Entropy Generation in MHD Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface under the Impact of Thermal Radiation and Heat Source/Sink

In this article, we have briefly examined the entropy generation in magnetohydrodynamic (MHD) Eyring–Powell fluid over an unsteady oscillating porous stretching sheet. The impact of thermal radiation and heat source/sink are taken in this investigation. The impact of embedded parameters on velocity function, temperature function, entropy generation rate, and Bejan number are deliberated through graphs, and discussed as well. By studying the entropy generation in magnetohydrodynamic Eyring–Powell fluid over an unsteady oscillating porous stretching sheet, the entropy generation rate is reduced with escalation in porosity, thermal radiation, and magnetic parameters, while increased with the escalation in Reynolds number. Also, the Bejan number is increased with the escalation in porosity and magnetic parameter, while increased with the escalation in thermal radiation parameter. The impact of skin fraction coefficient and local Nusselt number are discussed through tables. The partial differential equations are converted to ordinary differential equation with the help of similarity variables. The homotopy analysis method (HAM) is used for the solution of the problem. The results of this investigation agree, satisfactorily, with past studies.


Introduction
A non-Newtonian Fluid has exclusive features: it illustrates both the properties of liquid and solid, as the relationship between the shear stress and the shear rate becomes non-linear. In everyday life, industries, and technologies non-Newtonian fluids are used frequently. Non-Newtonian fluid flow problems in different dimensions, through a porous stretching sheet with heat transfer and magnetohydrodynamic effects, have plentiful and inclusive applications in several engineering and industrial sectors. They include heat exchanger design, glass blowing melt spinning, production of glass fibers, fiber and wire coating, industrialization of rubber and plastic sheets, etc. Eyring-Powell

Mathematical Modeling
Consider a two-dimensional (2-D) incompressible boundary layer flow of Eyring-Powell fluid over an oscillatory stretching sheet concurring with plane y. In this Cartesian coordinate system, x is parallel to the stretching sheet, and y is perpendicular to the stretching sheet. The stretching sheet is kept porous, and the flow is supposed in an unsteady state. The magnetic field is applied in the y direction. It is assumed that T w > T ∞ , where T w is the surface temperature and T ∞ is the temperature as the distance from the surface tends to infinity. The physical model of the problem is shown in Figure 1.

Mathematical Modeling
Consider a two-dimensional (2-D) incompressible boundary layer flow of Eyring-Powell fluid over an oscillatory stretching sheet concurring with plane y. In this Cartesian coordinate system, x is parallel to the stretching sheet, and y is perpendicular to the stretching sheet. The stretching sheet is kept porous, and the flow is supposed in an unsteady state. The magnetic field is applied in the y direction. It is assumed that w T T ∞ > , where w T is the surface temperature and T ∞ is the temperature as the distance from the surface tends to infinity. The physical model of the problem is shown in Figure 1. The above-stated problem satisfies all the conditions [29].
In Equations (1)-(3), u and v are the velocity components in the direction of x and y , respectively. Also, ν is the kinematic viscosity, σ is the electrical conductivity, 牋 and牋 C α are the fluid materials, ρ is the density, p C is the specific heat, k is the thermal conductivity, r Q is the heat source, and r q is the radiative heat flux. r q is defined as The above-stated problem satisfies all the conditions [29].
∂u ∂x In Equations (1)-(3), u and v are the velocity components in the direction of x and y, respectively. Also, ν is the kinematic viscosity, σ is the electrical conductivity, α and C are the fluid materials, ρ is the density, C p is the specific heat, k is the thermal conductivity, Q r is the heat source, and q r is the radiative heat flux. q r is defined as where σ * (Stefan-Boltzmann constant) and k * (absorption coefficient). Expending T 4 by Taylor series expansion, we obtained [28] Neglecting higher terms from Equation (5) and substituting in Equation (3), we have With the following boundary conditions we introduce the following dimensionless variables for the non-dimensionalization of the flow problem.
An influential tool in fluid mechanics is the thought of dimensional analysis and scaling laws; by looking at the physical properties present in a system, we may estimate their size and hence which, for example, might be neglected. In some cases, the system may not have a fixed natural length scale (timescale) while the solution depends on space (time). It is then necessary to construct a length scale (timescale) using space (time) and the other dimensional quantities. In a study of partial differential equations, particularly fluid dynamics, similarity variables is a form of solution which is similar to itself if the independent and dependent variables are appropriately scaled.
In the observation of above defined dimensionless variables, Equation (1) satisfied (2) and (6), and can be written as with the following boundary conditions In the above equations, P = 1 µαC and ň = x 2 c 3 2νC 2 are the fluid parameters, β = ν kc is the porosity parameter, D = ξ c depicts the oscillating frequency, M = σB 0 c is the magnetic field, ε = νQ r kcρC p is the heat source/sink, Pr = µC p k is the Prandtl number, and K r = 4σ * T 3 ∞ T kk * is the radiation parameter. Equation (11) is subject to the constraints ňP << 1.

