Aspects of Chemical Entropy Generation in Flow of Casson Nanofluid between Radiative Stretching Disks

The appropriate utilization of entropy generation may provoke dipping losses in the available energy of nanofluid flow. The effects of chemical entropy generation in axisymmetric flow of Casson nanofluid between radiative stretching disks in the presence of thermal radiation, chemical reaction, and heat absorption/generation features have been mathematically modeled and simulated via interaction of slip boundary conditions. Shooting method has been employed to numerically solve dimensionless form of the governing equations, including expressions referring to entropy generation. The impacts of the physical parameters on fluid velocity components, temperature and concentration profiles, and entropy generation number are presented. Simulation results revealed that axial component of velocity decreases with variation of Casson fluid parameter. A declining variation in Bejan number was noticed with increment of Casson fluid constant. Moreover, a progressive variation in Bejan number resulted due to the impact of Prandtl number and stretching ratio constant.


Introduction
Engineering systems' efficiency decreases in the presence of irreversibilities. Heat transfer and fluid flow are irreversible processes and their irreversibility may be articulated in terms of entropy generation. Since rotating disks associated with non-Newtonian fluid flow and heat transfer have many important applications, such as in liquid-metal pumping, rotor-stator systems, oil recovery, hydraulic presses, centrifugal machinery, various electronic disks, shrouded-disk rotation, rotating motors, boilers, and plastic films and artificial fibers, they have received increasing attention over the last decade.
Gorder et al. [1] studied viscous fluid flow between stretching disks. Soid et al. [2] numerically analyzed the heat transfer characteristics induced by shrinking and stretching disks under the influence aspects in unsteady nanofluid flow coffered by a moving surface. Kumar et al. [33] evaluated the effects of entropy generation for the flow of viscoelastic nano-material under the influence of transient convective dissipation. Another numerical investigation for determination of entropy generation for nanofluid has been explored by Rana et al. [34]. Aghakhani et al. [35] worked on the thermal aspects for natural convective flow alumina/water nanoparticles in presence of entropy generation features. Seyyedi et al. [36] examined the magnetic field and entropy generation impact to examine the heat transfer analysis in L-shaped enclosures. Salimi et al. [37] incorporated the features of entropy generation and heat sink for 3D jet flow under the assumptions of local thermal non-equilibrium constraints. The flow of chemically reactive non-Newtonian liquid due to stretched surface in presence of entropy generation phenomenon was recently considered by Khan et al. [38]. Mustafa [39] investigated slip effects in nanofluid flow induced by a rotating disk. Arikoglu et al. [40] investigated the entropy generation features in slip flow of viscous fluid due to rotating disk.
Recently, special interest has developed towards the flow of nanoparticles because of their diverse industrial and commercial applications, like energy generation, improvement of the thermal extrusion phenomenon, development of manufacturing processes, etc. In addition to this, the consumption of available energy and reduced energy loss is another novel aspect useful in various engineering applications and other industries to improve the thermal efficiency of systems. The utilization of entropy generation enables the minimization of available energy loss of performance systems. Keeping such motivations in mind, the present investigation presents the effects of entropy generation in flow of Casson nanofluid induced by stretching disks.
After carefully examining the above cited work, we note that entropy generation features in thermally developed flow of Casson nanofluid induced by two porous stretching disks have not been reported yet. Therefore, the current analysis aims to fill this gap. Additionally, the interesting features of magnetic field, heat absorption/generation and chemical reaction are also incorporated. The present research is an extension of that of Arikoglu et al. [40]. It integrated the Casson nanofluid model. Also, the novel features of thermal radiation, heat source/sink, and chemical reaction are incorporated. The analysis has been performed over porous stretching disks in contrast to simple stretching disks, which addresses modern engineering applications in the fields of material engineering, biomedical separation devices, petroleum engineering, distillation towers, jet engines, and atmospheric flows. The governing equations for current flow situations are constituted and tackled by employing the famous numerical shooting technique. Different flow parameters are graphically impacted with relevant physical significant.

