#
Go-MoS_{2}/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}has low thermal conductivity, there is a good chemical inertness in alumina that could maintain the stability of HNF. Additionally, Takabi et al. [4] researched HNF flow containing Al

_{2}O

_{3}-Cu nanoparticles in a sinusoidal corrugated enclosure.

_{2}nanoparticles.

## 2. Mathematical Formulation

_{1}and c

_{2}are constants and L is the characteristic length. ${B}_{0}$ is the magnetic field applied in the opposite flow direction and the radiative heat flux is defined as ${q}_{r}=\frac{-4{\sigma}^{*}}{3{k}^{*}}\frac{\partial {T}^{4}}{\partial r}$. The impacts of Stefan blowing, stagnation point, thermal radiative, Joule heating, and velocity slip are contemplated. The following assumptions are taken

- The flow is steady, laminar, and 2D-dimensional.
- The flow is incompressible.
- The cylinder is shrinking with uniform velocity along the x-direction.

**Continuity Equation**

**Momentum Equation**

**Temperature Equation**

**Concentration Equation**

_{2}O) and nanoparticles, while Table 2 provides the physical relations of the HNF. Here, the nanoparticle volume fraction of Graphene oxide (Go) and Molybdenum disulfide (MoS

_{2}) are symbolized by ${\phi}_{1}$ and ${\phi}_{2}$ respectively as follows

_{f}, Nusselt number Nu and Sherwood number Sh (referring to Waini et al. [23]).

## 3. Numerical Method

## 4. Results and Discussion

## 5. Conclusions

- Since water is used as the base fluid in our model, the Schmidt number is fixed.
- Surface velocity is always less than zero.
- The Prandtl number has a range from 1.7 to 13.7 which is based on the base fluid.

- There is an augmentation in temperature as well as concentration profiles due to the presence of Stefan blowing, but the velocity profile falls.
- The magnetic and curvature parameters cause a reduction in the temperature and concentration profiles and a rise in the velocity profile.
- The temperature profile rises due to an increase in Eckert number and thermal radiation parameter.
- The velocity profile rises as the magnetic field increases but decreases the thermal and concentration boundary layer.
- The heat transfer rate is improved by the thermal radiation parameter and shrinking case when ε < 0.
- With higher Stefan blowing, heat transmission increases but the mass transfer rate decreases in the presence of 2% of nanoparticles/HNF.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HNF | hybrid nanofluid |

