# Impacts of Stefan Blowing on Hybrid Nanofluid Flow over a Stretching Cylinder with Thermal Radiation and Dufour and Soret Effect

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

_{2}O

_{3}–Cu/H

_{2}O) flow over a stretching cylinder with radius ‘a’, as shown in Figure 1. Here, stretching cylinder taken along z-axis and r-axis is perpendicular to it. The free stream velocity and surface velocity are w

_{e}= 2cz and w

_{w}= 2bz, where $b>0$ and $c>0$ are constants. Stefan blowing, Soret–Dufour and thermal radiation effects are considered.

**Continuity Equation**

**Momentum Equation**

**Temperature Equation**

**Concentration Equation**

_{2}O

_{3}) and copper (Cu). The hybrid nanoparticle volume fraction (Al

_{2}O

_{3}–Cu) $\phi $ (referring to Waini et al. [9]) can be written as:

_{f}), Nusselt number (Nu), and Sherwood number (Sh) (referring to Waini et al. [8] and Waqas et al. [24])

## 3. Numerical Method

## 4. Results and Discussion

_{2}O

_{3}(${\phi}_{1}$) and copper Cu (${\phi}_{2}$) changes from 0 to 0.02 (2%). In Table 3, the present results of ${f}^{\u2033}\left(1\right)$ and $-2{\theta}^{\prime}\left(1\right)$ are in comparison with those of Wang [4] and Waini [8] for different values of Re, and we found that the results display good agreement.

_{f}and Nu when ${\phi}_{1}$ = 0.02, Sc = Sb = Sr = Du = Rd = 0 and Pr = 6.2 for different values of ε, ${\phi}_{2}$ and Re are given in Table 4. In Table 5, the numerical values of skin friction, Nusselt number, and Sherwood number are presented for different values of Sb, Sr, Du, Rd, ε, and ${\phi}_{2}$.

_{f}is found to be zero because the surface velocity is equal to free stream velocity. For ε < 1, C

_{f}is positive because the surface velocity is greater than the free stream velocity and vice versa is found in case of ε > 1. From Figure 7b, the Nusselt number is increasing while the value of ${\phi}_{1}$ and ${\phi}_{2}$ is increasing (up to 2%). Figure 7c, the Sherwood number increases slightly alongside the increase in the nanoparticle volume fraction of ${\phi}_{1}$ and ${\phi}_{2}$ (up to 2%).

## 5. Conclusions

- The Schmidt number is fixed in our model as water is taken as the base fluid.
- The Stefan blowing effect only arises at the impermeable surface and perpendicular to the flow direction.
- The value of the Prandtl number is dependent on the base fluid, so it ranges from 1.7 to 13.7.
- The main fallout of the current study is listed below as follows:
- As Stefan blowing intensifies, the thickness of the velocity, thermal, and concentration boundary layers grows. As a result, the hate transfer rate declines while the mass transfer rate rises.
- The convective heat transfer and mass transfer rate is improved by up to 20% with the inclusion of HNF 2%.
- Concentration (temperature) boundary layer enhanced (declines) by evaluating the Dufour and Soret numbers.
- The inclusion of thermal radiation improved the heat transfer rate as the stretching parameter increases.
- Higher values of Soret number reduce the mass transfer rate, while Stefan blowing parameter has a contrary impact on it.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HNF | Hybrid nanofluid |

