**1. Introduction**

Recently in few years it has been scrutinized that the analysis of the thin film flow has pointedly contributed in di fferent areas like industries, engineering, and technology, etc. All problems related to thin film flow have varied applications in di fferent fields. Some common usage of thin film associated with daily life are wire and fiber coating, extrusion of polymer and metal from die, crystal growing, plastic sheets drawing, plastic foam processing, manufacturing of plastic fluid, artificial fibers, and fluidization of reactor. Thin polymer films have abundant applications in engineering and technology. Several trade and biomedical sectors are associated with caring and functional coatings, non-fouling bio surfaces, advanced membranes, biocompatibility of medical implants, microfluidics, separations, sensors and devices, and many more. In the light of the above applications, this issue brings the attention of the researchers to improve the development of such type of study. At the initial stage the thin film flow problems discuss for viscous fluid flow and with the passage of time it turned to some Non-Newtonian Fluids. Crane [1] discussed at first time the viscid fluid motion in linear extended plate. Dandapat [2] has discussed the flow of the heat transmission investigation of the viscoelastic fluids over an extended surface. Wang [3] investigate time depending flow of a finite thin layer fluid upon a stretching plate. Ushah and Sridharan [4] used horizontal sheet for the same said problem and prolonged it to the thin partial with the analysis of heat transfer. Liu and Andersson [5] discuss heat transmission investigation of the thin partials motion upon an unsteady extending

piece. Aziz et al. [6] investigate the problem related to heat transfer of thin layer flow with thermal radioactivity. Tawade et al. [7] discussed the same problem with external magnetic field.

Furthermore, it is observed that the thin layer flow has vital roles in di fferent fields of science. Andersson [8] is considered to be a pioneer who used the Power law model and investigated the liquid film flow viscid fluids flow upon a time dependent extending piece. Waris et al. [9] have investigated nanofluid flow with variable viscosity and thermal radioactive pass a time dependent and extending Sheet. Anderssona et al. [10] discussed the Heat transfer investigation in liquid film upon time dependent stretching plate. Chen [11,12] examined the same problem using the Power-law model. Wang et al. [13] used HAM Method to discuss the problem as examined by Chen. Saeed et al. [14] recently investigated the thin layer flow of casson Nano fluid with thermal radioactivity. The disk they took was rotating and the flow was three-dimensional. Shah et al. [15] discussed the same problem using Horizontal rotating disk. Khan et al. [16,17] studied the heat transfer investigation of the inclined magnetic field with Graphene Nanoparticles. Ullah et al. [18] studied the Brownian Motion and thermophoresis properties of the nanofluids thin layer flow of the Reiner Philippo ff fluid upon an unstable stretching sheet. Shah et al. [19] investigate the thin film flow of the Williamson fluid upon an unsteady stretching surface.

Non-Newtonian fluids have abundant uses in the field of energy and technology. Plastic, food products, wall paint, greases, lubricant oil, drilling mud, etc are some common examples. Sisko fluid is also very important non-Newtonian fluids. At a very low shear rate, the Sisko flow has the same behavior as the Power-Law fluid. This property was used experimentally to fit the data of the flow of lubricating greases and also to model the flow of the whole human saliva [20]. Munir et al. studied Sisko fluid with mixed convection heat transfer. Siddiqui et al. [21,22] studied the thin film flow of Sisko fluid. Khan et al. [23] investigated the flow of boundary layer of the Sisko fluid upon a stretching surface. Molati et al. [24] discussed the MHD problem related to sisko fluid. Malik et al. [25] studied the sisko fluid with heat transfer and using convective boundary conditions.

In the past few years, the study of heat transfer and radiative flow through the stretch sheet has taken the attention from many scientists because of its large applications in engineering and industrial processes [26–31]. Rubber production, colloidal suspension, production of glass socks, metal spinning and drawing of plastic film, textile and paper production, use of geothermal energy, food processing, plasma studies, and aerodynamics are some practical examples of such flows. Radiation is often encountered in frequent engineering problems. Keeping in view its applications, Sheikholeslami et al [32–35] presented the application of radioactive nanofluid flow in di fferent geometries. Recently some researchers studied the problems related to heat transfer. Poom et al. [36] examined the casson nanofluid with MHD radiative flow and heat source using rotating channel. Khan et al. [37] discussed the similar HMD problem as above with time dependent and thin film flow of the three fluid models Oldroyed-B, Maxwell, and Je ffry. Ishaq et al. [38] discussed the MHD e ffect of unsteady porous stretching surface taking Nanofluid film flow of Eyring Powell fluid. Muhammad et al. [39] discussed the MHD rotating flow upon a stretching surface with radioactively Heat absorption. Hsiao [40–43] discussed non-Newtonian fluid flow with MHD.

