The Brownian and Thermophoretic Analysis of the Non-Newtonian Williamson Fluid Flow of Thin Film in a Porous Space over an Unstable Stretching Surface

Abstract

This paper explores Liquid Film Flow of Williamson Fluid over an Unstable Stretching Surface in a Porous Space . The Brownian motion and Thermophoresis effect of the liquid film flow on a stretching sheet have been observed. This research include, to focus on the variation in the thickness of the liquid film in a porous space. The self-similarity variables have been applied to convert the modelled equations into a set of non-linear coupled differential equations. These non-linear differential equations have been treated through an analytical technique known as Homotopy Analysis Method (HAM). The effect of physical non-dimensional parameters like, Eckert Number, Prandtl Number, Porosity Parameter, Brownian Motion Parameter, Unsteadiness Parameter, Schmidt Number, Thermophoresis Parameter, Dimensionless Film Thickness, and Williamson Fluid Constant on the liquid film size are investigated and conferred in this endeavor. The obtained results through HAM are authenticated, from its comparison with numerical (ND-Solve Method). The graphical comparison of these two methods is elaborated. The numerical comparison with absolute errors are also been shown in the tables. The physical and numerical results using h curves for the residuals of the velocity, temperature and concentration profiles are obtained.

1. Introduction

In the existing literature most of the study is related to Newtonian Fluids and very little attention is paid to the Non-newtonian fluids. Therefore Williamson Fluid has been selected from the class of non-newtonian shear thickening and shear thinning fluids, which has many uses in the field of industry and engineering. The flow of Pseudoplastic Fluids experimentally describe by Williamson [1] with verified results. The analytical study of Williamson Fluid can be found in the investigation of Dapra and Scarpi [2]. Thermophoresis (also Thermomigration, Thermodiffusion, the Soret Effect, or the Ludwig-Soret Effect) is a phenomenon observed in mixtures of mobile particles where the different particle types exhibit different responses, to the force of a temperature gradient. The term Thermophoresis most often applies to aerosol mixtures, but may also commonly refer to the phenomenon in all forms of matter. The term Soret Effect normally applies to liquid mixtures, which behave in different, less well-understood mechanisms than gaseous mixtures. Thermophoresis may not apply to thermomigration in solids, especially multi-phase alloys. The phenomenon is observed at the scale of one millimeter or less. An example that may be observed by the naked eye with good lighting is when the hot rod of an electric heater is surrounded by tobacco smoke, the smoke goes away from the immediate vicinity of the hot rod. As the small particles of air nearest the hot rod are heated, they create a fast flow away from the rod, down the temperature gradient. They have acquired higher kinetic energy with their higher temperature. When they collide with the large, slower-moving particles of the tobacco smoke they push the latter away from the rod. The force that has pushed the smoke particles away from the rod is an example of a Thermophoretic Force. Brownian motion or Pedesis is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving atoms or molecules in the gas or liquid. Transfer of heat energy play an important role in almost all of the industrial processes. It is used to save energy and reduce processing time in industrial processes. It is also used to raise the thermal rating and increase the working life of equipment. The Liquid Film Flow of Williamson Fluid in a Porous Space over an Unstable Stretching Surface has focused the interest of several researchers because of its many uses in the fields of engineering and industries. The hydrodynamics of a thin liquid film over an unsteady stretching sheet is studied by Wang et al. [3] and Cramer et al. [4] for the first time. The effect of surface mass transfer mixed convection flow is explored by Selim et al. [5]. Das [6] analyzed the impact of thermal radiation on MHD slip flow over a flat plate with variable fluid properties. The effects of radiation and heat transfer on MHD flow of Viscoelastic Liquid and heat transfer over a stretching sheet is studied by Siddeshwar et al. [7]. Nadeem and Hussain [8] solved the problem of flow and heat transfer analysis of Williamson Nanofluid . Hassanien et al. [9] worked on Variable viscosity and thermal conductivity effects on heat transfer by natural convection from a cone and a wedge in porous media. Aziz et al. [10] considered thin film flow and heat transfer on an unsteady stretching sheet with internal heating. Qasim et al. [11] used Buongiorno’s model to investigate heat and mass transfer in Nanofluid. Mahesh et al. [12] studied Heat and Mass Transfer in Nanofuid over an unsteady stretching surface.

Ellahi et al. studied Nanofluid over different phenomena mentioned in [13,14,15,16,17]. A detailed data on thin film Williamson Nanofluid Flow with Varying Viscosity and Thermal Conductivity on a Time-Dependent Stretching Sheet is given by Khan et al. [18]. The present research is the study of liquid film flow of Williamson Fluid in a porous medium over an unsteady stretching sheet with the combined effect of Thermophoresis and Brownian motion. The self-similarity variables has been used to convert the modelled equations into a set of non-linear coupled differential equations. The flow of fluid in a porous medium has also a significant role in the field of engineering and especially in Bio-engineering. The purification of liquids through filtration, human lungs, blood filtration are the application of porous media. The flow of fluid in a porous medium on a stretching sheet can be seen in [19,20]. These non-linear differential equations has been tackled through a powerful analytical method known as Homotopy Analysis Method (HAM) [21,22,23,24,25,26,27,28]. The relevant work can also be found in [29,30,31,32,33,34,35]. The effect of physical non-dimensional parameters like Porosity Parameter, Unsteadiness Parameter, Prandtl Number, Schmidt Number, and Dimensionless Film thickness on the liquid film size has been investigated and discussed. The results achieved by the HAM and numerical ND-Solve method are compared and presented in the form of figures and tables with absolute error to make understandable for readers.

