Entropy Generation and Heat Transfer Analysis in MHD Unsteady Rotating Flow for Aqueous Suspensions of Carbon Nanotubes with Nonlinear Thermal Radiation and Viscous Dissipation Effect

The impact of nonlinear thermal radiations rotating with the augmentation of heat transfer flow of time-dependent single-walled carbon nanotubes is investigated. Nanofluid flow is induced by a shrinking sheet within the rotating system. The impact of viscous dissipation is taken into account. Nanofluid flow is assumed to be electrically conducting. Similarity transformations are applied to transform PDEs (partial differential equations) into ODEs (ordinary differential equations). Transformed equations are solved by the homotopy analysis method (HAM). The radiative source term is involved in the energy equation. For entropy generation, the second law of thermodynamics is applied. The Bejan number represents the current investigation of non-dimensional entropy generation due to heat transfer and fluid friction. The results obtained indicate that the thickness of the boundary layer decreases for greater values of the rotation parameter. Moreover, the unsteadiness parameter decreases the temperature profile and increases the velocity field. Skin friction and the Nusselt number are also physically and numerically analyzed.


Introduction
Applications of nanofluids in technology and science are increasing day by day, and they play an important role in various machinery and engineering applications such as detergents, microchip technology, transferences, micromechanical systems and biomedical applications. In light of these applications, researchers have applied modern techniques and have modified the base fluids by adding ultra-fine solid particles. A fast-growing field of research is micro channel cooling, floor heating and heat renewal systems in various industries, which have been flourishing in the current era. performance by using various shapes of nano-additives. Alsarraf et al. [59] explored nanofluid flow with different nanoparticle shapes in a mini-channel heat exchanger using a two-phase mixture model. Moradikazerouni et al. [60] investigated the effects of five different channel forms of a micro-channel heat sink in forced convection, with application to cooling a supercomputer circuit board.
The aim of the current research is to obtain an analytical solution using the homotopy analysis method (HAM) for an unsteady, MHD, and the incompressible rotating flow of carbon nano tubes nanofluid over a shrinking surface with nonlinear thermal radiation and viscous dissipation effect. Here, we consider three types of nanofluids: CuO-water, Ag-water and Au-water, where water is used as the base fluid. The impact of the first order chemical reaction is also deliberated. The problem is formulated, solved and the corresponding results are examined in detail. Finally, the impact of the physical parameters on temperature and concentration profiles are presented and analyzed.

Mathematical Formulation of the Problem
Single-walled carbon nanotube nanofluid unsteady laminar incompressible three-dimensional rotating flow is considered over a shrinking surface. The Cartesian coordinates are chosen in the x, y, and z dimensions. The nanofluid rotates with an angular velocity about the z-axis, which is denoted by Ω(t). The surface velocity is represented by u w (x, t) and given as u w (x, t) = bx (1−δt) in the x direction, v w (x, t) in y direction and w w (x, t) in the z direction, and w w (x, t) is the wall mass flux velocity. The nanofluid flow is assumed to be thermally conductive. Radiative and viscous dissipation effects are taken into account.
Using all these assumptions, the governing equations are written as [9,25,44,45]: The boundary conditions are: where x, y and z are the directions of the velocity components; Ω denotes a constant angular velocity; µ n f denotes the nanofluid dynamic viscosity; ρ n f denotes the nanofluid density α n f ; T represents the temperature of the nanofluid; T w and T ∞ are the wall and the outside surface temperatures, respectively. The radiative heat flux in Equation (5) can be shown as [9,25]: where the Stefan-Boltzmann constant and the mean absorption coefficient are denoted by σ * and k * , respectively. By substituting Equation (7) into Equation (4), it can be written as [6][7][8][9]25]: Other parameters with nanoparticle volume fraction are mathematically presented as [6,7,22,23]: In Equation (9), for the base fluid the volumetric heat capacity is denoted as ρC p f and for CNTs as ρC p CNT . The thermal conductivity of the nanofluid, base fluid, and CNTs are denoted as k n f , k f and k CNT , respectively. The nanoparticle volume fraction is denoted by φ; the density viscosity of CNTs and the base fluid are represented by ρ CNT and ρ f , respectively.
Similarity transformations [26] are introduced as: Using Equation (10) and Equations (2)-(6), we obtain: The transformed boundary conditions are written as: where Ω = ω b is the rotation parameter, λ = δ b is the unsteadiness parameter and R = 16σ * T 3 ∞ 3k n f k * is the radiation parameter. The Prandtl number is denoted by Pr = α n f ν n f , the Eckert number is denoted by Ec = u 2 w c p (T−T ∞ ) and the temperature ratio parameter is denoted by θ w = T w T α .

