The study of the entropy generation in a thin film flow with variable fluid properties past over a stretching sheet

Abstract

This research inspects the liquid film flow of the nanofluid in a permeable medium with the consequence of thermal radiation over a stretching sheet. The viscidness and thermal conduction of the nanofluid varies with temperature in such a manner that the thermal conductivity considered in direct relation while the viscosity considered inversely proportional to the temperature field. The invariable magnetic field applies vertically to the flow field in the existence of entropy generation. For the above-mentioned nanofluid study, Buongiorno’s model is used. The leading equations are changed into a set of third- and second-order nonlinear coupled differential equations. These nonlinear ordinary differential equations are solved using the optimal approach of homotopy analysis method. The physical appearance of the modelled parameters based on the liquid film thickness is mainly focused. Furthermore, the influence of embedded parameters like variable viscosity parameter $\mathrm{\Lambda },$ Prandtl number $\mathrm{Pr},$ Schmidt number $\mathrm{Sc},$ Brinkman number $\mathrm{Br},$ Brownian motion constraint $\mathrm{Nb},$ thermophoresis constraint $\mathrm{Nt},$ magnetic parameter $M,$ thermal radiation parameter $\mathrm{Nr},$ Reynolds number $\mathrm{Re},$ diffusion coefficient $\lambda ,$ non-dimension temperature variation $\chi$ and non-dimension concentration variation $\mathrm{\Omega }$ is observed on the velocity pitch, temperature gradient and concentration sketch. The consequence of parameters due to entropy generation and Bejan number has also been observed in this work. The important physically quantities of skin friction coefficient, the local Nusselt number and Sherwood number have also been studied. Residual error and optimal values have been calculated for the range of each physical parameter. The present work is compared with the published work and the comparison has been shown physically and numerically.

Introduction

Liquid film flow is the significant natural phenomena. The raised extensive interest and its huge applications in the area of industries, engineering and technology in a last few years clarify its importance in nature. The practical application related to the investigation of thin film flow is an inspiring relationship between fluid mechanics and fluid dynamic. The most popular technological uses can establish in the freezing of electronic procedures, ¹ absorption, ² vaporization, ³ ink-jet printing ⁴ and self-assembly of nanoparticles. ⁵ The key role of liquid film flows is in coating phenomena. To maintain the ideal rate and best performance in fibre and wire coating industry, the liquid film plays a vital role to control friction within the heat and mass transfer. ⁶ The stretching materials like rubber and plastic materials have also a key role in the field of industry and engineering. Keeping in view all of the above applications, Wang ⁷ was the central to explore liquid film flow on a time-dependent stretched surface. The similar research was further extended by Usha and Sridharan. ⁸ The heat transfer consequence of the liquid film investigation on the parallel sheet was considered by Liu and Andersson. ⁹ To solve the problem using numerical methods and also deliberate the limited flow constraints. Aziz et al. ¹⁰ studied the consequence of inner heat production on stream in a thin fluid film at a time depending extending sheet. Recently, Tawade et al. ¹¹ investigated thin liquid flow over a time depended extending surface in the existence of thermal radioactivity and magnetic field. They used Newton–Raphson and Runge-Kutta Fehlberg methods for the solution problem. Andersson ¹² was the pioneer to study the flow of a thin film of non-Newtonian fluids in a time-dependent starching sheet by considering the power law model. Khan et al. ¹³ examined the steady flow of liquid film on an extended/contracted surface. They studied the variable fluid properties of the liquid film using the concept of variable thickness of the film. Qasim et al. ¹⁴ studied the nanofluid thin film on a time-dependent stretching sheet taking Buongiorno’s model. Liu and Andersson have examined the time-dependent flow of a thin film over a stretching surface.

Entropy generation is one of the main factors of fluid flow and has very common applications in physical science. Several researchers have done a lot of work on it in different fields of science and technology. In the literature, vary less work is available in case of thin films nanofluid flow. Bejan^15,16 was the first who inspected the entropy production in essential convective heat transmission. Liu and Lo ¹⁷ investigated entropy production in mixed convection magnetohydrodynamic (MHD) flow in perpendicular channel using numerical techniques. Nano liquids are the suspensions of nanoparticles and base liquid utilized for the upgrade of heat exchange and thermal conduction.

Heat transmission and entropy production in a fully developed mixed convection flow using nanofluids in perpendicular channel universal was examined by nanofluids. Srinivasacharya and Bindu ¹⁸ entropy production in a micropolar fluid flow over an inclined plate. Qing et al. ¹⁹ investigated the entropy production on the MHD Casson nanofluid flow on a permeable stretching/shrinking sheet. Abolbashari et al.^20,21 worked on logical modelling of entropy production for Casson nanofluid flow persuaded by an extending surface. Freidoonimehr and Rahimi ²² investigated the properties of thermophoresis and Brownian motion on nanofluid heat transmission and entropy production. Afridi et al. ²³ examined the entropy generation in the fluid flow over an extending surface with convection. Bhatti et al. ²⁴ studied the numerical reproduction of entropy production with thermal radioactivity on MHD Carreau nanofluid near a shrinking sheet. Abas et al. ²⁵ studied the entropy production on nanofluid flow through a horizontal Riga plate. Khan et al. ²⁶ have examined the impact of the entropy generation considering the flow of the Powell–Eyring fluid between the two rotating discs. In their study, they analysed the magnetic and thermo-diffusion effects.

Chauhan and Olkha ²⁷ have studied the Slip flow and heat transfer of a second grade fluid in a porous medium over a stretching sheet considering power-law model. Liu ²⁸ studied flow and heat transmission of an electric conducted fluid of second grade in a permeable surface on an extending sheet having a transverse magnetic field. Siddeshwar and Mahabaleshwar ²⁹ studied consequence of radioactivity and heat transfer on MHD flow of visco-elastic fluid and heat transfer on a stretched surface.

The heat transfer and magnetic effect are mainly using in the different technologies and industrial fields. In the recent literature, Soomro et al. ³⁰ and Haq et al.^31–33 examined the nanofluid flow for the rapid heat transfer using the different geometries.

In the field of science and engineering, most of mathematical problems are multifaceted in their nature and the exact solution is almost very difficult or even not possible. So for the solution of such problems, numerical and analytical methods are used to find the approximate solution. One of important and popular technique for the solution of such type problems is the homotopy analysis method (HAM). It is a substitute method and its main advantage is applying to the nonlinear differential equations without discretization and linearization. Liao^34–39 was the first one to investigate this scheme for the solution of nonlinear equations and generally proved that this method is rapidly convergent to the approximated solutions. Also, this method provides series solutions in the form of functions of a single variable. Solution with this method is important because it involves all the physical parameters of the problem and we can easily discuss its behaviour. Due to its fast convergence, many researchers Abbasbandy et al.,^40–42 Rashidi and Pour, ⁴³ Waris et al. ⁴⁴ and Noor et al. ⁴⁵ used this technique to solve highly nonlinear and coupled equations.

