Nanofluid thin film flow of Sisko fluid and variable heat transfer over an unsteady stretching surface with external magnetic field

Abstract

This research paper investigates two dimensional liquid film flow of Sisko nanofluid with variable heat transmission over an unsteady stretching sheet in the existence of uniform magnetic field. The basic governing time-dependent equations of the nanofluid flow phenomena with Sisko fluid are modeled and reduced to a system of differential equations with use of similarity transformation. The significant influence of Brownian motion and thermophoresis has been taken in the nanofluids model. An optimal approach is used to obtain the solution of the modeled problems. The convergence of the Homotopy Analysis Method (HAM) method has been shown numerically. The variation of the skin friction, Nusselt number and Sherwood number, their influence on liquid film flow with heat and mass transfer have been examined. The influence of the unsteadiness parameter $\mathit{St}$ over thin film is explored analytically for different values. Moreover for comprehension, the physical presentation of the embedded parameters, like $\beta$ , magnetic parameter $M$ , stretching parameter $\xi$ and Sisko fluid parameters $\epsilon$ , Prandtl number Pr, thermophoretic parameter $\mathit{Nt}$ , Brownian motion parameter $\mathit{Nb}$ , Schmidt number $\mathit{Sc}$ have been represented by graph and discussed.

Introduction

The flow analysis of thin film flow has got important loyalty due to its enormous application and uses in the field of engineering and technology, industries in a several years. Thin film flow problems realized in many fields, shifting from specific situations in the flow in human lungs to lubricant problems in industrial, which is perhaps one of the largest subfield of thin liquid flow problems. Investigation of thin liquid flow of practical applications is an interesting interaction between fluid mechanics, structural mechanics and theology. Cable, fibber undercoat is one of its significant applications. Extrusion of polymer and metal, striating of foodstuff, constant forming, elastic sheets drawing and fluidization of the device, exchanges and chemical treating apparatus are several well-known uses. In observation of these applications, it becomes an important topic for researchers to develop the study of liquid film on stretching the surface. The flow of liquid film was first studied for viscous flow and further it is extended to Non-Newtonian fluid. Sandeep and and Malvandi ¹ have described thin film fluid flow with heat transfer of non-Newtonian nano liquids. Wang ² has described thin film liquid over stretching the unsteady surface. Usha and Sridharan ³ have studied the movement of finite thin liquid over the unsteady stretching sheet. Liu and Andersson ⁴ have discussed the motion of film flow with heat transfer through a stretching surface. Aziz et al. ⁵ have studied for the production of inside heat during thin fluid film flow over a stretching sheet. Tawade et al. ⁶ have described thin film fluid flow with heat transmission in the existence of thermal radiation, for the solution of nonlinear equation; they used Runge–Kutta–Fehlberg and Newton Raphson method. Anderssona et al. ⁷ have studied liquid film flow with heat transfer on stretching sheet. Most researchers^8–11 have studied fluid flow of power laws and fluid film through an unsteady stretching surface for more different cases. Megahe ¹² has investigated thin Casson fluid flow with heat transfer on unsteady stretching sheet with the influence of the slip velocity in the existence of heat flux and viscid dissipation. Recently, Khan et al. ¹³ and Tahir et al. ¹⁴ investigated nanofluid thin film with new modifications.

Due to eccentric features of nano liquids that makes them proficient in various applications, nano fluids are used in pharmaceutical procedures, hybrid powered engines, fuel cells, microelectronics and currently it is mostly used in the field of nanotechnologies. From earlier studies, nano fluids have been found to retain dimensions smaller than 100 nm. Nano liquids are thinned liquid holdups of nanoparticles through at least one of their principal enriched thermophysical goods such as thermal diffusivity, conductivity, viscosity and warmth transmission convective coefficients equated to base liquids like oil or water. An extensive diversity of industrial processes includes the transmission of thermal energy. The development of high enactment thermal systems for heat transfer augmentation has become widespread nowadays. After this, researchers ¹⁵ have described nanofluid flow through the unsteady stretching sheet using Buongiorno’s model. Hayat et al. ¹⁶ have studied Maxwell nano liquid. Malik et al. ¹⁷ have studied Erying–Powell nanofluid through a stretching with the mixed convective flow of MHD. Nadeem et al. ¹⁸ have investigated Maxwell liquid flow through a vertical stretching surface in occurrence of nanoparticle. Raju et al. ¹⁹ have investigated MHD nano non-Newtonian liquid through a cone in the presence of free convective heat and mass transfer. Rokni et al. ²⁰ have studied nano fluid flow with heat transfer through the plates. Nadeem et al. ²¹ have found numerical solution of nano non-Newtonian liquid flow on stretching sheet. Shehzad et al. ²² have been investigated Jaffrey nano liquid MHD flow in the occurrence of convective boundary conditions. Sheikholeslami et al. ²³ have studied nano liquid flow with magnetic field in existence of heat transfer. Mahmoodi and Kandelousi ²⁴ have studied nanonfluid flow with heat transfer for cooling application. Recently, Shah et al.^25–28 have studied nanofluid flow with impacts of thermal radiation and hall current with rotating system. The more current investigational and theoretical study of Sheikholeslami on nanofluids using dissimilar phenomena, with modern application, possessions and properties with usages of diverse approaches can be studied in Sheikholeslami,^29,30 Sheikholeslami and Houman, ³¹ Sheikholeslami ³² and Nasir et al. ³³

Non-Newtonian fluid model has a lot of industrial applications. Many materials have great importance in the industries such as molten plastic, food products, wall paint, greases, lubricant oil, drilling mud etc, have non-Newtonian fluid behaviour. Non-Newtonian fluid, according to the researchers, is most applicable to the research. Sisko liquid is one of the important subclass which has more significance as its occurrence in many engineering operations. Practical application has produced an interest in searching the solvability of differential equation governing in flow of Non-Newtonian liquids, which have numerous uses in engineering field, applied mathematics and computer science. Munir et al. ³⁴ have discussed bidirectional flow of Sisko fluid through a stretching sheet. Olanrewaju et al. ³⁵ have studied Sisko unsteady free convective fluid flow in the existence of heat transfer passing through a flat plate. Khan et al. ³⁶ have studied Sisko fluid, steady flow with heat transfer in annular pipe. Khan and Shahzad ³⁷ have described Sisko boundary layer fluid flow through the stretching surface. Patel et al. ³⁸ have investigated laminar Sisko fluid boundary flow. Darji and Timol ³⁹ have studied Sisko fluid unsteady natural convective boundary layer flow. Siddiqui et al. ⁴⁰ have described Sisko fluid film on vertical belt for drainage system. Khan et al. ⁴¹ have discussed Sisko fluid in annular pipe. Sar et al. ⁴² have been described boundary layer equations of Sisko fluid. Marinca et al. ⁴³ have studied Maxwell fluid flow through a porous stretching plates. Moallemi et al. ⁴⁴ have found an exact solution for Sisko fluid flow in a pipe.

