In this article, the threedimensional Magnetohydrodynamics flow of a nanofluid over a horizontal nonlinearly stretching sheet in bilateral directions under boundary layer approximation is addressed. A twophase 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 nonlinear partial differential equations along with the boundary conditions is transformed into coupled nonlinear ordinary differential equations. Numerical solutions of the selfsimilar 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.
Keywords
 nanofluid
 thermophoresis
 Brownian motion
 thermal radiation
 powerlaw index
MSC 2010
 76D10
 65L10
 35Q79
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, heattreated materials travelling between a feed roll and a windup 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), Waqas et al. (2016), 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 and Bhargava (2012) studied the nanofluid flow towards a nonlinearly stretched surface. Nadeem and Lee (2012) studied the steady nanofluid flow over an exponentially stretched sheet. Considering the convective boundary condition (CBC), Mustafa et al. (2013) discussed nanofluid flow over an exponentially stretched surface. Using the Jeffrey's model for nonNewtonian fluid, Nadeem et al. (2014) investigated nanofluid flow over a stretching surface to obtain a numerical solution. Magnetohydrodynamics (MHD) plays a great role in several areas particularly in geophysics, medicine, engineering and many others [Hayat et al. (2016)]. Magneto nanofluids have significant applications in drug delivery, cancer therapy, enhancement of magnetic resonance and magnetic cell separation (Hayat et al. 2016). In 2013, Khan et al. analysed the effects of a magnetic field, applied externally on nanofluid flow due to a stretching sheet considering a nonsimilar solution of the problem. Influences of the magnetic field on Cuwater flow were analysed by Sheikholeslami et al. (2014). Using the Buongiorno's model, in 2016, Sheikholeslami and his coauthors described the unsteady flow of nanofluid in the presence of a magnetic field. Ma et al. (2019) studied the consequences of the magnetic field on forced convection AgMgO nanofluid flow in a channel with heat transfer. Many other research papers on nanofluid with magnetic field effects can be found in the studies conducted by Ma et al. (2019) and Bhatti et al. (2018).
Thermal radiation affects the heat transfer characteristics significantly particularly in control of heat transfer, space technology and hightemperature processes (Das et al. 2014). Hayat et al. (2016) 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 (Hayat et al. 2016).
The investigations described above are twodimensional. Threedimensional problem being more realistic and has already drawn the attention of several researchers. Wang (1984) first addressed the threedimensional flow due to a bidirectional stretching sheet. Ariel (2007) obtained the solution for a threedimensional flow caused by a stretching surface using a perturbation technique. Liu and Andersson (2008) analysed the heat transport of a 3D flow due to a stretching surface. Ahamad and Nazar (2011) discussed the MHD 3D flow of a viscoelastic fluid due to a stretching surface. In 2013, Hayat and his coauthors considered 3D OldroydB fluid flow past a stretching surface with a CBC. Considering the exponential temperature distribution, 3D flow and heat transfer of a viscous fluid were investigated by Liu et al. (2013). Nadeem et al. (2013) analysed the 3D Casson fluid flow due to a linearly stretching sheet in the presence of a magnetic field. Khan et al. (2014) obtained analytical and numerical solutions for a threedimensional flow caused by a nonlinearly stretching sheet. Khan et al. (2015) investigated threedimensional flow of a nanofluid past a nonlinearly stretching sheet and highlighted its applications to the solar energy industry. Existence of two solutions was reported by Raju et al. (2016) for flow due to a nonlinearly stretching sheet. Considering thermal radiation and heat generation/absorption, Shehzad et al. (2016) analysed 3D MHD flow of a Jeffrey nanofluid over a bidirectional stretching sheet.
The combined effects of MHD and thermal radiation on threedimensional nonlinear 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 threedimensional nanofluid flow caused by a nonlinearly 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 powerlaw 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 nonlinear 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.
Let us consider a 3D steady boundary layer nanofluid flow past a nonlinearly stretching surface. In a usual notation, the governing equations of the problem can be written as
Using the Rosseland approximation, the expression for radiative heat flux can be written as
The boundary conditions can be written as
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
Let us define the similarity variable and similarity transformations as
Using these transformations in Eqs (1)–(5) and also using the relation given by Eq. (6), we get the following selfsimilar equations
The boundary condition becomes
The differential Eqs (11)–(14) are coupled and nonlinear. Equations (11)–(14), together with the conditions (15a–15d), form a twopoint 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
For
For
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 Khan et al. (2015) and Raju et al. (2016) in the absence of the magnetic field and thermal radiation.
Comparisons for the values of
λ  

