Influence of a Darcy-Forchheimer porous medium on the flow of a radiative magnetized rotating hybrid nanofluid over a shrinking surface

In this paper, the heat transfer properties in the three-dimensional (3D) magnetized with the Darcy-Forchheimer flow over a shrinking surface of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Cu + Al_{2} O_{3} /$$\end{document}Cu+Al2O3/ water hybrid nanofluid with radiation effect were studied. Valid linear similarity variables convert the partial differential equations (PDEs) into the ordinary differential equations (ODEs). With the help of the shootlib function in the Maple software, the generalized model in the form of ODEs is numerically solved by the shooting method. Shooting method can produce non-unique solutions when correct initial assumptions are suggested. The findings are found to have two solutions, thereby contributing to the introduction of a stability analysis that validates the attainability of first solution. Stability analysis is performed by employing if bvp4c method in MATLAB software. The results show limitless values of dual solutions at many calculated parameters allowing the turning points and essential values to not exist. Results reveal that the presence of dual solutions relies on the values of the porosity, coefficient of inertia, magnetic, and suction parameters for the specific values of the other applied parameters. Moreover, it has been noted that dual solutions exist in the ranges of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{s} \le F_{sc}$$\end{document}Fs≤Fsc, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \ge M_{C}$$\end{document}M≥MC, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S \ge S_{C} ,$$\end{document}S≥SC, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{C} \le K$$\end{document}KC≤K whereas no solution exists in the ranges of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{s} > F_{sc}$$\end{document}Fs>Fsc, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M < M_{c}$$\end{document}M<Mc, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S < S_{c}$$\end{document}S<Sc, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{C} > K$$\end{document}KC>K. Further, a reduction in the rate of heat transfer is noticed with a rise in the parameter of the copper solid volume fraction.

www.nature.com/scientificreports/ Energy sustainability and improvement of the performance of thermal devices are the focus of research in different industrial and engineering areas, such as power generation, microelectronics, and air-conditioning. A creative range of thermodynamics has been important for the sustainability of energy 13 . The systems of cooling work on a fluid flow by a force diffusion in the presence and absence of convective transfer of heat during such engineering processes. As a result, the liquid thermal conductivity is worth improving for a better engineering operation. The development of nanofluids is caused by the spread of simple nanoparticles to common liquids, such as glycol, vegetable oil and water, or by the mixing of water with glycol. The nanoparticles are known as metal oxides ( Al 2 O 3 , Fe 2 O 3 , CuO), carbon (CNTs, MWCNT), nitride, and metal carbide. Numerous researchers looked at different types of nanoparticle mixtures, such as metals (Al, Cu, Fe), nanoparticle semiconductors, and metal oxides ( Al 2 O 3 , CuO). There are significant references to nanofluid in the books and articles of Minea 14 , Minkowycz et al. 15 , and Mebarek-Oudina et al. 16 . On the other hand, Fan and Wang 17 , Zhao et al. 18 , Mahian et al. 19,20 , Acharya et al. 21 , Buongiorno et al. 22 , Acharya et al. 23 , and Khan et al. 24 have written detailed articles on nanofluids.
In the past few years, the production of enhanced heat transfer fluids has earned substantial attention from scientists and scholars. Hybrid nanofluid is called a modern kind of nanofluid which is therefore used to expand the performance of heat transfer. According to Choi and Eastman 25 , "nanofluid is the mixture of solid nanoparticles in the base fluid. " Now, hybrid nanofluid can be explained as a mixture of nanoparticles in the regular nanofluid where the particles of nanofluid should be different. The computational model of Devi and Devi 26 is more accurate from the various models of the hybrid nanofluid as they have compared their results with the experimental outcomes of Suresh et al. 27 and obtained in the outstanding agreements. This model is used by numerous scholars such as Yan et al. 28 34 examined hybrid nanofluid over the radiated horizontal exponential surface. They examined the two solutions depending on the suction and stretching/shrinking parameter. The same problem was extended by Yan et al. 28 in which they examined the fluid flow with the effect of the Joule heating and magnetic parameter over the exponential horizontal surface. They have found double solutions depends on the magnetic field.
Many of the hybrid nanofluid flow problems described in the above paragraphs were studied for two-dimensional flow and also did not recognize the impact of multiple solutions for the rotating flow model. In the current examination, we have considered the rotational flow of the hybrid nanofluid of 3D. There are two main objectives of the present study to be considered, one of them is to find the multiple solutions of the model and the second one is to perform the stability analysis of multiple solutions. To the best of the authors' knowledge, such analysis on multiple solutions has not yet been performed in the published literature.

