MHD 3-dimensional nanoﬂuid ﬂow induced by a power-law stretching sheet with thermal radiation, heat and mass ﬂuxes

In this article, the three-dimensional Magnetohydrodynamics ﬂow of a nanoﬂuid over a horizontal non-linearly stretching sheet in bilateral directions under boundary layer approximation is addressed. A two-phase model has been used for the nanoﬂuid. The inﬂuences of thermophoresis, Brownian motion and thermal radiation on heat and mass transfers are considered. Two different cases for the heat and mass transfers are studied. In the ﬁrst case, uniform wall temperature and zero nanoparticles ﬂux due to thermophoresis are considered. In the second case, prescribed heat and mass ﬂuxes at the boundary are considered. By using the appropriate transformations, a system of non-linear partial differential equations along with the boundary conditions is transformed into coupled non-linear ordinary differential equations. Numerical solutions of the self-similar equations are obtained using a Runge–Kutta method with a shooting technique. Our results for special cases are compared with the available results in the literature, and the results are found to be in good agreement. It is observed that the pertaining parameters have signiﬁcant effects on the characteristics of ﬂow, heat and mass transfer. The results are presented and discussed in detail through illustrations.


Introduction
Boundary layer flow over a stretching surface has various trade and technical applications, such as metal and polymer extrusion, paper production, and many others.
Also, it has important applications in engineering and manufacturing processes. Such processes are wire and fibre covering, foodstuff dispensation, heat-treated materials travelling between a feed roll and a wind-up roll or materials manufactured by extrusion, glass fibre production, cooling of metallic sheets or electronic chips, crystal growing, drawing of plastic sheets and so on. The quality of the final product with the desired features in these processes depends on the degree of freezing in the process and the process of stretching.
Flow past a linearly stretching sheet was first studied analytically by Crane (1970). Gupta and Gupta (1977) mentioned that in all realistic situations, the stretching mechanism is not linear. Later on, several investigators, Hamad and Pop (2010), Turkyilmazoglu (2013), Pal and Mandal (2015), , studied such type of flow problems either with heat or with mass transfer effects.
The main characteristic of nanofluid (a combination of nanoparticles having size 1-50 nm and liquid) is to improve the base fluid's thermal conductivity. Usually, ethylene glycol, oil, water, and so on are taken as base fluids that have limited heat transfer properties due to their low thermal conductivity. By adding nanoparticles into the base fluid, Masuda et al. (1993) first detected the change in viscosity and thermal conductivity of such fluids. The abnormal increase in the thermal conductivity occurs by the dispersion of nanoparticle, which was first observed by Choi and Eastman (1995). In case of nanofluids, Buongiorno (2006) studied the convective heat transfer and indicated that the Brownian motion, thermophoresis as the most important means for the unpredicted heat transfer augmentation. Later on, different aspects of nanofluid flow were addressed by several researchers. Considering heat and mass transfer, Das (2012) pointed out the slip effects in nanofluid flow. By the finite element method, Rana  Thermal radiation affects the heat transfer characteristics significantly particularly in control of heat transfer, space technology and high-temperature processes (Das et al. 2014).  reported the combined effects due to Joule heating and solar emission on MHD thixotropic nanofluid flow. Radiative flows in companies with the external magnetic fields have wider applications in nuclear engineering, power technology, astrophysics, power generation and so on ).
The investigations described above are two-dimensional. Three-dimensional problem being more realistic and has already drawn the attention of several researchers. Wang (1984) first addressed the three-dimensional flow due to a bidirectional stretching sheet. Ariel (2007) obtained the solution for a three-dimensional flow caused by a stretching surface using a perturbation technique. Liu  The combined effects of MHD and thermal radiation on three-dimensional non-linear convective Maxwell nanofluid flow owing to a stretching sheet were reported by Hayat et al. (2017). The literature review reveals the fact that there is not so much work has been done on three-dimensional nanofluid flow caused by a non-linearly stretching sheet.
Motivated by this, in this article, we investigate the radiative heat transport of the MHD flow of a nanofluid past a stretching sheet with power-law velocity in two directions. The mathematical representations of the nanofluid model proposed by Buongiorno (2006) reflecting the contribution of nanoparticle volume fraction were used. This model also includes the effects of thermophoresis and the Brownian motion. The main objective here is to analyse the heat transport features of the flow of a nanofluid owing to the collective influences of magnetic field and radiation. Two different cases are considered: In Case I, uniform surface temperature is considered. Due to thermophoresis, a flux of nanoparticles volume fraction at the sheet is assumed to be zero. This assumption controls the volume fraction of nanoparticles submissively rather than actively (Mansur and Ishak 2013; Nield and Kuznetsov 2014); on the other hand, in Case II, heat and mass fluxes are considered at the sheet. Suitable similarity transformations are considered to transform the highly non-linear partial differential equations into ordinary ones. Numerical results are obtained by a Runge-Kutta method with a shooting technique. For a clear understanding of the flow and heat transport characteristics, the obtained results are analysed in detail with their physical interpretation through Tables and Figures. It is observed that the physical parameters affect the heat and mass transport significantly.

