Entropy Generation on Mhd Eyring–powell Nanofluid through a Permeable Stretching Surface

In this article, entropy generation of an Eyring–Powell nanofluid through a permeable stretching surface has been investigated. The impact of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also taken into account. The governing flow problem is modeled with the help of similarity transformation variables. The resulting nonlinear ordinary differential equations are solved numerically with the combination of the Successive linearization method and Chebyshev spectral collocation method. The impact of all the emerging parameters such as Hartmann number, Prandtl number, radiation parameter, Lewis number, thermophoresis parameter, Brownian motion parameter, Reynolds number, fluid parameter, and Brinkmann number are discussed with the help of graphs and tables. It is observed that the influence of the magnetic field opposes the flow. Moreover, entropy generation profile behaves as an increasing function of all the physical parameters.


Introduction
In recent years, nanofluid has received more and more attention by various scientists due to its numerous applications in engineering and various industrial processes.A nanofluid is comprised of a base fluid with tiny (nanometer) sized nanoparticles, such as carbides or carbon nanotubes, oxides, and metals, whereas traditional base liquids involve ethylene glycol, oil, and water.A nanofluid is very helpful in enhancing thermal conductivity and convection of heat transfer coefficient when it is analyzed with the base fluid.In modern technology, nanomaterials are becoming increasingly important in the performance of various heat exchangers, such as microelectronics, optical modulators, and chemical production.Magneto-nanofluids are also remarkable for their use in various applications, such as tunable optical fiber filters, magneto-optical wavelength filters, optical modulators, and optical switches.In biomedical engineering, magneto-nanoparticles are also very helpful in cancer therapy, sink-float separation, hyperthermia, magnetic resonance imaging (MRI), magnetic cell separation, drug delivery, and magnetic drug targeting.In particular, heat transfer and convective flow are influenced by the features of nanofluids, such as thermal conductivity and viscosity.Conventional heat transfer in various Newtonian and non-Newtonian fluids, such as ethylene glycol, oil, water, etc., holds a poor rate of heat transfer.However, the thermal conductivity of these kinds of fluids plays a significant He obtained the exact solutions for the governing flow problem.He also presented multiple solutions for the non-MHD stretching plate problem.A few recent studies on MHD can be found in [31][32][33][34], with a few additional studies referenced therein.According to the best of our knowledge, no such attempt has been made on studying entropy generation of a MHD Eyring-Powell fluid through a permeable stretching sheet.
With motivation from the above analysis in mind, the aim of the present study was to analyze the entropy generation of a MHD Eyring nanofluid over a stretching surface.The governing flow problem comprises of the momentum equation, energy equation, and nanoparticle concentration equation, which are further transformed into ordinary differential equations using similarity transformation variables.The reduced ordinary coupled differential equations are solved numerically with the help of the Successive linearization method (SLM) and Chebyshev spectral collocation method.This paper is organized as follows: after the introduction in Section 1, Section 2 consists of the mathematical formulation of the problem, Section 3 deals with the physical quantities, Section 4 explains the methodology of the problem, Section 5 characterize the entropy generation analysis, and finally Section 6 is devoted to the numerical results and discussion.

