1 Introduction

The fluid flow along a stretching sheet has been greatly concerned due to its various applications in engineering and industry, e.g., glass fibre and paper production, hot rolling, plastic sheet extrusion, spinning of fibers, and manufacturing of polymer. Crane^[1] examined the stretched flow problems. Abbasbandy et al.^[2] described the numerical solution of the magnetohydrodynamic (MHD) Oldroyd-B fluid for the Falkner-Skan flow. Rashidi et al.^[3] inspected a third-grade fluid flow accompanied by the magnetic field through a stretching sheet. Sheikholeslami et al.^[4] evaluated the heat transfer phenomenon of a nanofluid flow along a stretching surface. Lin et al.^[5] studied the effects of a stretching sheet and viscous dissipation on a pseudoplastic fluid. Pourmehran et al.^[6] analyzed the flow of a nanofluid along a stretching sheet. Turkyilmazoglu^[7] studied the micropolar fluid flow through a stretching surface in a porous space.

The fluids, in which Newton's law of viscosity does not hold, are non-Newtonian fluids. Ketchup, drilling muds, toothpastes, paints, and cosmetic products are some non-Newtonian fluids. Because of the different properties, these fluids cannot be defined by a single relationship. Rate, integral, and differential types are the three categories of non-Newtonian fluids. The Maxwell fluid shows the impact of the relaxation time only. Due to its drawback, a new model, named the Oldroyd-B fluid, is presented. This model predicts the relaxation time and the retardation time. Zhang et al.^[8] demonstrated a thin film Oldroyd-B fluid flow along a stretching sheet. Sajid et al.^[9] explored the stagnation point flow of a mixed convection Oldroyd-B fluid. Hayat et al.^[10] illustrated the MHD Oldroyd-B fluid flow with homogeneous and heterogeneous reactions. Shehzad et al.^[11] discussed the radiative heat transfer through the Oldroyd-B fluid flow with magnetic field. Hayat et al.^[12] described a three-dimensional (3D) Oldroyd-B fluid flow subjected to magnetic field. Niu et al.^[13] studied the viscoelastic effects in the Oldroyd-B fluid flow through a porous medium. Li et al.^[14] discussed the steady Oldroyd-B fluid flow with a porous medium. Hayat et al.^[15] analyzed an incompressible Oldroyd-B fluid flow through a stretching surface.

The phenomenon of heat transfer is useful in both practical and theoretical aspects. It has many applications in glass blowing, metallurgy, continuous casting, and polymer technology. Various engineering processes have taken place at very high temperature. Thermal radiation plays a pivotal role in such procedures. The effects of thermal radiation is useful when the difference between the surface temperature and the ambient temperature is high. Kayalvizhi et al.^[16] illustrated the MHD effects of thermal radiation on the viscous fluid flow. Yasin et al.^[17] illustrated the combined results of heat and mass transfer by a radiative stretched surface. Narayana and Babu^[18] investigated the MHD effects on the radiative Jeffrey fluid flow. Dogonchi et al.^[19] demonstrated the MHD effects of thermal radiation on a nanofluid. Zeeshan et al.^[20] examined the strength of thermal radiation in the stretched flow of the ferromagnetic fluid. Bhattacharyya et al.^[21] analyzed the heat transfer of a micropolar fluid flow with thermal radiation. Sheikholeslami et al.^[22] interpreted the nanofluid flow with the effect of thermal radiation.

The melting process is of great importance in widespread problems in technologies and geophysics. Melting heat transfer problems can be categorized according to the temperature ranges such as the melting temperature range of metals or other materials, the atmospheric temperature range in cold climates, and the refrigeration temperature range. Melting heat transfer is significant in the construction of semi conductor materials, the thawing of frozen ground, the melting of permafrost, and magma solidification. Epstein and Cho^[23] investigated the effects of melting heat transfer through a flat plate. Ahmad and Pop^[24] explored the viscous fluid with porous space and melting heat transfer. Awais et al.^[25] examined the Burger fluid flow with a melting process. Hayat et al.^[26] demonstrated the heat transfer phenomenon due to the melting in a nanofluid flow. Hayat et al.^[27] illustrated the effects of chemical reactions in the flow of carbon nanotubes with melting heat transfer.

The combination of free convection and forced convection is termed as mixed convection. These flows are recognized by an outer forcing system and inner volumetric forces at the same time. They have many applications in engineering and industry, e.g., boilers, automobile demisters, solar energy system, heat exchangers, and combustion chamber wall cooling in gas turbines. Rashidi et al.^[28] described a mixed convective nanofluid flow with the MHD effects by a sinusoidal channel. Hayat et al.^[29] presented the mixed convective nanofluid through a porous medium. Sajid et al.^[30] explored the effects of the magnetic field Oldroyd-B fluid flow with mixed convection. Hayat et al.^[31] discussed the effects of mixed convection on the flow with nano-sized particles. Abbasi et al.^[32] demonstrated a double stratified mixed convective flow of the Maxwell fluid.

In 1992, the homotopy analysis method (HAM) was firstly developed by Liao^[33]. He further modified this method by introducing the auxiliary parameters named as the convergence control parameter in 1997. The convergence control parameter provides a simple way to examine and authorize the convergence of a series solution. This method is always authentic, and does not directly depend on small or large physical parameters. It gives an appropriate way to make sure the convergence of the approximation series. A nonlinear problem is converted into an infinite linear sub-problem with convenience to select the related equation type and the base functions for the solutions. The HAM has been successfully used to solve nonlinear differential equations^[33-41].