Most mathematical problems in the field of engineering and science are complex in its nature and the exact solution of that type of problem is very tough. To find an approximate solution, numerical and analytical technique is used. The homotopy analysis technique, which is one of the widespread and important techniques for the solution of such type of problem. Liao in (1992)^45,46 has investigated it for the solution of such type of problems and it is proved that this method is fast converge to the approximated solution. The solution of this method is a series solution in the form of a single variable function, due to its fast convergence, most researchers Rashidi and Pour ⁴⁷ have used this technique to solve coupled and highly nonlinear equation. The recent use of HAM for nonlinear system of differential equations can be seen in Ishaq et al., ⁴⁸ Nasir et al., ⁴⁹ Hammed et al. ⁵⁰ and Muhammad et al. ⁵¹

In this work, we studied two-dimensional thin film flow of Sisko nanofluid with variable heat transmission over an unsteady stretching sheet in the existence of uniform magnetic field (MHD). The basic governing time-dependent equations of the nanofluid flow phenomena with Sisko fluid are modeled and reduced to a system of differential equations with use of similarity transformation.

Problem formulation

Consider an electrically conducting and time depending nanoliquid film flow of Sisko fluids during spreading surface. The elastic surface starts as of inhibit slit, which remains fixed at succession of a accommodate arrangement, The Cartesian coordinate system oxyz be there accustomed in a well-defined way that ox is equal to the plate and oy is flat to the surface. The surface of the flow is stretched to apply two equal and opposing forces along the x-axis and keeps the origin motionless. The physical model of flow is shown in Figure 1. The x-axis is taking beside the spreading sheet through a stress velocity

$${\overrightarrow{U}}_w\left(x,t\right)=\frac{\gamma x}{1-\varsigma t},$$

(1)

where $\varsigma ,\gamma$ represents stretching parameters and y-axis is vertical to it. The temperature and concentration of the wall of the liquids are taken and defined as

$${\mathrm{T}}_w\left(x,t\right)={\mathrm{T}}_0+{\mathrm{T}}_r\left(\frac{\gamma x^2}{2{\upsilon }_f}\right){\left(1-\varsigma t\right)}^{-1.5}$$

(2)

$$C_w\left(x,t\right)=C_0+C_r\left(\frac{\gamma x^2}{2{\upsilon }_f}\right){\left(1-\varsigma t\right)}^{-1.5}$$

(3)

Figure 1.

Geometry of the demonstrated problem.

Here ${\upsilon }_{\tilde{f}}$ denotes the kinematic viscidness of the fluid and ${\mathrm{T}}_0,{\mathrm{T}}_r$ represents the slit and position temperatures correspondingly, while $C_0\text{ and }{C}_r$ are the slit and a uniform volume section of thin nanoparticles. The external magnetic field is defined as

$$\overrightarrow{\mathrm{B}}(t)={\overrightarrow{\mathrm{B}}}_0{\left(1-\varsigma t\right)}^{-0.5}$$

(4)

where ${\overrightarrow{\mathrm{B}}}_0(t)$ denotes the applied magnetic field.

The theological model that illustrates the Sisko fluid^39–43 as

$$\overrightarrow{T}=-p\overrightarrow{I}+\overrightarrow{S}$$

(5)

$$\overrightarrow{S}=\left[a+b{\left|\sqrt{\mathit{tr}\left({\overrightarrow{A}}_1^2\right)}\right|}^{n-1}\right]{\overrightarrow{A}}_1$$

(6)

Here $P\overrightarrow{I}$ is a stress of isotropic, $\mathrm{S}$ is an extra tensor of stress, $n$ represents a power index, ${\overrightarrow{A}}_1$ is the stress of a Rivlin Ericksen tensor and ${\overrightarrow{A}}_{\mathrm{1}}\mathrm{=\nabla v+}{\left(\mathrm{\nabla v}\right)}^{\mathrm{T}}.$

The continuity equation^33,38

$$\mathit{div}\left(V\right)=0$$

(7)

For two-dimensional nanofluid, equation (7) reduced to the form of as

$$\frac{\partial \overrightarrow{u}}{\partial x}+\frac{\partial \overrightarrow{v}}{\partial y}=0$$

(8)

The momentum equation is written as

$$\rho \frac{D\overrightarrow{V}}{\mathit{Dt}}=\mathit{div}\overrightarrow{T}+\overrightarrow{J}\times \overrightarrow{B}+\rho \overrightarrow{f}$$

(9)

where $\rho$ denotes density of the fluid, $\overrightarrow{V}$ denotes velocity and $\overrightarrow{V}=\left(u(x,y),v(x,y),0\right)$ , $\overrightarrow{T}$ denotes Cauchy stress tensor and $\text{ g}$ is the body (gravitational) force. The term $\overrightarrow{J}\times \overrightarrow{B}$ is called the Lorentz Force, in which current density is denoted by $\overrightarrow{J}$ and the magnetic field by $\overrightarrow{B}$ . $\overrightarrow{J}$ is defend as $\overrightarrow{J}=\sigma \left(\overrightarrow{E}+\overrightarrow{V}\times \overrightarrow{B}\right)$ where it is called Ohm law. Where ‘ $\sigma$ ’ is used for electrical conductivity and $\overrightarrow{E}$ represents electric field and is assumed to be $\overrightarrow{E}=0$ . Using all above assumption, equation (9) reduced to as^34–42

$$\frac{\partial \overrightarrow{u}}{\partial t}+\overrightarrow{u}\frac{\partial \overrightarrow{u}}{\partial x}+\overrightarrow{v}\frac{\partial \overrightarrow{u}}{\partial y}=\frac{a}{\rho }\frac{{\partial }^2\overrightarrow{u}}{\partial y^2}-\frac{b}{\rho }\frac{\partial }{\partial y}{\left(-\frac{\partial \overrightarrow{u}}{\partial y}\right)}^n-\sigma {\overrightarrow{\mathrm{B}}}_{0}2\left(t\right)\overrightarrow{u}$$

(10)

Here $\overrightarrow{u}\text{ and }\overrightarrow{v}$ are the velocity vectors constituents alongside x and y-axis correspondingly, $\rho$ show the density of liquid.

