1 Introduction

There are several materials like food stuffs (ketchup,mayonnaise,alcoholic beverages,chocolates in liquefies form,ice creams,yogurt,etc.),biological products (vaccines,blood,syrups, synovial fluid,etc.),and chemical products (tooth pastes,cosmetics,shampoos,paints,pharmaceutical chemicals,etc.),which do not obey Newtonian’s law of viscosity. These are characterized as the non-Newtonian fluids. The investigation of such fluids is very significant because of their relevance to practical applications in industry and engineering. Nowadays,the non- Newtonian fluids are categorized into three classes,namely,the differential,rate,and integral types. Previous information shows that much attention has been given to the differential type fluids. Because constitutive equations in the differential type fluids are much easier,and one can explicitly express the shear stresses in terms of velocity components. However,this is not the case in rate type fluids. The existed literature indicates that little attention has been given to the flows of rate type fluids. The Maxwell fluid is the simplest subclass of rate type fluid. Only the relaxation time is described by this fluid model. On the other hand,an Oldroyd-B fluid has a measurable relaxation and retardation times,and under general flow conditions, it can capture the viscoelastic features of dilute polymeric solutions. For instance,Hayat et al.^[1] presented exact solutions for flow problems of an Oldroyd-B fluid. The flow of generalized Oldroyd-B fluid due to a constantly accelerating plate has been studied by Vieru et al.^[2]. Qi and Jin^[3] examined unsteady helical flows of a generalized Oldroyd-B fluid with the fractional derivative. The oscillating motion of an Oldroyd-B fluid between two infinite circular cylinders is studied by Fetecau et al.^[4]. It is concluded that the amplitude of transient oscillations is smaller in magnitude for the case of Oldroyd-B fluid than that of the Newtonian fluid. Fetecau et al.^[5] also presented energetic balance for the Rayleigh-Stokes problem of an Oldroyd-B fluid. Jamil et al.^[6] investigated unsteady helical flows of an Oldroyd-B fluid. Zheng et al.^[7] computed exact solutions for MHD flows of generalized Oldroyd-B fluids due to an infinite accelerating plate. Zheng et al.^[8] studied magnetohydrodynamic flows of generalized Oldroyd-B fluids with slip effects. Hayat et al.^[9] computed exact solutions in generalized Oldroyd-B fluids. Niu et al.^[10] developed the viscoelastic effects on thermal convection of an Oldroyd-B fluid in opentop porous media. Hayat et al.^[11] examined three-dimensional convective flows of Oldroyd-B fluids over the stretching surface. Shehzad et al.^[12] studied the three-dimensional mixed convective radiative flow of an Oldroyd-B fluid. The flow of Oldroyd-B fluid with nanoparticles and thermal radiation has been investigated by Hayat et al.^[13]. Ramzan et al.^[14] computed three dimensional flows of Oldroyd-B fluids in presence of Newtonian heating. Hayat et al.^[15] examined the flow of an Oldroyd-B fluid subject to homogeneous-heterogeneous reactions and Cattaneo-Christov heat flux.

Flows over the stretching surface have promising applications in several engineering processes. For example,extrusion of molten polymers through a slit die has a vital role in the production of plastic sheets and wire drawing. Moreover,glass production and paper production are some novel applications of the flows over stretched surfaces. Since the pioneering work of Crane^[16] on the flows over the stretching surface,various researchers are engaged in studying such work under different aspects. It is also noticed that the stretching velocity may not be linear in all the cases. Hence,few researchers have examined the flows induced by the exponentially stretching surface. For example,Magyari and Keller^[17] studied the heat and mass transport process over an exponentially stretching surface. The stagnation point flow and heat transfer due to an exponentially stretching/shrinking sheet have been presented by Bhattacharyya and Vajravelu^[18]. Thermally stratified magnetohydrodynamic flows induced by an exponentially stretching sheet have been analyzed by Mukhopadhyay^[19]. Pramanik^[20] presented radiative flows of Casson fluids past an exponentially stretched surface. Hayat et al.^[21] developed the convective flow of the nanofluid over an exponentially stretching sheet. The impact of the second-order slip on the nanofluid flow past an exponentially shrinking/stretching sheet using Buongiorno’s model has been presented by Rahman et al.^[22]. Nagalakshmi et al.^[23] studied effects of Hall current on the boundary layer flow induced by an exponentially stretching surface. Khan et al.^[24] presented the viscoelastic flow by an exponentially stretching sheet by considering Cattaneo-Christov heat flux. Mustafa et al.^[25] examined radiative flows induced by a bi-directional exponentially stretching surface.