MHD | Magnetohydrodynamics |

## Nomenclature

w,u | velocity components taken along z- and r-axis (m·s^{−1}) |

w_{w} | surface velocity |

T_{w} | surface temperature |

C_{w} | surface concentration |

w_{e} | free stream velocity |

T_{∞} | ambient temperature |

C_{∞} | ambient concentration |

a | cylinder radius (m) |

Du | Dufour number |

Sr | Soret number |

D | mass diffusivity (m^{2}·s^{−1}) |

k^{*} | mean absorption coefficient (c·m^{−1}) |

q_{r} | heat flux (kg·m^{2}·s^{−3}) |

k_{T} | thermal diffusion ratio |

C_{s} | concentration susceptibility |

C_{p} | specific heat (kg^{−1}·J) |

T | temperature of the fluid (K) |

C | concentration of the fluid |

k | thermal conductivity |

Pr | Prandtl number |

Rd | thermal radiation parameter |

Sc | Schmidt number |

Re | local Reynolds number |

C_{f} | skin friction coefficient |

Nu | Nusselt number |

Sh | Sherwood number |

Sb | Stefan blowing parameter |

Greek Symbols | |

ν | kinematic viscosity of the fluid (m^{2}·s^{−1}) |

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

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

μ | dynamic viscosity of the fluid (m^{2}·s^{−1}) |

α | thermal diffusivity (m^{2}·s^{−1}) |

ε | stretching parameter |

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. [Google Scholar]
- Crane, L.J. Flow past a stretching plate. Z. Angew. Phys. ZAMP
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Hayat, T.; Nadeem, S. Heat transfer enhancement with Ag–CuO/water hybrid nanofluid. Results Phys.
**2017**, 7, 2317–2324. [Google Scholar] [CrossRef] - Wang, C.Y. Stagnation flow on a cylinder with partial slip—An exact solution of the Navier–Stokes equations. IMA J. Appl. Math.
**2007**, 72, 271–277. [Google Scholar] [CrossRef] - Maskeen, M.M.; Zeeshan, A.; Mehmood, O.U.; Hassan, M. Heat transfer enhancement in hydromagnetic alumina–copper/water hybrid nanofluid flow over a stretching cylinder. J. Therm. Anal. Calorim.
**2019**, 138, 1127–1136. [Google Scholar] [CrossRef] - Rehman, S.; Anjum, A.; Farooq, M.; Hashim; Malik, M.Y. Melting heat phenomenon in thermally stratified fluid reservoirs (Powell–Eyring fluid) with joule heating. Int. Commun. Heat Mass Transf.
**2022**, 137, 106196. [Google Scholar] [CrossRef] - Salmi, A.; Madkhali, H.A.; Nawaz, M.; Alharbi, S.O.; Malik, M.Y. Non-Fourier modeling and numerical simulations on heat and transfer in tangent hyperbolic nanofluid subjected to chemical reactions. Int. Commun. Heat Mass Transf.
**2022**, 134, 105996. [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] - 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] - Waini, I.; Ishak, A.; Pop, I. Hybrid Nanofluid Flow over a Permeable Non-Isothermal Shrinking Surface. Mathematics
**2021**, 9, 538. [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] - Ali, A.; Kanwal, T.; Awais, M.; Shah, Z.; Kumam, P.; Thounthong, P. Impact of thermal radiation and non-uniform heat flux on MHD hybrid nanofluid along a stretching cylinder. Sci. Rep.
**2021**, 11, 20262. [Google Scholar] [CrossRef] - Rangi, R.R.; Ahmad, N. Boundary layer flow past a stretching cylinder and heat transfer with variable thermal conductivity. Appl. Math.
**2012**, 3, 205–209. [Google Scholar] [CrossRef] [Green Version] - Sankar, M.; Venkatachalappa, M.; Shivakumara, I.S. Effect of magnetic field on natural convection in a vertical cylindrical annulus. Int. J. Eng. Sci.
**2006**, 44, 1556–1570. [Google Scholar] [CrossRef] - Siddiqui, B.K.; Batool, S.; Mahmood ul Hassan, Q.; Malik, M.Y. Repercussions of Homogeneous and Heterogeneous Reactions of 3D Flow of Cu-Water and AL
_{2}O_{3}-Water Nanofluid and Entropy Generation Estimation along Stretching Cylinder. Ain Shams Eng. J.**2022**, 13, 101493. [Google Scholar] [CrossRef] - Lim, Y.J.; Shafie, S.; Isa, S.M.; Rawi, N.A.; Mohamad, A.Q. Impact of chemical reaction, thermal radiation and porosity on free convection Carreau fluid flow towards a stretching cylinder. Alex. Eng. J.
**2022**, 61, 4701–4717. [Google Scholar] [CrossRef] - Hayat, T.; Shehzad, S.A.; Alsaedi, A. Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid. Appl. Math. Mech.
**2012**, 33, 1301–1312. [Google Scholar] [CrossRef] - 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] - Shaheen, N.; Alshehri, H.M.; Ramzan, M.; Shah, Z.; Kumam, P. Soret and Dufour effects on a Casson nanofluid flow past a deformable cylinder with variable characteristics and Arrhenius activation energy. Sci. Rep.
**2021**, 11, 19282. [Google Scholar] [CrossRef] - Hayat, T.; Asad, S.; Alsaedi, A.; Alsaadi, F.E. Radiative Flow of Jeffrey Fluid through a Convectively Heated Stretching Cylinder. J. Mech.
**2014**, 31, 69–78. [Google Scholar] [CrossRef] - Jagan, K.; Sivasankaran, S.; Bhuvaneswari, M.; Rajan, S. Effect of non-linear radiation on 3D unsteady MHD nanoliquid flow over a stretching surface with double stratification. In Applied Mathematics and Scientific Computing; Birkhäuser: Cham, Switzerland, 2019; pp. 109–116. [Google Scholar]
- Gholinia, M.; Armin, M.; Ranjbar, A.A.; Ganji, D.D. Numerical Thermal Study on CNTs/C
_{2}H_{6}O_{2}–H_{2}O Hybrid Base Nanofluid upon a Porous Stretching Cylinder under Impact of Magnetic Source. Case Stud. Therm. Eng.**2019**, 14, 100490. [Google Scholar] [CrossRef] - Sreedevi, P.; Sudarsana Reddy, P.; Chamkha, A. Heat and Mass Transfer Analysis of Unsteady Hybrid Nanofluid Flow over a Stretching Sheet with Thermal Radiation. SN Appl. Sci.
**2020**, 2, 1222. [Google Scholar] [CrossRef] - Waqas, H.; Raza Shah Naqvi, S.M.; Alqarni, M.S.; Muhammad, T. Thermal Transport in Magnetized Flow of Hybrid Nanofluids over a Vertical Stretching Cylinder. Case Stud. Therm. Eng.
**2021**, 27, 101219. [Google Scholar] [CrossRef] - Fang, T.; Jing, W. Flow, heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 3086–3097. [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] - Gowda, R.J.; Kumar, R.N.; Rauf, A.; Prasannakumara, B.C.; Shehzad, S.A. Magnetized flow of Sutterby nanofluid through Cattaneo-Christov theory of heat diffusion and Stefan blowing condition. Appl. Nanosci.
**2021**. [Google Scholar] [CrossRef]

