Hybrid nanofluid flow towards a stagnation point on a stretching/shrinking cylinder

This paper examines the stagnation point flow towards a stretching/shrinking cylinder in a hybrid nanofluid. Here, copper (Cu) and alumina (Al2O3) are considered as the hybrid nanoparticles while water as the base fluid. The governing equations are reduced to the similarity equations using a similarity transformation. The resulting equations are solved numerically using the boundary value problem solver, bvp4c, available in the Matlab software. It is found that the heat transfer rate is greater for the hybrid nanofluid compared to the regular nanofluid as well as the regular fluid. Besides, the non-uniqueness of the solutions is observed for certain physical parameters. It is also noticed that the bifurcation of the solutions occurs in the shrinking regions. In addition, the heat transfer rate and the skin friction coefficients increase in the presence of nanoparticles and for larger Reynolds number. It is found that between the two solutions, only one of them is stable as time evolves.

dispersing one type of nanoparticle in the aforementioned fluids to enhance their thermal conductivity. Khanafer et al. 23 and Oztop and Abu-Nada 24 have utilized nanofluids to study the heat transfer enhancement in a rectangular enclosure. However, the researchers found that the thermal properties of the nanofluid could be improved with the addition of more than a single nanoparticle in the base fluid and named it 'hybrid nanofluid' . The experimental studies that considered the hybrid nano-composite particles have been conducted by several researchers, for example, Turcu et al. 25 and Jana et al. 26 . Hybrid nanofluid is an advanced fluid that incorporates more than one nanoparticle which has the capacity of raising the heat transfer rate because of the synergistic effects 27 .
Furthermore, the studies of hybrid nanofluid were extended to the boundary layer flow problem. For instance, Devi and Devi 28,29 started to examine the advantages of utilizing hybrid nanofluid over a stretching surface. They found that the heat transfer rate was intensified in the presence of the hybrid nanoparticles. In their studies, the new thermophysical model was introduced and validated with the experimental data of Suresh et al. 30 . Furthermore, Waini et al. 31 examined the stability of the multiple solutions of the flow over a stretching/shrinking surface in a fluid containing hybrid nanoparticles. They discovered that only one of the solutions is stable and thus physically reliable as time evolves. Besides, Waini et al. [32][33][34][35][36][37] in a series of papers have extended the problem to different surfaces. Moreover, the effects of MHD and viscous dissipation have been studied by Lund et al. 38 , considering Cu-Fe 3 O 4 /H 2 O hybrid nanofluid in a porous medium. Additionally, the problem of hybrid nanofluid flow with the effect of different physical parameters was also considered by several authors [39][40][41][42][43][44][45] .
Thus, the objective of this paper is to examine the hybrid nanofluid flow towards a stagnation point on a stretching/shrinking cylinder. Here, copper (Cu) and alumina (Al 2 O 3 ) are considered as the hybrid nanoparticles, while water as the base fluid.

Mathematical Model
Consider a hybrid nanofluid flow towards a stagnation point on a stretching/shrinking cylinder with radius a as illustrated in Fig. 1. Here, (z,r) is the cylindrical polar coordinates which assigned in the axial and radial directions, respectively. The flow is assumed to be symmetric about the z = 0 plane and also axisymmetric about the z-axis, with the stagnation line is at z = 0 and r = a. The surface velocity of the cylinder is given as w w (z) = 2bz where the static cylinder is denoted by b = 0, whereas the cylinder is stretched or shrunk when b > 0 or b < 0, respectively. Meanwhile, the free stream velocity is taken as w e (z) = 2cz where c > 0. Moreover, the surface temperature T w and the ambient temperature T ∞ are constant, where T w > T ∞ . Also, it is assumed that the shape of the nanoparticle is spherical and its size is uniform, while the agglomeration is disregarded since the hybrid nanofluid is formed as a stable composite. Therefore, the equations that govern the hybrid nanofluid flow are (see Wang 8 , Lok and Pop 12 ): subject to: www.nature.com/scientificreports www.nature.com/scientificreports/ where w and u represent the velocity components along the z-and r-axes, and T represents the temperature of the hybrid nanofluid. Further, the thermophysical properties of the hybrid nanofluid are defined in Table 1 24,28,31 . Besides, the physical properties of Al 2 O 3 , Cu, and water are provided in Table 2 24,31 . Here, Al 2 O 3 and Cu volume fractions are given by ϕ 1 and ϕ 2 and the subscripts n1 and n2 correspond to their solid components, respectively. Meanwhile, the fluid, nanofluid, and the hybrid nanofluid are designated by the subscripts f, nf, and hnf, respectively.
An appropriate transformation is introduced as follows (see Wang 8 , Lok and Pop 12 ): Employing these definitions, Eq. (1) is identically fulfilled. Then, the following similarity equations are obtained: where (′) represents the differentiation with respect to η, Re = ca 2 /2ν f represents the Reynolds number, and µ f represents the Prandtl number. Besides, the stretching/shrinking parameter symbolized by ε = b/c with ε > 0 and ε < 0 are for stretching and shrinking cylinder, respectively, while the static cylinder is denoted by ε = 0.
The skin friction coefficient C f and the Nusselt number Nu are defined as: Thermophysical Properties Nanofluid Hybrid nanofluid