Physical parameters in dimensionless form are normally happening in the variety of problems. How to control them by defining their range of validity is a big issue. In this article, a mathematical approach is offered to identify the correct range of physical parameters implementing the analytic approximate HAM as used by Turkyilmazoglu.^46–48

According to the above important discussion, the aim of the present work is to extend the idea of Qasim et al. ¹⁴ for further study by including the entropy generation terminology in the nanofilm flow and using the clue of varying viscosity and varying thermal conductivity terms. The study is also enlarged by the inclusion of porous media and thermal conductivity effects. For simplicity, the modelled equations have been considered in steady state. ¹³

Problem formulation

The nanofluid stream over a contracting/extending sheet is pondered in such a form, to the point that its viscosity and thermal conductivity are fluctuating while its density stays consistent. The Cartesian directions are established in such a way that ox is identical and y is vertical to the sheet. The constant sheet at $y=0$ moves with velocity of magnitude $U_w=\mathrm{bx},$ such that for $b>0$ the sheet is stretched and for $b<0$ the sheet is contracted as characterized in Khan et al., ¹³ Qasim et al. ¹⁴ and Liao ³⁹

The temperature field in the flow is defined as $T_w=T_0-T_{\mathrm{ref}}(b{x}^2/2\upsilon )$ , on the surface are assumed to vary with the distance $x$ from the slit, in which $T_0$ is the temperature at the slit and $T_{\mathrm{ref}}$ represents the constant reference temperature such that $0\le T_{\mathrm{ref}}\le T_0$ . Similarly, the concentration distribution $C_w=C_0-C_{\mathrm{ref}}(b{x}^2/2\upsilon )$ is fluctuating with distance $x$ from the slit. The nanofluid concentration at the surface of the slit is demonstrated by $C_0$ and $C_{\mathrm{ref}}$ signify the constant reference concentration. Using the Reynolds model for the variable fluid properties as mentioned in Waris et al., ⁴⁴ Noor et al. ⁴⁵ and ignoring the body forces in the flow field. Taking these assumptions, the continuity equation, the fundamental boundary principal equation, heat transmitting gradient and concentration filed can be identified as

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

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{1}{\rho }\frac{\partial }{\partial y}(\mu (T)\frac{\partial u}{\partial y})-\frac{\sigma {B_0}^{2}u}{\rho }$$

(2)

$$\begin{array}{l}\rho c_p\left[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right]=\frac{\partial }{\partial y}\left[k(T)\frac{\partial T}{\partial y}\right]+\mu (T){\left(\frac{\partial u}{\partial y}\right)}^2\\ +\tau \left[D_B\left(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}\right)+\frac{D_T}{T_h}{\left(\frac{\partial T}{\partial y}\right)}^2\right]-\frac{\partial q_r}{\partial y}+\sigma {B_0}^2{u}^2\end{array}$$

(3)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_h}\frac{{\partial }^{2}T}{\partial y^2}$$

(4)

The appropriate boundary condition for the flow configuration is given as^13,14,44,45

$$u=U_w;v=V_w;T=T_w;C=C_w\mathrm{at}y=0$$

(5)

$$\frac{\partial u}{\partial y}=\frac{\partial T}{\partial y}=\frac{\partial C}{\partial y}=0,v=u\frac{\mathrm{dh}}{\mathrm{dx}},\mathrm{at}y=h$$

(6)

where $u\mathrm{and}v$ are the velocity components alongside $x-\mathrm{and}y-\mathrm{axis}$ , respectively; the temperature gradient within the border, coat is denoted by $T$ while $T_h$ shows the fluid temperature in the free surface and similar consideration $C,C_h$ are chosen for the concentration profile.

According to the Reynolds model, the dimensionless temperature-dependent viscosity can be expressed as $\mu ={\mu }_0{e}^{-m(T-T_h)}$ where ${\mu }_0$ specifies the reference temperature viscidness and the strength of the dependency power between $\mu$ and $T$ is represented by $m$ and $\mathrm{\Lambda }=m(T_w-T_h)$ . The variable thermal conduction is denoted as $K=K_0{e}^{c(T-T_h)},$ such that $\epsilon =c(T_w-T_h).$

Here $\rho$ is the point density of the fluid, the specific heat is identified by $c_p$ , $D_B$ designates Brownian diffusion coefficient, $\tau =(\rho c{)}_p/(\rho c{)}_f$ represents the relation of nanoparticle and base fluid heat capability, the thermal diffusion ratio is denoted by $k_t$ , $q_r=-(4\sigma /3k)(\partial T^4/\partial y)$ shows the radiative heat inconsistency in which $\sigma$ represents the Stefan–Boltzmann limit and $k$ specifies the mean absorption influence. The conversion of the temperature is measured in such a technique that $T^4$ is linearly associated with temperature. Increasing $T^4$ by Taylor series nearly $T_h$ and omitting upper terms, we develop $T^4\cong 4{\mathrm{TT}}_h^3-3T_h^4$ and finally we found

$$\frac{\partial q_r}{\partial y}=-\frac{16\sigma T_h^3}{3k}\frac{{\partial }^{2}T}{\partial y^2}$$

(7)

Introducing the succeeding similarity transformations

$$\begin{array}{c}\psi =x\sqrt{\upsilon b}f\left(\eta \right),u=\frac{\partial \psi }{\partial y}=\mathrm{bxf}\prime \left(\eta \right),\hfill \\ v=-\frac{\partial \psi }{\partial x}=-\sqrt{\upsilon b}f\left(\eta \right),\eta =\sqrt{\frac{b}{\upsilon }}y\hfill \\ T(x,y)=T_0-T_{\mathrm{ref}}\left(\frac{b{x}^2}{2\upsilon }\right)\mathrm{\Theta }(\eta ),\hfill \\ C(x,y)=C_0-C_{\mathrm{ref}}\left(\frac{b{x}^2}{2\upsilon }\right)\varphi (\eta ),M=\frac{\sigma {B_0}^2}{\rho b},S=\sqrt{\frac{{V_0}^2}{b\upsilon }}\hfill \\ \mathrm{Nt}=\frac{\tau D_T(T_w-T_h)}{\upsilon T_h},\mathrm{Nb}=\frac{\tau D_B(C_w-C_h)}{\upsilon },\hfill \\ \mathrm{Sc}=\frac{\upsilon }{D_B},\mathrm{Ec}=\frac{{U_w}^2}{c_p(T_w-T_h)},\mathrm{Pr}=\frac{\upsilon }{\alpha },\mathrm{Nr}=\frac{16{\sigma }^{*}{T_h}^3}{3k^{*}k}\hfill \end{array}$$

(8)

in which prime identifies the derivative with respect to $\eta$ and $\psi$ is the stream function, $h$ identifies the film width and $\upsilon ={\mu }_0/\rho$ indicates the kinematics viscosity. Using the similarity alteration equations (7) and (8) into equations (1)–(6) accomplishes the continuity equation and the remaining equalities change to the scheme of nonlinear differential equations as

$$f″\prime +(1+\mathrm{\Lambda }\mathrm{\Theta })[\mathrm{ff}″-{(f\prime )}^2-\mathrm{Mf}\prime ]=0$$

(9)

$$\begin{array}{c}\mathrm{Pr}(1+\mathrm{\Lambda }\mathrm{\Theta })[f\mathrm{\Theta }\prime -2f\prime \mathrm{\Theta }+\mathrm{Nt}{(\mathrm{\Theta }\prime )}^2+\mathrm{Nb}\varphi \prime \mathrm{\Theta }\prime +M{(f\prime )}^2]\hfill \\ +\mathrm{Brf}″+(1+\epsilon \mathrm{\Theta }+\mathrm{Nr})(1+\mathrm{\Lambda }\mathrm{\Theta })\mathrm{\Theta }″=0\hfill \end{array}$$

(10)

$$\varphi ″+\mathrm{Sc}[f\varphi \prime -2f\prime \varphi ]+\frac{\mathrm{Nt}}{\mathrm{Nb}}\mathrm{\Theta }″=0$$

(11)

The non-dimensional thickness of the liquid film $\beta$ is defined as $\beta =(b/\upsilon {)}^{1/2}h$ . The thickness $\beta$ of the liquid film is the unknown variable determine the boundary of the problem.