2. Mathematical Formulation of Model

Suppose a two dimensional incompressible Liaquid Film Flow of Williamson Fluid on a Porous Unsteady Stretching Sheet with thermal radiation, where heat and mass are transferred simultaneously. The coordinate axes are chosen in such away that the x-axis is parallel to the plate while the y-axis is perpendicular to it. The stretching velocity of the sheet is in the direction of the x-axis which have magnitude $U_w=\frac{\alpha x}{1-\gamma t}$, in which $\alpha >0$ is the stretching velocity constraint and $\gamma \in [0,1]$. The temperature $T_w(x,t)=T_0-T_{ref}\left(\frac{\alpha x^2}{2\nu }\right){(1-\gamma t)}^{-1.5}$, where $T_0$ elaborates the temperature at the surface and $T_{ref}$ depicts the reference temperature. Similarly, $C_w(x,t)=C_0-C_{ref}\left(\frac{\alpha x^2}{2\nu }\right){(1-\gamma t)}^{-1.5}$ is the volume concentration, where $C_0$ illustrates the concentration at the surface and $C_{ref}$ shows the reference concentration. The time dependent term $\frac{\alpha x^2}{\nu (1-\gamma t)}$, indicates the local Reynold number which reliant on the stretching velocity $U_w(x,t)$. Initially the sheet is fixed with the origin and then an external force is applied to stretch the surface in the positive x-axis at the rate $\frac{\alpha }{(1-\gamma t)}$ in time t with velocity $U_w(x,t)$, where $\gamma \in [0,1]$. Now use the above conditions, to get the following equations as:

Continuity Equation,

$$\begin{array}{c}\hfill \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,\end{array}$$

(1)

Momentum Equation,

$$\begin{array}{c}\hfill \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{\partial }^{2}u}{\partial y^2}+2^{0.5}\mathsf{\Gamma }\nu \frac{{\partial }^{2}u}{\partial y^2}\frac{\partial u}{\partial y}-\frac{\nu \varphi }{K}u,\end{array}$$

(2)

Energy Equation,

$$\begin{array}{cc}\hfill & \frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{\partial }^{2}T}{\partial y^2}+\tau [D_B(\frac{\partial C}{\partial y}\frac{\partial T}{\partial y})+\frac{D_T}{T_{\infty }}{(\frac{\partial T}{\partial y})}^2]+\frac{\nu }{C_p}[{(\frac{\partial u}{\partial y})}^2+2^{0.5}\mathsf{\Gamma }{(\frac{\partial u}{\partial y})}^3],\end{array}$$

(3)

Concentration Equation,

$$\begin{array}{c}\hfill \frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}(\frac{{\partial }^{2}T}{\partial y^2}),\end{array}$$

(4)

along with the BCs,

$$\begin{array}{c}u=U_w,T=T_w,v=0,C=C_w,y=0,\hfill \\ \frac{\partial u}{\partial y}=\frac{\partial C}{\partial y}=\frac{\partial T}{\partial y}=0,v=\frac{dh}{dt}=0,y=h\left(t\right).\hfill \end{array}$$

(5)

u and v are the flow velocities along x and y axis respectively, the Specific heat at constant pressure is represented by $C_p$, the Thermal diffusivity of the base fluid is indicated by $\alpha =\frac{k}{\left(\rho c\right)p}$, $\mathsf{\Gamma }>0$ is the Time constant, the Fluid density is represented by $\rho c$, $\tau =\frac{\left(\rho c\right)p}{\left(\rho c\right)f}$ and the local nanoparticle Volume fraction is denoted by C . Also the Thermophoretic diffusion coefficient is indicated by $D_T$, $\rho$ is the Density, while the Brownian diffusion coefficient is shown by $D_B$. T is the local Temperature and the Film thickness is denoted by $h\left(t\right)$.

Now define the following similarity transformations as:

$$\begin{array}{c}\xi ={(\frac{\alpha }{\nu (1-\gamma t)})}^{0.5}y,\hfill \\ \psi (x,y,t)={(\frac{\nu \alpha }{1-\gamma t})}^{0.5}xf\left(\xi \right),\hfill \\ T(x,y,t)=T_0-T_{ref}(\frac{\alpha x^2}{2\nu }){(1-\gamma t)}^{-1.5}\theta \left(\xi \right),\hfill \\ C(x,y,t)=C_0-C_{ref}(\frac{\alpha x^2}{2\nu }){(1-\gamma t)}^{-1.5}\varphi \left(\xi \right).\hfill \end{array}$$

(6)

$\psi (x,y)$ is the Stream function which is defined as: $u=\frac{\partial \psi }{\partial y}$, $v=-\frac{\partial \psi }{\partial x}$. $\beta$ is Non-dimensional film thickness and is described as $\beta ={(\frac{\alpha }{\nu (1-\gamma t)})}^{0.5}\left(h\left(t\right)\right)$ [29,30].

