An innovative technique, using a mixture of nanoparticles and a base fluid, was first introduced by Choi^[1]. The fluid, which consists of solid nanoparticles and a base pure fluid, is known as a nanofluid. Nanofluids are of great interest in enhancing the thermal conductivity and convective heat transfer coefficient. Buongiorno^[2] developed an analytical model for the convective transport in nanofluids by considering the Brownian diffusion and thermophoresis. He explained the abnormal convective heat transfer enhancement observed in the nanofluids, and concluded that the Brownian diffusion and thermophoresis were the most important nanoparticle/base fluid slip mechanisms. Nanomaterials are more effective in micro/nano electromechanical devices, advanced cooling systems, and large scale thermal management systems via evaporators, heat exchangers, and industrial cooling applications. Such fluids are very stable without extra issues of erosion, sedimentation, non-Newtonian properties, and additional pressure drops. This is because of the tiny size and the low volume fraction of the nanoparticles required for the thermal conductivity enhancement. Nanofluids in the presence of a magnetic field are important in many applications such as optical modulators, magneto-optical wavelength filters, tunable optical fibre filters, and optical switches. Magneto nanofluids are especially useful in biomedicine, sink float separation, cancer therapy, etc. Magneto nanofluids are useful to guide the particles up the blood stream to a tumor with magnets. This is due to the fact that the magnetic nanoparticles are regarded to be more adhesive to tumor cells than non-malignant cells. Such particles absorb more power than microparticles in alternating the current magnetic fields tolerable in humans, e. g. , cancer therapy. Numerous applications involving nanofluids include drug delivery, hyperthermia, and contrast enhancement in magnetic resonance imaging and magnetic cell separation. The flows of nanofluids under different aspects have seldom been investigated^{[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]}.

The fluid flow over a stretching surface has gained the attention of many researchers due to its important applications in engineering processes, e. g. , polymer extrusion, drawing of plastic films and wires, glass fiber and paper production, manufacture of foods, crystal growing, and liquid films in condensation process. The flow induced by a surface stretched with a linear velocity has been widely studied. However, such consideration is not realistic when the process like the plastic sheet production is considered^[15], where a nonlinear stretching surface velocity is involved. To our knowledge, the flow and heat transfer characteristics over a nonlinearly stretching sheet have seldom been studied^{[16, 17, 18, 19, 20, 21]}. It is quite realistic, because in the plastic production and metallurgical processes, the surface velocity is always nonlinear. The rate of the heat transfer between the stretching surface and the fluid flow is important for the desired quality of the end product. Moreover, in a no-slip flow, the flow velocity is zero at the solid-fluid interface, whereas the flow velocity at the solid wall in the presence of a slip is non-zero. The fluids exhibiting the boundary slip have important technological applications such as polishing of the artificial heart valves and internal cavities. With a slip at the wall, the flow behavior and the shear stress in the fluid are quite different from the no-slip flow. The fluid flow and heat and mass transfer at the micro-scale have been widely studied^{[22, 23, 24, 25, 26]}. This is due to the wide applications in micro-electro-mechanical systems (MEMS). Such systems have relevance to the consideration of the velocity slip and the temperature and concentration jumps.

In the present study, we investigate the flow and heat and mass transfer phenomena over a nonlinearly stretching sheet with the slip effects. An incompressible viscous and electrically conducting nanofluid is considered. The problems are solved by the homotopy analysis method^{[27, 28, 29, 30, 31, 32, 33]}. The effects of different parameters on the velocity, temperature, and concentration profiles are sketched and analyzed. In addition, the heat and mass transfer rates are examined.

Let us consider the steady two-dimensional flow of a nanofluid towards a nonlinear stretching surface. The x-axis is taken along the stretching surface in the motion direction, and the y-axis is perpendicular to it. A transverse magnetic field acts in a transverse direction to the flow. It is assumed that the induced magnetic and electric field effects are negligible. The effects of the Brownian motion and thermophoresis are presented. The boundary layer flow is governed by the following equations:

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The subjected boundary conditions are

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where u and v are the velocity components along the x- and y-directions, respectively. n is the power-law index. ν, ρ, and σ are the kinematic viscosity, the density, and the electrical conductivity of the fluid, respectively. T , T_∞, C_w, and C_∞ are the fluid temperature, the ambient fluid temperature, the constant wall concentration, and the ambient fluid concentration, respectively. τ = (ρc)_p/(ρc)_f is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the fluid, DB is the Brownian diffusion coefficient, D_T is the thermophoretic diffusion coefficient, α = k/(ρc)_f is the thermal diffusivity, β₁ is the slip constant, β₂ is the thermal slip parameter, and β₃ is the concentration slip parameter.

We now introduce the following similarity transformations:

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Now, Eq. (1) is satisfied automatically, and Eqs. (2)-(5) after using Eq. (6) can be reduced as follows:

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where is the Hartman number, is the Prandtl number, is the Brownian motion parameter, is the thermophoresis parameter, is the Schmidt number, is the local slip parameter, is the local thermal slip parameter, and is the local concentration slip parameter.

