In this article, the magnetohydrodynamic flow and heat transmission of a micropolar hybrid nanofluid between two surfaces inside a rotating system are examined. The Hall current and thermal radiations are also considered for the flow system. In this article, a base fluid is taken as water, while graphene oxide (*GO*) and copper (*Cu*) are applied as hybrid nanoparticles. The flow is assumed to be in a steady state. The governing partial differential equations along with boundary conditions for the modeled problem are transformed into a system of non-linear ordinary differential equations using similarity transformations. The set of resultant ordinary differential equations is solved by using an optimal approach. The main focus of this study is to examine the magnetohydrodynamic heat transmission and hybrid nanofluid flow in a rotating system between two parallel plates by taking into account the thermal radiations and Hall current impacts. Various physical parameters are discussed in detail graphically in this article. The main outcome of this study indicates that the augmented values of the magnetic parameter increase the velocity profile and decrease the rotational velocity profile.

## I. INTRODUCTION

A large number of fluids, for instance, water, ethylene-glycol, and oil, acquire small thermal conductivity for heat transmission, which is considered as a barrier for the development of the thermal flow system. The modern world is a witness of significant progress in the fabrication of various components and devices, which are used in industry and engineering applications. For instance, in the industry, there are some devices that start increasing their temperature with the passage of time due to the resistance of electricity. Because of this electrical resistance, the heat bearing capacity of such devices reduced and ultimately gave rise to some technical fault. In order to reduce such technical faulty risks, there is a need for heat dissipation from these components and devices. So, to manage the proper heat transmission, the industrialists use some fluids, such as water, air, and lubricants. However, such pure fluids do not fulfill the requirements at the industrial level. For the fulfillment of these requirements, researchers have been endeavoring to keep the flow of fluid and heat transmission within the design limit by using various techniques. One of such techniques is the addition of nanoparticles to various pure fluids due to their smaller thermal conductivity. Moreover, the characteristics of nanofluids can also be customized for a particular use, as desired. The number of nanoparticles was recommended first by Choi^{1} for the enhancement of thermal characteristics for a normal fluid. A number of scientists and researchers had diverted their attention to studying the subject of heat transfer by using either a nanofluid or a pure fluid. Ellahi *et al.*^{2} discussed a mixed convective fluid flow for nanoparticles past a wedge inside a permeable medium. In this study, they have taken into consideration the impacts of different shapes of particles. Dogonchi and Ganji^{3} have studied electrically conducting heat transmission for a nanofluid and buoyancy flow past a stretched surface by taking into consideration the Brownian motion. In this analysis, they have noted the growth in profiles of velocity and temperature for augmentation in radiation parameters. They have also noticed in this study that with the growth of the magnetic parameter, the skin friction parameter is augmented. Dogonchi and Ganji^{4} have also examined the magnetohydrodynamic (MHD) flow of a nanofluid and heat transmission between two surfaces by taking into account Joule heat effects. In this study, they have noticed an augmentation in the Nusselt number, concentration, and temperature profiles with an increase in the magnitude of the Schmidt number. Oudina and Bessaïh^{5} studied numerically the convection of heat transmission for nanoparticles by using a cylindrical shaped annulus. Gourari *et al.*^{6} examined numerically the natural convective flow among coaxial inclined cylinders. Oudina^{7} studied convection heat transmission for the titania nanofluid by using various base fluids. Raza *et al.*^{8} studied the magnetohydrodynamic flow for molybdenum disulfide nanoparticles inside a conduit by considering the effects of various shapes. In this investigation, they analyzed numerically the influence of nanoparticles upon the magnetohydrodynamic flow. In this study, they pointed out that with augmenting values of solid volume fraction, there is an augmentation in the Nusselt number. Moreover, they also revealed in this study that the velocity profile increased from the bottom wall until the center of the channel and dropped down subsequently for augmented values of wall expansion ratio.

