Entropy Generation in MHD Second-Grade Nanoﬂuid Thin Film Flow Containing CNTs with Cattaneo-Christov Heat Flux Model Past an Unsteady Stretching Sheet

: Entropy generation plays a signiﬁcant role in several complex processes, extending from cosmology to biology. The entropy generation minimization procedure can be applied for the optimization of mechanical systems including heat exchangers, elements of nuclear and thermal power plants, ventilation and air-conditioning systems. In order to present our analysis, entropy generation in a thin ﬁlm ﬂow of second grade nanoﬂuid holding single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) with a Cattaneo–Christov heat ﬂux model is studied in this article. The ﬂow is considered passing a linearly extending surface. A variable magnetic ﬁeld with aligned angle ε is functioned along the extending sheet. With the aid of the homotopy analysis method (HAM), the ﬂuid ﬂow model is elucidated. The impressions of embedded factors on the ﬂow are obtainable through ﬁgures and discussed in detail. It is observed that the velocity proﬁle escalated with the increasing values of volume fraction of nanoparticles and second grade ﬂuid parameter. The higher values of volume fraction of nanoparticles, second grade ﬂuid parameter, non-linear heat source / sink, and thermal radiation parameter intensiﬁed the temperature proﬁle. Surface drag force escalated with heightening values of nanoparticles volume fraction, unsteadiness, ﬁlm thickness, magnetic, and second grade ﬂuid parameters. Entropy generation increased with enhancing values of magnetic parameter, Brinkman number, and Reynolds number.


Introduction
The hydrodynamics thin liquid moving over a linearly extending sheet is of significant importance in manufacturing processes. Starching sheets have tremendous applications in various fields of engineering and industries, such as food processing, tinning of copper wires, plastic sheets' extrusion, polymer surfaces drawing, continuous casting, and metal plates cooling, rotating heat exchangers, aeronautical engineering, nuclear reactor cooling system, solar collectors, homeo-therapy treatment, conductivity transfer. The Oldroyd-B thin film flow containing silver and copper nanoparticles was analyzed by Zhang et al. [23,24]. They found that the silver nanoparticles nanofluid had superior thermal conductivity associated to copper nanoparticles nanofluid. In another article they observed the thin film flow power law nanofluid with velocity slip and variable magnetic field influence. The detailed review about nanofluid relating to its description, preparation, thermophysical assets and applications' fields with different effects can be found in Refs. [25][26][27][28][29][30].
There are various uses for transfer of heat in manufacturing and engineering procedures which include microelectronics, fuel cells, nuclear reactors, and refrigeration centers. The underlying concept of all these operations being that the importance of thermal conductivity is believed to be unchanged. Such quality, indeed, correlates with heat and many other considerations. The thermal conductivity in MHD power-law fluid considering Soret and Dufour influences was discussed by Pal and Chatterjee [31]. Vajravelu et al. [32] exploring the flow of fluid with variable fluid properties. The improved version of Fourier's law by Cattaneo and Christov is called the Cattaneo-Christov (C-C) heat flux model [33]. This model is used for the control of the thermal boundary layer. The analytical approach for viscoelastic fluid with velocity slip conditions considering the heat flux model of C-C was investigated by Han et al. [34]. Mustafa [35] numerically and analytically analyzed the rotating nanofluid flow along with heat flux model of C-C past an extending sheet. The same case over an exponentially extending surface was argued by Khan et al. [36]. Lu et al. [37] observed the squeezing nanofluid flow containing carbon nanotubes (CNTs) along with C-C model of heat flux. The MHD flow of Williamson fluid using heat flux model of C-C was probed by Ramzan et al. [38]. Ramzan et al. [39] discussed the MHD fluid flow with C-C model of heat flux. In another article, Ramzan et al. [40] extended the same study for third grade fluid with heat flux model of C-C. Alshomrani and Ullah [41] investigated the Forchheimer flow of CNTs. Shah et al. [42] observed the flow of nanofluid containing CNTs with C-C model of heat flux. Now Newtonian fluids are in diverse types which have the multipart rheological properties dependent on their viscosity conduct as a function of shear rate, stress, deformation history etc. Each of them has its individual features so, there is no solitary mathematical model which can describe the fluid flow performance of all non-linear fluids. One of them is second grad fluid model [43] which is important sub type of Now Newtonian fluid. Rajagopal et al. [44] investigated second grade fluid flow over an infinite permeable plate with the suction properties. An exact solution is obtained, and it is found that the presence of solution is connected to the indication of material moduli and in stamped complexity to the classical Newtonian, liquid arrangements can be shown for the blowing problem. Alamri et al. [45] probed the MHD second-grade fluid past an extending sheet using C-C model of heat flux.
The review of literature shows that the thin film fluids are limited, and this topic is even tapered when we think about the nanofluids thin film flows. There are very few surveys accessible to address the thin film flows containing CNTs. The present model explores the second-grade thin film flow of nanofluid containing single-walled and multi-walled carbon nanotubes past an extending sheet. A variable magnetic field with aligned angle ε is functioned along the extending sheet. Entropy generation in a second-grade thin film flow of nanofluid comprising SWCNTs and MWCNTs with heat flux model of C-C is not yet been discussed (see Table 1). HAM is used to present the semi-analytical solution. The variation in second grade thin film flow due to physical parameters is displayed through figures. The rates of surface drag force and heat transfer are accessible through figures also.

