Rotating 3D Flow of Hybrid Nanofluid on Exponentially Shrinking Sheet: Symmetrical Solution and Duality

This article aims to study numerically the rotating, steady, and three-dimensional (3D) flow of a hybrid nanofluid over an exponentially shrinking sheet with the suction effect. We considered water as base fluid and alumina (Al2O3), and copper (Cu) as solid nanoparticles. The system of governing partial differential equations (PDEs) was transformed by an exponential similarity variable into the equivalent system of ordinary differential equations (ODEs). By applying a three-stage Labatto III-A method that is available in bvp4c solver in the Matlab software, the resultant system of ODEs was solved numerically. In the case of the hybrid nanofluid, the heat transfer rate improves relative to the viscous fluid and regular nanofluid. Two branches were obtained in certain ranges of the involved parameters. The results of the stability analysis revealed that the upper branch is stable. Moreover, the results also indicated that the equations of the hybrid nanofluid have a symmetrical solution for different values of the rotation parameter (Ω).


Introduction
For a number of industrial uses, such as the manufacture of rubber pads, the flow of fluid on the shrinking sheet must be considered. In the production phase, the moving surface is supposed to be compressed to its plane and the shrinking sheet connects with the surrounding fluid both thermally and mechanically. The behavior of a shrinking surface may occur in several materials with specific strengths. Initially, Sakiadis [1] proposed the idea of a flow of the boundary layer on a stretching sheet. Later, Crane [2] modified the concept of Sakiadis and applied it to both exponential and linear stretching surfaces. Recently, the flow over a stretching sheet has received a lot of consideration. Several recent studies [3][4][5][6][7][8] have been conducted in this respect, in which numerous impacts have been examined. Due to the high demand and applications of the shrinking surface, we have considered a 3D flow on the shrinking sheet in this research. structure of many products of material processing in the industries can be improved by the concept of multiple solutions". Analysis of the stability is necessary to determine the stable branch when multiple branches occur. Many researchers have stated that only a stable branch has a physical significance which means that only a stable branch can be used in practical applications. Weidman [32] recently discussed the possibility that more than one solution could also be stable. According to him, "since the triple solutions appear to be upper branches (which cannot be continued to their lower branches), then those solutions will also be stable". Therefore, multiple solutions/branches were considered in this study along with their stability analysis due to their important applications.
After evaluating the published literature, the motivation of this work is to examine the heat transfer characteristics of the rotating, steady, and 3D flow of the hybrid nanofluid. According to our best knowledge, no such study has been carried out for a hybrid nanofluid especially for the multiple solutions/branches.

Mathematical Description of the Problem
The flow of a three-dimensional hybrid nanofluid on an exponentially elastic shrinking surface is considered in a rotating frame of reference x, y, z. The velocity of the surface is u w = −ae x l , where a is the characteristic velocity of the surface (See Figure 1). Momentum boundary layers of a hybrid nanofluid flow with energy equations without viscous dissipation and thermal radiation can be described as ∂u ∂x The related boundary conditions (BCs) (2)(3)(4)(5) are where ρc p hn f µ hn f , , k hn f and, ρ hn f are the corresponding heat capacity, dynamic viscosity, thermal conductivity, and density of hybrid nanofluid. Moreover, subscript hn f shows the thermophilic properties of hybrid nanofluid. Further, w 0 > 0 indicates the suction, and w 0 < 0 indicates the injection, and Ω = Ω 0 e − x l is the local rotation parameter. The thermophysical properties are given in Tables 1 and 2.
Symmetry 2020, 12, x FOR PEER REVIEW 3 of 14 a stable branch has a physical significance which means that only a stable branch can be used in practical applications. Weidman [32] recently discussed the possibility that more than one solution could also be stable. According to him, "since the triple solutions appear to be upper branches (which cannot be continued to their lower branches), then those solutions will also be stable". Therefore, multiple solutions/branches were considered in this study along with their stability analysis due to their important applications. After evaluating the published literature, the motivation of this work is to examine the heat transfer characteristics of the rotating, steady, and 3D flow of the hybrid nanofluid. According to our best knowledge, no such study has been carried out for a hybrid nanofluid especially for the multiple solutions/branches.

Mathematical Description of the Problem
The flow of a three-dimensional hybrid nanofluid on an exponentially elastic shrinking surface is considered in a rotating frame of reference , , . The velocity of the surface is = − , where is the characteristic velocity of the surface (See Figure 1). Momentum boundary layers of a hybrid nanofluid flow with energy equations without viscous dissipation and thermal radiation can be described as The related boundary conditions (BCs) (2-5) are where , , and , are the corresponding heat capacity, dynamic viscosity, thermal conductivity, and density of hybrid nanofluid. Moreover, subscript ℎ shows the thermophilic properties of hybrid nanofluid. Further, 0 indicates the suction, and 0 indicates the injection, and = Ω is the local rotation parameter. The thermophysical properties are given in Tables 1 and 2.

