Study of Entropy Generation with Multi ‐ Slip Effects in MHD Unsteady Flow of Viscous Fluid Past an Exponentially Stretching Surface

: The current study aims to probe the impacts of entropy in a hydromagnetic unsteady slip flow of viscous fluid past an exponentially stretching sheet. Appurtenant similarity variables are employed to transmute the governing partial differential equations into a system of non ‐ linear differential equations, which are analytically solved by utilizing the homotopy analysis method (HAM). Moreover, a shooting technique with fourth–fifth order Runge–Kutta method is deployed to numerically solve the problem. The impact of the physical parameters that influence the flow and heat transmission phenomena are sketched, tabulated and discussed briefly. Additionally, the impact of these parameters on entropy generation is thoroughly discussed by plotting graphs of the local entropy generation number and the Bejan number.


Introduction
Initially, the analytical solution for a boundary layer flow problem over a linearly stretching sheet was explored by Crane [1]. Since then, numerous studies have been carried out on this topic [2][3][4][5]. Such investigations have many diverse applications in crystal growth, polymer extrusion, the spinning of fibers, condensation process, metallic sheet cooling, etc. However, the stretching rate of the surface may not always be linear in practical cases. There may be cases where the sheet may stretch with an exponential order. Motivated by this idea, Magyari and Keller [6] investigated the flow and heat transmission of a Newtonian fluid over an exponentially stretching sheet with an exponential temperature distribution. Elbashbeshy [7] discussed the characteristics of the heat transmission of viscous fluid over a permeable exponentially stretching sheet and attained similarity solutions. Al-Odat et al. [8] examined the impacts of magnetic field on the thermal boundary layer of a viscous fluid past an exponentially stretching surface. El-Aziz [9] assumed the impacts of viscous dissipation on a mixed convective micropolar fluid flow that was induced by an exponentially stretching surface. Bidin and Nazar [10] applied the implicit finite difference scheme to obtain to similarity solutions of radiative boundary layer flow of a Newtonian fluid over an exponentially stretching surface. Ishak [11] numerically investigated the steady hydro magnetic flow of a viscous fluid past a stretching surface in the presence of thermal radiation and found that thermal radiation plays a vital role in the augmentation of fluid temperature. Mukhopadhyay and Gorla [12] assumed the velocity and thermal slip conditions at the surface of a exponentially stretching sheet and explored the flow and heat transmission characteristics. They concluded that velocity and thermal slip parameters have decreasing impact on temperature. Sahoo and Poncet [13] explored the influence of partial slip on the flow and heat transmission of third-grade fluid past a continuously stretching surface. Bhattacharyya and Vajravelu [14] examined the stagnation point flow of a viscous fluid past an exponentially contracting sheet and attained similar solutions. Mukhopadhyay et al. [15] studied a Casson fluid flow that was induced by a stretching of sheet with an exponential order. Jyothi [16] examined the effects of thermophoresis on a dissipative Magneto-hydrodynamics viscous fluid flow due to an exponentially stretching surface and observed that thermophoresis had a decreasing effect on a species concentration profile. Reddy [17] considered a hydromagnetic Casson fluid flow past an inclined exponential surface in the presence of thermal radiation and a chemical reaction, and they carried out a comprehensive parametric analysis of the problem. Patil et al. [18] obtained non-similar solutions of a mixed convective flow of a Newtonian fluid past a stretching surface in an exponential order. Jusoh et al. [19] analyzed the MHD rotating flow of a ferrofluid past a permeable, exponentially stretching/shrinking sheet and obtained dual solutions.
Flow and heat transmission procedures are subjected to energy losses that are irreversible. According to Bejan [20], these energy losses can be measured by using the quantity known as the entropy generation. He remarked that by recognizing the elements that produce entropy, energy losses can be reduced. This idea was adopted by engineers and investigators to explicate entropy impacts in various geometrical configurations. However, Tayamol [21] was the first to examine the entropy generation impacts in a viscous flow past a stretching sheet that was immersed in permeable media. Aiboud and Saouli [22] explored the entropy production in a hydromagnetic viscoelastic fluid past a linearly stretching sheet. Butt et al. [23] investigated the entropy generation effects in a mixed convective flow of a second-grade fluid past a stretching surface and found that the viscoelastic parameter has a significant impact on entropy production. Later on, Butt et al. [24] considered the slip effects on a stretchable surface and studied the entropy effects in the magneto hydrodynamic flow of a viscous fluid. They remarked that the presence of a slip at the surface reduces the production of entropy. The effects of slip and heat generation/absorption effects on a nanofluid flow past a linearly stretching sheet were examined by Noghrehabadi et al. [25]. Butt and Ali [26] carried out a study related to entropy effects in a viscous fluid flow past an unsteady stretching surface. Rashidi et al. [27] considered the stagnation point flow of a Newtonian fluid past a porous stretching sheet that was immersed in spongy media and analyzed the entropy impacts. The entropy effects of a threedimensional Newtonian fluid over an exponential sheet were considered by Afridi and Qasim [28]. A mathematical investigation of an asymmetrical wavy motion of blood under entropy generation effects was presented by Riaz et al. [29]. Zaib, A. et al. explored the aligned magnetic flow comprising of nanoliquid over a radially stretching surface with entropy generation [30]. Recently, Butt et al. [31] theoretically explored the entropy impacts of a Casson nanofluid flow past an unsteady stretching surface. The found that unsteadiness had an enhancing effect on the generation of entropy effects.
Motivated by the stated facts, efforts have been devoted to scrutinize the impacts of entropy generation in the magnetohydrodynamic unsteady slip flow of a viscous fluid passing over an exponentially stretching sheet. To the best of our knowledge, entropy impacts, with the effect logs of an MHD unsteady slip flow have not been presented yet, and the results that are communicated here are new. The homotopy analysis method (HAM) and the shooting technique with the fourth-fifth order Runge-Kutta method were employed to solve the considered problem. The impact of the pertinent parameters on flow and heat and mass transmission features as well as on entropy generation are presented via graphs and tables, and they are explained briefly.

