Cattaneo-Christov Heat Flux Model for Second Grade Nanofluid Flow with Hall Effect through Entropy Generation over Stretchable Rotating Disk

The second grade nanofluid flow with Cattaneo-Christov heat flux model by a stretching disk is examined in this paper. The nanofluid flow is characterized with Hall current, Brownian motion and thermophoresis influences. Entropy optimization with nonlinear thermal radiation, Joule heating and heat absorption/generation is also presented. The convergence of an analytical approach (HAM) is shown. Variation in the nanofluid flow profiles (velocities, thermal, concentration, total entropy, Bejan number) via influential parameters and number are also presented. Radial velocity, axial velocity and total entropy are enhanced with the Weissenberg number. Axial velocity, tangential velocity and Bejan number are heightened with the Hall parameter. The total entropy profile is enhanced with the Brinkman number, diffusion parameter, magnetic parameter and temperature difference. The Bejan number profile is heightened with the diffusion parameter and temperature difference. Arithmetical values of physical quantities are illustrated in Tables.


Introduction
The enhancement of heat transfer utilizing nanofluids, specifically in solar collectors, has been gaining much attention among researchers. The necessity of heat transfer improvement by ordinary fluids, like ethylene glycol, water, kerosene oil, etc., cannot be achieved. The researchers have conducted many experiments in order to develop the thermal transfer rate. Erosion and blockage are the major disadvantages in the drop of higher pressure and heat transfer rates. To reduce such problems, nanofluids are introduced. The suspension of particles of size 1-100 nm in base fluids can improve the thermal conduction in nanofluids. Using nanoparticles suspension, the thermophysical properties of conventional fluids was first proposed by Choi [1]. The applications of nanofluids are energy storage, heat exchangers, chemical industry, refrigeration process, power production, etc. Choi and Eastman [1] introduced the idea of augmenting fluids thermal conductivity. The radiation influence on nanofluid flow was discussed by Farooq et al. [2]. Sajjid et al. [3] investigated the magnetohydrodynamic (MHD) Fe 3 O 4 nanofluid flow with radiation effect. The thermal and mass transmission in a nanofluid flow with chemical reaction and thermal radiation influences was presented by Sreedevi et al. [4]. The flow of silver and copper based nanofluid with radiation impact was determined by Qayyum et al. [5].

Problem Modeling
The second grade nanofluid flow by stretchable rotating disk is assumed. The heat model of C-C is also taken in the nanofluid flow. The Hall current influence is considered in this nanofluid flow. Furthermore, EGM is considered with heat generation/absorption, Joule heating and non-linear thermal radiation. At z = 0 the disk rotates with angular velocity α1. The ambient and disk temperatures are T∞ and Tw respectively. Similarly, the ambient and surface concentrations are C∞ and Cw. Geometry of the fluid is displayed in Figure 1.  The continuity, momentum, energy and concentration equations are taken as [40]: Coatings 2020, 10, 610 4 of 23 where u, v and w are the components of velocity in r, θ and z directions, respectively, ν f is the kinematic viscosity, β 1 is the material parameter, k f is the thermal conductivity, f is the density, cp is the specific heat, Q is the heat absorption/generation, σ f is the electrical conductivity, ( cp) f is the heat capacitance, D T is the thermophoretic diffusion coefficient and D B is the Brownian diffusion coefficient. Similarity transformations are defined as [40] The dimensionless forms of the leading equations are with where We = β 1 / h 2 is the Weissenberg number, Re = α 1 h 2 /ν f is the Reynolds number, A = a/α 1 is the stretching parameter, Pr = ( cp)ν f /k f is the Prandtl number, q = Q/ cpα 1 is the heat absorption/generation parameter, is the Eckert number, Rd = 16σ*T ∞ 3 /3k f k* radiation parameter and λ = rα 1 is the thermal relaxation parameter.

Skin Friction and Nusselt Number
Skin frictions along radial and tangential directions are Dimensionless forms are Re Heat transfer rate is defined as where q w is the wall heat flux, which is defined as The dimensionless form is

Entropy Generation
Equation (21) is reduced as The Bejan number is defined as are the temperature and concentration ratios, respectively.

HAM Solution
Linear operators are Initial guesses are taken as with where P 1 , P 2 , P 3 . . . , P 9 are called constants.

HAM Solution
Linear operators are Initial guesses are taken as where P1, P2, P3… P9 are called constants.