The boundary constraints of the problem in the dimensionless form are as follows

$$\begin{array}{c}f(0)=S,f\prime (0)=1,\mathrm{\Theta }(0)=1,\varphi (0)=1\hfill \\ f(\beta )=f″(\beta )=0,\mathrm{\Theta }\prime (\beta )=\varphi \prime (\beta )=0\hfill \end{array}$$

(12)

The physical constraints after generalization: where $\mathrm{Pr}=\rho \upsilon c_p/k$ designates the Prandtl number, $\mathrm{Sc}=\upsilon /D_B$ is used for Schmidt number, $D_B$ illustrates the Brownian diffusion quantity, $\mathrm{Br}=\mathrm{Pr}\mathrm{Ec}={\mu }_0{U}_W^2/k(T_W-T_h)$ denotes the product of Prandtl and Eckert numbers and known as Brinkman number, $\mathrm{Nb}$ directs the Brownian motion constraint, $\mathrm{Nt}$ signifies the thermophoresis factor, $M$ is the magnetic parameter, $\mathrm{Nr}$ indicates the thermal radiation parameter and S presents the steadiness parameter.

The attentive physical quantities are the skin friction coefficient $C_f$ , the local Nusselt number $N{u}_x$ and the local Sherwood number $S{h}_x$ , which are defined as

$$C_{\mathrm{fx}}=\frac{{\tau }_w}{\frac{1}{2}\rho U_w^2},N{u}_x=\frac{q_{w}x}{k(T_w-T_0)},S{h}_x=\frac{q_{m}x}{D_B(C_w-C_0)}$$

where

$${\tau }_w=\mu {\left(\frac{\partial u}{\partial y}\right)}_{y=0},q_w=-k{\left(\frac{\partial T}{\partial y}\right)}_{y=0},q_m=-D_B{\left(\frac{\partial C}{\partial y}\right)}_{y=0}$$

are the shear stress, heat and mass fluxes at the surface, respectively. Using the associated expressions for dimensionless skin friction coefficient $C_f$ , Nusselt number and Sherwood number are defined as

$${\mathrm{Re}}_x^{1/2}{C}_{\mathrm{fx}}=-f″(0),{\mathrm{Re}}_x^{-1/2}N{u}_x=-\theta \prime (0),{\mathrm{Re}}_x^{1/2}S{h}_x=-\varphi \prime (0)$$

where

$$R{e}_x=\frac{U_{w}x}{\upsilon }$$

is the local Reynolds number based on the stretching velocity.

Entropy generation analysis

The volumetric entropy production of viscid fluid is defined as

$$\begin{array}{c}\multicolumn{1}{c}{{S″\prime }_{\mathrm{gen}}=\frac{K(T)}{T_0^2}[{\left(\frac{\partial T}{\partial y}\right)}^2+\frac{16\sigma T_{\infty }^3}{3k}{\left(\frac{\partial T}{\partial y}\right)}^2]+\frac{\mu (T)}{T_0}{\left(\frac{\partial u}{\partial y}\right)}^2}\\ \multicolumn{1}{c}{+\frac{\mathrm{RD}}{C_0}{\left(\frac{\partial C}{\partial y}\right)}^2+\frac{\mathrm{RD}}{T_0}(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{\partial C}{\partial x}\frac{\partial T}{\partial x})+\frac{\sigma B_0^2{u}^2}{T_0}}\end{array}$$

(13)

The mention equation specifies, entropy production has two features, first is the conduction consequence which is also named heat transmission irreversibility (HTI) and the second one fluid friction irreversibility (FFI) and diffusive irreversibility (DI). Whereas the last two terms are due to the magnetic and porosity terms. The features entropy production can be expressed as

$$S″{\prime }_0=\frac{K_0{(\mathrm{\Delta }T)}^2}{L^2{T}_0^2}$$

(14)

The entropy generation number $N_G=S″{\prime }_{\mathrm{gen}}/S″{\prime }_0$ is used to demonstrate the non-dimensional system of entropy generation rate. The $N_G$ is achieved from the ratio of the actual entropy generation rate in equation (13) and characteristic entropy generation rate in equation (14)

$$\begin{array}{l}{N}_G=R_e\left(1+\epsilon \theta +Nr\right){\left({\theta }^{\prime }\right)}^2+\frac{R_{e}Br}{\mathrm{\Omega }(1+\mathrm{\Lambda }\theta )}{\left(f^{″}\right)}^2\\ +\frac{R_{e}Br}{\mathrm{\Omega }}M{\left(f^{\prime }\right)}^2+R_e\lambda {\left(\frac{\chi }{\mathrm{\Omega }}\right)}^2{\left({\varphi }^{\prime }\right)}^2+R_e\lambda \left(\frac{\chi }{\mathrm{\Omega }}\right){\theta }^{\prime }{\varphi }^{\prime }\end{array}$$

(15)

where $\mathrm{Br}$ denotes the Brinkman quantity, $\chi$ and $\mathrm{\Omega }$ are the non-dimensional temperature gradient and concentration variance, respectively, and $\lambda$ represents the diffusion quantity. These facts are specified in the subsequent form

$$R_e=\frac{b{L}^2}{\upsilon },\mathrm{\Omega }=\frac{\mathrm{\Delta }T}{T_0},\chi =\frac{\mathrm{\Delta }C}{C_0},\lambda =\frac{\mathrm{RD}{C}_0}{k}$$

(16)

The relative importance of the sources of entropy generation is an important factor for the engineers and this important measurement is the Bejan number

$$\mathrm{Be}=\frac{\frac{k(T)}{T^2}[{\left(\frac{\partial T}{\partial y}\right)}^2+\frac{16\sigma T_{\infty }^3}{3k}{\left(\frac{\partial T}{\partial y}\right)}^2]}{\frac{\mu (T)}{T}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{1}{T}\left(\sigma B_0^2{u}^2\right)}$$

(17)

The similarity transformation alters the Bejan number $\mathrm{Be}$ in the form

$$\mathrm{Be}=\frac{(1+\epsilon \theta +\mathrm{Nr}){(\theta \prime )}^2}{\frac{\mathrm{Br}}{\mathrm{\Omega }(1+\mathrm{\Lambda }\theta )}{(f″)}^2+\frac{\mathrm{Br}}{\mathrm{\Omega }}M{(f\prime )}^2}$$

(18)

Solution by HAM

Equations (8)–(10) and the resulting boundary restrain equation (11) are estimated by HAM. The solutions surrounded the auxiliary constraints $\hslash$ which control and adjustment to converge of the results.

