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Impact of Thermal Radiation on MHD GO-Fe_{2}O_{4}/EG Flow and Heat Transfer over a Moving Surface

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## Abstract

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## Highlights

- The stagnation point of heat transfer and hybrid nanofluid flow using the Powell-Eyring model toward stretching/shrinking surface problem is studied. The metal particles used are GO and Fe
_{2}O_{4}and Ethylene Glycol (EG) as base fluid. The analysis of entropy when the surface shrunk is highlighted. The thermal radiation and magnetic effects are also investigated.

## Abstract

_{2}O

_{4}) with the base fluid chosen as ethylene glycol (EG) is analyzed, including the effects of radiation and magnetic influence. The hybrid nanofluid flow is assumed to be asymmetric because it flows along a horizontal shrinking surface in response to external inducements. The mathematically modelled partial differential equations (PDEs) form is then derived into ordinary differential equations (ODEs) by implementing a proper similarity transformation to the PDEs. The mathematical formulation is then algorithmically estimated employing the bvp4c solver in MATLAB. The parameters’ effects on the skin friction measurement, local Nusselt number, entropy generation, velocity profile, and temperature profile are investigated and explained. This finding illustrated that the skin friction is augmented between 13.7% and 66.5% with the magnetic field, velocity slips, and the concentration of GO particles. As for the heat transmission ratio, only thermal radiation and velocity slip effects will affect the heat upsurge with the range of 99.8–147% for taken parameter values. The entropy for the shrinking case is found to increase between 16.6% and 43.9% with the magnetic field, velocity slip, and Eckert number.

## 1. Introduction

_{2}O

_{3}/Blood fluid subject to Joule heating and other few impacts and concluded that the Hartmann number reduced the temperature of the fluid. Other previous papers that investigated Joule heating covering liquid movement were the exploration of Joule heating on MHD movement of Walters-B nanofluid close to extending slab, adopting the Runge–Kutta–Fehlberg technique by Gholinia et al. [12], Joule heating reactions in porous materials by Alaidrous and Eid [13], where they discussed electromagnetic radiative non-Newtonian nanofluid by optimal homotopy analytical method, Joule heating analysis of Cattaneo–Christov Heat Flux on CuO-Cu-Al

_{2}O

_{3}/H

_{2}O flow over an inclined contracting plate by Jangid et al. [14], and the exploration of MHD bioconvection of nano liquid flow on an elongating wedge by Ferdows et al. [15].

_{2}O within a wavy microchannel [16] and the analysis of Ag/H

_{2}O inside a duplex-layered sinusoidal heat sink [17] are some of the studies on conventional nanofluids. Qureshi [18] compared the proportion of heat transmission between ZrO

_{2}–Cu/EO mixture nanofluid and ZrO

_{2}-EO nanofluid. The outcomes of this investigation revealed that mixture nanofluid is a superior heat conductor compared to traditional nanofluid. Mashayekhi et al. [19] investigated Cu-Al

_{2}O

_{3}/H

_{2}O within a binary-layered microchannel with heat absorption and found that the wall’s structure affects the heat transmission rate significantly. Hussain et al. [20] did a study about the reaction and heat transfer on GO-MoS

_{2}/H

_{2}O hybrid nanofluid on a curved surface with shape factors. They found that shape factors have a more noticeable impact on a hybrid nanofluid. The involvement of GO was examined in [21] with the effect of radiation, and it was concluded that the concentration of GO augments the momentum and thermal effects of the liquid. Another research that studied GO hybrid nanofluid compares GO/EO and GO-Fe

_{3}O

_{4}with thermal and magnetic effects [22]. Few other studies show the usage of Fe

_{2}O

_{4}particles in heat transfer problems [23,24,25].

