**1. Introduction:**

In the last few decades, scientists have given great attention to thin film flow. The basic idea behind such an important concept is the applications and mechanism of thin film flow. Thin films are ubiquitous in nature and their mechanism is important to understand, because it has a wide range of practical uses. The traditional theory of Newtonian fluids is mainly focused on the linear relationship between stress and strain tensor or on their rates. Newtonian fluids are hardly pointing towards the doctrine of linear relation between the stress and strain tensor. In the same way, fluids that do not agree with the linear distribution between the stress and strain tensor are known as non-Newtonian fluids. There are two main classes of non-Newtonian fluids, visco-inelastic and visco-elastic fluids. The subsequent has great importance due its dual nature. Visco-elastic fluids, due to their viscosity, show an elastic behavior up to some degree. In such fluids, energy is stored in the form of strain energy. One cannot ignore the strain in these types of fluids, because it is responsible to recover the original state. These fluids came under the definition of Newtonian fluids, when there is a first order derivative of all the tensor formed from the velocity field, while fluids having higher order derivative tensors present come under the definition of non-Newtonian fluids. Some of these fluids are second order, third order, and fourth order fluids. Third order liquid, which is due to its sequential nonlinear constraints, has a variety of applications.

The study of non-Newtonian fluids from a theoretical point of view is too complicated and we need a mathematical relation to briefly illuminate the relation between shear stress and the shear rate. Therefore, a variety of non-linear relations have been suggested for the study of these fluids. As a consequence, to deliberate such a categorizing demeanor, various models have been initiated and discussed in [1]. Non-Newtonian fluids are not so simple that a single mathematical relation can explain the whole scenario. Therefore, for this purpose, several models have been initiated and developed to briefly explain the nature of such fluids. Among all, Ellis fluid model, Sisko fluid model, the Carreau viscosity model, Ostwald-de Waele model, the cross-viscosity model, Carreau-Yasuda model, Powell-Eyring model, and Reiner-Philippoff fluid model are the most important models in explaining the nature of such fluids. These models have interesting properties in their own, in which the Ostwald-de Waele model is considered to be the basic model, normally known as the Power law model. The subsequent of all models discussed here is a time-independent three-parameter model, behaving like non-Newtonian fluids under intermediate shear rates and Newtonian fluids at extreme shear rates. Consequently, due to their dual nature, Reiner-Philippoff fluid has many applications in engineering sciences and other technologies.

A variety of fluids are available in nature, in which nanofluids are the most interesting fluids, due to their variety of applications. The most commonly uses of nanofluids are metal oxides, oxide ceramics and chemically stable metals, like Alumina, Silica, Zirconia, Titania, aluminum oxide, copper oxide, gold, copper and various forms of allotropes of carbon and metal carbides. Water, oils, polymeric solutions, lubricants, bio-fluids, and glycols are normally used as base fluids. Nanofluids are two-phased mixtures designed by spreading nanometer-sized particles, in which base fluid size ranging up to 100 nm. Nanoparticles play a key part in heat transfer analysis. On the other hand, applications of liquid film flow grow day by day. The most common uses of these flows are in heat exchange processes, techniques of coating, industrial and distillation processes, and many more. The applied usage of the liquid film flow is a fascinating interaction amongst fluid mechanics, structural mechanics, and technology. Some of the practical usages are polymer and metal extrusion, foodstuff processing, plastic sheets depiction, casting and fluidization of the reactor.

In the assessment of these applications, researchers have taken a keen interest in the study of liquid film flow on unstable surfaces. Stretching sheets at the beginning were treated as linear surfaces. Such a phenomenon encountered in many industrial processes, like, in cooling, extraction of polymer sheets and plastic sheets, etc. In these industrial processes the stretching sheet contacts with the fluid both mechanically and thermally. Sakiadis [2] work is considered to be the pioneering work in the study of boundary-layer flow over non-stationary and rigid surfaces. But in polymer industries stretching sheet play a key role, which is explained by Crane [3] in his famous work. After the work of Crane on stretching sheet and its numerous applications in the polymer industry, researchers have shown great interest in it. Stretching sheet problems have been investigated by different researchers under different physical parameters with their variations, like viscosity and thermal conductivity, magnetic and electric fields (MHD), thermal radiation, viscous dissipation, and chemical reactions etc.

