Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface

: The main intention of this article is to examine the heat transmission of the flow of Eyring– Powell fluid over an unstable oscillatory porous stretching surface. The effect of thermal radiation on the fluid flow is investigated, where the flow is actuated by the unbounded flexible surface which is extended occasionally to and fro in its plane. The rudimentary leading equations are changed to differential equations through the use of applicable similarity variables. An optimal and numerical approach has been used to find the solution of the modeled problems. The convergence of the homotopy analysis method has been shown numerically. The homotopy analysis method predictions of the structures formed are in close agreement with the obtained results from the numerical method. Comparisons between HAM and numerical methods are shown graphically as well as numerically. The convergence of this method has been shown numerically. The impacts of the skin friction and heat flux are shown through a table. The influences of the porosity, oscillation, thermal radiation, and heat absorption/generation are the main focus of this work. The consequences of emerging parameters are demonstrated through graphs.


Introduction
Boundary layer fluid flow problems in different dimensions through a stretching sheet with heat transfer and magnetohydrodynamic effects have plentiful and inclusive applications in several engineering and industrial sectors. They include glass blowing melt spinning, heat exchanger designing, fiber and wire coating, production of glass fibers, industrialization of rubber and plastic sheets, etc. In addition, the action of thermal radiation is very vital in calculating heat transmission in the polymer treating industry. In investigations of all these applications, many investigators deliberate the flow of different fluid models over a stretching sheet. Sakiadis [1] has studied boundary layer flow over a flat surface. Crane [2] has obtained the closed-form solution for the flow instigated by the stretching of a flexible parallel sheet moving periodically. Gupta and Gupta [3] have extended this work by considering suction/blowing at the surface of the sheet. The dissemination of chemically reactive species over a moving continuous sheet was deliberated by Anderson et al. [4]. Pop [5] deliberated time-dependent flow over a stretched surface. The impact of heat transmission on secondgrade fluid over a stretching sheet was explored by Cortell et al. [6]. Areal [7] studied an asymmetric viscoelastic fluid flow past a stretching sheet for different purposes in the field of fluid. Rashdi et al. [8,9] studied entropy generation in magneto hydrodynamic Eyring-Powell and Carreau nanofluid through a permeable stretching surface. Hayat et al. [10][11][12][13] have studied boundary layer flow using different phenomena.
There are no solitary constitutive equations of non-Newtonian fluid which clarify all the distinctive aspects of the compound rheological fluids. The Eyring-Powell model [14] an important subclass of these, modeled from the kinetic theory of liquids instead of the experimental relations. Recently, Prasad [15] studied heat transfer and momentum in Eyring-Powell fluid over a nonisothermal stretching sheet. Noreen et al. [16] examined the peristaltic flow of magnetohydrodynamic Eyring-Powell fluid in a channel. Ellahi [17] recently completed a numerical study of magnetohydrodynamic generalized Couette flow of Eyring-Powell fluid with heat transfer and the slip condition. Ellahi et al. [18] examined the shape effects of spherical and nonspherical nanoparticles in mixed convection flow over a vertical stretching permeable sheet. Other related studies concerning Eyring-Powell fluid can be seen in [19][20][21][22][23][24][25].
Thermal radiation is the procedure in which energy is released in the form of electromagnetic radiation by a surface in all directions. Thermal radiation has numerous uses in the areas of engineering and heat transfer analysis. In the case of conduction and convection, energy transmission amongst objects depends almost entirely on the temperature. For natural free convection, or when variable property effects are included, the power of temperature difference may be slightly larger than one, and can reach two. Tawade et al. [26] investigated a thin liquid flow through a stretching surface with the influence of thermal radiation and a magnetic field. A brief discussion was given on physical parameters in his work. Ellahi et al. [27] examined the boundary layer magnetic flow of nano-Ferroliquid under the influence of low oscillation over a stretchable rotating disk. Zeeshan et al. [28] studied the effect of a magnetic dipole on viscous ferrofluid past a stretching surface with thermal radiation. The Hall effect on Falkner-Skan boundary layer fluid flow over a stretching sheet was examined by Maqbool et al. [29]. The enhancement of heat transfer and heat exchanger effectiveness in a double-pipe heat exchanger filled with porous media was examined by Shirvan et al. [30]. Ramesh et al. [31] studied the Casson fluid flow near the stagnation point over a stretching sheet with variable thickness and radiation. Other related studies concerning stretching sheets can be seen in [32][33][34]. Bakier and Moradi et al. [35,36] studied the influence of thermal radiation on assorted convective flow on an upright surface in a permeable medium. Chaudhary et al. [37] investigated the thermal radiation effects of fluid on an exponentially extending surface.
The aim of the current research is to investigate the heat transmission of Eyring-Powell fluid over an unsteady oscillatory porous stretching surface. The homotopy analysis method is used in the present work for the solution of modelled equations which are nonlinear and coupled. The homotopy analysis method is a substitute method and its main advantage is in its application to nonlinear differential equations without discretization and linearization. In 1992, Liao [38][39][40] was the first to investigate this technique for the solution of that type of problem and generally proved that this method is rapidly convergent to the approximated solutions. Solutions using this technique are significant because they involve all the physical parameters of the problem and we can easily discuss their behavior. Due to its fast convergence, many researchers [41][42][43][44] have used this procedure to solve highly nonlinear combined equations. The effects of all the embedding parameters have been studied graphically. Khan et al. [45] studied the flow and heat transfer of Eyring-Powell fluid over an oscillatory stretching sheet with thermal radiation. Shah et al. [46,47] studied the effects of Hall current on three-dimensional non-Newtonian nanofluids and micropolar nanofluids in a rotating frame. Hameed et al. [48] investigated the combined magnetohydrodynamic and electric field effect on an unsteady Maxwell nanofluid flow over a stretching surface under the influence of variable heat and thermal radiation. Recently Muhammad et al. [49] studied the rotating flow of magneto hydrodynamic carbon nanotubes over a stretching sheet with the impact of non-linear thermal radiation and heat generation/absorption

