It is generally an accepted fact that non-Newtonian fluids are more industry oriented as compared to Newtonian fluids. Glue, coal water, custard, ketchup, inks, cosmetics, toothpaste and jellies are some examples of non-Newtonian fluids. Unlike Newtonian fluids, no single relation can be predicted for non-Newtonian fluids as each non-Newtonian fluid possesses varying properties of viscosity and elasticity, which makes the mathematical modeling of these fluids more complicated as compared to Newtonian fluids. Many researchers are involved in exploring the new dimensions in this area.^[1–5] Each non-Newtonian fluid model holds different features to exhibit a physical phenomenon. Eyring Powell fluid model^[6–8] is one such model that can be extracted from kinetic theory of gases instead of Power law model. However, at low and high sheer rates, it exhibits the Newtonian behavior rather than pseudo-plastic systems’ behavior. At different polymer concentrations, Eyring Powell model is considered to be more precise and reliable in estimating the fluid time scale.^[9]

In the present era of industrial revolution, Choi’s pioneering work^[10] in nanofluids characterized by their ability to upsurge the thermal conductivity of base liquid, has opened the gates for followers to work in new dimensions. Choi found that heat transfer rate can be doubled by adding a small amount of nano particles. The novel characteristics of thermophoresis and Brownian motion of such fluids also make them potentially practical. Applications of nanofluids include cooling of micro chips, nano-drug delivery, and cancer therapy. Many researchers are exploring new dimensions and adding valuable contributions toward nanofluids. Some of these include exploration by Sohail and Saleem^[11] that explores a time dependent flow on a rotating cone of Eyring Powell nanofluid under the impact of mixed convection. Waqar et al.^[12] studied three-dimensional heat generation/absorption flow of an Oldroyd-B nanofluid. Khan et al.^[13] focused 3D nanofluid flow in lateral directions over a nonlinearly stretched sheet. Jalilpour et al.^[14] focused MHD stagnation point nanofluid flow in the presence of prescribed heat flux and heat generation/absorption past a porous stretching sheet. In recent literature, various dimensions have been explored in the presence of nanofluids.^[15–20]

In general, four heating processes are available in the literature namely (i) prescribed surface heat flux, (ii) prescribed wall temperature, (iii) Newtonian heating, and (iv) conjugate/convective boundary conditions. In today’s modern world, demand for compact and light weight devices for technological and engineering machinery urge researchers to explore more avenues in heat transfer equipment with enhanced efficiency. Due to this increasing demand of small and light weight units, researchers have been focusing on effects on interface between axial wall conduction and thermal boundary layers in fluids that directly influence the heat exchange performances. Convective boundary condition is the generalized concept of prescribed surface temperature and prescribed heat flux conditions. Convective boundary condition is derived from the amalgamation of Newton’s law of cooling and Fourier’s law of heat conduction. It can be reduced to both prescribed surface temperature and heat flux conditions by making Biot number tends to infinity and zero respectively. Many researchers are exploring new mathematical models involving convective heat condition. To mention few amongst these, the study by Ibrahim^[21] who examined the magnetohydrodynamic flow of nanofluid near the stagnation point under the influence of convective boundary condition. Ibanez^[22] studied MHD flow past a channel with permeable walls in attendance of convective and hydrodynamic slip boundary conditions. Mustafa et al.^[23] addressed the radiative Maxwell fluid flow with impact of convective condition. Recently, a variety of alluring problems highlighting effects of convective boundary condition are discussed, see Refs. [24–27].

In above referred studies, none of these have disused combined effects of non-linear thermal radiation and variable thermal conductivity with amalgamation of chemical reaction. This has motivated us to study the problem of Eyring Powell nanofluid flow in presence of heat and mass convective boundary conditions past a continuously moving surface. No such study has been carried out till now in the literature as far as our knowledge is concerned. Series solutions have been obtained using famed Homotopy Analysis method (HAM).^[28–31] Partial differential equations with high nonlinearity are changed into nonlinear ordinary differential equations using appropriate transformation. Graphs of Skin friction coefficient, local Nusselt and Sherwood numbers with mandatory conversation versus various parameters are also added.

We assume 2D steady flow of an incompressible Eyring Powell nanofluid past a surface moving with constant velocity u_w. Both constant velocity u_w and uniform free stream velocity have the same direction. Wall temperature T_w and free stream temperature are constant with the assumption in attendance of convective heat and mass boundary conditions (See Fig. 1).

