Computational Analysis of Darcy–Forchheimer Flow of Cu/Al–Al2O3 Hybrid Nanofluid in Water over a Heated Stretchable Plate with Nonlinear Radiation

The aim of this study is to examine the Darcy–Forchheimer flow = of H2O-based Al−Al2O3/Cu−Al2O3 hybrid nanofluid past a heated stretchable plate including heat consumption/ generation and non-linear radiation impacts. The governing flow equations are formulated using the Naiver–Stokes equation. These flow equations are re-framed by using the befitted transformations. The MATLAB bvp4c scheme is utilized to compute the converted flow equations numerically. The graphs, tables, and charts display the vicissitudes in the hybrid nanofluid velocity, hybrid nanofluid temperature, skin friction coefficient, and local Nusselt number via relevant flow factors. It can be seen that the hybrid nanofluid velocity decreased as the magnetic field parameter was increased. The hybrid nanofluid temperature tended to rise as the heat absorption/generation, nanoparticle volume friction, and nonlinear radiation parameters were increased. The surface drag force decreased when the quantity of the magnetic parameter increased. The larger size of the radiation parameter led to enrichment of the heat transmission gradient.


Introduction
Many scientists and engineers are attempting to improve the heat transmission efficiency since it has an extensive variety of applications in the industrial sectors. Common liquids, such as ethylene glycol, kerosene, water, oil, and polymer-based solutions are used in the heat transmission processes. They have a poor heat transmission rate because of their weaker heat conductivity. To solve this deficiency, experts from several disciplines have attempted to increase the heat conductivity. One of the most effective ways to address this problem is by dispersing nanoparticles across various base fluids. HNFs (hybrid nanofluids) are composed of two or more distinct kinds of nanoparticles in a base fluid. In addition, the HNFs have a heat transmission rate that is much greater than that of general nanofluids, see [1][2][3]. These HNFs may be used in a number of contexts, including in heat exchangers, engine cooling, extrusion processes, micro-manufacturing, drug delivery, energy production, etc. Ikram et al. [4] investigated the flow of H 2 O-based Ag − TiO 2 hybrid nanofluid in a microchannel. They demonstrated that HNF velocity tended to decrease as HNPVF values increased. The MHD flow of H 2 O-based Al 2 O 3 − Cu HNF past a SS was explored by Jawad et al. [5]. They found that the SFC was upgraded when the SVF of the nanoparticles was developing. Devi and Devi [6] elucidated the flow of hydromagnetic Cu − Al 2 O 3 HNF in water over a SS. They noticed that the larger HTG occurred in Cu − Al 2 O 3 HNF aircraft, propulsion devices, power plants, furnace designs etc. Yusuf et al. [30] probed the radiative flow of Cu − TiO 2 /H 2 O HNF on a SS with slip condition. They revealed that the EG number quickens when the quantity of the radiation parameter is increased. The MHD NF flow on a plate with radiation was examined by Mustafa et al. [31]. They found that the larger temperature ratio parameter improves the thermal profile. The unsteady 3D MHD flow of HNF with radiation was illustrated by Mabood et al. [32]. They demonstrated that raising the radiation parameter leads to increase the NFT. Kumar et al. [33] explored the radiative flow of Williamson fluid on an SS. They found that the HTG is reinforced when the radiation parameter is improved. The numerical modeling of water-based Ag/Cu NF with radiation was addressed by Qayyum et al. [34]. Patel and Singh [35] investigated the influence of non-linear radiative flow of micropolar NF through a non-linear heated SS. Lu et al. [36] scrutinized the MHD flow of Carreau NF over a SS with non-linear radiation. They demonstrated that the TR parameter leads to fortifying the LNN. The influence of non-linear radiative flow of WNF on a SS was probed by Danish Lu et al. [37]. They discovered that by enhancing the radiation parameter causes to decay the local Sherwood number. The MHD flow of Casson HNF past a SS with non-linear radiation was scrutinized by Abbas et al. [38]. Their outcomes show that the temperature distribution escalates with the higher values of the non-linear radiation parameter. Eswaramoorthi et al. [39] investigated 3D radiative flow of CNTs over a Riga plate. They concluded that the Bejan number heightens when improving the radiation parameter.
According to the aforementioned literature reviews, there is still a lack of research on the flow of a H 2 O based Al − Al 2 O 3 /Cu − Al 2 O 3 HNF past a stretchable plate with convective heating, heat consumption/generation, and non-linear radiation effects. Our research outcomes are used in many numerous technical and industrial applications, like gas turbine rotors, crystal growing, drawing of films, lubrication processes, glider aircraft, power generation, etc.
Finally, the main objective of our investigations is as follows: • To deliberate the implications of the model's design on the HNF flow through the stretchable plate. • How does the usage of HNF lead to affect the velocity and temperature of the fluid? • How is the HNF temperature impacted by heat generation/absorption and nonlinear radiation? • How is the heat transfer mechanism improved when convective heating conditions are present?

