A thin film flow of nanofluid comprising carbon nanotubes influenced by Cattaneo-Christov heat flux and entropy generation

This study aims to scrutinize the thin film flow of a nanofluid comprising of carbon nanotubes (CNTs), single and multi-walled i.e., (SWCNTs and MWCNTs), with Cattaneo-Christov heat flux and entropy generation. The time-dependent flow is supported by thermal radiation, variable source/sink, and magneto hydrodynamics past a linearly stretched surface. The obtained system of equations is addressed by the numerical approach bvp4c of the MATLAB software. The presented results are validated by comparing them to an already conducted study and an excellent synchronization in both results is achieved. The repercussions of the arising parameters on the involved profiles are portrayed via graphical illustrations and numerically erected tables. It is seen that the axial velocity decreases as the value of film thickness parameter increases. It is further noticed that for both types of CNTs, the velocity and temperature distributions increase as the solid volume fraction escalates.


Introduction
The flow and heat transfer phenomenon in thin fluid film past stretched surfaces has promising applications including continuous casting, extrusion of plastic sheets, drawing of polymer surfaces, foodstuff processing, annealing and tinning of copper wires, and cooling of metallic plates [1]. The maintenance of the extrudes' surface is vital in the extrusion process smooth surface with minimum friction and enough strength is necessary for the coating procedure. Additionally, all this highly rely on the flow and heat transfer properties of the thin film over stretched surfaces. Because of this, the analysis in such cases is quite essential. Wang's [2] pioneering work by deliberating the hydrodynamics of time-dependent thin fluid film flow past a stretching sheet invited researchers to work in this attractive industry-oriented theme. Andersson et al. [3] further developed Wang's idea for heat transfer analysis. This case is further presented in a more generalized form by Chung and Andersson [4]. The solution to the same problem is discussed analytically by Wang [5]. The thin film flow is later analyzed aqueous based nanotubes with homogeneous-heterogeneous reactions past a Darcy-Forchheimer three-dimensional flow is studied by Alshomrani and Ullah [35]. Saleem et al. [36] discussed the squeezing three-dimensional nanofluid flow comprising of nanotubes in a Darcy-Forchheimer medium with thermal radiation and heat generation/absorption. There are numerous explorations that discuss on the flow of nanofluid amalgamated with carbon nanotubes in various scenarios but there are fewer that address the thin film flow. Some more explorations focusing on carbon nanotube or nanofluid flow may be found in References [37][38][39][40] and many therein.
The literature review reveals that the flow of a thin film with the Newtonian/non-Newtonian fluids is scarce in the literature and this subject gets even narrower if we talk about the thin film flows of nanofluids. Very few explorations are available that discuss the thin film flows of nanofluid-comprising nanotubes. Keeping in mind the importance of hydrodynamic flows, the idea of nanoliquid thin films in comparatively new and fewer explorations are available in the literature (see Table 1). This presented model is solved numerically and will present an estimated solution. The other limitation of the flow is that it is discussed in 2D and can be extended to 3D with some more novel effects like homogeneous-heterogeneous reactions, etc. The model presented here is an amalgamation of C-C heat flux and entropy generation in the thin film flows of the nanofluids comprising of both types of nanotubes (SWCNTs/MWCNTs) and has not yet been discussed in the literature. The numerical solution of the problem is achieved. A comparison with an already established result in the limiting case is also given and an excellent agreement between both is found. This corroborates our presented results. The graphical illustrations and numerically calculated values of the physical parameters are also added to the problem. Table 1. The studies on nanoliquid film flow.

Mathematical Modeling
Let us assume a thin film flow of a nanoliquid flow comprising CNTs past a time dependent linearly stretched surface. The elastic sheet emerges from a slender slit at the Cartesian coordinate system's origin ( Figure 1). The surface moves along the x-axis (y = 0) with a velocity u w (x, t) = bx (1−αt) , with b and a being the constants in the y-direction and temperature T w (x, y). The stream function ξ is considered such that u = ξ y , and v = −ξ x . linearly stretched surface. The elastic sheet emerges from a slender slit at the Cartesian coordinate system's origin ( Figure 1). The surface moves along the x-axis (y = 0) with a velocity   ( , ) , (1 α ) w bx u x t t with b and a being the constants in the y-direction and temperature Tw(x, y). The stream function ξ is considered such that u = ξy, and v = −ξx.  The thin film is of width h(x, y). The flow is laminar and incompressible. A magnetic field B(x, t) = , is employed normal to the extended surface. The governing unsteady conservation equations [17] under the aforementioned assumptions are appended as follows: With the following corresponding boundary conditions The Cattaneo-Christov term is defined as The heat source/sink "q " is represented by The thermophysical attributes (specific heat C p , density ρ and thermal conductivity k) of the base fluid (H 2 O) and carbon nanotubes (SWCNTs /MWCNTs) are appended in Table 2. The hypothetical relations are characterized as follows: Using the similarity transformations The requirement of Equation (1) is fulfilled undoubtedly and Equations (2) and (3) yield Additionally, the boundary conditions of Equation (4) become The values of various non-dimensional parameters are defined as follows: Physical quantities like the Skin friction coefficient and the local Nusselt number are given as Additionally, in dimensionless form, as follows: Coatings 2019, 9, 296 6 of 16

