#
Role of Nanoparticles and Heat Source/Sink on MHD Flow of Cu-H_{2}O Nanofluid Flow Past a Vertical Plate with Soret and Dufour Effects

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## Abstract

**:**

## 1. Introduction

_{2}O nanofluid into a porous stretching/shrinking channel. Stagnation point flows of upper convex Maxwell fluid past a stretching plate are evaluated by Ibrahim and Negera [4]. In their valuable work, Kataria et al. [5] originated the concept of heat generation/absorption magnetohydrodynamic via fluid flow past a porous vertical plate. The problem is solved by employing the Laplace transform technique, and the physical significance of pertinent parameters is tested. Kumar et al. [6] analyzed the performance of Casson and Maxwell fluids past a stretching sheet with an internal heat source and sink. It was reported that the thermal and concentration fields of Maxwell fluid are highly influenced by the non-dimensional parameters, compared to Casson fluid. Swain et al. [7] surveyed the incompressible Newtonian fluid over a porous stretching sheet. The impact of the porous parameter acting as an aiding force is reported in this study. Muhammed et al. [8] analyzed the 3D stretched flow of viscous dissipation with prescribed heat and concentration fluxes. In this research study, a withal magnetic field is applied in the flow region, and mathematical equations with physical quantifiers are formulated. Jithender et al. [9] demonstrated unsteady MHD Casson flow for the geometrical model of a plate in oscillation motion vertically, and the numerical outputs were obtained by computing the finite element method. Basant et al. [10] used two concentric cylinders to model a vertical annular micro-channel under the influence of a radial magnetic field. Amira et al. [11] improved the idea of hybrid nanofluids expressing mathematical models of stretching and shrinking sheets, and flow equations were solved by means of appropriate similarity transformations. Jawad et al. [12] investigated heat transfer in a semi-porous channel with stretching walls using MHD nanofluids. The channel was filled with an incompressible copper–water nanofluid and the outputs were revealed graphically. Gireesha et al. [13] examined the rate of nanoparticle injection and chemical reaction from steady planar Couette flow through a permeable micro-channel using the Runge–Kutta–Fehlberg fourth order. Recently, an increasing number of studies concerning nanofluids have been described in [14,15,16,17,18]. The impression of Soret and Dufour effects boosts the transmission of heat and mass. These effects play an important role when there are density differences in the flow. Hayat et al. [19] generalized three-dimensional radiative flow with Soret and Dufour effects. Saritha et al. [20] analyzed Soret and Dufour combined impact on the MHD flow of a power-law fluid across a flat plate. On a flat plate, MHD boundary-layer slip flow with Soret and Dufour implications was reported by Reddy and Saritha [21]. The investigation flow model is filled with second grade fluid, electrically conducting through a magnetic field. Jyotsa et al. [22] have identified an advanced mathematical model of exponentially accelerated inclined plates and dimensionless quantifiers that are tested for radiation and magnetic strength. [23]. Rashidi et al. [24] conveyed a buoyancy effect on the MHD flow of nanofluid over a stretching sheet. Studies on how different MHD nanofluids transmit heat differently due to differences in shape were carried out by [25,26,27,28].

## 2. Problem Formulation

_{∞}. A transverse magnetic field that is perpendicular to the plate and has an intensity, B

_{0}, that is constant, is meant to be applied. When the magnetic Reynolds number is low, the result of the induced magnetic field, which is significant, is irrelevant. The fluid is assumed to contain water-based magnetic nanoparticles such as aluminum oxide Al

_{2}O

_{3}and cupper Cu. Figure 1 represents the physical model, Table 1 lists the thermophysical attributes of nanoparticles and thermal conductivity for spherical shaped nanoparticles are tabulated in Table 2.

**Table 2.**Thermal conductivity for spherical shaped nanoparticles [34].

Model | Shape of Nanoparticles | Thermal Conductivity |
---|---|---|

I | Spherical | ${k}_{nf}={k}_{f}\left(\frac{{k}_{s}+2{k}_{f}-2\phi \left({k}_{f}-{k}_{s}\right)}{{k}_{s}+2{k}_{f}+\phi \left({k}_{f}-{k}_{s}\right)}\right)$ |

II | Spherical | ${k}_{nf}={k}_{f}\left(\frac{{k}_{s}+2{k}_{f}-2\phi \left({k}_{f}-{k}_{s}\right)}{{k}_{s}+2{k}_{f}+\phi \left({k}_{f}-{k}_{s}\right)}\right)$ |

## 3. Numerical Procedure

_{i}are given in Appendix A.