The present study is modelled in such a way that it investigates the boundary layer flow of an Oldroyd-B fluid. Considering fluid model can capture the relaxation and retardation effects. The two-dimensional flow is created by an exponentially stretching surface. The series solution to the resulting problem is constructed,and convergence is shown using a homotopy analysis method (HAM)^{[26, 27, 28, 29, 30, 31, 32, 33, 34, 35]}. Main attention in the discussion is focused on the analysis of relaxation and retardation time effects. Key points of the present study are summed up in the concluding section.

2 Model development

A stretched flow of an incompressible Oldroyd-B fluid with exponential velocity and temperature distributions is presented. The governing equations in the absence of thermal radiation and viscous dissipation effects are

where u and v are the velocity components along the x- and y-directions,respectively,ν is the kinematic viscosity,Λ₁ and Λ₂ are the relaxation time and the retardation time,respectively, T is the temperature,and α is the thermal diffusivity. The boundary conditions are Introduce Equation (1) is satisfied automatically,and Eqs. (2) and (3) are reduced as follows: Moreover,the Prandtl number P_r,dimensionless relaxation time constant β₁,and retardation time constant β₂ are defined as The ratio of the conductive thermal resistance to the convective thermal resistance of the fluid is referred as the Nusselt number Nu_x. It gives the heat transfer rate at the surface,which is defined by where q_w denotes the wall heat flux. In a dimensionless form, where denotes the local Reynolds number. 3 Homotopic solutions

We choose the linear operators L₁ and L₂ and the initial guesses f₀(η) and θ₀(η) in the forms of

together with the properties where c_i (i = 1,2,· · · ,5) are the constants. We construct the zeroth- and mth-order problems as follows: where ħ_f and ħ_θ are the nonzero auxiliary parameters. The general solutions are in which f_M^* and θ_M^* denote the special solutions. 4 Convergence analysis

The HAM is employed to compute the solution of problems consisting of Eqs. (6)-(8). Auxiliary parameters ħ_f and ħ_θ play a key role in adjusting and controlling the convergence and rate of approximations for the functions f and θ. The ħ-curves of f"(η) and θ'(η) are plotted to get admissible values of ħ_f and ħ_θ at 9th-order of approximations. The admissible values of ħ_f and ħ_θ are −0.85 ≤ ħ_f ≤ −0.3 and −1.1 ≤ ħ_θ ≤ −0.3 (see Fig. 1). Further,the series solutions converge in the whole region of η (0 < η < ∞) when ħ_f = ħ_θ = −0.6 (see Table 1).

5 Results and discussion

Figures 2 and 3 are plotted to analyze the impact of relaxation time constant β₁ and the retardation time constant β₂ on the flow field f'. The effect of parameter β₁ on the function f' is illustrated in Fig. 2. For larger β₁,the values of f' and the boundary layer thickness decrease. This is because of the fact that a slower recovery process is observed for the higher relaxation time,which causes the boundary layer thickness to grow at a slower rate. Figure 3 depicts the effects of retardation time constant β₂ on the velocity function f'. Here,when β₂ is increased, the enhancement in the fluid flow and its boundary layer thickness is obtained.

Figures 4-6 are plotted for the effects of Prandtl number Pr,the relaxation time constant β₁,and the retardation time constant β₂ on the temperature field θ. Figure 4 shows the effects of Pr on the temperature. Reduction in the temperature field θ is observed for larger values of Pr. Influence of β₁ on θ can be seen in Fig. 5. There is a decrease in θ when β₁ is increased. Figure 6 represents the effect of β₂ on θ. It is observed that an increase in β₂ decays the temperature profile θ.

Table 2 is prepared to show the numerical values of surface heat transfer rate for different emerging parameters. This table shows that the Nusselt number decreases when there is an increase in the relaxation time constant β₁,and it increases when the retardation time constant β₂ and Prandtl number Pr are increased.

6 Concluding remarks

This study addresses an Oldroyd-B fluid flow generated by a stretched sheet with the exponential velocity and temperature distribution. Impacts of emerging parameters on the heat and fluid flow are examined. Following observations are made:

(ⅰ) The effects of β₂ are quite opposite to those of β₁ on the velocity profile f'.

(ⅱ) The temperature and thermal boundary layer thickness decay for the larger Prandtl number.

(ⅲ) The variations of β₁ and β₂ are qualitatively similar on the temperature profile.

(ⅳ) The Nusselt number rises for larger β₂ and Pr while it reduces by increasing β1.