Numerical Study of (Au-Cu)/Water and (Au-Cu)/Ethylene Glycol Hybrid Nanoﬂuids Flow and Heat Transfer over a Stretching Porous Plate

: The purpose of the study is to investigate the (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanoﬂuids ﬂow and heat transfer through a linear stretching porous plate with the effects of thermal radiation, ohmic heating, and viscous dissipation. Similarity transformations technique is used to transform a governing system of partial differential equations into ordinary differential equations. The NDSolve Mathematica program is used to solve the nonlinear ordinary differential equations. Furthermore, the results are compared with the results of homotopy analysis method. The impacts of relevant physical parameters on velocity, temperature, and the Nusselt number are represented in graphical form. The key points indicate that the temperature of (Au-Cu)/water and (Au-Cu)/Ethylene glycol hybrid nanoﬂuids is increased with the effects of Eckert number and magnetic ﬁeld. The (Au-Cu)/Ethylene glycol hybrid nanoﬂuid also has a greater rate of heat transfer than (Au-Cu)/Water hybrid nanoﬂuid. on nanoﬂuids ﬂow


Introduction
The NDSolve (wolfram language) function is a general numerical differential equation solver. The NDSolve can handle a wide range of partial differential equations as well as ordinary differential equations. The NDSolve can be used to find the unknown functions that must depend on a single variable. The differential-algebraic equation can be solved by the NDSolve technique. Several significant studies are presented by using the NDSolve technique. Hayat et al. [1] examined the numerical solution of Darcy Forchheimer's nanofluid flow over-stretching curved surface via the NDSolve Mathematica technique.
Ullah et al. [2] used differential transform method, Adomian decomposition method, Daftardar-Jafari method, optimal homotopy asymptotic method, homotopy perturbation method, and the NDSolve Mathematica program to examine the magnetohydrodynamic squeezing flow of viscous fluid with slip effect. Ahmad et al. [3] analyzed the unsteady squeezing flow and entropy generation of viscous Al 2 O 3 -water, TiO 2 -water, CuO-water, Ag-water, and Cu-water nanofluids with the help of the NDSolve Mathematica technique. The effects of magnet field, thermal radiation, Joule heating, and viscous dissipation are also studied. Qayyum et al. [4] used the NDSolve Mathematica technique to discuss the theoretical investigation and modeling of curved parabolized surface with secondorder slip condition. Li et al. [5] explored the entropy optimized viscous fluid with the help of Newton's built-in work on 3D boundary layer flow of a DO-(TiO 2 /SiC) hybrid nanofluid shooting technique (NDSolve). Silent features entropy generation, Bejan number, concentration, velocity, and temperature are deliberated. microchannel. Beg et al. [30] developed the mathematical model of steady-state mixed bio-convection nanofluid flow over a vertical wall and found the approximate solution of the proposed model.
The literature mentioned above observed that several researchers worked on hybrid nanofluid flow and porous medium. The goal of this research is to examine the numerical results of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids flow and heat transfer over a stretching porous plate. The effects of physical parameters are explained in graphical form.

Problem Formulations
The 2-D incompressible hybrid nanofluids over a linear stretching porous plate are considered. A uniform magnetic field parallel to the y-axis is applied. The electric field and magnetic field B are assumed to be negligible. It is also assumed that the flow influences thermal radiation, viscous dissipation, and ohmic heating. The nanoparticles and fluid are in equilibrium and no-slip occurs between them. The geometry of the problem is displayed as [31] (Figure 1). ∂u ∂x Here, u, v, T, T w , and T ∞ are velocities, the temperature of hybrid nanofluid, wall temperature, and far field temperature, respectively. B, σ, and q r are the magnetic field, electrical conductivity, and radiative heat flux, respectively. (sink) and radiation parameter, the heat transfer rate was diminished. Lund et al. [28] investigated magnetized Water-(Al2O3/Cu) hybrid nanofluid flow and heat transfer along with a stretching sheet with the presence of magnetohydrodynamic (MHD), porous medium, and viscous dissipation. Ting et al. [29] discussed the flow and heat characteristics of Water-Al2O3 nanofluid in a porous microchannel. Beg et al. [30] developed the mathematical model of steady-state mixed bio-convection nanofluid flow over a vertical wall and found the approximate solution of the proposed model. The literature mentioned above observed that several researchers worked on hybrid nanofluid flow and porous medium. The goal of this research is to examine the numerical results of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids flow and heat transfer over a stretching porous plate. The effects of physical parameters are explained in graphical form.