The important interested physical quantities in this problem are the local skin-friction coefficient C_f , the local Nusselt number Nu_x, and the local Sherwood number Sh_x, which are defined by

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where the surface shear stress τ_w, the wall heat flux q_w, and the wall mass flux h_m are given by

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By using the above equations, we get

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where Re_x = u_wx/ν is the local Reynolds number.

The initial approximations f₀(η), θ₀(η), and φ₀(η) and the auxiliary linear operators L_f , Lθ, and Lφ are taken as follows:

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with

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where c_i (i = 1, 2, · · · , 7) are the constants.

If p ∈ [0,1] indicates the embedding parameter, then the zeroth-order deformation problems are constructed as follows:

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where are the nonzero auxiliary parameters, and the nonlinear operators N_f , N_θ, and N_φ are given by

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When p = 0 and p = 1, we have

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When p varies from 0 to 1, approach from f₀(η), θ₀(η), and φ₀(η) to f(η), θ(η), and φ(η), respectively. According to the Taylor series, we have

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where the convergence depends upon . By proper choices of , the series (25) converges for p = 1. Therefore, we have

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The resulting problems at this order can be presented as follows:

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The general solutions (f_m, θ_m, φ_m) comprising the special solutions

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Now, the solutions of Eqs. (7)-(9) subject to the boundary conditions (10) are computed by means of the homotopy analysis method. The convergence of the series solutions is highly dependent upon the auxiliary parameters . For valid ranges of these parameters, we have sketched the h-curves at the 15th-order of approximations (see Fig. 1). We can see that the admissible values of are , and , respectively. Further, the series solutions converge in the whole region of when h_f = 0. 7, h_θ = −1. 4, and h_φ = −1. 6.

Fig. 1 Combined h-curves for velocity, temperature, and concentration

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Figures 2-4 exhibit the dimensionless velocity profiles for different values of the exponent n, the Hartman number Ha, and the velocity slip parameter γ . The effects of the exponent n on the velocity profile f′ can be seen in Fig. 2. It is shown that the values of f′ and the boundary layer thickness decrease when n increases. The effects of the Hartman number Ha on the velocity f′ are depicted in Fig. 3. The graph shows that the values of the velocity function f′ and the boundary layer thickness decrease when Ha increases. Since the magnetic field has the tendency to slow down the movement of the fluid particles, the velocity and the momentum boundary layer thickness show a tendency to decrease. From Fig. 4, we can see that larger values of the slip parameter γ correspond to the lower velocity. In fact, with an increase in the velocity slip parameter, the stretching velocity is partially transferred to the fluid. Therefore, the velocity profile decreases.

Fig. 2 Effects of n on f′

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Fig. 3 Effects of Ha on f′

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Fig. 4 Effects of γ on f′

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The effects of the exponent n, the Hartman number Ha, the Prandtl number Pr, the Brownian motion parameter N_b, the thermophoresis parameter N_t, the Schmidt number Sc, the velocity slip parameter γ, the thermal slip parameter δ, and the concentration slip parameter ε on the temperature profile θ are shown in Figs. 5-9.

Fig. 5 Effects of n on θ

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Fig. 6 Effects of Ha and Pr on θ

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Fig. 7 Effects of Nt and Sc on θ

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Fig. 8 Effects of γ and δ on θ

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Fig. 9 Effects of Nb and ε on θ

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The effects of n on the temperature are analyzed in Fig. 5. It is observed that the temperature and the thermal boundary layer thickness are increasing functions of n.

Figure6 illustrates the effects of the Hartman number Ha and the Prandtl number Pr on the temperature profile θ. Here, the Lorentz force is a resistive force which opposes the fluid motion. As a consequence, the heat is produced, and the thermal boundary layer thickness increases. It is also seen that there is a decrease in the temperature θ when the Prandtl number Pr increases. In fact, the thermal diffusivity decreases when Pr increases, and thus the heat diffuses away slowly from the heated surface.

Figure7 represents the effects of the thermophoresis parameter N_t and the Schmidt number Sc on the temperature profile θ. Increasing the values of N_t leads to the increase in the temperature profile. The temperature profile θ and the thermal boundary layer thickness decrease when the Schmidt number Sc increases. This is due to the fact that an increase in Sc reduces the molecular diffusivity.

Figure8 displays the effects of the velocity slip parameter γ and the thermal slip parameter δ on the temperature field θ. The temperature field θ is found to increase when γ increases while decreases when δ increases. With an increase in the velocity slip parameter, there is less collision of particles. Therefore, the rate of the heat transfer decreases, which consequently decreases the temperature. It is depicted that the temperature and the thermal boundary layer thickness decrease when the thermal slip parameter increases. With an increase in the thermal slip parameter, the heat transfer from the sheet to the adjacent fluid decreases. Therefore, the temperature of the fluid decreases.

The effects of the Brownian motion parameter N_b and the concentration slip parameter ε on the temperature profile are displayed in Fig. 9. The Brownian motion describes the random movement of the nanoparticles in the base fluid. With an increase in the Brownian motion parameter, the random motion of the particles increases, and consequently the temperature profile increases. It is also observed that the temperature profile θ decreases when ε increases. As particles are more congested due to the difference between the plate concentration and the fluid concentration, the heat transfer becomes less, and the temperature decreases.