Also $\frac{dh}{dt}=-\frac{\gamma }{2}\beta {\left(\frac{\nu }{\alpha }\right)}^{0.5}{(1-\gamma t)}^{-0.5}$. Now put the values in the above equations we get a system of nonlinear coupled boundary value problems as:

$$\begin{array}{c}\frac{d^{3}f\left(\xi \right)}{d{\xi }^3}+\lambda \frac{d^{2}f\left(\xi \right)}{d{\xi }^2}\frac{d^{3}f\left(\xi \right)}{d{\xi }^3}+f\left(\xi \right)\frac{d^{2}f\left(\xi \right)}{d{\xi }^2}-{(\frac{df\left(\xi \right)}{d\xi })}^2-S(\frac{df\left(\xi \right)}{d\xi }+\frac{\xi }{2}\frac{d^{2}f\left(\xi \right)}{d{\xi }^2})-K_r\frac{df\left(\xi \right)}{d\xi }=0,\\ \frac{d^2\theta \left(\xi \right)}{d{\xi }^2}+Prf\left(\xi \right)\frac{d\theta \left(\xi \right)}{d\xi }-2Pr\frac{df\left(\xi \right)}{d\xi }\theta \left(\xi \right)-Pr(\frac{S}{2}(3\theta \left(\xi \right)+\xi \frac{d\theta \left(\xi \right)}{d\xi }))+\\ PrNb\frac{d\varphi \left(\xi \right)}{d\xi }\frac{d\theta \left(\xi \right)}{d\xi })+PrNt{(\frac{d\theta \left(\xi \right)}{d\xi })}^2+Pr{E}_c({(\frac{d^{2}f\left(\xi \right)}{d{\xi }^2})}^2+\lambda {(\frac{d^{2}f\left(\xi \right)}{d{\xi }^2})}^3)=0,\end{array}$$

$$\begin{array}{c}\frac{d^2\varphi \left(\xi \right)}{d{\xi }^2}+Sc(\frac{d\varphi \left(\xi \right)}{d\xi }f\left(\xi \right)-2\frac{df\left(\xi \right)}{d\xi }\varphi \left(\xi \right)-\frac{S}{2}(3\varphi \left(\xi \right)+\xi \frac{d\varphi \left(\xi \right)}{d\xi })+\frac{Nt}{Nb}\frac{d^2\theta \left(\xi \right)}{d{\xi }^2}=0,\end{array}$$

(7)

along with transformed boundary conditions,

$$\begin{array}{c}\hfill \frac{d^{2}f\left(\beta \right)}{d{\xi }^2}=0,\frac{df\left(0\right)}{d\xi }=1,f\left(0\right)=0,f\left(\beta \right)=\frac{S\beta }{2},\frac{d\theta \left(\beta \right)}{d\xi }=0,\theta \left(0\right)=1,\frac{d\varphi \left(\beta \right)}{d\xi }=0,\varphi \left(0\right)=1.\end{array}$$

(8)

where

$$\begin{array}{c}\lambda =\mathsf{\Gamma }{U}_w{(\frac{2\alpha }{\nu (1-\gamma t)})}^{0.5},K_r=\frac{{\nu }^2\varphi (1-\gamma t)}{\alpha K},\\ S=\frac{\gamma }{\alpha },Pr=\frac{\nu \rho C_p}{k},E_c=\frac{U_w^2}{C_p(T_w-T_0)},\end{array}$$

$$\begin{array}{c}Sc=\frac{\nu }{D_B},Nb=\tau D_B(C_w-C_{\infty }),Nt=\frac{\tau D_T(T_w-T_{\infty })}{\nu T_{\infty }}.\end{array}$$

(9)

3. Materials and Methods

In this section high accuracy of the applied method is applied to system of nonlinear boundary value problems obtained from the new modeled phenomenon. As a result, we see that this method gives best approximation and takes very less time to produce good results.

Solution of Problem

For the solution of system 7 an analytical technique, called Homotopy Analysis Method (HAM) is used. To apply this method we first find the initial guesses $f_0\left(\xi \right),{\theta }_0\left(\xi \right),{\varphi }_0\left(\xi \right)$ from the following as:

Zeroth order problem:

$$\begin{array}{c}\frac{d^3{f}_0\left(\xi \right)}{d{\xi }^3}=0,f_0\left(0\right)=0,\frac{d{f}_0\left(0\right)}{d\xi }=1,\frac{d^2{f}_0\left(\beta \right)}{d{\xi }^2}=0,\\ \frac{d^2{\theta }_0\left(\xi \right)}{d{\xi }^2}=0,{\theta }_0\left(0\right)=1,\frac{d{\theta }_0\left(\beta \right)}{d\eta }=0,\end{array}$$

$$\begin{array}{c}\frac{d^2{\varphi }_0\left(\xi \right)}{d{\xi }^2}=0,{\varphi }_0\left(0\right)=1,\frac{d{\varphi }_0\left(\beta \right)}{d\xi }=0,\end{array}$$

(10)

which gives the solution as

$$\begin{array}{c}\hfill f_0\left(\xi \right)=\xi ,{\theta }_0\left(\xi \right)=1,{\varphi }_0\left(\xi \right)=1.\end{array}$$

(11)

The linear operators are chosen as ${\psi }_f=\frac{d^{3}f\left(\xi \right)}{d{\xi }^3}$, ${\psi }_{\theta }=\frac{d^2\theta \left(\xi \right)}{d{\xi }^2}$ and ${\psi }_{\varphi }=\frac{d^2\varphi \left(\xi \right)}{d{\xi }^2}$ with the following properties

$$\begin{array}{c}\hfill {\psi }_f(C_1+C_2\xi +C_3{\xi }^2)=0,{\psi }_{\theta }(C_4+C_5\xi )=0,{\psi }_{\varphi }(C_6+C_7\xi )=0,\end{array}$$