The primary estimates are selected for the given problem similar as in Khan et al. ¹³

$$\begin{array}{l}{f}_0(\eta )=S+\eta +\left(\frac{{\eta }^3}{\beta }-3{\eta }^2\right)\left(\frac{S}{2{\beta }^2}+\frac{1}{2}\beta \right),\\ {\theta }_0(\eta )=\frac{{\eta }^2}{2}-\beta \eta +1,{\varphi }_0(\eta )=\frac{{\eta }^2}{2}-\beta \eta +1\end{array}$$

(19)

The linear operators elected are ${\stackrel{⌢}{L}}_f,{\stackrel{⌢}{L}}_{\theta },{\stackrel{⌢}{L}}_{\varphi }$

$${\stackrel{⌢}{L}}_f=\frac{{\partial }^4}{\partial {\eta }^4},{\stackrel{⌢}{L}}_{\theta }=\frac{{\partial }^2}{\partial {\eta }^2},{\stackrel{⌢}{L}}_{\varphi }=\frac{{\partial }^2}{\partial {\eta }^2}$$

(20)

These linear operators have the consequent properties

$$\begin{array}{l}{\stackrel{⌢}{L}}_f(c_1+c_2\eta +c_3{\eta }^2+c_4{\eta }^4)=0,\\ {\stackrel{⌢}{L}}_{\theta }(c_5+c_6\eta )=0,{\stackrel{⌢}{L}}_{\varphi }(c_7+c_8\eta )=0\end{array}$$

(21)

Wherever $c_k(k=1\to 8)$ , the quantities include in general solution. The nonlinear operators $N_f,N_{\theta },N_{\varphi }$ are taken as

$$\begin{array}{c}{N}_f[f(\eta ;p),\theta (\eta ;p)]=\frac{{\delta }^{3}f(\eta ;p)}{\delta {\eta }^3}\hfill \\ +(1+\mathrm{\Lambda }\theta (\eta ;p))\hfill \\ [f(\eta ;p)\frac{{\delta }^{2}f(\eta ;p)}{\delta {\eta }^2}-{\left(\frac{\delta f(\eta ;p)}{\delta \eta }\right)}^2-M\frac{\delta f(\eta ;p)}{\delta \eta }]\hfill \end{array}$$

(22)

$$\begin{array}{c}{N}_{\theta }[f(\eta ;p),\theta (\eta ;p),\varphi (\eta ;p)]\hfill \\ =(1+\epsilon \theta (\eta ;p)+\mathrm{Nr})(1+\mathrm{\Lambda }\theta (\eta ;p))\frac{{\delta }^2\theta (\eta ;p)}{\delta {\eta }^2}\hfill \\ +\mathrm{Pr}(1+\mathrm{\Lambda }\theta (\eta ;p))\hfill \\ \left[\begin{array}{c}f(\eta ;p)\frac{\delta \theta (\eta ;p)}{\delta \eta }-2\theta (\eta ;p)\frac{\delta f(\eta ;p)}{\delta \eta }+\mathrm{Nt}{\left(\frac{\delta \theta (\eta ;p)}{\delta \eta }\right)}^2\hfill \\ \multicolumn{1}{c}{+\mathrm{Nb}\frac{\delta \theta (\eta ;p)}{\delta \eta }\frac{\delta \varphi (\eta ;p)}{\delta \eta }+M{\left(\frac{\delta f(\eta ;p)}{\delta \eta }\right)}^2}\end{array}\right]\hfill \\ +\mathrm{Br}\frac{{\delta }^{2}f(\eta ;p)}{\delta {\eta }^2}=0\hfill \end{array}$$

(23)

$$\begin{array}{c}{N}_{\varphi }[f(\eta ;p),\theta (\eta ;p),\varphi (\eta ;p)]=\frac{{\delta }^2\varphi (\eta ;p)}{\delta {\eta }^2}\hfill \\ +\mathrm{Sc}[f(\eta ;p)\frac{\delta \varphi (\eta ;p)}{\delta \eta }-2\varphi (\eta ;p)\frac{\delta f(\eta ;p)}{\delta \eta }]\hfill \\ +\frac{\mathrm{Nt}}{\mathrm{Nb}}\frac{{\delta }^2\theta (\eta ;p)}{\delta {\eta }^2}\hfill \end{array}$$

(24)

The fundamental solution process by HAM is clear in the literatures^29–40 the 0th order scheme form equations (9)–(11) as

$$(1-p)L_f[f(\eta ;p)-f_0(\eta )]=p{\hslash }_f{N}_f[f(\eta ;p),\theta (\eta ;p)]$$

(25)

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

(26)

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

(27)

The correspondent boundary constraints are

$$\begin{array}{c}{f(\eta ;p)|}_{\eta =0}=S,{\frac{\delta f(\eta ;p)}{\delta \eta }|}_{\eta =0}=-1,{\frac{{\delta }^{2}f(\eta ;p)}{\delta {\eta }^2}|}_{\eta =\beta }=0\hfill \\ {\theta (\eta ;p)|}_{\eta =0}=1,{\frac{\delta \theta (\eta ;p)}{\delta \eta }|}_{\eta =\beta }=0,\hfill \\ {\varphi (\eta ;p)|}_{\eta =0}=1,{\frac{\delta \varphi (\eta ;p)}{\delta \eta }|}_{\eta =\beta }=0\hfill \end{array}$$

(28)

Wherever $p\in [0,1]$ is the embedding constraint, ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi }$ represent to adjust for solution convergence. When $p=0\mathrm{and}p=1$ , we have

$$f(\eta ;1)=f(\eta ),\theta (\eta ;1)=\theta (\eta ),\varphi (\eta ;1)=\varphi (\eta )$$

(29)

Increasing $f(\eta ;p),\theta (\eta ;p),\varphi (\eta ;p)$ in Taylor’s series approximately $p=0$

$$\begin{array}{c}f(\eta ;p)=f_0(\eta )+\sum _{m=1}^{\infty }{f}_m(\eta )p^m\hfill \\ \theta (\eta ;p)={\theta }_0(\eta )+\sum _{m=1}^{\infty }{\theta }_m(\eta )p^m\hfill \\ \varphi (\eta ;p)={\varphi }_0(\eta )+\sum _{m=1}^{\infty }{\varphi }_m(\eta )p^m\hfill \end{array}$$

(30)

where

$$\begin{array}{l}{f}_m(\eta )={\frac{1}{m!}\frac{\delta f(\eta ;p)}{\delta \eta }|}_{p=0},{\theta }_m(\eta )={\frac{1}{m!}\frac{\delta \theta (\eta ;p)}{\delta \eta }|}_{p=0},\\ {\varphi }_m(\eta )={\frac{1}{m!}\frac{\delta \varphi (\eta ;p)}{\delta \eta }|}_{p=0}\end{array}$$

(31)

