# Impacts of Casson Model on Hybrid Nanofluid Flow over a Moving Thin Needle with Dufour and Soret and Thermal Radiation Effects

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## Abstract

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_{3}-Cu/Ethylene glycol nanofluid flow over a moving thin needle under MHD, Dufour–Soret effects, and thermal radiation. By utilizing the appropriate transformations, the governing partial differential equations are transformed into ordinary differential equations. The transformed ordinary differential equations are solved analytically using HAM. Furthermore, we discuss velocity profiles, temperature profiles, and concentration profiles for various values of governing parameters. Skin friction coefficient increases by upto 45% as the Casson parameter raised upto 20%, and the heat transfer rate also increases with the inclusion of nanoparticles. Additionally, local skin friction, a local Nusselt number, and a local Sherwood number for many parameters are entangled in this article.

## 1. Introduction

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_{3}-Cu/water flow while taking buoyancy effects into account. Kavya et al. [8] looked into the issue of the Williamson hybrid nanofluid in a stretching cylinder with water-based Cu and MoS4 nanoparticles. The results indicate a decline in the heat transfer properties of the Williamson hybrid nanofluid flow with increasing Prandtl numbers and curvature parameter values. Raju et al. [9,10,11] investigated the flow and heat transmission characteristics of a nanofluid in a contracting or expanding porous channel with different permeabilities under the influence of MHD and thermal radiation.

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_{3}-water) nanoliquid and (AA7072-AA7075/water) hybrid nanofluid over a moving thin needle. Additionally, the combined effect of the Dufour and Soret diffusions tend to increase the heat transfer coefficient; however, the mass transfer coefficient exhibits dual behaviors.

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_{3}-Cu/Ethylene glycol) nanofluids over a moving thin needle. Additional MHD, thermal radiation, and Dufour and Soret effects are included. The HAM technique in the Mathematica software has been used to solve the specified problem in this study. Furthermore, tables and graphs are used to present the noteworthy physical quantities varying with parameters.

## 2. Mathematical Formulation of the Problem

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_{3}-Cu/Ethylene glycol) nanofluid over a lateral moving thin needle with radius c = 0.1. In cylindrical coordinates, variables x and r represent axial and radial coordinates, respectively. The thin needle’s velocity ${U}_{w}$ can move in the same direction or on the opposite direction of the free stream velocity ${U}_{\infty}$. Furthermore, a magnetic field perpendicular to the direction of flow is applied. It is also assumed that the concentrations at the wall and in the area are constant. Furthermore, it is assumed that the fundamental equations contain terms for MHD, thermal radiation, and Soret and Dufour terms.

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_{3}and Cu, respectively.

## 3. Method of Solution

## 4. Results and Discussions

## 5. Conclusions

- The boundary layer that forms around the needle is equally tiny in size.
- The series solution is obtained with the proposed HAM method. HAM gives better approximations than other numerical methods.
- In total, 2% nanoparticles are included and the rate of heat transfer increases up to 45%.
- The Casson parameter tends to makes the flow field smaller. Physically, increasing the Casson parameter causes an enhancement in the fluids dynamic viscosity, which reduces the fluid motion and results in a decrease in the profile of velocity.
- As the magnetic field strength increases, the velocity profiles become narrower, because magnetic fields produce Lorentz forces that oppose motion.
- As a result of the thermal radiation being included, heat is transferred more quickly.
- As the radiation parameter is increased, the temperature within the boundary layer naturally increases, because thermal radiation is dominant over a thermal conduction heat transfer.
- The rates of mass and heat transfer as well as the Dufour and Soret effects are accelerated, because with the Dufour effect, the concentration gradient affects the flow of thermal energy flux, and higher Soret numbers correspond to a greater temperature gradient, resulting in a greater convective flow.
- The present results are useful in the cooling technology and thermal science community.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MHD | Magnetohydrodynamics | |

HAM | Homotopy Analysis Method | |

c | needle radius (m) | |

${U}_{w}$ | velocity at the surface (m.s^{−1}) | |

${C}_{w}$ | wall concentration | |

${T}_{w}$ | temperature at the surface (K) | |

${C}_{\infty}$ | ambient concentration | |

${T}_{m}$ | temperature of mass fluid (K) | |

${K}^{*}$ | coefficient of mean absorption (c.m^{−1}) | |

${C}_{s}$ | concentration susceptibility | |

${U}_{\infty}$ | free stream velocity (m.s^{−1}) | |

${C}_{p}$ | specific heat (kg^{−1}.J) | |

${D}_{m}$ | mass diffusion coefficient (m^{2}.s^{−1}) | |

${T}_{\infty}$ | ambient temperature (K) | |

${u}_{{}_{1},}{v}_{1}$ | velocity components along x and r directions (m.s^{−1}) | |