Mathematical Modeling
Consider the unsteady, two-dimensional incompressible flow of an electrically conducting viscous fluid due to an exponentially stretching surface as shown in Figure 1. It is assumed that the exponentially stretching surface is placed along the x-axis, and the y-axis is situated normal to the surface. The viscous fluid is considered to be confined in the area is applied normal to the surface where 0 B is the constant. Here, c is the dimensional constant and L represents the characteristics' length. The effects of the induced magnetic field are neglected as the value of the magnetic Reynolds number is small. The surface is stretched with the exponential velocity 0 ( , ) 1 , and the surface temperature is kept at The ambient fluid temperature is assumed to be \ T  . The effects of viscous and joule dissipations are assumed to be present. Under the application of the boundary layer approximation, the governing equations for continuity, momentum and heat are: The appurtenant boundary constraints are Here, ( , ) u v are the velocity components in the ( , ) x y directions, respectively, T is the temperature of the fluid, 0 T is the reference temperature, T  is the temperature far away from the surface,  represents the fluid density,  is the kinematic viscosity, k is the thermal conductivity,  is the electrical conductivity, p c represents the specific heat at constant pressure, o U is the reference velocity,  is the hydrodynamic slip parameter, o T denotes the reference temperature, o A is the dimensional constant temperature, and c is the constant with 1 ct  .
The following set of similarity transformation is used to non-dimensionalized Equations (1)-(3): By substituting Equation (5) into Equations (1)-(3), the continuity Equation (1) is identically satisfied, and Equations (2) and (3) take the form: The associated boundary conditions after applying Equation (5) take the form Here, is the Eckert number. The skin friction coefficient fx C and the local Nusselt number x Nu are defined as: where the shear stress w  and the heat flux w q at the surface are given as: By substituting Equation (10) into Equation (9) and by using Equation (5), the dimensionless form of the skin friction coefficient fx C and the local Nusselt number x Nu are: where Re is the local Reynolds number.

