The suspension of solid nanoparticles such as Cu, Ag, TiO₂ or Al₂O₃ in the base fluids like ethylene glycol, oil or water is known as nanofluid. An insertion of nanometer-sized metallic particles in the base fluids leads to an increase in the thermal conductivity of the ordinary base fluids. The nanofluids are expected to have better thermal efficiency than the base fluids. Such fluids can be implemented in the processes of electronic cooling equipment, nuclear reactors, transportation, vehicle computers and transformer cooling, cancer therapy, heating and cooling process of energy conversion, etc. Choi^[1] performed an experimental study and found that the addition of nanoparticles enhanced the thermal conductivity of the base fluids. Buongiorno^[2] provided a mathematical model to explore the effects of thermophoresis and Brownian motion in the flow of nanofluid. The boundary layer flow of nanofluid induced by linearly stretching the surface was addressed by Khan and Pop.^[3] They showed that the thermal boundary layer thickness is an increasing function of both thermophoresis and Brownian motion parameters. Makinde and Aziz^[4] investigated the boundary layer flow of nanofluid past a stretching sheet with convective boundary condition. They observed that convective heating at the sheet results in higher temperature and rate of heat transfer from the sheet. Mustafa et al.^[5] discussed the two-dimensional stagnation-point flow of nanofluid towards a stretching sheet. Turkyilmazoglu^[6] studied the heat and mass transfer analysis for MHD flow of viscous nanofluid with slip effect. He provided the closed form solutions of velocity, temperature, and concentration profiles. Rashidi et al.^[7] provided the second law analysis in the steady flow of nanofluid due to a rotating porous disk. Thermal and concentration stratification effects in the boundary layer flow of viscous nanofluid by a vertical plate were addressed by Ibrahim and Makinde.^[8] Sheikholeslami et al.^[9] explored the MHD flow of nanofluid between two horizontal parallel plates in a rotating system. The flow of viscous nanofluid between the concentric cylinders is discussed by Zeeshan et al.^[10] Marangoni convection flow and heat transfer in pseudoplastic nanofluids is investigated by Lin et al.^[11] The MHD flow and heat transfer of nanofluids in porous media with variable surface heat flux, radiation and chemical reaction is examined by Zhang et al.^[12] Recently, Hayat et al.^[13] examined the three-dimensional flow of second grade nanofluid in the presence of thermal radiation, chemical reaction, and heat source/sink effects. Here, the flow is caused due to an exponential stretching sheet.

The study of boundary layer flow induced by a stretching surface has gained considerable attention due to its applications in industries and technological processes. There are several examples of such applications which include extrusion of plastic sheets, drawing of plastic films, paper production, wire drawing, hot rolling, metal spinning, and glass blowing. There are many fluids in our daily life usage like certain oils, shampoos, sugar solution, tomato paste, mud, apple sauce, soaps, blood at low shear rate, chyme, personal care products and many others which do not satisfy the Newton’s law of viscosity. Such fluids fall into the category of non-Newtonian fluids. In view of the diverse characteristics of these fluids, many models of non-Newtonian fluids have been proposed in the literature. The Powell–Eyring fluid model^[14–20] is one of such models. This model has important features because its constitutive equations can be deduced from the kinetic theory of gases rather than the empirical relation as in the power-law model. Further, it can correctly reduce to Newtonian flow behavior for low and high shear rates, whereas the power-law model describes an infinite effective viscosity for low shear rate and thus limiting its range of applicability.

In the present analysis, we considered the three-dimensional boundary layer flow of Powell–Eyring nanofluid over a bidirectional stretching surface in the presence of thermal radiation. Prescribed surface heat flux and prescribed surface mass flux conditions are utilized at the surface. To the best of the authors’ knowledge, all the previous studies discussed the heat transfer through prescribed surface heat flux condition. There is not a single study in nanofluid dynamics that dealt with both the conditions. The main purpose here is to utilize such conditions in the three-dimensional boundary layer flow of Powell–Eyring nanofluid. Mathematical formulation of the present analysis is performed subject to both prescribed surface heat flux and prescribed surface mass flux conditions. The homotopy analysis method (HAM)^[21–29] is applied for the solution’s development. Temperature and nanoparticles’ concentration profiles are examined through the plots. The local Nusselt and local Sherwood numbers are computed numerically and analyzed.

