Computers, Materials & Continua DOI:10.32604/cmc.2021.017643 | |

Article |

Hybrid Nanofluid Flow with Homogeneous-Heterogeneous Reactions

1Fakulti Teknologi Kejuruteraan Mekanikal dan Pembuatan, Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya, 76100, Durian Tunggal, Melaka, Malaysia

2Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600, UKM Bangi, Selangor, Malaysia

3Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

*Corresponding Author: Anuar Ishak. Email: anuar_mi@ukm.edu.my

Received: 05 February 2021; Accepted: 11 March 2021

Abstract: This study examines the stagnation point flow over a stretching/shrinking sheet in a hybrid nanofluid with homogeneous-heterogeneous reactions. The hybrid nanofluid consists of copper (Cu) and alumina (Al2O3) nanoparticles which are added into water to form Cu-Al2O3/water hybrid nanofluid. The similarity equations are obtained using a similarity transformation. 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

Keywords: Homogeneous-heterogeneous reactions; stagnation point; hybrid nanofluid; shrinking sheet; dual solutions; stability analysis

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 [1–4]. On the other hand, Hiemenz [5] was the first researcher to consider the boundary layer flow of a stagnation point over a rigid surface. Then, Homann [6] extended the problem to the axisymmetric flow, while Wang [7] considered the flow on a shrinking sheet. Furthermore, homogeneous (bulk) and heterogeneous (surface) reactions on the stagnation point flow were examined by Chaudhary and Merkin [8,9]. A simple of these reactions with equal and different diffusivities for autocatalyst and reactant was introduced in their studies. Then, Merkin [10] extended the problem to the Blasius flow. The homogeneous and heterogeneous reactions have significant applications in the biochemical, catalysis, and combustion systems. Inspired by these studies, Khan and Pop [11] examined these effects on the flow towards a permeable surface. They noticed that dual solutions exist in the injection region but a unique solution is observed in the suction region. Besides, Kameswaran et al. [12] studied a similar problem by considering the magnetic field effects. They discovered that the skin friction coefficient and the concentration gradient increased with increasing of the magnetic parameter. Apart from that, several studies [13–15] involving homogeneous and heterogeneous reactions have been reported in the literature.

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 [17–23]. 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 [24–26]. 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. [28–30]. Other physical aspects were considered by several authors [31–38]. Also, the review papers can be found in [39–44].

Motivated by the above mentioned studies, this paper considers the homogeneous-heterogeneous reactions on hybrid nanofluid flow with Al2O3-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.

The stagnation point flow triggered by a stretching/shrinking sheet in Al2O3-Cu/water hybrid nanofluid is considered. In Fig. 1, the free stream and the surface velocities are given as

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,

subject to:

where the coordinates

To obtain similarity solution, the following variables are employed [33]:

where

It is noted that the continuity Eq. (3) is identically satisfied.

Now, Eqs. (4) to (7) respectively reduce to:

subject to:

where primes denote the differentiation with respect to

Note that,

Using Eq. (17), Eqs. (13) and (14) become:

subject to:

The skin friction coefficient

Using the similarity variables (9), one obtains

where

The stability of the dual solutions over time is studied. This analysis was first introduced by Merkin [45] and then followed by Weidman et al. [46]. Firstly, consider the new variables as follows:

Now, consider the unsteady form of Eqs. (4) to (7) while Eq. (3) remains unchanged. On using (22), one obtains:

subject to:

Then, the disturbance is applied to the steady solutions

The sign (positive or negative) of the eigenvalue

subject to:

The quantity

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. Tosolve 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 Al2O3 and Cu volume fractions are applied in a one-to-one ratio. For instance,

The values of species concentration on the surface

Further, Figs. 7 to 9 display the effect of

The effects of

The variations of the eigenvalues

The stagnation point flow of Al2O3-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

Acknowledgement: We acknowledge the Universiti Teknikal Malaysia Melaka and the Universiti Kebangsaan Malaysia (DIP-2020-001) for financial supports.

Funding Statement: This research was funded by Universiti Kebangsaan Malaysia (Project Code: DIP-2020-001).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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