Hybrid Nanofluid Flow with Homogeneous-Heterogeneous Reactions

: This study examines the stagnation point flow over a stretch-ing/shrinking sheet in a hybrid nanofluid with homogeneous-heterogeneous reactions. The hybrid nanofluid consists of copper (Cu) and alumina (Al 2 O 3 ) nanoparticles which are added into water to form Cu-Al 2 O 3 /water hybrid nanofluid. The similarity equations are obtained using a similarity trans-formation. Then, the function bvp4c in MATLAB is utilised to obtain the numerical results. The dual solutions are found for limited values of the stretching/shrinking parameter. Also, the turning point arises in the shrinking region (λ < 0 ) . Besides, the presence of hybrid nanoparticles enhances the heat transfer rate, skin friction coefficient, and the concentration gradient. In addition, the concentration gradient is intensified with the heterogeneous reaction but the effect is opposite for the homogeneous reaction. Furthermore, the velocity and the concentration increase, whereas the temperature decreases for higher compositions of hybrid nanoparticles. Moreover, the concentration decreases for larger values of homogeneous and heterogeneous reactions. It is consistent with the fact that higher reaction rate cause a reduction in the rate of diffusion. However, the velocity and the temperature are not affected by these parameters. From these observations, it can be concluded that the effect of the homogeneous and heterogeneous reactions is dominant on the concentration profiles. Two solutions are obtained for a single value of parameter. The temporal stability analysis shows that only one of these solutions is stable and thus physically reliable over time.


Introduction
Boundary layer flow produced by the stretching or shrinking surface was introduced by researchers many years ago. The pioneered work of the problems can be found in the literature [1][2][3][4]. On the other hand, Hiemenz [5] was the first researcher to consider the boundary layer In 1995, Choi and Eastman [16] introduced nanofluid, which is a mixture of the base fluid and a single type of nanoparticles, to enhance the thermal conductivity. Some works on such fluids can be found in [17][18][19][20][21][22][23]. Recently, some studies found that advanced nanofluid which consists of another type of nanoparticles and the regular nanofluid could improve its thermal properties, and this mixture is termed as 'hybrid nanofluid'. The prior experimental works using the hybrid nanoparticles have been done by several researchers [24][25][26]. Besides, the numerical studies on the flow of hybrid nanofluids were studied by Takabi and Salehi [27]. Moreover, the dual solutions of the hybrid nanofluid flow were examined by Waini et al. [28][29][30]. Other physical aspects were considered by several authors [31][32][33][34][35][36][37][38]. Also, the review papers can be found in [39][40][41][42][43][44].
Motivated by the above mentioned studies, this paper considers the homogeneousheterogeneous reactions on hybrid nanofluid flow with Al 2 O 3 -Cu hybrid nanoparticles. Different from the work reported by Ramesh et al. [33], the present study considers the stagnation point flow towards a stretching/shrinking sheet. Most importantly, in this study, two solutions are discovered for a single value of parameter, and the stability of these solutions over time is tested.

Mathematical Model
The stagnation point flow triggered by a stretching/shrinking sheet in Al 2 O 3 -Cu/water hybrid nanofluid is considered. In Fig. 1, the free stream and the surface velocities are given as u e (x) = cx and u w (x) = dx, respectively, with c > 0, whereas d < 0 and d > 0 indicate the shrinking and stretching sheets, respectively, and d = 0 indicates the static sheet. Meanwhile, the ambient and the surface temperatures are given by T ∞ and T w , respectively, and both are constants. The homogeneous-heterogeneous reactions are also taking into consideration. Following Chaudhary and Merkin [8] and Merkin [10], a simple homogeneous reaction and the first order of heterogeneous reaction can respectively be written as where these processes are assumed to be isothermal. Here, a and b are the chemical concentrations for species A and B, respectively, with the rate constants k 1 and k s . Thus, the equations that govern the hybrid nanofluid flow are [8,33]: subject to: where the coordinates (x, y) with corresponding velocities (u, v) are measured along the x-and y-axes, and T represents the temperature. Besides, D A and D B are the corresponding diffusion coefficients of species A and B, and a 0 > 0. Further, the thermophysical properties can be referred to in Tabs. 1 and 2. Note that ϕ 1 (Al 2 O 3 ) and ϕ 2 (Cu) are the nanoparticles volume fractions where ϕ hnf = ϕ 1 + ϕ 2 , and the subscripts n1 and n2 correspond to their solid components, while the subscripts hnf and f signify the hybrid nanofluid and the base fluid, respectively. To obtain similarity solution, the following variables are employed [33]: where ψ is the stream function defined as u = ∂ψ/∂y and v = −∂ψ/∂x. Then one obtains It is noted that the continuity Eq. (3) is identically satisfied.
Now, Eqs. (4) to (7) respectively reduce to: 1 Pr subject to: where primes denote the differentiation with respect to η. The physical parameters appear in Eqs. (12) to (15) are the Prandtl number Pr, the Schmidt number Sc, the ratio of the diffusion coefficients δ, the strength of the homogeneous K and the heterogeneous K s reactions, and the stretching/shrinking parameter λ, and they are given as: Note that, λ < 0 and λ > 0 are for the shrinking and stretching sheets, respectively, while λ = 0 designates the static sheet. As discussed by Chaudhary and Merkin [8], both diffusion coefficients (D A and D B ) are of comparable sizes, thus, these coefficients are assumed to be equal by taking δ = 1. Hence, one gets: Using Eq. (17), Eqs. (13) and (14) become: subject to: The skin friction coefficient C f and the local Nusselt number Nu x are defined as: Using the similarity variables (9), one obtains where Re x = u e x/ν f is the local Reynolds number.