Consider the laminar steady three-dimensional boundary layer flow of an incompressible Powell–Eyring nanofluid over a linear stretching surface. The flow is caused by a bidirectional stretching surface. Brownian motion and thermophoresis are taken into account. We consider the Cartesian coordinate system in such a manner that the sheet coincides with the xy-plane and the fluid occupies the space z ≥ 0. Let U_w (x) = ax and V_w (y) = by denote the surface stretching velocities along the x- and y-directions respectively (see Fig. 1). The Rosseland’s approximation is employed for the thermal radiation expression. The prescribed heat and mass flux conditions are imposed at the boundary. The thermophysical properties of fluid are taken as constant. The governing boundary layer equations of Powell–Eyring nanofluid are

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Geometry of the problem.

The boundary conditions for the present flow analysis are

By inserting Eq. (9) in Eq. (8), we obtain

Substituting Eq. (10) into Eq. (4), we have

We now consider the following transformations:

Now, equation (1) is automatically satisfied, and equations (2)–(7) and (11) yield

In the above expressions, ε, δ₁, and δ₂ are the Powell–Eyring fluid parameters, α is the ratio parameter, Rd is the radiation parameter, Pr is the Prandtl number, N_b is the Brownian motion parameter, N_t is the thermophoresis parameter, Le is the Lewis number, and prime stands for differentiation with respect to η. These parameters can be expressed as follows:

The dimensionless form of local Nusselt number Nu_x and local Sherwood number Sh_x are given by

The initial guesses and auxiliary linear operators for homotopic solutions are

The above operators have the following properties:

Here, p denotes the embedding parameter, ħ_f, ħ_g, ħ_θ, and ħ_ϕ the non-zero auxiliary parameters and N_f, N_g, N_θ, and N_ϕ the nonlinear operators. Setting p = 0 and p = 1, we have

When p varies from 0 to 1 then , , , and vary from the initial guesses f₀ (η), g₀ (η), θ₀ (η), and ϕ₀ (η) to the final solutions f(η), g(η), θ (η), and ϕ (η), respectively. The Taylor series expansion gives the following expressions:

The convergence of the above series strongly depends upon ħ_f, ħ_g, ħ_θ, and ħ_ϕ. Considering that ħ_f, ħ_g, ħ_θ, and ħ_ϕ are chosen in such a manner that Eqs. (36)–(39) converge at p = 1 then we have

The mth-order problems have the solutions

Equations (40)–(43) depend upon the non-zero auxiliary parameters ħ_f, ħ_g, ħ_θ, and ħ_ϕ which play a vital role in controlling and adjusting the convergence of the series solutions. For this purpose, we sketched the ħ-curves at 21st order of approximations that give us the proper ranges of these auxiliary parameters. Figures 2 and 3 show that the suitable ranges of these auxiliary parameters are − 1.6 ≤ ħ_f ≤ − 0.1, − 1.5 ≤ ħ_g ≤ − 0.1, − 1.4 ≤ ħ_θ ≤ − 0.2, and − 1.35 ≤ ħ_ϕ ≤ − 0.7. Further, the presented series solutions are convergent in the whole domain of η when ħ_f = − 0.8 = ħ_g and ħ_θ = − 0.9 = ħ_ϕ. Table 1 shows that the 34th order of deformations are sufficient for the convergent series solutions.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. The ħ-curves for the functions f(η) and g(η).

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. The ħ-curves for the functions θ (η) and ϕ(η).

Table 1.

Table 1.

Table 1. Convergence of series solutions for different order of deformations when Rd = ε = 0.3 = δ1 = Nt, Pr = 1.0 = Le, and α = 0.2 = δ2 = Nb. . Order of approximations −f″(0) −g″(0) −θ ″(0) ϕ ″(0) 1 0.9246 0.1388 0.0650 0.1600 5 0.9247 0.1303 0.2969 0.4692 10 0.9247 0.1303 0.3614 0.5519 15 0.9247 0.1303 0.3766 0.5662 25 0.9247 0.1303 0.3834 0.5752 34 0.9247 0.1303 0.3844 0.5766 45 0.9247 0.1303 0.3844 0.5766 60 0.9247 0.1303 0.3844 0.5766

Table 1. Convergence of series solutions for different order of deformations when Rd = ε = 0.3 = δ1 = Nt, Pr = 1.0 = Le, and α = 0.2 = δ2 = Nb. .