Formulation of the Problem
Consider a two-dimensional incompressible boundary layer flow of Eyring-Powell fluid over an oscillating stretched sheet concurring with plane y (Figure 1). In the Cartesian coordinate system, x is beside the sheet and y is vertical to the sheet. The fluid flow is assumed to be in an unsteady state and the stretching sheet is kept porous. Let T w denote the surface temperature and T ∞ denote the temperature of the fluid as the distance from the surface tends to infinity. It is assumed that T w > T ∞ . The Cauchy stress tensor for Eyring-Powel fluid [15][16][17][18][19][20][21][22][23][24][25] is where  ij for Eyring-Powel fluid is Here, α and c denote the Eyring-Powel fluid constants. We expand the term 1 1 sinh i j u cx        as below: Using the boundary layer approximations, continuity, energy equations, and momentum are as: where   denotes the Stefan-Boltzmann constant and k is the absorption coefficient.
Expanding Equation (7) by Taylor series, we obtain By neglecting higher terms from Equation (8), we get In observation of Equations (7) and (8), Equation (6) The subjected boundary condition for the flow phenomena [45] is where the dimensionless variables are given as In observation of the dimensionless variables defined above, Equations (5) and (10) reduce to with boundary conditions In observation of Equation (12), Equation (16) takes the following forms:

Solution by HAM
Liao was the first person who used the basic idea of topology called homotopy and derived a new method known as the homotopy analysis method. He used two homotopic functions to derive this technique. Two functions are called homotopic functions when one of them can be continuously distorted into the other. Let  : such that, and . , This mapping f is then called homotopic.
In order to solve Equations (13) and (14) with the boundary Condition (15), we use the HAM according to the following process. The preliminary guesses are The linear operators are taken as These operators have the following properties: The zero-order problems from Equations (13) and (14) are The equivalent boundary conditions are  (26) where 01  is the embedding parameter. When 0  and 1,  we have ( ;1) ( ) and G( ;1) ( ).
Expanding ( ; ) F  and G( ; )  by Taylor series, Setting 1  in (29), we obtain The q th-order problem satisfies the following: with the conditions Here,