Mathematical Formulation
Consider the MHD boundary layer flow of an Eyring-Powell nanofluid over a permeable stretching surface near a stagnation point at y " 0. The MHD flow occurs in the domain at y ą 0. The fluid is electrically conducting due to an externally applied magnetic field, although the induced magnetic charge is very small, and is thereby taken to be zero.A cartesian coordinate is chosen in a way such that the x-axis is considered along the direction of the sheet whereas the y-axis is considered normal to it (see Figure 1).Suppose that r T w and r C w are the temperature and nanoparticle fraction at the sheet, respectively, while the temperature and nano-particle fraction at infinity are r T 8 and r C 8 , respectively.The velocity of the sheet is considered along the x-direction.shrinking or stretching two-three dimensional objects.He obtained the exact solutions for the governing flow problem.He also presented multiple solutions for the non-MHD stretching plate problem.A few recent studies on MHD can be found in [31][32][33][34], with a few additional studies referenced therein.According to the best of our knowledge, no such attempt has been made on studying entropy generation of a MHD Eyring-Powell fluid through a permeable stretching sheet.With motivation from the above analysis in mind, the aim of the present study was to analyze the entropy generation of a MHD Eyring nanofluid over a stretching surface.The governing flow problem comprises of the momentum equation, energy equation, and nanoparticle concentration equation, which are further transformed into ordinary differential equations using similarity transformation variables.The reduced ordinary coupled differential equations are solved numerically with the help of the Successive linearization method (SLM) and Chebyshev spectral collocation method.This paper is organized as follows: after the introduction in Section 1, Section 2 consists of the mathematical formulation of the problem, Section 3 deals with the physical quantities, Section 4 explains the methodology of the problem, Section 5 characterize the entropy generation analysis, and finally Section 6 is devoted to the numerical results and discussion.

Mathematical Formulation
Consider the MHD boundary layer flow of an Eyring-Powell nanofluid over a permeable stretching surface near a stagnation point at = 0.The MHD flow occurs in the domain at > 0.
The fluid is electrically conducting due to an externally applied magnetic field, although the induced magnetic charge is very small, and is thereby taken to be zero.A cartesian coordinate is chosen in a way such that the x-axis is considered along the direction of the sheet whereas the y-axis is considered normal to it (see Figure 1).Suppose that and are the temperature and nanoparticle fraction at the sheet, respectively, while the temperature and nano-particle fraction at infinity are and , respectively.The velocity of the sheet is considered along the x-direction.
The nonlinear radiative heat flux can be written as The governing equations of the MHD Eyring-Powell nanofluid model can be written as [35] B r u Bx `Br v By " 0, The nonlinear radiative heat flux can be written as and their respective boundary conditions are where b " a is considered for the present study.
and using Equation ( 8) in relation to Equations ( 1)-( 7), we get Their corresponding boundary conditions are where Pr " v a M "

Physical Quantities of Interest
The physical quantities of interest for the governing flow problem are the local Nusselt number and local Sherwood number which can be written as where q w and q m are described as Entropy 2016, 18, 224 5 of 14 With the help of dimensionless transformation in Equation ( 8), we have x " ´p1 `Nr q θ 1 p0q , Sh r " Sh x Re 1 2 x " ´φ1 p0q , where Sh r and Nu r are the dimensionless Sherwood number and local Nusselt number, respectively, and Re x " r u w x{ν is the local Reynolds number.

Numerical Method
We apply the Successive linearization method to Equation ( 9) with their boundary conditions in Equation (12), by setting [19,25] where f I are unknown functions which are obtained by iteratively solving the linearized version of the governing equation and assuming that f I p0 ď N ď I ´1q are known from previous iterations.
Our algorithm starts with an initial approximation f 0 which satisfies the given boundary conditions in Equation ( 13) according to SLM.The suitable initial guess for the governing flow problem is We write the equation in general form as where and where L and N are the linear and non-linear part of Equation (9).By substituting Equation ( 18) into Equation ( 9) and taking the linear terms only, we get and the corresponding boundary conditions becomes We solve Equation ( 23) numerically by the Chebyshev spectral collocation method.For numerical implementation, the physical region r0, 8q is truncated to r0, Γs ; we can take Γ to be sufficient large.With the help of subsequent transformations this region is further transformed into r´1, 1s, and we have We define the following discretization between the interval r´1, 1s.Now, we can apply Gause-Lobatto collocation points to define the nodes between r´1, 1s by Entropy 2016, 18, 224 6 of 14 with pN `1q number of collocation points.The Chebyshev spectral collocation method is based on the concept of differentiation matrix D. This differentiation matrix maps a vector of the function values G " r f pΩ 0 q , . . ., f pΩ N qs T the collocation points to a vector G 1 is defined as the derivative of p order for the function f pΩq can be written as The entries of matrix D can be computed by the method proposed by Bhatti et al. [25].Now, applying the spectral method, with derivative matrices on linearized Equations ( 23) and ( 24), we get the following linearized matrix system and the boundary conditions takes the following form where In the above equation A s,I´1 ps " 0, 1, . . .3q are pN `1q ˆpN `1q diagonal matrices with A s,I´1 `ΩJ ˘on the main diagonal and After employing Equation (31), the solutions for f I are obtained by iteratively solving Equation (30).We obtain the solution for f pζq from solving Equation (31) and Equations (10) and (11) are now linear; therefore we will apply Chebyshev pseudo-spectral method directly, and by doing so we get BH " S, (33) with their corresponding boundary conditions boundary conditions where H " `θ `ΩJ ˘, φ `ΩJ ˘˘, B is the set of linear coupled equations of temperature and nanoparticle concentration, S is a vector of zeros, and all vectors in Equation ( 33) are converted to a diagonal matrix.We imposed the boundary conditions in Equations ( 34) and (35) on the first and last rows of B and S, respectively.

