Entropy Generation in MHD Conjugate Flow with Wall Shear Stress over an Infinite Plate: Exact Analysis

The current work will describe the entropy generation in an unsteady magnetohydrodynamic (MHD) flow with a combined influence of mass and heat transfer through a porous medium. It will consider the flow in the XY plane and the plate with isothermal and ramped wall temperature. The wall shear stress is also considered. The influences of different pertinent parameters on velocity, the Bejan number and on the total entropy generation number are reported graphically. Entropy generation in the fluid is controlled and reduced on the boundary by using wall shear stress. It is observed in this paper that by taking suitable values of pertinent parameters, the energy losses in the system can be minimized. These parameters are the Schmitt number, mass diffusion parameter, Prandtl number, Grashof number, magnetic parameter and modified Grashof number. These results will play an important role in the heat flow of uncertainty and must, therefore, be controlled and managed effectively.


Introduction
In compound responses the heat transfer procedure is dependably joined by the mass transfer progression. Perhaps it was expected that the investigation of joined heat and mass transfer is supportive to obtaining the best comprehension of various nominal transfer procedures. In a porous medium the convective heat transfer over a plate has numerous uses, for example, for nuclear reactors, oil production, thermal insulation systems and in separation processes in chemical engineering. Ranganathan and Viskanta [1] researched the boundary layer mixed convective fluid inserted in a porous medium over a vertical plate. They guaranteed that the viscus impacts are important and cannot be dismissed. The effect of thermal radiation and chemical reaction on the fluid with external heat source over a stretching surface was talked about by Mohan Krishna et al. [2], who observed that the chemical reaction parameter became important when the mass transfer rate was growing. Recently Gupta et al. [3] investigated the heat transfer for incompressible nanofluid over an inclined

Flow Analysis
Assume the unsteady unidirectional MHD free convectional flow of an incompressible viscous fluid over a vertical unbounded plate. As shown in Figure 1, the x has been taken along the plate and y axis is perpendicular to the plate correspondingly. At the start, the fluid and plate both are at relaxation with T W (constant wall temperature). After a while, along the x axis the plate interrupts the fluid by time dependent shear stress f (t). Equivalently the temperature of the fluid is decreased or increased to T ∞ (T W + T ∞ ) t t 0 when t ≤ t 0 and afterwards for t > t 0 is continued at T W . An unchanging magnetic field of strength B 0 is applied normal ally to the flow path. Considering the fluid is in the y > 0 porous half space, the flow is laminar and ignoring the viscous dissipation by using Boussinesq's approximation, by Butt et al. [24] the governing equations of the flow are given. The continuity, momentum, energy and concentration equations for the boundary layer flow are written as,

∂Φ ∂x
∂Φ ∂y First Law Analysis Second Law Analysis here Φ(y, t) in x direction is the fluid velocity, B 0 the applied magnetic field, µ is the fluid viscosity, q r is Radiative heat flux, K permeability of porous medium, T(y, t) is the fluid temperature, v kinematic viscosity, β T coefficient of thermal expansion, g gravitational acceleration, non trivial shear stress is τ(y, t), σ electric conductivity of the fluid, c p heat capacity at constant pressure, k thermal conductivity, ρ constant density and S G volumetric rate of local entropy generation, C concentration, R the gas constant and D mass diffusivity are defined by Bejan [25]. ( ,0) , ( ,0) ; 0 (0, ) = 0 (0, t) = + ( -); 0 < < ( ) (0, ) , (0, ) ; 0, Under the Rosseland approximation for optically thick fluid [26,27] the radiation heat flux is given by, The corresponding boundary and initial condition are as follows by assuming there are no slip acts in the middle of plate and fluid so, Under the Rosseland approximation for optically thick fluid [26,27] the radiation heat flux is given by, where σ * and K R are the Stefan-Boltzmann constant and the mean spectral absorption coefficient respectively. Considering that the temperature variance in the flow stays necessarily minor, at that point Equation (8) can be implemented by expanding to a linearized T 4 into a Taylor series around T ∞ and ignoring higher order terms, we then have, Using Equation (9) into Equation (8) and substituting the achieved aftermath in Equation (5) we have, where Pr, v and Nr are defined by By familiarizing the following dimensionless variables Into Equation (2), Equation (4) and Equation (10) also releasing the star notations, we get ∂Φ ∂t Pr e f f ∂T ∂t ∂C ∂t where Pr e f f = Pr 1+Nr , is the effective prandlt number and Are the magnetic parameter, the characteristic time, for the porous medium the inverse permeability parameter, modified Grashof number, the Grashof number and Schmidt number, correspondingly. Equivalent dimensionless conditions are, Φ(y, t) = 0, T(y, 0) = T ∞ , C(y, 0) = 0; ∀y ≥ 0 T(0; t) = 1; t> 1, T(0; t) =t ; 0 < t ≤ 1