_{2}O

_{3}/ethylene glycol fluid [29] and the effects of MHD and heat source/sink of Powell–Eyring fluid on a stretching surface [30]. One of the crucial components that is needed in the heat transfer process is the base fluid. However, using the conventional base fluid without adding nanoparticles gives poor thermal conductivity. As a result, the researchers started to add particles to the base fluids. Some fluids that are mainly used for this process are water, engine oil, kerosene oil, and ethylene glycol. A few studies on ethylene glycol as the base fluid have been completed earlier, such as MHD flow of Cu/EG nanofluid with heat generation/absorption effect [31], Cu-Fe3O4/EG-based hybrid nanofluid slip flow on the elongating surface [32], and the mixed convective flow of SWCNT-MWCNT/EG hybrid nanofluid [33].

_{6}Al

_{4}V/C

_{2}H

_{6}O

_{2}-H

_{2}O was studied by Roja [34], where an increase in radiation led to an increase in entropy and decrease in the temperature of the liquid. Rashad et al. [35] looked into MHD Powell–Eyring Cu-Fe

_{3}O

_{4}/EG within a permeable medium. They detected that as the value of the magnetic field and radiation were enhanced, the rapidity of the fluid is improved, although the radiation effect alone enhanced the temperature within the boundary layer. Many scholars solved the problem of the radiation effect in heat transfer by studying the laminar flow of Cu/H

_{2}O within an upright channel [36], MHD flow of Cu/H

_{2}O between parallel plates [37], and MHD flow of GO/H

_{2}O in a porous channel [38] for nanofluids and MHD flow of CuO-Al

_{2}O

_{3}/H

_{2}O over two different geometries [39], MHD Al

_{2}O

_{3}-Ag/H

_{2}O flow over a stretching sheet [40], and MHD Williamson MoS

_{2}-ZnO/EG flow over a permeable stretching sheet [41] for hybrid nanofluids.

_{2}O

_{3}/EG hybrid nanofluid through a rotating channel was analytically solved and has been reported that a stronger magnetic field leads to growth of entropy generation, as the magnetic field produces a lot of resistance creating more Joule heating in the system [42]. The radiation effect also increases entropy in the system because the higher intensity of radiation increases the temperature of the liquid. Similar results on the effect of magnetic and radiation intensities on the entropy of the system can be seen in [43]. The entropy of non–Newtonian Carreau hybrid nanofluid was studied in [44], and it was concluded that a hybrid nanoparticle system produces more entropy than one that uses only one nanomaterial. The study of entropy and curvature effect of Cu–Al

_{2}O

_{3}/Blood in [45] showed that a higher concentration of the nanoparticles generates higher entropy because a higher concentration increases the flow’s momentum and enhances the randomness of the particles. They also stated that entropy in the system of hybrid nanofluid is high, while the entropy produced from only the base fluid system is lower. Entropy of hybrid nanofluids was further analyzed by Ghali et al. [46] and Hayat et al. [47].

_{2}O

_{4}/EG hybrid nanofluid’s flow and heat transfer over the expanding and contracting surfaces with thermal radiation effect. Fe

_{2}O

_{4}is widely used as a catalyst in the industry due to its magnetic behavior. Whereas, GO, used in many applications due to its characteristics, is a very promising particle to boost the catalyst. Hence, the combination of GO-Fe

_{3}O

_{4}with ethylene glycol is promising for a thermal heating and cooling system with minimum residue left in the system for a solar battery, radiator, and device sensors with minimal cost. Several studies focus on stretching surfaces with entropy analysis, but only a little information is available when the surface shrinks with entropy analysis. The unstable boundary layer partial differential equations are converted into a system of ordinary differential equations using the proper similarity transformation. The collected results for velocity, temperature, skin friction coefficient, local Nusselt number, and entropy generation are reviewed and highlighted in tables and graphs.

## 2. Mathematical Modelling

## 3. Flow Stability

## 4. Results and Discussion

_{2}O

_{4}nanoparticles and ${\varphi}_{2}$ represents GO nanoparticles. The concentration for both particles is 0.01.

