Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet

This research focuses on entropy generation rate per unit volume in magneto-hydrodynamic (MHD) mixed convection boundary layer flow of a viscous fluid over an inclined stretching sheet. Analysis has been performed in the presence of viscous dissipation and non-isothermal boundary conditions. The governing boundary layer equations are transformed into ordinary differential equations by an appropriate similarity transformation. The transformed coupled nonlinear ordinary differential equations are then solved numerically by a shooting technique along with the Runge-Kutta method. Expressions for entropy generation (Ns) and Bejan number (Be) in the form of dimensionless variables are also obtained. Impact of various physical parameters on the quantities of interest is seen.


Introduction
Hydromagnetic (MHD) boundary layer flow over a stretching surface has vital importance because of its ever-increasing usage in industrial applications such as the production of paper, glassware, cable coatings, fabrication of adhesive tapes, artificial fibers, metal spinning, roofing shingles, fine-fiber mats, and metallic plates to mention a few among others. In all such technological processes, the kinematics of stretching and heat transfer have a significant impact in the improvement of final product quality through control of the rate of cooling. Boundary layer flow over a moving continuous flat surface was studied for the first time by Sakiadis [1]. Crane [2] obtained the closed form similarity solution of the boundary layer flow over a linear stretching sheet. After the work of Crane [2], several researchers revisited this problem addressing different aspects. Here we mention some very recent studies [3][4][5][6][7][8][9][10][11][12][13][14][15] for the convenience of readers. All these studies have been restricted to the situations in which flow is induced by horizontally/vertically stretching sheet. Haung et al. [16] investigated the heat transfer analysis in the boundary layer flow of a viscous fluid caused by an inclined stretching sheet. Qasim et al. [17] studied the effects of thermal radiation on mixed convection flow of viscoelastic fluid along an inclined sheet. Sravanthi [18] studied the MHD slip flow over an exponentially stretching inclined sheet by taking into account the Soret-Dufour effects. MHD boundary layer flow over an inclined stretching sheet under the effects of mixed convection, heat generation/absorption and blowing/suction was reported by Eldahad and Aziz [19].
In all the above investigations, the objective was to understand the flow behavior, i.e., to find the velocity profile and temperature distribution (first law analysis) in the boundary layer region. Moreover, to understand the energy losses during the boundary layer flow, it makes sense to perform a second law analysis. The losses of energy in the system during the process are proportional to the entropy generation. Entropy production is the measure of irreversibility in the heat transfer problems. The study of entropy generation within the system is important as it helps to trace the sources which destroy available energy. Therefore, by knowing these factors or sources one can minimize the entropy in order to retain the quality of energy, which is commonly referred to as entropy generation minimization (EMG). At present the research topic of entropy generation minimization has acquired special status amongst scientists worldwide who are re-examining all energy consuming, converting and producing systems and developing new techniques in order to remove all sources that destroy the available work. Bejan [20][21][22] calculated for the first time the volumetric entropy generation rate in a fluid flow process. After the seminal work of Bejan, many researchers have performed thermodynamic analysis of flows and heat transfers in order to minimize entropy production [23][24][25][26][27][28][29][30].
Despite the many studies on the second law analysis of boundary layer flows, the entropy generation analysis of a boundary layer flow over an inclined impermeable stretching sheet under the effects of magnetic field and viscous dissipation has not been reported in the literature. Motivated by the abovementioned facts, the present research has been undertaken to perform an irreversibility analysis of MHD mixed convection flow over an inclined stretching sheet.

