# Analysis of Heat Transfer of Mono and Hybrid Nanofluid Flow between Two Parallel Plates in a Darcy Porous Medium with Thermal Radiation and Heat Generation/Absorption

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{2}/H

_{2}O) flow and the hybrid nanofluid (MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}) flow between the parallel plates and their heat transport property. The heat transport phenomenon is analyzed with the magnetic field, thermal radiation, heat source/sink, suction/injection effect, and porous medium. In the present model, the plate situated above is in the movement towards the lower plate, and the latter is stretching with a linear velocity. The prevailing PDEs depicting the modeled problem with the aforementioned effects are transformed via similarity transformations and solved via the “bvp4c” function, which is an inbuilt function in MATLAB software. The control of the factors on the fields of velocity and temperature, heat transfer rate, velocity boundary layer patterns, and streamlines is investigated. The solution profiles are visually shown and explained. Furthermore, the Nusselt number at the bottom plate is larger for the (MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}) hybrid nanofluid than for the (MoS

_{2}/H

_{2}O) nanofluid flow. In the presence of suction/injection, the streamlines appear to be denser. In addition, the magnetic field has a thinning consequence on the velocity boundary layer region. The results of this study apply to several thermal systems, engineering, and industrial processes, which utilize nanofluid and hybrid nanofluid for cooling and heating processes.

## 1. Introduction

_{2}and SiO

_{2}) are dispersed in a base fluid (e.g., H

_{2}O and C

_{2}H

_{6}O

_{2}), it is called the hybrid nanofluid. The component materials’ characteristics can be acquired while forming a hybrid nanofluid. When compared to individual nanoliquids, hybrid nanofluids have higher chemical stability, thermal conductivity, physical strength, mechanical resistance, and so on. Thermophysical and chemical features of hybrid nanomaterials are noteworthy, which are not seen in the individual components.

- Metal nanomaterials. Alumina/nickel (Al
_{2}O_{3}/Ni), alumina/copper (Al_{2}O_{3}/Cu), alumina/chromium (Al_{2}O_{3}/Cr), alumina/iron (Al_{2}O_{3}/Fe), magnesia/iron (MgO/Fe), and magnesium/carbon nanotube (Mg/CNT). - Ceramic nanomaterials. Ferric oxide/carbon nanotubes (Fe
_{3}O_{4}/CNT), nickel/silica (Ni/SiO_{2}), silica/alumina (SiO_{2}/Al_{2}O_{3}), alumina/titanium oxide (Al_{2}O_{3}/TiO_{2}), and silicon carbide/alumina (SiC/Al_{2}O_{3}). - Polymer nanomaterials. polyester/titanium oxide (TiO
_{2}), polymer/hydroxides, and polymer/carbon nanotubes (CNT).

_{2}and SiO

_{2}nanoparticles past a wedge and a cone and inferred the role of time lag during the transport of heat. A theoretical analysis of the influence of suction, injection, heat generation, and magnetohydrodynamic effects on a Williamson hybrid nanofluid (i.e., (MoS4–Cu)/water) over a stretching cylinder was done by Kavya et al. [10]. Raju et al. [11] inspected the heat transport property of two types of ternary hybrid nanofluids in an expanding or contracting porous channel. They considered ternary hybrid nanofluids with two different combinations of nanoparticles, graphene, carbon nanotubes, and aluminium oxide; and copper, silver, and copper oxide. Upadhya et al. [12] studied the entropy generation of an incompressible, steady Casson, micropolar, and hybrid nanofluid over a curved stretching sheet. They considered the hybrid nanofluid with silica SiO

_{2}and aluminium oxide Al

_{2}O

_{3}nanoparticles dispersed in water. They concluded that the micropolar fluid shows higher entropy generation compared to the Casson and hybrid nanofluid. Ullah et al. [13] investigated the thermal radiation and thermal slip parameter effects on the flow of hybrid nanoliquid past a stretchable rotating disk. They considered the hybrid nanoliquid with a combination of AA7072 and AA7075 nanoparticles and water.

_{2}O

_{3}, in the flow with water as a working fluid in the middle of two parallel plates was scrutinized by Khashi’ie et al. [16]. They discussed their model with the assumption that the flow was induced due to the movement of the upper plate and the deformation of the lower plate. Kapen et al. [17] conducted an analysis to scrutinize the consequence of injection on hybrid nanofluid flow (Cu-Al

_{2}O

_{3}/water) between two stationary parallel plates and performed the stability analysis of their solution. The two-dimensional squeezing unsteady MHD Casson fluid flow between two parallel plates with nonlinear radiation was scrutinized by Kumar et al. [18]. Shah et al. [19] conducted an analysis to scrutinize the effect of the Hall current and electric field on a flow of nanofluid with micropolar nature between two parallel and rotating plates. Li et al. [20] studied the axisymmetric transient squeezing flow of the Newtonian non-conducting fluid between the two circular horizontal plates in a porous medium.