(12)

where $C_i,i=1-7$ are constants. The resultant non-linear operators ${\aleph }_f,{\aleph }_{\theta }$ and ${\aleph }_{\varphi }$ are chosen as:

$$\begin{array}{c}{\aleph }_f(\xi ;\wp )=\frac{d^{3}f\left(\xi \right)}{d{\xi }^3}+f\left(\xi \right)\frac{d^{2}f\left(\xi \right)}{d{\xi }^2}+\lambda \frac{d^{2}f\left(\xi \right)}{d{\xi }^2}\frac{d^{3}f\left(\xi \right)}{d{\xi }^3}-{(\frac{df\left(\xi \right)}{d\xi })}^2-S(\frac{df\left(\xi \right)}{d\xi }+\frac{\xi }{2}\frac{d^{2}f\left(\xi \right)}{d{\xi }^2})-K_r\frac{df\left(\xi \right)}{d\xi },\end{array}$$

(13)

$$\begin{array}{c}{\aleph }_{\theta }[f(\xi ;\wp ),\theta (\xi ;\wp ),\varphi (\xi ;\wp )]=\frac{d^2\theta \left(\xi \right)}{d{\xi }^2}+Prf\left(\xi \right)\frac{d\theta \left(\xi \right)}{d\xi }-2Pr\frac{df\left(\xi \right)}{d\xi }\theta \left(\xi \right)-Pr(\frac{S}{2}(3\theta \left(\xi \right)+\xi \frac{d\theta \left(\xi \right)}{d\xi }))+\end{array}$$

$$\begin{array}{c}PrNb\frac{d\varphi \left(\xi \right)}{d\xi }\frac{d\theta \left(\xi \right)}{d\xi })+PrNt{(\frac{d\theta \left(\xi \right)}{d\xi })}^2+E_c({(\frac{d^{2}f\left(\xi \right)}{d{\xi }^2})}^2+\lambda {(\frac{d^{2}f\left(\xi \right)}{d{\xi }^2})}^3),\end{array}$$

(14)

$$\begin{array}{c}{\aleph }_{\varphi }[f(\xi ;\wp ),\theta (\xi ;\wp ),\varphi (\xi ;\wp )]=\frac{d^2\varphi \left(\xi \right)}{d{\xi }^2}+Sc(\frac{d\varphi \left(\xi \right)}{d\xi }f\left(\eta \right)-2\frac{df\left(\xi \right)}{d\xi }\varphi \left(\xi \right)-\end{array}$$

$$\begin{array}{c}\frac{S}{2}(3\varphi \left(\xi \right)+\xi \frac{d\varphi \left(\xi \right)}{d\xi }))+\frac{Nt}{Nb}\frac{d^2\theta \left(\xi \right)}{d{\xi }^2}.\end{array}$$

(15)

The basic idea of HAM is described in [21,22,23,24,25,26,27,28],

Zeroth-order problems:

$$\begin{array}{c}\hfill (1-\wp ){\psi }_f\left[f(\xi ;\wp )-f_0\left(\xi \right)\right]=\wp {\hslash }_f{\aleph }_f\left[f(\xi ;\wp )\right],\end{array}$$

(16)

$$\begin{array}{c}\hfill (1-\wp ){\psi }_{\theta }\left[\theta (\xi ;\wp )-{\theta }_0\left(\xi \right)\right]=\wp {\hslash }_{}{\aleph }_{\theta }[f(\xi ;\wp ),\theta (\xi ;\wp ),\theta (\xi ;\wp )],\end{array}$$

(17)

$$\begin{array}{c}\hfill (1-\wp ){\psi }_{\varphi }\left[\varphi (\xi ;\wp )-{\varphi }_0\left(\xi \right)\right]=\wp {\hslash }_{\varphi }{\aleph }_{\varphi }[f(\xi ;\wp ),\theta (\xi ;p),\varphi (\xi ;\wp )].\end{array}$$

(18)

The equivalent BCs are:

$$\begin{array}{c}f(0;\wp )=0,\frac{df(0;\wp )}{d\xi }=1,\frac{d^{2}f(\xi ;\wp )}{d{\xi }^2}=0,\end{array}$$

$$\begin{array}{c}\theta (0;\wp )=1,\frac{d\theta (0;\wp )}{d\xi }=0,\varphi (0;\wp )=1,\frac{d\varphi (\beta ;\wp )}{d\xi }=0.\end{array}$$

(19)

where $\wp \in [0,1]$ is the imbedding parameter, ${\hslash }_f,{\hslash }_{\theta }$ and ${\hslash }_{\varphi }$ are used to control the convergence of the solution. When $\wp =0$ and $\wp =1$, then:

$$\begin{array}{c}\hfill f(\xi ;1)=f\left(\xi \right),\theta (\xi ;1)=\theta \left(\xi \right),\varphi (\xi ;1)=\varphi \left(\xi \right).\end{array}$$

(20)

Expanding $f(\xi ;\wp ),\theta (\xi ;\wp )$ and $\varphi (\xi ;\wp )$ in Taylor’s series about $\wp =0$ as:

$$\begin{array}{c}f\left(\xi \right)=f_0\left(\xi \right)+{\sum }_{m=0}^{m=\infty }{f}_m\left(\xi \right){\wp }^m,\\ \theta \left(\xi \right)={\theta }_0\left(\xi \right)+{\sum }_{m=0}^{m=\infty }{\theta }_m\left(\xi \right){\wp }^m,\end{array}$$

$$\begin{array}{c}\varphi \left(\xi \right)={\varphi }_0\left(\xi \right)+{\sum }_{m=0}^{m=\infty }{\varphi }_m\left(\xi \right){\wp }^m.\end{array}$$