Similar to the variations in the parameters for the stretching sheet, the same phenomenon of stretching problem is studied for different geometries and is developed from time to time. Siddiqui et al. [4,5] investigated non-Newtonian fluids on a moving built with a sloping plane in one direction for thin film flow. Tawade et al. [6] studied the effect of magnetic field upon a thin fluid stream passes over a temperamental stretching sheet with heat. They used two different numerical

approaches, Newton-Raphson and RK-Felberg. They briefly explained and provided a detailed survey of different physical parameters.

Beside all these, the implemented techniques are not to be ignored. In literature, the boundarylayer equations obtained for stretching flow are solved by different approaches. Among all, numerical and perturbation approaches have been adopted and applied by many researchers successfully. In practice, numerical techniques are too difficult to apply due to the high non-linearity of the model equations, whereas perturbation techniques are not always applicable. Perturbation techniques need a small or large parameter to be presented in the equation, which is not always available to us. To overcome this situation, some new techniques were developed and implemented by the researchers successfully. Sajid and Hayat [7] used HAM and HPM to thin film flow of Sisko fluid and Oldroyed-6 constant. The effect of thermal radiations of blending convection stream over a steeping surface in a permeable channel is studied by Bakier [8]. Nargis and Tahir [9] have given a more detailed survey of grade third fluid on a moving belt in the direction of a slanted plane. Stretching problem in permeable medium with thermal effects of a slanted plate is investigated by Moradi et al. [10]. Chaudhary et al. [11] re-examined thermal radiation impacts of liquid on exponentially extending surfaces. Eldabe et al. [12] examined convection, radiation, and synthetic effects of MHD visco-elastic fluid flow in a permeable channel on a horizontal stretching sheet. Das [13] has investigated some important properties of thermal radiation and thermophoresis of MHD blended convective flow. Recently, Hsiao [14] has examined the heat and mass trasfer effects of Maxwell fluid. MHD flow of different models, like Powell–Eyring nanofluid and other non-Newtonian fluids on stretching surfaces are briefly explained in [15–17]. Crane [3] for the first time studied the flow of gummy liquid in a stretched surface. The effect of heat exchange on an extending sheet for viscoelastic liquids is discussed by Dandapat [18]. Wang [19] for the first time studied finite liquid film on an unsteady stretching domain. The problem discussed by Wang [19] has discussed by Usha and Sridharan [20] with a survey on different parameters. For a heat transfer analysis of liquid film fluid, numerical results for different parameters were obtained by Liu and Andersson [21]. The repercussion in the thin liquid film on an unsteady stretching sheet due to the inner heat production was examined by Aziz et al. [22].

The thin liquid film flow of non-Newtonian fluids has a lot of practical features. Consequently, it becomes a common solute in engineering and other technologies. Andersson et al. [23,24] investigated the non-Newtonian thin liquid films at a time depending stretching sheet by taking the Power law model in consideration. After this pioneer investigation of Andersson, scientists have given more attention to stretching problems by using the Power law model, for more detail see [25,26]. Other models also came in discussion during this era. Megahed et al. [27] examined the thin liquid film flow of Casson liquid for viscous promulgation with slip velocity and the transmission of variable heat transition. The same scenario was discussed by Abolbashari et al. [28] for nano particles with the generation of entropy. Buongiorno's model for nano fluid thin film on a temperamental extending stretched sheet was recently investigated by Qasim et al. [29]. A steady flow of liquids through a porous medium is studied by Ariel [30]. Ariel got a high non-linear coupled boundary value problem for the geometry under consideration and applied numerical methods to obtain an appropriate solution. Sahoo et al. [31] investigated heat exchange analysis with a uniform oblique magnetic field for non-Newtonian fluids. They successfully applied finite difference and Broyden's methods for the concatenation of the field of the velocity. Aiyesimi et al. [32,33] examined the thin liquid film flow of an MHD grade third fluid and obtained some interesting results by using perturbation techniques with a brief survey impact of slip parameters and magnetic parameters. Third grade fluid and its approximate analytical solution by using OHAM for three different kinds of flow has discussed by Islam and Shah et al. [34,35]. Makinde [36] studied the same geometry with isothermal effects for hydro-dynamically third order liquid film flow. The approximate solution for velocity and temperature was obtained by a Hermitepade method. A brief discussion and explanation was given by Yao and Liu [37] of the second order fluid over flat plates for unsteady flows. Erdogan et al. [38] examined the properties of unsteady flow of the