Entropy Generation
The entropy generation under the aforementioned assumptions is given as below: where all terms defined in Equation (15) portray the usual meaning. The entropy generation N G is defined as where S 0 and S gen are the characteristic entropy generation rate and the entropy generation rate. The parameters defined in the above equation are given as

Results and Discussion
This section is devoted to witnessing the impression of numerous parameters on the involved profiles whilst keeping in view their physical significance. The MATLAB built-in function bvp4c is utilized to address the differential Equations (9), (10), and (16) with the associated boundary conditions of Equation (11). To solve these, first we have converted the 2nd and 3rd order differential equations to the 1st order by introducing new parameters. The tolerance for the existing problem is fixed as 10 −6 . The initial guess we yield must satisfy the boundary conditions asymptotically and the solution as well.
The results show the influence of solid volume fraction (ϕ), dimensionless film thickness (λ), magnetic parameter (M), unsteadiness parameter (S), radiation parameter (R), thermal relaxation parameter (γ), and non-uniform heat source/sink parameter on the velocity, temperature and entropy generation profiles. Further, the numerical values for the Skin friction and Nusselt number are given in Tables 3  and 4 Coatings 2019, 9, x FOR PEER REVIEW 7 of 15   Figures 4 and 5 depict the behavior of axial velocity and the temperature field for the growth estimates of the film thickness parameter λ. It is found that both velocity and temperature profiles diminish for increasing values of the film thickness parameter λ. In fact, the more the film thickness, the lesser the fluid motion. This is because of the fact that higher values of film thickness dominate the viscous forces, eventually diminishing the fluid velocity. Similar behavior is observed for the temperature field.    Figures 4 and 5 depict the behavior of axial velocity and the temperature field for the growth estimates of the film thickness parameter λ. It is found that both velocity and temperature profiles diminish for increasing values of the film thickness parameter λ. In fact, the more the film thickness, the lesser the fluid motion. This is because of the fact that higher values of film thickness dominate the viscous forces, eventually diminishing the fluid velocity. Similar behavior is observed for the temperature field.  4 and 5 depict the behavior of axial velocity and the temperature field for the growth estimates of the film thickness parameter λ. It is found that both velocity and temperature profiles diminish for increasing values of the film thickness parameter λ. In fact, the more the film thickness, the lesser the fluid motion. This is because of the fact that higher values of film thickness dominate the viscous forces, eventually diminishing the fluid velocity. Similar behavior is observed for the temperature field.   The effect of the magnetic parameter M on the velocity and temperature fields can be visualized in Figures 6 and 7. Figure 6 displays the impact of the magnetic parameter M on axial velocity. The is the axial velocity of the declining function of the magnetic parameter M. Physically, by enhancing the magnetic parameter M, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid's motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.    The effect of the magnetic parameter M on the velocity and temperature fields can be visualized in Figures 6 and 7. Figure 6 displays the impact of the magnetic parameter M on axial velocity. The is the axial velocity of the declining function of the magnetic parameter M. Physically, by enhancing the magnetic parameter M, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid's motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.  The effect of the magnetic parameter M on the velocity and temperature fields can be visualized in Figures 6 and 7. Figure 6 displays the impact of the magnetic parameter M on axial velocity. The is the axial velocity of the declining function of the magnetic parameter M. Physically, by enhancing the magnetic parameter M, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid's motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.   The effect of the magnetic parameter M on the velocity and temperature fields can be visualized in Figures 6 and 7. Figure 6 displays the impact of the magnetic parameter M on axial velocity. The is the axial velocity of the declining function of the magnetic parameter M. Physically, by enhancing the magnetic parameter M, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid's motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.    Figures 8 and 9 show the effect of the unsteadiness parameter S on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter S, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter S. Therefore, the thermal and momentum boundary layer thicknesses decrease.     Figures 8 and 9 show the effect of the unsteadiness parameter S on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter S, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter S. Therefore, the thermal and momentum boundary layer thicknesses decrease.  Figures 8 and 9 show the effect of the unsteadiness parameter S on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter S, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter S. Therefore, the thermal and momentum boundary layer thicknesses decrease.     Figures 8 and 9 show the effect of the unsteadiness parameter S on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter S, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter S. Therefore, the thermal and momentum boundary layer thicknesses decrease.   Figure 10 determines the consequence of the thermal relaxation parameter γ on the temperature of the fluid. It is concluded that the temperature diminishes for increased values of the thermal relaxation parameter γ. The temperature tends to be sharper near the boundary as the value of γ is higher than the points on the growth in the wall slope of the temperature profile.  Figure 10 determines the consequence of the thermal relaxation parameter γ on the temperature of the fluid. It is concluded that the temperature diminishes for increased values of the thermal relaxation parameter γ. The temperature tends to be sharper near the boundary as the value of γ is higher than the points on the growth in the wall slope of the temperature profile.  Figure 11 demonstrates the impact of the radiation parameter R on the temperature profile. It is comprehended that the temperature field is an increasing function of the radiation parameter R. It is also concluded that the thermal boundary layer thickness for both carbon nanotubes is increased. In fact, larger estimates of the radiation parameter reduce the mean absorption coefficient and enhance the radiative heat flux's divergence. Due to this, the temperature of the fluid is upsurged. Figure 11. The illustration of R versus θ(η).