_{2}O

_{3}-Cu/water) flow through a stretching surface in a porous medium. Mishra and Kumar [34] studied the impact of a porous medium on the heat- and mass-transmission property of a nanofluid flow past a wedge. Yaseen et al. [35] published a comparative study describing the heat-transport property of hybrid nanofluid (MoS

_{2}–SiO

_{2}/kerosene oil) and (MoS

_{2}/kerosene oil) nanofluid flow amid the two disks in a rotating state in a porous medium characterized by the Darcy–Forchheimer relation. Ullah et al. [36] investigated the significance of entropy generation in the flow of Ethylene glycol/water nanofluid in a rotating frame in a Darcy–Forchheimer porous medium. Hayat et al. [37] investigated the entropy optimization in the nonlinear mixed convective unsteady magnetohydrodynamic flow of nanomaterials in porous space. Li et al. [38] inspected the flow of MHD third-grade liquid through Darcy–Forchheimer’s porous space with homogeneous–heterogeneous reactions. Ullah [39] performed the theoretical investigation of MHD nanofluid over a rotating and stretching disk in a Darcy–Forchheimer porous medium with zero-mass flux.

_{2}/H

_{2}O nanofluid flow and the MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}hybrid nanofluid flow between the two parallel plates. The novelty of the article is to comparatively study the heat-transport property of MoS

_{2}/H

_{2}O nanofluid flow and the MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}hybrid nanofluid flow in the middle of two parallel plates. Furthermore, as a novelty, the combined effects of the Darcy porous medium, heat absorption/generation, and radiation are considered at the same time to comparatively study the heat-transport property of the MoS

_{2}/H

_{2}O nanofluid flow and the MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}hybrid nanofluid.

- Importance of the hybrid nanofluid flow amid two parallel plates in a porous medium;
- Comparison of the flow behavior of MoS
_{2}/H_{2}O nanofluid flow and the MoS_{2}–SiO_{2}/H_{2}O–C_{2}H_{6}O_{2}hybrid nanofluid flow; - Visualization of thermal behavior of flow when heat source/sink and thermal radiation is inevitable;
- The difference in heat-transfer rates of MoS
_{2}/H_{2}O nanofluid flow and the MoS_{2}–SiO_{2}/H_{2}O–C_{2}H_{6}O_{2}hybrid nanofluid flow at the lower and upper plates.

## 2. Mathematical Modeling

#### 2.1. Model Development

_{1}and T

_{2}, respectively. Furthermore, the current model deals with the fluid suction/injection, and ${v}_{w}=-\frac{{V}_{0}}{1-\alpha t}$ is the wall mass velocity; where ${V}_{0}>0$ for suction, ${V}_{0}<0$ for injection, and ${V}_{0}=0$ corresponds to an impermeable plate. In addition, the lower plate displaces with linear velocity, ${u}_{w}=\frac{bx}{1-\alpha t}$, where $t<\frac{1}{\alpha}$ and the lower plate is stretching/shrinking.

_{p}—heat capacity, k—thermal conductivity, $B\left(t\right)$—magnetic field strength, ${\varphi}^{*}$ indicates porosity of the porous medium, k

_{o}—permeability of the porous medium, $\lambda $ is the stretching/shrinking parameter,${Q}_{o}$—heat absorption/generation coefficient, and ‘b’ denotes the stretching/shrinking rate of the lower plate.

_{r}) is defined as (see Ref. [7]):

_{2}/H

_{2}O (nanofluid) flow and the MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid) flow. In the former fluid, the MoS

_{2}nanoparticles are individually disseminated in the water to prepare the mono-nanofluid (MoS

_{2}/H

_{2}O). In the latter case, initially MoS

_{2}nanoparticles and later SiO

_{2}nanoparticles are disseminated in the mixture, which is the combination of two fluids, water (H

_{2}O) and ethylene glycol (C

_{2}H

_{6}O

_{2}), to produce the hybrid nanofluid MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}. The base fluid mixture in the hybrid nanofluid is taken as 50% water and 50% ethylene glycol. The mathematical correlations for the nanofluid and hybrid nanofluid properties and their properties are mentioned in Table 1 and Table 2. For the evaluation of the thermophysical properties mentioned in Table 1, we adopt the correlations by Devi and Devi [41], which are feasible and correct based on the experimental validation. These correlations are built based on the physical assumptions. The subscripts are used as follows: “f—base fluid, hnf —hybrid nanofluid, and nf—nanofluid”.

#### 2.2. Transforming the Governing Equations Using Similarity Transformation

_{0}is the reference temperature for hybrid nanofluid flow.

