1 Introduction

One of the most important emerging developments of the 21^st century is nanotechnology. It has been commonly used in manufacturing and nanometersized materials have special physical and chemical properties. With its increasingly significant and complex effect on a wide range of industries, including biotechnology, oil, electronics and consumer goods, it promises to change our lives in this decade. In many engineering processes with applications in industries such as extrusion, melt-spinning, heat rolling, wire drawing, glass-fiber processing, plastic and rubber sheet manufacturing, cooling of a large metal plate in a bath that may be an electrolyte, etc., flow over a stretching surface is an important issue. The authors in [1] investigate the elastic deformation effects on the boundary layer flow of an incompressible second grade two phase nanofluid model over a stretching surface in the presence of suction and partial slip boundary condition.

In industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a windup roller, which is located at a finite distance away [2]. There are many applications in engineering and industry for boundary layer flow behavior over a stretching surface [3]. Having a low heat transfer in a fluid would cause limited heat transfer and can lead to limited heat transfer efficiency. Due to the high thermal conductivity of metal particles, adding them to a fluid would increase the thermal conductivity and also heat transfer of the resultant mixture fluid. Choi's [4] initial analysis of the term "nanofluid" identified a liquid suspension containing ultra-fine particles. Nanoparticles (e.g., Copper (Cu), Silver (Ag), Alumina (Al₂O₃), Titanium (TiO₂)) range from 1 to 100 nm in diameter [5]. The base liquid's thermal conductivity is improved by (10% – 50%) if it is suspended by a low volumetric fraction (less than 5%) of nanoparticles [6, 7, 8]. In Ref. [9], the authors examined improvements in the thermal conductivity of fluids (such as oil, water, and ethylene glycol mixture) which are poor heat transfer by suspending nano/micro or large particle materials in these fluids. Kuznetsov and Nield [10] investigated the effect of nanoparticles on the natural convection boundary-layer flow through a vertical plate, using a model in which Brownian motion and thermophoresis are represented.

During the recent decades, convective heat transfer of nanofluids is a hot topic of academic and industrial research due to its various applications in industrial processes such as thermal heating, power generation and chemical processing [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].

In many applications in the polymer and metallurgy industries, hydro-magnetic techniques are used [23]. Hence, the influence of the magnetic field has attracted significant interest in recent years due to its high applications in physics, chemistry, and engineering [24]. MHD free convection flow of Sodium Alginate nanofluid on a solid sphere with prescribed wall temperature is investigated by [25]. The authors examine the free convection flow of Casson nanofluid in the prsence of magnetic field. The relevant partial differential equations are first converted into non-dimensional equations by using appropriate transformation and then computed by utilizing the Keller box method. MHD peristaltic transport of copper-water nanofluid in an artery with mild stenosis for different shapes of nanoparticles is studied in [26]. The characteristics of MHD, heat sink/source, and convective boundary conditions in chemically reactive radiative Powell-Eyring nanofluid flow via Darcy channel using a nonlinearly settled stretching sheet/surface are used in Rasool and Shafiq [27]. In ref. [28], the authors concern with the examination of heat transfer rate, mass and motile micro-organisms for convective second grade nanofluid flow.

Considering these facts, and motivated by above discussed papers in the field of nanofluids. There is a physical justification to study the prsent article, there is enhance in the dimensionless of the stream function f, temperature θ, and volume of nanoparticles ϕ, respectively, (see Fig. 1 in ref. [19]). Due to the existing of Prandtl number in the momentum equation in ref. [19]. All the above previous research did not address this point. So, we studied the MHD flow of nanofluid over a stretched surface on heat and mass transfer to extend to the work of Abd Elazem [19]. Because of the importance this study for engineers and researchers in nearly every branch in engineering and science. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. The governing equations of the present study are solved numerically solved by using the Chebyshev pseudospectral technique [29, 30, 31]. The effect of different parameters on temperature, concentration, local skin friction f″(0), reduced Nusselt number, and reduced Sherwood number is investigated through tables and graphs.

Figure 1

Physical diagram of the flow geometry

2 Analysis

A uniform magnetic field of force B_o is imposed in the y–direction according to Ref. [8] (see Fig. 1). For the current study, the basic steady conservation of mass, momentum, thermal energy and nanoparticles equations becomes:

subject to the boundary conditions:

Where, u and v are the velocity components along the axes x and y, respectively, ρ_f is the density of the base fluid, ν is the kinematic viscosity, σ* is the electrical conductivity, p is the fluid pressure, α=kf(ρc)f is the thermal diffusivity, D_B is the Brownian diffusion coefficient, D_T is the thermophoretic diffusion coefficient, τ=(ρc)p(ρc)f  is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid with ρ being the density, c is the volumetric volume expansion coefficient and ρ_p is the density of the particles, c_p is the fluid specific heat at constant pressure, μ_f is the viscosity of the base fluid and s is suction (or injection) parameter, respectively. T is the temperature of the fluid, C is the fraction of the volume of nanoparticles, T_w is the temperature of the stretching surface, C_w is the fraction of the volume of nanoparticles on the stretching surface, T_∞ is the ambient temperature and C_∞ is the fraction of the volume of ambient nanoparticles. Under the related work [10].