Mathematical Formulation of the Problem
Consider the steady two dimensional boundary layer flow induced by the inclined stretching sheet having the linear stretching velocity u w (x) = cx. A magnetic field B o is applied normally to the stretching sheet. Figure 1 shows the physical model and coordinates system. For boundary layer flow the continuity, momentum and energy equations take the following forms: ∂u ∂x In the above equations, u and v represent velocity components in the directions of the x and y axes, T ∞ shows the free stream temperature, g denotes gravitational acceleration, α* is the thermal diffusivity, ρ is the density of the fluid, v is the kinematic viscosity of the fluid, B o depicts the imposed magnetic field, C p represents the specific heat of the fluid at constant pressure, α is the inclination of the stretching sheet and β represents the thermal coefficient. Here we consider the thermal expansion coefficient [31,32] and the surface temperature of the stretching sheet of the form T w (x) = T ∞ + ax 2 [33][34][35] in order to have a self-similarity equation.
The relevant boundary conditions are: in which T w and u w respectively represent the temperature and velocity of the stretching boundary. We look for a solution through the following transformation: Incompressibility condition (1) is now automatically satisfied, whereas the other equations lead to the following expressions: 1 Pr The dimensionless numbers appearing in Equations (7) and (8) are: While the boundary conditions, Equations (4) and (5) take the following form: The skin friction coefficient C f and the Nusselt number Nu x which are defined as: where τ w is the shear stress and q w is the heat flux and k is the thermal conductivity. Using variables (6), we obtain:

Irreversibility Analysis
In the presence of magnetic field, the entropy generation rate per unit volume is given by:

Irreversibility Analysis
In the presence of magnetic field, the entropy generation rate per unit volume is given by: In the above equation Φ represents the viscous dissipation function, J is the current density and ∇ ≡î ∂ dx +ĵ ∂ dy +k ∂ dz is the Del operator. In the present article it is supposed that the magnitude of QV is negligible compared to the electric current density J and the electric force E has no significant effects as compared to the magnetic force V × B. By taking these approximations and the Prandtl boundary layer approximation as well Equation (14) reduces to: It is clearly seen from the above equation that there are three sources of entropy generation. The first source is the heat transfer which is due the presence of a temperature gradient in the problem, the second source is the viscous dissipation, which is due to the fluid friction, and the last source is the magnetic field which cause Joule dissipation irreversibility. The entropy generation number is the dimensionless form of volumetric entropy generation rate, which represent the ratio between volumetric entropy generation rate ( . S gen ) and characteristic entropy generation ( .

S o ).
Using Equation (6) the entropy generation given in Equation (15) reduces to: where .
S o = kc v and Ω = T ∞ T w −T ∞ represent the characteristic entropy generation and dimensionless temperature parameter, respectively.
In many engineering problems engineers need to measure the relative importance of the sources of entropy generation, and for this important measurement the Bejan number is defined as: Conductive irreversibity Viscous irreversibility + Magnetic irreversibility By using similarity variables Be takes the following form:

Results and Discussion
In this section our concern is to analyse the effects of important physical parameters on the quantities of interest. Therefore   friction is responsible for the rise of temperature. Figure 2c presents the variation of entropy generation number for changing values of M. It is observed that Ns is increasing as a function of M which is due to the resistive nature of the magnetic force, so we can achieve the main goal of the second law of thermodynamics that is the minimization of entropy generation by reducing the magnetic field. Figure 2d demonstrates the variation of Bejan number Be with an increase in M, where it is observed that Be decreases as M increases. The graph depicts that with an increasing value of M, viscosity and magnetic irreversibility dominate the heat transfer irreversibility at the surface of the stretching sheet. The magnetic and viscosity irreversibility completely rule the thermal irreversibility for large values of M (M > 2) at the surface. However, far away from the stretching sheet the dominancy effects are reversed. Figure 2a displays the impact of the magnetic field parameter M on the velocity profile. It is seen that the velocity decreases by increasing M. This is happening because the applied transverse magnetic field generates a Lorentz force, which decelerates the fluid velocity. Figure 2b displays the effect of M on the fluid temperature. It is seen that temperature rises with rising values of M. Physically, the presence of M sets up a retarding force (Lorentz force), which boosts the fluid friction and this fluid friction is responsible for the rise of temperature. Figure 2c presents the variation of entropy generation number for changing values of M. It is observed that Ns is increasing as a function of M which is due to the resistive nature of the magnetic force, so we can achieve the main goal of the second law of thermodynamics that is the minimization of entropy generation by reducing the magnetic field. Figure