Convergence Investigation
By using the auxiliary parameters f  , g  , θ  and φ  , we analyzed the convergence regions for f'(ζ), g(ζ) and θ(ζ) of the modeled system of equations. At the 25th deformation order, the −  curves are presented here (see Figure 2). Convergence regions for f"

Results and Discussion
Variation in the second grade nanofluid flow due to influential variables on f '(ζ), g(ζ), θ(ζ), ϕ(ζ), N G (ζ) and Be(ζ) are illustrated in Figures 3-29. The parameters are taken fixed as Re = 0.9, We = 0.3, components escalate with a greater Weissenberg number. Figures 11-13 illustrate the change in f '(ζ), g(ζ) and θ(ζ) via the magnetic field parameter. A higher magnetic parameter reduces the velocity components, while a reverse impact of the magnetic parameter on the thermal field is observed. The heightening magnetic field produces higher resistive force to the flow of fluid, which drops the motion of the fluid flow. Thus, the velocity components decline. On the other hand, the higher resistive force increases the electrons collision, which produces more heat to fluid flow. Therefore, the thermal field rises with the higher values of magnetic parameter. Figures 14 and 15 exhibit the change in velocity components (f (ζ), g(ζ)) via a stretching parameter. The momentum boundary layer escalates with higher values of stretching parameter and, consequently, the velocity profile f (ζ) heightens. On the other hand, the velocity profile g(ζ) declines with a higher stretching parameter. This influence is due to the fact that the higher values of stretching parameter reduce the angular velocity of the fluid flow. Figure 16 displays the change in thermal field via Brownian motion and thermophoresis parameters. The rising values of Brownian motion and thermophoresis parameters intensify the thermal field. The rising thermophoretic force pushes the fluid particles to move form heated to cold regions and, consequently, the temperature field increases. A similar impact is also depicted against the Brownian motion parameter. Figure 17 shows the change in thermal profile via thermal relaxation parameter. Higher values of relaxation parameter decline the temperature profile. With higher values of thermal relaxation parameter, the material particles need more potential to transmit energy to its surrounding particles. Additionally, this behavior is less for the C-C model as compared to Fourier's law. Figure 18 illustrates the change in thermal field via temperature difference parameter. An escalating conduct is detected in thermal field by heightening the temperature difference parameter. By increasing the temperature difference parameter, the temperature at the wall increases, and then the ambient temperature also increases. Consequently, the temperature field heightens. Figure 19 clarifies the change in thermal field via heat generation/absorption parameter. Clearly, the increasing heat parameter increases the temperature profile. The heat generation/absorption parameter acts like a heat generator. The increasing generation/absorption increases the thermal field of the fluid flow. Figure 20 indicates the change in thermal profile via the Prandtl number. A declining impact is detected via increasing Prandtl number. Figure 21 indicates the variation in concentration field via thermophoresis parameter. As the increasing thermophoresis parameter increases the thermal field (see Figure 16), consequently, the concentration of the fluid flow also increases. The opposite impact of Brownian motion parameter is depicted against the concentration profile (see Figure 22). The change in concentration field via the Schmidt number is displayed in Figure 23. A declining impact is observed here. The concentration distribution in inversely related with the Schmidt number. The intensifying estimations of the Schmidt number reduce the thickness of the boundary layer flow. The concentration distribution therefore declines.                     Coatings 2020, 10, 610 10 of 22 Figure 9. We on f(ζ). Figure 9. We on f (ζ).

Entropy Optimization and Bejan Number
Figures 24 and 25 depict the impact of Brinkman number on the total entropy profile and the Bejan number profile, respectively. The total entropy profile escalates with a higher Brinkman number, while the opposite trend is observed on the Bejan number profile. The heat rises in the fluid moving in related region heightens with greater Brinkman number. Therefore, the total entropy profile rises with the Brinkman number, and the Bejan number declines (see Figure 25). Figures 26 and 27 illustrate the change in the total entropy profile and the Bejan number profile, via greater values of the diffusion parameter. Both total entropy and the Bejan number profiles escalate with larger values of diffusion parameter. The rising diffusion parameter increases the nanoparticle's diffusion rate. Thus, both profiles escalate with a greater diffusion parameter. Figures 28 and 29 display the change in total entropy and Bejan number profiles via the magnetic parameter. The total entropy profile escalates, while the Bejan number profile deescalates with a higher magnetic parameter. According to the Lorentz force, the total entropy of the system heightens, while the Bejan number reduces with higher magnetic parameter. So, the total entropy profile escalates, while the Bejan number profile deescalates with a higher magnetic parameter. Figures 30 and 31 illustrate the change in total entropy and the Bejan number profiles via temperature difference parameters. Both profiles escalate with higher temperature difference parameters. Figures 32 and 33 show the change in total entropy and Bejan number profiles via the Weissenberg number. Increasing the Weissenberg number reduces the fluid flow viscosity. This behavior escalates both total entropy and the Bejan number profiles. Figures 34 and 35 indicate the variation the total entropy and Bejan number profiles via the Hall parameter. The total entropy profile declines, while the Bejan number profile escalates with the higher Hall parameter. The Hall parameter has a direct effect on the Lorentz force term and current density of the nanofluid. Thus, the electrical conductivity of the nanofluid heightens with the higher Hall parameter and, consequently, the total entropy profile is reduced. The opposite impact of Hall parameter on the Bejan number is displayed in Figure 35. entropy profile escalates, while the Bejan number profile deescalates with a higher magnetic parameter. According to the Lorentz force, the total entropy of the system heightens, while the Bejan number reduces with higher magnetic parameter. So, the total entropy profile escalates, while the Bejan number profile deescalates with a higher magnetic parameter. Figures 30 and 31 illustrate the change in total entropy and the Bejan number profiles via temperature difference parameters. Both profiles escalate with higher temperature difference parameters. Figures 32 and 33 show the change in total entropy and Bejan number profiles via the Weissenberg number. Increasing the Weissenberg number reduces the fluid flow viscosity. This behavior escalates both total entropy and the Bejan number profiles. Figures 34 and 35 indicate the variation the total entropy and Bejan number profiles via the Hall parameter. The total entropy profile declines, while the Bejan number profile escalates with the higher Hall parameter. The Hall parameter has a direct effect on the Lorentz force term and current density of the nanofluid. Thus, the electrical conductivity of the nanofluid heightens with the higher Hall parameter and, consequently, the total entropy profile is reduced. The opposite impact of Hall parameter on the Bejan number is displayed in Figure 35.   entropy profile escalates, while the Bejan number profile deescalates with a higher magnetic parameter. According to the Lorentz force, the total entropy of the system heightens, while the Bejan number reduces with higher magnetic parameter. So, the total entropy profile escalates, while the Bejan number profile deescalates with a higher magnetic parameter. Figures 30 and 31 illustrate the change in total entropy and the Bejan number profiles via temperature difference parameters. Both profiles escalate with higher temperature difference parameters. Figures 32 and 33 show the change in total entropy and Bejan number profiles via the Weissenberg number. Increasing the Weissenberg number reduces the fluid flow viscosity. This behavior escalates both total entropy and the Bejan number profiles. Figures 34 and 35 indicate the variation the total entropy and Bejan number profiles via the Hall parameter. The total entropy profile declines, while the Bejan number profile escalates with the higher Hall parameter. The Hall parameter has a direct effect on the Lorentz force term and current density of the nanofluid. Thus, the electrical conductivity of the nanofluid heightens with the higher Hall parameter and, consequently, the total entropy profile is reduced. The opposite impact of Hall parameter on the Bejan number is displayed in Figure 35.