Where ψ is a stream function provided by

Well then, Eq. (1) identically satisfied. For converting Eqs. (1) – (5) with the boundary conditions (6) into the following nonlinear ordinary differential equations a similarity solution in Ref. [10] was implemented.

with the boundary conditions:

Where primes indicate differentiation for η and Pr, Nb, Nt, Ec, M, and Le are Prandtl number, Brownian motion parameter, thermophoresis parameter, Eckert number, magnetic parameter, and Lewis number, respectively. The physical parameters below are described by:

Here, gravitational acceleration, volumetric expansion coefficient of the fluid, nanoparticle volume fraction at the surface, ambient nanoparticle volume fraction attained as y tends to be infinite, and local Rayleigh number, respectively, are also the symbols g, β, ϕ_w, ϕ_∞, and Ra_x. In addition, f, θ, and ϕ are the dimensionless of the stream function, temperature, and volume of nanoparticles respectively. It distinguishes the local skin friction C_f, the reduced Nusselt number Nur and the reduced Sherwood number Shr^[10]

where, Re_x is the local Reynolds number based on the stretching velocity u_w(x).

3 Results and Discussion

3.1 Validation of the numerical solution

The results for the local skin friction f″(0) are compared with those obtained by Abd Elazem [19] for different values of Pr, s, and η_max in Table 1. It is noticed that the comparison shows excellent agreement for each values of Pr, s, and η_max. Also, dimensionless similarity functions f (η), θ(η) and ϕ(η) are matching with the previously published [19] as shown in Fig. 2. Therefore, it is confident that the present results are very accurate.

Figure 2

Comparison of the proposed study with previous work published by Abd Elazem [19] in the case of Pr = Le = 1, Nb = Nt = 0.1, s = 0.1, and M = Ec = 0

Table 1

Comparison test results for local skin friction f″(0) when (a) Nb = Nt = 0.1, Le = 10, M = Ec = 0.0, and s = 1 at different values of Pr

(b) Nb = Nt = 0.1, Pr = Le = 10, and M = Ec = 0.0 at different values of s

(c)Nb = Nt = 0.1, Pr = 10, Le = 100, M = Ec = 0.0, and s = 5 at different values of η_max

(a) Pr	1	3	10	103	105
Abd Elazem [19]	−1.24162	−0.591478	−0.291396	−0.10255	−0.100026
Present results	−1.24162	−0.591478	−0.291396	−0.10255	−0.100026
(b) s	−10.0	−5.0	−0.5	−5.0	−10.0
Abd Elazem [19]	−0.0645743	−0.116767	−0.270266	−0.500422	−0.825446
Present results	−0.0645743	−0.116767	−0.270266	−0.500422	−0.825446
(c) ηmax	1	3	5	10	20
Abd Elazem [19]	−1.22092	−0.615552	−0.53052	−0.500422	−0.49848
Present results	−1.22092	−0.615552	−0.53052	−0.500422	−0.49848

3.2 Results for temperature and solid volume fraction profiles

Equations (9) – (11) have been numerically solved with the boundary conditions (12) using the Chebyshev pseudospectral technique. It is found that as the distance increases from the solid boundaries, both the temperature and the concentration profiles start at unity near the wall and reach to vanish. In the case of Nt = Nb = 0.1, Pr = Le = 1, s = 0.01 and M = 1, Figure 3 serves to highlight the present numerical results for Ec = −0.05, −0.01, 0.0, 0.01 on temperature profiles. It is demonstrated that with an increase in Ec, the temperature profiles and thermal boundary layer thickness are enhanced. Physically, the ohmic heating effect due to the effects on electromagnetic work is found to be produced an increase in the fluid temperature, and thus a decrease in the surface temperature gradient. Further, it is found that the effect of viscous heating leads to an increase in the temperature; this effect is more pronounced in the presence of the magnetic filed. It is acknowledged that an increase in Nt, the temperature profile accelerates also, the temperature profile raises the elevation of the Eckert number as shown in Fig. 4 (positive Ec values correspond to wall heating, while the opposite is true for Ec negative values).