The purpose of this manuscript is to model and analyze thin film flow of Sisko fluid on time dependent stretching surface in the presence of the constant magnetic field (MHD). Here the thin liquid fluid flow is assumed in two dimensions. The governing time-dependent equations of Sisko fluid are modelled and reduced to ODEs by use of Similarity transformation with unsteadiness non-dimensionless parameter *St*. To solve the model problem, we used Homotopy Analysis Method-(HAM) which is one of the strongest and most time-saving methods. In addition, the heat transfer rates of thermal radiation are studied and analyzed. Shang [44] used another strong technique as Lie Algebra for solution of such type problem.

#### **2. Basic Equations**

The stated governing equation and heat equation are given below [23,24]:

$$d\dot{v}V = 0,\tag{1}$$

$$
\rho \frac{dV}{dt} = -\nabla p + \text{div}\mathbf{S} \tag{2}
$$

For Sisko fluid *S* is given as [25–28]

$$\overrightarrow{S}^{\prime} = \left[ a + b \left| \sqrt{\frac{1}{2} tr\_{\!\!\! 1}^{\prime 1}} \right|^{n-1} \right]\_{1} \tag{3}$$

where

$$A\_1 = \begin{pmatrix} \gcd V \end{pmatrix} + \begin{pmatrix} \gcd V \end{pmatrix}^T \tag{4}$$

wherever *a*, *b*, *n* are the material constants which are distinct for dissimilar fluids. If we take a = 0, b = 1 and n = 0 in the Sisko fluid model then we obtained the Power-law fluid model. If we take a = 1, b = 0 and n = 1 in the Sisko fluid model then we obtained the stress–strain relationship of Newtonian fluid.

Because of two-dimensional fluid flow the velocity and the stress profile are presumed

$$
\overrightarrow{\dot{V}}\_{\cdot} = \begin{bmatrix} \overrightarrow{u}\_{\cdot}(\mathbf{x}, \mathbf{y}), \overrightarrow{v}\_{\cdot}(\mathbf{x}, \mathbf{y}), \mathbf{0} \end{bmatrix}, \overrightarrow{\overset{\rightarrow}{S}}{\text{S}}\_{\cdot} = \overrightarrow{\overset{\rightarrow}{S}}(\mathbf{x}, \mathbf{y}), \mathbf{T}\_{\cdot} = \mathbf{T}(\mathbf{x}, \mathbf{y}) \tag{5}
$$

where →*u*&<sup>→</sup>*v* are representing velocity components.

Inserting Equation (5) into Equations (1) and (2), the momentum and continuity equations reduce to the form as:

$$
\overrightarrow{u}\_x + \overrightarrow{v}\_y = 0 \tag{6}
$$

 as

$$\begin{split} \rho \left( \stackrel{\scriptstyle \rightarrow}{u}\_{1} + \stackrel{\scriptstyle \rightarrow}{u}\stackrel{\scriptstyle \rightarrow}{u}\_{x} + \stackrel{\scriptstyle \rightarrow}{v}\stackrel{\scriptstyle \rightarrow}{u}\_{y} \right) &= \, -p\_{1x} + a \stackrel{\scriptstyle \rightarrow}{(\stackrel{\scriptstyle \rightarrow}{u}\_{xx} + \stackrel{\scriptstyle \rightarrow}{u}\_{yy})} + 2b \frac{\partial}{\partial x} \Big[ \stackrel{\scriptstyle \rightarrow}{u}\_{x} \Big| 4 \left( \stackrel{\scriptstyle \rightarrow}{u}\_{x} \right)^{2} + \left( \stackrel{\scriptstyle \rightarrow}{u}\_{y} + \stackrel{\scriptstyle \rightarrow}{v}\_{x} \right)^{2} \Big| \stackrel{\scriptstyle \frac{\scriptstyle \rightarrow}{2}}{\,} \Big] \\ &+ b \frac{\partial}{\partial y} \Big[ \left( \stackrel{\scriptstyle \rightarrow}{u}\_{y} + \stackrel{\scriptstyle \rightarrow}{v}\_{x} \right) \Big| 4 \left( \stackrel{\scriptstyle \rightarrow}{u}\_{x} \right)^{2} + \left( \stackrel{\scriptstyle \rightarrow}{u}\_{y} + \stackrel{\scriptstyle \rightarrow}{v}\_{x} \right)^{2} \Big| \stackrel{\scriptstyle \frac{\scriptstyle \cdot}{2}}{\,} \Big] \end{split} \tag{7}$$