Problem Formulations
The 2-D incompressible hybrid nanofluids over a linear stretching porous plate are considered. A uniform magnetic field parallel to the y-axis is applied. The electric field and magnetic field B are assumed to be negligible. It is also assumed that the flow influences thermal radiation, viscous dissipation, and ohmic heating. The nanoparticles and fluid are in equilibrium and no-slip occurs between them. The geometry of the problem is displayed as [31] (Figure 1).  The radiative heat flux is transformed into the following form using the Rosseland approximation: where k * is the mean absorption and δ * is the Stefan-Boltzmann. Moreover, the temperature within the flow was expected to be low, thus T 4 was stated as a linear function of temperature. As a result of applying the Taylor series and ignoring higher-order terms, T 4 is obtained as: The similarity transformations are: By using Equations (4)-(7), Equations (1)-(3) become: The relevant boundary conditions are described as a fellow: where R represents the radiation parameter, m is the injection parameter, Pr denotes the Prandtl number, Ec, represents the Eckert number, k represents the porous medium, and M represents the magnetic parameter. Table 1 displays the thermophysical characteristics of water and nanoparticles. The local Nusselt number (Nu) is given as: where, q w (x) = −k n f ∂T ∂y y=o .

Results and Discussion
The effects of different physical parameters such as radiation, injection, Prandtl number, Eckert number, and magnetic parameter on velocity and temperature profiles are investigated. The heat transfer rate is also represented graphically. Figure 2 shows that the velocity of (Au-Cu)/Water hybrid nanofluid is greater than (Au-Cu)/Ethylene glycol hybrid nanofluid. Figure 3 explains the velocity distribution of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids for the dissimilar value of φ 1 and φ 2 . It is noted from Figure 3 that the velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids decrease with the increase of φ 1 and φ 2 . This occured because the Au and Cu nanoparticles possess higher thermal conductivity than the water and Ethylene glycol. The velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids within the influences of M is displayed in Figure 4. As seen in Figure 4, the velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids have an inverse relation with M. Physically, the Lorentz force generated by M acted against the motion of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids. The velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids flow with a variation of m, as shown in Figure 5. It is noted from Figure 5 that the velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids decrease with the enhancement of m. This is due to the removal of nanofluid particles through porous surface that reduced the boundary layer thickness. Figure 6 shows the velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids for various values of K. As seen in Figure 4, velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids have a direct relation with K. Physically porous medium reduces the resistance to hybrid nanofluids and thus enhances the velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids. Figure 7 shows that the temperature of (Au-Cu)/Water hybrid nanofluid is greater than (Au-Cu)/Ethylene glycol hybrid nanofluid. This is due to the thermal conductivity of (Au-Cu)/Water hybrid nanofluid, which is more than (Au-Cu)/Ethylene glycol hybrid nanofluid. Figure 8 shows the effects of φ 1 and φ 2 on temperature distribution of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids. The temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids is the inverse function of φ 1 and φ 2 . The reason for this is that temperature distribution is eventually slowed from stretching porous plates. Figure 9 shows the temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids with the influences of M. It was observed that the temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids flow enhances with increasing the numerical value of M. Physically, the Lorentz force heating in the temperature equation serves as an extra heat source to the thermal boundary layer. Figure 10 depicts the impact of m on the temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids. Figure 10 indicates that as m is increased, the temperature of the (Au-Cu)/Water and (Au-Cu)/Ethylene hybrid nanofluids decreases. The reason for this is that the fluid density of fluid decreases when increasing the numerical value of m, which results in the temperature of hybrid nanofluids decreasing. Figure 11 illustrates the temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids for dissimilar values of R. The temperature profiles of the hybrid nanofluids are seen to decrease as R values are increased. As T ∞ reduces as R decreases, the Rosseland radiation absorptivity drops, and the thickness of the thermal boundary layer decreases. Figure 12 presents the temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids boosted with the impacts of Ec. Physically, the fraction of heat is shown to be deposited in hybrid nanofluids. The rate of heat transfer in (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids is plotted in Figures 13 and 14. It is noted that in the case (Au-Cu)/Water hybrid nanofluid the heat transfer rate has a direct and inverse relationship with R and m, respectively. In case of (Au-Cu)/Ethylene glycol hybrid the rate of heat transfer has a direct relationship with R and m. The heat transfer rate of (Au-Cu)/Ethylene glycol hybrid nanofluids is higher than the heat transfer rate of (Au-Cu)/Water hybrid nanofluids. To check the accuracy of numerical solution, the results obtained by the NDSolve method are compared with the homotopy analysis method. Figures 15 and 16 are presented as a comparison of the velocity and temperature of (Au-Cu)/Ethylene glycol hybrid nanofluid. Figure 17 presents the error decay for 10th-order approximation. Tables 2 and 3 show an excellent comparison of velocity and temperature profiles of (Au-Cu)/Ethylene hybrid nanofluid. To ensure the validity of our findings, we compared the results of our study ( f or θ (0)) with the published results of Chen [32], Zaimi et al. [33], and Naramgari and Sulochana [34] for m = 0, Ec = 0, M = 0, φ 1 , φ 2 = 0 as shown in Table 4. The obtained results are found in good agreement with the published results.
has a direct and inverse relationship with R and m, respectively. In case of (Au-Cu)/Ethylene glycol hybrid the rate of heat transfer has a direct relationship with R and m. The heat transfer rate of (Au-Cu)/Ethylene glycol hybrid nanofluids is higher than the heat transfer rate of (Au-Cu)/Water hybrid nanofluids. To check the accuracy of numerical solution, the results obtained by the NDSolve method are compared with the homotopy analysis method. Figures 15 and 16 are presented as a comparison of the velocity and temperature of (Au-Cu)/Ethylene glycol hybrid nanofluid. Figure 17 presents the error decay for 10 th -order approximation. Tables 2 and 3 show an excellent comparison of velocity and temperature profiles of (Au-Cu)/Ethylene hybrid nanofluid. To ensure the validity of our findings, we compared the results of our study ( ′(0) ) with the published results of Chen [32], Zaimi et al. [33], and Naramgari and Sulochana [34] for = 0, = 0, = 0, 1 , 2 = 0 as shown in Table 4. The obtained results are found in good agreement with the published results.  has a direct and inverse relationship with R and m, respectively. In case of (Au-Cu)/Ethylene glycol hybrid the rate of heat transfer has a direct relationship with R and m. The heat transfer rate of (Au-Cu)/Ethylene glycol hybrid nanofluids is higher than the heat transfer rate of (Au-Cu)/Water hybrid nanofluids. To check the accuracy of numerical solution, the results obtained by the NDSolve method are compared with the homotopy analysis method. Figures 15 and 16 are presented as a comparison of the velocity and temperature of (Au-Cu)/Ethylene glycol hybrid nanofluid. Figure 17 presents the error decay for 10 th -order approximation. Tables 2 and 3 show an excellent comparison of velocity and temperature profiles of (Au-Cu)/Ethylene hybrid nanofluid. To ensure the validity of our findings, we compared the results of our study ( ′(0) ) with the published results of Chen [32], Zaimi et al. [33], and Naramgari and Sulochana [34] for = 0, = 0, = 0, 1 , 2 = 0 as shown in Table 4. The obtained results are found in good agreement with the published results.

