1 Introduction

Flow problems attributed to linear, non-linear or exponentially stretching surfaces have attained global interest owing to their practical applications in polymer industry and engineering fields. Such technical processes include polymers containing filaments or cooling of the continuous strips, artificial fibers, glassblowing, wire drawing, extrusion of rubber sheets, blood vessels, water pipes, irrigation channels, sewer pipes etc. Moreover, polymer industry has a fundamental role in the mechanism of molten polymers. Materials made up of extrusion processes encountered by windup and feed roll or travelled trough conveyer-belts relate the distinctiveness characteristics of stretching surfaces^[1]. The finishing of the product rigorously depends on the cooling and stretching rates in an industrial process. A boundary layer estimation is generally used to find out the solutions. Sakiadis^[2-3] initially presented the concept of boundary layer flow of viscous fluid passing a flat plate. After that, many researchers worked out on the Sakiadis flow model through different aspects. Crane^[4] obtained an exact closed-form solution of an incompressible viscous liquid bounded by a linear stretching surface. Gupta and Gupta^[5] presented the stretching flow over a surface with suction/blowing. Suction and surface slip conditions were implemented by Wang^[6] to investigate the problem of viscous liquid due to the stretching surface. Rashidi and Pour^[7] explored the flow characteristics of viscous fluid bounded by the stretching surface.

Animal blood, mud, ketchup, paint, shampoo and greases, paper pulps, slurries, and certain oils are a few examples related to non-Newtonian fluids. Liquids of such features do not belong to the category of Newtonian fluids and describe the non-linear relationship between the shear rate and shear stress. The rheological distinctiveness of such fluids has importance in paper coating, fossil fuels, plasma, and polymers. The modeled expressions of non-Newtonian fluids are higher-order and very complex. In general, non-Newtonian liquids are categorized into three classes, namely, integral, differential and rate types. To investigate the characteristics of such fluids, various models are presented in the literature. Amongst the non-Newtonian fluid models, the Powell-Eyring model has attained special attention of the researchers. This attention is due to its distinguishing characteristics in modern science. This model results from the theory of kinetics, and it can behave as Newtonian liquids for both high and low shear rates. In 1944, the model was first proposed by Powell and Eyring^[8]. Hayat et al.^[9] reported the analytical solutions to observe the flow and heat transfer characteristics of Powell-Eyring liquid due to the moving sheet with convective conditions. Javed et al.^[10] used the Keller box numerical scheme to address the flow characteristics of Powell-Eyring fluid. Flow behavior due to the stretching cylinder was predicted by Hayat et al.^[11]. They considered the Powell-Eyring nanofluid in the presence of magnetic field and thermal radiation.

Transfer of heat due to natural convection commonly occurs in various industrial processes and physical problems such as fiber insulation, geothermal systems, copper materials, refrigeration, chemical reactors, packed beds, heat exchangers, petroleum reservoirs, atomic plants, and space satellites. Fourier^[12] was the first who developed the famous theory of heat conduction which examined the performance of heat transport through various aspects. This theory was used tremendously to explore the heat transport mechanism^[13-25]. The foremost disadvantage related to this model is the energy equation of parabolic type which specifies the early disturbance immediately experienced via medium under inspection. To overcome such issues, Cattaneo^[26] modified Fourier's heat conduction theory with the addition of a new term named as the relaxation time. Oldroyd's upper-convected formulation was used to develop the frame-indifferent rule by Christov^[27]. This mechanism is of fundamental concern in bio-engineering procedures, electronic devices, heat reduction in nuclear plants, liquid pasteurization like milk, and hybrid generators. Tibullo and Zampoli^[28] discussed the structural stability and uniqueness of the energy expressions with the model presented by Cattaneo and Christov. Recently, several researchers have made significant attempts related to the Cattaneo-Christov flux in Refs. [29]-[35].

Chemical reaction is of significant interest in hydrometallurgical and chemical industries like food processing and nourishment handling. Moreover, configuration of smog characterizes the homogeneous chemical reaction of the first order. Chemical reaction along with the ambient fluid plays a vital role in the generation or absorption of diffusion species and improves the quality of final product. Such processes acquire places in polymer production, manufacturing of glassware or ceramics, etc. A numerical exploration of magnetohydrodynamic(MHD) nanofluids with chemical reaction was reported by Mabood et al.^[36]. Consequence of chemical reaction in flows of pseudoplastic nanofluids through the asymmetric channel was shown by Hayat et al.^[37]. Numerical outcomes of chemical reaction in Williamson nanofluids owing to the stretching surface were presented by Krishnamurthy et al.^[38]. The analytical study of chemically reactive Maxwell nanofluid flows via the bidirectional surface was given by Hayat et al.^[39]. Hayat et al.^[40] demonstrated the flow behavior of Jeffrey fluid due to the non-linear surface which stretched in the radial direction.