Fig. 1

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	Fig. 1 Flow diagram.

The Cauchy stress tensor in an Eyring–Powell model^[6] is governed by the relation:

Considering variable thermal conductivity with as defined in Ref. [32]. The nonlinear radiative heat flux q_r via Rosseland’s approximation is

The case λ = 0 relates to the flow over an immobile surface because of free stream velocity. However, λ = 1 points out the moving plate in the fluid. The flow of fluid and plate moving in the same direction is represented by the case . Here, we consider the case .

The skin friction coefficient , local Nusselt number Nu_x, and Sherwood number Sh_x can be written as:

In non-dimensional forms Skin friction, local Nusselt and Sherwood numbers are

Liao^[36] suggested Homotopy Analysis method in 1992 for the construction of series solutions of differential equations with high nonlinearity. This method has advantages over the contemporary methods because of the following reasons:

(i) This method is independent of selection of large or small parameters.

(ii) Convergence of series solution in this method is guaranteed.

(iii) Ample choice for the selection of base functions and linear operatos is available in this method.

Initial guess estimates required for series solutions in Homotopy analysis method are defined as:

3.1 Zeroth-Order Problem

The problem at zeroth order is assembled as

with nonlinear operators , , and are given by

Here, non-zero auxiliary parameters are , , and , with p is an embedding parameter such that . Having values p = 0 and p = 1, we get

If we change p from 0 to 1, the values of the functions and will fluctuate from initial guesses , , to the final solutions , , respectively. With the help of Taylor’s series, Eqs. (26) to (28) take the form

The value of auxiliary parameters , , and are selected in such a manner that the series (29)–(31) converge at p = 1, i.e.,

Series solution requires convergence region. Auxiliary parameters , , and play a key role in achieving this goal. To find this region, Fig. 2 is drawn to show -curves when ϵ = 0.4, δ = 0.4, λ = 0.4, Pr = 1, Nt = 0.8, Nb = 0.2, Rd = 0.4, Rc = 0.3, , α = 0.2, and Le = 1.0. The endurable ranges of the auxiliary parameters and are , and respectively. Table 1 shows the convergence of HAM solution. It is observed that 29^th order of approximation is good enough for convergent series solution. The values obtained in the Table are consistent with the tolerable ranges in the graph.

Fig. 2

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	Fig. 2 -curves of f, θ, and ϕ.

Table 1

Table 1

Table 1 Series solutions’ convergence for varied order of approximations when ϵ = 0.4, δ = 0.4, λ = 0.4, Nb= 0.2, Le = 1.0, , Nt = 0.8, Pr = 1.0, Rd = 0.4, Rc = 0.3, , α = 0.2, , and . . Order of approximations 1 0.110 39 0.086 39 0.077 65 5 0.071 16 0.078 46 0.073 37 10 0.066 38 0.074 97 0.076 93 15 0.065 68 0.073 71 0.078 79 20 0.065 56 0.073 24 0.079 52 25 0.065 54 0.073 15 0.079 67 29 0.065 54 0.073 12 0.079 72 30 0.065 54 0.073 12 0.079 72

Table 1 Series solutions’ convergence for varied order of approximations when ϵ = 0.4, δ = 0.4, λ = 0.4, Nb= 0.2, Le = 1.0, , Nt = 0.8, Pr = 1.0, Rd = 0.4, Rc = 0.3, , α = 0.2, , and . .

This segment emphasis on the discussion of graphical illustrations of different emerging parameters on all dimensionless distributions.

Figures 3 and 4 illustrate the impact of fluid parameter ϵ on the velocity field for two separate values of λ. It is clear from Fig. 3 that velocity and momentum boundary layer thickness escalate with growing values of ϵ, when λ = 1 (Sakiadis flow).^[32] However, the velocity decreases with increase in value of ϵ for λ = 0.4. The impact of variable thermal conductivity parameter α on temperature field is displayed in Fig. 5. It is clear from the figure that temperature distribution shows increasing tendency when value of α is increased. This is because of an accepted fact that liquids with higher thermal conductivity possesses higher temperature. The effects of Biot numbers γ₁ and γ₂ with , on temperature and concentration distributions have been portrayed in Figs. 6 and 7. Here, it is observed that temperature increases rapidly for but a low increment is witnessed for . It means that for smaller values of heat transfer Biot numbers , temperature profile increases rapidly but for larger values of , the temperature profile increases slowly. Similar behavior in case of mass transfer Biot number γ₂ is witnessed in Fig. 7. figure 8 is drawn to illustrate the consequence of Prandtl number Pr on temperature distribution. From the definition of Prandtl number , we see that for higher Prandtl number, thermal diffusivity must have a smaller value. So, a gradual upsurge in Prandtl number results in reduction in boundary layer thickness and temperature profile. From Fig. 9, it is obvious that an increase in Brownian motion parameter Nb, enhances collision of particles and thus boosts the temperature profile and its related boundary layer thickness. Effect of thermophoresis parameter Nt on temperature field is shown in Fig. 10. It is observed that an upsurge in values of Nt leads to an increase in the temperature profile and its allied boundary layer thickness.