$$\begin{aligned} \rho \left( \overrightarrow{\boldsymbol{v}}\_{t} + \overrightarrow{\boldsymbol{u}} \, \overrightarrow{\boldsymbol{v}}\_{x} + \overrightarrow{\boldsymbol{v}} \, \overrightarrow{\boldsymbol{v}}\_{y} \right) &= -p\_{1x} + a \left( \overrightarrow{\boldsymbol{v}}\_{xx} + \overrightarrow{\boldsymbol{v}}\_{yy} \right) + b \frac{\partial}{\partial x} \Big[ \left( \overrightarrow{\boldsymbol{u}}\_{y} + \overrightarrow{\boldsymbol{v}}\_{x} \right) \Big| \mathbf{4} \{ \overrightarrow{\boldsymbol{u}}\_{x} \}^{2} + \left( \overrightarrow{\boldsymbol{u}}\_{y} + \overrightarrow{\boldsymbol{v}}\_{x} \right)^{2} \Big]^{\frac{n-1}{2}} \\ 2b \frac{\partial}{\partial y} \Big[ \overrightarrow{\boldsymbol{v}}\_{y} \Big| \mathbf{4} \{ \overrightarrow{\boldsymbol{u}}\_{x} \}^{2} + \left( \overrightarrow{\boldsymbol{u}}\_{y} + \overrightarrow{\boldsymbol{v}}\_{x} \right)^{2} \Big]^{\frac{n-1}{2}} \end{aligned} \tag{8}$$

If *a* = 0, then above equations become the power law fluid and when *b* = 0, then it reduced Newtonian fluid. Introduce the dimensionless variable as:

$$\mu = \frac{\overrightarrow{u}}{\overrightarrow{U}}, v = \frac{\overrightarrow{v}}{\overrightarrow{U}}, t = \frac{\textbf{t}}{\overrightarrow{U}}, x = \frac{\textbf{x}}{L}, y = \frac{y}{L} and p = \frac{p}{\rho \overrightarrow{sU}} \tag{9}$$

Equations (6) and (7) are written as

$$\begin{aligned} \frac{\partial \frac{\partial}{\partial t}}{\partial t} + u \frac{\partial \upsilon}{\partial x} + v \frac{\partial \upsilon}{\partial y} &= -\frac{\partial p\_1}{\partial y} + \varepsilon\_1 \left( \frac{\partial^2 \upsilon}{\partial x^2} + \frac{\partial^2 \upsilon}{\partial y^2} \right) + \\\ \varepsilon\_2 \frac{\partial}{\partial x} \bigg[ \left( \frac{\partial u}{\partial y} + \frac{\partial \upsilon}{\partial x} \right) \bigg] 4 \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)^2 \bigg]^{\frac{u-1}{2}} \\\ + 2 \varepsilon\_2 \frac{\partial}{\partial y} \bigg[ \frac{\partial \upsilon}{\partial y} \bigg] 4 \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)^2 \bigg]^{\frac{u-1}{2}} \end{aligned} \tag{10}$$

The dimensionless parameter ε1 and ε2 are defined as

$$
\varepsilon\_1 = \frac{a/\rho}{\overrightarrow{LU}} \text{ and } \varepsilon\_2 = \frac{b/\rho}{\overrightarrow{LU}} \text{\(\overrightarrow{U}\)}{\text{L}\overrightarrow{U}}^{n-1} \tag{11}
$$

The above equation after using the boundary layer approximations become as

$$\rho \left( u\_t + u u\_x + v u\_y \right) = -p\_{1\_x} + a u\_{yy} + b \frac{\partial}{\partial y} \left( \left| u\_y \right|^{n-1} u\_y \right) \tag{12}$$

$$0 \,\, = \, -p\_{1\_y} \tag{13}$$

#### **3. Mathematical Formulation of the Problem**

Consider a time depending and electric conducting the flow of thin layer of the Sisko fluid with impact of thermal radiations during spreading surface where *x*-axis is in parallel to the slot where *y*-axis shown in the figure is orthogonal to the surface as given below in Figure 1.