Mathematical Modeling
We have assumed the laminar and incompressible thin film flow of second grade fluid containing carbon nanotubes (CNTs) over a linearly extending sheet with Cattaneo-Christov model of heat flux. The stretching sheet is positioned at the starting point of a coordinate system (x, y) as displayed in Figure 1. The stretching sheet moves along x-direction with a velocity U w (x, t) = bx (1−αt) where α and b are constants, and temperature is considered as T w (x, y). The width of the thin film flow is h(x, y). Along the extending surface, a variable magnetic field of B(x, t) = B 0 (1−at) 1 2 is applied with an allied angle ε. The stream function ξ is taken such that u = ∂ξ ∂y and v = − ∂ξ ∂x .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 27  3  2  3  3  2  2  2  1  2  2  2  3  2  2 cos , The boundary conditions are defined as: 0, , 0, as .  The Cauchy stress tensor in a second-grade fluid is [43,44] where α 1 and α 2 are material parameters, µ is dynamic viscosity, p is the pressure, I is the identity tensor, A 1 and A 2 are Rivlin-Ericksen tensors defined above, d/dt is the material time derivative, and V is the velocity of the fluid. The Clausius-Duhem inequality is verified and Helmholtz free energy is minimum in equilibrium for the fluid locally at rest when [43,44] µ ≥ 0, α 1 ≥ 0, α 1 + α 2 = 0.
When α 1 + α 2 = 0, the second grade fluid equation reduces to viscous fluid. Applying the above assumptions, the governing equations can be specified as [46] ∂u ∂x v ∂u ∂y + u ∂u ∂x + ∂u ∂t = υ n f ρc p n f v ∂T ∂y + u ∂T ∂x + ∂T ∂t + λ 2 The boundary conditions are defined as: In the above equations, u and v are velocity components along x− and y− directions respectively, υ n f is the kinematic viscosity of the nanofluid, α 1 is the non-dimensionless second grade fluid parameter, ρ n f is the density of the nanofluid, σ n f is the electrical conductivity of the nanofluid, c p is the specific heat on the nanofluid, λ 2 is the Cattaneo-Christov parameter, k n f is the thermal conductivity of the nanofluid, σ * is the Stephan-Boltzmann constant, k * is the mean absorption coefficient, and q is the non-uniform heat source/sink term.
q is defined as According to Xue model [47] The similarity variables are defined as Using Equation (11), the continuity equation is obvious, and Equations (6) and (7) yield with Appl. Sci. 2020, 10, 2720 6 of 21 In the above equations, φ signifies nanoparticles volume fraction, λ represents the film thickness, S indicates the unsteadiness parameter, α indicates the dimensionless second grade fluid parameter, M is the magnetic parameter, Pr is the Prandtl number, Rd represents the thermal radiation parameter, A * , B * are the heat source/sink parameters, Ec is the Eckert number, and γ is the thermal relaxation parameter.
The dimensionless parameters are defined as