MHD | Magnetohydrodynamics |

## Nomenclature

u,v | velocity components along x- and r-axes (m s^{−1}) |

L | characteristic length (m) |

w_{w} | surface velocity (m s^{−1}) |

u_{e} | free stream velocity (m s^{−1}) |

T_{w} | surface temperature (K) |

C_{w} | surface concentration (mol m^{3}) |

T_{∞} | ambient temperature (K) |

D | mass diffusivity (m^{2} s^{−1}) |

C_{∞} | ambient concentration (mol m^{3}) |

C_{s} | concentration susceptibility |

q_{r} | heat flux (kg m^{2} s^{3}) |

k_{T} | thermal diffusion ratio |

C_{p} | specific heat (kg^{−1} J) |

k^{*} | mean absorption coefficient (cm^{−1}) |

T | the temperature of the fluid (K) |

a | cylinder radius (cm) |

k | thermal conductivity (W m^{−1} K^{−1}) |

C | fluid concentration (mol m^{3}) |

Rd | thermal radiation parameter |

B_{0} | strength of magnetic field (A m^{−1}) |

Ec | Eckert number |

M | Magnetic parameter |

Pr | Prandtl number |

Sc | Schmidt number |

C_{f} | skin friction coefficient |

Sb | Stefan blowing parameter |

A | velocity slip parameter |

Re | local Reynolds number |

Nu | local Nusselt number |

Sh | local Sherwood number |

Greek symbols | |

ν | kinematic viscosity (m^{2} s^{−1}) |

σ | electrical conductivity (S m^{−1}) |

μ | dynamic viscosity (m^{2} s^{−1}) |

ρ | the density of the fluid (kg m^{−3}) |

ε | shrinking parameter (<0) |

σ^{*} | Stefan-Boltzmann constant (W m^{−2} K^{−4}) |

Subscripts | |

∞ | ambient |

f | base fluid |

nf | nanofluid |

hnf | hybrid nanofluid |

## References

- Choi, S.U.; Eastman, J.A. Enhancing Thermal Conductivity of Fluids with Nanoparticles; Argonne National Lab (ANL): Argonne, IL, USA, 1995.
- Turcu, R.; Darabont, A.; Nan, A.; Aldea, N.; Macovei, D.; Bica, D.; Vekas, L. New polypyrrole-multiwall carbon nanotubes hybrid materials. J. Optoelectron. Adv. Mater.
**2006**, 8, 643–647. [Google Scholar] - Jana, S.; Salehi Khojin, A.; Zhong, W.H. Enhancement of fluid thermal conductivity by the addition of single and hybrid nano-additives. Thermochim. Acta
**2007**, 462, 45–55. [Google Scholar] [CrossRef] - Takabi, B.; Salehi, S. Augmentation of the heat transfer performance of a sinusoidal corrugated enclosure by employing hybrid nanofluid. Adv. Mech. Eng.
**2014**, 6, 147059. [Google Scholar] [CrossRef] - Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys. ZAMP
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Waini, I.; Ishak, A.; Pop, I. Hybrid nanofluid flow towards a stagnation point on a stretching/shrinking cylinder. Sci. Rep.
**2020**, 10, 9296. [Google Scholar] [CrossRef] - Wang, C.Y. Stagnation Flow towards a Shrinking Sheet. Int. J. Non-Linear Mech.
**2008**, 43, 377–382. [Google Scholar] [CrossRef] - Waini, I.; Ishak, A.; Pop, I. Hybrid Nanofluid Flow on a Shrinking Cylinder with Prescribed Surface Heat Flux. Int. Numer. Methods Heat Fluid Flow
**2020**, 31, 1987–2004. [Google Scholar] [CrossRef] - Awaludin, I.S.; Ahmad, R.; Ishak, A. On the stability of the flow over a shrinking cylinder with prescribed surface heat flux. Propuls. Power Res.
**2020**, 9, 181–187. [Google Scholar] [CrossRef] - Ali, A.; Marwat, D.N.K.; Asghar, S. Viscous Flow over a Stretching (Shrinking) and Porous Cylinder of Non-Uniform Radius. Adv. Mech. Eng.
**2019**, 11, 168781401987984. [Google Scholar] [CrossRef] [Green Version] - Jagan, K.; Sivasankaran, S.; Bhuvaneswari, M.; Rajan, S.; Makinde, O.D. Soret and Dufour effect on MHD Jeffrey nanofluid flow towards a stretching cylinder with triple stratification, radiation and slip. Defect Diffus. Forum
**2018**, 387, 523–533. [Google Scholar] [CrossRef] - Sankar, M.; Kiran, S.; Sivasankaran, S. Natural Convection in a Linearly Heated Vertical Porous Annulus. J. Phys. Conf. Ser.
**2018**, 1139, 012018. [Google Scholar] [CrossRef] - Rashid, U.; Liang, H.; Ahmad, H.; Abbas, M.; Iqbal, A.; Hamed, Y.S. Study of (Ag and TiO
_{2})/Water Nanoparticles Shape Effect on Heat Transfer and Hybrid Nanofluid Flow toward Stretching Shrinking Horizontal Cylinder. Results Phys.**2021**, 21, 103812. [Google Scholar] [CrossRef] - Girish, N.; Sankar, M.; Makinde, O.D. Developing Buoyant Convection in Vertical Porous Annuli with Unheated Entry and Exit. Heat Transf.