1  0  −1  −1  −1  0  0  0 
0.5  −1.22474  −1.2247431  −1.2247429  −0.612372  −0.6123721  −0.6123720  
1.0  −1.414214  −1.4142141  −1.4142140  −1.414214  −1.4142140  −1.4142140  
3  0  −1.624356  −1.6243560  −1.6243560  0  0  0 
0.5  −1.989422  −1.9894221  −1.9894220  −0.994711  −0.9947110  −0.9947110  
1.0  −2.297186  −2.2971860  −2.2971860  −2.297186  −2.2971860  −2.2971860 
Grid independence test has been performed and presented through Table 2, which shows that the values of
Comparisons of the values of
1  0  −1  −1  0  0 
0.5  −1.2247429  −1.2247429  −0.6123720  −0.6123720  
1.0  −1.4142140  −1.4142140  −1.4142140  −1.4142140  
3  0  −1.6243560  −1.6243560  0  0 
0.5  −1.9894220  −1.9894220  −0.9947110  −0.9947110  
1.0  −2.2971860  −2.2971860  −2.2971860  −2.2971860 
To obtain indepth 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
From Figures 3(a–f), it is clear that the fluid velocity in the direction of
The effect of
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(a–d). 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
The effects of the Prandtl number
Almost negligible effect in the temperature is seen due to changes in the Brownian motion parameter
To provide a clear understanding of the flow field, the streamlines are presented for several values of
Figures 11(a,b) indicate that the coefficient of the skin friction (local) in the directions of
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 −
Values of −
N_{t}  

− 
− 
− 
− 
− 
− 
− 
− 

0.3  1.04746  −0.44891  1.29932  −0.55685  1.52625  −0.65416  1.72719  −0.744224 
0.6  0.97822  −0.838474  1.18042  −1.01179  1.35314  −1.15983  1.4979  −1.28392 
0.9  0.914269  −1.17549  1.07387  −1.08369  1.20209  −1.54554  1.30267  −1.67486 
1.2  0.855295  −1.46622  0.978737  −1.67784  1.07114  −1.83624  1.13802  −1.9509 
1.5  0.80098  −1.71639  0.894035  −1.91579  0.958127  −2.05313  1.00002  −2.14289 
1.8  0.751003  −1.93115  0.818763  −2.10539  0.860844  −2.2136  0.88466  −2.27486 
The current study provides solutions (numerical) for a steady MHD threedimensional nanofluid flow past a bidirectional nonlinearly 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:
A rise in the magnetic strength causes a reduction in the resistance and the rate of heat transport.
Both the thermal and the concentration boundary layer thicknesses increase with an increase in the magnetic parameter in addition to the radiation parameter.
The volume fraction of the nanoparticles is a decreasing function of the Brownian motion parameter.
In the presence of thermophoresis, both temperature and volume fraction of the nanoparticles increase significantly in the boundary layer.
Comparisons for the values of f″(0) and g″(0) in the absence of magnetic field
λ  

1  0  −1  −1  −1  0  0  0 
0.5  −1.22474  −1.2247431  −1.2247429  −0.612372  −0.6123721  −0.6123720  
1.0  −1.414214  −1.4142141  −1.4142140  −1.414214  −1.4142140  −1.4142140  
3  0  −1.624356  −1.6243560  −1.6243560  0  0  0 
0.5  −1.989422  −1.9894221  −1.9894220  −0.994711  −0.9947110  −0.9947110  
1.0  −2.297186  −2.2971860  −2.2971860  −2.297186  −2.2971860  −2.2971860 
Comparisons of the values of f″(0) and g″(0) in the absence of the magnetic field for different grid sizes
1  0  −1  −1  0  0 
0.5  −1.2247429  −1.2247429  −0.6123720  −0.6123720  
1.0  −1.4142140  −1.4142140  −1.4142140  −1.4142140  
3  0  −1.6243560  −1.6243560  0  0 
0.5  −1.9894220  −1.9894220  −0.9947110  −0.9947110  
1.0  −2.2971860  −2.2971860  −2.2971860  −2.2971860 
Values of −θ′(0) and −ϕ′(0) for different values of thermophoresis parameter Nt and Prandtl number Pr with n = 4.0, λ = 0.9, Nb = 0.7, Pr = 2.0, R = 0.5, Sc = 15.0
N_{t}  

− 
− 
− 
− 
− 
− 
− 
− 

0.3  1.04746  −0.44891  1.29932  −0.55685  1.52625  −0.65416  1.72719  −0.744224 
0.6  0.97822  −0.838474  1.18042  −1.01179  1.35314  −1.15983  1.4979  −1.28392 
0.9  0.914269  −1.17549  1.07387  −1.08369  1.20209  −1.54554  1.30267  −1.67486 
1.2  0.855295  −1.46622  0.978737  −1.67784  1.07114  −1.83624  1.13802  −1.9509 
1.5  0.80098  −1.71639  0.894035  −1.91579  0.958127  −2.05313  1.00002  −2.14289 
1.8  0.751003  −1.93115  0.818763  −2.10539  0.860844  −2.2136  0.88466  −2.27486 
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