Surface Drag Force
The skin friction coefficient is defined as where Using Equation (11), Equations (16) and (17) are reduced as

Heat Transfer Rate
The heat transfer rate is defined as Using Equation (11), the dimensionless form of Equations (19) and (20) are reduced as where

Entropy Generation
Entropy generation for the above mentioned suppositions is Using Equation (11), the dimensionless form of Equation (20) is BrRe where where N G , α 2 and Br, Re indicate the rate of entropy optimization rate, gradient of temperature, Brinkman number, and Reynold's number, respectively.

HAM Solution
To solve Equations (12) and (13) with (14) by using HAM, it is assumed that f = V and θ = H. Initial guesses and linear operators are where Ω i (i = 1, 2, 3, . . . , 7) are arbitrary constants. 0 th − order problems Assume that W ∈ [0, 1] be the rooted factor and V and H are the non-zero supporting factors, then Appl. Sci. 2020, 10, 2720 8 of 21 k th − order problems k th − order problems are When W = 0 and W = 1, we can write By Taylor's expansion The series (38) and (39) converge at W = 1, then where

Convergence of HAM
Auxiliary factors f and θ play an imperative character in convergence regions for velocity and temperature functions. Here, in Figure 2 at 25th order of approximations, the − curves are schemed. The ranges for velocity and temperature profiles are −4.0 ≤ f ≤ 3.0 and −0.04 ≤ θ ≤ 0.04 respectively. Table 2 show numerically the convergent of homotopy analysis method.     [46] and Xu et al. [48], which addressed the nanofluids flow through a thin film under the magnetic impact. Thus, to compare our work, the special effects of all fixed factors are neglected. Great coordination is proven among the established comparison. It should be noted that the red colour represents the increasing values of unsteadiness parameter S , black colour represents ref. [46], yellow colour represents ref. [48], and blue colour represents the present values.    Figure 3 is designed to visualize the accuracy of the model provided by contrasting it with Sandeep [46] and Xu et al. [48], which addressed the nanofluids flow through a thin film under the magnetic impact. Thus, to compare our work, the special effects of all fixed factors are neglected. Great coordination is proven among the established comparison. It should be noted that the red colour represents the increasing values of unsteadiness parameter S, black colour represents ref. [46], yellow colour represents ref. [48], and blue colour represents the present values.

Velocity and Temperature Functions
The consequences of nanoparticles volume fraction φ on velocity

