An optimal analysis for Darcy–Forchheimer three-dimensional Williamson nanofluid flow over a stretching surface with convective conditions

Abstract

The augmented thermal conductivity is significant in betterment of heat transfer behavior of fluids. A number of other physical quantities such as density, viscosity, and specific heat play the key role in fluid flow behavior. Investigators have shown that the nanofluids have not only superior heat conductivity but also have better convective heat transfer capability than the base fluids. In this article, the analysis of three-dimensional Williamson fluid has been carried out under investigation. The fluid flow is taken over a linear porous stretching sheet under the influence of thermal radiation. The transformed system of equations has been solved by homotopy analysis method. The impact of embedded parameters on the fluid flow has shown graphically. The velocity profile in x-direction is decreased with the augmented stretching, Williamson, coefficient of inertia, and porosity parameters. The velocity profile in y-direction is increased with the enlarged stretching parameter, while reduced with the augmented Williamson, coefficient of inertia, and porosity parameters. The temperature profile is increased with the enlarged stretching, radiation, thermophoresis, parameter and Brownian motion parameters, and Biot number while decreased with the increased Prandtl number. The concentration profile is increased with the increased thermophoresis parameter and Biot numbers, while decreased with the enlarged stretching and Brownian motion parameters.

Introduction

There are abundant applications of non-Newtonian fluid in the field of geophysics, biological sciences, chemical industries, petroleum industries, and so on. Such flows appear in polymer processing, biological fluids, plastic manufacturing, food processing, ice and magma flows. The three-dimensional studies of nanofluid flow over a stretching surface have got researchers attention. The theory of pseudo plastic fluid has been presented by Williamson. ¹ He called this type of fluid as pseudo fluid due to exceptional properties from viscous and ideal plastic fluids. Ariel ² presented the three-dimensional steady laminar flow of incompressible viscous fluid over a stretching surface using homotopy perturbation method. Nadeem et al. ³ examined the magnetohydrodynamic (MHD) Casson fluid over an exponentially shrinking sheet. The two-dimensional flow of Williamson fluid under low Reynolds number and long wavelength has been studied by Nadeem and Akram. ⁴ Pop and Na ⁵ examined the incompressible micro polar fluid flow over a stretching sheet. Sakiadis ⁶ examined the three-dimensional viscous fluid flow over a stretching sheet. The three-dimensional Williamson fluid flow over a stretching surface has been examined by Malik et al. ⁷ Gupta and Gupta ⁸ examined the process of suction and blowing together with mass and heat transfer rate over a stretching surface. El-Aziz ⁹ examined the effect of radiation by studying the fluid flow over an unsteady stretched surface. The effect of thermal radiation on the fluid flow over vertically porous stretched surface has been analyzed by Mukhopadyay. ¹⁰ The change in concentration and mass transfer rates of the fluid flow over a horizontal stretched sheet has been numerically investigated by Shateyi and Motsa. ¹¹ The influence of thermal-diffusion and diffusion-thermo on three-dimensional MHD fluid flow over a permeable stretched surface has been analyzed by El-Eziz. ¹² The effect of thermal radiation on viscous fluid flow over non-linear stretched surface has been examined by Cortell. ¹³ Ishak et al. ¹⁴ numerically studied the heat transfer phenomena in a fluid flow. Dawar et al. ¹⁵ analytically examined the effect of thermal radiation on MHD Eyring–Powell fluid flow over a porous stretched surface. The MHD Casson nanofluid and radiative heat transfer in a rotating channel have been examined by Dawar et al. ¹⁶ Using effective thermal conductivity model, Shah et al. ¹⁷ examined the electrical magnetite Casson feerofluid over a stretching/shrinking sheet. The effect of thermal radiation on MHD Jeffery fluid has been examined by Abro et al. ¹⁸ Sheikholeslami and Houman ¹⁹ examined the effect of Coulomb force on nanofluid heat transfer in a porous enclosure. The influences of Brownian motion on fluid flow and convective heat transfer in a porous cavity have been examined by Sheikholeslami and Houman. ²⁰ Considering Darcy model, Sheikholeslami ²¹ examined the effect of Lorentz forces on fluid flow in a porous cylinder. Waqas et al. ²² numerically studied the MHD Carreau nanofluid flow model over an exponentially stretched surface. Hayat et al. ²³ deliberated the stagnation point of MHD flow over non-linear stretched surface under the impact of non-uniform heat generation/absorption. Hayat et al. ²⁴ deliberated the relation of Darcy–Forchheimer with water-based carbon nanotubes flow in a rotating disk. Waqas et al. ²⁵ analyzed the effect of magnetic field and mixed convection of an Oldroyd-B nanofluid stratified flow. Hayat et al. ²⁶ examined the MHD nanofluid flow over non-linear stretched surface. The analysis of Newtonian heating in a stagnation point flow of Burgers’ fluid has been presented by Hayat et al. ²⁷ The joule heating impacts on MHD boundary layer flow of Burgers’ fluid over a stretched surface have been analyzed by Hayat. ²⁸ The effect of internal heat generation/absorption and Newtonian heating in Erying–Powell fluid over a stretched surface has been examined by Hayat et al. ²⁹ The effect of thermal radiation and thermo-diffusion on Williamson fluid flow over a porous stretched/shrinked sheet has been examined by Bhatti and Rashidi. ³⁰ The effect of chemical reaction and thermal radiation on MHD flow of nanofluid gyrotactic microorganisms has been examined by Bhatti et al. ³¹ Bhatti et al. ³² presented the numerical method for solving stagnation point flow over a permeable shrinked sheet. Alharbi et al. ³³ analytically studied the MHD flow of Erying–Powell fluid with entropy generation. Sheikholeslami et al. ³⁴ examined the heat transfer behavior of nanoparticle enhanced phase change material solidification through an enclosure with V-shaped fins. Sheikholeslami ³⁵ presented new computational approach for energy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media. The others related studies can be found in the literature.^36–38 Waqas et al.,^39–42 Bhatti et al.^43,44 recently studied non-Newton fluid using different models with different effects. Sulochana et al. ⁴⁵ studied Darcy–Forchheimer Williamson Nanofluid fluid with nonlinear radiation. Hayat et al. ⁴⁶ and Shah et al.^47,48 recently studied nanofluid flow in rotating system.