(21)

where

$$\begin{array}{c}{f}_m\left(\xi \right)={\left.\frac{1}{m!}\frac{d^{m}f(\xi ;\wp )}{d{\xi }^m}\right|}_{\wp =0},\\ {\theta }_m\left(\xi \right)={\left.\frac{1}{m!}\frac{d^m\theta (\xi ;\wp )}{d{\xi }^m}\right|}_{\wp =0},\end{array}$$

$$\begin{array}{c}{\varphi }_m\left(\xi \right)={\left.\frac{1}{m!}\frac{d^m\varphi (\xi ;\wp )}{d{\xi }^m}\right|}_{\wp =0}.\end{array}$$

(22)

The secondary constraints ${\hslash }_f,{\hslash }_{\theta }$ and ${\hslash }_{\varphi }$ are selected in such away that the series (21) converges at $\wp =1$. Use $\wp =1$ in (21) to get:

$$\begin{array}{c}f\left(\xi \right)=f_0\left(\xi \right)+{\sum }_{m=0}^{m=\infty }{f}_m\left(\xi \right),\\ \theta \left(\xi \right)={\theta }_0\left(\xi \right)+{\sum }_{m=0}^{m=\infty }{\theta }_m\left(\xi \right),\end{array}$$

$$\begin{array}{c}\varphi \left(\xi \right)={\varphi }_0\left(\xi \right)+{\sum }_{m=0}^{m=\infty }{\varphi }_m\left(\xi \right).\end{array}$$

(23)

The $m^{th}$ -order problem satisfies the following:

$$\begin{array}{c}{\psi }_f\left[f_m\left(\xi \right)-{\chi }_m{f}_{m-1}\left(\xi \right)\right]={\hslash }_f{R}_m^f\left(\xi \right),\\ {\psi }_{\theta }\left[{\theta }_m\left(\xi \right)-{\chi }_m{\theta }_{m-1}\left(\xi \right)\right]={\hslash }_{\theta }{R}_m^{\theta }\left(\xi \right),\end{array}$$

$$\begin{array}{c}{\psi }_{\varphi }\left[{\phi }_m\left(\xi \right)-{\chi }_m{\varphi }_{m-1}\left(\xi \right)\right]={\hslash }_{\varphi }{R}_m^{\varphi }\left(\xi \right).\end{array}$$

(24)

The boundary conditions for this problem are:

$$\begin{array}{c}\frac{d^2{f}_m\left(\beta \right)}{d{\xi }^2}=0,\frac{d{f}_m\left(0\right)}{d\xi }=1,f_m\left(0\right)=0,f_m\left(\beta \right)=\frac{S\beta }{2},\frac{d{\theta }_m\left(\beta \right)}{d\xi }=0,{\theta }_m\left(0\right)=1,\end{array}$$

$$\begin{array}{c}\frac{d{\varphi }_m\left(\beta \right)}{d\xi }=0,{\varphi }_m\left(0\right)=1.\end{array}$$

(25)

Here

$$\begin{array}{c}{R}_m^f\left(\xi \right)=\frac{d^3{f}_{m-1}}{d{\xi }^3}+\lambda {\sum }_{k=0}^{m-1}\frac{d^2{f}_{m-1-k}}{d{\xi }^2}\frac{d^3{f}_k}{d{\xi }^3}+\left[f_{m-1}\frac{d^2{f}_{m-1}}{d{\xi }^2}-{\sum }_{k=0}^{m-1}\frac{d{f}_{m-1-k}}{d\xi }\frac{d{f}_k}{d\xi }-S\left(\frac{d{f}_{m-1}}{d\xi }+\frac{\xi }{2}\frac{d^2{f}_{m-1}}{d{\xi }^2}\right)\right]-\\ \mathrm{Kr}\frac{d{f}_{m-1}}{d\xi },\end{array}$$

(26)

$$\begin{array}{c}{R}_m^{\theta }\left(\xi \right)=\frac{d^2{\theta }_{m-1}}{d{\xi }^2}+Pr\left[-\frac{S}{2}\left(3{\theta }_{m-1}+\xi \frac{d{\theta }_{m-1}}{d\xi }\right)-2{\sum }_{k=0}^{m-1}{\theta }_{m-1-k}\frac{d{f}_k}{d\xi }+{\sum }_{k=0}^{m-1}{f}_{m-1-k}\frac{d{\theta }_k}{d\xi }\right]+\\ E_c\left[{\sum }_{k=0}^{m-1}\frac{d^2{f}_{m-1-k}}{d{\xi }^2}\frac{d^2{f}_k}{d{\xi }^2}+\lambda {\sum }_{k=0}^{m-1}\frac{d^2{f}_{m-1-k}}{d{\xi }^2}{\sum }_{\ell =0}^k\frac{d^2{f}_{k-1}}{d{\xi }^2}\frac{d^2{f}_{\ell }}{d{\xi }^2}\right]+Nt{\sum }_{k=0}^{m-1}\frac{d{\theta }_{m-1-k}}{d\xi }\frac{d{\theta }_k}{d\xi }\end{array}$$

$$\begin{array}{c}+Nb(\frac{d{\theta }_{m-1}}{d\xi }\frac{d{\varphi }_{m-1}}{d\xi }),\end{array}$$

