Impacts of Nanoparticle Shape on Al 2 O 3 -Water Nanofluid Flow and Heat Transfer over a Non-Linear Radically Stretching Sheet

The results of this article can be useful in science and technology advancement, such as nanofluidics, micro mixing and energy conversion. The pur-pose of this article is to examine the impacts of nanoparticle shape on Al 2 O 3 -water nanofluid and heat transfer over a non-linear radically stretching sheet in the existence of magnetic field and thermal radiation. The different shapes of Al 2 O 3 nanoparticles that have under contemplation are column, sphere, hexahedron, tetrahedron, and lamina. The governing partial differential equations (PDEs) of the problem are regenerated into set of non-linear ordinary differential equations (ODEs) by using appropriate similarity transformation. The bvp4c program has used to solve the obtained non-linear ordinary differential equation (ODEs). The Nusselt number for all shapes of Al 2 O 3 nanoparticle shapes in pure water with Pr 6.2 = is presented in graphical form. It has reported that the heat transfer augmentation in lamina shapes nanoparticles is more than other shapes of nanoparticle. The relation of thermal boundary layer with shapes of nanoparticles, solid volume fraction, magnetic field and thermal radiation has also presented with the help of graphical representation. It is also demonstrated that lamina shape nanoparticles have showed large temperature distribution than other shapes of nanoparticles.


Introduction
The fluid flows over stretching sheet have gained considerable attention in fields How to cite this paper: Rashid, U. and Ibrahim, A. (2020) Impacts of Nanoparticle Shape on Al 2 O 3 -Water Nanofluid Flow and Heat Transfer over a Non-Linear Radically Stretching Sheet. Advances in Nanoparticles, 9, 23-39. https://doi.org/10.4236/anp.2020.91002 of engineering due to its extensive use, such as bundle wrapping, hot rolling, extrusion of sheet material, glass fiber, wire rolling and extrusion of polymer sheet [1]. The fluid behaviour and various physical aspects are associated with the stretching sheet having been discussed by different authors [1] [2] [3] [4]. Recently, the boundary layer flow of nanofluid over a stretching sheet has become very interesting topic among researchers. The steady boundary layer flow, nanoparticle volume fraction and heat transfer in nanofluid over a linear stretching surface were analysed by khan and pop [5]. The nano boundary layer flow over a stretching sheet by applying differential transform method (DTM) was studied by Rashidi and Erfani [6]. Numerical solution of nanofluid flow in permeable rotating sheet was studied by Sheikholeslami and Ganji [7].
Due to peculiar properties, nanofluids are significant in numerous applications in heat transfer including microelectronics, hybrid powered engines, fuel cells, and pharmaceutical processes [8]. Nanofluid, containing nanoparticles was introduced by Choi et al. [9] and discussed the fluid which contains nanoparticles that were suspended in basic fluid, such as ethylne glycol, propylen glycol, water etc. Nanoparticle having high thermal conductive metals, such as, copper, aluminum, silicon or silver helps to intensify the thermal conductivity of such mixtures, which consequently improves over all the energy transport capability. Nadeem and Lee [10] introduced the nanofluid flow over an exponentially stretching sheet. Rana and Bhargave [11] extended the work, and studied the laminar boundary layer flow of a nanofluid over stretching sheet. The effects of magnetic field on a nanofluid over a stretching sheet have been investigated by Sheikholeslami and Chamkha [12]. Model of stagnation point flow of nanofluid over a stretching sheet was developed by Ul Haq et al. [13]. Mixed convection boundary layer fluid flow along a stretching sheet in porous medium was numerically discussed by Mukhopadhayay Som et al. [14].
The study of magnetic effects of nanofluid flow has gained vast attention of engineering and sciences because of its extensive significant industrial applications such as metallurgical process and polymer industry [15]. Sheikholeslami et al. [16] numerically discussed the magnetic field effect on natural convection heat transfer of water-cu and water-cuo nanofluid. Yadav et al. examined the magnetic field effect on sunset of nanofluid convection [17]. Xuan et al. [18] studied the effect of magnetic field on heat transfer of nanofluid flowing through a microchannel. Rashid [19] has been numerically studied the magnetic field effect on steady laminar flow over vertical plate. Ashorynejad et al. [15] have examined the magnetic field effect on natural convection of water-Ag nanofluid between two coaxial circular cavities. Ghasemi et al. [20] numerically investigated study on natural convection heat transfer in an inclined enclosure filled with a water-Cuo nanofluid. Mahumoudi et al. [21] numerically discussed magnetic field effect on the natural convection of water-Cuo nanofluid in triangular enclosure. Hamad [22] had investigated analytical solution of the magnetic field effect on natural convection of nanofluid over stretching sheet. Sheikholeslami nanofluid between two parallel plates was conducted by Esfe et al. [35].
The shape of nanoparticles is very significant to change thermal conductivity of nanofluid. The present research focuses on to investigate the impacts of nanoparticle shape on Al 2 O 3 -water nanofluid and heat transfer over a non-linear radically stretching sheet in the presence of magnetic field and thermal radiation. There are five shapes of nanoparticles which are under consideration; column, sphere, hexahedron, tetrahedron and lamina. Numerical solutions of nonlinear ordinary differential equations (ODE's) are solved by bvp4c program. The effects of empirical shape factor, solid volume fraction, magnetic field and radiation parameter are discussed in detail.