Effects of the Prandtl Number
The impact of Prandtl number Pr on the fluid temperature is depicted in Figure 3a. The graph depicts that the temperature decreases with large values of the Pr number. Physically, a fluid with high Pr has low thermal diffusivity, which in turn reduces the temperature of the fluid. Figure 3b reveals that the dimensionless entropy generation Ns increases with the increasing value of Pr. This happens because with an increase in Pr increases the temperature gradients in the boundary layer, which in turn increases the Ns. Figure 3c demonstrates the influence of Pr on the Bejan number. The plot shows that for a fixed value of η the Bejan number decreases when Pr increases. It is also clear that near the surface of the stretching sheet, Be is close to zero, which means that the magnetic field intensity and fluid friction both play more important roles in the total entropy generation. Further, it is observed that far away from the stretching surface, the heat transfer contribution in the total entropy plays a substantial part.

Effects of the Prandtl Number
The impact of Prandtl number Pr on the fluid temperature is depicted in Figure 3a. The graph depicts that the temperature decreases with large values of the Pr number. Physically, a fluid with high Pr has low thermal diffusivity, which in turn reduces the temperature of the fluid. Figure 3b reveals that the dimensionless entropy generation Ns increases with the increasing value of Pr. This happens because with an increase in Pr increases the temperature gradients in the boundary layer, which in turn increases the Ns. Figure 3c demonstrates the influence of Pr on the Bejan number. The plot shows Entropy 2017, 19, 10 6 of 11 that for a fixed value of η the Bejan number decreases when Pr increases. It is also clear that near the surface of the stretching sheet, Be is close to zero, which means that the magnetic field intensity and fluid friction both play more important roles in the total entropy generation. Further, it is observed that far away from the stretching surface, the heat transfer contribution in the total entropy plays a substantial part.
The impact of Prandtl number Pr on the fluid temperature is depicted in Figure 3a. The graph depicts that the temperature decreases with large values of the Pr number. Physically, a fluid with high Pr has low thermal diffusivity, which in turn reduces the temperature of the fluid. Figure 3b reveals that the dimensionless entropy generation Ns increases with the increasing value of Pr. This happens because with an increase in Pr increases the temperature gradients in the boundary layer, which in turn increases the Ns. Figure 3c demonstrates the influence of Pr on the Bejan number. The plot shows that for a fixed value of η the Bejan number decreases when Pr increases. It is also clear that near the surface of the stretching sheet, Be is close to zero, which means that the magnetic field intensity and fluid friction both play more important roles in the total entropy generation. Further, it is observed that far away from the stretching surface, the heat transfer contribution in the total entropy plays a substantial part.  Figure 4a illustrates the effect of the mixed convection parameter λ on velocity. It is anticipated that velocity increases by increasing the mixed convection parameterλ. Physically with an increase in λ the buoyancy force increases, which results in an enhancement of the velocity profile. Figure 4b shows the effect of the buoyancy parameter on the temperature profile. It is seen that the thermal convective parameter decreases the temperature of the fluid. Figure 4c represents the variation of entropy generation number Ns versus different values of λ. Ns decreases near the boundary when λ increases and goes to zero asymptotically as we move away from the boundary. Figure 4d displays the effect of buoyancy parameter on the Bejan number. Be increases with the increasing value of λ.

Effects of the Mixed Convective Parameter
Also, the contribution of fluid friction and magnetic field in the entropy generation dominate the heat transfer effect at the surface. Further, it is perceived that far away from the boundary the effective source of the production of entropy is the conductive irreversibility.  Figure 4a illustrates the effect of the mixed convection parameter λ on velocity. It is anticipated that velocity increases by increasing the mixed convection parameter λ. Physically with an increase in λ the buoyancy force increases, which results in an enhancement of the velocity profile. Figure 4b shows the effect of the buoyancy parameter on the temperature profile. It is seen that the thermal convective parameter decreases the temperature of the fluid. Figure 4c represents the variation of entropy generation number Ns versus different values of λ. Ns decreases near the boundary when λ increases and goes to zero asymptotically as we move away from the boundary. Figure 4d displays the effect of buoyancy parameter on the Bejan number. Be increases with the increasing value of λ. Also, the contribution of fluid friction and magnetic field in the entropy generation dominate the heat transfer effect at the surface. Further, it is perceived that far away from the boundary the effective source of the production of entropy is the conductive irreversibility. increases and goes to zero asymptotically as we move away from the boundary. Figure 4d displays the effect of buoyancy parameter on the Bejan number. Be increases with the increasing value of λ.

