**1. Introduction**

The liquid coolants are mainly employed to keep the operating temperature of different equipment in the specified range by transferring heat from them. Presently, the investigations show that the thermal conductivities of different fluids used in liquid coolants are much smaller when compared to those of solid metals. The natural laminar and convective heat energy transfer is a significant process in engineering and industry due to its numerous applications. Currently, a lot of research is in progress to investigate the different heat energy transfer characteristics of the newly developed fluid called "hybrid nanofluid" due to an extensive range of engineering and technological uses, like medical manufacturing, microfluidics, transportation, generator cooling, naval structures, and solar heating. A hybrid nanofluid refers to a combination in a base fluid of two different types of nanoparticles. Therefore, if nanoparticles materials are correctly picked, then they will enhance each other's positive aspects. The metallic nanoparticles, such as aluminum, copper, and zinc, have high thermal conductivities. However, the use of these metallic nanoparticles for nanofluid applications is limited because of their reactivity and stability. On the other hand, ceramic nanoparticles have lower thermal conductivity relative to metal

nanoparticles, but they have many desirable possessions, such as chemical inertness and stability. In the case of closed cavities, heat transfer study appeared in several applications and has been widely deliberated in the literature [1–3]. In open cavities, the processes of natural convection give significant results by simulating more complex geometries at the open end. In open cavities natural convection is related to different engineering systems, such as the cooling of electrical equipment, room air conditioning, and solar thermal central receiver systems etc. [4–6]. Skok et al. [4] have undertaken an experimental survey of the open cavity natural convective flow numerically. They found some good agreements between the experimental data and the numerical results. For larger values, their results are very significant. Chan and Tien [7] studied the two-dimensional (2-D) natural convection flow in narrow open cavities and conducted relative studies while using open square cavities with protracted domains. The small thermal conductivities of the base fluids used in the natural convection are showed to be an essential obstacle for increasing the heat energy transformation rate beyond a definite limit. As a ground breaking work, Choi et al. added solid nanoparticles to the base fluid to enhance its thermal conduction, and called it nanofluids [8]. Mahmoudi et al. [9] numerically deliberated the enhancement of natural convective heat energy transfer flow. Sheremet et al. performed the numerical investigation of non-steady natural convection in a permeable open corrugated cavity through a two-phase model [10]. They showed that the Nusselt number reduces by enhancing the number of iterations. Presently, Tassaddiqet al. [11]investigated couple stress magneto-hydrodynamic nanofluid thin film flow over an exponential stretching sheet with joule heating and viscous dissipation. Tahar and Chamkha [12] discussed the flow of a hybrid nanofluid through horizontal and confocal elliptical cylinders with natural convection enhancement. Moreover, it has been determined that the key aim of including the nanomaterials in a transferor liquid is to augment its thermal conductivity. It shall also be noted that stable nanofluids have significant attributes, such as small nanoparticle concentration and higher thermal conductivities. Hence, the majority of the studies in the past have been carried out to attain high thermal conductivity through the use of a single nanomaterial [13–16]. Currently, many articles have considered the topic of hybrid nanofluids [17–20]. Mixed nanomaterials display important chemical and thermo-physical characteristics that do not happen in a single component. Hybrid nanomaterials are mainly categorized into three types [21–23]. In current years, numerous numerical and experimental works that are associated to hybrid nanofluids have been published, and their results display that they are more suitable than the conventional nanofluids. Suresh et al. [24] discussed a copper-alumina nanocomposite powder, which was mixed by using thermochemical method, and prepared a hybrid nanofluid through a two-step procedure. In another study, Suresh et al. [25] examined the heat energy transfer characteristics of hybrid nanofluid (alumina-copper)/water metal nanomaterials and polymer nanomaterials. Nadeem et al. [26] deliberated the MHD Maxwell nanofluid flow through a stretch sheet. Rockney et al. [27] studied the MHD nanofluid flow involving heat energy transfer through two plates. Shehzad et al. [28] investigated a nanofluidic flow by using the Jaffrey fluid model with MHD convective boundary conditions. Mahmood et al. discussed the flow of nanofluids for cooling purposes [29]. Nanofluid flow through a porous medium by incorporating the heat conduction through channels have explored by Fakour et al. [30]. Hatami et al. [31] described the laminar flow of nanofluids through rotating disks. Nadeem et al. [32] investigated the non-orthogonal and nanofluid non-Newtonian flows with heat energy transformation. Sheikholeshlami et al. [33] thoroughly investigated the nanofluid flow through a semi-porous channel. Akbar et al. [34] studied the viscosity and buoyancy impacts during the nanofluid MHD flow over a stretch surface. Fakour et al. [35] have undertaken the nanofluid flow through vertical channels. Maskeen et al. [36] have examined nanofluid flow by using water as base fluid and investigated the enrichment of heat energy transfer through the stretching sheet. Akilu et al. [37] deliberated the flow with the thermo-physical properties of water-based composite nanofluids. Hayat et al. [38] considered the Newtonian nanofluid flow through a cylinder along with the heating impacts. Further study can be read in [39–42].