#### 2.3. Nusselt Numbers

## 3. Numerical Method

## 4. Results and Discussion

_{2}/H

_{2}O, and hybrid nanofluid, MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}, is presented in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. In the figures, “solid lines embody the solution for hybrid nanofluid (i.e., designated by hnf), and the dotted lines embody the solution for nanofluid (i.e., designated by nf)”. For the computations of the results, the parameters were fixed as: ${\phi}_{1}={\phi}_{2}=\delta =0.1$, $\lambda =2$, ${R}_{d}=\eta =1$, $M=2$, $Q=0.3$, $S=0.5$, $Sq=3.2$, $Da=0.04$, $Pr=6.2$ (for mono nanofluid), and $Pr=29.82$ (for hybrid nanofluid), and any discrepancy from the aforementioned values is mentioned at the suitable place in the figure or table.

#### 4.1. Velocity Profile

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid) and MoS

_{2}/H

_{2}O (nanofluid). It is observed that near the lower plate, MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid) has greater velocity, but after the transition point, a reverse pattern is witnessed, i.e., MoS

_{2}/H

_{2}O (nanofluid) has higher velocity near the upper plate.

#### 4.2. Temperature Profile

_{d}). It is witnessed that the $\theta (\eta )$ rises with the increment in the radiation parameter (R

_{d}). An increase in thermal radiation leads to a decline in the coefficient of heat absorption, which elevates the fluid temperature. Thus, the increased quantity of heat transmitted in the area as a result of enhanced radiation raises the temperature $\theta (\eta )$.

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid) and MoS

_{2}/H

_{2}O (nanofluid). It is observed that MoS

_{2}/H

_{2}O (nanofluid) flow has a higher temperature in comparison to the flow of MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid).

#### 4.3. Streamlines and Velocity Boundary Layer Pattern

#### 4.4. Nusselt Numbers

_{d}) cause the Nusselt numbers at both the plates ($N{u}_{x1}^{*}$ and $N{u}_{x2}^{*}$) to fall.

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid) flow in comparison to MoS

_{2}/H

_{2}O (nanofluid) flow. However, contrary behavior is seen at the upper plate.

## 5. Conclusions

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}(hybrid nanofluid) and MoS

_{2}/H

_{2}O (nanofluid) between two parallel plates”. Modeling of the flow is done with the assumptions of “heat-generation/absorption effect, porous medium, radiation, and injection/suction effect”. The principal equations are handled by the “bvp4c” function of the MATLAB software.

- The injection effect and the shrinking of the lower plate aid the hybrid nanofluid flow.
- The thermal radiation parameter and heat sink/source parameter have a positive correlation with the thermal field.
- The hybrid nanofluid flow has a higher Nusselt number at the lower plate than the nanofluid.
- The streamlines become denser under the influence of suction and injection effects.
- The presence of a magnetic field has a thinning consequence on the velocity boundary layer region.
- The results of this study apply to several thermal systems, engineering, and industrial processes, which utilize nanofluid and hybrid nanofluid for cooling and heating processes.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Roman Letters | |||

b | constant | v_{w} | Velocity of mass flux (m/s) |

B_{o} | Magnetic induction (W/m^{2}) | $\left(x,y\right)$ | Cartesian coordinates (m) |

$B\left(t\right)$ | Magnetic field (kg/(s^{2}·m^{2})) | ||

Da | Darcy number | Greek symbols | |

$f{}^{\prime}(\eta )$ | Dimensionless velocity | $\alpha $ | Constant |

$h\left(t\right)$ | Distance between plates (m) | λ | Stretching/shrinking parameter |

k | Thermal conductivity (W/mK) | $\delta $ | Temperature-ratio parameter |

${K}^{*}$ | Mean absorption coefficient (m^{−1}) | ${\varphi}^{*}$ | Porosity of the porous medium |

k_{o} | Permeability of the porous medium (m^{2}) | ${\phi}_{1}$ | Solid volume fraction of MoS_{2} |

M | Magnetic-field parameter | ${\phi}_{2}$ | Solid volume fraction of SiO_{2} |

$N{u}_{x}^{*}$ | Nusselt number | $\nu $ | Kinematic viscosity (m^{2}/s) |

Pr | Prandtl number | $\mu $ | Dynamic viscosity (kg m^{−1} s^{−1}) |

$Q$ | Heat-source/sink parameter | $\rho $ | Density (kg/m^{3}) |

q_{r} | Radiative-heat flux (W/m^{2}) | $\psi $ | Stream function |

R_{d} | Radiation parameter | ${\sigma}^{*}$ | Stefan–Boltzmann constant (W. m^{−2}. K^{−4}) |

Re_{x} | Local Reynolds number | $\sigma $ | Electrical conductivity ((s^{3}. m^{2})/kg) |