Effects of the Mixed Convective Parameter
Also, the contribution of fluid friction and magnetic field in the entropy generation dominate the heat transfer effect at the surface. Further, it is perceived that far away from the boundary the effective source of the production of entropy is the conductive irreversibility.   Figure 5a shows that Ns increases with the increasing value of Ec, so by decreasing the Eckert number Ec one can achieve the main objective of the second law of thermodynamics, the minimization of entropy production rate per unit volume. The effect of increasing the Ec on the entropy production is prominent near the surface of the stretching sheet, while it has no significant effect in the main flow region. The effect of Ec on Be is plotted in Figure 5b. It is observed that an increase in Ec decreases the Bejan number Be. On the surface of the stretching sheet the effects of magnetic and viscous irreversibility dominate the conductive irreversibility, whereas far away from the stretching sheet the contribution of heat transfer in the generation of entropy is prominent.   Figure 5a shows that Ns increases with the increasing value of Ec, so by decreasing the Eckert number Ec one can achieve the main objective of the second law of thermodynamics, the minimization of entropy production rate per unit volume. The effect of increasing the Ec on the entropy production is prominent near the surface of the stretching sheet, while it has no significant effect in the main flow region. The effect of Ec on Be is plotted in Figure 5b. It is observed that an increase in Ec decreases the Bejan number Be. On the surface of the stretching sheet the effects of magnetic and viscous irreversibility dominate the conductive irreversibility, whereas far away from the stretching sheet the contribution of heat transfer in the generation of entropy is prominent.

Effects of the Dimensionless Temperature Parameter
The variation of Ns with the increasing value of the dimensionless temperature parameter is portrayed in Figure 6a. This figure shows that Ns increases with the increasing value of Ω. It is verified that Ns asymptotically goes to zero as we move away from the boundary. Figure 6b shows the variation of Be with different values of Ω. It is clearly seen that near the stretching sheet the fluid friction and magnetic field irreversibility dominate. It is also observed that with increasing value of Ω the Be decreases and for small values of Ω the effect due to heat transfer dominates far away from the stretching sheet. minimization of entropy production rate per unit volume. The effect of increasing the Ec on the entropy production is prominent near the surface of the stretching sheet, while it has no significant effect in the main flow region. The effect of Ec on Be is plotted in Figure 5b. It is observed that an increase in Ec decreases the Bejan number Be. On the surface of the stretching sheet the effects of magnetic and viscous irreversibility dominate the conductive irreversibility, whereas far away from the stretching sheet the contribution of heat transfer in the generation of entropy is prominent.

Effects of the Dimensionless Temperature Parameter
The variation of Ns with the increasing value of the dimensionless temperature parameter is portrayed in Figure 6a. This figure shows that Ns increases with the increasing value of Ω. It is verified that Ns asymptotically goes to zero as we move away from the boundary. Figure 6b shows the variation of Be with different values of Ω. It is clearly seen that near the stretching sheet the fluid friction and magnetic field irreversibility dominate. It is also observed that with increasing value of

Conclusions
The main conclusions of the present study are: (1) It is observed that the velocity magnetic parameter M has a decreasing effect on the velocity while the thermal convective parameter λ has an increasing effect on it.  Table 1).

Conclusions
The main conclusions of the present study are: (1) It is observed that the velocity magnetic parameter M has a decreasing effect on the velocity while the thermal convective parameter λ has an increasing effect on it. (2) The thickness of the thermal boundary layer increases as the values of magnetic parameter and Eckert number increase, while a decreasing effect has been observed with the variation of thermal convective parameter and Prandtl number. There is a decrease in local Nusselt number for large values of M and Ec (see Table 1).