**Table 1.**Thermophysical attributes of base fluid (H

_{2}O) and nanoparticles (Al

_{2}O

_{3}and Cu) (referring to Waini et al. [8]).

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

ρ (kg m^{−3}) | 3970 | 8933 | 997.1 |

$k$ (W m^{−1} K^{−1}) | 40 | 400 | 0.613 |

${C}_{P}$ (J kg^{−1} K^{−1}) | 765 | 385 | 4179 |

**Table 2.**Thermophysical attributes of nanofluid and HNF (referring to Waini et al. [8]).

Properties | Nanofluid | HNF |
---|---|---|

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

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

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

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

**Table 3.**Comparison of ${f}^{\u2033}\left(1\right)$ and $-2{\theta}^{\prime}\left(1\right)$ when Pr = 6.2, $\epsilon =0$, and Sb = Sr = Du= Rd = 0, ${\phi}_{1}={\phi}_{2}=0$.

Re | ${\mathit{f}}^{\u2033}\left(1\right)$ | $-2{\mathit{\theta}}^{\prime}\left(1\right)$ | ||
---|---|---|---|---|

Wang [4] | Waini et al. [8] | Present | Present | |

0.2 | 0.78604 | 0.786042 | 0.786040 | 1.508638 |

1 | 1.48418 | 1.484183 | 1.484186 | 2.793424 |

10 | 4.16292 | 4.162920 | 4.162921 | 7.701474 |

**Table 4.**Comparison values of $\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right){C}_{f}$ and Nu when Pr = 6.2, and Sb = Sr = Du = Rd = 0.