non-Newtonian fluids, with a brief description on Poiseuille, Couette, and generalized form of Couette flow. Abdulhameed et al. [39] successfully applied Laplace transformation, perturbation techniques, and separation of variable methods for the clarity of unsteady non-Newtonian fluids over an oscillating plate. Huan was the first to settle variational rules for nano thin film-lube [40] with the help of the method of semi-inverse [41–45]. Kapitza [46], Yih [47], Krishna and Lin [48], Anderson and Dahl [49], and Cheng et al. [50] considered thin film flow problems with distinct geometric expressions. As time passes, the thin film flow applications in engineering sciences increase day by day and as a result, the researchers extended the work to a new world. Recently, coating and fiber applications of thin film flow are described and discussed in [51–54]. The geometry and other physical constraints have fixed, but some impurities have been introduced to the study as discussed in [55,56], to improve and enhance the heat transfer analysis.

An interesting and remarkable behavior of time-dependent non-Newtonian fluid is its pseoudo-plasticity, which vanishes with the expansion of shear rate. Many models have initiated as discussed earlier in the investigation of the behavior of such fluids. Among all, in Reiner-Philippoff fluid model [57], researchers have shown great interest. In 1965, Kapr and Gupta [58] have studied Reiner-Philippoff fluid two-dimensional flow in a linear channel. Different approaches have been adopted with different geometrical aspects to discuss this famous model.

In 1994, Tsung-Yen [59] investigated the boundary-layer flow problems by using this model. The basic idea of boundary-layer theory was introduced by L. Prandlt in Heidelberg, Germany, in August 1904, at the third International Congress of Mathematicians. It is the region in the fluid flow, developing at large Reynolds numbers. This region is strongly affected by inertial forces and viscous forces. Boundary layer theory is very important and has a variety of dimensions and visual perception of interest, and has been studied for a long interval of time. In 2009, this model was investigated by Yam [60] for the boundary flow past a stretching wedge. During this era Patel and Timol [61] used the technique of similarity solution for three dimensional boundary layer type equations for non-Newtonian fluids. Ahmad [62] examined the Reiner-Phillippoff fluid flow based nano-liquids past a stretching sheet. Recently, Ahmad et al. [63] discussed the same model with the same geometry with shifting and thickness in the stretching sheet.

Discussion has shown that different problems arise due to varying geometry as well as the fluid behavior, and different approaches are adopted by the researchers to meet their needs. Most of the problems that arise are highly non-linear and it is a difficult job to handle such problems with the usual available techniques in literature. Nowadays, perturbation techniques [64,65] are in the main stream for dealing with such problems. These methods work in the presence of small or large scale parameters. These parameters are not always available to us in applied sciences and we cannot apply these techniques to these types of problems. To deal with such problems, we use non-perturbative techniques like the "Lyapunov's artificial small parameter method" [66], the method of d-Expansion, and the adomians decomposition method (ADM) [67]. Various approaches have been adopted by researchers to find the solution to their problems. An exact solution in literature is very rare. This is because of the complexity of the geometry of the problem. That is why we often see numerical approaches to find the approximate solution. Among all, homotopy analysis method (HAM) [68–70] is the one by virtue of which we can find the approximate solution. In 1992, Liao [71] for the first time developed and implemented this method and found solution in the form of series in a single variable. Liao also discussed the convergence of this proposed method and found a rapid convergence. HAM has some interesting points of interest. Most importantly, this method is independent of whether a given non-linear problem contains any small or large parameters or not. In HAM, we can modify and control the region of convergence, where necessary, and is helpful in selecting distinct sets of base operations, which approximate a non-linear problem with less effort.

The goal of our current investigations is to obtain the thin liquid film flow of Reiner-Philippoff fluid over a stretching sheet with heat transfer and thermal radiations. Boundary-layer equations are obtained from the physical demonstrated geometry. Thermophoresis effects and Brownian motion are also encircled with different physical parameters. A similarity solution is obtained with the help of new variables, due to which a complicated model is transformed into simple coupled ordinary differential equations. An analytical approach is adopted for the solution of the reduced system. HAM is implemented with initial guess as required for the implementation of the technique, due to its fast convergence. With the variation of different physical parameters, the results are plotted, tabulated, and discussed in detail. The physical significance of Sherwood number and skin friction is presented by tables.