A reasonable study has been done on non-Newtonian fluid flows passing a stretching surface, but still many works need to be done in order to examine the analysis of the Cattaneo-Christov theory. Therefore, the major emphasis of current investigation is to discuss the impact of the Cattaneo-Christov model in flows of Powell-Eyring fluids over the stretching sheet. The present study is done with consideration of heat source/sink and constructive/destructive chemical reaction. The consideration of heat absorption or generation cannot be omitted in the problems of dissociating and chemical reactive liquids. It plays a major role in the processes of particle deposition of chemical reactors, under-ground wastage, electronic chips, and semi-conductor waters. The obtained governing system is highly non-linear and coupled. We adopt a numerical technique to tackle such problem. The mathematical formulation of the present analysis is presented in Section 2. The numerical solution of boundary value problem is given in Section 3. In Section 4, we discuss the numerical results of the emerging physical parameters. The main findings of present study are summarized in Section 5.

2 Problem formulation

We consider laminar flows of Powell-Eyring fluids due to the stretching sheet with application of the Cattaneo-Christov theory. The presence of pressure gradient is ignored. Heat generation/ absorption and chemical reactions are present. The sheet has the constant temperature and concentration. The governing equations for continuity, momentum, temperature, and concentration are^{[10, 29]}

(1)

(2)

(3)

(4)

with the prescribed boundary conditions^{[30, 32]},

(5)

where u and v stand for the velocity components along x-and y-directions, respectively, υ is the kinematic viscosity, ρ is the density, β and c are the fluid parameters, λ₁ is the heat flux relaxation time, λ₂ is the mass flux relaxation time, α is the thermal diffusivity, D is the diffusion co-efficient, T is the temperature, T_w is the wall temperature, T_∞ is the ambient liquid temperature, Q is the heat generation/absorption co-efficient, C is the concentration, C_w is the wall concentration, C_∞ is the ambient liquid concentration, k₀ is the reaction rate, and u=u_w(x)=ax is the stretching velocity.

To reduce Eqs.(2)-(4) in a dimensionless form, the following transformations are used^{[30, 32]}:

(6)

Equation(1) is trivially satisfied while the other expressions yield

(7)

(8)

(9)

(10)

where and are the fluid parameters, is the Prandtl number, De₁ =λ₁ a is the Deborah number with respect to the heat flux relaxation time, is the heat generation/absorption parameter, is the Schmidt number, De₂ =λ₂a is the Deborah number with respect to the mass flux relaxation time, and is the chemical reaction parameter.

3 Numerical solution

The highly non-linear coupled system of Eqs.(7)-(9) is solved numerically with the finite difference method via boundary conditions(10). The order of Eqs.(7)-(10) is reduced by substituting

(11)

The reduced flow model is

(12)

(13)

(14)

(15)

To solve the above mentioned system, we discretize the domain [0, ∞] uniformly by choosing the proper step size h. Simpson's rule^[41] with the help of Milne^[42] is implemented to integrate Eq.(11). The second-order central finite difference approximations are used to discretize Eqs.(12)-(14) at the grid point η =η_i. We compute

(16)

(17)

(18)

The iterative procedure based on the successive over relaxation(SOR) parameter method is used to solve the system(16)-(18) with Eq.(15). For the acceleration and the accuracy in the solution of the iterative procedure, we use an algorithm mentioned by Syed et al.^[43]. We stop the iterative procedure, if the given criterion meets four successive iterations,

(19)

in which e_tol is the error tolerance and computed to at least 10^-12 during over executions.