Sq | Squeezing parameter | $\rho {C}_{p}$ | Heat capacity (J/m^{3}K) |

S | Suction/injection parameter | $\eta $ | Similarity variable |

t | Time (s) | ${\zeta}_{i}\left(i=1-5\right)$ | Constant |

T | Temperature (K) | ||

T_{o} | Reference temperature (K) | Subscripts | |

T_{1} | Lower plate temperature (K) | ||

${T}_{2}$ | Upper plate temperature (K) | f | Base fluid |

$\theta $ | Dimensionless temperature | nf | Nanofluid |

$\left(u,v\right)$ | Components of velocity (m/s) | hnf | Hybrid nanofluid |

${u}_{w}$ | Stretching velocity (m/s) | ||

${V}_{o}$ | Constant | Superscripts | |

${V}_{h}$ | Velocity of the upper plate moving towards/away from the lower plate (m/s) | ${}^{\prime}$ | $\mathrm{Derivative}\mathrm{w}.\mathrm{r}.\mathrm{to}\eta $ |

## Appendix A. Derivation of the Flow Problem

#### Appendix A.1. Derivation of Continuity Equation

#### Appendix A.2. Derivation of Momentum Equations

#### Appendix A.3. Derivation of Energy Equation

#### Appendix A.4. Derivation of Boundary Conditions

## References

- Choi, S.U.S. Enhancing Thermal Conductivity of Fluids with Nanoparticles. Am. Soc. Mech. Eng. Fluids Eng. Div.
**1995**, 231, 99–105. [Google Scholar] - Choi, C.; Yoo, H.S.; Oh, J.M. Preparation and Heat Transfer Properties of Nanoparticle-in-Transformer Oil Dispersions as Advanced Energy-Efficient Coolants. Curr. Appl. Phys.
**2008**, 8, 710–712. [Google Scholar] [CrossRef] - Taylor-Pashow, K.M.L.; Della Rocca, J.; Huxford, R.C.; Lin, W. Hybrid Nanomaterials for Biomedical Applications. Chem. Commun.
**2010**, 46, 5832–5849. [Google Scholar] [CrossRef] [PubMed] - Masteri-Farahani, M.; Movassagh, J.; Taghavi, F.; Eghbali, P.; Salimi, F. Magnetite–Polyoxometalate Hybrid Nanomaterials: Synthesis and Characterization. Chem. Eng. J.
**2012**, 184, 342–346. [Google Scholar] [CrossRef] - Mehryan, S.A.M.; Izadpanahi, E.; Ghalambaz, M.; Chamkha, A.J. Mixed Convection Flow Caused by an Oscillating Cylinder in a Square Cavity Filled with Cu–Al
_{2}O_{3}/Water Hybrid Nanofluid. J. Therm. Anal. Calorim.**2019**, 137, 965–982. [Google Scholar] [CrossRef] - Ullah, I.; Hayat, T.; Alsaedi, A. Optimization of Entropy Production in Flow of Hybrid Nanomaterials through Darcy–Forchheimer Porous Space. J. Therm. Anal. Calorim.
**2021**, 147, 5855–5864. [Google Scholar] [CrossRef] - Yaseen, M.; Kumar, M.; Rawat, S.K. Assisting and Opposing Flow of a MHD Hybrid Nanofluid Flow Past a Permeable Moving Surface with Heat Source/Sink and Thermal Radiation. Partial Differ. Equations Appl. Math.
**2021**, 4, 100168. [Google Scholar] [CrossRef] - Ullah, I.; Jan, R.U.; Khan, H.; Alam, M.M. Improving the Thermal Performance of (ZnO-Ni/H
_{2}O) Hybrid Nanofluid Flow over a Rotating System: The Applications of Darcy Forchheimer Theory. Waves Random Complex Media**2022**, 1–17. [Google Scholar] [CrossRef] - Garia, R.; Rawat, S.K.; Kumar, M.; Yaseen, M. Hybrid Nanofluid Flow over Two Different Geometries with Cattaneo-Christov Heat Flux Model and Heat Generation: A Model with Correlation Coefficient and Probable Error. Chin. J. Phys.
**2021**, 74, 421–439. [Google Scholar] [CrossRef] - Kavya, S.; Nagendramma, V.; Ahammad, N.A.; Ahmad, S.; Raju, C.S.K.; Shah, N.A. Magnetic-Hybrid Nanoparticles with Stretching/Shrinking Cylinder in a Suspension of MoS
_{4}and Copper Nanoparticles. Int. Commun. Heat Mass Transf.**2022**, 136, 106150. [Google Scholar] [CrossRef] - Raju, C.S.K.; Ahammad, N.A.; Sajjan, K.; Shah, N.A.; Yook, S.J.; Kumar, M.D. Nonlinear Movements of Axisymmetric Ternary Hybrid Nanofluids in a Thermally Radiated Expanding or Contracting Permeable Darcy Walls with Different Shapes and Densities: Simple Linear Regression. Int. Commun. Heat Mass Transf.
