Regular perturbation solution of Couette flow (non-Newtonian) between two parallel porous plates: a numerical analysis with irreversibility

The unavailability of wasted energy due to the irreversibility in the process is called the entropy generation. An irreversible process is a process in which the entropy of the system is increased. The second law of thermodynamics is used to define whether the given system is reversible or irreversible. Here, our focus is how to reduce the entropy of the system and maximize the capability of the system. There are many methods for maximizing the capacity of heat transport. The constant pressure gradient or motion of the wall can be used to increase the heat transfer rate and minimize the entropy. The objective of this study is to analyze the heat and mass transfer of an Eyring-Powell fluid in a porous channel. For this, we choose two different fluid models, namely, the plane and generalized Couette flows. The flow is generated in the channel due to a pressure gradient or with the moving of the upper lid. The present analysis shows the effects of the fluid parameters on the velocity, the temperature, the entropy generation, and the Bejan number. The nonlinear boundary value problem of the flow problem is solved with the help of the regular perturbation method. To validate the perturbation solution, a numerical solution is also obtained with the help of the built-in command NDSolve of MATHEMATICA 11.0. The velocity profile shows the shear thickening behavior via first-order Eyring-Powell parameters. It is also observed that the profile of the Bejan number has a decreasing trend against the Brinkman number. When ηi → 0 (i = 1, 2, 3), the Eyring-Powell fluid is transformed into a Newtonian fluid.


Introduction
The study in non-Newtonian fluids gets much attention of researchers due to its great importance in medical sciences, industries, and biological outcomes. It plays a vital role in bubble absorption, boiling, and polymer and plastic foam manufacturing. The relation between the stress and strain is more complex in non-Newtonian fluids as compared with that in Newtonian fluids. Further, these fluids deviate from Newton's law of viscosity. The nature of the non-Newtonian fluids cannot be understood from the single constitutive equation due to its complexity. Due to this fact, various non-Newtonian constitutive models have been presented by scientists and researchers such as the Sisko-fluid model, the Casson fluid model, the FENE-P fluid model, the Power-law fluid model, the viscoelastic fluid model, the micropolar fluid model, and the Eyring-Powell fluid model [1][2][3][4][5][6][7][8][9][10] . Ogunseye and Sibanda [11] worked on the Eyring-Powell fluid model subject to various boundary constraints. First, this fluid model is derived from the kinetic theory of fluids rather than the empirical relation. Second, this fluid model can be reduced into a Newtonian fluid in a certain shear rate. The model of an Eyring-Powell fluid can be used to formulate the flow of modern industrial materials such as ethylene glycol and powder graphite. The heat and mass transfer of an Eyring-Powell fluid in the channels and pipes has been studied by many researchers. Ali et al. [12] discussed the heat and mass transfer of an Eyring-Powell fluid in a pipe under the effect of viscous dissipation. They used the perturbation and shooting methods to handle the nonlinear boundary-value problems. The effect of variable viscosity on the Eyring-Powell fluid was analyzed by Nazeer et al. [13][14] . They used numerical and analytical techniques to solve the non-dimensional equations. The flow and heat transfer of an Eyring-Powell fluid in a porous channel was investigated by Khan et al. [15] . They used the homotopy analysis method and the shooting method based on Newton's Raphson method to obtain the solution to the problem. Riaz et al. [16] analyzed the effect of heat and mass transfer of an Eyring-Powell fluid within a rectangular complaint channel. They used the perturbation method to find the analytical expression of velocity and temperature fields. Waqas et al. [17] used the generalized Fourier's and Fick's laws to discuss the heat and mass transfer of an Eyring-Powell fluid over a stretching cylinder.
Recently, the major challenge in various industrial processes is the loss of energy. The attention of many researchers has been shifted towards the entropy generation in the flow of non-Newtonian fluids. The applications of entropy generation can be found in heat exchanger pumps and electronic cooling systems. Scientists and engineers are working to find methods on how to control the wastage of useful energy. For this, the second law of thermodynamics is the best, accurate, and eminent tool for optimizing a given system. According to Bejan [18] , various investigations have been presented and available in the literature to minimize the entropy generation based on the second law of thermodynamics. For this, Sivaraj and Sheremet [19] analyzed the free convection flow with the effect of entropy generation of magnetohydrodynamic (MHD) ferrofluids in a conduit which was heated from the bottom side. They used the finite difference scheme to solve the given nonlinear partial differential equations, and concluded that the mean heat transfer rate and mean entropy generation have the inverse relation via the magnetic parameter. The effect of the periodic magnetic field on the natural convection flow and entropy generation on a ferrofluid was investigated by Mehryan et al. [20] . The finite element method was used to solve the highly nonlinear boundary value problem. They observed that the mean heat transfer rate and the mean entropy generation of the uniform magnetic field are lower than the periodic magnetic field. Ganesh et al. [21] used the second-order thermal slip condition and entropy generation to investigate the buoyancy-driven flow of viscous fluid over a stretching surface.
Motivated by the above work, the research on entropy generation on the plane and generalized Couette flows of an Eyring-Powell fluid in a porous channel was carried out for the first time. We consider two flow phenomena, namely, the plane and generalized Couette flows, and calculate the velocity distribution, the temperature field, the entropy, and the Bejan numbers, separately with the help of the analytical and numerical approaches. For the analytical approach, we pick the perturbation method to solve both the systems. The built-in command of NDSolve in MATHEMATICA is chosen to validate the analytical solution. The computational results of both flow phenomena are discussed in a graphical form.