The secondary restrictions ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi }$ are chosen in a manner that the series equations (25)–(27) converge at $p=1$ ; switching $p=1$ in equations (25)–(27), we get

$$\begin{array}{c}f(\eta )=f_0(\eta )+\sum _{m=1}^{\infty }{f}_m(\eta )\hfill \\ \theta (\eta )={\theta }_0(\eta )+\sum _{m=1}^{\infty }{\theta }_m(\eta )\hfill \\ \varphi (\eta )={\varphi }_0(\eta )+\sum _{m=1}^{\infty }{\varphi }_m(\eta )\hfill \end{array}$$

(32)

The mth order scheme satisfies the subsequent

$$\begin{array}{c}{L}_f[f_m(\eta )-{\chi }_m{f}_{m-1}(\eta )]={\hslash }_f{R}_m^f(\eta )\hfill \\ L_{\theta }[{\theta }_m(\eta )-{\chi }_m{\theta }_{m-1}(\eta )]={\hslash }_{\theta }{R}_m^{\theta }(\eta )\hfill \\ L_{\varphi }[{\varphi }_m(\eta )-{\chi }_m{\varphi }_{m-1}(\eta )]={\hslash }_{\varphi }{R}_m^{\varphi }(\eta )\hfill \end{array}$$

(33)

The consistent boundary situations are

$$\begin{array}{c}{f}_m(0)={f\prime }_m(0)={\theta }_m(0)={\varphi }_m(0)=0\hfill \\ {f″}_m(\beta )={\theta }_m(\beta )={\varphi }_m(\beta )=0\hfill \end{array}$$

(34)

Here

$$\begin{array}{c}{R}_m^f(\eta )=\frac{{\delta }^3{f}_{m-1}}{\delta {\eta }^3}\hfill \\ +[\sum _{k=0}^{m-1}{f}_{m-1-k}\frac{{\delta }^2{f}_k}{\delta {\eta }^2}-\sum _{k=0}^{m-1}\frac{\delta f_{m-1-k}}{\delta \eta }\frac{\delta f_k}{\delta \eta }-M\frac{\delta f_{m-1}}{\delta \eta }]\hfill \\ +\mathrm{\Lambda }[\sum _{k=0}^{m-1}{\theta }_{m-1-k}[\sum _{j=0}^k{f}_{k-j}\frac{{\delta }^2{f}_j}{\delta {\eta }^2}-\sum _{j=0}^k\frac{\delta f_{k-j}}{\delta \eta }\frac{\delta f_j}{\delta \eta }]-M\sum _{k=0}^{m-1}{\theta }_{m-1-k}\frac{\delta f_k}{\delta \eta }]\hfill \end{array}$$

(35)

$$\begin{array}{cc}\multicolumn{1}{c}{R_m^{\theta }(\eta )=}& (1+\mathrm{Nr})\frac{{\delta }^2{\theta }_{m-1}}{\delta {\eta }^2}+(\epsilon +\mathrm{\Lambda }(1+\mathrm{Nr}))\sum _{k=0}^{m-1}{\theta }_{m-1-k}\frac{{\delta }^2{\theta }_k}{\delta {\eta }^2}+\epsilon \mathrm{\Lambda }\sum _{k=0}^{m-1}{\theta }_{m-1-k}\sum _{j=0}^k{\theta }_{k-j}\frac{{\delta }^2{\theta }_j}{\delta {\eta }^2}\hfill \\ \multicolumn{1}{c}{}& +\mathrm{Br}\frac{{\delta }^2{f}_{m-1}}{\delta {\eta }^2}+\mathrm{Pr}\left[\begin{array}{c}\sum _{k=0}^{m-1}{f}_{m-1-k}\frac{\delta {\theta }_k}{\delta \eta }-2\sum _{k=0}^{m-1}\frac{\delta f_{m-1-k}}{\delta \eta }{\theta }_k+\mathrm{Nt}\sum _{k=0}^{m-1}\frac{\delta {\theta }_{m-1-k}}{\delta \eta }\frac{\delta {\theta }_k}{\delta \eta }\hfill \\ \multicolumn{1}{c}{+\mathrm{Nb}\sum _{k=0}^{m-1}\frac{\delta {\theta }_{m-1-k}}{\delta \eta }\frac{\delta {\varphi }_k}{\delta \eta }+M\sum _{k=0}^{m-1}\frac{\delta f_{m-1-k}}{\delta \eta }\frac{\delta f_k}{\delta \eta }}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& +\mathrm{\Lambda }\mathrm{Pr}\sum _{k=0}^{m-1}{\theta }_{m-1-k}\left[\begin{array}{c}\sum _{j=0}^k{f}_{k-j}\frac{\delta {\theta }_j}{\delta \eta }-2\sum _{j=0}^k\frac{\delta f_{k-j}}{\delta \eta }{\theta }_j+\mathrm{Nt}\sum _{j=0}^k\frac{\delta {\theta }_{k-j}}{\delta \eta }\frac{\delta {\theta }_j}{\delta \eta }\hfill \\ \multicolumn{1}{c}{+\mathrm{Nb}\sum _{j=0}^k\frac{\delta {\theta }_{k-j}}{\delta \eta }\frac{\delta {\varphi }_j}{\delta \eta }+M\sum _{j=0}^k\frac{\delta f_{k-j}}{\delta \eta }\frac{\delta f_j}{\delta \eta }}\end{array}\right]\hfill \end{array}$$

(36)

$$\begin{array}{l}{R}_m^{\varphi }(\zeta )=\frac{{\delta }^2{\varphi }_{m-1}}{\delta {\eta }^2}\\ +Sc\left[{\displaystyle \sum _{k=0}^{m-1}{f}_{m-1-k}\frac{\delta {\varphi }_k}{\delta \eta }}-2{\displaystyle \sum _{k=0}^{m-1}\frac{\delta f_{m-1-k}}{\delta \eta }{\varphi }_k}\right]+\frac{Nt}{Nb}\frac{{\delta }^2{\theta }_{m-1}}{\delta {\eta }^2}\end{array}$$

(37)

where

$${\chi }_m=\{\begin{array}{c}0,\mathrm{if}p\le 1\hfill \\ \multicolumn{1}{c}{1,\mathrm{if}p>1}\end{array}$$

(38)

HAM solution convergence

The convergence of equations (23)–(25) exclusively be influenced by the auxiliary limitations ${\hslash }_f,{\hslash }_{\mathrm{\Theta }}$ and ${\hslash }_{\varphi }$ . It is a selection in a manner that controls and converges the series result. The probability section of $\hslash$ curves of $f″(0),\vartheta \prime (0),\mathrm{and}\varphi \prime (0)$ for $30\mathrm{th}$ order estimated HAM result. The operative section of $\hslash$ is $-1.7<{\hslash }_f<-0.4,-1.4<{\hslash }_{\theta }<-0.4\mathrm{p}-1.4<{\hslash }_{\phi }<-0.4$ .

Results and discussion

The fluid of viscous flow behaviour is deliberated by describing the whole parameters. Figure 1 explains the geometry of the problem. Figures 2–4 show the $h$ curves of $f(\eta ),\theta (\eta )\mathrm{and}\varphi (\eta )$ for the 20th order estimated HAM solution. The behaviour of the dimensionless film thickness $\beta$ during fluid motion can be perceived from Figure 5. Rise in the parameter $\beta$ reduces the fluid motion because it is the decreasing function of the velocity and thickness of the liquid film. Physically, the viscous forces growing with larger values of $\beta$ cause the fluid motion and as a result, fluid velocity falls down.