**2022**, 135, 106110. [Google Scholar] [CrossRef] - Upadhya, S.M.; Raju, S.V.S.R.; Raju, C.S.K.; Shah, N.A.; Chung, J.D. Importance of Entropy Generation on Casson, Micropolar and Hybrid Magneto-Nanofluids in a Suspension of Cross Diffusion. Chin. J. Phys.
**2022**, 77, 1080–1101. [Google Scholar] [CrossRef] - Ullah, I.; Hayat, T.; Alsaedi, A.; Asghar, S. Dissipative Flow of Hybrid Nanoliquid (H
_{2}O-Aluminum Alloy Nanoparticles) with Thermal Radiation. Phys. Scr.**2019**, 94, 125708. [Google Scholar] [CrossRef] - Singh, K.; Rawat, S.K.; Kumar, M. Heat and Mass Transfer on Squeezing Unsteady MHD Nanofluid Flow between Parallel Plates with Slip Velocity Effect. J. Nanosci.
**2016**, 2016, 9708562. [Google Scholar] [CrossRef] - Salehi, S.; Nori, A.; Hosseinzadeh, K.; Ganji, D.D. Hydrothermal Analysis of MHD Squeezing Mixture Fluid Suspended by Hybrid Nanoparticles between Two Parallel Plates. Case Stud. Therm. Eng.
**2020**, 21, 100650. [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] - Tiam Kapen, P.; Gervais Njingang Ketchate, C.; Fokwa, D.; Tchuen, G. Linear Stability Analysis of (Cu-Al
_{2}O_{3})/Water Hybrid Nanofluid Flow in Porous Media in Presence of Hydromagnetic, Small Suction and Injection Effects. Alexandria Eng. J.**2021**, 60, 1525–1536. [Google Scholar] [CrossRef] - Sathish Kumar, M.; Sandeep, N.; Rushi Kumar, B.; Saleem, S. Effect of Aligned Magnetic Field on MHD Squeezing Flow of Casson Fluid between Parallel Plates. Defect Diffus. Forum
**2018**, 384, 1–11. [Google Scholar] [CrossRef] - Shah, Z.; Islam, S.; Gul, T.; Bonyah, E.; Altaf Khan, M. The Electrical MHD and Hall Current Impact on Micropolar Nanofluid Flow between Rotating Parallel Plates. Results Phys.
**2018**, 9, 1201–1214. [Google Scholar] [CrossRef] - Li, Y.M.; Ullah, I.; Ameer Ahammad, N.; Ullah, I.; Muhammad, T.; Asiri, S.A. Approximation of Unsteady Squeezing Flow through Porous Space with Slip Effect: DJM Approach. Waves Random Complex Media 2022. [CrossRef]
- Ullah, I. Heat Transfer Enhancement in Marangoni Convection and Nonlinear Radiative Flow of Gasoline Oil Conveying Boehmite Alumina and Aluminum Alloy Nanoparticles. Int. Commun. Heat Mass Transf.
**2022**, 132, 105920. [Google Scholar] [CrossRef] - Ben Henda, M.; Waqas, H.; Hussain, M.; Khan, S.U.; Chammam, W.; Khan, S.A.; Tlili, I. Applications of Activation Energy along with Thermal and Exponential Space-Based Heat Source in Bioconvection Assessment of Magnetized Third Grade Nanofluid over Stretched Cylinder/Sheet. Case Stud. Therm. Eng.
**2021**, 26, 101043. [Google Scholar] [CrossRef] - Mishra, A.; Kumar, M. Thermal Performance of MHD Nanofluid Flow Over a Stretching Sheet Due to Viscous Dissipation, Joule Heating and Thermal Radiation. Int. J. Appl. Comput. Math.
**2020**, 6, 123. [Google Scholar] [CrossRef] - Rashidi, M.M.; Babu, M.J.; Sandeep, N.; Ali, M.E. MHD Squeezing Flow of Nanofluid between Parallel Plates in the Presence of Aligned Magnetic Field. J. Comput. Theor. Nanosci.
**2016**, 13, 8700–8708. [Google Scholar] [CrossRef] - Ullah, I.; Ullah, R.; Alqarni, M.S.; Xia, W.-F.; Muhammad, T. Combined Heat Source and Zero Mass Flux Features on Magnetized Nanofluid Flow by Radial Disk with the Applications of Coriolis Force and Activation Energy. Int. Commun. Heat Mass Transf.
**2021**, 126, 105416. [Google Scholar] [CrossRef] - Sharma, R.; Hussain, S.M.; Raju, C.S.K.; Seth, G.S.; Chamkha, A.J. Study of Graphene Maxwell Nanofluid Flow Past a Linearly Stretched Sheet: A Numerical and Statistical Approach. Chin. J. Phys.
**2020**, 68, 671–683. [Google Scholar] [CrossRef] - Ge-Jile, H.; Shah, N.A.; Mahrous, Y.M.; Sharma, P.; Raju, C.S.K.; Upddhya, S.M. Radiated Magnetic Flow in a Suspension of Ferrous Nanoparticles over a Cone with Brownian Motion and Thermophoresis. Case Stud. Therm. Eng.