**Figure 1.** Physical Sketch of the model problem.

It is stretching flow surface also the origin is immovable because equivalent and opposed forces are acting along the *X*-axis. The *x*-axis is taking with stress velocity along the spreading surface.

$$\mathcal{U}(\mathbf{x},t) = \frac{c\mathbf{x}}{1 - bt} \tag{14}$$

In which *c* and *b* are constants. The *y*-axis is making a right angle to it. The term *cx*<sup>2</sup> <sup>υ</sup>(<sup>1</sup>−*bt*) is local Reynolds number, the surface velocity *<sup>U</sup>*(*<sup>x</sup>*, *t*). The heat and mass transfer simultaneously here is defined as *Ts*(*<sup>x</sup>*, *t*) = *T*◦ − *Tre f*... *cx*<sup>2</sup> 2υ (1 − *bt*) 1 2 , which is surface temperature. Here *T*0 is temperature at the slit, *Tre f* is reference temperature such that 0 ≤ *Tre f* ≤ *T*0. also *Ts*(*<sup>x</sup>*, *t*) defines temperature of the sheet, obtain from the *T*0 at the slit in 0 ≤ *c* ≤ 1.

First the slit is static with center and after an exterior power is implied to bounce the slit in the positive *x* -axis at the rate *c* 1−*bt*, where *c* ∈ [0, 1]. An exterior transmission magnetic force is given normally to the extending sheet which is presumed to be variable type and selected as

$$B(t) \;=\; B\_0(1 - bt)^{-0.5}\tag{15}$$

The fluid flow is assumed unsteady laminar and incompressible. The elementary boundary governing equations after using assumption are reduced as

$$u\_x + v\_y = \begin{array}{c} 0 \end{array} \tag{16}$$

$$u\_t + uu\_x + vu\_y = \frac{a}{\rho} u\_{yy} - \frac{b}{\rho} \frac{\partial}{\partial y} (-u\_y)^u - \sigma \stackrel{\rightarrow}{B}^2(t) u \tag{17}$$

$$u(T\_t) + u(T\_x) + v(T\_y) = \frac{k}{\rho \mathbf{C}\_p} (T\_{yy}) - \frac{1}{\rho \mathbf{C}\_p} (q\_{r\_y}) \tag{18}$$

Here *qr* approximation of Rosseland, where *qr* is define as [29–32],

$$q\_{r\_y} = - \left(\frac{4\sigma^\*}{3k^\*}\right) T\_y^4 \tag{19}$$

Here *T* represents the temperature fields, σ<sup>∗</sup> is the Stefan–Boltzmann constant, *K*∗ is the mean absorption coefficient, *k* is the thermal conductivity of the thin film. Expanding *T*<sup>4</sup> using Taylor's series about *T*∞ as below

$$T^4 = T^4\_{\text{oo}} + 4T^3\_{\text{oo}}(T - T\_{\text{oo}}) + 6T^2\_{\text{oo}}(T - T\_{\text{oo}})^2 + \dots \tag{20}$$

Neglecting the higher order terms

$$
\Delta T^4 \cong -3T\_{\phi\phi}^4 + 4T\_{\phi\phi}^3 T\_\prime \tag{21}
$$

Using Equation (21) in Equation (20) we ge<sup>t</sup> the following

$$q\_{r\_y} = -\frac{16T\_{\infty}^\* \sigma^\*}{3K^\*} T\_{yy^\*}^4 \tag{22}$$

Using Equation (22) in Equation (18) we ge<sup>t</sup> the following

$$T\_t + \mu T\_x + \upsilon T\_y = \frac{k}{\rho \mathbb{C}\_p} T\_{yy} - \frac{1}{\rho \mathbb{C}\_p} \left(\frac{16T\_{\infty}^\* \sigma^\*}{3K^\*} T\_{yy}^4\right) \tag{23}$$

The accompanying boundary conditions are given by

$$\begin{array}{rcl} \stackrel{\rightarrow}{u} = \stackrel{\rightarrow}{U}\_{\cdot \cdot} \stackrel{\rightarrow}{v} = 0, \, T = \, T\_{\text{s} \cdot} \text{ at } \mathbf{y} = \, 0, \\\ \stackrel{\rightarrow}{u}\_{y} = \, T\_{\text{y}} = \, 0 \, \text{at } \mathbf{y} = \, \mathbf{h}, \end{array} \tag{24}$$