Mathematical Formulation
The MHD DFF of H 2 O based Cu − Al/Al 2 O 3 HNF past a stretchable plate is investigated. Let u and v are the HNF velocity factors along the x and y axes. A stable magnetic field of magnitude B 0 is activated in the flow direction and resultant magnetic field is disregarded due to small size of Reynolds number. The outcomes of heat generation/absorption and non-linear radiation are also taken into account. Moreover, the sheet and free stream HNFT's are denoted as T w and T ∞ < T w , respectively. The physical schematic of the flow model are displayed in Figure 1. The governing mathematical model can be defined as follows based on the preceding assumptions, see Devi and Devi [6]: The initial and boundary conditions are expressed as Define the variables Implementing the aforementioned adjustments (5) in (2) and (3), we get the following simplified equations: The correlated boundary conditions are where The SFC and the LNN are defined as:

Numerical Solutions
The re-framed expressions (6) and (7) with the correlated boundary restraints (8) are solved numerically by implement the MATLAB bvp4c approach. Initially the higher order problem is transformed into a first order ODE form, see Prabakaran et al. [40]. In this regard, we consider the followings: with the constraints are, To solve the above problem numerically, we use the MATLAB bvp4c method with maximal residual error is 10 −5 and size of the step is 0.05.

Results and Discussion
The primary goal of this section is to delivers the effect of various emerging flow parameters on HNFV, HNFT, SFC and LNN. Table 1 exhibits the thermal properties of aluminum, copper, aluminum oxide, and water. Table 2 shows the mathematical expressions of thermal properties of the HNF. The SFC of water based Cu − Al 2 O 3 and Al − Al 2 O 3 HNF for various values of M, f w, Fr, φ 1 , φ 2 and λ was presented in Table 3. It is perceived that the SFC diminishes when raises the values of Fr, M, f w and λ and it improves when strengthening the quantity of φ 1 and φ 2 for both HNFs. Table 4 presents the LNN for distinct values of Γ, Rd, Hg, f w, Fr and φ 2 for both HNFs. It is viewed that the HTR raises when enriching the values of Rd, Γ, f w, and φ 2 and the opposite effect attains for the larger size of Hg and Fr for both HNFs. Table 5 exhibits the comparison of θ (0) with Rd = M = Hg = f w = 0 to Devi and Devi [6] for distinct values of Pr and are found in agreeable accord.  Table 2. Thermophysical properties of Hybrid nanofluid.

Properties Hybrid Nanofuid
Density Viscosity Thermal conductivity Electrical conductivity

Conclusions
The steady, 2D, non-linear radiative Darcy-Forchheirmer flow of H 2 O based hybrid nanofluid past a stretchable plate with the presence of heat absorption/generation and magnetic field was investigated. Two different mixture of hybrid nanofluid, namely Cu − Al 2 O 3 and Al − Al 2 O 3 are taken into account. The governing flow models are re-changed by implementing the suitable transformations and solved by using MATLAB bvp4c code. Some remarkable observations of our findings are given below.
• The hybrid nanofluid velocity profile decrepitude's for larger quantity of Fr (Forchheirmer), M (magnetic field parameter) and f w (suction/injection parameter). • The larger values of Rd (radiation parameter) improve the hybrid nanofluid fluid temperature. • The hybrid nanofluid has a larger heat transfer rate than the ordinary fluid.
• The more presence Fr (Forchheirmer number), M (magnetic field parameter) and f w (suction/injection parameter) causes to reduce the skin friction coefficient. • The Rd (radiation parameter) and Γ (temperature ratio parameter) lead to enriching the heat transfer rate. • The Cu − Al 2 O 3 hybrid nanofluid have higher heat transfer rate than the Al − Al 2 O 3 hybrid nanofluid.