Entropy Analysis
After the application of the boundary layer approximation, the expression for the local volumetric rate of entropy generation G S for the viscous fluid in the presence of a magnetic field is defined as: Entropy due Entropy due to Entropy due to to magnetic field Heat Transfer viscous dissipation The above expression shows that the key factors that are responsible for the production of entropy effects are heat transfer, viscous dissipation, and the presence of a magnetic field.
The dimensionless form of the expression for the local entropy generation defined in Equation (12) can be obtained by using Equation (5) as: where G N is the ratio of the local entropy generation rate to the characteristic entropy generation rate and is called the local entropy generation number. Here, is the local Reynolds number, and

Br
Ec  is the Brinkman number. Equation (13) can also be written in the form: Here, H G N denotes the entropy generation number due to heat transfer, Another useful quantity is the Bejan number, which is given as: Entropy due to heat transfer = . Local entropy generation This expression helps to identify whether the heat transfer entropy effects dominate the viscous dissipation and magnetic field or vice versa. It can be seen that the value of the Bejan number varies between 0 and 1. When the value of Be is greater than 0.5, the entropy effects that are caused by heat transfer dominate the fluid friction and magnetic field effects. On the other hand, if the value of the Bejan number is less than 0.5, the entropy effects due to fluid friction and magnetic field dominate the transfer entropy effects. When 0.5 Be  , the contributions of both the entropy effects are equal.

Homotopy Analysis Method
The analytical method known as the homotopy analysis method (HAM) is a powerful technique that is helpful in solving highly nonlinear equations. This method has been employed to obtain the solution of the problems related to fluid dynamics and heat transmission (see [23, 24, 26, and 27]). In this article, Equations (6) and (7) with the boundary conditions of Equation (8) were solved by the HAM by considering the following linear operators and linear guesses: The above operator possesses the following properties: The zeroth order deformation problem can be written as: Here, The non-linear operators are defined as: The rest of the details of the method can be seen in the literature (see [23,24,26]). The solution of the differential Equations (6) and (7) with the boundary conditions of Equation (8) can be written in the form of infinite series as: From the above procedure, it can be noted that the series solutions that are mentioned in Equation (25)

Numerical Method
In order to counter check the results that were computed by the HAM, the nonlinear differential Equations (6) and (7) were numerically solved by applying shooting technique with fourth-fifth order Runge-Kutta-Fehlberg method. For this, Equations (6) and (7) were transformed in to a set of first order differential equations. For this purpose, let and the associated initial conditions are where 1 2 and r r are the unknown values. In order to solve the above system of Equations (27) and (28)  The convergence criterion was taken to be 5 10  , and a step size of 0.001 was selected.