(27)

$$\begin{array}{c}{R}_m^{\varphi }\left(\xi \right)=\frac{d^2{\varphi }_{m-1}}{d{\xi }^2}+Sc\left[{\sum }_{k=0}^{m-1}{f}_{m-1-k}\frac{d{\varphi }_j}{d\xi }-2{\sum }_{k=0}^{m-1}\frac{d{f}_{m-1-k}}{d\xi }{\varphi }_k-\frac{S}{2}\left(3{\varphi }_{m-1}+\xi \frac{d{\varphi }_{m-1}}{d\xi }\right)\right]\end{array}$$

$$\begin{array}{c}+\frac{Nt}{Nb}\frac{d^2{\theta }_{m-1}}{d{\xi }^2},\end{array}$$

(28)

where

$$\begin{array}{c}\hfill {\chi }_m=\begin{array}{c}0,if\wp \le 1\\ 1,if\wp >1.\end{array}\end{array}$$

(29)

4. Representation of Achieved Results in the Form of Figures and Tables

In this section the results achieved by HAM are shown in the form of figures and tables. The convergence of the series given in (21), $f\left(\eta \right)$, $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ entirely depend upon the auxiliary parameters ${\hslash }_f,{\hslash }_{\theta }$ and ${\hslash }_{\varphi }$ which are called ℏ-curves. It is selected in such a way that it controls and converges the series solution. The probable selection of ℏ can be found by plotting ℏ-curves of $f^{\prime \prime }\left(0\right),{\theta }^{\prime }\left(0\right),{\varphi }^{\prime }\left(0\right)$. The valid region of ℏ is $-1.5<{\hslash }_f<-0.5,-1.5<{\hslash }_{\theta }<-0.5$ and $-1.5<{\hslash }_{\varphi }<-0.5$. Here $\eta =\xi$ is chosen.

5. Results and Discussion

In this paper the Liquid Film Flow of Non-newtonian Williamson Fluid over an Unstable Stretching Surface in a Porous Space has been investigated. Thermophoresis and Brownian Motion Effect has been countered to the liquid film flow. The governing equations have been transformed through suitable similarity variables into nonlinear coupled differential equations with physical conditions. The solution of the coupled problem has been obtained by using an analytical approach called, HAM. The solution of the coupled problem and fast convergence of this method is mainly focused. This paper has examined the consequences of governing parameters on the transient velocity, temperature, and concentration profiles. Figure 1 illustrates the geometry of model used. Comparisons are carried out between the obtained results and the results achieved by numerical N-Desolve method for velocity, temperature, and concentration profiles (shown in Figure 2, Figure 3 and Figure 4). The effects of physical parameters appear in the problem, are shown graphically and discussed. Figure 5, Figure 6 and Figure 7, elaborate the behavior of the non-dimensional unsteady parameter S for velocity, temperature and concentration field during fluid motion in a porous medium past over a Unsteady Stretching Sheet. The unsteady parameter S is inversely related to the stretching constant of the velocity field, where as it is directly related to the stretching constants of the temperature and concentration fields. Therefore, when the values of S are increasing the values of the velocity field are decreasing while the values of the temperature and concentration fields increase. Physically, unsteadiness S produce buoyancy forces in the way of the flow field. These forces resist the fluid flow and therefore, the velocity field falls and the temperature distribution as well as the concentration profile is boosted. The effect of the Williamson Fluid constant $\lambda$ on the velocity field is illustrated in Figure 8. The velocity is found to reduce when $\lambda$ is augmented. Because rise in relaxation time causes higher resistance to the fluid flow and as a result reduces the velocity field. Also increase in $\lambda$ increase the temperature due to increase in resistance to the fluid flow as shown in Figure 9. Non-dimensional porosity parameter $Kr$ have direct relation to viscosity parameters. So a rise in non-dimensional porosity parameter reduces fluid motion as explained in Figure 10. Physically, larger values of $Kr$ generate larger open space and create hurdle to flow and as a result the flow field is retarded. The resistance force produces larger values of $Kr$ which increase the temperature and concentration profiles shown in Figure 11 and Figure 12.

The effect of Prandtl number $Pr$ has been shown in the Figure 13, describing that for larger values of $Pr$ decreases the temperature $\theta \left(\eta \right)$. The increase in Prandtl number reduces the thermal boundary layer due to which the temperature decreases.The influence of the Schmidt number $Sc$ is depicted in Figure 14 and Figure 15, showing that temperature and concentration fields decrease when the parameter $Sc$ increases because Schmidt number $Sc$ is reciprocal to the molecular diffusivity. It indicates that as the values of of the Eckert number $Ec$ increase the fluid temperature also increases while its converse effect has been observed in the solute concentration illustrated in Figure 16 and Figure 17. Physically, $Ec$ is connected with the viscous dissipation term in the equation of energy, therefore, larger values of $Ec$ should lead to increase the quantity of heat being produced by the shear forces in the fluid and as a result raises the fluid temperature. Figure 18 and Figure 19, illustrate the effects of Brownian motion parameter $Nb$ on the dimensionless temperature and concentration profiles. The fluid temperature increases as the value of increase of Brownian motion parameter increase while converse effect on the solute concentration. The increase in the value of thermophoresis parameter, increase both temperature and concentration as illustrated in Figure 20 and Figure 21. The fluid flow is also falling when the thickness of film is increased. Larger values of thickness $\beta$ generate the friction force and as a result the flow motion falls down. Increase in the film thickness deliver more fluid in the boundary layer region and cooling effect is produced, which absorbs the heat transfer from the sheet to the fluid and temperature profile drops down. Concentration has vital application in thermal conductivity and chemical reactions. The concentration profile $\varphi \left(\xi \right)$ is reliant on film size $\beta$ and increases with larger values of $\beta$ indicated in Figure 22, Figure 23 and Figure 24. The h-curves of $f^{\prime \prime }\left(0\right),{\theta }^{\prime }\left(0\right)$, and ${\varphi }^{\prime }\left(0\right)$ for the 4th-order HAM approximated solution are elaborated in Figure 25, Figure 26 and Figure 27. Figure 28, Figure 29, Figure 30 and Figure 31 indicate h curves of the residuals for velocity, temperature and concentration profiles respectively.