Waini et al. [8] | Present Result | |||||
---|---|---|---|---|---|---|

${\phi}_{2}$ | Re | $\epsilon $ | $\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right){C}_{f}$ | Nu | $\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right){C}_{f}$ | Nu |

0 | 0.2 | 0 | 0.873892 | 1.632938 | 0.873890 | 1.632940 |

0.01 | - | - | 0.946854 | 1.712795 | ||

0.02 | 0.5 | 0.2 | 1.021036 | 1.792922 | 1.021036 | 1.792928 |

1 | 1.457949 | 2.509315 | 1.457940 | 2.509317 | ||

2 | 0.5 | 1.092842 | 4.177081 | 1.092840 | 4.177085 |

**Table 5.**Numerical values of $\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right){C}_{f}$, Nu and Sh when Pr = 6.2, Re = 1 and ${\phi}_{1}$ = 0.02.

Sb | Sr | Du | Rd | $\mathit{\epsilon}$ | ${\mathit{\phi}}_{2}$ | $\left(\raisebox{1ex}{$\mathrm{Re}\mathit{z}$}\!\left/ \!\raisebox{-1ex}{$\mathit{a}$}\right.\right){\mathit{C}}_{\mathit{f}}$ | Nu | Sh |
---|---|---|---|---|---|---|---|---|

0.0 | 0.1 | 0.1 | 0.1 | 0.3 | 0.02 | 1.312229 | 5.235561 | 2.147327 |

0.1 | - | - | - | - | - | 1.356625 | 3.852460 | 2.181614 |

1.0 | - | - | - | - | - | 0.899479 | 0.015974 | 1.434119 |

2.0 | - | - | - | - | - | 0.699197 | 0.000416 | 1.031123 |

- | 0.2 | - | - | - | - | 0.687271 | 0.000026 | 1.061832 |

- | 0.3 | - | - | - | - | 0.676204 | −0.000335 | 1.090929 |

- | 0.4 | - | - | - | - | 0.637198 | −0.000383 | 1.123820 |

- | - | 0.2 | - | - | - | 0.658677 | −0.001201 | 1.120188 |

- | - | 0.3 | - | - | - | 0.658589 | −0.001785 | 1.20365 |

- | - | 0.4 | - | - | - | 0.658499 | −0.002355 | 1.120542 |

- | - | - | 0.2 | - | - | 2.083692 | 21.114771 | 0.253986 |

- | - | - | 0.4 | - | - | 1.810421 | 20.757020 | 0.505700 |

- | - | - | 0.6 | - | - | 1.690819 | 22.816878 | 0.465192 |

- | - | - | - | 0.1 | - | 2.124273 | 22.499878 | 0.458991 |

- | - | - | - | 0.2 | - | 0.711615 | −0.003096 | 1.103116 |

- | - | - | - | 0.3 | - | 1.690819 | 22.816878 | 0.465192 |

- | - | - | - | - | 0.01 | 0.657825 | −0.004825 | 1.122316 |

- | - | - | - | - | 0.015 | 0.657825 | −0.004825 | 1.122316 |

- | - | - | - | - | 0.02 | 0.657825 | −0.004825 | 1.122316 |

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## Share and Cite

**MDPI and ACS Style**

Narayanaswamy, M.K.; Kandasamy, J.; Sivanandam, S.
Impacts of Stefan Blowing on Hybrid Nanofluid Flow over a Stretching Cylinder with Thermal Radiation and Dufour and Soret Effect. *Math. Comput. Appl.* **2022**, *27*, 91.
https://doi.org/10.3390/mca27060091

**AMA Style**

Narayanaswamy MK, Kandasamy J, Sivanandam S.
Impacts of Stefan Blowing on Hybrid Nanofluid Flow over a Stretching Cylinder with Thermal Radiation and Dufour and Soret Effect. *Mathematical and Computational Applications*. 2022; 27(6):91.
https://doi.org/10.3390/mca27060091

**Chicago/Turabian Style**

Narayanaswamy, Manoj Kumar, Jagan Kandasamy, and Sivasankaran Sivanandam.
2022. "Impacts of Stefan Blowing on Hybrid Nanofluid Flow over a Stretching Cylinder with Thermal Radiation and Dufour and Soret Effect" *Mathematical and Computational Applications* 27, no. 6: 91.
https://doi.org/10.3390/mca27060091