**2020**, 49, 2551–2576. [Google Scholar] [CrossRef] - Chu, Y.-M.; Nisar, K.S.; Khan, U.; Daei Kasmaei, H.; Malaver, M.; Zaib, A.; Khan, I. Mixed Convection in MHD Water-Based Molybdenum Disulfide-Graphene Oxide Hybrid Nanofluid through an Upright Cylinder with Shape Factor. Water
**2020**, 12, 1723. [Google Scholar] [CrossRef] - Khan, U.; Zaib, A.; Ishak, A.; Waini, I.; Abdel-Aty, A.-H.; Sheremet, M.A.; Yahia, I.S.; Zahran, H.Y.; Galal, A.M. Agrawal Axisymmetric Rotational Stagnation-Point Flow of a Water-Based Molybdenum Disulfide-Graphene Oxide Hybrid Nanofluid and Heat Transfer Impinging on a Radially Permeable Moving Rotating Disk. Nanomaterials
**2022**, 12, 787. [Google Scholar] [CrossRef] - Wahid, N.S.; Arifin, N.M.; Pop, I.; Bachok, N.; Hafidzuddin, M.E.H. MHD stagnation-point flow of nanofluid due to a shrinking sheet with melting, viscous dissipation and Joule heating effects. Alex. Eng. J.
**2022**, 61, 12661–12672. [Google Scholar] [CrossRef] - Alarabi, T.H.; Rashad, A.M.; Mahdy, A. Homogeneous–Heterogeneous Chemical Reactions of Radiation Hybrid Nanofluid Flow on a Cylinder with Joule Heating: Nanoparticles Shape Impact. Coatings
**2021**, 11, 1490. [Google Scholar] [CrossRef] - Khashi’ie, N.S.; Waini, I.; Arifin, N.M.; Pop, I. Unsteady Squeezing Flow of Cu–Al
_{2}O_{3}/Water Hybrid Nanofluid in a Horizontal Channel with Magnetic Field. Sci. Rep.**2021**, 11, 14128. [Google Scholar] [CrossRef] - Jagan, K.; Sivasankaran, S. Soret & Dufour and Triple Stratification Effect on MHD Flow with Velocity Slip towards a Stretching Cylinder. Math. Comput. Appl.
**2022**, 27, 25. [Google Scholar] [CrossRef] - Yashkun, U.; Zaimi, K.; Abu Bakar, N.A.; Ishak, A.; Pop, I. MHD Hybrid Nanofluid Flow over a Permeable Stretching/Shrinking Sheet with Thermal Radiation Effect. Int. J. Numer. Methods Heat Fluid Flow
**2020**, 31, 1014–1031. [Google Scholar] [CrossRef] - Pal, D.; Mandal, G.; Vajravalu, K. Soret and Dufour Effects on MHD Convective–Radiative Heat and Mass Transfer of Nanofluids over a Vertical Non-Linear Stretching/Shrinking Sheet. Appl. Math. Comput.
**2016**, 287–288, 184–200. [Google Scholar] [CrossRef] - Waini, I.; Khan, U.; Zaib, A.; Ishak, A.; Pop, I. Inspection of TiO
_{2}-CoFe_{2}O_{4}Nanoparticles on MHD Flow toward a Shrinking Cylinder with Radiative Heat Transfer. J. Mol. Liq.**2022**, 361, 119615. [Google Scholar] [CrossRef] - Aladdin, N.A.L.; Bachok, N.; Rosali, H.; Wahi, N.; Abd Rahmin, N.A.; Arifin, N.M. Numerical Computation of Hybrid Carbon Nanotubes Flow over a Stretching/Shrinking Vertical Cylinder in Presence of Thermal Radiation and Hydromagnetic. Mathematics
**2022**, 10, 3551. [Google Scholar] [CrossRef] - Eswaramoorthi, S.; Jagan, K.; Sivasankaran, S. MHD Bioconvective Flow of a Thermally Radiative Nanoliquid in a Stratified Medium Considering Gyrotactic Microorganisms. J. Phys. Conf. Ser.
**2020**, 1597, 012001. [Google Scholar] [CrossRef] - Uddin, M.J.; Alginahi, Y.; Bég, O.A.; Kabir, M.N. Numerical Solutions for Gyrotactic Bioconvection in Nanofluid-Saturated Porous Media with Stefan Blowing and Multiple Slip Effects. Comput. Math. Appl.
**2016**, 72, 2562–2581. [Google Scholar] [CrossRef] - Lund, L.A.; Omar, Z.; Raza, J.; Khan, I.; Sherif, E.-S.M. Effects of Stefan Blowing and Slip Conditions on Unsteady MHD Casson Nanofluid Flow over an Unsteady Shrinking Sheet: Dual Solutions. Symmetry
**2020**, 12, 487. [Google Scholar] [CrossRef] [Green Version] - Ali, B.; Hussain, S.; Abdal, S.; Mehdi, M.M. Impact of Stefan Blowing on Thermal Radiation and Cattaneo–Christov Characteristics for Nanofluid Flow Containing Microorganisms with Ablation/Accretion of Leading Edge: FEM Approach. Eur. Phys. J. Plus
**2020**, 135, 821. [Google Scholar] [CrossRef] - Rana, P.; Makkar, V.; Gupta, G. Finite element study of bio-convective Stefan blowing Ag-MgO/water hybrid nanofluid induced by stretching cylinder utilizing non-Fourier and non-Fick’s laws. Nanomaterials
**2021**, 11, 1735. [Google Scholar] [CrossRef] - Raza, A.; Khan, S.U.; Khan, M.I.; Farid, S.; Muhammad, T.; Khan, M.I.; Galal, A.M. Fractional order simulations for the thermal determination of graphene oxide (GO) and molybdenum disulphide (MoS
_{2}) nanoparticles with slip effects. Case Stud. Therm. Eng.**2021**, 28, 101453. [Google Scholar] [CrossRef]