Properties Hybrid Nanofluid
Dynamic viscosity Density Thermal conductivity Heat capacity We will employ similarity variable (6) in Equations (1)-(4) in order to obtain the similarity solutions Using the relationship between Equations (1) and (6), we obtain which leads to Substituting the stream function relationship with Equations (6)- (7) in Equations (2)-(5) yields Pr along with BCs where prime represents the differentiation with respect to η, Pr = ϑ α is Prandtl, and Ω = Ω 0 l a is the constant dimensionless rotation parameter.
Physical quantities are the skin friction coefficient and local Nusselt, which are expressed as By substituting Equation (6) in Equation (14), the following is obtained where Re x = al ϑ e x l is the local Reynold number.

Temporal Stability Analysis
The findings of the boundary layer problem (9-11) demonstrate that multiple branches occur. An analysis of stability is then carried out which has been performed by many researchers [33][34][35][36]. For the stability study, the unsteady forms of equations are supposed to be used. Henceforth, Equations (2)-(4) can be expressed as where t indicates the time. By considering t in terms of τ, the current similarity variables (6-7) for the unsteady flow are as follows.
with the following BCs The stabilization of heat transfer solutions and steady-state flow solutions f 0 (η), g 0 (η), and θ 0 (η) can be obtained by setting τ → 0. Therefore, the functions and H(η, τ) = H 0 (η) can be written in Equations (25)- (27). Thus, the following problem of linearized eigenvalue can be expressed as: subject to the following BCs: Note that the smallest eigenvalue (ε) can be determined by easing the boundary condition [38,39]. In this analysis, the condition F 0 (η) → 0 was relaxed and Equations (29)-(31) were solved along with a new relaxed BC F 0 (0) = 1 for a fixed value of the applied parameters.

Numerical Method
The three-stage Labatto III-A method is adopted to solve the above system of Equations (9)- (12) numerically with the help of a bvp4c solver in the MATLAB software. Shampine et al. [40] provides a detailed explanation of this method and how it works in MATLAB. We kept the tolerance at 10 −5 to obtain good accuracy in the solutions. According to Raza et al. [41] and Lund et al. [42], "this collocation polynomial and formula offers a C 1 continuous solution in which mesh error control and selection are created on the residual of the continuous solution. The tolerance of relative error is fixed 10 −5 for the current problem. The suitable mesh determination is created and returned in the field sol.x. The bvp4c returns solution, called as sol.y., as a construction. In any case, values of the solution are gotten from the array named sol.y relating to the field sol.x". To determine the stable solution, the three-stage Labatto III-A method is also used to obtain the values of the smallest eigenvalue. For a better understanding, the algorithm of the method is illustrated in Figure 2.

Results and Discussion
The system of nonlinear ODEs (9-11) along with BCs (12) was solved numerically by employing the three-stage Labatto III-A method. We have kept fixed = 6.2 for the water-based hybrid nanofluid at room temperature 25 °C. The Using Keller box method, the obtained results were compared with the published results of Rosali et al. [43] as shown in Table 3 for the coefficients of the skin friction ′′(0) and ′(0) for numerous values of the local rotating parameter Ω and found to be in excellent agreement. Therefore, it can be concluded that the current numerical technique can be employed with considerable confidence to solve the considered problem. Table 4

Results and Discussion
The system of nonlinear ODEs (9-11) along with BCs (12) was solved numerically by employing the three-stage Labatto III-A method. We have kept fixed Pr = 6.2 for the water-based hybrid nanofluid at room temperature 25 • C. The Using Keller box method, the obtained results were compared with the published results of Rosali et al. [43] as shown in Table 3 for the coefficients of the skin friction f (0) and g (0) for numerous values of the local rotating parameter Ω and found to be in excellent agreement. Therefore, it can be concluded that the current numerical technique can be employed with considerable confidence to solve the considered problem. Table 4 was constructed for the values of the f (0) g (0) and −θ (0) of hybrid nanofluid. Table 3. Comparison of f (0) and g (0) for various values of Ω when = 2,