${q}_{r}$ | radiative heat flux (kg.m^{2}.s^{−3}) | |

${K}_{T}$ | ratio of thermal diffusion | |

${C}_{f}$ | Skin friction | |

$S{h}_{x}$ | Sherwood number | |

C | fluid concentration | |

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

Greek symbols | ||

$\nu $ | kinematic viscosity (m^{2}.s^{−1}) | |

$\alpha $ | Casson parameter | |

$\rho $ | fluid density (kg.m^{−3)} | |

$\epsilon $ | velocity ratio parameter | |

$\mu $ | dynamic viscosity of a fluid (m^{2}.s^{−1}) | |

${\sigma}^{*}$ | Stefan-Boltzmann constant (W.m^{−2}.K^{−4}) | |

Subscripts | ||

hnf | hybrid nanofluid | |

‘ | differentiation w.r.t. $\eta $ | |

nf | nanofluid | |

f | fluid | |

$\infty $ | ambient |

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**Figure 3.**Impacts of (

**a**) Casson parameter $\alpha $ on ${f}^{\prime}\left(\eta \right)$ and (

**b**) Magnetic parameter M on ${f}^{\prime}\left(\eta \right)$.

**Figure 4.**Impacts of (

**a**) Dufour parameter Du on $\theta \left(\eta \right)$ and (

**b**) Radiation parameter Rd on $\theta \left(\eta \right)$.

**Figure 6.**Influence of (

**a**) $\alpha $ on ${C}_{f}{\left(R{e}_{x}\right)}^{0.5}$, (

**b**) $\alpha $ on $N{u}_{x}{\left(R{e}_{x}\right)}^{-0.5}$, and (

**c**) $\alpha $ on $S{h}_{x}{\left(R{e}_{x}\right)}^{-0.5}$.

**Figure 7.**Influence of (

**a**) M on ${C}_{f}{\left(R{e}_{x}\right)}^{0.5}$, (

**b**) M on $N{u}_{x}{\left(R{e}_{x}\right)}^{-0.5}$, and (

**c**) M on $S{h}_{x}{\left(R{e}_{x}\right)}^{-0.5}$.

**Figure 8.**Impacts of (

**a**) Rd on $N{u}_{x}{\left(R{e}_{x}\right)}^{-0.5}$ and (

**b**) Rd on $S{h}_{x}{\left(R{e}_{x}\right)}^{-0.5}$.

**Figure 9.**Impacts of (

**a**) Du on $N{u}_{x}{\left(R{e}_{x}\right)}^{-0.5}$ and (

**b**) Du on $S{h}_{x}{\left(R{e}_{x}\right)}^{-0.5}$.

**Table 1.**Thermophysical properties of nanoparticles and base fluid (Sulochana et al. [17]).

Properties | Al_{2}O_{3} | Cu | Ethylene Glycol |
---|---|---|---|

$\rho $ (kg.m^{−3}) | 3970 | 8933 | 1190 |

$k$ (W.m^{−1}.K^{−1}) | 40 | 400 | 0.258 |

${C}_{P}$ (J.kg^{−1}.K^{−1}) | 765 | 385 | 2400 |

$\sigma $ ($\Omega $.m^{−1}) | $1.502\times {10}^{-10}$ | $5.96\times {10}^{7}$ | $10.7\times {10}^{-5}$ |

Properties | Nanofluid | Hybrid Nanofluid |
---|---|---|

Density | ${\rho}_{nf}={\varphi}_{1}{\rho}_{s1}+(1-{\varphi}_{1}){\rho}_{f}$ | $\begin{array}{l}{\rho}_{hnf}=[{\varphi}_{1}{\rho}_{s1}+(1-{\varphi}_{2}){\rho}_{f}](1-{\varphi}_{1})\\ +{\varphi}_{2}{\rho}_{s2}\end{array}$ |

Heat capacity | $\begin{array}{l}{(\rho {C}_{p})}_{nf}=(1-{\varphi}_{1}){(\rho {C}_{p})}_{f}\\ +{\varphi}_{1}{(\rho {C}_{p})}_{s1}\end{array}$ | $\begin{array}{l}{(\rho {C}_{p})}_{hnf}=[(1-{\varphi}_{2}){(\rho {C}_{p})}_{f}+{\varphi}_{1}{(\rho {C}_{p})}_{s1}]\\ (1-{\varphi}_{1})+{\varphi}_{2}{(\rho {C}_{p})}_{s2}\end{array}$ |

Dynamic viscosity | ${\mu}_{nf}=\frac{{\mu}_{f}}{{(1-{\varphi}_{1})}^{2.5}}$ | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{(1-{\varphi}_{1})}^{2.5}{(1-{\varphi}_{2})}^{2.5}}$ |

Thermal conductivity | $\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{s1}-2{\varphi}_{1}({k}_{f}-{k}_{s1})+2{k}_{f}}{{k}_{s1}+{\varphi}_{1}({k}_{f}-{k}_{s1})+2{k}_{f}}$ | $\frac{{k}_{hnf}}{{k}_{f}}=\frac{{k}_{s2}+{k}_{nf}(2-2{\varphi}_{2})+2{\varphi}_{2}{k}_{s2}}{{k}_{s2}-{\varphi}_{2}{k}_{s2}+{k}_{nf}(2+{\varphi}_{2})}$ |