Figure 3

Effect of Ec on temperature profiles for Nb = Nt = 0.1, Pr = Le = 1, s = 0.01, and M = 1

Figure 4

Effects of Ec and Nt on temperature profiles for Nb = 0.1, Pr = Le = 1, s = 0.01, and M = 1

The temperature decreases as Pr increases. This is in agreement with the physical fact that the thermal boundary layer thickness decreases with increasing Pr. Further, according to Fig. 5, the numerical results for the profiles of θ(η) for (a) Ec = −0.01 and (b) Ec = 0.01 when Pr = 0.7, 1, 10, 10⁵ at Nt = Nb = 0.1, Le = 1, s = 0.01, and M = 1. Physically the thermal boundary-layer thickness, as predicted, is less than the boundary-layer thickness of the momentum when Pr ≫ −1. Also, the temperature function decreases (see Fig. 5). At high values of the Prandtl number Pr (values of Pr ≫ −1, decrease conductivity, and increase pure convection). Besides the typical matching of temperature profiles [19] at values (10 < Pr ≤ 10⁵). Notwithstanding this, it should also be noticed that with the rise in Ec from −0.01 to 0.01, there is a slightly greater difference in Ec = 0.01 if Pr = 0.7, 1 (see Fig. 5).

Figure 5

Effects of Ec and Pr on temperature profiles for Nb = Nt = 0.1, Pr = Le = 1, s = 0.01, and M = 1

By contrast, it is clearly shown that an increase in M and Ec increases the thermal boundary layer thickness and the temperature profiles. Physically, application of a transverse magnetic field to an electrically conducting fluid creates a resistive-type force called the Lorentz force. This conclusion meets the logic that the magnetic field exerts a retarding force on the free-convection flow in the boundary layer and increase its temperature (see Figure 6).

Figure 6

Effects of Ec and M on temperature profiles for Nb = Nt = 0.1, Pr = Le = 1, s = 0.01, and M = 1

In the case of Le > 1, the thickness of the boundary-layer for mass friction function is smaller than the thermal boundary-layer thickness (see Fig. 7). Variations in the concentration function increased with Ec growing far from the boundary. Concentration function profiles generally decrease with the rise in Lewis number as in Fig. 7. It can be seen that an increment in Lewis number decreases the solid volume fraction of nanofluid profiles. Physically, This is due to the fact that mass transfer rate of nanofluid increases as Lewis number increases. It also reveals that the concentration gradient at surface of the stretching sheet increases. Moreover, the concentration at the surface of stretching sheet decreases as Lewis number increases. Finally, Figure 8 shows the variation of Concentration profiles with Nb and Ec when Nt = 0.1, Pr = 10, Le = 1.5, s = 0.01, and M = 3. It's clear that the thickness of the boundary layer for mass friction function decreased as Nb increased. Besides, the concentration profiles decrease with the increase of the Eckert number. Physically, the Brownian motion parameter helps to measure the strength of the Brownian diffusion of the nanoparticles in the flow field. Due to the Brownian diffusion, the nanoparticles tend to move away from the surface of the sheet and as result a decrease in nanoparticle volume fraction is encountered within the boundary layer region.

Figure 7

Effects of Ec and Le on concentration profiles for Nt = Nb = 0.1, Pr = 1, s = 0.01, and M = 1

Figure 8

Effects of Ec and Nb on concentration profiles for Nt = 0.1, Pr = 10, Le = 1, s = 0.01, and M = 1

4 Conclusion

The effects of various physical parameters on nanofluids that flow past a stretching surface were explored. This study is very important for engineers and researchers in nearly every branch in engineering and science. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. Numerically a system of nonlinear ordinary differential equations was solved using Chebyshev's pseudospectral technique at specified physical parameters. From previous results, it can be concluded that:

1. It has been found that the present results show that the reduced number of Nusselt is a decreasing function at a fixed value of Ec = 0.01, while the reduced number of Sherwood is an increasing function for variation of Nt with Nb, at Pr = Le = 10, M = 1, and s = 0.01.

2. It has been noted that the current results indicate that the reduced number of Nusselt and the reduced number of Sherwood are monotonous functions for variation of M with Ec when Nb = Nt = 0.1, Pr = Le = 1, and s = 0.01.

3. The magnetic field, Joule heating, Eckert number and thermophoresis parameter improves the temperature field and increases the thermal boundary layer thickness while the Prandtl number Pr reduces the temperature field and decreases the thermal boundary layer thickness where the Prandtl number Pr does not affect the temperature profile (10 ≺ Pr ≤ 10⁵).

4. Changing the parameter Nb as known at values (0.1, 0.3, 0.5) decreases the profile of the concentration. Furthermore, the Eckart number (from −0.01 ≤ Ec ≤ 0.01) has a direct impact on the concentration profile, where the Eckart number reduces the profile of concentration.