Figure 1.

Geometry of the problem.

Figure 2.

The graph of ${\hslash }_f$ curve for $f″(0)$ , when $\mathrm{Pr}=1,\mathrm{\Lambda }=0.001\epsilon =S=\beta =M=\mathrm{Nr}=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.01$ .

Figure 3.

Show the $\hslash$ curve for $\theta \prime (0),$ when $\mathrm{Pr}=1,\mathrm{\Lambda }=0.001\epsilon =S=\beta =M=\mathrm{Nr}=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.01$ .

Figure 4.

The graph of $\hslash$ curve for $\varphi \prime (0),$ when $\mathrm{Pr}=1,\mathrm{\Lambda }=0.001\epsilon =S=\beta =M=\mathrm{Nr}=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.01$ .

Figure 5.

Deviations in velocity, pitch $f(\eta )$ for dissimilar numbers of $\beta$ , when $\mathrm{Pr}=10,\mathrm{\Lambda }=0.2,\epsilon =\mathrm{S}=1,M=0.3,\mathrm{Nr}=0.1,\mathrm{Ec}=1,\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.5$ .

The characteristics of magnetic parameter $M$ on the velocity field are shown in Figure 6. The velocity distribution inversely varies with magnetic factor $M$ . Increasing magnetic parameter $M$ decreases velocity field when it is close to the sheet. This is because of the fact that the rise in the magnetic constraint $M$ develops the resistive force to the flow is called the Lorentz force. This force has a tendency to slow the velocity of the flow in the boundary layer, and another force called Carioles force shows reverse effect on the velocity. Figure 7 describes the characteristics of magnetic parameter $M$ on temperature distribution which are increasing for higher values and decreases for the small values of $M$ . Basically, $M$ very dependent on the Lorentz force, where this force becomes weaker for weak values of magnetic parameter and this weakness leads to decrease the temperature distribution. The characteristics of the thermal radiation parameter $\mathrm{Nr}$ allied to temperature distribution have been deliberated in Figure 8. The temperature distribution through the boundary layer growths with the rising number of $\mathrm{Nr}$ . The physics behind this purpose is that the improved radiation growths which improves the amount of heat energy transfers to the fluid and leads to increase the temperature distribution.

Figure 6.

Deviations in velocity, pitch $f(\eta )$ for unalike numbers of $M$ , where $\mathrm{Pr}=10,\mathrm{\Lambda }=\epsilon =1,\beta =S=1,\mathrm{kr}=\mathrm{Nr}=0.5,\mathrm{Ec}=0.6,\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.5$ .

Figure 7.

Deviations in the temperature field $\mathrm{\Theta }(\eta )$ for dissimilar numbers of $M$ , where $\mathrm{Pr}=\mathrm{\Lambda }=\epsilon =\beta =S=1,\mathrm{Nr}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.5,\mathrm{Ec}=0.6$ .

Figure 8.

Deviations in the temperature field $\mathrm{\Theta }(\eta )$ for dissimilar numbers of $\mathrm{Nr}$ $\epsilon =1,\beta =S=1,M=0.1,\mathrm{Ec}=0.6,\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=\mathrm{Pr}=,\mathrm{\Lambda }=0.5$ .

The effect of $\mathrm{Pr}$ on temperature distribution is shown in Figure 9. Temperature distribution varies inversely with $\mathrm{Pr}$ . It is clear that temperature field and the thickness of the thermal boundary layer are reduced with high values of $\mathrm{Pr}$ . Physically, the fluids having a small number of $\mathrm{Pr}$ has larger thermal diffusivity and this effect is opposite for higher Prandtl number. Due to this fact, large $\mathrm{Pr}$ causes the thermal boundary layer to decrease. The effect is even more distinct for the small number of $\mathrm{Pr}$ ; subsequently, the thermal boundary layer thickness is relatively large.

Figure 9.

Deviations in the temperature field $\mathrm{\Theta }(\eta )$ for dissimilar numbers of $\mathrm{Pr}$ , where $\mathrm{\Lambda }=\epsilon =\beta =S=1,M=0.5,\mathrm{Ec}=0.6,\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=\mathrm{Nr}=0.5$ .

Figure 10 represents the consequence of viscosity factor $\mathrm{\Lambda }$ on the velocity profile. It is observed that with increasing viscosity parameter, velocity field tends to reduce, because for larger values of viscosity parameter, the viscous forces become weak and as a result inertial forces develop stronger. Generally, viscosity has an inverse relation with the velocity. Fluid of thick film like honey, oil, polymers, and so on has high viscosity but are slow in motion as compared to the thin liquids. The effects of viscosity constraint $\mathrm{\Lambda }$ over the temperature field is exposed in Figure 11, which gives the same result discussed for Figure 10, that is, viscosity decreases with high temperature. By utilizing temperature, the forces of attractions between atoms and molecules of the fluid are vanishing and the physical properties of the fluid are also changed. The effect of $\mathrm{Nb}$ on concentration profile is shown in Figure 12. The increasing values of $\mathrm{Nb}$ decrease the fluid concentration. This is due to that increasing $\mathrm{Nb}$ rises the kinetic energy of the nanoparticles and as a result the concentration profile declines or one can say that rise of Brownian motion parameter $\mathrm{Nb}$ concentration distribution fall down. The performance of the Brinkman number $\mathrm{Br}$ related with temperature profile is shown in Figure 13. Since the viscous dissipation term is associated with Brinkman number $\mathrm{Br}$ , in the equation of energy, therefore, greater values of the $\mathrm{Br}$ automatically increase the temperature profile. Thermophoresis parameter $\mathrm{Nt}$ of temperature distribution and concentration field is shown in Figures 14 and 15. Both ( $\mathrm{Nt}$ and $\mathrm{Nb}$ ) increase the temperature field when to increase these parameters. The same effects of $\mathrm{Nt}$ concentration field are shown in Figure 15. The influence of the parameter $\mathrm{Sc}$ is shown in Figure 16. Schmidt number $\mathrm{Sc}$ is a dimensionless number and it is the ratio of momentum diffusivity and mass diffusivity. So for large values of $\mathrm{Sc}$ , the concentration increases and decreases for small values.

Figure 10.

Deviations in the velocity ground $f(\eta )$ for dissimilar numbers of $\mathrm{\Lambda }$ , where $\mathrm{Pr}=10,\epsilon =1,\beta =S=1,M=0.5,\mathrm{Ec}=0.6,\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=\mathrm{Nr}=0.5$ .

Figure 11.

Deviations in the temperature field $\mathrm{\Theta }(\eta )$ for dissimilar numbers of $\mathrm{\Lambda }$ , where $\mathrm{Pr}=\epsilon =1,\beta =S=1,M=0.5,\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=\mathrm{Nr}=0.5,\mathrm{Ec}=0.6$ .

Figure 12.

Deviations in the concentration field $\varphi (\eta )$ for dissimilar numbers of $\mathrm{Nb}$ , where $\mathrm{Pr}=0.4,\epsilon =1,\mathrm{\Lambda }=S=1,\beta =0.4,M=0.5,\mathrm{Nt}=\mathrm{Sc}=\mathrm{Nr}=0.6,\mathrm{Ec}=0.7$ .