Table 1. Shows the Nomenclature.

Alphabet	Defined as	Alphabet	Defined as
x	horizontal coordinate $\left(m\right)$	$T_r$	Reference temperature
y	vertical coordinate $\left(m\right)$	$T_0$	initial temperature of the fluid (K)
u	horizontal velocity component (m/s)	T	temperature (K)
v	vertical velocity component (m/s)	$U_w$	Velocity of the stretching sheet
S	Unsteadiness parameter	$T_1$	final temperature of the fluid (K)
$T_w$	Temperature at the sheet	T	temperature (K)
K	thermal diffusivity (m${}^2$)	t	time (s)
S	Unsteadiness parameter	f	Dimensionless Velocity
b	stretching parameter(constant)	k${}^{\prime }$	permeability coefficient of the porosity
$f\left(\xi \right)$	nondimensional variable for velocity	$C_w$	Nanoparticle volume fraction at sheet
$Pr$	Prandtl number	$Ec$	Eckert number
$Kr$	nondimensional porosity parameter	Sc	Schmidt number
$Nb$	Brownian motion parameter	$D_B$	Brownian diffusion coefficient
$C_p$	specific heat at constant pressure (kJ kg${}^{-1}$ K${}^{-1}$)	$Nt$	Thermophoresis parameter
Greek symbols	Defined as	Greek Symbols	Defined as
$\varphi$	Dimensionless nanoparticle volume fraction	$\xi$	Similarity variable
$\nu$	kinematic viscosity of the fluid	${\rho }_f$	Density of base fluid
$\rho$	density (kg m${}^{-3}$)	${\left(\rho c\right)}_p$	Heat capacity of the nanoparticle material
$\mathsf{\Gamma }$	Time constant	$\alpha$	Thermal diffusivity of the base fluid
${\rho }_p$	Nanoparticle mass density	${\left(\rho c\right)}_f$	Heat capacity of the base fluid
$\nu$	Kinematic viscosity of the base fluid	$\lambda$	Williamson fluid constant
$\beta$	non-dimensional film thickness	$\theta$	Dimensionless temperature
$\psi$	non-dimensional stream function	${(}^{\prime })$	differentiation w. r. t. $\xi$

illustrates the symbols used in the manuscript. In

Table 2. HAM, Numerical Solution and their absolute Error are shown for $f\left(\eta \right)$, when h = −0.47, $\lambda =0.2,\mathrm{kr}=0.2,Pr=1,\mathrm{Ec}=0.6,\mathrm{Nb}=0.4,\mathrm{Nt}=0.5,S=0.2,\beta =1$ and $Sc=0.5$.

$\mathit{\eta }$	Numerical Solution for	HAM Solution for	Absolute Error
	$\mathit{f}\left(\mathit{\eta }\right)$	$\mathit{f}\left(\mathit{\eta }\right)$
0	0	0	0
0.1	0.0952761	0.0955184	$2.4\times {10}^{-4}$
0.2	0.182152	0.182958	$8.1\times {10}^{-4}$
0.3	0.262045	0.263561	$1.5\times {10}^{-3}$
0.4	0.336198	0.338462	$2.3\times {10}^{-3}$
0.5	0.405709	0.408698	$2.9\times {10}^{-3}$
0.6	0.471556	0.47522	$3.6\times {10}^{-3}$
0.7	0.534618	0.5389	$4.3\times {10}^{-3}$
0.8	0.595686	0.600535	$4.6\times {10}^{-3}$
0.9	0.655482	0.660862	$5.4\times {10}^{-3}$
1	0.714666	0.720558	$5.8\times {10}^{-3}$

,

Table 3. HAM, Numerical Solution and their absolute Error are elobarated for $\theta \left(\eta \right)$, when h = −0.5, $\lambda =0.2,\mathrm{kr}=0.2,Pr=0.5,\mathrm{Ec}=0.6,\mathrm{Nb}=0.4,\mathrm{Nt}=0.5,S=0.2,\beta =1$ and $Sc=0.7$.