Thermal energy equations

$$\overrightarrow{V}.\nabla T=\nabla .\left[\frac{k(T)}{(\rho c_p)f}\nabla T\right]+\tau \left[D_B\nabla C.\nabla T+\left(\frac{D_T}{T_0}\right)\nabla T.\nabla T\right]$$

(11)

T is the local temperature, $\tau =\frac{{\left({\rho }_c\right)}_p}{{\left({\rho }_c\right)}_f}$ illustrate ratio of base liquid and nanoparticles thermal capacities, $c_p$ represent the heat capacitance of liquid, direct the Brownian diffusion constant $D_B$ , while the thermophoretic diffusion constant is denoted by $D_T$ , $T_0$ denotes the temperature of the liquid which is detached from the sheet. Equation (11) after using assumption is reduced to the simplified form is

$$\begin{array}{l}\frac{\partial T}{\partial t}+\overrightarrow{u}\frac{\partial \mathrm{T}}{\partial x}+\overrightarrow{v}\frac{\partial \mathrm{T}}{\partial y}=\frac{1}{\rho c_p}\frac{\partial }{\partial y}\left[K(T)\frac{\partial \mathrm{T}}{\partial y}\right]\\ +\tau \left[D_B\left(\frac{\partial C}{\partial y}\frac{\partial \mathrm{T}}{\partial y}\right)+\frac{D_T}{{\mathrm{T}}_{\infty }}{\left(\frac{\partial \mathrm{T}}{\partial y}\right)}^2\right]\end{array}$$

(12)

Here $K(T)=K_0{e}^{c(T-T_0)}$ the elastic thermal conductivity in which $\lambda =c(T_w-T_0)$ is used for elastic thermal conductivity restriction.

Mass Transfer equation

$$\overrightarrow{V}.\nabla C=D{}_B\nabla 2C+\frac{D_T}{T_0}{\nabla }^{2}T$$

(13)

Equation (13) after using assumption which is reduced to the simplified form is

$$\left[\frac{\partial C}{\partial t}+\overrightarrow{u}\frac{\partial C}{\partial x}+\overrightarrow{v}\frac{\partial C}{\partial y}\right]=D_B\frac{{\partial }^{2}C}{\partial y^2}+\left(\frac{D_T}{{\mathrm{T}}_{\infty }}\right)\frac{{\partial }^2\mathrm{T}}{\partial y^2}$$

(14)

The appropriate boundary conditions for flow configuration are taken as

$$\begin{array}{l}\overrightarrow{u}=U_w,\overrightarrow{v}=0,\hspace{1em}\mathrm{T}={\mathrm{T}}_w,C=C_w\text{\hspace{1em}at}\hspace{1em}\hspace{1em}y=0\text{\hspace{1em}and\hspace{1em}}C>0\\ \frac{\partial \overrightarrow{u}}{\partial x}=0,\hspace{1em}\frac{\partial \mathrm{T}}{\partial x}=0,\hspace{1em}\frac{\partial C}{\partial x}=0,\hspace{1em}\overrightarrow{v}=\frac{\mathit{dh}}{\mathit{dt}}\mathrm{\hspace{1em}at} y=h\end{array}$$

(15)

Familiarizing the succeeding similarity transformations

$$\begin{array}{l}\psi =x{\left(\frac{\upsilon \gamma }{1-\varsigma t}\right)}^{\frac{1}{2}}f\left(\eta \right),\overrightarrow{u}=\frac{\partial \psi }{\partial y}=\frac{\gamma x}{(1-\varsigma t)}{f}^{\prime }\left(\eta \right),\\ \overrightarrow{v}=-\frac{\partial \psi }{\partial x}=-{\left(\frac{\upsilon \gamma }{1-\varsigma t}\right)}^{\frac{1}{2}}f\left(\eta \right)\\ \eta ={\left(\frac{\gamma }{\upsilon (1-\varsigma t)}\right)}^{\frac{1}{2}}y,h={\left(\frac{\upsilon \left(1-\varsigma t\right)}{\gamma }\right)}^{\frac{1}{2}},\\ \theta (\eta )=\frac{\mathrm{T}-{\mathrm{T}}_0}{{\mathrm{T}}_w-{\mathrm{T}}_0},\varphi (\eta )=\frac{C-C_0}{C_w-C_0}\end{array}$$

(16)

Here $\Psi$ show the stream function, $h(t)$ denotes thickness of fluid film and $\upsilon =\frac{\mu }{\rho }$ is the kinematics viscosity. The dimensionless film thickness is

$$\beta ={\left(\frac{\varsigma }{\upsilon \left(1-\varsigma t\right)}\right)}^{\frac{1}{2}}h(t)$$

(17)

This in term of derivative is

$$\frac{\mathit{dh}}{\mathit{dt}}=-\frac{c\beta }{2}{\left(\frac{\upsilon }{\varsigma \left(1-\varsigma t\right)}\right)}^{\frac{1}{2}}$$

(18)

Using the similarity transformation equation (16) into equations (10)–(14), realizes the continuity equality; the leftover equations are transformed to system of non-linear differential equations as

$$\begin{array}{c}\epsilon f^{‴}+n\xi {\left(-f^{″}\right)}^{n-1}{f}^{‴}+f{f}^{″}-{\left(f^{\prime }\right)}^2\\ -\mathit{St}\left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)-M{f}^{\prime }=0\end{array}$$

(19)

$$\begin{array}{c}\frac{\left(1+\lambda \theta \right)}{\mathrm{Pr}}\theta ″+f\theta \prime -2f^{\prime }\theta -\frac{\mathit{St}}{2}\left(3\theta +\eta {\theta }^{\prime }\right)\\ +\mathit{Nt}{\left(\theta \prime \right)}^2+\mathit{Nb}\theta \prime {\varphi }^{\prime }=0\end{array}$$

(20)

$$\varphi ″+\mathit{Sc}\left[f\varphi \prime -2f^{\prime }\varphi -\frac{\mathit{St}}{2}\left(3\varphi +\eta {\varphi }^{\prime }\right)\right]+\frac{\mathit{Nt}}{\mathit{Nb}}\theta ″=0$$

(21)

The boundary conditions of the problem are

$$\begin{array}{c}f(0)=0,f\prime (0)=1,\theta (0)=1,\varphi (0)=1\\ f(\eta )=\frac{S\beta }{2},f^{″}(\eta )=0,{\theta }^{\prime }(\eta )=0,{\varphi }^{\prime }(\eta )=0\end{array}$$

(22)

The physical constraints after generalization are obtained as, $\text{St}=\frac{\gamma }{\epsilon }$ is the non-dimensional measure of unsteadiness, $\varsigma =\frac{a}{\rho \nu }$ is Sisko fluid parameter, $\xi =\frac{b}{\rho \nu }{\left({\left(\frac{\gamma x}{\left(1-\varsigma t\right)\sqrt{\nu }}\right)}^{\frac{3}{2}}\right)}^{n-1}$ is stretching parameter and $M\text{=}\frac{{\sigma }_f{B}_0^2}{b{\rho }_f}$ represents the magnetic, $\lambda =m(\mathrm{T}-{\mathrm{T}}_0)$ represents the variable viscosity parameter, $\mathrm{Pr}=\frac{\rho \upsilon c_p}{k}$ is Prandtl number, $\mathit{Nt}=\frac{\tau D_w\left({\mathrm{T}}_w-{\mathrm{T}}_{\infty }\right)}{\upsilon {\mathrm{T}}_{\infty }}$ represents thermophoresis constraint, $\text{ Nb=}\frac{\tau D_B(C_w-C_{\infty })}{\upsilon }$ shows Brownian motion limitation, $\text{ Sc=}\frac{\upsilon }{D_B}$ denotes Schmidt number and all of these are well-defined individually.