Figure 13.

Variations in the velocity field $f(\eta )$ for dissimilar values of $\mathrm{Nb}$ , where $\mathrm{Pr}=\epsilon =1,\mathrm{\Lambda }=\beta =S=1,\mathrm{kr}=M=0.5,\mathrm{Nt}=\mathrm{Sc}=\mathrm{Nr}=0.5,\mathrm{Ec}=0.7$ .

Figure 14.

Deviations in the temperature field $\theta (\eta )$ for dissimilar numbers of $\mathrm{Nt}$ , where $\mathrm{Pr}=\epsilon =1,\mathrm{\Lambda }=\beta =S=\mathrm{Nb}=1,M=0.5,\mathrm{Sc}=\mathrm{Nr}=0.5,\mathrm{Ec}=0.7$ .

Figure 15.

Deviations in the concentration field $\varphi (\eta )$ for unalike numbers of $\mathrm{Nt}$ , where $\mathrm{Pr}=\epsilon =1,\mathrm{\Lambda }=\beta =S=\mathrm{Nb}=1,M=0.5,\mathrm{Sc}=\mathrm{Nr}=0.5,\mathrm{Ec}=0.7$ .

Figure 16.

Deviations in the concentration field $\varphi (\xi )$ happen for unalike numbers of $\mathrm{Sc}$ , when $\mathrm{Pr}=0.4,\epsilon =\mathrm{\Lambda }=1,\beta =S=\mathrm{Nb}=0.4,M=0.5,\mathrm{Nt}=\mathrm{Nr}=0.5,\mathrm{Ec}=0.6$ .

Figure 17 represents the effect of magnetic parameters $M$ and $\mathrm{Br}$ on the entropy generation field $N_G$ . The entropy generation increases with the larger values of Brinkman number $\mathrm{Br}$ . The increasing effect of the $\mathrm{Br}$ at the surface of the sheet is low and gradually rises in the flow field towards the free surface of the liquid film. Actually, the heat produced by viscous dissipation due to lower conduction rises the entropy generation. Whereas the $N_G$ decreases for larger values of $M$ . Because the resistance strength is produced due to the Lorentz force in the magnetic field and the result are same as in Afridi et al. ²³

Figure 17.

Deviations in the entropy function $N_G(\eta )$ for unalike numbers of $M$ are solid red and dashed red lines when $\mathrm{Br}=1.1$ and for $\mathrm{Br}$ in solid blue and dashed blue lines when $M=0.3$ , where the remaining parameters have been kept fixed as $\mathrm{Pr}=1,\mathrm{\Lambda }=\epsilon =\beta =S=0.4,R=0.6,\mathrm{Nt}=\mathrm{Nb}=\chi =0.7,\mathrm{Sc}=0.4,\mathrm{Nr}=\mathrm{\Omega }=\lambda =0.5$ .

Figure 18 demonstrates the effect of Prandtl number $\mathrm{Pr}$ and Reynolds number $\mathrm{Re}$ on the non-dimensional parameter $N_G$ of entropy regime. The larger valves of the $\mathrm{Pr}$ rise the temperature gradient and as a result, the $N_G$ boosts up. Likewise, the entropy regime increases for larger values of $\mathrm{Re}$ and the results are same as Afridi et al. ²³ and Bhatti et al. ²⁴

Figure 18.

Deviations in entropy function $N_G(\eta )$ for unalike numbers of $\mathrm{Re}$ are red dashed lines when $\mathrm{Pr}=3.1$ and for Pr in solid blue and dashed blue lines when Re = 0.1, where the remaining parameters have been kept fixed as $\mathrm{\Lambda }=\epsilon =\beta =S=0.5,M=R=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\chi =0.7,\mathrm{Sc}=0.4,\mathrm{Nr}=\mathrm{\Omega }=\lambda =0.4$ .

The effect of the magnetic parameter $M$ and Brinkman number $\mathrm{Br}$ versus Bejan number $\mathrm{Be}$ is shown in Figure 19. It is observed that the larger amount of the magnetic parameter boost up the Bejan number and the larger amount of the Brinkman number declines the Bejan number.

Figure 19.

Deviations in Bejan number $\mathrm{Be}$ for unlike numbers of $M\mathrm{and}\mathrm{Br}$ when $\mathrm{\Lambda }=\epsilon =\beta =S=0.5,M=R=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\chi =0.7,\mathrm{Sc}=0.4,\mathrm{Nr}=\mathrm{\Omega }=\lambda =0.4$ .

The closed agreement of the present work and Qasim et al. ¹⁴ has been displayed in Figures 20–22 for the velocity, temperature and concentration fields, respectively.

Figure 20.

Represent the relationship of Qasim et al. ¹⁴ and present result for velocity $f(\eta )$ , when $\beta =1,\mathrm{\Lambda }=0,M=0,\mathrm{Kr}=0,\epsilon =0,\mathrm{Nr}=0,S=0,\mathrm{Pr}=1,\mathrm{Nt}=1,\mathrm{Nb}=0.5,\mathrm{Sc}=0.5,\mathrm{Ec}=0.5,h=-0.2$ .

Figure 21.

Represent the relationship of Qasim et al. ¹⁴ and present result for temperature pitches $\theta (\eta )$ , when $\beta =1,\mathrm{\Lambda }=0,M=0.3,\epsilon =0,\mathrm{Nr}=0,S=0,\mathrm{Pr}=1,\mathrm{Nt}=1,\mathrm{Nb}=0.5,\mathrm{Sc}=0.5,\mathrm{Ec}=0.5,h=-0.2$ .

Figure 22.

Represent the relationship of Qasim et al. ¹⁴ and present result for concentration pitches $\varphi (\eta )$ , $\beta =1,\mathrm{\Lambda }=0,M=0.3,\epsilon =0,\mathrm{Nr}=0,S=0,\mathrm{Pr}=1,\mathrm{Nt}=1,\mathrm{Nb}=0.5,\mathrm{Sc}=o.5,\mathrm{Ec}=0.5,h=-0.2$ .

The parametric range of all the physical constraints, velocity, temperature and concentration profiles has been obtained and plotted in Figures 23–27. In these figures, the desired error is achieved by the HAM method with the testing of the maximum and minimum values the physical parameters involved in the problem. The values of the parameters are selected to control the residual error and the minimum error achieved is up to ${10}^{-3}$ while the maximum error is up to ${10}^{-7}$ . The domain in each sub region is taken from the previous data used throughout this analysis. If the present findings are used for engineering purposes, the correct physical parameters up to the desired accuracy must be chosen from these ranges.

Figure 23.

Represent the range of the parameters $\mathrm{\Lambda },\beta$ for the desired residual error of HAM.

Figure 24.

Represent the range of the parameters $M,\mathrm{Nb}$ for the desired residual error of HAM.

Figure 25.

Represent the range of the parameters $\mathrm{Nr},\mathrm{Pr}$ for the desired residual error of HAM.

Figure 26.

Represent the range of the parameter $\mathrm{Sc}$ versus velocity field $f\prime (\eta )$ for the desired residual error of HAM.