The present analysis discusses the effect of melting heat transfer on a two-dimensional (2D) Oldroyd-B fluid flow. The effects of the heat generation or absorption and thermal radiation due to a stretched surface are examined. A mixed convection flow is also considered. The behaviors of different involved parameters are examined and discussed through plots.

2 Model development

We assume that the steady Oldroyd-B fluid flow is over a stretching sheet along the x-axis with the velocity u_s(x) =cx. The heat transfer aspects are examined through melting, thermal radiation, and heat generation or absorption. The temperature at the melting surface is T_m, the free stream temperature is T_∞, and T_∞ > T_m. With the employed boundary layer approximation, we can write the flow equations as follows:

(1)

(2)

(3)

The appropriate boundary conditions are

(4)

(5)

where the velocity component u is along the x-direction, and the velocity component v is along the y-direction. λ₁ is the relaxation time, g is the gravitation acceleration, λ₂ is the retardation time, ν is the kinematic viscosity of the fluid, β is the thermal expansion coefficient, ρ is the density, σ is the electrical conductivity, k is the thermal conductivity, C_p is the specific heat, T is the fluid temperature, Q₀ is the heat generation or absorption coefficient, c_s is the heat capacity of the solid surface, T₀ is the fixed or initial temperature, and λ is the latent heat of the fluid.

The transformations are^[25]

(6)

The continuity equation holds identically. Then, Eqs. (2)-(5) become

(7)

(8)

(9)

where prime represents the derivative with respect to η, β₁ and β₂ are the dimensionless relaxation and retardation time constants, respectively, Ha is the Hartman number, Gr is the Grashof number, Re is the local Reynolds number, Pr is the Prandtl number, r is the radiation parameter, γ is the heat generation/absorption parameter, M is the dimensionless melting parameter, and

When

the Oldroyd-B fluid can be reduced to the Newtonian fluid.

The local Nusselt number is

(10)

The dimensionless local Nusselt number is

(11)

where represents the local Reynolds number.

3 Solution procedure

We consider the linear operators and the initial guesses for the unknown functions as follows:

(12)

(13)

where

(14)

3.1 Zeroth-order deformation equations

(15)

(16)

where p ∈ [0,1] denotes the embedding parameter. The nonlinear operators and the non-zero auxiliary parameters are

(17)

(18)

with the following boundary conditions:

(19)

(20)

3.2 mth-order deformation equations

(21)

(22)

where

(23)

(24)

(25)

(26)

The general solutions (f_m(η), θ_m (η)) consisting of the special solutions (f_m^*(η), θ_m^* (η)) are

(27)

where c_i (i=1, 2, 3, 4, 5) are constants.

4 Convergence analysis

The HAM gives us a choice to obtain the convergence series solutions. The convergence area of the series solution is controlled by the auxiliary parameters. The proper ranges of and are (see Figs. 1 and 2)

Table 1 illustrates the convergence of the series solutions for the velocity f"(0) and the temperature θ'(0). Here, the convergence for the velocity f"(0) and the temperature θ'(0) is achieved at the 17th-order of approximations.

5 Discussion 5.1 Dimensionless velocity profiles

Figures 3 and 4 present the effects of β₁ and β₂ on the velocity profile f'(η). Figure 3 depicts the decreasing behavior of the velocity for the rising values of the relaxation time constant β₁. From the figure, we can see that, the bigger β₁ is, the stronger the viscous forces in the fluid flow are. The retardation time constant β₂ has an opposite effect on the velocity field (see Fig. 4).

Figure 5 illustrates the effects of the Hartman number Ha on the velocity profile. From the figure, we can see that, when Ha increases, the velocity decreases. The fluid motion decreases when the magnetic field is used, which causes the velocity profile to reduce. The effects of the melting parameter are shown in Fig. 6. There is an enhancement in the velocity field f'(η) when the melting parameter enhances. The variations of λ on the velocity f'(η) are sketched in Fig. 7. The velocity field enhances when the values of λ increase.

5.2 Dimensionless temperature profiles

Figures 8 and 9 are plotted for the heat generation parameter γ and the melting parameter M on the temperature θ (η). From the figures, we can see that, the temperature profile θ(η) reduces either when the heat generation increases or the melting parameter increases.

Figure 10 illustrates the effects of Pr on the temperature θ(η). From the figure, we can see that, there is an increase in the temperature when Pr increases. Figure 11 indicates the temperature distribution θ(η) for larger values of the radiation parameter r. From the figure, we can see that the temperature distribution reduces when r increases.

5.3 Nusselt number

Figure 12 illustrates the effects of the Hartman number Ha on the Nusselt number via γ. From the figure, we can see that the magnitude of the Nusselt number increases when Ha increases.

Figure 13 demonstrates the effects of γ on the Nusselt number via Pr. From the figure, we can see that, when the heat generation/absorption parameter increases, the Nusselt number increases.

The effects of the Prandtl number on the surface heat transfer rate via the radiation parameter are shown in Fig. 14. From this figure, we can see that the magnitude of the surface heat transfer rate decreases when r and Pr increase.

6 Main results

The radiative Oldroyd-B fluid flow with heat generation/absorption and melting heat transfer is studied. The key points of the given problem are as follows:

(ⅰ) The behaviors of the Deborah numbers β₁ and β₂ are opposite on the velocity profile.

(ⅱ) There is decay in the velocity when the Hartman number enhances.

(ⅲ) The temperature field increases when Pr increases.

(ⅳ) The temperature decreases when the melting parameter increases.

(ⅴ) The surface heat transfer rate increases when γ increases.

(ⅵ) The Maxwell model is taken when λ₂=0, whereas a viscous fluid model is obtained when λ₁=λ₂ =0.