Mathematical formulations with geometry
Let us consider the two-type flows, namely, the plane and generalized Couette flows of the Eyring-Powell fluid in a horizontal porous channel, as shown in Fig. 1. The distance between the lower plate and the upper plate is 2H, in which the lower and upper plates are set at y = −H, and y = H, respectively. It is assumed that the lower plate is maintained with the temperature θ 1 , and the upper plate has a temperature θ 2 . The dimensionless equations for the given flow problem with the boundary conditions are given as follows. The component of the extra tensor of an Eyring-Powell fluid [22] is given by where μ is called the dynamic velocity, and γ and C are known as the material fluid parameters. Use the Taylor's series expansion as The continuity equation is The equation of motion is The energy equation is Substituting Eq. (2) into Eqs. (4)-(5), we obtain the new forms of Eqs. (4)-(5) as ∂ ∂y μ ∂u ∂y where ρ, p, k, and C P indicate the density, the pressure, the thermal conductivity, and the specific heat, respectively. The boundary conditions are given by where U 0 , θ 1 , and θ 2 are called the velocity of moving plate and the temperatures of the lower and upper walls of the channel, respectively. To convert the above system of equations into a dimensionless form, we use the following dimensionless quantities: With the help of the above equation, we obtain the dimensionless equations with the boundary conditions as where

Plane Couette flow
In 1890, a French physicist Maurice Couette presented the concept of flow between two concentric cylinders to measure the viscosity of the fluids with the help of viscosimeter. This flow is commonly used to account the dynamic viscosity between the two-concentric cylinder and parallel plates. The plane Couette flow gives the simplest and easiest model to understand the phenomenon of heat and mass transfer between co-axial infinite (finite) cylinders or parallel channels. This type of flow is very important in lubrication, food processing, and polymer solutions. The Couette flow is also known as a simple shear flow because it predicts the uniform shear stress distribution. In this flow, the velocity shows a linear relationship for the laminar flow's regime. In the plane Couette flow, both the boundaries (channel) have a different velocity. In our model, the fluid flows between two parallel porous plates due to the motion of the upper plate with the uniform velocity U 0 (moving along the x-axis direction), and the pressure gradient is zero. The equations of motion and energy for the present flow situation are given by Since an exact solution of the above developed system is not possible, to find the approximate analytical solution, we use the regular perturbation method taking ε as a perturbation parameter. For the continuous and convergent solution, we should have 0 < ε 1 [23] . The small values of the Brinkman number and the first-order non-dimensional Eyring-Powell parameter facilitate us to find the approximate analytical solution of the given problem. For this, we assume the velocity and temperature in the term of formal power series of ε n (n = 0, 1, 2, · · · ).
Substituting Eq. (17) into Eqs. (14)- (16), we obtain the following zeroth-and first-order systems of ordinary equations with boundary conditions. O(ε 0 ): O(ε 1 ): Using the values of the constants of integration into Eq. (12) and some simplification, we obtain To find the values of constants of integration, we have the imposed boundary conditions (23) and obtain Combining the solution of the velocity profile, we obtain the final expression of the velocity profile in the original parameters as . (26) Putting the values of the constants of integration and making some simplification, we obtain Solving the system of the temperature profile of order ε, we have Combining the solution of the velocity profile, we obtain the final version of the temperature profile as follows: . (29)

Entropy generation and Bejan number of plane Couette flow
The normalized form of the total entropy number is given by The first term in the above equation is due to heat generation and is denoted by N s1 , while the second term comes due to viscous dissipation and is denoted by N s2 , i.e., The analytical expressions of both terms are given by The total entropy number is defined by The relation for the Bejan number is defined by