Physical Quantities of Interest
For engineering interest, the skin fraction coefficient C f and local Nusselt number Nu is defined as In observation of Equation (9), Equation (12) can be written as

Entropy Generation and Bejan Number
For the above-stated problem, the local entropy generation rate can be defined as [10] Executing Equation (8), the above equation becomes where The Bejan number is defined as

Solution by HAM
Homotopy analysis method was introduced by Liao [30][31][32] for the first time. He used one of the elementary ideas of topology, called homotopy, to derive this method. He used two homotopic functions in the derivation of this technique. The functions are called homotopic function when one of them can be continuously distorted into another. Assume that Z 1 , Z 2 are two functions which are continuous, and X, Y are two topological spaces where Z 1 & Z 2 map from X to Y, then Z 1 is said to be homotopic to Z 2 if there is a continuous function , Thus, The mapping is called homotopic. In order to solve Equations (9) and (10) with the boundary conditions Equation (11), we use HAM [38][39][40][41] with the succeeding process.
The initial suppositions are chosen as follows: The linear operators are taken as L f and L g : which have the following succeeding properties: where n i (i = 1 − 5) are constants.
The resultant non-linear operators, NL f and NL g , are specified as The zero-order problems from Equations (9) and (10) are The equivalent boundary conditions are When Ω = 0 and Ω = 1, we have Expanding f (ϕ; Ω) and g(ϕ; Ω) by Taylor series, where Setting Ω = 1 in (28), we obtain The qth-order problem satisfies the following: with the conditions Here, where the overall homotopic series solutions in general form are specified as

HAM Convergence
Whenever we calculate the series solution of the velocity function and temperature function by using HAM, the parameters h f and h g , which are called assisting parameters, appear. The function of these parameters is to adjust the convergence of these solutions. At the 5th order approximation, the h-curves of f (0) and g (0) are plotted in Figures 2 and 3, respectively. The convergence region of the velocity function is −0.8 ≤ h f ≤ −0.1, and the convergence region of the temperature function is −1.0 ≤ h g ≤ −0.5.    Tables 1 and 2 Tables 1 and 2 depict the influence of various parameters on skin fraction coefficient C f x and local Nusselt number Nu x . These tables express the best agreement with our previous study. The impacts of emerging parameters on skin fraction coefficient are presented in Table 1. From the tabulated values, we see that the skin fraction coefficient reduces with the escalation in β, P, and ň. The impacts of emerging parameters on local Nusselt number is presented in Table 2. The local Nusselt number reduces with the escalation in Pr, while it escalates with the escalation in ε and K r .

