1 Introduction

Materials obeying Newton's law of viscosity are known as the viscous fluids.There is a direct and linear relationship of such fluids between the shear stress and the shear strain rate (i.e.,the Hookian relationship).There are several materials (such as yoghurt,pasta,ketchup,shampoo,certain oils,mud,paints,blood at low shear rate,clay coatings,polymer melts,and greases) which have nonlinear relationship between the shear stress and the rate of strain.These materials are characterized as non-Newtonian fluids.The flows of non-Newtonian fluids in recent times have wide coverage in polymer devolatisation,fermentation,bubble absorptions,composite processing,boiling,bubble columns,plastic foam processing,and several others.Further,the flows of non-Newtonian fluids induced by a moving surface occur in paper production,plastic films,polymer extru sion,extrusion of plastic sheets,food processing,etc.Hence,the analysis of non-Newtonian fluids at present has received interest of scientists and researchers (both experimentalists and theoreticians).Such fluids vary in nature,and their constitutive equations take different forms (see^{[1, 2, 3, 4, 5]}and several references therein).

In nature,there exist a large number of chemical reactions which have important applications.Some of the reactions proceed very slowly or do not react at all in the absence of a catalyst.The interaction between the homogeneous and heterogeneous chemical reactions is very intricate involving different rates of the chemical species of production and consumption both on the catalyst surface and within the fluid.Many chemical reactions such as combustion,catalysis,and biomedical systems consist of homogeneous and heterogeneous reactions simultaneously.Reactions such as propane oxidation over platinum and methane/ammonia have great significance in the chemical industry^{[6, 7]}.Chaudhary and Merkin^[8]discussed the homogeneousheterogeneous effects in the forced convection stagnation point flow of viscous fluid.Khan and Pop^[9]analyzed such effects in the viscous flow in the region of stagnation point on an infinite permeable wall.To the best knowledge of these authors,there is no attempt yet for predicting the characteristics of homogeneous-heterogeneous reaction in flows of non-Newtonian fluids.Hence,the objective here is to fill such void.Therefore,the present analysis is prepared to analyze the homogeneous-heterogeneous reaction effects in the two-dimensional boundary layer flow of a Jeffrey fluid induced by a stretching surface.The homogeneous-heterogeneous reaction is considered subject to the same reactant and auto catalyst diffusion coefficients.The governing momentum and concentration equations are modeled.The exact solution for the velocity is presented.The series solution for the concentration equation is given by the homotopy analysis method (HAM)^{[10, 11, 12, 13, 14, 15]}.The convergent series solution for the concentration equation is also examined.The variation of different parameters on the physical quantities is analyzed.

2 Mathematical formulation

Let us investigate the incompressible two-dimensional boundary layer flow of a Jeffrey fluid induced by an impermeable stretching sheet.Here,we consider the homogeneous-heterogeneous reaction model proposed in Ref.^[8].A Cartesian coordinate system is selected in a way that the x-axis is parallel to the stretching sheet while the y-axis is in the normal direction to the sheet.The heat released by the reaction is assumed negligible.The homogeneous reaction for the cubic autocatalysis can be represented as follows:

where r denotes the rate,while an isothermal and first-order reaction on the catalyst surface is taken in the form of In the above expressions,a and b denote the concentrations of the chemical species A and B,while k₁ and k_s denote the rate constants.These equations ensure that there exists no reaction rate both at the boundary layer outer edge and in the external flow region.The relevant equations through the aforementioned assumptions are where u and v represent the components of velocity along the x-and y-axes,respectively,λ₁ represents the retardation time,λ₂ represents the ratio of the relaxation time to the retardation time,ν represents the kinematic viscosity,and D_A and D_B are the diffusion coefficients of the species A and B,respectively.The applied boundary conditions are defined as follows: in which U_w and a₀ are the constants. Substituting into Eqs. (4)-(7),we get where f is the dimensionless stream function,β is the material parameter,K is the strength of the homogeneous reaction measurement,Sc represents the Schmidt number,δ represents the diffusion coefficients ratio,Ks represents the strength of the heterogeneous reaction measurement,and Re represents the Reynolds number.The definitions of these quantities are The exact solution of Eq. (9) subject to the relevant conditions is^[16] with It is often the case that the diffusion coefficients of chemical species A and B have the same value.This argument facilitates us to take the diffusion coefficients D_A and D_B as the same,i.e.,δ=1 and thus Now,Eqs. (10) and (11) yield with the boundary conditions

3 Solution procedure

The initial guess g₀ and the auxiliary linear operator in the HAM solutions are chosen in the following forms:

with where C_i (i=1,2) are the arbitrary constants.