Entropy Generation
For viscous fluid flow in a magnetic field the volumetric rate of local entropy generation S G .
where entropy generation due to heat transfer is 1, 2 is the entropy generation due to a magnetic field, 3 is entropy generation owed by mass transfer and 4 is entropy generation owed by fluid friction. Now applying dimensionless variables in Equation (17), we have, Similarly where S 0 is characteristic entropy generation rate, its value is Using the value of S 0 in S G , we get where λ is the concentration difference, M d is the mass diffusion parameter, Ω is the dimensionless temperature difference and the brinkman number is denoted by B r , which are defined under,

Solution of the Problem
To solve Equation (13) to Equation (15) under the conditions (16), by applying Laplace transform method and develop the below mentioned differential equations, With boundary conditions, Using Equation (25) in Equation (23) we get, Its inverse Laplace transform is Here −Pr e f f y 2 4t ) .
Here erfc(.) is used for the complementary error function and erf (.) for error function of Gauss [15].
Is the equivalent heat transfer rate called Nusselt number. Now using Equation (25) in Equation (24) we get the solution in the form, its Laplace transform is in the form And Is the equivalent Sherwood number or mass transfer rate. Using Equation (25) in Equation (22), we have, Its corresponding Laplace inverse is, are the corresponding convective and mechanical parts of velocity. Since Equations (27) and (35) this is noted that T(y, t) is correct for all +ve values of Pr e f f but for Pr e f f = 1, the convective part of velocity is not valid. Therefore, by putting Pr e f f = 1 in Equation (14) to get the convective part of velocity using same process we have the following result,

Constant Temperature on the Plate
Near the isothermal plate, the velocity and rate of heat transfer can be show for the flow as, Equation (40) is not effective for Pr e f f = 1, thus by taking Pr e f f = 1 in Equation (14) and assuming a similar technique, we get:

Special Cases
There is a more general form of the velocity in Section 4. Therefore, to distinguish the physical understanding of the problem, we will show some special cases for its solutions with limiting solutions. Therefore, we discuss some special cases for ramped and isothermal plate, as in the literature its technical applicability is well known. As discussed in several books and articles, initially shear stress is produced due to friction between fluid particles and due to fluid viscosity. Indeed, fluids at rest cannot resist a shear stress; i.e., when a shear stress is applied to the static fluid, the fluid will not remain at rest, but will move because of the shear stress. This idea of shear stress and in particular for the accelerating and arbitrary shear stresses as discussed in the following two cases, where the traditional shear failure always occurs. Particularly, in high temperature in a high grade asphalt pavement, accelerating or arbitrary shear stress is needed.

Case-1
Here we consider f (t) = f t b (b > 0) where the plate put on an accelerating shear stress to the fluid and so the mechanical part becomes, For M = 0 the equivalent result is, Which is the same as with Corina et al. [19] (Equation (32). Furthermore, when K p = 0, Equation (45) gives

Case-2
Here we taking the arbitrary function f (t) = f H(t), where H(.) is used for unit step function and f is a dimensionless constant. The shear stress is applied to the fluid after some time. The convective mechanical part of the velocity becomes as follows Moreover, if we put M = 0 in Equation (47) we have Which is quite the same as Corina et al. [28] (Equation (28)) with the modification of K p Now if we take both K p = 0 and M = 0, Equation (47) has the form

Results and Discussion
To analyze the physical understanding and the flow behavior of the results taken from dimensionless velocity, temperature and the corresponding irreversibility analysis, a chain of numerical calculation has taken out for several values of embedded parameters.