Results and Discussion
The solutions of non-linear Ordinary differential Equations (6) and (7) with the boundary constraints of Equation (8) were compared with existing literature under limiting conditions. Table 3 was drawn to compare the numerical values of '(0)   with those mentioned by Magyari and Keller [6], El-Aziz [9], Nazar [10] and Ishak [11] Table 5      is presented in Figure 4a. A minute boost in the fluid temperature was seen as the value of o A increased. However, this boost was very miniature as compared to the variation in temperature for other flow parameters. Figure 4b presents the effects of Pr on the temperature distribution ( )   . A decline in the thermal boundary layer thickness was seen with a rising Pr value. On the other hand, Figure 4c shows that with the rise in values of Ec , the viscous dissipation effects were augmented, which caused an increase in the fluid temperature.  Figure 5 shows the contribution of each type of entropy source within the boundary layer thickness. It is quite evident from the figure that entropy production was maximum at the surface of the exponentially stretching sheet. Furthermore, the contribution of the heat transfer entropy effects was maximum as compared to the entropy impacts due to viscous dissipation and the magnetic field. These entropy effects were quite significant within the boundary layer region. However, in the far away regime, these effects were insignificant.  Figure 8a presents the effects of M on Be . It can be seen that the entropy effects due to heat transfer were significant near the vicinity of the exponentially stretching sheet. With a rise in the value of M , the heat transfer entropy effects started to decrease a bit. However, within the boundary layer region and the far away regime, the entropy effects due to heat transfer became significantly dominant over the entropy effects due to the viscous dissipation and the magnetic field with an increase in M . Figure 8b shows that in the absence of slip effects (i.e., when 0   ), the impacts of entropy because of the viscous dissipation and the magnetic field were prominent. As the value of the slip parameter  increased, the value of the Bejan number increased. This indicates that the entropy effects due to heat transfer started to become dominant over the viscous dissipation and magnetic field entropy effects, as illustrated in the figure. The impact of A on Be is displayed in Figure 8c. It can be seen that the value of Be lied within the range 0. 5 1 Be   in the neighborhood of the exponentially stretching surface. This shows that the heat transfer entropy effects were prominent in this region, and these effects became stronger with increase in the value of A. Moreover, it was observed that the entropy effects due to the viscous dissipation and the magnetic field became significantly stronger within the boundary layer and the far away region as the value of A increased. Thus, there was a dominance of the viscous dissipation and magnetic field entropy effects over the heat transfer entropy effects in these regions. Figure 9a depicts  Be . This shows that rise in the value of o A resulted a slight boost in the value of Be . Additionally, it was seen that the heat transfer entropy effects were prominent at the surface of the exponentially stretching sheet, and the viscous dissipation and magnetic field entropy effects were significant in the faraway region. Figure 9b indicates that the entropy effects due to the viscous dissipation and the magnetic field became dominant over the heat transfer entropy effects with an increase in the value of the group parameter / Br  . Figure 9c reveals the impact of increasing Re L on Be . The value of Be decreased very slightly near the exponentially stretching sheet with an increase in the Reynolds number Re L . Within the boundary layer, a slight increasing behavior was observed as the value of Re L increased. Again, a decreasing effect was seen on Be in the far away regime with rise in the value of Re L . Moreover, it was witnessed that heat transfer entropy effects were prominent at the surface, whereas the entropy due to the viscous dissipation and the magnetic field was significant in the faraway region.

Conclusions
In this study, the impacts of entropy generation in an MHD unsteady slip flow over an exponentially stretching sheet were explored. The problem was solved by employing both the HAM and the shooting method, and the outcomes were briefly discussed. It was observed that the resistive force reduced the fluid velocity with the rise in the magnetic field parameter M and caused the temperature to augment. Consequently, an increase in entropy was witnessed, as it significantly depended upon the fluid friction. Thus, the presence of magnetic field plays a notable role in the production of entropy effects. Moreover, as the value of the magnetic field parameter M increased, the Lorentz force became strong, which resisted the fluid movement and caused the decline in velocity '( ) f  . The heat transfer rate from the exponentially stretching surface increased with a rise in the slip parameter  , the unsteadiness parameter A , the temperature exponent 0 A , and the Prandtl number Pr . On the contrary, reverse impacts were observed in the case of M and Ec . It can be deduced that the presence of a slip at the surface of exponentially stretching sheet depreciates the entropy effect, whereas the application of an externally applied magnetic field augments the entropy effects. Moreover, unsteady flow also has an increasing effect on entropy production. The heat transmission entropy effects are dominant at the exponentially stretching sheet, whereas the viscous dissipation and magnetic field entropy impacts dominate the heat transmission entropy impacts in the faraway region. The value of G N rises with an increase in the unsteadiness parameter A near the surface and declines within the boundary layer regime. In the case of the temperature exponent o A , an opposite behavior was observed as compared to the unsteadiness parameter. The heat transmission entropy effects are dominant at the exponentially stretching sheet, whereas the viscous dissipation and magnetic field entropy impacts dominate the heat transmission entropy impacts in the faraway region. Finally, we conclude that the heat transmission entropy impacts are the major cause of entropy production.