If p ∈^{[0, 1]}represents the embedding parameter,then _g represents the non-zero auxiliary parameter.The zeroth-order deformation problem is

where the nonlinear operator N_g is defined by For p = 0 and p = 1,we have With the variation of p from zero to one,(η;p) goes from the initial solution g₀(η) to the final solution g(η).Using the Taylor series expansion,we can write The auxiliary parameters are so properly chosen that the series (28) converge at p = 1 and thus The mth-order deformation problems are given by The general solution of Eqs. (30)-(32) is in which g_m^*(η) denotes the special solution.

4 Convergence analysis

Obviously,the homotopy solutions contain the auxiliary parameter (_g).The series solution convergence of the problem depends upon the auxiliary parameter.To verify the convergence region,we sketch the -curve for the series solution (see Fig. 1).The admissible value of _g through Fig. 1 is −2.3≤_g≤−0.6.

5 HAM-based MATHEMATICA package BVPh 2.0

We also compute the solution of nonlinear ordinary differential equation (19) by the MATHEMATICA package BVPh 2.0 using the boundary condition (20).We find the minimum squared residual error 2.010 75× 10⁻⁹ for the concentration profile (g(η)) at the 20th-order of approximations.Fig 2 is plotted for the solution of concentration profile.The total residual error corresponding to different orders of approximations is displayed in Fig. 3.It is noted that the error decreases when the order of approximation increases.

6 Discussion

Now,we analyze the graphical results of various pertinent parameters on the axial velocity and concentration fields.The behavior of λ₂ on the axial velocity distribution is shown in Fig. 4.The velocity is found to decrease when λ₂ increases.The behavior of the Deborah number (β) on the axial velocity field is displayed in Fig. 5.Here,we observe that the axial velocity field and the associated thickness of the boundary layer increase when the Deborah number is increased.The characteristics of strength of the homogeneous reaction K on the concentration distribution is sketched in Fig. 6.Higher values of the strength of homogeneous reaction K result in the reduction of the concentration distribution.The variation in the strength of the heterogeneous reaction K_s on the concentration field is plotted in Fig. 7.Here,we can see that the concentration field enhances for larger K_s.The influence of the Schmidt number Sc on the concentration field is sketched in Fig. 8.Higher values of the Schmidt number result in the enhancement of the concentration distribution.The characteristics of the strength of homogeneous reaction K and λ₂ on g'(0) are shown in Fig. 9.It is revealed that g'(0) decreases when the strengths of the homogeneous reaction K and the heterogeneous reaction K_s are increased.The behavior of the heterogeneous reaction Ks and the Schmidt number Sc on g(0) is shown in Fig. 10.Here,g(0) enhances for higher values of the Schmidt number Sc.Further,it is noted that g(0) has small values with the enhancement of the strength of heterogeneous reaction K_s.Figure 11 displays g(0) for the effects of the Schmidt number Sc and the homogeneous reaction K.It is concluded that g(0) shows the increasing behavior for higher values of the Schmidt number Sc.However,g(0) decreases by increasing K.

Table 1 presents the series solution convergence.Here,we conclude that the series solution converges at the 34th-order of approximations for g'(0).

7 Conclusions

We examine the characteristics of homogeneous-heterogeneous reaction in the twodimensional flow of Jeffrey fluid induced by an impermeable stretching sheet.The behaviors of various pertinent parameters are discussed and analyzed through graphs.The following conclusions can be made through the analysis of the graphs.

(i) The characteristics of the Deborah number (or the material parameter)β and λ₂ on the axial velocity distribution are opposite.

(ii) The concentration distribution decreases with an increase in the strength of the homogeneous reaction K.

(iii) The effect of the strength of heterogeneous reaction K_s on the concentration profile is opposite to that of the strength of homogeneous reaction.

(iv) The value of concentration profile enhances for larger values of the Schmidt number.

(v) The value of g(0) increases by increasing the Schmidt number Sc whereas a reverse trend is seen for increasing K and K_s.

Acknowledgements The authors are grateful for the comments of reviewers.