The local skin–friction coefficient $\overrightarrow{C}f$ is defined as follows

$${\overrightarrow{\text{C}}}_f=\frac{{\left({\overrightarrow{S}}_{\mathit{xy}}\right)}_{y=0}}{\rho {\overrightarrow{U}}_w^2}$$

(23)

where

$${\overrightarrow{S}}_w=\left(a\frac{\partial \overrightarrow{u}}{\partial y}+b\frac{\partial \overrightarrow{u}}{\partial y}{\left|\frac{\partial \overrightarrow{u}}{\partial y}\right|}^{n-1}\right)$$

(24)

$${\overrightarrow{\text{C}}}_f\sqrt{{\text{Re}}_x}=\left(\xi f^{″}(0)-{\left[-f^{″}(0)\right]}^n\right)$$

(25)

where ${\text{Re}}_x$ is called local Reynolds number, defined as $\text{Re}x=\frac{{\overrightarrow{U}}_{w}x}{\nu }$ . The Nusselt number is defined as $\mathrm{N}u=\frac{h{Q}_w}{\widehat{k}\left(T_0-T_h\right)}$ $Q_w$ is the heat flux and $Q_w=-\widehat{k}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}$ . The Sherwood number is defined as $\mathit{Sh}=\frac{h{J}_w}{D_B\left(C_0-C_h\right)},$ $J_w$ is the mass flux and $J_w=-D_B{\left(\frac{\partial C}{\partial y}\right)}_{y=0}$ . Here the dimensionless form of $\mathrm{N}u$ and $\mathit{Sh}$ is obtained as

$$\mathrm{N}u=-\left(1+\lambda \Theta \right){\Theta }^{\prime }(0),\mathit{Sh}=-{\Phi }^{\prime }\left(0\right)$$

(26)

Solution by HAM

Here we use the homotopy analysis method to solve the equations (19)–(21) consistent with the boundary restrain (equation (20)). The solutions enclosed the secondary parameters $\mathit{\hslash }$ which standards and switches to the combination of the solutions.

The preliminary guesses are selected as follows

$${\tilde{f}}_0(\eta )=\eta ,{\tilde{\theta }}_0(\eta )=1\text{\hspace{1em}and\hspace{1em}}{\tilde{\varphi }}_0(\eta )=1$$

(27)

The linear operatives represented by $L_{\tilde{f}},L_{\tilde{\theta }}\text{ and }{L}_{\tilde{\varphi }}$

$$L_{\tilde{f}}\left(\tilde{f}\right)=\tilde{f}‴,L_{\tilde{\theta }}\text{(}\tilde{\theta }\text{)=\hspace{0.17em}}\tilde{\theta }\text{″\hspace{1em} and \hspace{1em}}{L}_{\tilde{\varphi }}\left(\tilde{\varphi }\right)=\tilde{\varphi }″$$

(28)

Which have the subsequent applicability

$$\begin{array}{l}{L}_{\tilde{f}}\left(e_1+e_2\eta +e_3{\eta }^2\right)=0,\\ L_{\tilde{\theta }}\left(e_4+e_5\eta \right)=0\text{\hspace{0.17em}\hspace{1em} and \hspace{1em}}{L}_{\tilde{\varphi }}\left(e_6+e_7\eta \right)=0\end{array}$$

(29)

where $e_i(i=1->7)$ the coefficients involve in general solution:

The corresponding non-linear operators $N_{\tilde{f}},N_{\tilde{\theta }}\text{ and }{N}_{\tilde{\varphi }}$ are carefully chosen of the form

$$\begin{array}{l}{N}_{\tilde{f}}\left[\tilde{f}\left(\eta ;\zeta \right)\right]=\epsilon \frac{{\partial }^3\tilde{f}}{\partial {\eta }^3}+n\xi {\left(-\frac{{\partial }^2\tilde{f}}{\partial {\eta }^2}\right)}^2\frac{{\partial }^3\tilde{f}}{\partial {\eta }^3}+\tilde{f}\frac{{\partial }^2\tilde{f}}{\partial {\eta }^2}\\ -{\left(\frac{\partial \tilde{f}}{\partial \eta }\right)}^2-\text{St}\left(\frac{\partial \tilde{f}}{\partial \eta }+\frac{\eta }{2}\frac{{\partial }^2\tilde{f}}{\partial {\eta }^2}\right)-M\frac{\partial \tilde{f}}{\partial \eta }\\ \end{array}$$

(30)

$$\begin{array}{l}{N}_{\tilde{\theta }}\left[\tilde{f}\left(\eta ;\zeta \right),\theta \left(\eta ;\zeta \right),\varphi \left(\eta ;\zeta \right)\right]=\frac{\left(1+\lambda \tilde{\theta }\right)}{\mathrm{Pr}}\frac{{\partial }^2\tilde{\theta }}{\partial {\eta }^2}+\tilde{f}\frac{\partial \tilde{\theta }}{\partial \eta }\\ -2\frac{\partial \tilde{f}}{\partial \eta }\tilde{\theta }-\frac{\mathit{St}}{2}\left(3\tilde{\theta }+\eta \frac{\partial \tilde{\theta }}{\partial \eta }\right)+\mathit{Nt}{\left(\frac{\partial \tilde{\theta }}{\partial \eta }\right)}^2+\mathit{Nb}\frac{\partial \tilde{\theta }}{\partial \eta }\frac{\partial \tilde{\varphi }}{\partial \eta }\\ \end{array}$$

(31)

$$\begin{array}{l}{N}_{\tilde{\varphi }}\left[\tilde{f}\left(\eta ;\zeta \right),\tilde{\theta }\left(\eta ;\zeta \right),\tilde{\varphi }\left(\eta ;\zeta \right)\right]=\frac{{\partial }^2\tilde{\varphi }}{\partial {\eta }^2}\\ +\mathit{Sc}\left[\tilde{f}\frac{\partial \tilde{\varphi }}{\partial \eta }-2\frac{\partial \tilde{f}}{\partial \eta }\tilde{\varphi }-\frac{\mathit{St}}{2}\left(3\tilde{\varphi }+\eta \frac{\partial \tilde{\varphi }}{\partial \eta }\right)\right]+\frac{\mathit{Nt}}{\mathit{Nb}}\frac{{\partial }^2\tilde{\theta }}{\partial {\eta }^2}\\ \end{array}$$

(32)

The 0th-order scheme forms equations (8)–(10) as

$$\left(1-\zeta \right)L_{\tilde{f}}\left[\tilde{f}\left(\eta ;\zeta \right)-{\tilde{f}}_0\left(\eta \right)\right]=p{\mathit{\hslash }}_{\tilde{f}}{N}_{\tilde{f}}\left[\tilde{f}\left(\eta ;\zeta \right)\right]$$