Entropy Generation Analysis
The volumetric entropy generation of the Eyring-Powell nanofluid is given by [36] In the above equation, the entropy generation consists of three effects: (i) conduction effect (also known as heat transfer irreversibility, (HTI)); (ii) fluid friction irreversibility (FFI); and (iii) diffusion (also known as diffusive irreversibility, (DI)).The entropy generation characteristics can be written as With the help of Equation ( 8), the entropy generation in dimensionless form can be written as These numbers are given in the following form

Results and Discussion
This section deals with the theoretical and graphical behavior of different physical quantities that are obtained in the present flow problems.The computational software Matlab has been utilized to investigate the novelties of all the physical parameters, such as the Hartmann number, fluid parameter, Prandtl number, radiation parameter, Lewis number, thermophoresis parameter, Brownian motion parameter, Reynolds number, and Brinkmann number.In particular, we discuss their influence on velocity profile, temperature profile, nanoparticle concentration profile, and entropy profile.For this purpose, Figures 1-10 are drawn, where Figure 1 shows the geometry of the problem.Table 1 shows the numerical computation of the Nusselt number and Sherwood number for different values of the Prandtl number, radiation parameter, Brownian motion parameter, thermophoresis parameter, and Lewis number.Table 2 represents a numerical comparison with the existing published results [26] by taking γ " M " 0 as a special case of our study.From this table, we can see that our results are in excellent agreement, which confirms the validity of our present methodology.Figures 2 and 3 are provided for the velocity profile against Hartman number ( ), fluid parameter ( ), and suction/injection parameter ( ). Figure 2 elucidates that when the Hartmann number ( ) increases then it opposes the flow which causes a reduction in the fluid velocity.In fact, this is due to the existence of the Lorentz force which originated when the magnetic field was applied.However, we can observe that suction/injection parameter ( ) does not provide any resistance to the flow, and hence the velocity of the fluid tends to rise when the suction/injection parameter increases.Figure 3 shows that when the fluid parameter ( ) rises then it tends to oppose the flow, which causes a reduction in the velocity profile.Figures 4 and 5 are provided for the temperature profile against Brownian motion parameter, thermophoresis parameter, Prandtl number, and radiation parameter.It can be observed from Figure 4 that the Brownian motion parameter ( ) enhances the temperature profile and boundary layer thickness, however, the temperature profile behaves in a similar way when the thermophoresis parameter ( ) increases.An enhancement in the thermophoresis parameter produces a force which leads to the movement of nanoparticles from a hot region to cold region, and hence the temperature profile and thermal boundary layer thickness increases.It can be observed from Figure 5 that the radiation parameter ( ) enhances the temperature profile.In fact, this happens due to the increment in radiation parameter causing a reduction in the mean absorption coefficient, which, as a result, leads to an increase in the radiative heat transfer.Here we can also observe that larger values of the Prandtl number reduces the temperature profile and the boundary layer thickness.An increment in the Prandtl number coincides with weaker thermal diffusivity.It is worth mentioning that those fluids which hold weaker thermal diffusivity have lower temperatures.This type of thermal diffusivity reveals a reduction in the boundary layer thickness and temperature profile.