**2021**, 25, 100915. [Google Scholar] [CrossRef] - Raju, C.S.K.; Ibrahim, S.M.; Anuradha, S.; Priyadharshini, P. Bio-Convection on the Nonlinear Radiative Flow of a Carreau Fluid over a Moving Wedge with Suction or Injection. Eur. Phys. J. Plus
**2016**, 131, 409. [Google Scholar] [CrossRef] - Ullah, I.; Alghamdi, M.; Xia, W.F.; Shah, S.I.; Khan, H. Activation Energy Effect on the Magnetized-Nanofluid Flow in a Rotating System Considering the Exponential Heat Source. Int. Commun. Heat Mass Transf.
**2021**, 128, 105578. [Google Scholar] [CrossRef] - Nandeppanavar, M.M.; Vaishali, S.; Kemparaju, M.C.; Raveendra, N. Theoretical Analysis of Thermal Characteristics of Casson Nano Fluid Flow Past an Exponential Stretching Sheet in Darcy Porous Media. Case Stud. Therm. Eng.
**2020**, 21, 100717. [Google Scholar] [CrossRef] - Shah, Z.; Dawar, A.; Islam, S.; Khan, I.; Ching, D.L.C. Darcy-Forchheimer Flow of Radiative Carbon Nanotubes with Microstructure and Inertial Characteristics in the Rotating Frame. Case Stud. Therm. Eng.
**2018**, 12, 823–832. [Google Scholar] [CrossRef] - Shafiq, A.; Rasool, G.; Khalique, C.M. Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow. Symmetry
**2020**, 12, 741. [Google Scholar] [CrossRef] - Ahmad, S.; Ali, K.; Rizwan, M.; Ashraf, M. Heat and Mass Transfer Attributes of Copper–Aluminum Oxide Hybrid Nanoparticles Flow through a Porous Medium. Case Stud. Therm. Eng.
**2021**, 25, 100932. [Google Scholar] [CrossRef] - Mishra, A.; Kumar, M. Numerical Analysis of MHD Nanofluid Flow over a Wedge, Including Effects of Viscous Dissipation and Heat Generation/Absorption, Using Buongiorno Model. Heat Transf.
**2021**, 50, 8453–8474. [Google Scholar] [CrossRef] - Yaseen, M.; Rawat, S.K.; Kumar, M. Cattaneo–Christov Heat Flux Model in Darcy–Forchheimer Radiative Flow of MoS
_{2}–SiO_{2}/Kerosene Oil between Two Parallel Rotating Disks. J. Therm. Anal. Calorim.**2022**, 147, 10865–10887. [Google Scholar] [CrossRef] - Ullah, I.; Hayat, T.; Aziz, A.; Alsaedi, A. Significance of Entropy Generation and the Coriolis Force on the Three-Dimensional Non-Darcy Flow of Ethylene-Glycol Conveying Carbon Nanotubes (SWCNTs and MWCNTs). J. Non-Equilib. Thermodyn.
**2022**, 47, 61–75. [Google Scholar] [CrossRef] - Hayat, T.; Ullah, I.; Alsaedi, A.; Momani, S. Entropy Optimization in Nonlinear Mixed Convective Flow of Nanomaterials through Porous Space. J. Non-Equilib. Thermodyn.
**2021**, 46, 191–203. [Google Scholar] [CrossRef] - Li, Y.M.; Ullah, I.; Alam, M.M.; Khan, H.; Aziz, A. Lorentz Force and Darcy-Forchheimer Effects on the Convective Flow of Non-Newtonian Fluid with Chemical Aspects. Waves Random Complex Media
**2022**. [Google Scholar] [CrossRef] - Ullah, I. Activation Energy with Exothermic/Endothermic Reaction and Coriolis Force Effects on Magnetized Nanomaterials Flow through Darcy–Forchheimer Porous Space with Variable Features. Waves Random Complex Media
**2022**. [Google Scholar] [CrossRef] - Ahmad, S.; Farooq, M.; Javed, M.; Anjum, A. Slip Analysis of Squeezing Flow Using Doubly Stratified Fluid. Results Phys.
**2018**, 9, 527–533. [Google Scholar] [CrossRef] - Devi, S.P.A.; Devi, S.S.U. Numerical Investigation of Hydromagnetic Hybrid Cu Al
_{2}O_{3}/Water Nanofluid Flow over a Permeable Stretching Sheet with Suction. Int. J. Nonlinear Sci. Numer. Simulat.**2016**, 17, 249–257. [Google Scholar] [CrossRef] - Upadhya, M.S.; Raju, C.S.K. Implementation of Boundary Value Problems in Using MATLAB®. In Micro and Nanofluid Convection with Magnetic Field Effects for Heat and Mass Transfer Applications Using MATLAB; Elsevier: Amsterdam, The Netherlands, 2022; pp. 169–238. [Google Scholar] [CrossRef]
- Ahmad, S.; Nadeem, S. Thermal Analysis in Buoyancy Driven Flow of Hybrid Nanofluid Subject to Thermal Radiation. Int. J. Ambient Energy
**2020**. [Google Scholar] [CrossRef] - Hayat, T.; Sajjad, R.; Alsaedi, A.; Muhammad, T.; Ellahi, R. On Squeezed Flow of Couple Stress Nanofluid between Two Parallel Plates. Results Phys.
**2017**, 7, 553–561. [Google Scholar] [CrossRef]

