathematical and Computational Applications*. 2022; 27(6):98.
https://doi.org/10.3390/mca27060098

**Chicago/Turabian Style**

Jagan, Kandasamy, and Sivanandam Sivasankaran.
2022. "Three-Dimensional Non-Linearly Thermally Radiated Flow of Jeffrey Nanoliquid towards a Stretchy Surface with Convective Boundary and Cattaneo–Christov Flux" *Mathematical and Computational Applications* 27, no. 6: 98.
https://doi.org/10.3390/mca27060098