Magnetohydrodynamics (MHD) is the study of the dynamics of such fluids, which are conducted electrically, such as plasmas, electrolytes, and liquid metals. It was first introduced by Alfven.^{9} The basic theme of MHD is that the magnetic field induces a current through a moving conducting fluid as a result of which forces are produced in the fluid. These forces are discussed by various investigators in different contexts. Magnetohydrodynamics has various engineering functions, for instance, crystal intensification, cooling of the reactor, plasma, MHD sensor, power generation, and magnetic drug targeting. For a given field, MHD is directly proportional to the strength of the magnetic induction. Hayat *et al.*^{10} discussed the entropy generation of the MHD flow for nanoparticles past a non-linear stretched medium. In this study, they investigated nanoconcentration and temperature by taking into account the Brownian motion along with the thermophoresis effects. They have also established that the flow is strongly restricted by the magnetic influence. Farooq *et al.*^{11} discussed the MHD flow for the Maxwell fluid using nanoparticles past a stretched surface. It is to be noticed that whenever magnetic forces get stronger, Hall effects are produced because in such a situation, the Hall current cannot be overlooked. The idea of this current was first introduced by Hall.^{12} Pop and Soundalgekar^{13} discussed the Hall current for a steady hydromagnetic flow of a viscous fluid. Ahmad and Zueco^{14} studied mass and heat transmission in a rotating porous conduit by using Hall effects. Aziz^{15} investigated Hall effects for the flow of nanofluid heat transfer past a stretched surface. Awais *et al.*^{16} have also investigated the effects of viscous dissipation for a convective Jeffery flow using the Hall current.

In the last few decades, there was a need to model such a fluid, which consists of micro-shaped pieces. Ultimately, the struggle of researchers gave rise to the beginning of a new type of fluid known as a micropolar fluid. It is a group of those fluids, which consists of microstructure molecules. This term was first noticed by Eringen.^{17,18} The efforts of Origen transformed the analysis of a rheological complicated fluid by bringing out the theory of micropolar fluids, the examples of such fluids are polymeric, paints, and colloidal solutions. This term then became a field of vibrant explorations. Since these types of fluids consist of micro-rotation vectors due to this characteristic, these fluids help in modeling blood flow, which is flowing in the blood vessels. Mirzaaghaian and Ganji^{19} carried out an examination of heat transfer for micropolar fluids through a porous surface. They analyzed the significance of various physical parameters upon stream function and temperature profile. Kumar *et al.*^{20} investigated a non-Fourier heat flux model for a micropolar liquid flow across a coagulated sheet. They have solved the modeled problem using shooting and Runge Kutta Fehlberg methods and have shown that their results were in good agreement with already existing results. Moreover, the buoyancy and the primary slip parameters have also shown to be increasing functions of velocity fields in this study. Ramadevi *et al.*^{21} investigated a MHD mixed convective flow of a micropolar fluid over a stretching surface using a modified Fourier heat flux model. They have established in this investigation that the velocity profile increased, while temperature and concentration profiles decreased with an increase in the primary slip parameter. Kumar *et al.*^{22} discussed the simultaneous solutions for first- and second-order slips of a micropolar fluid flow across a convective surface in the presence of Lorentz force and variable heat source/sink. Mehmood *et al.*^{23} studied numerically the micropolar Casson fluid over a stretched plate using some internal heating phenomenon. In this study, they discussed the impacts of various physical parameters for the strong and weak concentrations of velocity and temperature profiles. They have shown that the heat flux and skin friction dropped down for comparatively weak concentrations near the wall. Siddiqa *et al.*^{24} examined a periodic natural convective flow for a micropolar fluid by using the radiation effects. In this study, they reduced the modeled equations to a suitable form by introducing a stream function and then solved these equations numerically by applying the finite difference method along with the Keller-box scheme. They obtained exceptional results in comparison to previous studies. Srinivasacharya and Bindu^{25} analyzed the heat transmission and entropy generation for the flow of a micropolar fluid inside an annulus by using a magnetic field. In this analysis, they kept both the walls at the same velocity using the idea of suction/injection. They used the Chebyshev spectral collection method for the solution of the modeled problem. In this study, the least quantity of entropy generation was noted at the outer boundary, while at the inner boundary, the result was totally different, that is, the maximum entropy generation was noticed at this surface. Kumar *et al.*^{26} examined the physical aspects of an unsteady electrically conducting free convective stagnation point flow of a micropolar fluid past a stretching surface. Their results revealed that the heat source/sink and thermal radiation parameters have a tendency to augment the fluid temperature. Mabood *et al.*^{27} discussed numerically the mass and heat transmission for the micropolar fluid flow passing a stretching sheet in a permeable channel. In this analysis, a non-uniform source of heat and the Soret effect along with a magnetic field was applied to the fluid flow.