**Figure 17.**Streamline patterns for different values of suction/injection parameter (S). (

**a**) S = −0.5; (

**b**) S = 0; (

**c**) S = 0.5.

**Figure 18.**Velocity boundary layer pattern in the (

**a**) absence of magnetic field and (

**b**) presence of magnetic field.

**Table 1.**Thermophysical properties of mono and hybrid nanofluid (see Devi and Devi [41]).

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

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

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

Thermal conductivity | $\frac{{k}_{nf}}{{k}_{f}}=\left[\frac{{k}_{s1}+2{k}_{f}-2{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}\right]$ | $\begin{array}{l}\frac{{k}_{hnf}}{{k}_{nf}}=\left[\frac{{k}_{s2}+2{k}_{nf}-2{\phi}_{2}\left({k}_{nf}-{k}_{s2}\right)}{{k}_{s2}+2{k}_{nf}+{\phi}_{2}\left({k}_{nf}-{k}_{s2}\right)}\right]\\ \mathrm{where}\\ \frac{{k}_{nf}}{{k}_{f}}=\left[\frac{{k}_{s1}+2{k}_{f}-2{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}\right]\end{array}$ |

Electrical conductivity | $\begin{array}{c}\frac{{\sigma}_{nf}}{{\sigma}_{f}}=1+\frac{3(\sigma -1){\phi}_{1}}{2+\sigma -(\sigma -1){\phi}_{1}}\\ \mathrm{where}\\ \sigma ={\sigma}_{s1}/{\sigma}_{f}\end{array}$ | $\begin{array}{l}\frac{{\sigma}_{hnf}}{{\sigma}_{nf}}=\left[\frac{{\sigma}_{s2}+2{\sigma}_{nf}-2{\phi}_{2}\left({\sigma}_{nf}-{\sigma}_{s2}\right)}{{\sigma}_{s2}+2{\sigma}_{nf}+{\phi}_{2}\left({\sigma}_{nf}-{\sigma}_{s2}\right)}\right]\\ \mathrm{where}\\ \frac{{\sigma}_{nf}}{{\sigma}_{f}}=\left[\frac{{\sigma}_{s1}+2{\sigma}_{f}-2{\phi}_{1}\left({\sigma}_{f}-{\sigma}_{s1}\right)}{{\sigma}_{s1}+2{\sigma}_{f}+{\phi}_{1}\left({\sigma}_{f}-{\sigma}_{s1}\right)}\right]\end{array}$ |

Heat capacitance | $\begin{array}{c}{\left(\rho {C}_{p}\right)}_{nf}={\phi}_{1}{\left(\rho {C}_{p}\right)}_{s1}\\ +(1-{\phi}_{1}){\left(\rho {C}_{p}\right)}_{f}\end{array}$ | $\begin{array}{cc}{\left(\rho {C}_{p}\right)}_{hnf}& ={\phi}_{2}{\left(\rho {C}_{p}\right)}_{s2}\\ & +(1-{\phi}_{2})\left[{\phi}_{1}{\left(\rho {C}_{p}\right)}_{s1}+\left(1-{\phi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}\right]\end{array}$ |

Properties/Constituents | H_{2}O | H_{2}O + EG (50:50) | MoS_{2} | SiO_{2} |
---|---|---|---|---|

${C}_{p}(\mathrm{J}/\mathrm{kgK})$ | 4179 | 3288 | 397.746 | 730 |

$k(\mathrm{W}/\mathrm{mK})$ | 0.613 | 0.425 | 34.5 | 1.5 |

$\rho ({\mathrm{kg}/\mathrm{m}}^{3})$ | 997.1 | 1056 | 5060 | 2650 |

$\sigma {(\Omega \mathrm{m})}^{-1}$ | 0.05 | 0.00509 | $2.09\times $10^{4} | $1.0\times $10^{−18} |

Pr | 6.2 | 29.86 |

**Table 3.**Comparison of values of $f{}^{\u2033}\left(1\right)$ and $f{}^{\u2033}\left(0\right)$ when $Sq={\phi}_{1}={\phi}_{2}=\frac{1}{Da}=Q={R}_{d}=0$, $\lambda =1$ for various M.

f″(0) | f″(1) | ||||||
---|---|---|---|---|---|---|---|

M | S | Khashi’ie et al. [16] | Hayat et al. [44] | Present | Khashi’ie et al. [16] | Hayat et al. [44] | Present |

0 | 0.5 | −7.4111525 | −7.411153 | −7.41115256 | 4.7133028 | 4.713303 | 4.71330278 |

1 | 0.5 | −7.5916177 | −7.591618 | −7.5916177 | 4.7390165 | 4.739017 | 4.7390165 |