(33)

$$\left(1-\zeta \right)L_{\tilde{\theta }}\left[\tilde{\theta }\left(\eta ;\zeta \right)-{\tilde{\theta }}_0\left(\eta \right)\right]=p{\mathit{\hslash }}_{\tilde{\theta }}{N}_{\tilde{\theta }}\left[\tilde{f}\left(\eta ;\zeta \right),\tilde{\theta }\left(\eta ;\zeta \right),\tilde{\varphi }\left(\eta ;\zeta \right)\right]$$

(34)

$$\left(1-\zeta \right)L_{\tilde{\varphi }}\left[\tilde{\varphi }\left(\eta ;\zeta \right)-{\tilde{\varphi }}_0\left(\eta \right)\right]=\zeta {\mathit{\hslash }}_{\tilde{\varphi }}{N}_{\tilde{\varphi }}\left[\tilde{f}\left(\eta ;\zeta \right),\tilde{\theta }\left(\eta ;\zeta \right),\tilde{\varphi }\left(\eta ;\zeta \right)\right]$$

(35)

The correspondent boundary constrains are

$$\begin{array}{l}{\tilde{f}(\eta ;\zeta )|}_{\eta =0}=0,{\frac{\partial \tilde{f}(\eta ;\zeta )}{\partial \eta }|}_{\eta =0}=1,{\frac{{\partial }^2\tilde{f}(\eta ;\zeta )}{\partial {\eta }^2}|}_{\eta =\beta }=0\\ {\tilde{\theta }(\eta ;\zeta )|}_{\eta =0}=1,{\frac{\partial \tilde{\theta }(\eta ;\zeta )}{\partial \eta }|}_{\eta =\beta }=0,{\tilde{\varphi }(\eta ;\zeta )|}_{\eta =0}=1,\\ {\frac{\partial \tilde{\varphi }(\eta ;\zeta )}{\partial \eta }|}_{\eta =\beta }=0\end{array}$$

(36)

where $\zeta \in [0,1]$ is the embedding constraint, ${\mathit{\hslash }}_{\tilde{f}},{\mathit{\hslash }}_{\tilde{\theta }}\text{,}{\mathit{\hslash }}_{\tilde{\varphi }}$ is used to regulate solution convergence. When $\zeta =0\text{ and }\zeta =1$ we have

$$\tilde{f}(\eta ;1)=\tilde{f}(\eta ),\tilde{\theta }(\eta ;1)=\tilde{\theta }(\eta )\text{\hspace{1em} and \hspace{1em}}\tilde{\varphi }(\eta ;1)=\tilde{\varphi }(\eta )$$

(37)

Expanding the velocity, temperature and concentration field, $\tilde{f}(\eta ;\zeta ),\tilde{\theta }(\eta ;\zeta )\text{ and }\tilde{\varphi }(\eta ;\zeta )$ in Taylor’s series approximately $\zeta =0$

$$\begin{array}{l}\tilde{f}\left(\eta ;\zeta \right)={\tilde{f}}_0(\eta )+{\displaystyle {\sum }_{n=1}^{\infty }{\tilde{f}}_n(\eta ){\zeta }^n}\\ \tilde{\theta }\left(\eta ;\zeta \right)={\tilde{\theta }}_0(\eta )+{\displaystyle {\sum }_{n=1}^{\infty }{\tilde{\theta }}_n(\eta ){\zeta }^n}\\ \tilde{\varphi }\left(\eta ;\zeta \right)={\tilde{\varphi }}_0(\eta )+{\displaystyle {\sum }_{n=1}^{\infty }{\tilde{\varphi }}_n(\eta ){\zeta }^n}\end{array}$$

(38)

$$\begin{array}{l}{\tilde{f}}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \tilde{f}(\eta ;\zeta )}{\partial \eta }|}_{p=0},{\tilde{\theta }}_n(\eta )={\frac{1}{n!}\frac{\partial \tilde{\theta }(\eta ;\zeta )}{\partial \eta }|}_{p=0}\text{\hspace{1em} and }\\ {\tilde{\varphi }}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \tilde{\varphi }(\eta ;\zeta )}{\partial \eta }|}_{p=0}\end{array}$$

(39)

The dependable boundary constrains are

$$\begin{array}{l}{\tilde{f}}_n(0)={\widetilde{f^{\prime }}}_n(0)={\tilde{\theta }}_n(0)={\tilde{\varphi }}_n(0)=0\\ {\widetilde{f^{″}}}_n(\beta )={\widetilde{{\theta }^{\prime }}}_n(\beta )={\widetilde{{\varphi }^{\prime }}}_n(\beta )=0\end{array}$$

(40)

Here

$$\begin{array}{l}{R}_n^{\tilde{f}}\left(\eta \right)=\epsilon {\widetilde{f^{‴}}}_{n-1}+n\xi {\displaystyle \sum _{k=0}^{n-1}{\left(-{\widetilde{f^{″}}}_{n-1-k}\right)}^{n-1}}{\widetilde{f^{‴}}}_k\\ +{\displaystyle \sum _{k=0}^{n-1}{\tilde{f}}_{n-1-k}{\widetilde{f^{″}}}_k}-{\displaystyle \sum _{k=0}^{n-1}{\widetilde{f^{\prime }}}_{n-1-k}{\widetilde{f^{\prime }}}_k}\\ -\mathit{St}\left({\widetilde{f^{\prime }}}_{n-1}+\frac{\eta }{2}{\widetilde{f^{″}}}_{n-1}\right)-M{\widetilde{f^{\prime }}}_{n-1}\end{array}$$

(41)

$$\begin{array}{l}{R}_n^{\tilde{\theta }}\left(\eta \right)=\frac{1}{\mathrm{Pr}}\left({\widetilde{{\theta }^{″}}}_{n-1}+\lambda {\displaystyle \sum _{k=0}^{n-1}{\tilde{\theta }}_{n-1-k}{\widetilde{{\theta }^{″}}}_k}\right)+{\displaystyle \sum _{k=0}^{n-1}{\tilde{f}}_{n-1-k}{\widetilde{{\theta }^{\prime }}}_k}\\ -2{\displaystyle \sum _{k=0}^{n-1}{\widetilde{f^{\prime }}}_{n-1-k}{\tilde{\theta }}_k}-\frac{\mathit{St}}{2}\left(3{\tilde{\theta }}_{n-1}+\eta {\widetilde{{\theta }^{\prime }}}_{n-1}\right)\\ +\mathit{Nt}{\displaystyle \sum _{k=0}^{n-1}{\widetilde{{\theta }^{\prime }}}_{n-1-k}{\widetilde{{\theta }^{\prime }}}_k}+\mathit{Nb}{\displaystyle \sum _{k=0}^{n-1}{\widetilde{{\theta }^{\prime }}}_{n-1-k}{\widetilde{{\varphi }^{\prime }}}_k}\end{array}$$