The influence of non-uniform heat source/sink parameters A* and B* on the temperature distribution is shown in Figures 12 and 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters.   Figure 11 demonstrates the impact of the radiation parameter R on the temperature profile. It is comprehended that the temperature field is an increasing function of the radiation parameter R. It is also concluded that the thermal boundary layer thickness for both carbon nanotubes is increased. In fact, larger estimates of the radiation parameter reduce the mean absorption coefficient and enhance the radiative heat flux's divergence. Due to this, the temperature of the fluid is upsurged.  Figure 11 demonstrates the impact of the radiation parameter R on the temperature profile. It is comprehended that the temperature field is an increasing function of the radiation parameter R. It is also concluded that the thermal boundary layer thickness for both carbon nanotubes is increased. In fact, larger estimates of the radiation parameter reduce the mean absorption coefficient and enhance the radiative heat flux's divergence. Due to this, the temperature of the fluid is upsurged. Figure 11. The illustration of R versus θ(η).
The influence of non-uniform heat source/sink parameters A* and B* on the temperature distribution is shown in Figures 12 and 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters.  The influence of non-uniform heat source/sink parameters A* and B* on the temperature distribution is shown in Figures 12 and 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters. Figure 11. The illustration of R versus θ(η).
The influence of non-uniform heat source/sink parameters A* and B* on the temperature distribution is shown in Figures 12 and 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters.         . Figure 16. The illustration of Rex versus NG(η). Table 3 is erected to envision the precision of the presented model by comparing it with Sandeep [17] who discusses the flow of nanofluids past a thin film under the influence of the magnetic field. To make a comparison, we have neglected the impacts of the volume fraction, electrical conductivity, and thermal relaxation parameters. Excellent alignment is achieved between both results. Table 4 shows the estimates of the Skin friction coefficient for different parameters. It is seen that the Skin friction coefficient increases for growing values of the magnetic parameter, solid volume fraction, unsteadiness parameter, and film thickness. Table 5 demonstrates the numerical values of Nusselt number for numerous parameters. It is determined that the Nusselt number increases with augmented values of the dimensionless film thickness, radiation parameter, solid volume fraction, and unsteadiness parameter, while it diminishes for growing values of non-uniform heat source/sink.   Table 3 is erected to envision the precision of the presented model by comparing it with Sandeep [17] who discusses the flow of nanofluids past a thin film under the influence of the magnetic field. To make a comparison, we have neglected the impacts of the volume fraction, electrical conductivity, and thermal relaxation parameters. Excellent alignment is achieved between both results. Table 4 shows the estimates of the Skin friction coefficient for different parameters. It is seen that the Skin friction coefficient increases for growing values of the magnetic parameter, solid volume fraction, unsteadiness parameter, and film thickness. Table 5 demonstrates the numerical values of Nusselt number for numerous parameters. It is determined that the Nusselt number increases with augmented values of the dimensionless film thickness, radiation parameter, solid volume fraction, and unsteadiness parameter, while it diminishes for growing values of non-uniform heat source/sink.

Conclusions
The thin film flow of nanofluid comprising of CNTs of both types (SWCNTs/MWCNTs) is studied whilst keeping in view the important applications of CNTs in many engineering applications. The flow is supported by the additional effects like C-C heat flux and entropy generation. The model is solved numerically with the support of the MATLAB software function bvp4c. The highlights of the existing study are

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Velocity and temperature distributions are mounting functions of the solid volume fraction for both types of CNTs in case of the thin film flow.

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For growing estimates of the thin film thickness parameter, the axial velocity diminishes.

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The velocity and temperature distributions show an opposite trend for the strong magnetic field in a thin film flow model.

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Larger estimates of heat source/sink parameter lead to an increase in the temperature of the fluid.

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The temperature of the fluid is decreased for higher values of the thermal relaxation parameter.

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With an increase in the estimates of film thickness, the magnetic parameter and the Skin friction coefficient show mounting behavior.

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The Nusselt number shows declining behavior for growing values of non-uniform heat source/sink.

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Entropy generation in the case of thin film flow is higher for larger estimates of the Brinkman number and the magnetic parameter.