Velocity and Temperature Functions
The consequences of nanoparticles volume fraction φ on velocity f (η) and temperature θ(η) of the thin film second grade fluid flow containing SWNCTs and MWCNTs are shown in Figures 4 and 5. Both velocity f (η) and temperature θ(η) profiles heighten with greater values of volume fraction of nanoparticles. The relationship between nanoparticles volume fraction and convective flow is directly proportional. Therefore, the higher estimations of nanoparticles volume fraction φ increase fluid velocity f (η) and temperature θ(η) for both cases of CNTs. The deviation in velocity function f (η) against magnetic parameter M is displayed in Figure 6. Physically, the increasing magnetic field boosts up the Lorentz forces which increases the resistive force to fluid motion and consequently f (η) reduces.  Figure 8. A similar impact is observed for the temperature profile. The variations in velocity f (η) and temperature θ(η) fields via dimensionless second grade fluid par α are shown in Figures 9 and 10. It is perceived that higher estimations of dimensionless second grade fluid par α escalate f (η) and θ(η) of the fluid flow. Infect there is inverse relation between the viscosity of fluid and α. Therefore, the greater values of α moderate the viscosity of the fluid, and accordingly the fluid velocity upsurges. Alike impact is also perceived in temperature profile. The variations in f (η) and θ(η) via unsteadiness parameter S are displayed in Figures 11 and 12. Both f (η) and θ(η) reduce with higher estimations of unsteadiness parameter S. Physically, due to the greater unsteadiness parameter, the buoyancy effect actions on the flow and declines it. Thus, the momentum and thickness of the thermal boundary layer diminish. The deviation in temperature field θ(η) via heat source/sink parameters A * and B * is displayed in Figures 13 and 14. Actually, heat source/sink parameters A * and B * act like a heat creator. The greater heat source/sink, the greater heat will produce to the fluid flow. Therefore, temperature profile heightens via A * and B * . The variation in θ(η) via thermal radiation parameter Rd is displayed in Figure 15. θ(η) increases with the heightens in thermal radiation parameter Rd. This effect is because of the datum that the higher estimations of Rd deescalates the mean absorption coefficient and escalates the divergence of radiative heat flux. Therefore, the thermal profile escalates with the rise in Rd. The influence of thermal relaxation parameter γ on θ(η) of the fluid flow is publicized in Figure 16. It is depicted that higher estimations of thermal relaxation parameter γ reduces θ(η). From here, it is concluded that the particles require much additional time for transferring of heat to its neighborhood particles and so the heat of the fluid reduces. Therefore, the higher estimations of γ reduce θ(η). fluid flow. Therefore, temperature profile heightens via * A and * B . The variation in ( ) θ η via thermal radiation parameter Rd is displayed in Figure 15. ( ) θ η increases with the heightens in thermal radiation parameter Rd . This effect is because of the datum that the higher estimations of Rd deescalates the mean absorption coefficient and escalates the divergence of radiative heat flux. Therefore, the thermal profile escalates with the rise in Rd . The influence of thermal relaxation parameter γ on ( ) θ η of the fluid flow is publicized in Figure 16. It is depicted that higher estimations of thermal relaxation parameter γ reduces ( ) θ η . From here, it is concluded that the particles require much additional time for transferring of heat to its neighborhood particles and so the heat of the fluid reduces. Therefore, the higher estimations of γ reduce ( ) θ η .

Skin Friction Coefficient
The engineers take interest to calculate some physical quantities in the flow/circulation of the fluid in some parts of the machinery, like the friction between the extending surface and the nanofluid and thermal transmission rate between the fluids and some solid parts of the mechanical system in the machinery. The friction between the nanofluid and extending sheet also affect the temperature of the nanofluid. This friction between the nanofluid and extending sheet is known as skin friction. Figure 17 is displayed to investigate the surface drag force in a thin film flow containing both SWCNTs and MWCNTs against the dimensionless parameters like nanoparticles volume fraction, unsteadiness, film thickness, magnetic, and second grade fluid. The surface drag Figure 16. Variation in θ(η) via γ.

Skin Friction Coefficient
The engineers take interest to calculate some physical quantities in the flow/circulation of the fluid in some parts of the machinery, like the friction between the extending surface and the nanofluid and thermal transmission rate between the fluids and some solid parts of the mechanical system in the machinery. The friction between the nanofluid and extending sheet also affect the temperature of the nanofluid. This friction between the nanofluid and extending sheet is known as skin friction. Figure 17 is displayed to investigate the surface drag force in a thin film flow containing both SWCNTs and MWCNTs against the dimensionless parameters like nanoparticles volume fraction, unsteadiness, film thickness, magnetic, and second grade fluid. The surface drag force C f of the thin film flow containing both SWCNTs and MWCNTs heightens against nanoparticles volume fraction, unsteadiness parameter, film thickness, magnetic field parameter, and second grade fluid parameter. It is also depicted that the impact of dimensionless parameters on SWCNTs is dominant compared to MWCNTs. It should be noted that all the three colours (i.e., black, blue, and red) represent the variation in skin friction coefficient due to dimensionless parameters.

Heat Transfer Rate
The engineers also take interest to calculate the rate of heat exchange between the flowing fluids and the solid parts of the surface in contact with fluids. This rate is called Nusselt number which is the ratio between convective thermal transmission and conductive thermal transmission at the specific point in the flow. Figure 18 is displayed to investigate the variation in thermal transfer rate of the thin film flow containing both SWCNTs and MWCNTs via dimensionless parameters. The thermal transmission rate of the thin film flow containing SWCNTs and MWCNTs heightens with the higher estimations of nanoparticles volume fraction, film thickness, and unsteadiness parameter while declines with the higher estimations of heat source and sink. It should be noted that all the three colours (i.e., black, blue, and red) represent the variation in local Nusselt number due to dimensionless parameters.