(42)

$$\begin{array}{l}{R}_n^{\tilde{\varphi }}\left(\eta \right)={\widetilde{{\varphi }^{″}}}_{n-1}+\mathit{Sc}[{\displaystyle \sum _{k=0}^{n-1}{\tilde{f}}_{n-1-k}{\widetilde{{\varphi }^{\prime }}}_k-}2{\displaystyle \sum _{k=0}^{n-1}{\widetilde{f^{\prime }}}_{n-1-k}{\tilde{\varphi }}_k}\\ -\frac{\mathit{St}}{2}\left(3{\tilde{\varphi }}_{n-1}+\eta {\widetilde{{\varphi }^{\prime }}}_{n-1}\right)]+\frac{\mathit{Nt}}{\mathit{Nb}}{\widetilde{{\theta }^{″}}}_{n-1}\end{array}$$

(43)

where

$${\chi }_n=\{\begin{array}{cc}0,& \text{if }\zeta \le \text{1}\\ 1,& \text{if }\zeta >\text{1}\end{array}$$

(44)

Convergence of HAM solution

The convergence of equation (33)–(35), wholly be particulared by the secondary restrictions ${\mathit{\hslash }}_{\tilde{f}},{\mathit{\hslash }}_{\tilde{\theta }}\text{ and }{\mathit{\hslash }}_{\tilde{\varphi }}$ . It is a choice in a way to controls and merge the series answer. The probability sector of $\mathit{\hslash }$ are designed $\mathit{\hslash }$ curves of $\widetilde{f^{″}}\left(0\right),\widetilde{{\theta }^{\prime }}\left(0\right)\text{\hspace{0.17em} and \hspace{0.17em}}\widetilde{{\varphi }^{\prime }}\left(0\right)$ for $20th$ order approximated HAM solution. The effective region of $\mathit{\hslash }$ is $-1.5<{\mathit{\hslash }}_{\tilde{f}}<0.0,-1.5<{\mathit{\hslash }}_{\tilde{\theta }}<0.0\text{ \&}-2.5<{\mathit{\hslash }}_{\tilde{\varphi }}<\mathrm{0.0.}$ The convergence of the HAM technique by $\mathit{\hslash }$ -curves for velocity, temperature and concentration fields has been represented in Figures 2–4 respectively. Tables 1 and 2 display the numerical values of HAM solutions at different approximation using different values of parameters for the different power index. It is clear from the table that homotopy analysis technique is a speedily convergent techniques. The theoretical details and convergence of h curved are proved in [1.1] by a theorem as:

Figure 2.

The $\mathit{\hslash }$ - curves of $f\left(\eta \right)$ for dissimilar value of power index $n=0,1,2,3.$

Figure 3.

The combined $\mathit{\hslash }$ -curve of $\theta \text{(}\eta \text{)}$ and $\varphi \text{(}\eta \text{).}$

Figure 4.

Effect of $\beta$ on $f\left(\eta \right)$ for dissimilar value of power index $n=0,1,2,3.$

Table 1.

The convergence table of HAM up to 25th-order approximations for n = 0 and n = 1 when $Sc=0.5,Nt=\lambda =St=M=0.1,h=-0.5,Pr=Nb=1.$

Order of approximation	$\begin{array}{l}{f}^{″}(0)\\ n=0\end{array}$	$\begin{array}{l}{\theta }^{\prime }(0)\\ n=0\end{array}$	$\begin{array}{l}{\varphi }^{\prime }(0)\\ n=0\end{array}$	$\begin{array}{l}{f}^{″}(0)\\ n=1\end{array}$	$\begin{array}{l}{\theta }^{\prime }(0)\\ n=1\end{array}$	$\begin{array}{l}{\varphi }^{\prime }(0)\\ n=1\end{array}$
1	−0.0600000	−0.107500	0.0537500	−0.0600000	−0.107500	−0.0537500
5	−0.115692	−0.106849	−0.0935671	−0.0598283	−0.107022	−0.0936935
10	−0.119261	−0.106807	−0.0961163	−0.0598283	−0.107003	−0.0962917
15	−0.119313	−0.106806	−0.0961530	−0.0598283	−0.107003	−0.0963318
20	−0.119316	−0.106806	−0.0961552	−0.0598283	−0.107003	−0.0963322
23	−0.119316	−0.106806	−0.0961554	−0.0598283	−0.107003	−0.0963323
25	−0.119316	−0.106806	−0.0961554	−0.0598283	−0.107003	−0.0963324

Table 2.

The convergence table of HAM up to 25th-order approximations for n = 2 and n = 3 when $Sc=0.5,Nt=St=M=0.1,Pr=Nb=1$ .

Order of approximation	$\begin{array}{l}{f}^{″}(0)\\ n=2\end{array}$	$\begin{array}{l}{\theta }^{\prime }(0)\\ n=2\end{array}$	$\begin{array}{l}{\varphi }^{\prime }(0)\\ n=2\end{array}$	$\begin{array}{l}{f}^{″}(0)\\ n=3\end{array}$	$\begin{array}{l}{\theta }^{\prime }(0)\\ n=3\end{array}$	$\begin{array}{l}{\varphi }^{\prime }(0)\\ n=3\end{array}$
1	−0.0600000	−0.107500	0.0537500	−0.0600000	−0.107500	0.0537500
5	−0.108644	−0.106860	0.0935736	−0.115370	−0.106849	0.0935672
10	−0.107774	−0.106837	0.0960198	−0.117830	−0.106811	0.0959988
15	−0.107761	−0.106836	0.0961796	−0.117699	−0.106810	0.0961558
20	−0.107760	−0.106836	0.0961819	−0.117692	−0.106810	0.0961582
23	−0.107760	−0.106836	0.0961821	−0.117692	−0.106810	0.0961583
25	−0.107760	−0.106836	0.0961821	−0.117692	−0.106810	0.0961583

Theorem 1.1: Suppose that $g(h)$ be a continuous function on $\left[a,b\right]$ and all derivatives of $f:\left[a,b\right]\to \text{R}$ exist and have a common $M$ such that $\underset{x\in |a,b|}{\max }|f^{(k)}(x)|\le M$ for all $k$ : Furthermore, assume that $G_n\left(x,\alpha \right)$ be the Taylor polynomial of degree $n$ for $f(x)$ about some $\alpha \in \left(a,b\right)$ , say $\alpha =g(h),$ then $\forall \epsilon >0$ and $\gamma \in \left(a,b\right)$ there exists $\text{N}\in \text{N}$ and interval $\left(c,d\right)$ so that

$$\forall h\in \left(c,d\right)\text{\hspace{0.17em} and \hspace{0.17em}}n\ge \text{N}:\left|f(\gamma )-G_n\left(\gamma ,g(h)\right)\right|<\epsilon .$$