Electrical Conductivity | $\frac{{\sigma}_{nf}}{{\sigma}_{f}}=\frac{{\sigma}_{s1}-2{\varphi}_{1}({\sigma}_{f}-{\sigma}_{s1})+2{\sigma}_{f}}{{\sigma}_{s1}+{\varphi}_{1}({\sigma}_{f}-{\sigma}_{s1})+2{\sigma}_{f}}$ | $\frac{{\sigma}_{hnf}}{{\sigma}_{f}}=\frac{{\sigma}_{s2}+{\sigma}_{nf}(2-2{\varphi}_{2})+2{\varphi}_{2}{\sigma}_{s2}}{{\sigma}_{s2}-{\varphi}_{2}{\sigma}_{s2}+{\sigma}_{nf}(2+{\varphi}_{2})}$ |

**Table 3.**Convergences of series ${\varphi}_{1}=0.01$, ${\varphi}_{2}=0.01$, $\epsilon =1$, $\alpha =1$, Pr = 7, Sr = 0.1, c = M = 0.1, Rd = Du = 0.2, and Sc = 0.6.

Order | $-{\mathit{f}}^{\u2033}(\mathit{c})$ | $-{\mathit{\theta}}^{\prime}(\mathit{c})$ | $-{\mathit{\varphi}}^{\prime}(\mathit{c})$ |
---|---|---|---|

1 | 0.50000 | 1.00001 | 1.00001 |

5 | 0.50002 | 1.00005 | 1.00004 |

10 | 0.50004 | 1.00011 | 1.00008 |

15 | 0.50006 | 1.00015 | 1.00012 |

20 | 0.50009 | 1.0002 | 1.00016 |

25 | 0.50011 | 1.00025 | 1.0002 |

30 | 0.50013 | 1.0003 | 1.00024 |

35 | 0.50016 | 1.00035 | 1.00029 |

40 | 0.50018 | 1.0004 | 1.00033 |

**Table 4.**${C}_{f}{\left(R{e}_{x}\right)}^{0.5}$, $N{u}_{x}{\left(R{e}_{x}\right)}^{-0.5}$, and $S{h}_{x}{\left(R{e}_{x}\right)}^{-0.5}$ number values computed for various physical parameters.

$\mathit{\alpha}$ | M | Du | Rd | ${\mathit{C}}_{\mathit{f}}{\left(\mathit{R}{\mathit{e}}_{\mathit{x}}\right)}^{0.5}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\left(\mathit{R}{\mathit{e}}_{\mathit{x}}\right)}^{-0.5}$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}{\left(\mathit{R}{\mathit{e}}_{\mathit{x}}\right)}^{-0.5}$ |
---|---|---|---|---|---|---|

1 | 0.1 | 0.2 | 0.2 | −2.30857 | −1.92005 | −1.75259 |

1.4 | −1.63701 | 1.46098 | 1.46092 | |||

1.7 | 1.46086 | 1.04996 | 1.04996 | |||

2 | 1.04991 | 1.04987 | 1.04983 | |||

1 | 0.1 | 0.2 | 0.2 | −2.30857 | −2.37217 | −2.43422 |

0.2 | −2.4948 | 1.46113 | 1.46059 | |||

0.3 | 1.46006 | 1.45955 | 1.05001 | |||

0.4 | 1.0496 | 1.0492 | 1.04881 | |||

1 | 0.1 | 0.2 | 0.1 | −2.30857 | −2.30857 | −2.30857 |

0.2 | −2.30857 | 1.17214 | 1.46113 | |||

0.3 | 1.71912 | 1.9514 | 1.0528 | |||

0.4 | 1.05001 | 1.04792 | 1.04628 | |||

1 | 0.1 | 0.1 | 0.2 | −2.30857 | −2.30857 | −2.30857 |

0.2 | −2.30857 | 1.44608 | 1.46113 | |||

0.3 | 1.4762 | 1.49127 | 1.05029 | |||

0.4 | 1.05001 | 1.04974 | 1.04946 |

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**MDPI and ACS Style**

Reddy, V.S.; Kandasamy, J.; Sivanandam, S.
Impacts of Casson Model on Hybrid Nanofluid Flow over a Moving Thin Needle with Dufour and Soret and Thermal Radiation Effects. *Math. Comput. Appl.* **2023**, *28*, 2.
https://doi.org/10.3390/mca28010002

**AMA Style**

Reddy VS, Kandasamy J, Sivanandam S.
Impacts of Casson Model on Hybrid Nanofluid Flow over a Moving Thin Needle with Dufour and Soret and Thermal Radiation Effects. *Mathematical and Computational Applications*. 2023; 28(1):2.
https://doi.org/10.3390/mca28010002

**Chicago/Turabian Style**

Reddy, Vinodh Srinivasa, Jagan Kandasamy, and Sivasankaran Sivanandam.
2023. "Impacts of Casson Model on Hybrid Nanofluid Flow over a Moving Thin Needle with Dufour and Soret and Thermal Radiation Effects" *Mathematical and Computational Applications* 28, no. 1: 2.
https://doi.org/10.3390/mca28010002