_{2}O

_{4}/EG drives $\u03f5$ to increase skin friction. Velocity slip also helps in boosting heat transfer because, as shown in the table, the Nusselt number’s value increases when $\u03f5$ intensifies. In addition, $\u03f5$ elevates the production of disturbance in the system if its value increases. The increment in the rapidity slippage causes the molecules that make up the hybrid nano liquid to lose their stored thermal energy and thus lose their temperature near the boundary layer of the surface. This event physically exceeds the surface friction forces and swells the heat transference process next to the boundary layer of the surface. The slight change in skin friction coefficient, concerning parameter effects Eckert number $Ec$ and thermal radiation $Rd,$ is because these parameters do not affect the momentum of fluid. Instead, the change in the value of the local Nusselt number is evident because these effects are closely related to the temperature of the liquid. Table 4 shows that as the value of $Ec$ increases, the value of the Nusselt number decreases because $Ec$ will prevent the heat from being transferred to the surface. However, when Rd enhances, more heat is carried in the flow, so the value of the Nusselt number will improve. This phenomenon implies that $Ec$ prevents heat transmission while $Rd$ supports the transfer of heat. Noticeably, the strengthening of $Ec$ leads to the increment of entropy. In contrast, when the value of $Rd$ augments, there is a minimal reduction in the entropy generation. It is important to know the behavior of entropy for shrinking surfaces as, based on Adesanya et al. [57], the entropy generation must be regulated for sustainable and efficient manufacturing. For the Biot parameter $Bi$, the results show that it does not have any effect on all three values of skin friction coefficient, local Nusselt number, and entropy generation. The physical reasons are that when the Eckert number exceeds the required value, the temperature differences of the fluid diminish, meaning that the temperature gradient gradually dwindles, which, in turn, reduces the heat transference rate of the hybrid nano liquid. While the relatively high thermal radiative value reduces the conventional liquid pair and raises the interfacial distances between its molecules, the molecules begin to lose the internally stored thermal energy, and thus the total heat transition rate of the hybrid nano liquid increases.

_{2}O

_{4}/EG fluid at contracting surface $\alpha =-1.4$. It is evident that $\u03f5$ increases the velocity of fluid as $\u03f5$ increases. A reference to Stoke’s Law is needed to clarify this condition, where $\u03f5$ will increase if the density of the fluid is higher leading to a higher velocity gap between the liquid and the surface. $\u03f5$ is also expected to create an extra interruption and speed up the fluid’s velocity. The temperature of the liquid decreases as $\u03f5$ goes higher, and as the temperature of the fluid drops, its viscosity climbs. This spectacle leads to a reduction in the boundary layer thickness. This incident is physically due to the fact that the rate of rapidity slippage inside the liquid is inversely proportional to the root of the viscosity of the conventional liquid, which leads to the occurrence of that physical phenomenon that causes the effects mentioned above.

_{2}O

_{4}/EG and the thickness of the thermal boundary layer decrease. The massive size of the Biot number is said to activate the cooling process on the system’s surface, hence cooling down the particles. So, this implies that with the presence of graphene oxide particles in the fluid, Bi enhances the heat diffusivity from the liquid to the surface. The opposite of the physical effect occurs with the Eckert quantity. The role of viscosity increases with the boost in the Biot number near the boundary layer, so the apparent diminishing in temperatures occurs as a result of the increment in heat transition rates approaching this layer, so its thermal thickness rises, and its stored thermal energy is lost. Temperatures begin to decrease until relative stability occurs in the internal viscosity of the nano liquid, at which the temperature stabilizes away from the boundary layer.