Results and Discussion
By utilising the package bvp4c in MATLAB software, Eqs. (11), (12), and (18) subjected to Eqs. (15) and (19) are solved numerically. In particular, bvp4c is a finite-difference code that implements the three-stage Lobatto IIIa formula [49]. This is a collocation formula that provides a continuous solution with fourth-order accuracy. Mesh selection and error control are based on the residual of the continuous solution. The effectiveness of this solver ultimately counts on our ability to provide the algorithm with an initial guess for the solution. Because the present problem may have multiple (dual) solutions, the bvp4c function requires an initial guess of the solution for Eqs. (11), (12), and (18). Using this guess value, the velocity, temperature and the concentration profiles must satisfy the boundary conditions (15) and (19) asymptotically. Determining an initial guess for the first solution is not difficult because the bvp4c method will converge to the first solution even for poor guesses. However, it is rather difficult to determine a sufficiently good guess for the second solution of Eqs. (11), (12), and (18). Also, this convergence issue is influenced by the value of the selected parameters. In this study, the relative tolerance was set to 10 −10 . To solve this boundary value problem, it is necessary to first reduce the equations to a system of first-order ordinary differential equations. Further, the effect of several physical parameters on flow behaviour is examined. The total composition of Al 2 O 3 and Cu volume fractions are applied in a one-to-one ratio. For instance, 1% of Al 2 O 3 (ϕ 1 = 1%) and 1% of Cu (ϕ 2 = 1%) are mixed to produce 2% of Al 2 O 3 -Cu hybrid nanoparticles volume fractions, i.e., ϕ hnf = 2%.  The values of species concentration on the surface g(0) are compared with Khan and Pop [11] for various values of K and K s when ϕ hnf = 0 (regular fluid), Sc = 1, and λ = 0 (static sheet). It is found that the results are comparable for each K and K s considered, as shown in Tab. 3.
Also, Tab. 4 shows the comparison of Re  Further, Figs. 7 to 9 display the effect of ϕ hnf on f (η), θ (η) and g (η) when λ = −1.2, 6.2, Sc = 1, and K = K s = 0.5. It is seen that both branch solutions of f (η) and g (η) show an increasing pattern for higher compositions of ϕ hnf . Meanwhile, the temperature θ (η) declines for both branch solutions. Physically, the addition of the nanoparticles increases the viscosity of the fluid and thus slowing down the fluid motion. Also, the added nanoparticles dissipate energy in the form of heat and consequently exert more energy which enhances the temperature.
The effects of K and K s on the concentration g (η) when λ = −1.2, Pr = 6.2, Sc = 1 and ϕ hnf = 2% are depicted in Figs. 10 and 11, respectively. The decreasing pattern of concentration on both branch solutions is observed. It is consistent with the fact that the reaction rate increases as the values of K and Ks increase, which cause the reduction in the diffusion rate. The variations of the eigenvalues γ against λ when ϕ hnf = 2% are portrayed in Fig. 12. For the positive values of γ , it is noted that e −γ τ → 0 as time evolves (τ → ∞). In contrast, for the negative values of γ , e −γ τ → ∞, which shows an increasing of disturbance over time. These behaviours show that the first solution is stable and physically reliable, while the second solution is unstable in the long run.

Conclusion
The stagnation point flow of Al 2 O 3 -Cu/water hybrid nanofluid over a stretching/shrinking sheet was studied. Both homogeneous and heterogeneous reactions were considered. Findings revealed that dual solutions appeared for some ranges of the shrinking strength λ. The solutions were obtained up to a certain critical value of λ(< 0), in the shrinking region. The increment of the skin friction coefficient Re 1/2 x C f , local Nusselt number Re −1/2 x Nu x , and the concentration gradient at the surface g (0) were observed with the rise of the nanoparticles volume fractions ϕ hnf . The values of g (0) decreased for larger values of K, but increased with the rise of K s . Finally, it was discovered that two solutions were obtained for a single value of parameter. Further analysis showed that only one of these solutions is stable and thus physically reliable over time.