The study of heat transmission between two permeable plates is one of the most prominent subjects for research in the current time. Hence, many investigations have been conceded to review the heat transmission properties between two surfaces. Mustafa *et al.*^{28} studied heat and mass transmission between two surfaces. Mustafa’s outcomes highlighted an augmentation in the Nusselt number through an enhancement in the Prandtl number. Alizadeh *et al.*^{29} discussed the MHD flow for a micropolar fluid through a conduit filled with nanoparticles subjected to thermal radiations. It is observed in this investigation that augmentation in the values of the Nusselt number is totally dependent on the nanoparticle volume fraction and thermal radiations. Mehmood and Ali^{30} discussed the hydromagnetic flow with the transfer of heat between two stretched surfaces. They concluded in this study that the temperature of the system is reduced with the application of a magnetic field. Chamkha *et al.*^{31} discussed heat transmission and the flow of nanoparticles between the stretched sheet and permeable surface inside a rotating system. In this analysis, they have shown that the Reynolds number along with the nanofluid volume fraction grew due to heat transmission at the surface in both suction and injection cases. Dogonchi *et al.*^{32} analyzed heat transmission in graphene oxide nanoparticles through a permeable conduit using the effects of thermal radiations. It was seen in this study that with augmentation in the values of the Reynolds number, the skin friction was automatically raised.

Recently, a new class of fluids is introduced, named as hybrid nanofluid. The thermal characteristics and potential of heat diffusion properties of the fluid flow are upgraded in this class of fluids. That is why, this class of fluids is distinguished by its thermal characteristics and prospective applications, which essentially provide enrichment for heat transport properties. A hybrid substance is a material that joins together both physical as well as chemical characteristics of different materials and then provides these characteristics in a standardized form. Hybrid nanofluids have pretty good applications in various fields of heat transfer, for instance, transportations, manufacturing, and medical sciences. Various research articles have been published to study the concepts of hybrid nanofluids. This class of fluids is formed by suspension of unlike nanoparticles in composite/mixture type of base fluids as explained by Sarkar *et al.*^{33} Suresh *et al.*^{34} conceded an experimental study. In this study, it is examined that the thermal conductivity as well as the viscosity of the resultant mixture augmented with growth in the volume fraction of the nanofluid. Chamkha *et al.*^{35} studied numerically the time-dependent natural convective hybrid nanofluid in a semicircular container. Gorla *et al.*^{36} analyzed heat effects for source/sink on a hybrid nanofluid in a permeable square container. Sidik *et al.*^{37} discussed various types of hybrid nanofluids in detail. In this study, they also discussed various techniques for the preparation of hybrid fluids and factors that affect the performance of these fluids along with various applications. Shah *et al.*^{38} have investigated the electrical and thermal properties of the nanofluid flow over a starching surface.

In nature, most of the modeled problems are highly non-linear and are very complicated to solve, even sometimes, it is not possible to determine the exact solution to such problems. In order to determine the solution to such problems, one needs to use an appropriate numerical or analytical technique. Among various such techniques, the homotopy analysis method is also very popular for the solution of complex problems. It can be applied to nonlinear ordinary differential equations without making use of linearization or discretization. This method was first introduced by Liao^{39,40} for the solution of various non-linear problems. Liao has also proved that this technique is very fast convergent and presents a series solution in functional form. This method provides the solution in such a way that consists of all parameters for a given problem, and hence the behavior of these parameters can be discussed easily.