Mathematical Analysis
In order to model the equations for flow analysis, we consider a steady two-dimensional axisymmetric flow of an incompressible, electrically conducting Casson nanofluid between two stretchable disks. The magnetic field effects with magnetic field strength B 0 are applied vertically to the surface shown in Figure 1. Here (u, v, w) and (r, θ, z) are assumed to be velocity components and cylindrical coordinates. The flow is described in cylindrical coordinates (r, θ, z) where z is chosen as the perpendicular axis. The lower disk is static at the plane z = 0. The governing equations for current flow problem are [1,[5][6][7]; Entropy 2020, 20, x 4 of 16 Figure 1. Schematic diagram and coordinate system.
where µisthe dynamic viscosity, denotes density, is the parameter of Casson fluid, is the pressure, is the electrical conductivity, 0 is the magnetic field strength, is the permeability of porous medium, is the temperature, is the concentration, is the thermal diffusivity, reflects the specific heat at constant pressure, 0 is the heat generation parameter, is the thermophoretic diffusion coefficient, is the Brownian diffusion coefficient, is the ratio of heat capacity, 0 is the reaction constant, is the mean temperature,while is the Brownian motion coefficient. In view of Rossel and approximation, the radiative heat flux ( ) is written as: where µ isthe dynamic viscosity, ρ denotes density, β is the parameter of Casson fluid, p is the pressure, σ is the electrical conductivity, B 0 is the magnetic field strength, k is the permeability of porous medium, T is the temperature, C is the concentration, α is the thermal diffusivity, c p reflects the specific heat at constant pressure, Q 0 is the heat generation parameter, D T is the thermophoretic diffusion coefficient, D B is the Brownian diffusion coefficient, τ is the ratio of heat capacity, k 0 is the reaction constant, T m is the mean temperature, while Nb is the Brownian motion coefficient. In view of Rossel and approximation, the radiative heat flux (q r ) is written as: where k * is the coefficient of Rossel and mean absorption, σ * is the constant of Stefan-Boltzmann.
By expanding T 4 about free stream temperature T 0 as follows: For further analysis, the temperature gradient within the flow is assumed to be small and subsequently the higher order terms can be ignored i.e., By using Equation (9) in Equation (7), we have Now let us introduce similarity transforms as follows [5][6][7]: By using similarity transforms in Equations (1) to (5), the following transformed governing equations are obtained In the above expression, γ is stretching ratio, λ is heat source parameter, Rd is thermal radiation parameter, Re is Reynolds number, Pr is Prandtl number, Le is the Lewis number, M is parameter of magnetic, Nb is parameter of Brownian motion, D B is Brownian diffusion parameter, Nt is thermophoretic parameter, λ 1 and λ 2 are slip lengths, S is porosity parameter, and D T is thermophoresis diffusion coefficient.

Entropy Generation Equation
The expressions for local entropy generation volumetric rate for Casson nanofluid are given by where µ and K are the viscosity and thermal conductivity, T 0 is the reference temperature, and Φ is the viscous dissipation. In Equation (17), the term Φ can be written as: Using Equation (18) in Equation (17) yields The characteristics entropy generation is defined as After inserting Equation (11) into Equation (17), the following equation for entropy generation number will be obtained where In the above expression, Br is the Brinkman number and Ec is the Eckert number. The Bejan number Be is another alternative irreversibility distribution parameter. This is the ratio between entropy generations due to heat transfer to the total entropy generations. In a dimensionless form, Bejan number is given as follows:

Be =
Entropy generation due to heat trans f er Total entropy generation ,

Results and Discussion
The system of Equations (12)- (14), along with boundary conditions, Equation (16), has been solved numerically by employing the shooting method. The results are validated by comparing present numerical values with those generated by Mustafa [39], and presented in Table 1. A convincible accuracy of results has been found between both studies. In this section, physical explanation of flow parameters, like the stretching ratio γ, Reynold number, Reynold number Re, Prandtl number Pr, magnetic parameter M, thermophoretic parameter Nt, thermal radiation parameter Rd, Lewis number Le, and Brownian motion parameter Nb, is presented graphically. Each parameter is varied while other parameters are kept constant, i.e., M = 1, β = 0.1, γ = 0.3, S = 0.5, Nt = 0.5, λ = 0.5, Le = 1.5, Nb = 0.7, K 1 = 0.3, and Pr = 0.5.  Figures 2 and 3, velocity profiles increase for rising values of porosity parameter S. The physical justification of such an increasing trend may be attributed to the involvement of the permeability of the porous medium. Figure 4 shows that the axial velocity component is a growing function of magnetic parameter M. It is seen that the thickness of thermal boundary layer surges with the implementation of magnetic field. Figure 5 shows that radial velocity component is decreasing function of magnetic parameter due to stretching disks. Physically, a change in magnetic number is associated with the Lorentz force, which is of resistive nature and subsequently decreases the nanoparticles' velocity. Figures 6 and 7 show that increment of Casson fluid parameter β leads to a progressive velocity distribution. The physical aspect of such trend is due to involvement of yield stress which is associated with Casson fluid parameter β. The graphical explanations presented in Figures 8 and 9 reveal that the axial and radial velocity component increases by varying stretching ratio γ.
an increasing trend may be attributed to the involvement of the permeability of the porous medium. Figure 4 shows that the axial velocity component is a growing function of magnetic parameter . It is seen that the thickness of thermal boundary layer surges with the implementation of magnetic field. Figure5 shows that radial velocity component is decreasing function of magnetic parameter due to stretching disks. Physically, a change in magnetic number is associated with the Lorentz force, which is of resistive nature and subsequently decreases the nanoparticles' velocity. Figures6  and 7 show that increment of Casson fluid parameter leads to a progressive velocity distribution. The physical aspect of such trend is due to involvement of yield stress which is associated with Casson fluid parameter . The graphical explanations presented in Figures 8 and 9 reveal that the axial and radial velocity component increases by varying stretching ratio .    Figure 4 shows that the axial velocity component is a growing function of magnetic parameter . It is seen that the thickness of thermal boundary layer surges with the implementation of magnetic field. Figure5 shows that radial velocity component is decreasing function of magnetic parameter due to stretching disks. Physically, a change in magnetic number is associated with the Lorentz force, which is of resistive nature and subsequently decreases the nanoparticles' velocity. Figures6  and 7 show that increment of Casson fluid parameter leads to a progressive velocity distribution. The physical aspect of such trend is due to involvement of yield stress which is associated with Casson fluid parameter . The graphical explanations presented in Figures 8 and 9 reveal that the axial and radial velocity component increases by varying stretching ratio .                     Figures10-13 show the influence of Prandtl number , thermophoretic parameter , and heat source/sink parameter on temperature profiles ( ). Figure 10 shows a declining temperature distribution for apparent values of Prandtl number . Such a decline in temperature profile due to peak values of Prandtl number is due toa weaker thermal diffusivity. Therefore, proper values of Prandtl number play a frequent role in controlling the heating and cooling processes. Figure 11 depicts the consequence of thermophoretic parameter on . The thermophoresis phenomenon has a significant contribution in many industries. The thermophoresis is a migration process of heated fluid particles towards the cold region, due to which the temperature increases. From Figure  12, an increasing temperature profile is resulted for heat source parameter ( > 0). However, for heat sink case ( < 0), opposite observations are obtained. Physically, an improved temperature due to heat source is associated with addition of heat to the system. However, for the heat sink case, heat is removed from the system which reduces the temperature.  Figures10-13 show the influence of Prandtl number , thermophoretic parameter , and heat source/sink parameter on temperature profiles ( ). Figure 10 shows a declining temperature distribution for apparent values of Prandtl number . Such a decline in temperature profile due to peak values of Prandtl number is due toa weaker thermal diffusivity. Therefore, proper values of Prandtl number play a frequent role in controlling the heating and cooling processes. Figure 11 depicts the consequence of thermophoretic parameter on . The thermophoresis phenomenon has a significant contribution in many industries. The thermophoresis is a migration process of heated fluid particles towards the cold region, due to which the temperature increases. From Figure  12, an increasing temperature profile is resulted for heat source parameter ( > 0). However, for heat sink case ( < 0), opposite observations are obtained. Physically, an improved temperature due to heat source is associated with addition of heat to the system. However, for the heat sink case, heat is removed from the system which reduces the temperature.   Figure 10 shows a declining temperature distribution θ for apparent values of Prandtl number Pr. Such a decline in temperature profile due to peak values of Prandtl number is due toa weaker thermal diffusivity. Therefore, proper values of Prandtl number play a frequent role in controlling the heating and cooling processes. Figure 11 depicts the consequence of thermophoretic parameter Nt on θ. The thermophoresis phenomenon has a significant contribution in many industries. The thermophoresis is a migration process of heated fluid particles towards the cold region, due to which the temperature increases. From Figure 12, an increasing temperature profile is resulted for heat source parameter (λ > 0). However, for heat sink case (λ < 0), opposite observations are obtained. Physically, an improved temperature due to heat source is associated with addition of heat to the system. However, for the heat sink case, heat is removed from the system which reduces the temperature. Figures 14-17 predict the influence of the Lewis number Le, thermophoretic parameter Nt, Brownian motion parameter Nb, and reaction parameter K 1 on concentration profiles φ(η). Figures 14 and 15 show that concentration profiles increase with the increase in Le and Nt. Physically, the Lewis number is associated with the mass diffusion coefficient. The higher variation in Lewis number corresponds to low mass diffusion which declines the concentration of nanoparticles. Figures 16 and 17 show decreasing behavior of concentration profile by increasing the value of Brownian motion parameter Nb and chemical reaction parameter K 1 .The Brownian constant Nb occupies a reverse relation with dimensionless concentration Equation (14), which means that the maximum values assigned to N b retarded the concentration distribution. Figures 18 and 19 aim to report the influence of Casson fluid parameter β and stretching ratio constant γ on entropy generation number N G . Figure 18 revealed that an improved total entropy generation distribution is examined when Casson fluid parameters get maximum values. Figure 19 examined that entropy generation rates decrease the function of stretching ratio constant γ. distribution for apparent values of Prandtl number . Such a decline in temperature profile due to peak values of Prandtl number is due toa weaker thermal diffusivity. Therefore, proper values of Prandtl number play a frequent role in controlling the heating and cooling processes. Figure 11 depicts the consequence of thermophoretic parameter on . The thermophoresis phenomenon has a significant contribution in many industries. The thermophoresis is a migration process of heated fluid particles towards the cold region, due to which the temperature increases. From Figure  12, an increasing temperature profile is resulted for heat source parameter ( > 0). However, for heat sink case ( < 0), opposite observations are obtained. Physically, an improved temperature due to heat source is associated with addition of heat to the system. However, for the heat sink case, heat is removed from the system which reduces the temperature.        Figures14-17 predict the influence of the Lewis number , thermophoretic parameter , Brownian motion parameter , and reaction parameter 1 on concentration profiles ( ). Figures  14 and 15 show that concentration profiles increase with the increase in and . Physically, the Lewis number is associated with the mass diffusion coefficient. The higher variation in Lewis number corresponds to low mass diffusion which declines the concentration of nanoparticles. Figures 16 and 17 show decreasing behavior of concentration profile by increasing the value of Brownian motion parameter and chemical reaction parameter 1 .The Brownian constant occupies a reverse relation with dimensionless concentration Equation (14), which means that the maximum values assigned to retarded the concentration distribution. Figures 18 and 19 aim to report the influence of Casson fluid parameter and stretching ratio constant on entropy generation number . Figure 18 revealed that an improved total entropy generation distribution is examined when Casson fluid parameters get maximum values. Figure 19 examined that entropy generation rates decrease the function of stretching ratio constant .             Figure 21 shows that Bejan number decreases with . Figures 22 and 23 elucidate that the Bejan number increases with stretching ratio constant and . The Bejan number is a very important dimensionless number in the entropy generation which reflects the ratio between entropy generations associated with heat transfer to total entropy generation. Usually, the Bejan number reveals numerical values between 0 and 1.In the case that the Bejan number assigns a value close to 1, it means that entropy generation associated with heat transfer is more dominant. To increase the cooling system in the flow of conducting liquids, the Prandtl number can be utilized appropriately.    Figure 21 shows that Bejan number Be decreases with β. Figures 22 and 23 elucidate that the Bejan number Be increases with stretching ratio constant γ and Pr. The Bejan number is a very important dimensionless number in the entropy generation which reflects the ratio between entropy generations associated with heat transfer to total entropy generation. Usually, the Bejan number reveals numerical values between 0 and 1.In the case that the Bejan number assigns a value close to 1, it means that entropy generation associated with heat transfer is more dominant. To increase the cooling system in the flow of conducting liquids, the Prandtl number can be utilized appropriately.  Figure 21 shows that Bejan number decreases with . Figures 22 and 23 elucidate that the Bejan number increases with stretching ratio constant and . The Bejan number is a very important dimensionless number in the entropy generation which reflects the ratio between entropy generations associated with heat transfer to total entropy generation. Usually, the Bejan number reveals numerical values between 0 and 1.In the case that the Bejan number assigns a value close to 1, it means that entropy generation associated with heat transfer is more dominant. To increase the cooling system in the flow of conducting liquids, the Prandtl number can be utilized appropriately.