## 5. Conclusions

- The fluid velocity will be faster in the range of 9.7–54.3% when the values of magnetic field and slip velocity parameters are optimized for the studied values of these parameters.
- The temperature of the fluid will be hotter by a percentage ranging between 76.6% and 215% with the increment in the value of the thermal radiation parameter effect, the Eckert number, and the concentration of GO, within the limits of the selected values for the parameters under study.
- Skin friction will be elevated with the high magnetic field, velocity slip, and concentration of GO between 13.7% and 66.5% with these parameters’ values incrementations.
- Thermal radiation and velocity slip can be enhanced to improve the heat transfer rate with the range of 99.8–147% for taken parameters’ range.
- Magnetic field, velocity slip, and Eckert number support the production of entropy (with 16.6–43.9%) when their values are raised within the range for these parameters.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclatures

$Bi$ | Biot number | ${T}_{w}$ | the fluid temperature of the surface |

${B}_{r}$ | Brinkman number | ${T}_{\infty}$ | ambient temperature |

${C}_{f}$ | skin friction coefficient | $b$ | initial stretching rate |

${C}_{p}$ | specific heat $\left(Jk{g}^{-1}{K}^{-1}\right)$ | Greek symbols | |

$Ec$ | Eckert number | $\varphi $ | the volume fraction of the nanoparticles |

${E}_{G}$ | dimensional entropy $\left(J/K\right)$ | $\rho $ | density $Kg{m}^{-3}$ |

${h}_{f}$ | heat transfer coefficient | ${\sigma}^{*}$ | Stefan Boltzmann constant |

$k$ | thermal conductivity $\left(W{m}^{-1}{K}^{-1}\right)$ | $\psi $ | stream function |

${k}^{*}$ | absorption coefficient | $\u03f5$ | velocity slip parameter |

$Rd$ | radiation parameter | $\mu $ | dynamic viscosity of the fluid ($kg{m}^{-1}{s}^{-1}$) |

${N}_{G}$ | dimensionless entropy generation | $\nu $ | kinematic viscosity of the fluid (${m}^{2}{s}^{-1}$) |

$N{u}_{x}$ | local Nusselt number | $\theta $ | dimensionless temperature |

$Pr$ | Prandtl number $\left(\nu /\alpha \right)$ | $\chi $, $\varsigma $ | material parameters |

${q}_{r}$ | radiative heat flux | $\alpha $ | stretching and shrinking parameter |

${q}_{w}$ | wall heat flux | Subscripts | |

$Re$ | Reynolds number | $f$ | base fluid |

$v$ | velocity component $\left(m{s}^{-1}\right)$ | $0$ | surface |

${U}_{w}$ | the velocity of the stretching sheet | $nf$ | nanofluid |

${v}_{w}$ | mass flux constant velocity | $s$ | particles |

$x,y$ | dimensional space coordinates $\left(m\right)$ | $GO$ | graphene oxide nanoparticle |

$T$ | fluid temperature | Fe_{2}O_{4} | iron dioxide nanoparticle |

$M$ | magnetic parameter | $EG$ | ethylene glycol |

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**Figure 3.**(

**a**) Skin friction ${C}_{f}R{e}_{x}^{1/2}$ for varied $M$; (

**b**) the Nusselt number $N{u}_{x}R{e}_{x}^{-1/2}$ for varied $M;$ (

**c**) entropy generation ${N}_{G}R{e}^{-1}$ for varied $M$ and $\chi =0.1$, $\varsigma =0.1$, $Rd=0.1$, $Ec=0.01$, $\u03f5=0.1$, $Bi=0.1$.

**Figure 4.**(

**a**) Skin friction ${C}_{f}R{e}_{x}^{1/2}$ for varied ${\varphi}_{2}$; (

**b**) the Nusselt number $N{u}_{x}R{e}_{x}^{-1/2}$ for varied ${\varphi}_{2}$; (

**c**) entropy generation ${N}_{G}R{e}^{-1}$ for varied ${\varphi}_{2}$ and $\mathrm{M}=0.1$, $\mathsf{\chi}=0.1$, $\mathsf{\varsigma}=0.1$, $Rd=0.1$, $Ec=0.01$, $\u03f5=0.1$, $Bi=0.1$.