Mathematical Model
We have considered two dimensional, steady and laminar boundary layer flow in water-based nanofluid, having various shapes of Al 2 O 3 nanoparticles, pass over a non-linear radically stretching sheet with influences of magnetic field and thermal radiation. The under contemplated shapes of nanoparticles are column, sphere, hexahedron, tetrahedron and lamina. The radically stretching sheet is placed at z = 0, transverse magnetic field and radiation field are applied along the z-axis, which are shown in Figure 1. Furthermore, a cylindrical co-ordinate system (r, θ, z) has been used. The flow is in rotational symmetry, so the physical quantities are independent of θ. The components of velocities u and w are direction of r and z respectively. The partial governing equations of the axisymmetric flow are 0, u u w r r z The nanofluid flow is happened due to stretching sheet, there is no role of pressure gradient in the fluid flow field. The above equation after applying the boundary layer approximation has reduced as shown in [37].
With the boundary conditions related to problem are given by where k S represents the thermal conductivity of solid. The f k represents the thermal conductivity of base fluid. The m represents the shape factor and its numerical values are given in Table 1. Furthermore, thermophysical properties of liquid and solid nanoparticles are presented in Table 2.
By using the nonlinear Rosseland approximation, the radiation flux converted into form That the temperature within the flow, such as that 4 T may be expressed as a temperature liner function. Hence by expanding 4 T by Taylor series and neglecting higher order terms, we obtained the following relation 4 3 4 By Employing Equations (11) and (12), Equation (7) converted into ( ) The similarity transformation of Equations (5)-(9) and stokes stream function ( ) , r z ψ are defined in the following form By the Equation (15) into Equations ((5)-(6)) and Equation (13), the Equation (5) identical satisfied Equation ( (6), (13)) and their related boundary value conditions reduce as ( ) ( ) Here M is the Magnetic parameter, it is defined as ( ) The quantity of particle interest in this problem is the Nusselt number defined as ( ) where ( ) w q x is the wall heat flux given by Using Equation (6)     performance of heat transfer in nanoparticles is such as Lamina > Column > Tetrahedron > Hexahedron > Sphere.
The graphical depictions are used to illustrate the relation of appropriate parameters on velocity profiles and thermal boundary layers. The solid volume fraction is very significant parameter for nanofluid. Figure 9 illustrates that the         boundary layer thickness of nanoparticles. It is noted that the temperature raised with intensifying value of radiation parameter, because in the presences of thermal radiation implies an immense enlarging in the radiative heat which encourage thermal state of nanofluid initiate temperature to intensify. Thermal boundary layer thickness of lamina shape nanoparticles observed more animated by thermal radiation effect.

Conclusions
The shapes effect of nanoparticles effects on boundary layer flow and heat transfer in Al 2 O 3 -water nanofluid over a non-linear radically stretching sheet have been designed in this study. The effects of nanoparticles shape, solid volume fraction, magnetic field and thermal radiation on thermal boundary layer and heat transfer rate with value of prandtle number (Pr = 6.2) have discussed in details. Non-linear thermal radiation is taken into consideration. The following results have been proven: • The lamina shape of nanoparticles acts as principle in lead of disturbance on thermal boundary layer thickness. • The tetrahedron shape of nanoparticles acts amidst role in disturbance of thermal boundary layer thickness.