Result and discussion

The current work focuses on the interpretation of the nano liquid film motion through modelled parameters. The graphical explanation of these parameters has been displayed in Figures 4–13. Figure 4 shows the influence of film thickness $\beta$ for dissimilar values of power index $n=0,1,2$ and 3 on velocity profile. It is seeming that the $f\left(\eta \right)$ falls with higher values of $\beta .$ Actually, liquid film produces opposition to film flow and makes to slow down the liquid velocity of nano particles with greater values of $\beta$ . This reduction increased when power index $n$ is increased. The effect of Sisko fluid parameter $\epsilon$ for dissimilar values of power index $n=0,1,2$ and 3 on velocity profile is shown in Figure 5. The large values of Sisko fluid parameter $\epsilon$ rise the liquid fluid motion of nanoparticles, but when the power index goes toward increase, then this effect is seen opposite and in case of $n=2\text{ and }n=3$ the velocity field reduces. The characteristics of magnetic parameter $\mathrm{M}$ on velocity field are shown in Figure 6. From the mathematical formulation, it is clear that the velocity distributions $f(\eta )$ are inversely varied with magnetic parameter $\mathrm{M}$ along the y-direction. Increasing magnetic parameter $\mathrm{M}$ decreases the velocity field (Figure 6) when it is closed to the surface. This influence of $\mathrm{M}$ on velocity field is because of the fact that the rise in the $\mathrm{M}$ progresses friction force for the movement is named the Lorentz force. Lorentz force has the similarity to reduce the velocity of flow in the boundary sheet. Increasing values of the power index $n=0,1,2$ to 3, the reduction is increased. Figure 7 shows the influence of unsteady constraint $\mathit{St}$ on the $f(\eta )$ for dissimilar values of power index $n=0,1,2$ and 3. It is observed that the velocity profile $f(\eta )$ directly varies with unsteadiness parameter $\mathit{St}$ . Increasing $S$ rises the fluid motion. It perceived that solution depended on the unsteadiness parameter $\mathit{St}$ , the solution exists only when $\mathit{St}\in \left[02\right].$ Increasing $\mathit{St}$ rises the nanofluid motion. It is clear that changing power index having the same reaction to the unsteady parameter, that is, the increased value of the power index rises the velocity distribution. The effect of the unsteadiness parameter $\mathit{St}$ on the heat profile $\theta (\eta )$ is presented in Figure 7. It is seen that $\theta (\eta )$ directly varies with unsteadiness parameter ‘ $\mathit{St}$ ’. Increasing unsteadiness parameter $\mathit{St}$ rises the temperature, which in result increase kinetic energy of the fluid, so the liquid film motion rises. It is detected that the influence of unsteadiness parameter $\mathit{St}$ for the different power index $n=0,1,2$ and 3 having the same effect on $\theta (\eta )$ and same effect is observed for concentration profile. The effects of stretching parameter $\xi$ for dissimilar values of power index $n=0,1,2$ and 3 on velocity profile are shown in Figure 9. In case of $n=1$ , it is clear from Figure 9 that velocity profile increases for large values of stretching parameter $\xi$ . It is observed that when values of power index are varied, then the effect of stretching parameter $\epsilon$ changes and for $n=3$ this effect is totally opposite to that which the velocity field $f(\eta )$ reduces. When $n=0$ the stretching parameter $\epsilon$ becomes zero, the thermophoresis parameter $\mathit{Nt}$ faces depreciation in contrast with temperature profile. This phenomenon is described by Figure 10. The thermophoresis limitation supports in growing the surface temperature. An irregular moment of nano-suspended particles in the fluid represent the Brownian motion. Due to this irregularity in motion, nano-suspended particles produce kinetic energy and the temperature increases, as a result the thermophoretic force is produced. This force causes intensity in the fluid to move away from the surface of the stretching sheet. Subsequently, the temperature inside the boundary layer rises as $\mathit{Nt}$ raises. The free surface temperature is increased. The Brownian motion constraint is illustrated in Figure 12 The reality is that arbitrary motion of particles of the fluid generates the collision of the particles. Increase in the value of Brownian motion constraint Nb results increase in heat of the fluid, consequently causing reduction in free surface nanoparticle volume friction and it is displayed in Figure 12 represents that by changing the $\mathit{Sc}$ , the dimensional $\theta (\eta )$ intensifications. Figure 12 shows that for different measures of $\mathit{Sc}$ , dimensional $\theta (\eta )$ reduces. It is observable that flow portion is more in the horizontal direction by increasing $\mathit{Sc}$ . It is because that the Schmidt parameter is ratio of momentum and concentration, diffusivities and increases in $\mathit{Sc}$ , creates to decline thickness of liquid and therefore $\theta (\eta )$ decreases. The opposite effect of $\mathit{Sc}$ is found in the concentration profile (Figure 12). The impact of $\mathit{Pr}$ for numerous measures is explained in Figure 13. It is presented that increasing rate of $\mathit{Pr}$ , reduces boundary layer size, which subsequently reduces the velocity. Actually, the $\mathit{Pr}$ , a dimensionless amount, is the ratio of kinematic viscosity to thermal diffusivity. When the value of momentum diffusivity is more than thermal diffusivity, the value of Prandtl quantity is increased. Thus, the heat spread at the surface grows with growing amount of the $\mathit{Pr}$ while the mass transmission is determined as the $\mathit{Pr}$ growing. The purpose is that with the greater values of $\mathit{Pr}$ amount, the thermal boundary layer reduces. The significance is uniformly clearer for slight $\mathit{Pr}$ quantity since the thermal boundary layer thickness is comparatively more. The concentration field shows a reducing performance beside $\mathit{Pr}$ due to the thinning of the boundary layer (Figure 13).

Figure 5.

Effect of $\epsilon$ on $f\left(\eta \right)$ for dissimilar value of power index $n=0,1,2,3.$

Figure 6.

Effect of M on $f\left(\eta \right)$ for dissimilar value of power index $n=0,1,2,3.$

Figure 7.

Effect of St on $f\left(\eta \right)$ for dissimilar value of power index $n=0,1,2,3.$

Figure 8.

Effect of St on $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ .

Figure 9.

Effect of $\xi$ on $f\left(\eta \right)$ for dissimilar values of power index $n=1,2,3.$

Figure 10.

Influence of $\mathit{Nt}$ on $\varphi \left(\eta \right)$ and $\varphi \left(\eta \right)$ .

Figure 11.

The effect of Nb on $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ .

Figure 12.

Effect of Sc on $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ .

Figure 13.