Fig. 3

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	Fig. 3 Impact of ϵ on when λ = 1.0.

Fig. 4

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	Fig. 4 Impact of ϵ on when λ = 0.4.

Fig. 5

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	Fig. 5 Impact of α on .

Fig. 6

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	Fig. 6 Impact of γ1 on .

Fig. 7

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	Fig. 7 Impact of γ2 on .

Fig. 8

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	Fig. 8 Impact of Pr on .

Fig. 9

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	Fig. 9 Impact of Nb on .

Fig. 10.

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	Fig. 10. Impact of Nt on .

From Fig. 11, it is evident that temperature profile is growing function of non-linear thermal radiation parameter Rd. Actually, increasing values of Rd boosts the thermal boundary thickness. This is due to the fact that increase in thermal radiation parameter results in decrease in mean absorption coefficient, which eventually upsurges the divergence of the radiative heat flux. That is why the fluid’s temperature increases due to increase in rate of radiative heat transfer. Figure 12 is portrayed to show the effects of chemical reaction parameter Rc varying from non-destructive () to destructive () on concentration distribution. The solute concentration diminishes because of destructive chemical reaction; this eventually decreases the solutal boundary layer thickness by a small amount and the negative wall slope of the concentration distribution. An opposing impact is observed in case of non-destructive chemical reaction.

Fig. 11.

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	Fig. 11. Impact of Rd on .

Fig. 12.

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	Fig. 12. Impact of Rc on .

Figure 13 depicts the effects of fluid parameters δ and ϵ of Skin friction coefficient. Observations show that Skin friction coefficient escalates for higher values of ϵ. However, opposite behavior is witnessed in case of δ. The influence of thermophoresis parameter Nt and Brownian motion parameter Nb on Nusselt number is displayed in Fig. 14. For growing values of both Nb and Nt, decrease in Nusselt number is perceived. For growing values of Lewis number Le and Prandtl number Pr, Sherwood number shows an increasing tendency. This effect is shown in Fig. 15.

Fig. 13.

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	Fig. 13. Impact of ϵ and δ on .

Fig. 14.

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	Fig. 14. Impact of Nb and on .

Fig. 15.

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	Fig. 15. Impact of Le and Pr on .

Table 2 depicts the comparison of existing series solutions with Hayat et al.^[35] and Aziz^[36] in limiting case. An outstanding agreement is found amongst the three solutions.

Table 2.

Table 2.

Table 2. Comparison of values for with those of Hayat et al.[35] and Aziz[36] for different values in absence of nanofluid and . . Pr Present Ref. [35] Ref. [36] 0.1 0.05 0.0594 0.0594 0.0373 0.10 0.0594 0.0594 0.0594 0.20 0.848 0.848 0.848 0.40 0.1076 0.1076 0.1076 0.60 0.1182 0.1182 0.1182 0.80 0.1243 0.1243 0.1243 1.00 0.1283 0.1283 0.1283 0.72 0.05 0.0428 0.0428 0.0428 0.10 0.0747 0.0747 0.0747 0.20 0.1193 0.1193 0.1193 0.40 0.1700 0.1700 0.1700 0.60 0.1981 0.1981 0.1981 0.80 0.2159 0.2159 0.2159 1.00 0.2282 0.2282 0.2282

Table 2. Comparison of values for with those of Hayat et al.[35] and Aziz[36] for different values in absence of nanofluid and . .

Effects of convective boundary conditions on time independent boundary layer Eyring Powell nanofluid flow past a constantly moving surface in attendance of free stream velocity is discussed. Effects of non linear thermal radiation, chemical reaction, and variable thermal conductivity are also taken into account. Series solution for the said problem is obtained using Homotopy Analysis method (HAM). The prominent outcomes of this problem are: •

Temperature field is declining function of Pr.

The authors declare no competing interests.