The main aim of the current study is to discuss the magnetohydrodynamic flow and heat transmission for a micropolar hybrid nanofluid within a rotating system between two plates in the presence of the Hall current and thermal radiations, where the bottom plate is stretched, and the top plate is permeable. The solution of nonlinear coupled modeled equations will be carried out by the homotopy analysis method. The main ideas will also be supported with the help of sketching graphs for different parameters. A critical examination of these physical parameters over dimensionless velocity, rotational velocity, micro-rotations, and temperature profiles will also be presented through graphs.

## II. PROBLEM FORMULATION

Consider incompressible hybrid Casson nanofluid between two parallel plates, such that the bottom plate is stretched and is set at *y* = 0, while the top plate is set at *y* = *δ*. Here, the flow under consideration is steady, viscous, radiative, and also laminar. In this system, the coordinates are chosen in such a manner that both the plates and fluid are moving around the *y axis* with angular velocity Ω. Both the plates are at a distance “*δ*” from each other. The upper plate of the system is exposed to consistent suction and injection velocity of magnitude $v\u03030$. A magnetic field with strength $B\u03030$ is practiced to the modeled system along the *y axis*. Moreover, the effects of the Hall current are considered for the flow model. Flow is conducted electrically, and whenever the magnetic field grows stronger, the Hall current generates and affects the flow. These effects result in the generation of a force and produce flow across the *z axis*, which ultimately deflects the assumed flow in 3D.

Ohm’s law consisting of the Hall current is expressed as

where $J\u0303$ represents the current density, while $B\u0303=0,B0,0$ denotes the magnetic field, the electric field is expressed by $E\u0303$ such that $E\u0303=0,E0,0$, and *ω*_{e}is the oscillating frequency for electron. The electron number density is given by *n*_{e}. For feeble ionized molecules, Ohm’s law provides *J*_{y} = 0, while *J*_{x} and *J*_{z} are given by

where *m* = *ω*_{e}*t*_{e}.

Following the aforementioned suppositions, governing equations are composed as^{31,41,42}

Continuity equation:

Momentum equations:

Micropolar flow equation:

Energy equation:

where Ω is the angular velocity, *B*_{0} is the magnetic field, *T* is the temperature, and *q*_{r} is the radiative heat flux. N denotes the microrotation angular velocity. We use the Rosseland approximation to simplify *q*_{r} as

In Eq. (11), *σ*^{*} shows the Stefan Boltzmann constant such that *σ*^{*} = 5.6697 × 10^{−8} W m^{−2} K ^{−4}, whereas *k*^{*} is the coefficient of mean absorption. Accordingly, Eq. (10) is reduced to the following form:

where *ρ*_{hnf}, *μ*_{hnf}, (*ρCp*)_{hnf}, and *k*_{hnf} are respective notations for density, dynamic viscosity, heat capacitance, and thermal conductivity. For micropolar nanoparticles, these terms are defined in Table I.

ρ_{hnf} | (1 − ϕ_{Cu} − ϕ_{Go})ρ_{f} + ϕ_{Cu}ρ_{Cu} + ϕ_{Go}ρ_{Go} |

(ρC_{p})_{hnf} | (1 − ϕ_{Cu} − ϕ_{Go}) (ρC_{p})_{f} + ϕ_{Cu}(ρC_{p})_{Cu} + ϕ_{Go}(ρC_{p})_{Go} |

μ_{hnf} | $\mu f1\u2212\varphi Cu\u2212\varphi Go2.5$ |

$khnfkf$ | $kCu\varphi Cu+kGo\varphi Go\varphi Cu+\varphi Go+2kf+2kCu\varphi Cu+kGo\varphi Go\u22122\varphi Cu+\varphi Gokf\xd7kCu\varphi Cu+kGo\varphi Go\varphi Cu+\varphi Go+2kf\u2212kCu\varphi Cu+kGo\varphi Go+\varphi Cu+\varphi Gokf\u22121$ |