Figures 6 and 7 are provided for the concentration profile against the Brownian motion parameter, thermophoresis parameter, Lewis number, and suction/injection parameter.Figures 2 and 3 are provided for the velocity profile against Hartman number pMq, fluid parameter pγq , and suction/injection parameter pSq. Figure 2 elucidates that when the Hartmann number pMq increases then it opposes the flow which causes a reduction in the fluid velocity.In fact, this is due to the existence of the Lorentz force which originated when the magnetic field was applied.However, we can observe that suction/injection parameter pSq does not provide any resistance to the flow, and hence the velocity of the fluid tends to rise when the suction/injection parameter increases.Figure 3 shows that when the fluid parameter pγq rises then it tends to oppose the flow, which causes a reduction in the velocity profile.Figures 4 and 5 are provided for the temperature profile against Brownian motion parameter, thermophoresis parameter, Prandtl number, and radiation parameter.It can be observed from Figure 4 that the Brownian motion parameter pN b q enhances the temperature profile and boundary layer thickness, however, the temperature profile behaves in a similar way when the thermophoresis parameter pN t q increases.An enhancement in the thermophoresis parameter produces a force which leads to the movement of nanoparticles from a hot region to cold region, and hence the temperature profile and thermal boundary layer thickness increases.It can be observed from Figure 5 that the radiation parameter pN r q enhances the temperature profile.In fact, this happens due to the increment in radiation parameter causing a reduction in the mean absorption coefficient, which, as a result, leads to an increase in the radiative heat transfer.Here we can also observe that larger values of the Prandtl number reduces the temperature profile and the boundary layer thickness.An increment in the Prandtl number coincides with weaker thermal diffusivity.It is worth mentioning that those fluids which hold weaker thermal diffusivity have lower temperatures.This type of thermal diffusivity reveals a reduction in the boundary layer thickness and temperature profile.
Figures 6 and 7 are provided for the concentration profile against the Brownian motion parameter, thermophoresis parameter, Lewis number, and suction/injection parameter.Figure 6 shows that an increment in the Lewis number pL e q tends to reduce the concentration profile and its boundary layer thickness.Moreover, the concentration profile and boundary thickness behave in a similar way when the suction parameter pSq increases.From Figure 7, we can observe that an increment in the Brownian motion parameter pN b q tends to decrease the boundary layer thickness and nanoparticle concentration profile; however, the nanoparticle concentration profile behaves in an opposite way when the thermophoresis parameter pN t q increases.Figures 8-10 represent the entropy profile for the Reynolds number, Brinkmann number, radiation parameter, and Hartmann number.In Figure 8 we can easily notice that the entropy profile decreases due to the increment in radiation parameter pN r q; however, when ζ Ñ 8 then its behavior starts to change and becomes the opposite after certain values of ζ.It can be observed from Figures 9 and 10 that the entropy profile increases correspondingly with increasing values for the Reynolds number, Brinkmann number, and Hartmann number.

Conclusions
Entropy generation of an Eyring-Powell nanofluid through a permeable stretching surface has been investigated numerically.The impact of MHD and nonlinear thermal radiation are also taken into consideration.The solution of the governing flow problem has been obtained with the help of the Successive linearization method and Chebyshev spectral collocation method.The major outcomes are summarized below:

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The velocity of the fluid decreases due to an increment in the fluid parameter and Hartmann number.

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The entropy profile enhances all the physical parameters.

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The temperature profile increases due to an increment in the radiation parameter.

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The nanoparticle concentration increases for large values of the thermophoresis parameter.

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The nanoparticle concentration decreases due to a greater influence of the Lewis number.

Figure 1 .
Figure 1.Geometry of the problem.

Figure 1 .
Figure 1.Geometry of the problem.

Table 1 .
Numerical values of reduced Nusselt number and local Sherwood number for various values of Pr, N r , N b , L e , and N t .

Table 2 .
Comparison of f 2 p0q with existing published data for different values of stretching parameter α ą 0.