|Computers, Materials & Continua |
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: email@example.com
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 . 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.
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  was the first researcher to consider the boundary layer flow of a stagnation point over a rigid surface. Then, Homann  extended the problem to the axisymmetric flow, while Wang  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  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  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.  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  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 . 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. , 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.
2 Mathematical Model
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 and , respectively, with , whereas and indicate the shrinking and stretching sheets, respectively, and indicates the static sheet Meanwhile, the ambient and the surface temperatures are given by and , respectively, and both are constants. The homogeneous-heterogeneous reactions are also taking into consideration.
Following Chaudhary and Merkin  and Merkin , 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, and are the chemical concentrations for species and , respectively, with the rate constants and . Thus, the equations that govern the hybrid nanofluid flow are [8,33]:
where the coordinates with corresponding velocities are measured along the - and -axes, and represents the temperature. Besides, and are the corresponding diffusion coefficients of species and , and . Further, the thermophysical properties can be referred to in Tabs. 1 and 2. Note that (Al2O3) and (Cu) are the nanoparticles volume fractions where , and the subscripts and correspond to their solid components, while the subscripts and signify the hybrid nanofluid and the base fluid, respectively.
To obtain similarity solution, the following variables are employed :
where is the stream function defined as and . Then one obtains
It is noted that the continuity Eq. (3) is identically satisfied.
Now, Eqs. (4) to (7) respectively reduce to:
where primes denote the differentiation with respect to . The physical parameters appear in Eqs. (12) to (15) are the Prandtl number , the Schmidt number the ratio of the diffusion coefficients , the strength of the homogeneous and the heterogeneous reactions, and the stretching/shrinking parameter , and they are given as:
Note that, and are for the shrinking and stretching sheets, respectively, while designates the static sheet. As discussed by Chaudhary and Merkin , both diffusion coefficients ( and ) are of comparable sizes, thus, these coefficients are assumed to be equal by taking . Hence, one gets:
Using Eq. (17), Eqs. (13) and (14) become:
The skin friction coefficient and the local Nusselt number are defined as:
Using the similarity variables (9), one obtains
where is the local Reynolds number.
3 Temporal Stability Analysis
The stability of the dual solutions over time is studied. This analysis was first introduced by Merkin  and then followed by Weidman et al. . 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:
Then, the disturbance is applied to the steady solutions , , and of Eqs. (11), (12) and (18) by employing the following relations :
The sign (positive or negative) of the eigenvalue determines the stability of the solutions. Also, , , and are relatively small compared to , , and . On using Eq. (27), and after linearization, Eqs. (23) to (25) become:
The quantity is fixed in order to obtain in Eqs. (28) to (30). Without loss of generality we take , see also .
4 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 . 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, of Al2O3 and of Cu are mixed to produce of Al2O3-Cu hybrid nanoparticles volume fractions, i.e., .
The values of species concentration on the surface are compared with Khan and Pop  for various values of and when (regular fluid), , and (static sheet). It is found that the results are comparable for each and considered, as shown in Tab. 3. Also, Tab. 4 shows the comparison of and with Waini et al.  when for different values of and for Cu/water . It is noted that the present results are comparable with that mentioned literature. Next, Tab. 5 displays the values of , and for Al2O3-Cu/water with various values of parameters when and . The rising of tends to upsurge the values of these physical quantities. These behaviours also can be seen in Figs. 2 to 4. This finding is consistent with the fact that the added hybrid nanoparticles improve the heat transfer rate due to synergistic effects as discussed by Sarkar et al. . Besides, the increasing of leads to enhance the values of and , but the values of is declining with . Also, the rise of declines the values of , whereas the effect of larger is to enhance the values of . However, the values of and are not affected by these parameters. At the same time, Figs. 5 and 6 show the variations of against with the effect of and . On the other hand, dual solutions are obtained for a limited range of as shown in Figs. 2 to 6. The bifurcation of the solutions occurs at for all considered parameters.
Further, Figs. 7 to 9 display the effect of on , and when and . It is seen that both branch solutions of and show an increasing pattern for higher compositions of . Meanwhile, the temperature 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 and on the concentration when and 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 against when are portrayed in Fig. 12. For the positive values of , it is noted that as time evolves . In contrast, for the negative values of , , 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.
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 . The solutions were obtained up to a certain critical value of , in the shrinking region. The increment of the skin friction coefficient , local Nusselt number , and the concentration gradient at the surface were observed with the rise of the nanoparticles volume fractions . The values of decreased for larger values of , but increased with the rise of . 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.
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.
|This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.|