ρ_{hnf} | (1 − ϕ_{Cu} − ϕ_{Go})ρ_{f} + ϕ_{Cu}ρ_{Cu} + ϕ_{Go}ρ_{Go} |

(ρC_{p})_{hnf} | (1 − ϕ_{Cu} − ϕ_{Go}) (ρC_{p})_{f} + ϕ_{Cu}(ρC_{p})_{Cu} + ϕ_{Go}(ρC_{p})_{Go} |

μ_{hnf} | $\mu f1\u2212\varphi Cu\u2212\varphi Go2.5$ |

$khnfkf$ | $kCu\varphi Cu+kGo\varphi Go\varphi Cu+\varphi Go+2kf+2kCu\varphi Cu+kGo\varphi Go\u22122\varphi Cu+\varphi Gokf\xd7kCu\varphi Cu+kGo\varphi Go\varphi Cu+\varphi Go+2kf\u2212kCu\varphi Cu+kGo\varphi Go+\varphi Cu+\varphi Gokf\u22121$ |

The subjected boundary conditions for the modeled problem are

Recommended dimensionless quantities are given as

Using Eq. (14) into Eqs. (5)–(9) and (12), we noticed that Eq. (5) is identically satisfied, while the rest of the equations are reduced to the following form:

where *D*_{1} and *D*_{2} are constants and are defined as follows:

When the non-dimensional variables are used from Eq. (13) into Eq. (14), the boundary conditions are transformed to the following new form:

where $\alpha =\u2212v0a\u2009\u2009\delta $ is the suction/injection parameter. At this stage, we shall also declare that when *α* > 0, there will be injection flow at the upper surface and whenever *α* < 0, there will be suction flow at the same surface. Note also that

Here, *Ro*, *N*_{1}, *Ha*, *N*, *Re*, *Ec*, and *Pr* denote the rotation, coupling, magnetic, and radiation parameters, and Reynolds, Eckert, and Prandtl numbers, respectively. The thermophysical characteristics of the nanofluid are expressed in Table II.

### A. Skin friction coefficient and Nusselt number

Mathematically skin friction is defined as

where the Nusselt number is defined as

Making use of Table I and Eq. (13) in Eqs. (21) and (22), the above two equations are reduced to the following form:

## III. METHOD FOR SOLUTION

Here, we use homotopy analysis method (HAM) for the solution of the nonlinear equations. This technique is semi-analytical and presents solutions to the problem in functional form. In order to derive this technique, he has taken into consideration one of the elementary ideas of topology known as homotopy. We know that two functions are homotopic if one can be continuously distorted in other functions. For derivation of HAM, let us assume that $X\u2322$ and $Y\u2322$ are two different topological spaces. Assume also that $\psi \u23221$ and $\psi \u23222$ are two continuous mappings from $X\u2322$ to $Y\u2322$. It is obvious that there will be a homotopy between the above mappings if there exists *ψ* such that

Such that $\u2200\u2009x\xaf\u2208X\u2322$,

where *ψ* is called homotopic. Using boundary conditions, Eqs. (15)–(18) can be solved with the help of HAM. Initial guesses for solution of these stated equations are specified as follows:

where linear operators of transformed equations are expressed as

## IV. RESULTS

Making use of boundary conditions as specified in Eq. (19), modeled equations [Eqs. (14)–(17)] have been evaluated by HAM. Various physical parameters for micropolar hybrid nanofluids have been discussed in detail. Some dimensionless quantities are described in Table I. The thermophysical characteristics are summarized in Table II. A comparison between the present result and the results of Ref. 31 for the Nusselt Number is given in Table III. Figure 1 expresses the graphical view of the modeled problem. The effects of various physical parameters for the modeled problem are shown in Figs. 2–18.