The effects of emerging physical parameters namely fluid parameter ε, ratio parameter α, radiation parameter Rd, Prandtl number Pr, thermophoresis parameter N_t, and Brownian motion parameter N_b on the temperature profile θ (η) are shown in Figs. 4–9. Influence of fluid parameter ε on the temperature profile is analyzed in Fig. 4. From Fig. 4, it is noted that the temperature profile θ (η) and thermal boundary thickness are reduced when we increase the values of fluid parameter. Figure 5 shows the impact of ratio parameter α on the temperature profile θ (η). It can be seen that an increase in the ratio parameter reduces the temperature profile θ (η). From Fig. 6, it is observed that the temperature θ (η) and thermal boundary layer thickness are enhanced by the increase in radiation parameter Rd. For Rd = 0, there is no thermal radiation effects. Hence, the presence of thermal radiation enhances the temperature and thermal boundary layer thickness. An increase in Prandtl number Pr is observed from Fig. 7. The larger Prandtl number has a relatively lower thermal diffusivity. Therefore, a rapid increase in the Prandtl number Pr reduces the temperature and thermal boundary layer thickness. The influences of thermophoresis and Brownian motion parameters on the temperature profile θ (η) are shown in Figs. 8 and 9. An increase in the thermophoresis and Brownian motion parameters gives rise to the temperature and thermal boundary layer thickness. The presence of nanoparticles greatly enhances the thermal conductivity of the fluid. Here, the thermal conductivity of fluid is enhanced when we increase the values of thermophoresis and Brownian motion parameters. Such increase in thermal conductivity is responsible for the higher temperature and thicker thermal boundary layer thickness.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Impact of ε on θ(η).

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Impact of α on θ(η).

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. Impact of Rd on θ(η).

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. Impact of Pr on θ(η).

Fig. 8.

	Figure Option ViewDownloadNew Window
	Fig. 8. Impact of Nt on θ(η).

Fig. 9.

	Figure Option ViewDownloadNew Window
	Fig. 9. Impact of Nb on θ (η).

Figures 10–15 are sketched to explore the behavior of fluid parameter ε, ratio parameter α, Lewis number Le, Prandtl number Pr, thermophoresis parameter N_t, and Brownian motion parameter N_b on the nanoparticles concentration profile ϕ (η). From Fig. 10, it is noticed that the larger values of fluid parameter ε reduce the nanoparticles concentration field ϕ (η) and its related boundary layer thickness. It is clearly shown that both nanoparticles’ concentration ϕ (η) and its associated boundary layer thickness are reduced for larger values of ratio parameter α (see Fig. 11). Figure 12 is drawn to see the impact of Lewis number Le on the nanoparticles’ concentration profile ϕ (η). The Lewis number is inversely proportional to the Brownian diffusion coefficient due to which an increase in the Lewis number Le yields a decrease in Brownian diffusion coefficient, which results in a decrease in nanoparticles’ concentration and its related boundary layer thickness. Nanoparticles’ concentration profile ϕ (η) for different values of Prandtl number Pr is shown in Fig. 13. It is seen that the nanoparticles’ concentration profile ϕ (η) is a decreasing function of Prandtl number Pr. Figure 14 illustrates the variation of nanoparticles’ concentration ϕ (η) for various values of thermophoresis parameter N_t. It is examined that the nanoparticles’ concentration boundary layer thickness increases with the increasing value of N_t. Figure 15 displays the effect of Brownian motion parameter N_b on nanoparticles’ concentration ϕ (η). Large N_b leads to an increase in the nanoparticles’ motion and consequently the viscosity of nanofluid decreases. That is why the nanoparticles’ concentration and its related boundary layer thickness reduces.

Fig. 10.

	Figure Option ViewDownloadNew Window
	Fig. 10. Impact of ε on ϕ (η).

Fig. 11.

	Figure Option ViewDownloadNew Window
	Fig. 11. Impact of α on ϕ (η).

Fig. 12.

	Figure Option ViewDownloadNew Window
	Fig. 12. Impact of Le on ϕ (η).

Fig. 13.

	Figure Option ViewDownloadNew Window
	Fig. 13. Impact of Pr on ϕ (η).

Fig. 14.

	Figure Option ViewDownloadNew Window
	Fig. 14. Impact of Nt on ϕ (η).

Fig. 15.

	Figure Option ViewDownloadNew Window
	Fig. 15. Impact of Nb on ϕ (η).