The Effects on Velocity
In Figure 2, it has perceived that the velocity profile is reducing with increasing M in both ramped and isothermal wall temperature situations. Actually, it is because of an increase in magnetic field M which leads the frictional force to decrease and causes it to resist the flow of the fluid, therefore velocity decreases. The effect of Kp inverse permeability parameter over isothermal and ramped walls have been seen in Figure 3. It is noticed from the graph that velocity is decreasing with an increase in Kp. This effect occurs due to the porous medium, which is an increasing Kp that strengthens the resistance and consequently decreases the velocity. The influence of shear stress f is showed in Figure 4 where we noticed that when the value of f decreases, the velocity of fluid increases. In Figure 5 it is shown that when the value of effective prandlt number Pr e f f is decreased, the velocity increases for both isothermal and ramped walls. The influence of Grashof number Gr on the velocity is shown in Figure 6, where it has observed that velocity increase with increasing Gr. The influence of modified Grashof number Gm on the velocity is also increasing, it increases the velocity more rapidly than Gr showed in Figure 7. The effect of Schmidt number Sc is shown in Figure 8. It is detected that by increasing the value of Sc, the velocity decreases.

Mechanism of S N by Different Parameters
Properties of embedded parameters on S N are highlighted in Figures 9-18. The effect of Grashof number is shown in Figure 9. It is seen that S N increases with the increasing value of . The influence of wall shear stress is presented in Figure 10, where it is noticed that S N decreases with increasing , therefore the rate of entropy generation can be reduced by increasing the value of . In Figure 11, it is noticed that S N decreases after increasing the value of Pr eff . The effect of group parameter / r B Ω over S N is highlighted in Figure 12, where it is observed that S N is the increasing function of group parameter / r B Ω . The graphical result shows that group parameter has an important ability to control S N . The Brinkman group parameter / r B Ω regulates significance of viscous effects and it is also noticed that this parameter is associated with the fluid viscosity term. The brinkman group parameter / r B Ω appears directly proportional to the square of the velocity and an increase in it evidently accelerates flow and as a result entropy will increase. The effect of inverse permeability parameter p K is shown in Figure 13. It is seen that a decrease in S N occurs with an increase in p K for both ramped and isothermal walls temperature. The reason behind this fact is that the entropy generation is an increasing function of dissipative forces. Further, the impact of p K is more prominent at the stretching boundary whereas the effects die out in the region away from the boundary. In Figure 14, the value of magnetic parameter is increased with the increasing S N . The increasing graph of is shown in Figure 15. A rapid increase in S N occurs with the increasing value of . The effect of the Schmit number over the local entropy generation rate S N is highlighted in Figure 16. It is observed that S N increases with increasing value of . The effect of mass diffusion parameter d M is shown in Figure 17. It is detected that N is also improved. In Figure 18, the influence of group parameter 2 / λ Ω over S N is shown, which an increasing function of the group parameter.