**Figure 2.**Impacts of Sb on ${f}^{\prime}\left(\eta \right)$, $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

**Figure 4.**Impacts of M on ${f}^{\prime}\left(\eta \right)$, $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

**Figure 7.**Impacts of ${\phi}_{2}$ on ${f}^{\prime}\left(\eta \right)$ and $\theta \left(\eta \right)$.

Properties | Go | MoS_{2} | H_{2}O |
---|---|---|---|

ρ (kg m^{−3}) | 1800 | 5060 | 997.1 |

${C}_{P}$ (J kg^{−1} K^{−1}) | 717 | 397.21 | 4179 |

$k$ (W m^{−1} K^{−1}) | 5000 | 904.4 | 0.63 |

σ (S m^{−1}) | 6.30 × 10^{7} | 2.09 × 10^{4} | 0.05 |

Properties | Correlations of HNF |
---|---|

Density | ${\rho}_{hnf}=\left(1-{\phi}_{2}\right)\left[\left(1-{\phi}_{1}\right){\rho}_{f}+{\phi}_{1}{\rho}_{n1}\right]+{\phi}_{2}{\rho}_{n2}$ |

Heat Capacity | ${\left(\rho {C}_{p}\right)}_{hnf}=\left(1-{\phi}_{2}\right)\left[\left(1-{\phi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}+{\phi}_{1}{\left(\rho {C}_{p}\right)}_{n1}\right]+{\phi}_{2}{\left(\rho {C}_{p}\right)}_{n2}$ |

Dynamic Viscosity | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\phi}_{1}\right)}^{2.5}{\left(1-{\phi}_{2}\right)}^{2.5}}$ |

Thermal Conductivity | $\frac{{k}_{hnf}}{{k}_{nf}}=\frac{{k}_{n2}+2{k}_{nf}-2{\phi}_{2}\left({k}_{nf}-{k}_{n2}\right)}{{k}_{nf}+2{k}_{nf}+{\phi}_{2}\left({k}_{nf}-{k}_{n2}\right)}\mathrm{where}\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{n1}+2{k}_{f}-2{\phi}_{1}\left({k}_{f}-{k}_{n1}\right)}{{k}_{n1}+2{k}_{f}+{\phi}_{1}\left({k}_{f}-{k}_{n1}\right)}$ |