Physical Quantities of Interest
Skin friction in the x and y directions is denoted as C f x and C f y , respectively, and the Nusselt number is Nu x . These are defined as: , where τ wx and τ wy are the surface shear stress in the x and y directions, respectively; q w is the surface heat flux. These can be defined as: Using Equations (16) and (17), we obtain: The local Reynolds number is denoted by Re x = u w x/v .

Entropy Generation and Bejan Number
The dimensional local entropy rate per unit volume for a nanofluid is given by [42][43][44][45][46][47][48][49][50][51][52][53]: where ∇T = ∂T ∂x + ∂T ∂y + ∂T ∂z and Φ represent viscous dissipation. In this instance, We have three sub-generators of entropy, as deduced from Equation (20). The heat transfer dimensional entropy generator is represented by S h . Due to thermal radiation, the dimensional entropy generator is represented by S R and the inter-friction of the fluid layers is represented by S f . S g,c is defined as: Now, the non-dimensional Ns (Nusselt number) is defined as: To evaluate the non-dimensional Ns we use Equations (19) and (21) combined with Equations (10) and (22) to obtain: Here A = k n f k f , Br and Re L are the Brinkman and Reynolds numbers, respectively, and Ω is the non-dimensional temperature, which can be shown as: Equation (23) can be rewritten as: where the fluid friction, thermal radiation and heat transfer non-dimensional entropy generators are denoted by N h , N R and N f , respectively. The mathematical description of the Bejan number (Be) is: From Equation (26), it is clear that the Bejan number is limited to the unit interval [0, 1].

Solution Procedure
The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: Lf , Lĝ and Lθ are linear operators which are represented as. The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: They have the following applicability: where ( 1, 2, 3, ..., 9) k k = is constant.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: They have the following applicability: where ( 1, 2, 3, ..., 9) k k = is constant.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on The modeled Equations (11)-(13) with boundary conditions from Equation (14), toge the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysi is applied due to its outstanding results in boundary layer equations. Several researchers [46 used HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.
They have the following applicability:

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems effects on They have the following applicability: The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on Actually, increasing the rotation parameter enhances the kinetic energy, which consequently increases the velocity profile, whereas the transverse velocity ( The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on Actually, increasing the rotation parameter enhances the kinetic energy, which consequently increases the velocity profile, whereas the transverse velocity ( The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on Actually, increasing the rotation parameter enhances the kinetic energy, which consequently increases the velocity profile, whereas the transverse velocity ( The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on Actually, increasing the rotation parameter enhances the kinetic energy, which consequently increases the velocity profile, whereas the transverse velocity ( with higher values of the rotation parameter. Figures 3 and 4 The modeled Equations (11)-(13) with boundary conditions from Equation (14), tog the conditions from Equations (23) and (27), are solved with HAM. The homotopy analy is applied due to its outstanding results in boundary layer equations. Several researchers [4 used HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.
They have the following applicability:

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problem effects on  (23) and (27), are solved with HAM. The homotopy a is applied due to its outstanding results in boundary layer equations. Several research used HAM due to it fast convergence. The preliminary guesses are selected as follow They have the following applicability:

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled pro effects on   (23) and (27), are solved with HAM. The homo is applied due to its outstanding results in boundary layer equations. Several re used HAM due to it fast convergence. The preliminary guesses are selected as They have the following applicability:

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the mode effects on The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: They have the following applicability: where ( 1, 2, 3, ..., 9) k k = is constant.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on The modeled Equations (11)-(13) with boundary conditions from Equation (14), together with the conditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method is applied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have used HAM due to it fast convergence. The preliminary guesses are selected as follows: They have the following applicability: where ( 1, 2, 3, ..., 9) k k = is constant.