Stability Analysis
The existence of the non-uniqueness solutions of Eqs. (6) to (8) To study the stability of the solutions of Eqs. (1) to (3), the unsteady form of these equations are considered. Using (11) and following the same procedure as previous section, the equations transformed to: To examine the stability behaviour, the disturbance is imposed to the steady solution f = f 0 (η) and θ = θ 0 (η) of Eqs. (6) to (8) by using the following relations (see Weidman et al. 47 where γ indicates the unknown eigenvalue that determines the stability of the solutions, whereas F(η) and G(η) are small compared to f 0 (η) and θ 0 (η). The disturbance is taken exponentially as it demonstrates the rapid decline or development of the disturbance. By employing Eq. (15), Eqs. (12) and (13) Without loss of generality, the values of γ from Eqs. (16) to (18) are obtained for the case of F″(1) = 1 as proposed by Harris et al. 48 .

Numerical Method
The boundary value problem solver, bvp4c, available in the Matlab software is utilized for solving Eqs. (6) to (8), numerically. As described in Shampine et al. 49 , the aforementioned solver is a finite difference method that employs the 3-stage Lobatto IIIa formula. The selection of the initial guess and the boundary layer thickness η ∞ depend on the parameter values applied to obtain the required solutions. Moreover, several researchers 50-55 were also employing this solver for solving the boundary layer flow problems. First, Eqs. (6) and (7) are reduced to a system of ordinary differential equations of the first order. Equation (6) is written as:  Table 4. Values of (Rez/a)C f and Nu for Cu/water (ϕ 1 = 0) and Al 2 O 3 -Cu/water (ϕ 1 = 0.04) under different values of physical parameters. when Pr = 6.2. www.nature.com/scientificreports www.nature.com/scientificreports/   www.nature.com/scientificreports www.nature.com/scientificreports/ θ η with the boundary conditions: Then, Eqs. (19) to (21) are coded in Matlab software to obtain the required solutions.