Mathematical Formulation
MHD, steady, 3D flow of Cu − Al 2 O 3 /water hybrid nanofluid along heat transfer on the shrinking surface has been taken into account as revealed in Fig. 1. The sheet at z = 0 is velocity in x-axis direction i.e., u w (x) = −cx . Mass flux of velocity is w w (x) = −S cϑ f , the temperature within (outside) boundary layer is T w ( T ∞ ). The porous space is saturated by an incompressible fluid that describes the relationship of Darcy-Forchheimer. Liquid and surface are both rotating with 0 constant angular velocity of the z-axis taken to surface as normal. A field of uniform magnetic is positioned in the z-axis direction which is B = B 0 . These consequences in x-axis and z-axis directions having magnetic effects. As a result, the magnetic field directly effects the x-axis and z-axis directions. The Reynolds number of magnetic is assumed to be very low, and the field of induced magnetic is disregarded.
Hybrid nanofluid flow equations for temperature and momentum with boundary layer assumptions are expressed as follows 7,9 : (1) u x + v y + w z = 0 where u , v , and w are the corresponding velocity components in x, y, and z-axes, σ hnf is a hybrid nanofluid electrical conductivity, and q r = −4σ * 3k * ∂T 4 ∂z is the radiative flux where k * and σ * are the coefficient of mean absorption and Stefan-Boltzmann constant. Further, k hnf , ρc p hnf , ρ hnf and , µ hnf are the corresponding thermal conductivity, heat capacity, density, and dynamic viscosity of hybrid nanofluid. Moreover, K * is the porous medium permeability, F = C b x(K * ) 1/2 is the non-uniform inertia coefficient of medium, C b is the drag coefficient. Furthermore, the hnf subscription illustrates the hybrid nanofluid properties. The thermophysical properties are given in Tables 1 and 2.

Properties Hybrid nanofluid
Density Here prime represents the differentiation of η , 3k * k f is the radiation parameter, and S is the suction (injection) parameter when S > 0 ( S < 0).
The coefficient of skin friction and the Nusselt number are two physical quantities of interest that are expressed as follows.

Results and discussion
The system of ODEs (7-9) subject to BCs (10) has been numerically solved with the Maple computational software with the aid of shootlib function. We have compared the values of √ ReC fx and √ ReC fy with Zaimi et al. 40 's results for pure water (i.e., φ 1 = φ 2 = 0) over the stretching surface (i.e., f ′ (0) = 1 ) in Table 3. It is concluded from the results of Table 3 that the findings indicate excellent agreement with the previous study. It can also be concluded that the present code can be used confidently to investigate the problem under consideration in current work.
1      www.nature.com/scientificreports/ this is due to higher copper volume fraction values aiding in the reduction of boundary layer thickness, while it reduces in the second solution as both applied parameters increase. Further, g ′ (0) increases when ∅ 2 increases in the second solution, while no change is noticed in the behavior of g ′ (0) in the first solution. Moreover, g ′ (0) increases in the second solution when S increases by keeping a fixed value of ∅ 2 , the physical reason for this increase is that the suction creates less resistance in fluid flow. On the other hand, the reverse trend is noticed in the first solution. Reduced heat transfer rate ( −θ ′ (0) ) enhances in both solutions when S increases, while reverse trend is observed when ∅ 2 increases. It is then inferred that the importance of suction is essential in deciding the presence of non-unique solutions. The effect of porosity parameter K on the magnitude of f ′′ (0), g ′ (0), and −θ ′ (0) for numerous values of φ 2 have been plotted in Figures 11, 12 and 13. The corresponding critical values of φ 2 = 0, 0.02, 0.05 are  Reduced heat transfer ( −θ ′ (0) ) reduces in both solutions when ∅ 2 increases, while a reverse trend is observed when K increases in the first solution by keeping the constant values ∅ 2 . This is therefore determined that the porosity values are significant in evaluating the occurrence of non-unique solutions. The effects of volume fraction of copper ( ∅ 2 ) are examined in Figures 14, 15 and 16 for the distributions of velocity f ′ (η) , g(η) and temperature θ(η) . All profiles comply asymptotically with the BCs (10) and also it is noticed that there are two solutions. Figures 14 and 15 show that the thickness of hydrodynamic boundary layer declines with rising values of ∅ 2 for both solutions but it should be noted that dual nature exists in the second  . A special phenomenon is found for the thickness of thermal boundary layer where it enhances in both solutions as the magnitude of ∅ 2 rises (see Fig. 16). In contrast to the first solution, it should also be noticed that the second solution boundary layer has a greater thickness. Figure 17 illustrates that how the rotation parameter effects on profiles of velocity g(η) . Increased rotation parameter (�) values result in a higher velocity of hybrid nanofluid and higher thickness of momentum layer in both solutions. It is also observed that only a single solution exists when = 0.05 . Larger values of result in a higher rate of rotation in the second solution relative to the first solution. Consequently, the greater rotation results lead to the higher velocity of hybrid nanofluid and the higher thickness of momentum layer of all solutions. Figure 18 demonstrates the nature of the profiles of temperature for effects of radiation R . Such parameter appears only in the temperature Eq. (9) and is not attached to the momentum Eqs. (7)(8); thus, the changes in R do not induce any shifts in the profile of velocity. The thickness of boundary layers is shown to rise in all branches with an improvement in R which suggests that the temperature gradient at the surface is smaller with higher R  www.nature.com/scientificreports/ magnitudes. Radiation parameter tests spread of the thermal radiation owing to the conduction heatwave. Higher values of R, therefore indicate the predominance of thermal radiation on the conductions. As a consequence, because of R , the device emits a considerable volume of heat energy, which causes temperature to rise and indicates that the fluid temperature ( θ(η) ) is increasing due to high radiation presence. The governing Eqs. (26)(27)(28)(29) were solved by using the bvp4c solver in Matlab software. The governing equations provide an infinite range of eigenvalues. Smallest negative eigenvalues: ε < 0 implies that flow has an initial disruption development that may disrupt the flow and, ultimately, induce unstable flow. Besides that, the smallest positive eigenvalues; ε > 0 shows that there is just an initial decay of disturbance, are showing stable flow. In this regard, Table 4 sets out the values of the smallest eigenvalue where it can be easily seen that the first solution is the stable one.