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on  (14), together with nditions from Equations (23) and (27), are solved with HAM. The homotopy analysis method lied due to its outstanding results in boundary layer equations. Several researchers [46][47][48][49][50] have HAM due to it fast convergence. The preliminary guesses are selected as follows: and L g f LL  are linear operators which are represented as.
hey have the following applicability:

Results and Discussion
In this section, the physical outcome of dissimilar parameters of the modeled problems and their effects on f (η), g(η) and θ(η) are discussed in detail. The effect of Ω, β, φ and λ on the velocity profile is shown in Figures 1-8. The impact of Ω on f (η) and g(η) is presented in Figures 1 and 2. It can be seen that for larger values of Ω the velocity profile ( f (η)) is increased while g(η) is decreased. Actually, increasing the rotation parameter enhances the kinetic energy, which consequently increases the velocity profile, whereas the transverse velocity (g(η)) is reduced with higher values of the rotation parameter. Figures 3 and 4 represent the influence of φ on f (η) and g(η). The higher values of φ reduce the velocity profiles. This is because the increase in φ further increases the density of the nanofluid, and as a result slows down the fluid velocity profile. Figures 5 and 6 describe the effect of λ on f (η) and g(η). It was perceived that increases in λ reduce the velocity profile. It is also indicated from the figure that the velocity intensifies with increasing λ, whereas we observed the opposite influence of λ on the fluid velocity inside the nanofluid and the thickness of the layer. Figures 7 and 8 show the influence of β on f (η) and g(η). With an increase in β the velocity profile of the fluid film is decreased. It was also detected that an increase in β results in a decrease in the fluid velocity of the nanofluid and the layer thickness. The purpose behind this influence of β by the stimulation of a lingering body force, stated as the Lorentz force, is due to the existence of β in an electrically conducting nanofluid layer. The action of this force is perpendicular to both fields. Since β represents the ratio of the viscous force to the hydromagnetic body force, a larger value of β specifies a higher hydromagnetic body force, due to which the fluid flow is reduced. The Lorentz force theory states that β has a converse consequence on f (η) and g(η). Therefore, the greater values of β reduce f (η) and g(η).                  Figure 9 presents the impact of  on the     profile. Figure 9 shows that a decrease in  reduces the boundary layer thickness. It can be seen that when unsteadiness in the stretching increases, the thin film fluid temperature and the free surface temperature are consequently reduced.   Figure 9 presents the impact of  on the     profile. Figure 9 shows that a decrease in  reduces the boundary layer thickness. It can be seen that when unsteadiness in the stretching increases, the thin film fluid temperature and the free surface temperature are consequently reduced. The influence of the physical parameters λ, Ec, Rd and Pr on θ(η) is shown in Figures 9-13. Figure 9 presents the impact of λ on the θ(η) profile. Figure 9 shows that a decrease in λ reduces the boundary layer thickness. It can be seen that when unsteadiness in the stretching increases, the thin film fluid temperature and the free surface temperature are consequently reduced. Under consitions of stirring, it was revealed that greater values of λ cause the fluid temperature to fall radically, while the thickness of the thermal boundary layer is increased. The influence of Rd on θ(η) is shown in Figure 10. By increasing Rd, the temperature of the nanofluid boundary layer area increased. In fact, when Rd is raised, then it is obvious that it increases θ(η) in the boundary layer area in the fluid layer. It is shown in Figure 11 that θ(η) increases with a rise in Rd. Thermal radiation has a dominating role in the comprehensive surface heat diffusion when the coefficient of convection heat transmission is small. Increasing Rd then increases the temperature in the boundary layer area in the fluid layer. This increase leads to a drop in the rate of cooling for nanofluid flow. Therefore, the fluid θ(η) is increased. The graphical representation shows that θ(η) is increased when we increase the ratio strength and thermal radiation temperature. Thermal radiation has an important role in heat conduction when the coefficient of convection heat transmission is small. The impact of Pr on θ(η) given in Figure 11. It was observed that θ(η) decreases with larger values of Pr, while it rises for smaller values. The variation of θ(η) with respect to the variation of Pr is illustrated and shows that Pr specifies the ratio of momentum diffusivity to thermal diffusivity. It can be concluded that θ(η) decreases with increasing Pr. The nanofluids have a greater thermal diffusivity with a small Pr, but this influence does not hold for larger values of Pr; hence, θ(η) of a fluid displays a reducing behavior. Actually, the fluids having a smaller Pr have a greater thermal diffusivity, and this impact is the reverse for greater values of Pr. Based on this, a very large value of Pr causes the thermal boundary layer to drop. Figure 13 shows that with increasing Ec, θ(η) is enlarged, which is supported by the physics. By increasing Ec, heat stored in the liquid is dissipated, causing the temperature to be enhanced. θ(η) is increased with greater values of Ec and the thermal boundary layer thickness of the nanofluid becomes larger.  Figure 13 shows that with increasing Ec ,     is enlarged, which is supported by the physics. By increasing Ec , heat stored in the liquid is dissipated, causing the temperature to be enhanced.     is increased with greater values of Ec and the thermal boundary layer thickness of the nanofluid becomes larger.     (23) and (27)). The influence of Re, Br, Rd and Ω on Ns and Be are examined and displayed in Figures 14-20. Figures 14-17 examine one of the significant features of this study, i.e., volumetric entropy generation for Br and Re. The influence of Ns becomes increasingly important to all these parameters. Higher Ns and Be values are due to an increase of Br. The higher Ns and Be values are also generated by the role of Re. Ns and Be strongly depend on Re. We observed that increasing Re also increases Ns. As Re increases, hectic motion occurs, the fluid moves more vigorously and thus the impact of heat transfer and fluid friction on Ns and Be tends to increase entropy generation. Figure 18 shows that entropy generation is reduced with increased Rd. From Figure 20, it can be seen that Be increases with an increase in Rd. From Figures 19 and 20, it can be observed that Be is reduced near the lower plate of the channel where Ω is more intense; meanwhile, farther from the plate the drift is reversed due to further contribution from the irreversible heat transfer on Ns and Be, which reduces the nearby upper plate of the channel with an increase in Ω. Now, we analyze the impact of the parameters that perform a role in entropy generation and the Bejan number (Equations (23) and (27)). depend on Re . We observed that increasing Re also increases Ns . As Re increases, hectic motion occurs, the fluid moves more vigorously and thus the impact of heat transfer and fluid friction on Ns and Be tends to increase entropy generation. Figure 18 shows that entropy generation is reduced with increased Rd . From Figure 20, it can be seen that Be increases with an increase in Rd . From           Figure 21 it can be observed that the unsteady parameter increases C f x and C f y . However, this trend is reversed for greater values of Ω. It can be seen in Figure 22 that the skin friction coefficient reduces for increasing values of Ω. Figure 23 shows that for increasing values of φ, the skin friction coefficient increases. From the convergence of the series given in Equation (25), f (η), g(η), θ(η) depends entirely upon the auxiliary parameters f , g , θ and the so-called -curve. It is selected in such a way that it controls and converges on the series solution. The probable selection of can be found by plotting -curves of f (0), g (0), θ (0) for the 20th order approximated HAM solution, as shown in Figures 24 and 25. The valid region of is −0.1 < f < 0.3, −0.5 < g < 0.1, −0.5 < θ < 0.1.