## 4. Findings and Discussion

## 5. Conclusions

- ➢
- Fluid velocity rises with the Grashof number while it falls in the magnetic field.
- ➢
- The effects of Prandtl number and viscous dissipation are to improve the velocity and temperature.
- ➢
- The Dufour effect raises the velocity and temperature while reducing due to the Soret effect.
- ➢
- Thermal conductivity is enhanced by heat sources and radiation.
- ➢
- With chemical reaction and Schmidt number, concentration decreases.
- ➢
- Rate of heat transfer accelerated with Du values and retards with the values of Q and Ec.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

B_{0} | Applied magnetic field |

C | Non-dimensional concentration |

C_{p} | Specific heat (constant pressure) |

C^{*} | Species concentration |

C_{∞}^{*} | Free stream concentration |

C_{w}^{*} | Species concentration at wall |

Du | Dufour number |

g | Acceleration due to gravity |

Gr | Grashof number |

K | Permeability parameter |

k_{f} | Thermal conductivity of the base fluid |

k_{s} | Thermal conductivity of the nanoparticles |

k_{nf} | Thermal conductivity of the nanofluid |

k^{*} | Mean absorption coefficient |

M | Magnetic field parameter |

N_{r} | Radiation parameter |

Pr | Prandtl number |

Q | Heat generation parameter |

T | Non-dimensional temperature |

T^{*} | Temperature |

T_{∞}^{*} | Free-stream temperature |

t^{*} | Time |

T_{w}^{*} | Temperature at wall |

Sc | Schmidt number |

Sr | Soret number |

φ | Solid volume fraction of the nanoparticle |

δ | Chemical reaction parameter |

λ | Buoyancy parameter |

u^{*} | Velocity components along x^{*}−y^{*} direction |

u_{0} | Initial velocity |

nf | Nanofluid |

q_{r} | Radiative heat flux |

σ_{nf} | Electrical conductivity of the nanofluid |

β_{nf} | Thermal expansion coefficient of the nanofluid |

(x^{*}, y^{*}) | Dimensional co-ordinates |

ρ_{nf} | Nanofluid density |

μ_{f} | Viscosity of the base fluid |

## Appendix A

## References

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**Figure 1.**The physical model and coordinate system [29].

H_{2}O | Al_{2}O_{3} | Cu | |
---|---|---|---|

${c}_{p}\left({\mathrm{JKg}}^{-1}{\mathrm{K}}^{-1}\right)$ | 4179 | 765 | 8993 |

$\rho \left({\mathrm{Kgm}}^{-3}\right)$ | 997.1 | 3970 | 385 |

$k\left({\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}\right)$ | 0.613 | 40 | 401 |

$\beta \times {10}^{-5}\left({\mathrm{K}}^{-1}\right)$ | 21 | 401 | 1.67 |

**Table 3.**The Nusselt number $Nu=-\frac{{K}_{nf}}{{K}_{f}}{\left(\frac{\partial \theta}{\partial y}\right)}_{y=0}$, Pr = 7.

Q | Du | Ec | Nu |
---|---|---|---|

0.2 | 0.5 | 0.01 | 0.5765 |

0.5 | 0.2624 | ||

1.0 | −0.4015 | ||

1.0 | 0.5721 | ||

1.5 | 0.5701 | ||

0.02 | 0.5677 | ||

0.03 | 0.5601 |

**Table 4.**Numerical values of Skin friction (C

_{f}) and Nusselt number (${\theta}^{\prime}(0)$) with Pr = 6.2, φ = 0.05 and Ec = 0.01 for Cu-water nanofluid as $\beta \to \infty $.

Gr | M | λ | R | Present | Previous (Khan et al. [27]) | ||
---|---|---|---|---|---|---|---|

C_{f} | −${\mathbf{\theta}}^{\mathbf{\prime}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ | C_{f} | −${\mathit{\theta}}^{\mathbf{\prime}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ | ||||

5 | 1 | 0.2 | 1 | 0.7295 | 1.1654 | 0.7296 | 1.1653 |

10 | 2.6396 | 2.6395 | |||||

3 | −0.2742 | 1.0512 | −0.2741 | 1.0511 | |||

5 | −0.9281 | 1.0488 | −0.9283 | 1.0488 | |||

0.3 | 0.586 | 0.5867 | |||||

0.4 | 0.4377 | 0.4378 | |||||

2 | 1.0480 | 0.7974 | 1.0479 | 0.7973 | |||

3 | 1.2629 | 0.6620 | 1.2629 | 1.2630 |

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**MDPI and ACS Style**

Kune, R.; Naik, H.S.; Reddy, B.S.; Chesneau, C.
Role of Nanoparticles and Heat Source/Sink on MHD Flow of Cu-H_{2}O Nanofluid Flow Past a Vertical Plate with Soret and Dufour Effects. *Math. Comput. Appl.* **2022**, *27*, 102.
https://doi.org/10.3390/mca27060102

**AMA Style**

Kune R, Naik HS, Reddy BS, Chesneau C.
Role of Nanoparticles and Heat Source/Sink on MHD Flow of Cu-H_{2}O Nanofluid Flow Past a Vertical Plate with Soret and Dufour Effects. *Mathematical and Computational Applications*. 2022; 27(6):102.
https://doi.org/10.3390/mca27060102

**Chicago/Turabian Style**

Kune, Ramesh, Hari Singh Naik, Borra Shashidar Reddy, and Christophe Chesneau.
2022. "Role of Nanoparticles and Heat Source/Sink on MHD Flow of Cu-H_{2}O Nanofluid Flow Past a Vertical Plate with Soret and Dufour Effects" *Mathematical and Computational Applications* 27, no. 6: 102.
https://doi.org/10.3390/mca27060102