Mechanism of N S by Different Parameters
Properties of embedded parameters on N S are highlighted in Figures 9-18. The effect of Grashof number Gr is shown in Figure 9. It is seen that N S increases with the increasing value of Gr. The influence of wall shear stress f is presented in Figure 10, where it is noticed that N S decreases with increasing f , therefore the rate of entropy generation can be reduced by increasing the value of f . In Figure 11, it is noticed that N S decreases after increasing the value of Pr e f f . The effect of group parameter B r /Ω over N S is highlighted in Figure 12, where it is observed that N S is the increasing function of group parameter B r /Ω. The graphical result shows that group parameter has an important ability to control N S . The Brinkman group parameter B r /Ω regulates significance of viscous effects and it is also noticed that this parameter is associated with the fluid viscosity term. The brinkman group parameter B r /Ω appears directly proportional to the square of the velocity and an increase in it evidently accelerates flow and as a result entropy will increase. The effect of inverse permeability parameter K p is shown in Figure 13. It is seen that a decrease in N S occurs with an increase in K p for both ramped and isothermal walls temperature. The reason behind this fact is that the entropy generation is an increasing function of dissipative forces. Further, the impact of K p is more prominent at the stretching boundary whereas the effects die out in the region away from the boundary. In Figure 14, the value of magnetic parameter M is increased with the increasing N S . The increasing graph of Gm is shown in Figure 15. A rapid increase in N S occurs with the increasing value of Gm. The effect of the Schmit number Sc over the local entropy generation rate N S is highlighted in Figure 16. It is observed that N S increases with increasing value of Sc. The effect of mass diffusion parameter M d is shown in Figure 17. It is detected that when M d increased N S is also improved. In Figure 18, the influence of group parameter λ/Ω 2 over N S is shown, which an increasing function of the group parameter.                         λ Ω on .

Influences of Embedded Parameters on Bejan Number
The Bejan number is quite useful, as one can get evidence about dominancy of magnetic field and fluid friction through heat transfer entropy or vice versa. The influence of on the Bejan number is shown in Figure 19. The magnetic field leads to increasing the fluid friction and entropy with a decreasing value of . The heat transfer reunification comes to be dominant in the region near to the plate with an increasing value of , shown in Figure 20, while far away from the plate the friction Figure 18. Influence of λ/Ω 2 on Ns.

Influences of Embedded Parameters on Bejan Number
The Bejan number is quite useful, as one can get evidence about dominancy of magnetic field and fluid friction through heat transfer entropy or vice versa. The influence of f on the Bejan number is shown in Figure 19. The magnetic field leads to increasing the fluid friction and entropy with a decreasing value of f . The heat transfer reunification comes to be dominant in the region near to the plate with an increasing value of Gr, shown in Figure 20, while far away from the plate the friction of the fluid's irreversibility become powerful and hence the Bejan number is strengthened. Figure 21 showed that the fluid friction entropy and the magnetic field are improved with an increase in the group consideration B r /Ω. Figure 22 explains that the Bejan number because of the magnetic field and the fluid friction becomes minor with an increase in the K p nearby plate. In Figure 23 it is showed that the fluid friction entropy and magnetic field increased with a rise in the value of M, for individually ramped and isothermal plates. Figure 24 showed an increase in the Bejan number with a decrease in Pr e f f . The effect of the Schmit number Sc is showed in Figure 25, where it is observed that the Bejan number is decreasing with an increasing value of Sc. The effect of group parameter λ/Ω 2 over the Bejan number is showed in Figure 26. It is noted that Be decreases with an increase in λ/Ω 2 . In Figure 27 the influence of Gm is shown and it is observed that Be decreases with an increase in Gm. The effect of M d is exposed in Figure 28, where it is seen that the Bejan number decreases with an increasing value of M d .

Assumptions and Deductions
The effect of entropy generation for conjugate MHD unsteady flow through a porous medium near a vertical plate is deliberated. The exact solution for velocity profile is found by using the Laplace transform method. The Bejan number Be and number of local entropy generation Ns are discussed for various parameters. The effects are displayed for different embedded parameters. The main conclusions are: • Rises in, B r /Ω, Sc, Gm, λ/Ω 2 Md and M leads to decreases of the Bejan number for isothermal and ramped wall temperature individually.

•
The Entropy generation in the fluid can be controlled and reduced by the f constant wall shear stress. • Rises in B r /Ω, Sc, M d , Pr, Gr, M, K p and Gm increase Ns.
Author Contributions: A.K. and I.K. designed the study; D.K. and F.u.K. conducted the experiments with technical assistance from F.A. and T.A.A. analyzed the data and wrote the paper; A.K., I.K. and K.S.N. provided general assistance. All authors have read and approved the final submission.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no competing interests.