The numerical values of −f″(0), −g″(0), −θ″(0), and ϕ″(0) at different orders of approximations when Rd = ε = 0.3 = δ₁ = N_t, δ₂ = α = 0.2 = N_b, Le = 1.0 = Pr, ħ_f = − 0.8 = ħ_g, and ħ_θ = − 0.9 = ħ_ϕ are computed in Table 1. This table depicts that the values of − f″(0), −g″(0), −θ″(0), and ϕ″(0) started to repeat from 34th order of deformations. Therefore, 34th order of approximations are essential for convergent series solutions. Table 2 presents the comparison for various values of ratio parameter with homotopy perturbation method (HPM) and exact solutions. Table 2 shows an excellent agreement of HAM solutions with the existing homotopy perturbation method (HPM) and exact solutions in a limiting sense. This confirms the validity of HAM solutions. Table 3 shows the numerical values of local Nusselt and local Sherwood numbers for various values of δ₁, δ₂, ε, α, Rd, Le, Pr, N_t, and N_b. We have seen that the values of local Nusselt and Sherwood numbers are higher when the larger values of ratio parameter α and radiation parameter Rd are taken into account. An increase in the values of thermophoresis parameter N_t leads to lesser values of local Nusselt and Sherwood numbers.

Table 2.

Table 2.

Table 2. Comparative values of −f″(0) and −g″(0) for various values of α when ε = δ1 = δ2 = 0. . α −f″(0) −g″(0) HAM HPM[30] Exact[30] HAM HPM[30] Exact[30] 0 1 1 1 0 0 0 0.1 1.020260 1.017027 1.020260 0.066847 0.073099 0.066847 0.2 1.039495 1.034587 1.039495 0.148737 0.158231 0.148737 0.3 1.057955 1.052470 1.057955 0.243359 0.254347 0.243360 0.4 1.075788 1.070529 1.075788 0.349209 0.360599 0.349209 0.5 1.093095 1.088662 1.093095 0.465205 0.476290 0.465205 0.6 1.109947 1.106797 1.109947 0.590529 0.600833 0.590529 0.7 1.126398 1.124882 1.126398 0.724532 0.733730 0.724532 0.8 1.142489 1.142879 1.142489 0.866683 0.874551 0.866683 0.9 1.158255 1.160762 1.158254 1.016540 1.022922 1.016539 1.0 1.173722 1.178511 1.173721 1.173722 1.178511 1.173721

Table 2. Comparative values of −f″(0) and −g″(0) for various values of α when ε = δ1 = δ2 = 0. .

Table 3.

Table 3.

Table 3. Values of local Nusselt number (Rex)−1/2Nux and local Sherwood number (Rex)−1/2Sux for different values of the parameters ε, δ1, δ2, α, Rd, Le, Pr, Nt, and Nb. . ε δ1 δ2 α Rd Le Pr Nt Nb (Rex)−1/2Nux (Rex)−1/2Sux 0.0 0.3 0.2 0.2 0.3 1.0 1.0 0.3 0.2 0.4423 0.3485 0.5 0.4956 0.3862 1.2 0.5390 0.4181 0.3 0.1 0.2 0.2 0.3 1.0 1.0 0.3 0.2 0.4859 0.3680 0.3 0.4777 0.3733 0.5 0.4770 0.3728 0.3 0.3 0.1 0.2 0.3 1.0 1.0 0.3 0.2 0.4773 0.3732 0.8 0.4773 0.3732 1.4 0.4773 0.3732 0.3 0.3 0.2 0.0 0.3 1.0 1.0 0.3 0.2 0.4001 0.3352 0.4 0.5429 0.4047 0.6 0.6010 0.4323 0.3 0.3 0.2 0.2 0.0 1.0 1.0 0.3 0.2 0.4093 0.3731 0.5 0.5158 0.3734 0.8 0.5496 0.3736 0.3 0.3 0.2 0.2 0.3 0.9 1.0 0.3 0.2 0.4745 0.3431 1.4 0.4880 0.4860 1.6 0.4922 0.5387 0.3 0.3 0.2 0.2 0.3 1.0 0.7 0.3 0.2 0.3932 0.2803 0.9 0.4520 0.3431 1.5 0.5761 0.5133 0.3 0.3 0.2 0.2 0.3 1.0 1.0 0.2 0.2 0.5356 0.4328 0.5 0.4184 0.3325 0.7 0.3575 0.3044 0.3 0.3 0.2 0.2 0.3 1.0 1.0 0.3 0.2 0.5762 0.3132 0.4 0.4256 0.5628 0.5 0.3619 0.5980

Table 3. Values of local Nusselt number (Rex)−1/2Nux and local Sherwood number (Rex)−1/2Sux for different values of the parameters ε, δ1, δ2, α, Rd, Le, Pr, Nt, and Nb. .

The three-dimensional (3D) boundary layer flow of Powell–Eyring nanofluid in the presence of thermal radiation over a bidirectional stretching surface is investigated. Heat and mass flux conditions are imposed for the present analysis. The main points of this research are listed as follows.