Graphical Discussion
In this section, we have discussed the influences of different embedded parameters and dimensionless numbers on velocity function f (ϕ), temperature function g(ϕ), entropy generation rate N G , and Bejan number Be. These embedded parameters and dimensionless numbers are oscillating frequency D, porosity parameter β, fluid parameters P and ň, magnetic parameter M, heat source/sink ε, Prandtl number Pr, rotation parameter K r , and Reynolds number Re. To comprehend the influence of these parameters and dimensionless numbers, Figures 4-19 are schemed. Figure 4 displays the influence of oscillating frequency D on velocity function f (ϕ). It is observed that augmented values of D increased the flow motion. Actually, the higher value of oscillating frequency D increases the kinetic energy of fluid molecules which result in increases in the flow motion. The impact of porosity parameter β on velocity function f (ϕ) is shown in Figure 5, which has a dominating effect on the flow motion. Generally, the porosity creates resistance in the flow path, and declines the velocity of the flow motion. In fact, growing values of β show the large number of porous spaces, which create resistance in the flow path and reduce overall fluid motion. Basically, with the increase, the number of holes in the porous plates are increased. The nanoliquid particle aspect hurdles in, flowing over these holes. Hence, it is obvious that the increasing values of β reduce the velocity function f (ϕ).   Figure 8 shows the impact of magnetic field M on velocity function f (ϕ). Lorentz force theory says that the magnetic field has a reverse effect on velocity function. Hence, the higher values of M reduce f (ϕ). This important effect of M on velocity profile f (ϕ) is because of the fact that the increases in the M movements, or the friction force, is named the Lorentz force. It has the affinity to reduce the fluid velocity in the boundary sheet. Figure 9 shows the impact of heat source/sink ε on temperature function g(ϕ). Generally, the heat source/sink performs like a heat generator, which releases heat to the flow of fluid. Therefore, the enhancement in ε improves the temperature field g(ϕ). In addition, this helps to grow the thickness of the boundary layer. Figure 10 demonstrates the impact of oscillating frequency D on temperature function g(ϕ). Generally, the high oscillating frequency reduces the temperature function much more. Hence, the increasing values of D reduce g(ϕ). Figure 11 demonstrates the impact of radiation parameter K r on temperature function g(ϕ). Thermal radiation has a leading role in heat transmission when the coefficient of convection heat transmission is small. The enhancement in K r improves g(ϕ). Actually, when we increase thermal radiation parameter Rd, then it is apparent that it enhances the temperature in the boundary layer area in the fluid layer. This increase leads to a drop in the rate of cooling for nanofluid flow. Figure 12 shows the impact of Prandtl number Pr on temperature function g(ϕ). Physically, the nanofluids have a large thermal diffusivity with small Pr, but this effect is reversed for higher Pr, therefore, the temperature of liquid shows a decreasing behavior. Physically, the fluids having a small number of Pr have a larger thermal diffusivity, and this effect is opposite for higher Prandtl numbers. Due to this fact, a large value of Pr causes the thermal boundary layer to drop. Figures 13 and 14 demonstrate the influence of porosity parameter β on the entropy generation rate N G and Bejan number Be, respectively. From these figures, we observe that the porosity parameter β has a reversed impact on N G and Be. Figures 15 and 16 demonstrate the impact of magnetic parameter M on generation rate N G and Bejan number Be, respectively. Here, M has a reversed impact on N G and Be. That is, the enhancement in magnetic parameter reduces N G and Be. Figures 17 and 18 are plotted to describe the impact of radiation parameter K r on generation rate N G and Bejan number Be, respectively. From Figure 17, we observed that the increase in K r reduces N G , while Figure 18 depicts the reverse impact on Be. Figure 19 demonstrates the impact of (Re) on N G . From this figure, we observed that the increasing Reynolds number increases the entropy generation.

Conclusions
In this article, we investigated the entropy generation on MHD Eyring-Powell fluid over an unsteady oscillatory porous stretching sheet. The impact of thermal radiation and heat source/sink is taken into account. Also, this article is compared with one from our previous study, and found to

Conclusions
In this article, we investigated the entropy generation on MHD Eyring-Powell fluid over an unsteady oscillatory porous stretching sheet. The impact of thermal radiation and heat source/sink is taken into account. Also, this article is compared with one from our previous study, and found to agree satisfactorily. The concluding remarks of this study are listed below: • The velocity function reduces with the enhancement in magnetic field and porosity parameter,

Conclusions
In this article, we investigated the entropy generation on MHD Eyring-Powell fluid over an unsteady oscillatory porous stretching sheet. The impact of thermal radiation and heat source/sink is taken into account. Also, this article is compared with one from our previous study, and found to agree satisfactorily. The concluding remarks of this study are listed below:

•
The velocity function reduces with the enhancement in magnetic field and porosity parameter, and escalates with the enhancement in oscillating frequency and fluid parameters.

•
The temperature function reduces with the enhancement in oscillating frequency and Prandtl number, and escalates with the enhancement in heat source/sink and radiation parameter.

•
The entropy generation rate reduces with the escalation in porosity parameter, thermal radiation, magnetic field, and escalates with the enhancement in Reynolds number.

•
The Bejan number reduces with the enhancement in porosity parameter and magnetic field, and increases with the enhancement in thermal radiation parameter.