$\mathit{\eta }$	Numerical Solution for	HAM Solution for	Absolute Error
	$\mathit{\theta }\left(\mathit{\eta }\right)$	$\mathit{\theta }\left(\mathit{\eta }\right)$
0	1	1	$1.4\times {10}^{-8}$
0.1	0.907342	0.90737	$2.8\times {10}^{-5}$
0.2	0.82775	0.827596	$1.5\times {10}^{-4}$
0.3	0.759861	0.759482	$3.8\times {10}^{-4}$
0.4	0.702621	0.702067	$5.5\times {10}^{-4}$
0.5	0.655233	0.654586	$6.5\times {10}^{-4}$
0.6	0.617098	0.616438	$6.7\times {10}^{-4}$
0.7	0.587786	0.587169	$6.2\times {10}^{-4}$
0.8	0.567002	0.566451	$5.5\times {10}^{-4}$
0.9	0.554573	0.554078	$4.9\times {10}^{-4}$
1	0.550428	0.549956	$4.7\times {10}^{-4}$

and

Table 4. HAM, Numerical Solution and their absolute Error are depicted for $\varphi \left(\eta \right)$, when h = −0.6, $\lambda =0.2,\mathrm{kr}=0.2,Pr=1,\mathrm{Ec}=0.6,\mathrm{Nb}=1,\mathrm{Nt}=0.1,S=0.2,\beta =1$, and $Sc=0.5$.

$\mathit{\eta }$	Numerical Solution for	HAM Solution for	Absolute Error
	$\mathit{\varphi }\left(\mathit{\eta }\right)$	$\mathit{\varphi }\left(\mathit{\eta }\right)$
0	1	1	$2.9\times {10}^{-9}$
0.1	0.940581	0.941167	$5.9\times {10}^{-4}$
0.2	0.890316	0.8912	$8.8\times {10}^{-4}$
0.3	0.848224	0.849209	$9.8\times {10}^{-4}$
0.4	0.813465	0.81443	$9.6\times {10}^{-4}$
0.5	0.785325	0.786207	$8.8\times {10}^{-4}$
0.6	0.763201	0.76398	$7.8\times {10}^{-4}$
0.7	0.74659	0.747277	$6.9\times {10}^{-4}$
0.8	0.735082	0.735705	$6.2\times {10}^{-4}$
0.9	0.728352	0.728942	$5.9\times {10}^{-4}$
1	0.726151	0.726734	$5.8\times {10}^{-4}$

the results are compared, which are achieved by HAM and Numerical(ND-Solve method) for velocity, temperature and concentration profiles. The residuals gained by HAM are also depicted in

Table 5. Illustrates the residuals achieved by HAM for system of coupled differential equations forming in velocity, temperature and concentration profiles.

$\mathit{\eta }$	Residuals for	Residuals for	Residuals for
	$\mathit{f}\left(\mathit{\eta }\right)$	$\mathit{\theta }\left(\mathit{\eta }\right)$	$\mathit{\varphi }\left(\mathit{\eta }\right)$
0	$-2.0\times {10}^{-1}$	$-3.7\times {10}^{-1}$	$5.1\times {10}^{-2}$
0.1	$-1.2\times {10}^{-1}$	$-7.9\times {10}^{-2}$	$2.4\times {10}^{-2}$
0.2	$-4.9\times {10}^{-2}$	$3.2\times {10}^{-3}$	$2.2\times {10}^{-2}$
0.3	$-4.9\times {10}^{-4}$	$2.5\times {10}^{-2}$	$2.9\times {10}^{-2}$
0.4	$3.3\times {10}^{-2}$	$2.9\times {10}^{-2}$	$3.2\times {10}^{-2}$
0.5	$5.2\times {10}^{-2}$	$2.1\times {10}^{-2}$	$2.3\times {10}^{-2}$
0.6	$6.1\times {10}^{-2}$	$5.9\times {10}^{-3}$	$2.2\times {10}^{-3}$
0.7	$5.9\times {10}^{-2}$	$-1.3\times {10}^{-2}$	$-2.6\times {10}^{-2}$
0.8	$4.9\times {10}^{-2}$	$-3.3\times {10}^{-2}$	$-5.5\times {10}^{-2}$
0.9	$3.3\times {10}^{-2}$	$-4.6\times {10}^{-2}$	$-7.3\times {10}^{-2}$
1	$9.9\times {10}^{-3}$	$-4.9\times {10}^{-2}$	$-8.7\times {10}^{-2}$

.

6. Conclusions

The main conclusion of this endeavor is the study of liquid film in a porous medium considering non-Newtonian Williamson fluid on an unstable stretching surface. The effect of Thermophoresis and Brownian motion has been countered to the liquid film flow. The solutions of the problems have been achieved by using analytical technique, HAM for velocity, temperature and concentration fields respectively. The influences of all parameters included in the problem have been described and the solutions are displayed in the diagrams for checking their effects on velocity, temperature as well as concentration fields. The coupled problem has been solved by using an analytical method HAM. The h curves for the residuals of velocity, temperature and concentration have been sketched.

The main concluded points are derived as,

(1)

Increasing thickness parameter $\beta$ produce the friction force and as a result velocity of the fluid film falls down.

(2)

The larger values of $\beta$ transport more fluid in the boundary layer region and cooling effect is produced which absorbed the heat transfer from the sheet and as a result the temperature reduces.

(3)

The Eckert number $Ec$ is allied with the viscous dissipation term and lead to incrrease the quantity of heat being produced by the shear forces in the fluid. Therefore, larger values of $Ec$ raises the temperature field.

(4)

The larger values of Prandtl number $Pr$ reduces the thermal boundary layer due to which the temperature field reduces.

(5)

Higher values of Porosity parameter $Kr$ generate larger open space and create hurdle to flow and as a result the flow field reduces.