**Figure 5.**(

**a**)${f}^{\prime}\left(n\right)$ for varied $M$; (

**b**) $\theta \left(n\right)$ for varied $M$ and $\chi =0.1$, $\varsigma =0.1$, $Rd=0.1$, $Ec=0.01$, $\u03f5=0.1$, $Bi=0.1$, $\alpha =-1.3$.

**Figure 6.**(

**a**) ${f}^{\prime}\left(n\right)$ for varied $\u03f5$; (

**b**) $\theta \left(n\right)$ for varied $\u03f5$ and $\chi =0.1$, $\varsigma =0.1$, $Rd=0.1$, $Ec=0.01$, $M=0.1$, $Bi=0.1$, $\alpha =-1.4$.

**Figure 7.**(

**a**) ${f}^{\prime}\left(n\right)$ for varied ${\varphi}_{2};$ (

**b**) $\theta \left(n\right)$ for varied ${\varphi}_{2}$ and $M=0.1$, $\chi =0.1$, $\varsigma =0.1$, $Rd=0.1$, $Ec=0.01$, $\u03f5=0.1$, $Bi=0.1$, $\alpha =-1.4$.

**Figure 8.**$\theta \left(n\right)$ for varied Rd and $M=0.1$, $\chi =0.1$, $\varsigma =0.1$, $Ec=0.01$, $\u03f5=0.1$, $Bi=0.1$, $\alpha =-1.4$.

**Figure 9.**$\theta \left(n\right)$ for varied Ec and $M=0.1$, $\chi =0.1$, $\varsigma =0.1$, $Rd=0.1$, $\u03f5=0.1$, $Bi=0.1$, $\alpha =-1.4$.

**Figure 10.**$\theta \left(\eta \right)$ for varied Bi and $M=0.1$, $\chi =0.1$, $\varsigma =0.1$, $Rd=0.1$, $Ec=0.01$, $\u03f5=0.1$, $\alpha =-1.4$.

**Table 1.**The thermophysical features of hybrid nanofluids (Source: Devi and Devi et al. [50]).

Features | Hybrid Nanofluid |
---|---|

Density ($\rho $) | ${\rho}_{hnf}=\left(1-{\varphi}_{2}\right)\left[\left(1-{\varphi}_{1}\right){\rho}_{f}+{\varphi}_{1}{\rho}_{s1}\right]+{\varphi}_{2}{\rho}_{s2}$ |

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

Heat capacity ($\rho {C}_{p}$) | ${\left(\rho {C}_{p}\right)}_{hnf}=\left(1-{\varphi}_{2}\right)\left[\left(1-{\varphi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}+{\varphi}_{1}{\left(\rho {C}_{p}\right)}_{s1}\right]+{\varphi}_{2}{\left(\rho {C}_{p}\right)}_{s2}$ |

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

Thermophysical Properties | Graphene Oxide (GO) | Iron Dioxide (Fe_{2}O_{4}) | Ethylene Glycol (EG) |
---|---|---|---|

$k\left(W{m}^{-1}{K}^{-1}\right)$ Thermal conductivity | 5000 | 9.7 | 0.253 |

$\rho \left(kg{m}^{-2}\right)$ Density | 1800 | 5180 | 1115 |

${C}_{p}\left(Jk{g}^{-1}{K}^{-1}\right)$ Specific heat | 717 | 670 | 2430 |

α | Bachok et al. [55] | Wahid et al. [56] | Present Study | |||
---|---|---|---|---|---|---|

1st Solution | 2nd Solution | 1st Solution | 2nd Solution | 1st Solution | 2nd Solution | |