Results and Discussion
In the present study, the volume fractions of Cu are varied from 0 to 0.04 (0 ≤ ϕ 2 ≤ 0.04), while the volume fraction of Al 2 O 3 is kept fixed at ϕ 1 = 0.04 and water as the base fluid. Table 3 Table 4 shows the effects of ϕ 2 , Re, and ε on the skin friction coefficient (Rez/a)C f and the Nusselt number Nu for Cu/water (ϕ 1 = 0) and Al 2 O 3 -Cu/water (ϕ 1 = 0.04) when Pr = 6.2. It is observed that the values of (Rez/a)C f and Nu increase with the increasing values of ϕ 2 and Re. Besides, the values of (Rez/a)C f decrease, whereas Nu increases with increasing values of ε. Also, the heat transfer rate for Al 2 O 3 -Cu/water hybrid nanofluid is intensified if compared to Cu/water nanofluid.
The non-uniqueness of the solutions of Eqs. (6) to (8) is observed for some values of ε as can be seen in Figs. 2-5. For example, the variations of (Rez/a)C f and Nu against ε for several values of ϕ 1 and ϕ 2 when Re = 1 and = .
Pr 6 2 are displayed in Figs. 2 and 3. It is noticed that dual solutions are possible for ε c < ε < −1, and the solution is unique for ε ≥ −1. Besides, both branch solutions (first and second) merge up to certain critical values of ε, say ε c . Here, ε c1 = −1.54398, ε c2 = −1.54269, and ε c3 = −1.52549 are the critical values for the case of regular fluid (ϕ 1 = ϕ 2 = 0), Al 2 O 3 /water nanofluid (ϕ 1 = 0.04, ϕ 2 = 0), and Al 2 O 3 -Cu/water hybrid nanofluid (ϕ 1 = ϕ 2 = 0.04), respectively. In addition, the Nusselt number Nu enhances for both stretching and shrinking cases in the presence of nanoparticles. However, the skin friction (Rez/a)C f increases for fixed values of ε started from ε < 1, but these values decrease for ε > 1 and zero skin friction is observed for ε = 1. Comparing the three types of fluid, it is found that these physical quantities are intensified for hybrid nanofluid compared to the others. The observation is consistent with the fact that the added hybrid nanoparticles have the capacity of raising the heat transfer rate because of the synergistic effects as discussed by Sarkar et al. 27 .  Pr 6 2. The effect of increasing Re has a similar trend if compared to the effect of nanoparticles on the skin friction (Rez/a)C f . Physically, Reynolds number Re indicates the relative significance of the inertia effect compared to the viscous effect. As expected, the skin friction coefficients (the surface shear-stress) increases for increasing values of the Reynolds number Re. The upsurge of Re has a tendency to enhance the Nusselt number Nu for ε > −1.2, and it decreases for ε < −1.2, where the heat transfer occurs almost at the same rate as ε = −1.2. Also, it is noticed that the increment is more dominant for the stretching case (ε > 0). Besides, the range of ε for which the solution is in existence decreases as Re increases. As shown in Figs   The variations of the smallest eigenvalues γ against ε when ϕ 1 = ϕ 2 = 0.04 and = Re 1 are portrayed in Fig. 12. It is noticed that the values of γ are positive for the first solution, while it is negative for the second solution. Also, the values of γ approach to zero for both solutions when ε → ε c = −1.52549. Thus, this finding confirms that the first solution is stable and physically reliable while the second solution is not. Besides, it is concluded that the bifurcation of the solutions happens at the critical (minimum) value ε = ε c . Equations (6) to (8) admit dual solutions due to the reverse flow occurs in the boundary layer induced by the shrinking sheet. The occurrence of this phenomenon creates the separation of the boundary layer where the flow moves in the opposite direction as shown in Fig. 1(b). The stability of the solutions is indicated by the sign of the eigenvalue γ. As described in Eq. (15), there is an initial decay of disturbance when γ > 0 as time evolves, i.e. e −γτ → 0 as τ → ∞. Thus, the flow is stable in the long run when γ > 0. In contrast, the flow is unstable when γ < 0 since e −γτ → ∞ as → ∞. The latter shows an increase of disturbance as time evolves. This analysis shows that the first solutions are stable and thus physically reliable in the long run, while the second solutions are not and have no physical sense. Although such solutions are deprived of physical significance, they are nevertheless of interest so far as the differential equations are concerned. These solutions are of mathematical interest since they are also solutions to the differential equations. Similar equations may arise in other situations where the corresponding solutions could have more realistic meaning 56 .  www.nature.com/scientificreports www.nature.com/scientificreports/

Conclusion
In the present study, the hybrid nanofluid flow towards a stagnation point on a stretching/shrinking cylinder has been accomplished. The results were obtained through the bvp4c solver in Matlab software. The results validation was done for limiting cases where the current results were found to compare well with the existing results. The effects of the nanoparticles volume fractions (ϕ 1 and ϕ 2 ), stretching/shrinking parameter ε, and Reynolds number Re on the flow and heat transfer characteristics have been examined. From this investigation, we can draw the following conclusions: • The findings revealed that the heat transfer rate improved in the presence of hybrid nanoparticles.
• It was found that dual solutions are possible for certain physical parameters, where the bifurcation of the solutions occurred in the shrinking region (ε < 0). • The Nusselt number Nu enhanced with increasing values of the Reynolds number Re. The effect of Re was more dominant for the case of the stretching surface (ε > 0). • The velocity increased, but the temperature decreased with the rising of ϕ 2 and Re.
• Between the two solutions, only one of them is stable and physically reliable, while the other is unstable as time evolved.