Physical Quantities
The values of skin friction along radial and tangential directions are presented in Table 1. The higher Weissenberg and Reynolds numbers increase the skin friction, while the magnetic, Hall and stretching parameters have a reducing influence on the skin friction along the radial direction. The heightening magnetic parameter escalates the skin friction while opposite trend is observed via the Weissenberg number, Reynods number, Hall parameter and stretching parameter along a tangential direction. Table 2 represents the values of Nux, via influential parameters and numbers. The rising values of λ, heightens Nux, while increasing Re, Pr, q, Nb, Nt, Ec and θw reducing Nux. Table 1. Numerical values of skin friction along radial (Rer, Cfr) and tangential (Reθ, Cfθ) directions.

Physical Quantities
The values of skin friction along radial and tangential directions are presented in Table 1. The higher Weissenberg and Reynolds numbers increase the skin friction, while the magnetic, Hall and stretching parameters have a reducing influence on the skin friction along the radial direction. The heightening magnetic parameter escalates the skin friction while opposite trend is observed via the Weissenberg number, Reynods number, Hall parameter and stretching parameter along a tangential direction. Table 2 represents the values of Nu x , via influential parameters and numbers. The rising values of λ, heightens Nu x , while increasing Re, Pr, q, Nb, Nt, Ec and θ w reducing Nu x . Table 1. Numerical values of skin friction along radial (Re r , C fr ) and tangential (Re θ , C fθ ) directions.

We
Re M m A Re r C fr Re θ C fθ

Conclusions
The second grade nanofluid flow by a stretching disk is examined here. The nanofluid flow is characterized with Hall current, Brownian motion and thermophoresis influences. Entropy optimization with nonlinear thermal radiation, Joule heating and heat absorption/generation is also presented. Concluding remarks are mentioned below: • Velocity along the radial direction is enhanced with the Weissenberg number and stretching parameter, however, the conflicting influence is observed via the Reynolds number. • Velocity along the axial direction is heightened with the Hall parameter and Weissenberg number, while it is decreased with the magnetic parameter and the Reynolds number. • Velocity along the tangential direction is escalated with the Hall parameter, while it decays with the magnetic parameter, Reynolds number and stretching parameter.

•
The temperature profile is increased with the magnetic parameter, Brownian motion parameter, thermophoresis parameter, temperature difference and heat generation/absorption parameter, whereas it is reduced with the Hall parameter, thermal relaxation parameter and Prandtl number.

•
The concentration profile is increased with the thermophoresis parameter, while it is reduced with the Brownian motion parameter and the Schmidt number.

•
The total entropy profile is enhanced with the Brinkman number, diffusion parameter, magnetic parameter, temperature difference and Weissenberg number, while a declining influence is detected via the Hall parameter.

•
The Bejan number profile is heightened with diffusion parameter, temperature difference and Hall parameter, while it is diminished with the Brinkman number, magnetic parameter and Weissenberg number. Funding: This research received no external funding.