Conclusions
The exploration of nanoparticles preparations has introduced more deliberation in mechanical and industrial engineering owing to their probable use for increasing the continuous phase fluid thermal performance of cooling devices. A significant source of renewable energy is thermal radiation, which can be beneficial to govern overall population levels. In the present work, the second

Conclusions
The exploration of nanoparticles preparations has introduced more deliberation in mechanical and industrial engineering owing to their probable use for increasing the continuous phase fluid thermal performance of cooling devices. A significant source of renewable energy is thermal radiation, which can be beneficial to govern overall population levels. In the present work, the second law of thermodynamics is applied in terms of the impact of nanoparticles on non-dimensional entropy for rotating flow with suggested thermal radiation. Mathematical modeling is established by modeling five different types of nanoparticles with the purpose of achieving an appropriate

Conclusions
The exploration of nanoparticles preparations has introduced more deliberation in mechanical and industrial engineering owing to their probable use for increasing the continuous phase fluid thermal performance of cooling devices. A significant source of renewable energy is thermal radiation, which can be beneficial to govern overall population levels. In the present work, the second law of thermodynamics is applied in terms of the impact of nanoparticles on non-dimensional entropy for rotating flow with suggested thermal radiation. Mathematical modeling is established by modeling five different types of nanoparticles with the purpose of achieving an appropriate mechanism to enhance the thermal conductivity of continuous phase fluid. The following conclusions can be made: • The unsteadiness parameter decreases the temperature profile and increases the velocity field.

•
The thermal boundary layer thickness is reduced for larger values of the rotation rate parameter.

•
The heat transfer rate rises for greater values of Rd and θ w .

•
With increasing values of Pr, the heat profile θ(η) reduces.

•
The performance of Be is examined for the optimal values of the parameters at which Ns decreases. • Entropy generation is increased with the increase of Pr, Ec and radiative heat flux.

•
Velocity and temperature profiles decrease due to the increased unsteadiness parameter.