Heat Transfer Rate
The engineers also take interest to calculate the rate of heat exchange between the flowing fluids and the solid parts of the surface in contact with fluids. This rate is called Nusselt number which is the ratio between convective thermal transmission and conductive thermal transmission at the specific point in the flow. Figure 18 is displayed to investigate the variation in thermal transfer rate of the thin film flow containing both SWCNTs and MWCNTs via dimensionless parameters. The thermal transmission rate of the thin film flow containing SWCNTs and MWCNTs heightens with the higher estimations of nanoparticles volume fraction, film thickness, and unsteadiness parameter while declines with the higher estimations of heat source and sink. It should be noted that all the three colours (i.e., black, blue, and red) represent the variation in local Nusselt number due to dimensionless parameters.   Figure 19 displays the impact of magnetic parameter M on N G (η). The higher estimations of magnetic parameter M escalate the entropy generation rate. The higher magnetic parameter effect, the greater the Lorentz force which increases the resistance of thin film fluid motion, and consequently the temperature increases. This heightens the thermal transfer rate at the wall as shown in Figure 19. Since entropy optimization N G (η) is a function of temperature gradient, so the robust magnetic field escalates entropy optimization N G (η). Figure 20 depicts the impact of Brinkman number Br on entropy optimization N G (η). Here, entropy rate upsurges against higher estimations of Brinkman number. Actually Br is a heat created source within the fluid moving region. Heat generated together with the heat transmission from the wall increases the entropy optimization. The higher Brinkman number allows less thermal conductivity to the fluid flow and therefore N G (η) boosted.     ( ) G N η . Physically, with a higher Reynolds number, the inertial force in the system increases the viscous force which enhances the disturbance in the fluid movement and promotes an increase in entropy generation. Therefore, entropy optimization increases due to the contribution of heat transfer. Figure 22 indicates the variation in entropy generation via temperature difference parameter α . A decreasing impact is depicted here. Figure 20. Variation in N G (η) via Br. Figure 21 displays the variation in entropy profile via Reynolds number. The higher Reynolds number escalates N G (η). Physically, with a higher Reynolds number, the inertial force in the system increases the viscous force which enhances the disturbance in the fluid movement and promotes an increase in entropy generation. Therefore, entropy optimization increases due to the contribution of heat transfer. Figure 22 indicates the variation in entropy generation via temperature difference parameter α. A decreasing impact is depicted here.   Confirmed Figure 22. Variation in N G (η) via α.

Conclusions
An MHD thin film flow of second grade nanofluid holding SWCNTs and MWCNTs with heat flux model of Cattaneo-Christov and entropy generation is analyzed in this article. The flow is considered past a linearly extending sheet. The final remarks of the current analysis are 1.
Velocity profile heightens with the increase in nanoparticles volume fraction and second grade fluid parameter, whereas the declining impact is observed via magnetic parameter, film thickness, and unsteadiness parameter.

2.
Temperature profile heightens with the escalation in second grade fluid parameter, nanoparticles volume fraction, radiation parameter, and heat source/sink parameters while a reducing influence is observed via film thickness, unsteadiness parameter, and thermal relaxation parameter.

3.
Surface drag force escalates with the higher values of nanoparticles volume fraction, unsteadiness parameter, film thickness, magnetic parameter, and second grade fluid parameter.

4.
Thermal transfer rate surges with the heightening values of nanoparticles volume fraction, unsteadiness parameter, and film thickness, whereas a reducing effect is observed via heat source/sink parameters.

5.
The higher value of Reynolds number augmented the entropy optimization. 6.
Entropy generation increases with higer values of magnetic parameter and Brinkman number and Reynolds number. 7.
The entropy N G (η) declines with the rise of temperature difference ratio parameter.