Concluding Remarks
• It is noted that the porosity parameter, Casson fluid, and stretching ratio parameter upsurge

Concluding Remarks
• It is noted that the porosity parameter, Casson fluid, and stretching ratio parameter upsurge the radial velocity component.

Concluding Remarks
• It is noted that the porosity parameter, Casson fluid, and stretching ratio parameter upsurge the radial velocity component.

•
The radial component of velocities is increased due to the variation of the porosity parameter and Casson fluid parameter, while the impact of magnetic parameter is reverse.

Concluding Remarks
• It is noted that the porosity parameter, Casson fluid, and stretching ratio parameter upsurge the radial velocity component.

•
The radial component of velocities is increased due to the variation of the porosity parameter and Casson fluid parameter, while the impact of magnetic parameter is reverse. • Temperature profiles increase for thermophoretic and heat source parameters. • A declining nanoparticles temperature results from the Prandtl number and heat sink parameter. • Concentration profile shows increasing behavior for the Lewis number and thermophoretic parameter.

•
Increasing values of the stretching ratio and Prandtl number increase the Bejan number, while reverse behavior is observed for the Casson fluid parameter.

•
The observations from current analysis can be useful in thermal energy exchange processes, cooling processes, energy consumptions, thermodynamics applications, aircrafts, thermal extrusion systems etc.