. | . | Present results in the absence . | . |
---|---|---|---|

. | Re . | of N and Ha = 0, Ec = 0
. | Results of Ref. 31 . |

ϕ = 0% | 0.1 | 1.078 381 555 | 1.076 50 |

0.5 | 1.403 658 578 | 1.404 37 | |

1 | 1.813 003 76 | 1.816 03 | |

1.5 | 2.201 327 214 | 2.202 19 | |

ϕ = 5% | 0.1 | 1.298 621 376 | 1.298 72 |

0.5 | 1.619 052 807 | 1.619 31 | |

1 | 2.026 519 802 | 2.027 32 | |

1.5 | 2.422 591 74 | 2.424 41 | |

ϕ = 10% | 0.1 | 1.573 849 277 | 1.571 95 |

0.5 | 1.889 474 207 | 1.888 89 | |

1 | 2.292 857 255 | 2.293 26 | |

1.5 | 2.691 972 738 | 2.693 99 |

. | . | Present results in the absence . | . |
---|---|---|---|

. | Re . | of N and Ha = 0, Ec = 0
. | Results of Ref. 31 . |

ϕ = 0% | 0.1 | 1.078 381 555 | 1.076 50 |

0.5 | 1.403 658 578 | 1.404 37 | |

1 | 1.813 003 76 | 1.816 03 | |

1.5 | 2.201 327 214 | 2.202 19 | |

ϕ = 5% | 0.1 | 1.298 621 376 | 1.298 72 |

0.5 | 1.619 052 807 | 1.619 31 | |

1 | 2.026 519 802 | 2.027 32 | |

1.5 | 2.422 591 74 | 2.424 41 | |

ϕ = 10% | 0.1 | 1.573 849 277 | 1.571 95 |

0.5 | 1.889 474 207 | 1.888 89 | |

1 | 2.292 857 255 | 2.293 26 | |

1.5 | 2.691 972 738 | 2.693 99 |

## V. DISCUSSIONS

This work describes the MHD flow and heat transmission for a micropolar hybrid nanofluid inside a rotating system between two plates, where the lower surface is kept stretched, while the upper one is porous. Furthermore, the Hall current and thermal radiations are used to deal with the modeled problem. In this segment, we shall describe the effects of different embedding parameters by using different graphs. The detailed discussions for velocity, rotational velocity, microrotation velocity, and temperature profiles are given below.

### A. Velocity profile $f\u2032$($\zeta $)

In this paragraph, a detailed discussion is carried out for illustration of the impact of suction/injection parameter *α*, magnetic and rotation parameters *Ha* and *Ro*, respectively, upon the velocity profile. In Fig. 2, the impact of suction and injection parameters on the velocity profile is described. As discussed before, when *α* > 0, we have injection flow, while for *α* < 0, there is suction flow in the closed vicinity of the upper plate for the assumed flow system. One can examine easily that with growing values for *α*, there is a corresponding growth in the magnitude of the velocity profile. On the other hand, this phenomenon is reverse for declining values of suction parameter. In Fig. 3, we can observe that the velocity profile has slightly reduced with augmented values of *Ha* for the micropolar hybrid nanofluid. This observation can be seen more visible in the interval 0.2 ≤ *ζ* ≤ 0.9 as we are approaching the permeable wall. Figure 4 indicates the effects of rotation parameter *Ro* upon velocity. It is shown in this figure that there is twofold performance of the velocity profile regarding distinct values of *Ro*. In the interval 0.0 ≤ *ζ* ≤ 0.4, it is increasing, while in the interval 0.4 < *ζ* ≤ 1.0, this phenomenon is reversed because the velocity is obviously dropped down for augmented values of *Ro*. Figure 5 depicts the impact of *β* upon the velocity profile of the micropolar hybrid nanofluid. It can be examined that the magnitude of the velocity profile is augmented near the porous wall of the flow system as the values of *β* are increasing.