Electric conductivity | $\frac{{\sigma}_{hnf}}{{\sigma}_{nf}}=\frac{{\sigma}_{n2}+2{\sigma}_{nf}-2{\phi}_{2}\left({\sigma}_{nf}-{\sigma}_{n2}\right)}{{\sigma}_{nf}+2{\sigma}_{nf}+{\phi}_{2}\left({\sigma}_{nf}-{\sigma}_{n2}\right)}\mathrm{where}\frac{{\sigma}_{nf}}{{\sigma}_{f}}=\frac{{\sigma}_{n1}+2{\sigma}_{f}-2{\phi}_{1}\left({\sigma}_{f}-{\sigma}_{n1}\right)}{{\sigma}_{n1}+2{\sigma}_{f}+{\phi}_{1}\left({\sigma}_{f}-{\sigma}_{n1}\right)}$ |

**Table 3.**Comparison results where Sb = λ = M = Ec = Rd = Sc = K = 0 and ${\phi}_{1}={\phi}_{2}=0$ when Pr = 6.2.

ε | ${\mathit{f}}^{\u2033}\left(\mathit{\eta}\right)$ | ||
---|---|---|---|

Wang [7] | Waini et al. [8] | Present Result | |

−1 | 1.328820 | 1.328817 | 1.328820 |

−0.5 | 1.495670 | 1.495670 | 1.495670 |

0 | 1.232588 | 1.232588 | 1.232588 |

Sb | M | Ec | Rd | ε | ${\mathit{\phi}}_{2}$ | ${\mathit{R}\mathit{e}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$ | ${\mathit{R}\mathit{e}}_{\mathit{x}}^{-1/2}\mathit{N}\mathit{u}$ | ${\mathit{R}\mathit{e}}_{\mathit{x}}^{-1/2}\mathit{S}\mathit{h}$ |
---|---|---|---|---|---|---|---|---|

0 | 0.1 | 0.1 | 0.1 | −0.5 | 0.02 | 1.578609 | 1.269266 | 0.471888 |

0.1 | - | - | - | - | - | 1.369317 | 1.098009 | 0.438235 |

1.0 | - | - | - | - | - | 1.180348 | 0.291620 | 0.308434 |

2.0 | - | - | - | - | - | 1.067028 | 0.072320 | 0.237721 |

- | 0.1 | - | - | - | - | 1.067028 | 0.072320 | 0.237721 |

- | 0.2 | - | - | - | - | 1.161345 | 0.032563 | 0.234640 |

- | 0.3 | - | - | - | - | 1.205204 | 0.016897 | 0.237441 |

- | - | 0.1 | - | - | - | 1.221111 | 0.008869 | 0.238595 |

- | - | 0.2 | - | - | - | 1.172795 | −0.022768 | 0.240634 |

- | - | 0.3 | - | - | - | 1.197174 | −0.101285 | 0.238227 |

- | - | - | 0.2 | - | - | 1.139701 | −0.095667 | 0.24396 |

- | - | - | 0.4 | - | - | 1.139701 | −0.078478 | 0.243946 |

- | - | - | 0.6 | - | - | 1.139700 | −0.049477 | 0.243946 |

- | - | - | - | −0.1 | - | 0.970726 | 0.054617 | 0.273412 |

- | - | - | - | −0.2 | - | 1.014529 | 0.036342 | 0.267537 |

- | - | - | - | −0.3 | - | 1.098488 | −0.022513 | 0.256132 |

- | - | - | - | - | 0.01 | 1.115239 | −0.36832 | 0.254321 |

- | - | - | - | - | 0.015 | 1.115239 | −0.36832 | 0.254321 |

- | - | - | - | - | 0.02 | 1.115239 | −0.036832 | 0.254321 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Narayanaswamy, M.K.; Kandasamy, J.; Sivanandam, S.
Go-MoS_{2}/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation. *Math. Comput. Appl.* **2022**, *27*, 110.
https://doi.org/10.3390/mca27060110

**AMA Style**

Narayanaswamy MK, Kandasamy J, Sivanandam S.
Go-MoS_{2}/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation. *Mathematical and Computational Applications*. 2022; 27(6):110.
https://doi.org/10.3390/mca27060110

**Chicago/Turabian Style**

Narayanaswamy, Manoj Kumar, Jagan Kandasamy, and Sivasankaran Sivanandam.
2022. "Go-MoS_{2}/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation" *Mathematical and Computational Applications* 27, no. 6: 110.
https://doi.org/10.3390/mca27060110