Conclusions
This study investigates the effects of viscous dissipation and ohmic heating on the flow of (Au-Cu)/Water and (Au-Cu)/ Ethylene glycol hybrid nanofluids through a linear stretching porous plate. A well-known Mathematica software, NDSolve, is used to solve the nonlinear differential equations numerically. The numerical results are compared with the analytical results via the homotopy analysis method and excellent agreement is noted. The study revealed impacts of relevant physical parameters, i.e., radiation parameter, Prandtl number, injection parameter, Eckert number, and magnetic parameter on hybrid nanofluids flow and the characteristics of heat transfer. It is pointed out that the velocity of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids are decreased with increasing the value of m, M, R and velocity is increased with the impact of K. The temperature of (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids decreased with the influences of φ 1 = φ 2 , R and m, while the temperature of (Au-Cu)/water and (Au-Cu)/Ethylene glycol hybrid nanofluids is increased with the effects of Ec and M. The heat transfer rate of (Au-Cu)/Ethylene glycol hybrid nanofluid is also greater when compared to the (Au-Cu)/Water hybrid nanofluid. Acknowledgments: The authors would like to thank Taif University Researches Supporting Project (TURSP-2020/96), Taif University, Taif, Saudi Arabia, for their financial assistance. The authors also wish to express their gratitude to anonymous referees for their excellent comments and recommendations, which substantially improved the work.

Conflicts of Interest:
The authors declare no conflict of interest.