**1. 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 in various scenarios like magnetic impact [6,7], thermo-capillary impacts [8], and non-Newtonian fluids [9–12].

The above-mentioned studies on thin liquid films are limited to Newtonian and non-Newtonian fluids in the absence of nanofluids. In recent years, the subject of nanofluids, owing to their amazing characteristic of high thermal conductivity, has gained much attention of researchers and scientists. The seminal work by Choi and Eastman [13] introducing "nanofluids" has revolutionized the heat transfer processes. A nanofluid is an amalgamation of the solid metallic particles called "Nanoparticles" with a size of 1–100 nm and ordinary liquids. Nanofluids are the finest coolants with amazing applications including microelectronics, optical manufacturing and transportation [14]. There are studies that emphasize the thin film liquid flow of nanofluids. Lin et al. [15] numerically scrutinized the thin film Pseudo-plastic nano liquid flow with the impact of internal heat generation by utilizing R–K scheme and Newton's method. Later, Lin et al. [16] extended this study to the impacts of viscous dissipation and temperature reliant thermal conductivity. The nano-liquid thin film flow comprising graphene nanoparticles under the influence of aligned magnetic effect is discussed by Sandeep [17]. Zhang et al. [18] studied the Oldroyd-B nanofluid thin film flow analytically with two types of nanoparticles, i.e., silver and copper and found that nanofluid containing silver nanoparticles has a better thermal conductivity in comparison to the copper nanoparticles. Zhang et al. [19] also deliberated the power law nano liquid thin film flow with the slip using the differential transform method. The problem of nanofluid thin films flowing past an elastic stretched sheet is solved using the least square method (LSM) by Fakour et al. [20]. Ishaq et al. [21] deliberated the analytical solution of Powell-Eyring nano liquid thin film flow with thermal radiation past a permeable stretched surface. The flow of Darcy-Forchheimer nanofluid thin film comprising SWCNTs past an unsteady stretched surface is studied by Nasir et al. [22].

There are numerous applications of heat transfer in industrial and engineering processes. These include cooling towers, fuel cells, microelectronics, and nuclear reactors. The fundamental essence in all these processes is that the value of thermal conductivity is presumed to be a constant. However, this value varies with temperature and other factors. Pal [23] and Vajravelu et al. [24] observed that the thermal conductivity varies linearly as the temperature is altered from 0◦ to 400◦ F. Initially, the Fourier law of heat conduction has been used in the modeling of heat transfer applications but the system encounters an initial disturbance due to the "parabolic energy equation" which is referred to as "paradox in heat conduction". This shortcoming in the Fourier's model was addressed by Cattaneo [25] who introduced the thermal relaxation time in the Fourier law of heat conduction. Cattaneo's act helped to represent the temperature profile via the hyperbolic energy equation and heat transport propagation using thermal waves with a controlled speed. This heat transport mechanism is employed in diverse practical scenarios, ranging from nano-liquid flow models to skin burn injury models [26]. Moreover, several materials possess a large thermal relaxation time, such as biological tissues having a relaxation time of 91–100 s and sand of 21 s. To uphold the material invariant formulation, Christov altered the Maxwell-Cattaneo model by swapping the time derivative with Oldroyd's upper convected derivative. This improved version is nowadays being termed as the Cattaneo-Christov (C-C) heat flux model. Later, Han et al. [27] introduced an analytical solution for the viscoelastic material including the velocity slip boundary along with the C-C heat flux. Mustafa [28] analyzed the rotating flow of the Maxwell fluid with an upper convected derivative and C-C heat flux over a linearly stretched surface using both the analytical and numerical methods. A similar case was examined by Khan et al. [29] considering an exponentially stretched surface. The squeezed flow of the C-C heat flux with CNTs between two parallel disks is studied by Lu et al. [30]. Ramzan et al. [31] studied the flow of the Williamson fluid flow numerically with C-C heat flux associated with the convective boundary condition and homogeneous-heterogeneous reactions. The flow of the magnetohydrodynamics (MHD) second-grade fluid over a stretched cylinder with C-C heat flux is discussed by Alamri et al. [32]. Ramzan et al. [33,34] deliberated the Maxwell and third-grade fluid flows with homogeneous-heterogeneous reactions and C-C heat flux. The flow of 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 [37–40] 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.



<sup>√</sup>) means effect is present and (×) means effect is absent.
