The combined effect of magnetic field and viscous dissipation on the boundary layer flow over a permeable stretching sheet in a casson nanofluid with convective boundary condition

Introduction

A Nanofluid is a fluid containing nanometer sized particles, called Nanoparticles. These fluids are engineered colloidal suspension of nanoparticles in a base fluid. The nanoparticles used in nanofluids are characteristically made of metals, oxides, carbides, or carbon nanotubes. ordinary base fluids include water, ethylene Glycol and oil. Nanofluids have narrative properties that make them potentially useful in many applications in heat transfer, including microelectronics, fuel cells, pharmaceutical processes, and hybrid-powered engine, engine cooling/vehicle thermal management, domestic refrigerator, chiller, heat exchanger, in grinding, machining and in boiler gas temperature reduction. They demonstrate enhanced thermal conductivity and the convective heat transfer coefficient compared to the base fluid. Knowledge of the rheological behaviour of nanofluids is found to be very basic in deciding their suitability for convective heat transfer applications. The fluid flow over a stretching surface has significant applications such as extrusion, wire drawing, metal spinning, hot rolling, etc [1][2][3] . A wide variety of problems dealing with heat and fluid flow over a stretching sheet have been studied with both Newtonian and non-Newtonian fluids and with the addition of obligatory electric and magnetic fields, different thermal boundary conditions, and power law variation of the stretching velocity. A representative sample of the recent literature on the topic is provided by references [4][5][6][7][8][9][10][11][12] . After the poineering work by Sakiadis [13] , a large amount of literature is available on boundary layer flow of Newtonian and non-Newtonian fluids over linear and nonlinear stretching surface. The problem of natural convection in a regular fluid past a vertical plate is a conventional problem first studied theoretically by E. Pohlhausen in giving to an experimental study by Schmidt and Beckmann [14] . In the past few years, convective heat transfer in nanofluids has turn out to be a topic of major existing interest. Recently Khan and Pop [15] used the model of Kuznetsov and Nield [16] to study the boundary layer flow of a nanofluid past a stretching sheet with a constant surface temperature. Makinde, and Aziz [17] investigated the effect of a convective boundary condition on boundary layer flow, heat and mass transfer and nanoparticle fraction over a stretching surface in a nanofluid. The transformed non-linear ordinary differential equations governing the flow are solved numerically by the Runge-Kutta Fourth order method. lately, Pal et al. [20] investigated the influence of thermal radiation on mixed convection flow of nanofluid caused by nonlinearly stretching/shrinking sheet. Kandasamy and Periasamy [21] considered study, of free convection flow,heat and mass transfer of Newtonianfluid past nonlinearly stretching sheet with the effect of chemical reaction and magnetic field. The laminar boundary layer flow of electrically conducting fluid towards nonlinearly stretching sheet under the influence of first-order chemical reaction was hypothetically studied by Raptis and Perdikis [22] . On the other hand, the numerical and analytical solutions of steady-state boundary layerflow of micropolar fluid induced due to nonlinearly stretching sheet were established by Damseh et al. [23] and Magyari and Chamkha [24] , respectively. The joint effects of slip and chemical reaction on electrically conducting fluid over a nonlinearly porous stretching sheet were analyzed by Yazdi et al. [25] . Bhattacharyya and Layek [26] considered the velocity slip effects on boundary layer flow of viscous fluid past a permeable stretching sheet with chemical reaction. The steady two-dimensional boundary layer flow of Newtonian fluid due to stretching sheet saturated in nanofluid in the presence of chemical reaction is explored by Kameswaran et al. [27] . Aurangzaib et al. [28] investigated theoretically the brunt of thermal radiation on unsteady natural convection flow produced by stretching surface in the presence of chemical reaction and magnetic field. Shehzad et al. [29] reported the effects of magnetic field on mass transfer flow of Casson fluid past a permeable stretching sheet in the presence of chemical reaction. Pal andMandal [30] investigated the characteristics of mixed convection flow of nanofluid towards a stretching sheet under the influence of chemical reaction and thermal radiation. Similarity solutions for unsteady boundary flow of Casson fluid induced due to stretching sheet embedded in a porous medium in the presence of first order chemical reaction were obtained by Makanda and Shaw [31] . on the other hand, convective boundary condition plays a high-flying role in many engineering processes and industries such as gas turbines, material drying, textile drying, laser pulse heating, nuclear plants,transpiration cooling, and food process. It is because that the convective boundary condition applied at the surface is more practical and realistic. The two-dimensional laminar boundary layer flow of Newtonian fluid caused by porous stretching surface in the presence of Convective boundary condition is investigated numerically by Ishak [32] . Makinde and Aziz [33] analyzed steady incompressible flow of nanofluids towards stretching sheet with convective boundary condition using Boungiornomodel. Moreover, RamReddy et al. [34] incorporated the effects of Soret and investigated mixed convection flow due to vertical plate in a nanofluid under convective boundary condition. Das et al. [35] discussed the heat and mass transfer flow of hydromagnetic nanofluid past a stretching sheet place in a porous medium with convective boundary condition. The threedimensional laminar flow of Casson nanofluid due to stretching sheet in the presence of convective boundary condition is developed by Nadeem and Haq [36] .Malik et al. [37] investigated the effect of convective boundary condition past a stretching sheet in the presence of magnetic field. The two-dimensional electrically conducting flow of Casson nanofluid created by stretching sheet with convective boundary condition is performed by Hussain et al. [38] . In the above mentioned appraisal of literature,no work has been reported, and to trounce this void, we have considered the study apprehensive to the effect of a convective boundary condition and magnetic field effect on boundary layer flow, heat transfer and nanoparticle fraction over a permeable stretching sheet in a Casson nanofluid, with viscous dissipation.Further investigated the heat,flow and mass transfer charecteristics of the pertinent parameters numerically and displayed them in graphical form.