HAM Solution Convergence
When we compute the series solutions of the velocity and temperature functions in order to use HAM, the assisting parameters

Results and Discussion
In this section, we argue the special effects of the concerned parameters graphically. In all the graphs, the values of  and  are chosen such that the product K  should be much smaller than one. Figure 4 shows the effect of the rate of the relative amplitude of frequency and the stretching rate A on the time series of the velocity distribution. It is observed that the amplitude of the flow motion falls with large values of . A Figure 5 demonstrates the effect of A on the temperature profile. It is observed that the temperature profile G decreases as A increases. Actually, the amplitude of oscillation rises for large values of A which, in turn, decreases the temperature. The influence of the Prandtl number Pr on the temperature distribution is shown in Figure 6. The temperature distribution varies inversely with Pr It is clear that the temperature distribution decreases for large Pr and increases for small values of Pr . Physically, the fluids with small Pr have larger thermal diffusivity, and this effect is opposite for a higher Prandtl number Pr . Due to this fact, a large Pr causes the thermal boundary layer to decrease. The effect is even more distinct for small Pr since the thermal boundary layer thickness is relatively large. The impact of the thermal radiation parameter Rd is presented in Figure 7. Thermal radiation has a dominating role in the comprehensive surface heat transmission, when the coefficient of convection heat transmission is small. When we increase the thermal radiation parameter Rd, we see that it augments the temperature in the boundary layer area in the fluid layer. Figure 8 represents the influence of  on the velocity profile. It is noticed that the increasing value of  increases the velocity of the fluid during oscillation. The features of the porosity parameter  on the velocity field are shown in Figure  9, and have an imperative character in terms of the flow motion. The higher values of  increase the porous space; this produces resistance in the flow path and reduces the flow motion. In fact, growing values of  show a large number of porous spaces, which create resistance in the flow path and reduce overall fluid motion. Figure 10 shows the influence of the dimensionless fluid parameter  on the velocity profile. Large values of  speed up the flow motion and increase its oscillation. Figure 11 presents the influence of the heat source/sink  on the temperature profile. It is obvious from the figure that increases in the value enhance the temperature profile  of the flow. This occurs due to the fact that the different values of  perform like a heat generator, releasing heat energy to the flow. This helps to develop the thermal boundary layer thickness. Figures 12 and 13 show the effect of different values of  on the velocity and temperature profiles. Greater values of  increase the temperature and velocity profiles. Figures 14 and 15 show the comparison between HAM and numerical solutions using the NDsolve technique on velocity and temperature profiles, respectively. An excellent agreement is found between the homotopy analysis method and the ND-solve technique.         G 

Table Discussion
The physical quantities such as skin friction coefficient   Tables 3 and 4. Table 3 shows the comparison of the HAM and numerical solution for the velocity profile, while Table 4 shows the comparison of the HAM and numerical solution for the temperature profile.

Conclusions
In this article, we have analyzed an Eyring-Powell fluid over an oscillatory thermally conductive stretching sheet in the presence of thermal radiation and a heat source/sink. A coordinate transformation was used to transform the semi-infinite flow domain to a finite computational domain. The homotopy analysis method was used to solve the modeled problem. The main remarks from this study are as follows:


The amplitude of the velocity decreases with increasing A and porosity  , while it increases with an increase in dimensionless fluid parameters K and .

 
The temperature increases with increasing A , radiation parameter Rd , and heat source/sink  , while it decreases with increasing Prandtl number Pr and ratio of the oscillation frequency of the sheet to its stretching rate A .


The local Nusselt number increases with increasing Prandtl number Pr , heat source/sink  , dimensionless fluid parameter , and radiation parameter Rd , while it decreases with increasing porosity  and dimensionless fluid parameter  .
Author Contributions: Abdullah and Zahir Shah modelled the problem. Waris Khan solved the problem. Abdullah and Muhammad Idrees wrote the manuscript. Zahir Shah and Saeed Islam thoroughly checked the mathematical modeling and English language corrections. Saeed Islam and Taza Gul contributed in the results and discussions. All the corresponding authors finalized the manuscript after its internal evaluation.

Conflicts of Interest:
The authors declare no conflict of interest.