Results of [43]
Present Results  Table 4. Values of f (0), g (0) and −θ (0) for the various values of φ Cu , φ Al 2 O 3 when Pr = 6.2, S = 2.5, Ω = 0.01. The effects of φ Cu on the coefficients of skin friction f (0), g (0), and the heat transfer rate −θ (0) against various values of the suction parameter S are given in Figures 3-5. Here, we focus solely on multiple branches. The non-uniqueness of the branches is only possible when Ω = 0.01 (refer to Figure 6). Furthermore, dual branches for the Equations (9)         Moreover, in the case of Ω 0, ′(0) increases when Ω increases. Furthermore, it is revealed from Figure 7 that the skin friction increases with the decelerated flow. (i.e, Ω 0) and decreases with the accelerated flow (i.e, Ω 0). On the other hand, no solution is found when Ω = 0 for the fixed value of = 1. Furthermore, the symmetrical behavior of the branches is shown in these figures.   Moreover, in the case of Ω 0, ′(0) increases when Ω increases. Furthermore, it is revealed from Figure 7 that the skin friction increases with the decelerated flow. (i.e, Ω 0) and decreases with the accelerated flow (i.e, Ω 0). On the other hand, no solution is found when Ω = 0 for the fixed value of = 1. Furthermore, the symmetrical behavior of the branches is shown in these figures. The skin friction increases (decreases) by increasing the copper volume fraction in the upper (lower) branch. It is also examined that when the volume fraction of the copper enhances, the separation of the boundary layer expands. The heat transfer rate reduces when φ Cu increases in both branches, while it is increasing the function of the suction parameter. Figures 6-8 were plotted to demonstrate the effect of Ω on f (0), g (0), and −θ (0) against the fixed values of S. It was observed that skin friction coefficient f (0) and heat transfer −θ (0) are increasing functions of the rotating (Ω) parameter when suction S increases. It is also shown that f (0) increases for the higher values of rotational parameter Ω in both positive and negative sides. Moreover, in the case of Ω > 0, g (0) increases when Ω increases. Furthermore, it is revealed from Figure 7 that the skin friction increases with the decelerated flow. (i.e, Ω < 0) and decreases with the accelerated flow (i.e, Ω > 0). On the other hand, no solution is found when Ω = 0 for the fixed value of S = 1. Furthermore, the symmetrical behavior of the branches is shown in these figures.   Figure 9 was plotted to examine the effects of Ω on the hybrid nanofluid velocity ′( ). It was detected that the velocity of the hybrid nanofluid declines as the rotation (Ω) parameter is increased. Moreover, no oscillation behavior is found in ′( ) for the higher values of Ω. It happened due to various effects, such as the effects of the shrinking parameter, volume fraction, and suction.     Figure 9 was plotted to examine the effects of Ω on the hybrid nanofluid velocity ′( ). It was detected that the velocity of the hybrid nanofluid declines as the rotation (Ω) parameter is increased. Moreover, no oscillation behavior is found in ′( ) for the higher values of Ω. It happened due to various effects, such as the effects of the shrinking parameter, volume fraction, and suction.      Figure 9 was plotted to examine the effects of Ω on the hybrid nanofluid velocity ′( ). It was detected that the velocity of the hybrid nanofluid declines as the rotation (Ω) parameter is increased. Moreover, no oscillation behavior is found in ′( ) for the higher values of Ω. It happened due to various effects, such as the effects of the shrinking parameter, volume fraction, and suction.    When Ω is increased, the velocity of hybrid nanofluid contains duality in the behavior. For the negative and positive values of the rotation (Ω) parameter, the behavior of the velocity profile was found to have the same behavior. Physically, this indicates that there is a symmetrical solution to the hybrid nanofluid problem.

Upper Lower
Symmetry 2020, 12, x FOR PEER REVIEW 11 of 14 velocity profile was found to have the same behavior. Physically, this indicates that there is a symmetrical solution to the hybrid nanofluid problem. The values of the smallest eigenvalues for different values of and are shown in Table  5. The positive eigenvalue causes the initial decay of disturbance and thus stabilizes the flow. In contrast, the negative results of the smallest eigenvalue show that the flow is unstable. Table 5 shows that is positive for the upper branch, whereas is negative for the lower branch.

Conclusions
In this study, the flow of rotating, steady, and three-dimensional heat transfer of a hybrid nanofluid on a penetrable exponential shrinking surface together with the suction effect were investigated numerically. The governing PDEs have been converted to a system of ODEs using the suitable exponential similarity transformation. The three-stage Labatto III-A technique was then implemented for the solving of the system of ODEs. Numerical results indicate that the current outcomes of ′′(0) and ′(0) are in good agreement with the results previously published. The point-wise conclusions are the following: The values of the smallest eigenvalues ε for different values of S and φ Cu are shown in Table 5. The positive eigenvalue causes the initial decay of disturbance and thus stabilizes the flow. In contrast, the negative results of the smallest eigenvalue show that the flow is unstable. Table 5 shows that ε is positive for the upper branch, whereas ε is negative for the lower branch.

Conclusions
In this study, the flow of rotating, steady, and three-dimensional heat transfer of a hybrid nanofluid on a penetrable exponential shrinking surface together with the suction effect were investigated numerically. The governing PDEs have been converted to a system of ODEs using the suitable exponential similarity transformation. The three-stage Labatto III-A technique was then implemented for the solving of the system of ODEs. Numerical results indicate that the current outcomes of f (0) and g (0) are in good agreement with the results previously published. The point-wise conclusions are the following:

1.
In comparison to a viscous fluid, the heat transfer rate of the hybrid nanofluid is better in the attendance of hybrid nanoparticles.

2.
Two branches exist in the specific ranges of physical parameters.

3.
The upper branch remains stable while the lower branch is unstable.

4.
Rate of heat transfer upsurges for the advanced values of the mass suction in both branches.