−0.25 | 1.4022408 | - | 1.402240767 | - | 1.402240774 | - |

−0.5 | 1.4956698 | - | 1.495669720 | - | 1.495669732 | - |

−0.75 | 1.4892983 | - | 1.489298191 | - | 1.489298195 | - |

−1.15 | 1.0822315 | 0.1167022 | 1.082231123 | 0.116702139 | 1.082231123 | 0.116702132 |

−1.2 | 0.9324739 | 0.2336497 | 0.932473307 | 0.233649729 | 0.932473309 | 0.233649727 |

−1.2465 | 0.5842956 | 0.5542825 | 0.584281454 | 0.554296191 | 0.584281488 | 0.554296191 |

−1.24657 | 0.5639733 | - | 0.574525263 | 0.564003924 | 0.574525624 | 0.564009932 |

**Table 4.**Skin friction coefficient, local Nusselt number, and entropy generation, when $\alpha =-1.4$.

$\mathit{E}\mathit{c}$ | $\mathit{R}\mathit{d}$ | $\mathit{\u03f5}$ | $\mathit{B}\mathit{i}$ | ${\mathit{C}}_{\mathit{f}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{\frac{1}{2}}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{-\frac{1}{2}}$ | Entropy | |||
---|---|---|---|---|---|---|---|---|---|

1st Solution | 2nd Solution | 1st Solution | 2nd Solution | 1st Solution | 2nd Solution | ||||

0.01 | 0.1 | 0.1 | 0.1 | 1.072133669 | 0.750354986 | −0.063235727 | −0.043932841 | 4.359617604 | 2.129891634 |

0.8 | - | - | - | 1.072133757 | 0.750354987 | −5.05888402 | −3.514627291 | 26.322070129 | 12.730478987 |

1.0 | - | - | - | 1.072133758 | 0.750354987 | −6.323605114 | −4.393284109 | 38.677880812 | 18.694241172 |

- | 0.07 | - | - | 1.072133679 | 0.750354989 | −0.072036548 | −0.049763586 | 4.360764374 | 2.130420151 |

- | 0.08 | - | - | 1.072133680 | 0.750354987 | −0.068565825 | −0.047457329 | 4.360295382 | 2.130203896 |

- | - | 0.2 | - | 1.612105529 | 0.466433063 | −0.099002297 | −0.031028763 | 9.921324931 | 0.821946760 |

- | - | 0.3 | - | 1.816822281 | 0.370772639 | −0.027679856 | −0.027591472 | 12.630987007 | 0.519323731 |

- | - | - | 0.45 | 1.072133668 | 0.750354989 | −0.063235889 | −0.043932852 | 4.359617616 | 2.129891649 |

- | - | - | 0.5 | 1.072133668 | 0.750354987 | −0.063235894 | −0.043932843 | 4.359617616 | 2.129891638 |

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## Share and Cite

**MDPI and ACS Style**

Aminuddin, N.A.; Nasir, N.A.A.M.; Jamshed, W.; Ishak, A.; Pop, I.; Eid, M.R.
Impact of Thermal Radiation on MHD GO-Fe_{2}O_{4}/EG Flow and Heat Transfer over a Moving Surface. *Symmetry* **2023**, *15*, 584.
https://doi.org/10.3390/sym15030584

**AMA Style**

Aminuddin NA, Nasir NAAM, Jamshed W, Ishak A, Pop I, Eid MR.
Impact of Thermal Radiation on MHD GO-Fe_{2}O_{4}/EG Flow and Heat Transfer over a Moving Surface. *Symmetry*. 2023; 15(3):584.
https://doi.org/10.3390/sym15030584

**Chicago/Turabian Style**

Aminuddin, Nur Aisyah, Nor Ain Azeany Mohd Nasir, Wasim Jamshed, Anuar Ishak, Ioan Pop, and Mohamed R. Eid.
2023. "Impact of Thermal Radiation on MHD GO-Fe_{2}O_{4}/EG Flow and Heat Transfer over a Moving Surface" *Symmetry* 15, no. 3: 584.
https://doi.org/10.3390/sym15030584