Figure 27.

Represent the residual error for the temperature $\mathrm{\Theta }(\eta )$ and concentration $\varphi (\eta )$ fields.

Table 1 shows the extension of the present work with the published work. The validation of the HAM method has been attained with the numerical outputs in the form of average squared residual error for the thin film nanofluid problem in Table 2.

Table 1.

Literature on the nano-liquid film with variable fluid properties.

Authors	Models	Magnetic field	Variable fluid properties	Thermal radiation	Entropy generation
Khan et al. 13	Homogeneous	Not considered	Considered	Not considered	Not considered
Qasim et al. 14	Buongiorno	Considered	Not considered	Not considered	Not considered
Present	Buongiorno	Considered	Considered	Considered	Considered

Table 2.

The averaged squared residual error for the thin film nanofluid problem.

$m$	${\epsilon }_m^f$	${\epsilon }_m^{\mathrm{\Theta }}$	${\epsilon }_m^{\varphi }$
$2$	$2.2369\times {10}^{-4}$	$7.8623\times {10}^{-5}$	$6.7891\times {10}^{-3}$
$4$	$9.186\times {10}^{-5}$	$2.1968\times {10}^{-5}$	$9.8301\times {10}^{-4}$
$6$	$2.9627\times {10}^{-5}$	$9.8673\times {10}^{-6}$	$5.4328\times {10}^{-4}$
$8$	$7.0856\times {10}^{-6}$	$1.7684\times {10}^{-6}$	$8.0032\times {10}^{-5}$
$10$	$5.888\times {10}^{-6}$	$7.862\times {10}^{-7}$	$5.11328\times {10}^{-5}$

The convergence of the results is presented in Table 3 and it is detected that momentum and temperature gradient equality converges to 30th order of calculations while the concentration equality converges up to 31th order of calculations. The important physical quantities of skin friction coefficient, local Nusselt number and Sherwood number under the effect of embedded parameters have been displayed in Tables 4–6. The large quantities of magnetic parameter $M$ , variable viscosity $\mathrm{\Lambda }$ and thickness parameter $\beta$ increase the skin friction coefficient $-f″(0)$ , as shown in Table 4. Because the larger values of $M$ , $\mathrm{\Lambda }$ and $\beta$ increase the friction force to boost up the skin friction coefficient. The larger amount of the Prandtl number provides maximum cooling and as a result, the Nusselt number increases as shown in Table 5.The upsurges values of the variable viscosity parameter also prolong the cooling effect and enhance the $-\mathrm{\Theta }\prime (0)$ . The larger amount of the parameters $M$ and $\mathrm{Nr}$ increases the temperature field and the reverse effect of these two parameters have been observed in $-\mathrm{\Theta }\prime (0)$ . In Table 6, the influence of the $\mathrm{Sc},\mathrm{Nb}$ and $\mathrm{Nt}$ versus Sherwood number has been displayed. The contrasting significance of the above parameters has been obtained for the Sherwood number $-\varphi (0).$

Table 3.

The convergence of HAM problem solutions, when $h=-.2,\epsilon =1,\beta =1,\mathrm{\Lambda }=1,S=0.9,\mathrm{Sc}=0.5,M=0.5,\mathrm{\Lambda }=0.5,\mathrm{Nb}=0.5,\mathrm{Nt}=0.5,\mathrm{Pr}=0.5,\mathrm{Ec}=0.5,\mathrm{Nr}=0.5$ .

Order of calculations	$-f″(0)$	$-\vartheta \prime (0)$	$-\varphi \prime (0)$
5	1.12123	0.348104	0.163805
15	1.32839	0.339004	0.169680
20	1.33325	0.338867	0.169460
30	1.33420	0.338859	0.169389
31	1.33420	0.338859	0.169388
32	1.33420	0.338859	0.169387
33	1.33420	0.338859	0.169387

HAM: homotopy analysis method.

Table 4.

Sketch of the $-f″(0)$ when $\mathrm{Pr}=0.8,\mathrm{\Lambda }=\epsilon =\beta =S=\mathrm{Nr}=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.3,h=-0.5$ .

$M$	$\mathrm{\Lambda }$	$\beta$	$-f″(0)$
0.3	0.3	0.3	0.582966
0.4			0.618819
0.5			0.654423
0.3	0.4	0.3	0.619867
	0.5		0.656724
0.3	0.3	0.4	0.747594
		0.5	0.889790

Table 5.

Sketch of the $-\theta \prime (0)$ , when $\mathrm{Pr}=0.8,\mathrm{\Lambda }=\epsilon =\beta =S=\mathrm{Nr}=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.3,h=-0.5$ .

$M$	$\mathrm{\Lambda }$	$\mathrm{Nr}$	$\mathrm{Pr}$	$-\theta \prime (0)$
0.3	0.3	0.3	0.8	0.256216
0.4				0.255581
0.5				0.254952
	0.4			0.257399
	0.5			0.258328
		0.4		0.233718
		0.5		0.214823
			0.8	0.256216
			1.0	0.320070

Table 6.

Sketch of the $-\varphi \prime (0)$ , when $\mathrm{Pr}=0.8,\mathrm{\Lambda }=\epsilon =\beta =S=\mathrm{Nr}=\mathrm{Ec}=\mathrm{Nt}=\mathrm{Nb}=\mathrm{Sc}=0.3,h=-0.5$ .

$\mathrm{Sc}$	$\mathrm{Nb}$	$\mathrm{Nt}$	$-\varphi \prime (0)$
0.3	0.3	0.3	−0.086403
0.4			−0.029549
0.5			0.0273354
	0.4		−0.022165
	0.5		0.0163781
		0.4	−0.171872
		0.5	−0.257339

Conclusion

The nanofluid flow, selecting a thin layer over an extended/contracted sheet, has been demonstrated in this research. The published exertion of Khan et al. ¹³ and Qasim et al. ¹⁴ has been extended with the inclusion of entropy regime. The extension has also been enlarged with the attachment of variable viscosity and thermal conductivity. Also the porosity and thermal radiation terms included in the present work. The nonlinear equations have been solved using the optimal approach (optimal homotopy analysis method [OHAM]). The authentication of the work has been linked to the numerical method. The comparison of the present work and published work has been compared with the elimination of non-similar terms.

The concluded remarks are highlighted as:

• The flexible viscosity and thermal conductivity effects have been observed through graphs and discussed.

• The effects of Brownian motion and thermophoresis constraints have mainly focused and the observation have been notified on the variable thickness of fluid film with flexible fluid properties.

• The Entropy regime and the impact of physical parameters related to entropy have also been focused.

• The impact of the magnetic field and Brinkman number versus the Bejan number has been observed.

• The thermal radiation effects have been perceived on the flow pattern and it is observed that the variable thickness of the liquid film is more affected as compared to the uniform thickness of the layer.

• The impact of the various parameters versus the skin friction coefficient, Nusselt number and Sherwood number has been observed numerically.

• Residual error and optimal values have been calculated and the range of each physical parameter has also been observed.

• The recent research has been compared with the published work and much closed agreement has been achieved.

According to the prevailing literature, this is the first effort concerning the flexible, fluid properties considering nanofluid film with variable thickness.