### B. Rotational velocity profile *g*($\zeta $)

In this subsection, a detailed discussion is carried out for explaining the impact of suction/injection parameter *α*, magnetic parameter *Ha*, rotation parameter *Ro*, Reynolds number *Re*, and *β* upon the profile of rotational velocity. Figure 6 demonstrates the impact of the suction or injection parameter upon the rotational velocity profile regarding distinct values for *α*. It is shown that for growing values of *α*, there is an augmentation in *g*(*ζ*) in the closed vicinity of the permeable surface, while a reduction is observed in values of *g*(*ζ*) near the stretched surface for *α* < 0. Figure 7 indicates that growing values of *Ha* reduced velocity for the micropolar hybrid nanofluid. Figure 8 demonstrates the effects of rotational parameter *Ro* upon the velocity profile *g*(*ζ*) for a hybrid nanofluid. In this figure, we observe an augmentation in *g*(*ζ*) for growing values of *Ro*. Figure 9 specifies the effects of the Reynolds number *Re* upon *g*(*ζ*). Here, it is noticed that for augmented values of *Re*, there is an increase in the rotational velocity profile in the interval 0.0 ≤ *ζ* ≤ 0.9 but a minor change can also be seen in the small segment 0.9 < *ζ* ≤ 1.0, where a slight increase can be observed in the velocity profile. Figure 10 portrays the impact of *β* upon the rotational velocity profile for a micropolar hybrid nanofluid. It can be examined that the magnitude of *g*(*η*) is augmented near the porous wall of the flow system as the values for *β* are growing.

### C. Microrotation velocity profile *G*($\zeta $)

In this subsection, we shall discuss the impact of magnetic parameter *Ha* and Reynolds number *Re* upon the microrotation velocity profile *G*(*ζ*). Figures 11 and 12 show the effects of these physical parameters. Figure 11 depicts the impact of the magnetic parameter upon *G*(*ζ*). We observed that the velocity profile decreases in the closed vicinity of stretching plate for growing values of *Ha*. Slight augmentation can also be seen in values of *G*(*ζ*) in the interval 0.9 ≤ *ζ* ≤ 1.0. Figure 12 explains the performance of *G*(*ζ*) for different values of *Re*. Its reduction can be observed in values of *G*(*ζ*) in the interval 0 ≤ *ζ* ≤ 0.7, whereas an augmentation can also be seen for values of *G*(*ζ*) in the interval 0.7 < *ζ* ≤ 1.0.

### D. Temperature profile $\theta $($\zeta $)

Here, we will discuss the impacts of suction/injection parameter *α*, Eckert number *Ec*, magnetic parameter *Ha*, Prandtl number *Pr*, rotation parameter *Ro*, and Reynolds number *Re* upon the profile of temperature. These physical parameters are displayed in Figs. 13–18 and discussed in detail in adjacent lines. Figure 13 demonstrates the effects of *α* upon temperature profile *θ*(*ζ*). It is noticed that *θ*(*ζ*) is augmented for positive values of suction/injection parameter near porous wall of the modeled system. In Fig. 14, the impact of the Eckert number *Ec* upon the temperature profile is highlighted. It is observed that an increase in *Ec* gives rise to a thermal layer of boundary and produced internal energy that ultimately increase temperature, whereas small values of *Ec* reduce temperature. Figure 15 shows that jump up values of magnetic parameter give rise to the temperature profile. Actually, *Ha* depends upon the temperature gradient of the surrounding micropolar hybrid nanoparticles, and hence an increase in *Ha* will also increase the random motion of nanoparticles, which increases the internal energy and finally increases the values of *θ*(*ζ*). Figure 16 indicates the impact of *Pr* upon *θ*(*ζ*). It is noticed that growing values of *Pr* will increase the internal energy, and hence the temperature of the modeled problem will increase ultimately. Figure 17 shows the effects of the rotational parameter upon the temperature profile for a hybrid nanofluid. It is perceived that when *Ro* increases, the fluid motion also jumps up, and as a result, it gives rise to the temperature profile. In Fig. 18, we can see a drop in *θ*(*ζ*) with growing values of Reynolds number. Actually, an increase in the values of *Re* jumps down the internal energy of nanoparticles and ultimately drops the temperature of the fluid.