Ceramic materials like alumina (Al2O3) have numerous excellent possessions, such as chemical inertness and good stability. Yet, alumina's thermal conductivity is small when compared to metallic nanoparticles. Metallic nanoparticles, such as copper, have greater thermal conductivity. Yet, reactivity and stability are two significant factors that impede these metallic nanoparticles from being used. The inclusion of small quantities of copper particles in an alumina matrix will significantly increase the thermal possessions without affecting the nanofluid's stability. Jena et al. [43] clarified the synthesis of Cu-Al2O3 nanocomposites while using hydrogen reduction techniques from chemically formulated Cu-Al2O3 mixtures. Niihara [44] and Oh et al. [45,46] revealed the manufacture of Al2O3-Cu nanocomposite made from fine powder mixtures of Al2O3 and CuO nanoscale. The proposed nanocomposites had a new concept of material design, thermal properties, and enhanced mechanical dramatically.

The Darcy–Forchheimer law is the law for porous media flows with Reynolds numbers more than approximately 1 to 10, and the inertial impacts can also become important. Sometimes a term of inertia is applied to the equation of Darcy, defined as the word Forchheimer. The non-linear behavior of the pressure difference vs. flow data can be taken into account by this term. In 1856, Hennery Darcy has developed the flow of a steady fluid through porous media, when he was carrying out his work of moving the water through sand bags. Though his outcome was not very much reliable at that time because he did not deliver a positive consequence. Yet, based on his research, a Dutch scientist, known as Forchheimer, came in 1901. Forchheimer, gave his ideas and expressions more extensive. Forchheimer added the velocity term square in the momentum equations for calculating inertial forces as well as limit layer flow [47]. Muskat later added the name 'Forchheimer term' [48] Pal and Mondal [49] later investigated Darcy–Forchheimer's model over a stretch face, where they defined that the value of the electric field parameter increases with a lessening in the nanoparticles concentration sketch. Ganesh et al. [50] conducted a study on the nanofluid flow of Darcy–Forchheimer MHD over a stretched/shrinking sheet and determined that the temperature value rises with the presence of viscous-dissipation effects. Hayat et al. [51] investigated the Darcy–Forchheimer flow with heat flux and variable thermal conductivity between Cattaneo and Christov. Muhammad et al. [52] an updated Darcy–Forchheimer flow of Maxwell nanofluid model due to convective boundary conditions. They used the pores medium, and found that the concentration and temperature-profile of nanoparticles increased with the value of the porosity parameter. Jawad et al. [53] have been studied MHD Nanofluid Darcy–Forchheimer thin film flow with Navier's partial slip and joule dissipation. Uddin et al. [54] have been deliberated flow with nonlinear thermal Radiation of Darcy–Forchheimer Sisko nanomaterial. Mohamed et al. [55] have studied flow of Carreau nanofluid over a convectively heated nonlinear stretching surface in the presence chemically reactive species. Lahmar et al. [56] have investigated unsteady nanofluid squeezing flow with the effects of an inclined magnetic field and variable thermal conductivity. Mohamed et al. [57] have discussed FEM for blood-based SWCNTs flow with electromagnetic radiation through a circular cylinder in a porous medium. Mohamed et al. [58] have described SQLM for external yield stress effect on three-dimensional (3D) MHD nanofluid flow in a porous medium. Mohamed [59] has studied MHD boundary-layer flow of two-phase nanofluid model over an exponentially stretching sheet with a heat generation and Chemical reaction effect. Mohamed [60] has investigated unsteady flow of water-NPs over a stretching sheet in a saturated porous medium in the stagnation-point region with chemical reaction. Other related study can be read in [61–63].

The aim of this research work is the investigation of the thermal characteristics of the Darcy– Forchheimer hydromagnetic hybrid nanofluid flow through a stretching porous cylinder. The model equations with appropriate boundary conditions are solved analytically. Thermophoresis and Brownian motion impacts are mainly focused in this work. The impacts of modeled parameters over velocity, temperature, and concentrations profiles are graphically studied.
