Inspection of hybrid based nanofluid flow over a curved surface
Introduction
In the recent years, reasonable attention has been paid to the flow problems and heat transfer caused through continuous stretching/shrinking. The flow problems of linear stretching have been discussed comprehensively through many authors. The flow analysis due to stretching / shrinking has been prolonged to formulate several flow problems. Crane [1] was one of the first who highlighted the effects of viscous fluids on the boundary layer. His work was extended in various flow directions for non-Newtonian and Newtonian fluids. Chiam [2] also extended the work of Crane [1]. Chiam [2] studied the stagnation flow region at a stretching surface. Numerical solution of stagnation flow region at stretching surface was analyzed through Mahapatra and Gupta [3]. In their study, they found the similarity exact solutions of Navier Stokes equations. Wang [4] highlighted the stagnation flow region over a shrinking sheet for axisymmetric and in both dimensional cases. He also worked out on the unique solution and dual solutions values (specific) of the ratio of shrinking. The effects of viscous flow in the presence of suction at stretching surface and by considering surface slip are analyzed by Wang [5]. He also highlighted the axisymmetric stretching for uniqueness and existence. Stagnation region flow at a stretching surface was discussed by Mahapatra et al. [6]. They have studied the power-law fluid model and analyzed it analytically having magnetic hydrodynamics while they appealed that their results agreed well with numerical results. Sajid et al. [7] analyzed the viscous fluid at a stretching curved surface. They also highlighted the influence of dimensionless curvature and claimed that boundary layer thickness increases while skin friction coefficient declines. Natural convection over the vertical stretching cylinder was analyzed by Wang [8]. He discussed different flow behaviors on the vertically stretching cylinder. He claimed that false diffusions computations can be more valued when a reactive system is to be analyzed computationally. Considering quantitative prediction of response yields call for extra accurate modeling of diffusion of the reacting species, it's far crucial to know the right effective diffusivity. Khan et al. [9] also discussed about the flow of nanofluids by impinging tangent hyperbolic model at a stretching cylinder under the stagnation region. Many investigators discussed about the stretching curve channels with different effects of physical parameters. The present literature inspection on the stretched flows Ref. [10], [11], [12], [13], [14], [15] and exposes that the work isn't done on the exponentially curve channel.
Most demanding applications of nanofluids are much interesting in the fields of engineering and sciences. The uses of nanotechnology in modern science attracted researchers to analyze about nanofluid models with different aspects. Nanofluids are mixture of nanosized particles and basefluid. Choi [16] analyzed nanofluid. He highlighted that heat in the nanofluid gain higher as compare to simple base fluid. Basically, heat transfer and thermal conductivity are directly proportional to each other. Low thermal conductivity gave low heat transfer (e g compared with base fluids and nanofluids). Some experimentally, numerically and analytically have been discussed on nanofluids in the occurrence of natural convection and boiling heat transfer by Xuan and Li [17], convective heat transfers by Abu-Nada [18] and thermal conductivity by Kang et al. [19]. Several researchers highlighted nanofluids flow over stretched surface in the presence of different aspects (see Refs. [20], [21], [22], [23], [24], [25]).
Hybrid nanofluid is a mixture of two different nanosized particles with base fluids. Hybrid nanofluids have numerous applications in the field of heat transfer like as manufacturing, microelectronics, medical, microfluidics, microelectronics etc. When the nanosized particles are properly scattered, hybrid nanofluid can deal tremendous advantages other than the atypically high successful thermal conductivity. To improve it even better, Hybrid nanofluid has been introduced to analyze the properties of heat transfer. Suresh et al. [26] have discussed about the hybrid nanofluids experimentally and numerically. Devi and Devi [27] explored the impacts of hybrid nanofluid at stretching sheet. They also highlighted the influence of effecting Lorentz force and explored by numerically. Moghadassi et al. [28] have highlighted hybrid nanofluid impact on forced convective heat transfer. He solved this problem numerically. Sinz et al. [29] and Minea [30] have studied hybrid nanofluid numerically. Some latest studies on different flow geometries are listed in Refs. [31], [32], [33], [34], [35].
The point of this investigation is to analyze the influence of involving parameters of hybrid nanomaterial base fluid over an exponentially stretching curved surface. The mathematical flow model executed a system of partial differential equations. The system is first changed to a system of ordinary differential equations after employing the suitable similarity transformation. Dimensionless system is elucidated through numerical scheme bvp4c method. We also highlighted the impacts of suction/injection parameter, curvature parameter, thermal slip parameter, solid nanoparticle and stretching parameter on the hybrid nanofluid through graphs and tables. The expression of the skin friction and Nusselt numbers are highlighted to understand the flow properties. We are expected that our results gained through this analysis which will not only give helpful material for applications but also give a complement to the existing literature. To the most effective of our data, these effects will be original and new.
Section snippets
Mathematical formulations
Consider a steady flow of Cu − Al2O3/H2O over a permeable curved exponentially stretching surface. Where s is the arc length along the flow direction while r is normal to tangent vector at any point on the surface shown in Fig. 1(a). The geometry of the stretching is revealed in Fig. 1(b). It is considered that the surface is stretched along the velocity in the s − direction in which c is constant. Here, vw is due to porous surface which represents two cases if vw < 0 and vw > 0
Results and discussion
The system of non-linear ordinary differential Eqs. (14) and (15) with respect to the boundary conditions (16) is solved numerically through bvp4c method to analyze the different physical parameters, such as stretching parameter R0, curvature parameter K, solid nanoparticle of cooper Φ2, stretching parameter β and thermal slip parameter M. Effects of these physical parameters are highlighted in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9. Figs. 2 and 3 show the comparative
Final remarks
Transportation of hybrid nanoparticle with base fluid over an exponentially stretching curved surface has been analyzed. Numerical outcomes of the established mathematical problem are solved and analyzed. The significant physical parameters of the flow properties are highlighted through graphs and tables. We draw some significant observations which further support different fields which are given below.
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In the case of suction and injection, opposite behavior to be noted on the temperature
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