### E. Table discussion

In Table I, some dimensionless quantities are described for use in the modeled problem. The thermophysical properties of micropolar nanoparticles are discussed in Table II. It is important to reveal at this stage that for the current study, the volume fractions for copper and graphene oxide are utilized in equal magnitude, that is, (*φCu* = *φGo* = 0.5%). In order to support our result provided by the semi-analytical technique HAM, we compared these results with other available results in the literature, such as Ref. 31. Comparison confirmed that our outcome is in tremendous agreement with field values, which are described in Table III.

## VI. CONCLUSION

In this study, we examined the MHD flow and heat transmission of micropolar hybrid nanoparticles between two surfaces. The lower surface is assumed as stretchable and the upper surface as permeable. The modeled problem is transformed to a set of nonlinear ODEs. HAM is used to solve this non-linear system of ODEs. As a hybrid substance is a material that combines both physical and chemical characteristics of different materials and then provides these properties in a standardized and upgraded form, this model can be applied in the area of heat transmission, such as transportations and medical sciences. The impacts of various dynamic parameters upon velocity, rotational velocity, micropolar velocity, and temperature profiles have been explored in this work. After a detailed study of the article, it is noticed that:

There is a growth in the velocity profile for

*α*> 0, while the velocity is reduced with growing values of magnetic parameter and for negative values of*α*.A dual behavior for various values of rotational parameter

*Ro*is also noticed in velocity profile*f*′(*ζ*). In the interval 0.0 ≤*ζ*≤ 0.4, it increases, while in the interval 0.4 <*ζ*≤ 1.0, this phenomenon is reversed because velocity is obviously reducing for a corresponding augmentation in the values of*Ro*.Rotational velocity profile

*g*(*ζ*) seemed to be increased with augmented values of rotational parameter*Ro*. On the other hand, a reduction in*g*(*ζ*) is also observed with augmented values of*Ha*and*Re*.A decreasing behavior is also observed in microrotation velocity profile

*G*(*ζ*) for augmented values of magnetic parameter*Ha*and Reynolds number*Re*.It is also noticed that with augmentation in the values of Eckert number

*Ec*, magnetic parameter*Ha*, Prandtl number*Pr*, and rotation parameter*Ro*, there is a corresponding growth in temperature profile*θ*(*ζ*). A drop down is also seen in*θ*(*ζ*) with growing values of Reynolds number*Re*.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

The authors declare that they have no competing interest.

## NOMENCLATURE

### Symbol Physical Description

- $B\u0303o$
magnetic field (N mA

^{−1})*C*_{f}skin friction coefficient

*C*_{p}specific heat (Jkg

^{−1}K^{−1})- $E\u0303$
electric field intensity (NC

^{−1})*Ec*Eckert number

*Ha*magnetic parameter

*k*_{hnf}thermal conductivity of hybrid nanoparticles

*k*_{f}thermal conductivity of base fluid

*m*Hall parameter

*N*microrotation velocity (m/s

^{−1})*Nu*Nusselt number

*N*_{1}coupling parameter

*p*dimensional pressure

*Pr*Prandtl number

*q*_{r}radiative heat flux (Wm

^{−2})*Re*Reynolds number

*Rd*radiation parameter

*Ro*rotation parameter

*T*dimensional temperature (K)

*T*_{∞}temperature of top wall

*t*_{e}flow time

*v*_{o}injection velocity (m/s

^{−1})*x*,*y*dimensional Cartesian coordinates

*x*,*ζ*dimensionless Cartesian coordinates

### Greek symbols

### Subscript notations

## REFERENCES

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_{3}–Cu/water hybrid nanofluids using two step method and its thermo-physical properties