Effect of Pr on $\theta \left(\eta \right)$ and $\varphi \left(\eta \right).$

Table discussion

Tables 3, 4 and 5 depict the influence of skin friction, Sherwood number and Nusselt number due to altered parameters. The effects of $M,\xi ,\beta$ and $\mathit{St}$ on skin friction for the different behavior of $n=0,1,2\&3$ are shown in Table 3. It is observed that increasing rates of magnetic and unsteady parameter $M\&\mathit{St}$ increases the Skin–friction coefficient, while for dissimilar values of power index $n=0,1,2\text{\hspace{0.17em} and \hspace{0.17em}}3$ , it decreases. The higher value of stretching and the thickness parameter $\xi \&\beta$ , reduces the skin friction coefficient, while four different values of power index $n=0,1,2\text{\hspace{0.17em} and \hspace{0.17em}}3$ it is noted to increase. The effects of $M,\mathit{St},\beta$ and $\mathrm{Pr}$ on the Nusselt number for the different behavior of $n=0,1,2\text{\hspace{0.17em} and \hspace{0.17em}}3$ are shown in Table 4. It is observed that increasing rates of magnetic and unsteady parameter $M$ reduces the Nusselt number while the unsteady parameter $\mathit{St}$ and thickness parameter $\beta$ increase the Nusselt number. The effects of $\mathit{Nb},\mathit{Nt},\mathit{Sc},\mathrm{Pr}$ and $\mathit{St}$ on the Sherwood Number are shown in Table 5. From Table 5, it is noticed that local Sherwood number values increase due to increase in thermoporetic parameter. Increasing the value of Schmidt number decreases the Sherwood number where increasing unsteady parameter and Prandtl number causes to decrease the Sherwood number.

Table 3.

The Skin friction coefficient for dissimilar values of $M,\xi ,\beta$ and $St$ when $Sc=0.6,Nt=0.2,Pr=1,Nb=0.9$ .

$M$	$\xi$	$\beta$	$St$	$\begin{array}{l}{C}_f\\ n=0\end{array}$	$\begin{array}{l}{C}_f\\ n=1\end{array}$	$\begin{array}{l}{C}_f\\ n=2\end{array}$	$\begin{array}{l}{C}_f\\ n=3\end{array}$
0.1	0.5	0.1	1.5	−0.124726	−0.25778	−0.1708	−0.2478
0.5				−0.143755	−0.29719	−0.2034	−0.2849
1.0				−0.167483	−0.34634	−0.2460	0.3309
1.5	0.1			−0.038228	−0.38755	−0.1776	−0.3811
	0.5			−0.124726	−0.25778	−0.1708	−0.2478
	1.0			−0.249452	−0.25850	−0.1708	−0.2461
	1.5	0.1		−0.374178	−0.25908	−0.3434	−0.2445
		0.5		−0.508809	−1.10089	−0.7463	−0.8475
		1.0		−0.692997	−1.70742	−6.9860	−0.8868
		1.5	0.1	−0.500106	−1.35924	−1.8672	−0.8876
			0.5	−0.077021	−0.15907	−0.2217	−0.1536
			1.0	−0.100915	−0.20849	−0.1320	−0.2009

Table 4.

Variation in Nusselt umber for different values of $M,Pr,Rd$ and $St$ when $Sc=0.6,Nt=0.2,\beta =0.5,Nb=\mathrm{0.9.}$

$M$	$\beta$	$St$	$Pr$	$\begin{array}{l}N{u}_x\\ n=0\end{array}$	$\begin{array}{l}N{u}_x\\ n=1\end{array}$	$\begin{array}{l}N{u}_x\\ n=2\end{array}$	$\begin{array}{l}N{u}_x\\ n=3\end{array}$
0.1	0.1	1.5	1.5	0.626541	0.352783	0.352309	0.352250
0.5				0.626198	0.352676	0.352140	0.352063
1.0				0.625771	0.352543	0.351935	0.351830
1.5	0.1			0.626541	0.352783	0.352309	0.352250
	0.5			2.38501	1.52758	1.51452	1.49232
	1.0			4.78618	2.52231	2.26158	2.45957
	1.5	0.1		5.10531	2.34792	1.62477	1.84831
		0.5		0.407137	0.228747	0.228439	0.228416
		1.0		0.517063	0.290827	0.290433	0.290394
		1.5	1.5	0.626541	0.352783	0.352309	0.352250
			3.0	1.08812	0.324721	0.324254	0.324196
			5.0	1.40251	0.309182	0.308718	0.308659
			7.0	1.56869	0.301646	0.301183	0.301125

Table 5.

Variation in Sherwood number for different values of $Nb,Nt,Sc,Pr$ and $St$ when $S=0.8,\lambda =0.2,\beta =0.6,M=2.$

$Nb$	$Nt$	$Sc$	$St$	$Pr$	$\begin{array}{l}S{h}_x\\ n=0\end{array}$	$\begin{array}{l}S{h}_x\\ n=1\end{array}$	$\begin{array}{l}S{h}_x\\ n=2\end{array}$	$\begin{array}{l}S{h}_x\\ n=3\end{array}$
0.1	0.5	0.1	1.5	1.5	−1.35820	−1.35938	−1.3582	−1.358
0.5					−0.238811	−0.239002	−0.2388	−0.238
1.0					−0.098888	−0.098954	−0.0988	−0.098
1.5	0.1				0.0223188	0.0223610	0.0223	0.022
	0.5				−1.35820	−1.35938	−1.3582	−1.358
	1.0				−2.75366	−2.75610	−2.7537	−2.753
	1.5	0.1			−4.14542	−4.14910	−4.1455	−4.145
		0.5			−1.18991	−1.19086	−3.9667	−1.189
		1.0			−0.981683	−0.982342	−3.7453	−0.502
		1.5	0.1		−0.398281	−0.398452	−1.8105	−0.398
			0.5		−0.882057	−0.882792	−2.3059	−0.882
			1.0		−1.12055	−1.12151	−2.9193	−1.120
			1.5	1.5	−0.642239	−1.35938	−3.5263	−1.358
				3.0	−1.21912	−2.06224	−5.6130	−2.060
				5.0	−1.53875	−2.45134	−6.7706	−2.449
				7.0	−1.69378	−2.64001	−7.3323	−2.638

Conclusion

Concerning numerous applications in the engineering field and the study of industrial device, we present a mathematical model for describing the fluid film of non-Newtonian nano-fluids flow with flexible viscidness and thermal conductivity in the presence of uniform magnetic field. The model of the flow is supposed for Sisko nano-liquids over a stretching sheet. We strongly believe that this reconsideration efforts a worthy probability to the detectives to categorize the great thermal conductive, non-Newtonian fluid which afford good heat transfer equipment by great thermal conductive nanoparticles.

The key concluded remarks of this work are:

It is perceived that the large value of Magnetic parameter drops the velocity distribution of the nanofluid films.

The larger values of Brownian motion parameter rise the profile of temperature.

Thermal boundary layer thickness reduces with rise of Sc, where Nusselt number rises with a rise in radiation parameter.

The increasing values of Pr increase the surface temperature, where opposite effect is found for unsteady parameter St, that is, the large values of St reduce the surface temperature.

It is noticed that the temperature profile falls with the large numbers of thermophoresis parameter Nt and increases for small values.

The increasing values of Nb reduce the mass flux, where Nt increases the mass flux. The higher values of Sc reduce the mass flux, while it increases with increasing values of Sc.

The convergence of the HAM method with the variation of the physical parameters is observed numerically.

Non-dimensional velocity reduction in variable viscosity and magnetic constraints.