4 | 0.5 | −8.1103342 | −8.110334 | −8.11033423 | 4.8202511 | 4.820251 | 4.82025109 |

9 | 0.5 | −8.9100956 | −8.910096 | −8.91009566 | 4.9648698 | 4.96487 | 4.9648698 |

4 | 0 | −4.5878911 | −4.587891 | −4.5878911 | 1.8424469 | 1.842447 | 1.84244688 |

4 | 0.3 | −6.6656620 | −6.665662 | −6.66566187 | 3.6536948 | 3.653695 | 3.65369492 |

4 | 0.6 | −8.8514442 | −8.851444 | −8.85144422 | 5.3912475 | 5.391248 | 5.39124755 |

4 | 1 | −11.9485843 | −11.948584 | −11.94858428 | 7.5934262 | 7.593426 | 7.59342617 |

**Table 4.**Numerical values of heat-transfer coefficient of nanofluid (MoS

_{2}/H

_{2}O) and (MoS

_{2}–SiO

_{2}/H

_{2}O–C

_{2}H

_{6}O

_{2}) hybrid nanofluid.

MoS_{2}/H_{2}O | MoS_{2}−SiO_{2}/H_{2}O−C_{2}H_{6}O_{2} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Da | M | Sq | $\mathit{\lambda}$ | S | ${\mathit{\phi}}_{1}$ | ${\mathit{\phi}}_{2}$ | $\mathit{Q}$ | R_{d} | $\mathit{N}{\mathit{u}}_{\mathit{x}1}^{*}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}2}^{*}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}1}^{*}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}2}^{*}$ |

0.04 | 2 | 3.2 | 2 | 0.5 | 0.1 | 0.1 | 0.3 | 1 | −1.34916991 | −2.43760429 | −0.21121505 | −2.89742728 |

0.06 | −1.36245421 | −2.41385979 | −0.21886821 | −2.77005155 | ||||||||

0.1 | −1.37544707 | −2.39085605 | −0.22679853 | −2.64453893 | ||||||||

0.04 | 6 | −1.31504548 | −2.49970926 | −0.19808463 | −3.1328579 | |||||||

14 | −1.25838822 | −2.60698691 | −0.17464376 | −3.62612217 | ||||||||

2 | 3.6 | −1.13585234 | −2.6143865 | −0.12586435 | −3.57622461 | |||||||

4 | −0.95338562 | −2.7883278 | −0.08616728 | −4.15987165 | ||||||||

3.2 | −2 | −0.92386278 | −2.96095613 | −0.09282364 | −4.81304915 | |||||||

0 | −1.12295604 | −2.70537179 | −0.12698724 | −3.93950167 | ||||||||

2 | −0.1 | −0.79960932 | −2.95380186 | −0.06620933 | −4.65301925 | |||||||

0.3 | −1.13661608 | −2.61269763 | −0.1260337 | −3.56934147 | ||||||||

0.5 | 0.15 | 0.15 | −1.52834388 | −2.58065725 | −0.3057826 | −2.9868471 | ||||||

0.2 | 0.2 | −1.72972588 | −2.74265991 | −0.4340701 | −3.0839793 | |||||||

0.1 | 0.1 | −0.1 | −1.0896009 | −3.22226056 | −0.0312476 | −5.69328936 | ||||||

0.1 | −1.21377331 | −2.84089142 | −0.11532536 | −4.39973995 | ||||||||

0.3 | 3 | −3.63080799 | −4.81817893 | −1.31211613 | −5.20644542 | |||||||

5 | −5.99010125 | −7.2109876 | −3.064243 | −7.52676646 |

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**

Yaseen, M.; Rawat, S.K.; Shafiq, A.; Kumar, M.; Nonlaopon, K.
Analysis of Heat Transfer of Mono and Hybrid Nanofluid Flow between Two Parallel Plates in a Darcy Porous Medium with Thermal Radiation and Heat Generation/Absorption. *Symmetry* **2022**, *14*, 1943.
https://doi.org/10.3390/sym14091943

**AMA Style**

Yaseen M, Rawat SK, Shafiq A, Kumar M, Nonlaopon K.
Analysis of Heat Transfer of Mono and Hybrid Nanofluid Flow between Two Parallel Plates in a Darcy Porous Medium with Thermal Radiation and Heat Generation/Absorption. *Symmetry*. 2022; 14(9):1943.
https://doi.org/10.3390/sym14091943

**Chicago/Turabian Style**

Yaseen, Moh, Sawan Kumar Rawat, Anum Shafiq, Manoj Kumar, and Kamsing Nonlaopon.
2022. "Analysis of Heat Transfer of Mono and Hybrid Nanofluid Flow between Two Parallel Plates in a Darcy Porous Medium with Thermal Radiation and Heat Generation/Absorption" *Symmetry* 14, no. 9: 1943.
https://doi.org/10.3390/sym14091943