Figures 10-14 illustrate the effects of the exponent n, the Hartman number Ha, the Prandtl number Pr, the Brownian motion parameter N_b, the thermophoresis parameter N_t, the Schmidt number Sc, the velocity slip parameter γ, the thermal slip parameter δ, and the concentration slip parameter " on the dimensionless nanoparticle volume fraction profile φ.

Fig. 10 Effects of n and Ha on φ

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Fig. 11 Effects of Pr and Nb on φ

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Fig. 12 Effects of Nt and Sc on φ

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Fig. 13 Effects of γ and δ on φ

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Fig. 14 Effects of ε on φ

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The effects of the exponent n and the Hartman number Ha on the concentration profile are depicted in Fig. 10. It is observed that the mass fraction φ and the associated boundary layer increase when n increases while decrease when Ha increases.

The effects of the Prandtl number Pr and the Brownian motion parameter N_b on the con- centration profile are observed in Fig. 11. It is noted that the concentration profile φ decreases when the Brownian motion parameter N_b increases while increases when Pr increases.

Figure12 shows that the concentration profile increases when the thermophoresis parameter N_t increases, while the Schmidt number Sc decreases when φ increases.

The variations of the velocity slip parameter γ, the thermal slip parameter δ, and the concentration slip parameter ε on the dimensionless nanoparticle volume fraction profile φ can be seen in Figs. 13 and 14. It is noted that there is an increase in the concentration profile when γ increases. The concentration profile decreases when δ and ε increase. When the velocity slip parameter increases, the movement of the particles decreases, resulting in the increase in the concentration. Also, increasing the values of the thermal slip parameter and the concentration slip parameter leads to less mass transfer. Therefore, the concentration profile decreases.

Figure15 represents the variations of the skin friction coefficient f′′(0) with respect to the Hartman number Ha and the exponent n. The magnitude of the skin friction coefficient f′′(0) increases when n increases.

Fig. 15 Variations of skin friction coefficient with Ha for different n

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Figure16 demonstrates the variations of the local Nusselt number −θ′(0) with respect to the velocity slip parameter , the thermal slip parameter δ, and the concentration slip parameter ε. It is noticed from the graph that −θ′(0) decreases when the thermal slip parameter δ and the concentration slip parameter ε increase.

Fig. 16 Variations of local Nusselt number with γ for different δ and ε

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Figure17 illustrates the effects of the velocity slip parameter γ on the local Sherwood number φ′(0) for different values of the thermal slip parameter δ and the concentration slip parameter ". It is noticed from Fig. 17 that the local Sherwood number decreases when δ and ε increase.

Fig. 17 Variations of local Sherwood number with γ for different δ and ε

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Table1 shows the convergence of the series solutions. It is observed that the convergence for the velocity, the temperature, and the concentration is achieved at the 7th-order, the 30th-order, and the 35th-order of approximations, respectively.

Table 1 Convergence of homotopy analysis method for different orders of approximations when n = 2, Ha = 0. 3, Pr = 1, N_b = 0. 3, N_t = 0. 1, Sc = 0. 9, γ = 0. 7, δ = 0. 3, and ε = 0. 5

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In Table2, some numerical values of the skin friction coefficient are given. The tabular values show that the skin friction coefficient decreases when increases while increases when γ Ha and n increase.

Table 2 Numerical values of skin friction coefficient for different values of Ha, n, and γ

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Table3 includes the values of the local Nusselt number and the Sherwood number. It is noted that the Nusselt number decreases when Ha, N_b, N_t, Sc, γ, and δ increase while increases when n, Pr, and ε increase. However, the Sherwood number increases when n, Nb, Sc, and δ increase while decreases when Ha, Pr, N_t, γ , and ε increase.

Table 3 Numerical values of local Nusselt and Sherwood numbers for different values of Ha, n, Pr, N_b, N_t, Sc, γ, δ, and ε

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The magnetohydrodynamic (MHD) flow of a nanofluid over a nonlinearly stretching sheet is studied. The effects of different parameters on the velocity, temperature, and concentration profiles are analyzed. It is observed that the values of the temperature and concentration profiles for nonlinear stretching sheets are larger than those of the linear stretching sheets. When we increase the velocity slip parameter, the stretching velocity is partially transferred to the fluid, and consequently the velocity profile decreases. Also, the temperature and thermal boundary layer thickness decrease when the thermal slip parameter increases. This is due to the fact that increasing the values of the thermal slip parameter decreases the heat transfer from the sheet to the adjacent fluid. Less mass transfer occurs due to the increases in the thermal and concentration slip parameters, and thus the concentration profile decreases. Moreover, the magnitude of the skin friction coefficient increases when the Hartman number and the powerlaw index increase. Basically, the Hartman number yields the Lorentz force, which provides resistance to the flow. The rates of the heat and mass transfer increase when the power-law index increases. It must be noted that one can recover no slip effects when γ = δ = ε = 0.