Convective transport equations

consider steady two-dimensional boundary layer flow of a nanofluid past a stretching sheet with a linear velocity variation with the distance i.e. where c is a real positive number, is stretching rate, and is the coordinate measured from the location, where the sheet velocity is zero

We introduce the following dimensionless quantities (7) Where is the stream function with (10) subject to the following boundary conditions.

where primes denote differentiation with respect to and the five parameters appearing in Eqs. (9-12) are defined as follows. ……………(13)

Result and Discussion

Eqs. (8-10) subject to the boundary conditions, Eqs.( 11) and ( 12), were solved numerically using Runge-kutta -Fehlberg fourth order method. We now turn our attention to the discussion of graphical results that provide additional insights into the problem under investigation. Where as opposite trend is noticed for the effect of casson fluid parameter β on temperature profile, as depicted in fig4, i.e temperature in the boundary layer increases with increase in Casson fluid parameter, which is consistant with the results obtained By, M. Trivedi1 et al. [39] , Ullah et al. [40] , Sudipta Ghosh and Swati Mukhopadhyay [41] . The impact of both Nt and Nb on temperature profile is displayed in fig 5 .Here it is observed that temperature increases with increase of Nt as well as Nb. Enhancing Nt and Nb leads to faster random motion of nanoparticles in fluid flow, resulting in increasing thickness of thermal boundary layer, further enlarges temperature of Casson nanofluid abruptly. As in process of thermopheresis, heated nanoparticles close to the sheet move away from heated region to colder region Fig 6, displays the effect of Lewis number Le, on temperature profile. Lewis number expresses the relative contribution of thermal diffusion rate to species diffusion rate in the boundary layer regime.An increase of Lewis number will reduce thermal boundary layer thickness and will be accompanied with a decrease in temperature. Larger Le will suppress concentration values. I.e inhibit nanoparticle species diffusion. There will be much greater reduction in concentration bopundary layer thickness than thermal boundary layer thickness over an increment in Lewis Number. Fig. 7 illustrates the effect of Biot number on the thermal boundary layer. As expected, the stronger convection results in higher surface temperatures, causing the thermal effect to penetrate deeper into the quiescent fluid. The temperature profiles depicted in Fig. 8 show that as the Prandtl number increases, the thickness of the thermal boundary layer decreases as the curve become increasingly steeper. Fig 8 shows that the effect of magnetic parameter, on the temperature profiles is noticeable only in region close to the sheet as the curves tend to merge at larger distances from the sheet. It was observed in Fig. 7, that as the convective heating of the sheet is enhanced Bi increases, i.e the thermal penetration depth increases. Because the concentration distribution is driven by the temperature field, one expects that a higher Biot number would endorse a deeper penetration of the concentration. This expectation is certainly realized in Fig. 14, which displays a higher concentrations at higher values of the Biot number, which is consistant with the results of Gbadeyan, J.A et al. [42] When magnetic M is increased due to rising Lorentz forces, nanoparticle concentration profiles are increased, resulting in thickening of concentration boundary layer , is as shown in fig 15.

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