Generalized Couette flow
This physical model is similar to the previous flow model, i.e., the plane Couette flow. The difference between this model and the previous one is the pressure gradient. In the previous case, the fluid flows due to the motion of the upper plate, and the pressure gradient is zero. However, in this scenario, the pressure gradient is not zero. The upper plate is heated, and the lower one is cold, which is similar to the previous case. With the help of the perturbation technique, we obtain the system of each order as follows.
O(ε 0 ): O(ε 1 ) : Using the values of the constants of integration and making some simplification, we obtain To find the values of constants of integration, we impose the conditions and obtain Combining the solution of the velocity profile, we obtain the final version of the velocity profile as follows: By using the boundary conditions, we have the values of constants as follows: Solving the system of the temperature profile of order ε, we obtain Combining the solution of the velocity profile, we obtain the final version of the temperature profile as follows: T =χ 10 +χ 11 e −P eξ + Br χ 12 + χ 13 e −P eξ + χ 14 ξ + χ 15 e − Reξ (1+η 1 )

Entropy generation and Bejan number of generalized Couette flow
The entropy numbers for the generalized Couette flow given by The relation for the Bejan number is defined by Be = N s1 /(N s1 + N s2 ).

Comparison with previous study
We compare our results with the results of Gupta and Massoudi [24] for the case of uniform viscosity model. Gupta and Massoudi [24] discussed the plane Couette flow of generalized second grade fluid between two heated walls for the case of Newtonian m = 0 and non-Newtonian m = 0 fluids under the consideration of constant and variable viscosity models. They set their plates at y = 0 and y = H, respectively. For comparison purpose, we set our channels at y = 0 and y = H. We present the values of velocity and temperature gradients for the Newtonian case against the variations of pressure gradient and viscous dissipation parameters in Table 1, and note that our solutions and those of Gupta and Massoudi [24] agree with each other. Table 1 Comparison with Gupta and Massoudi [24] , when m = η1 = η2 = η3 = Re = P e = 0

Px
Br Gupta and Massoudi [24] Present  Fig. 2. From this figure, it is observed that the velocity of the fluid has a decreasing behavior versus the Eyring-Powell parameter, while an opposite behavior is seen via the Reynolds number. It is noted that when η i → 0 (i = 1, 2, 3), the flow will be Newtonian, and the maximum velocity will be achieved in this situation. Further, the fluid gets thicker by increasing the values of the Eyring-Powell parameter η i . As a result, the viscosity of the fluid decreases. Physically, the Eyring-Powell parameter predicts the shear-thickening effects. Figure 3   The reason is that in the case of the plane Couette flow, the flow is just due to the top lid and the pressure gradient is not taken, for the case of generalized Couette flow, the source of the fluid flowing is a constant pressure gradient and motion of the top lid. The velocity of the generalized Couette flow increases with the increasing values of the pressure gradient parameter. The physical justification is that when more pressure is applied on the fluid, more fluid will take place in the plates with a higher velocity. Figures 6 and 7 depict the effects of the pressure gradient parameter, the Eyring-Powell parameter, the Brinkman number, and the Peclet number on the temperature profiles. It is noted that the temperature is an increasing function of the Brinkman number, the pressure gradient parameter, and the Peclet number, but a decreasing function of only the Eyring-Powell parameter. The effects of the Eyring-Powell parameters, the Brinkman number, and the Peclet number on the total entropy number are displayed in Figs. 8 and 9. The total entropy number shows an increasing trend via the Peclet number, the Eyring-Powell parameter, and the Brinkman number, and an opposite behavior is observed via the second Eyring-Powell parameter. It is observed that the total entropy number gets the maximum values at the left wall and approaches zero at the end of the channel. The profiles of the Bejan number versus the length of the channel under the effects of the Reynolds number, the Brinkman number, the Peclet number, and the first-order Eyring-Powell parameter are shown in Figs. 10 and 11. The profiles show that the Bejan number is the maximum at the end of the channel and the minimum at the start of the channel. It is also shown that the

Conclusions
The analytical and numerical solutions of the plane Couette flow and the generalized Couette flow of an Eyring-Powell fluid through a porous channel are investigated. In the present study,  (iv) The entropy generation is the maximum at the lower wall and approaches zero at the top wall.
(v) The entropy number increases near the lower wall via the first-order Eyring-Powell parameter, the Brinkman number, and the Peclet number, while the entropy number decreases via the third-order Eyring-Powell parameter.
(vi) The profile of the Bejan number decreases via the Brinkman number in the entire channel.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.  (1 + η1) 2 ((1 + η1)P e − 4Re) " ,