Mathematical modelling
Let us consider a 3-D steady boundary layer nanofluid flow past a non-linearly stretching surface. In a usual notation, the governing equations of the problem can be written as where the components of velocity in the x, y and z directions are u, υ and w, respectively; kinematic viscosity is ν, electrical conductivity is σ , ρ f is the density, T denotes the temperature of the fluid, α denotes the thermal diffusivity, B 0 = a/(x + y) 1−n represents the variable magnetic field, the effectual heat capability of the nanoparticles is (ρc) p , and (ρc) f is the effectual heat capability of nanofluid; D B , D T are the diffusion coefficients due to the Brownian motion and the thermophoresis, respectively; q r represents the radiative heat flux, C is the nanoparticle volume fraction and T ∞ is the steady temperature of the inviscid free stream. Using the Rosseland approximation, the expression for radiative heat flux can be written as where σ s and k e are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. This investigation is restricted to the optically bulky fluid. For sufficiently small temperature difference linearisation of Eq. (6) can be performed. Using the Taylor's series expansion for T 4 about T ∞ and ignoring the higher-order terms, we get

A .Boundary conditions for fluid flow
The boundary conditions can be written as u w and υ w denote the velocities of stretching along the x and y directions, respectively, and n is a non-linearly stretching parameter and C ∞ is the uniform volume fraction for the nanoparticles in the inviscid free stream. B. Boundary conditions for heat, mass transport There are different types of heating process that specify the temperature distribution on the surface and ambient fluid. These are constant or prescribed wall temperature (CWT or PWT); constant or prescribed surface heat flux (CHF or PHF), CBC and Newtonian heating. In this problem, we have considered two cases of temperature distributions: CWT and CHF separately. Moreover, different methods are available for the active or passive control of nanoparticles, namely, constant or prescribed concentration at the wall (CHF or PWC); zero mass or normal flux of nanoparticles at the surface (ZMF); constant or prescribed mass flux (CMF or PMF). In this problem, two different boundary conditions, ZMF and CMF for mass transfer, are considered. Therefore, the conditions at the boundary for temperature, volume fraction for nanoparticles are Case I: Case II: where T w and C w are the temperature and concentration at the surface, respectively. Let us define the similarity variable and similarity transformations as (for Case-II). Using these transformations in Eqs (1)-(5) and also using the relation given by Eq. (6), we get the following self-similar equations Here, the magnetic parameter is M = represents the Brownian motion parameter and N t = represents the thermophoresis parameter, radiation parameter is R = is the Schmidt number, λ = b a denotes the ratio of rate of stretching along y-direction to that of x-direction.
The boundary condition becomes Dimensionless wall shear stresses, heat and mass fluxes are where τ zx and τ zy are shear stresses at the wall and q w represents the heat flux at the wall, j w denotes the mass flux which are described as Finally, we get where the Reynolds numbers (local) along the directions of x, y are Re x = u w (x+y) The differential Eqs (11)- (14) are coupled and non-linear. Equations (11)- (14), together with the conditions (15a-15d), form a two-point boundary value problem. By converting it into an initial value problem, those equations are numerically solved using the Runge-Kutta method and the shooting technique by choosing an appropriate value for n ∞ .   (ii) For λ = 1, the system leads to the axisymmetric nanofluid flow owing to a non-linearly stretched surface.