4 Results and discussion

The aim of recent investigation is to examine the salient characteristics of incompressible flows of Powell-Eyring liquids subject to the Cattaneo-Christov model. Numerical results are shown with graphs and tables. A detailed discussion with the physical interpretation is presented for the emerging dimensionless physical parameters. To obtain a better understanding of the flow characteristics, we choose to describe the shear stresses, the heat and mass transfer rates, the velocity, the temperature and concentration for different values of the physical parameters. Figure 1 is plotted to discuss the velocity profiles f(η) and f'(η) for increasing values of the fluid parameter ε. Both the velocity profiles enlarge for larger values of ε. Physically, the resistance between the fluid layers decreases for the larger fluid parameter which in turn enhances the velocity of fluid. Figure 2 is drawn to investigate the impact of ε on the temperature and concentration fields. It is noticed that temperature and concentration profiles decline for larger ε. Moreover, the boundary layer thickness rises in case of velocity field and falls in the scenario of temperature and concentration fields. Figure 3 is drawn to illustrate the impact of the fluid parameter δ. The larger values of δ correspond to the lower velocity profiles f(η), and they enlarge the temperature profiles θ(η) and the thermal boundary layer thickness. It is also observed from Figs. 2 and 3 that fluid parameters have opposite effects on temperature fields. The impact of different values of Prandtl and Deborah numbers on the temperature θ(η) is depicted in Fig. 4. Physically, the ratio of momentum diffusivity to thermal diffusivity is known as the Prandtl number. Weaker thermal diffusivity implies a larger Prandtl number. Because the diffusion rate decreases, conduction decreases while convection increases. No doubt fall in the temperature and thickness of layer is seen for enhanced values of Pr. The larger Deborah number De₁ causes a reduction in the temperature field and boundary layer thickness. The Deborah number results from the relaxation time due to the heat flux. Such relaxation time appears directly in the definition of De₁. The fluid which has less relaxation time of heat change leads to the larger temperature, and the fluid with the more relaxation time of heat change results in the lower temperature. In the present case, larger values of De₁ cause more heat change relaxation time which decays the temperature. Figure 5 is plotted to describe the behavior of heat generation/absorption parameter β on θ(η). The profiles rise for the larger heat generation parameter(β >0), and they decay for the lower heat absorption parameter(β < 0). More heat is added into the fluid for an enhancement in heat generation which amplifies the temperature profiles. The variations of the Schmidt number and Deborah number with respect to the mass flux relaxation time are elucidated in Fig. 6. The Schmidt number is the ratio of momentum diffusivity to mass diffusivity. Therefore, larger Sc causes a reduction in the mass diffusivity which leads to the reduction in concentration profiles. Moreover, increased values of De₂ reduce the concentration profiles. The Deborah number De₂ arises from the relaxation time of mass flux. Less relaxation time of mass flux leads to higher concentration, and the fluid with more mass flux relaxation time results in smaller concentration. Hence, more mass flux relaxation time occurs due to the larger value of De₂ which decreases the concentration profiles. Figure 7 describes the impact of chemical reaction parameter ζ. A reduction in ϕ(η) is observed in case of the destructive reaction rate(ζ >0) whereas a reverse trend is obtained for the generative reaction rate(ζ < 0). An enhancement in destructive chemical reaction has an ability to retard the diffusion rate and molecular diffusivity of solutal concentration. This phenomenon generates retarding impacts on the concentration field in case of ζ >0.

The numerical investigations are demonstrated in the tabular forms. Table 1 is drawn to reveal the impact of the fluid parameter, the Prandtl number, the Deborah number, and the heat generation/absorption parameters on the heat transfer rate. We note that the larger fluid parameter, the Prandtl number and the Deborah number magnify -θ'(0). Moreover, the heat transfer rate enlarges for β < 0 and it declines for β >0. Table 2 exposes the effects of the fluid parameter, the Schmidt number, and the mass flux Deborah number on the mass transfer rate. Here, -ϕ'(0) increases for intensified values of the fluid parameter, the Schmidt number, and the mass flux Deborah number. Furthermore, for ζ < 0, the mass transfer rate decreases while it enhances for ζ >0.

5 Conclusions

A numerical solution is developed to explore the characteristics of the Cattaneo-Christov model in laminar incompressible flows of Powell-Eyring liquids due to the stretching surface. Appropriate transformations are adopted to exchange the non-linear system of partial differential equations into the ordinary ones. We observe that fluid parameters ε and δ have quite opposite impacts on temperature profiles. The velocity f'(η) is increased in the case of larger ε. Larger Prandtl and heat flux Deborah numbers cause a decline in the temperature θ(η) with the interconnected boundary layer thickness. The Schmidt number and the mass flux Deborah number have similar impacts on concentration profiles. Heat generation and absorption have dissimilar effects on temperature fields. Larger values of Pr and De₁ increase -θ'(0), and increasing values of the Schmidt number and the mass flux Deborah number enhance -ϕ'(0).