Influence of buoyancy force on Ag-MgO/water hybrid nanofluid flow in an inclined permeable stretching/shrinking sheet
Introduction
A number of research papers have been published, both theoretical and experimental, with the flat plate in two particular positions - horizontal [1,2] and vertical [3,4]. However, there is limited research on the inclined flat plate. Thus, it is worthwhile to explore the flow on a flat plate inclined at various angles using boundary layer approximation as it is often come across in engineering devices, for example in inclination/acceleration sensors and solar water heaters. The investigation of MHD flow in an inclined plate was done by Aydin and Kaya [5]. A non-uniform heat sink/source and MHD effects over an inclined surface in a viscoelastic fluid have been scrutinized numerically by Ramesh et al. [6]. Afterwards, Ramesh et al. [7] reviewed the flow driven by an inclined stationary/moving plate. They noticed that the moving flat plate temperature is relatively lower than the stationary flat plate. A numerical computation of magnetohydrodynamic flow over an inclined plate in nanofluid have been examined by Goyal and Bhargava [8] and found that strengthen the value of inclination angle parameter cause the thermal boundary layer thickness to increase. Ilias et al. [9] have recently scrutinized the influence of magnetic nanofluids and heat transfer characteristic on unsteady flow through an inclined plate. A good investigation of flat plate with various inclination angle parameter can also be found in research paper [[10], [11], [12], [13]].
The emergence of nanotechnology industries yields to a major advance in cooling technologies. To improve the heat transfer efficiency, a small solid particle or also known as nanoparticle (1-100 nm) are added into a conventional fluid [14]. Thus, a range of research on nanofluid has been done to gain an understanding of heat transfer efficiency. The impact of nanoparticle past a stretching cylinder on heat transfer characteristic with slip has been studied by Pandey and Kumar [15]. Afterwards, Rashid et al. [16] explored the entropy analysis of nanofluid over a porous shrinking wall with magnetohydrodynamic effect analytically. Recently, Reddy and Sreedevi [17] have studied the chemical reaction effect on mixed convection flow of nanofluid caused by an inclined plate in porous media. Meanwhile Anuar et al. [18] investigated carbon nanotube (nanofluid) flow caused by a nonlinear deformable sheet with magnetohydrodynamic effect. Furthermore, the nanofluid's flow and its heat transfer characteristic in a various geometry and influences has been scrutinized by Soomro et al. [19], Krishna and Chamkha [20], Gangadhar et al. [21], Kebede et al. [22] and others.
However, technologists have determined the existence of new interesting advanced nanofluid notable as hybrid nanofluid. It is a combination of two distinct particles (nanosized) with base fluids and is capable on enhancing the rate of heat transfer owing to its synergistic effects [23]. The dynamic viscosity and thermal conductivity of nanoparticle (Ag–MgO) have been considered experimentally by Esfe et al. [24]. Rostami et al. [25] have numerically scrutinized the flow of SiO2-Al2O3/water nanofluid over a vertical plate and noticed that hybrid nanofluid admits a greater rate of heat transfer respect to regular nanofluid. Additionally, the thermophysical characteristics of TiO2–Cu/H2O hybrid nanofluid with the Lorentz force over a stretching sheet have been examined by Ghadikolaei et al. [26]. Further, the thermal conductivity of TiO2-CuO/water nanofluid driven by a moving/static wedge have been explored by Dinarvand et al. [27]. Afterwards, Waini et al. [28] explored the unsteady flow of hybrid nanofluid (Cu–Al2O3/water) past a stretching/shrinking sheet and noticed the existence of dual solution. Anuar et al. [29] considered convective hybrid nanofluid (Cu–Al2O3/water) in their studies of MHD deformable sheet with homogeneous-heterogeneous effect. Meanwhile, a comparative investigation was made by Hayat et al. [30] on the hybrid (MWCNTs–Cu/water) and mono (Cu/water) nanofluid over a curved sheet with slip effect. More recently, Wahid et al. [31] explored the heat generation of hybrid nanofluid (Cu–Al2O3/water) slip flow due to an exponentially stretching/shrinking sheet. They found out that an upsurges values of copper nanoparticle volume fraction could enhance the local Nusselt number. Such other investigations of hybrid nanofluid in various flow configuration have been discussed by many researchers, see for example the work of Shehzad et al. [32], Nadeem et al. [33], Sheikholeslami et al. [34], Reddy and Sreedevi [35], Upreti et al. [36] and many others.
Existing literature witness that no exploration in the past has been made in hybrid nanofluid flow on an inclined stretching or shrinking plate with stability analysis in the existence of suction. To fill this gap, a mathematical formulation of the current research is developed by referring to the research done by Afridi et al. [37]. Utilizing the similarity transformation method, the partial differential equations (PDEs) are transformed into an ODEs and solved numerically. Further, to assess the physical feasibility of the solutions obtained, a stability analysis has been taken into practice. The presence of non-unique solutions has attracted attention from many researchers and almost everyone has sought to carry out a stability analysis. Therefore, for the convenience of readers, we list here some recent literature on stability analysis (see [[38], [39], [40], [41], [42], [43]]).
Section snippets
Mathematical framework
Two-dimensional flow of hybrid nanofluid (Ag-MgO/water) over an inclined permeable deformable sheet (stretching or shrinking) with an angle of inclination α is investigated. The coordinates system and physical model of the present study are exemplified as in Fig. 1. With the velocity uw(x) = ax in which a is a constant velocity characteristic of the sheet, the surface is stretch or shrunk linearly in the x−direction whilst y−axis is normal to it. The surface temperature of the sheet is
Flow stability
For certain governing parameters, non-unique solutions are found to occur. Therefore, in determining which of the solutions attain are stable, a stability analysis is needed. For this method, Merkin's work [46] was used with time-dependent problem in mind. Hence the unsteady case for the present study is given as:where t refers to time and the corresponding conditions are:
Hence,
Analysis of results
The system of boundary value problem (6)–(8) has been analyzed numerically for inclined permeable stretching and shrinking sheet using the Matlab's built in function known as bvp4c solver. The present numerical solutions on reduced skin friction f″(0) (see Table 3) were validated with the studies of Roşca and Pop [49] for several values of s and ε when α = 90∘ (horizontal plate), λ = 0 (forced convection flow) and φ1 = φ2 = 0 (regular fluid). Also, comparison of heat transfer −θ′(0) for several
Conclusion
In the existing study, a numerical investigation has been conducted on the heat transfer analysis and boundary layer flow of Ag-MgO/water hybrid nanofluid driven by an inclined permeable stretching/shrinking sheet. Using similarity transformation, self-similar nonlinear ordinary differential equations are obtained and solved numerically using bvp4c in Matlab software. The findings are summarized as follows:
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Two solutions are noticed in both cases of inclined permeable stretching and shrinking
Credit author statement
Nur Syazana Anuar: Methodology, Formal analysis, Software, Writing - Original Draft; Norfifah Bachok: Supervision, Funding acquisition, Writing - Review & Editing; Ioan Pop: Conceptualization, Supervision, Writing - Review & Editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by the Fundamental Research Grant Scheme (FRGS) under Ministry of Education with project number FRGS/1/2018/STG06/UPM/02/4.
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