Testing of numerical results
To test the validity of our numerical results, an assessment is made with the available results from the open literature for some special cases and presented in Table 1. Table 1 shows a complete agreement of our results with those of  and Raju et al. (2016) in the absence of the magnetic field and thermal radiation.
Grid independence test has been performed and presented through Table 2, which shows that the values of f (0), g (0) are the same for two different grid sizes (0.01, 0.03).

Analysis of the results
To obtain in-depth features, the numerical results are presented graphically in Figures 2-12. Figures 2(a-f) depict the effects of magnetic field on velocity components along with the directions of x and y axes, temperature and concentration fields. Linearly (n = 1) and non-linearly stretching (n = 3) cases are considered. As the Lorentz force (which resists the movement of fluid) increases (increase in magnetic parameter M), the velocity in both directions decreases [Figures 2(a,b)]. This result is similar to that of Mahanthesh et al.  From Figures 5(a-d), it is seen that the temperature [ Figures 5(a,b)] and nanoparticle volume fraction [ Figures 5(c,d)] are increasing functions of thermophoresis parameter N t : The force responsible for the diffusion of nanoparticle owing to temperature ramp is recognised. Growth in the width of the boundary layer related to the temperature field is observed for higher values of the thermophoresis parameter N t [ Figures 5(a,b)]. Through a rise in N t, , the thermal conductivity of the nanofluid increases due to the company of nanoparticles. As a result, the concentration of the fluid increases through the rise in N t [ Figures. 5(c,d)].
The influences of radiation on temperature and nanoparticle volume fraction are portrayed in Figures 6(ad). It is observed that the temperature [ Figures 6(a,b)] and the nanoparticle volume fraction [ Figures 6(c,d)] both increase for a rise in R. As K e diminishes when the thermal radiation parameter R increases, which, in turn, causes to amplify the deviation of the heat flux due to radiation. The speed of heat transport due to radiation augments and consequently the fluid temperature increases. The width of the temperature boundary layer enlarges by a rise in heat transport to the liquid due to radiation.
The effects of the Prandtl number Pr on temperature and concentration fields of the nanofluid are presented in Figures 7(a-d). The thicknesses of both the boundary layers: thermal [ Figures 7(a,b)  . A significant decrease in concentration is noted for the non-linearly stretching sheet (n = 3) case compared to the linearly stretching sheet (n = 1) case. Schmidt number, Sc, is the relation of diffusivities related to momentum and mass. A higher value of Sc indicates the smaller value of coefficient D B of Brownian diffusion that opposes the nanoparticles to penetrate intensely into the liquid.
To provide a clear understanding of the flow field, the streamlines are presented for several values of n in Figures 10(a,b). Figures 11(a,b) indicate that the coefficient of the skin friction (local) in the directions of x, y increases with increasing λ , M and n. From Figure 11(c), it is clear that the Nusselt number in the CWT case increases due to an increase in λ and n but decreases with the magnetic field parameter M. However, the local Sherwood number in the ZMF case increases with the magnetic parameter r but decreases with the stretching ratio parameter λ and with the power-law index n. These changes are depicted in Figure 11(d).
The effects of N b , N t on the Nusselt number for the CWT case and local Sherwood number for ZMF case are portrayed in Figures 12(a,b), respectively. Both are found to increase with an increase in N b. On the other hand, the Nusselt number [ Figure 12(a)] decreases and Sherwood number [ Figure 12(b)] increases with the increasing values of N t . Table 3 shows that −θ (0), −φ (0)both decrease with the increasing values of the thermophoresis parameter. Also, −θ (0) increases with the increasing Prandtl number but −φ (0) decreases with Pr.

Concluding remarks
The current study provides solutions (numerical) for a steady MHD three-dimensional nanofluid flow past a bi-directional non-linearly stretching surface. Two types of conditions at the boundary for temperature and volume fraction for nanoparticles are considered. From the current investigation, the following observations are derived: 1. A rise in the magnetic strength causes a reduction in the resistance and the rate of heat transport.
2. Both the thermal and the concentration boundary layer thicknesses increase with an increase in the magnetic parameter in addition to the radiation parameter.
3. The volume fraction of the nanoparticles is a decreasing function of the Brownian motion parameter.
4. In the presence of thermophoresis, both temperature and volume fraction of the nanoparticles increase significantly in the boundary layer. T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k