The main aim of this work is to analyse three-dimensional radiative Williamson nanofluid over stretching surface with Darcy–Forchheimer medium. The fluid flow is taken over a linear porous stretching sheet under the influence of thermal radiation. The transformed system of equations has been solved by homotopy analysis method (HAM).^49–58 The impact of embedded parameters on the fluid flow has shown graphically.

Problem formulation

Consider the unsteady three-dimensional incompressible flow of Williamson fluid over a linear porous stretching sheet. The sheet is stretched along xy-direction (Figure 1). The motion of the fluid is caused stretching sheet and the fluid occupy the space $z>0$ . The Williamson nanofluid in porous media is designated by assuming the Darcy–Forchheimer model where saturating porous space obeys the Darcy–Forchheimer expression. According to these boundary layer approximations, the equations of continuity and momentum for the Williamson fluid are given as

$$\nabla ·V=0$$

(1)

$$\rho \frac{\mathrm{dV}}{\mathrm{dt}}=\nabla ·S+\rho b$$

(2)

where $\rho$ , $V$ , $S$ , $b$ , and $d/\mathrm{dt}$ are the density, velocity vector, Cauchy stress tensor, the specific body force vector, and material derivative, respectively.

Figure 1.

Physical representation of the fluid flow problem.

The mathematical equation for Williamson fluid is

$$S=p\mathbf{I}+\tau$$

(3)

$$\tau =\left[\frac{{\mu }_0-{\mu }_{\infty }}{1-\mathrm{\Gamma }{\gamma }^{*}}\right]A_1$$

(4)

where $\tau$ indicates the extra stress tensor; ${\mu }_0,{\mu }_{\infty }$ are the limiting viscosities at zero and at infinite shear rate, respectively; $\mathrm{\Gamma }$ is the time constant; $A_1$ is the Rivlin–Ericksen tensor; and ${\gamma }^{*}$ is defined as

$$\begin{array}{l}{\gamma }^{*}=\sqrt{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\pi }\\ \pi =trace(A_1^2)\end{array}$$

(5)

$${\gamma }^{*}=\sqrt{{\left(\frac{\partial u}{\partial x}\right)}^2+\frac{1}{2}{(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})}^2+{\left(\frac{\partial v}{\partial y}\right)}^2}$$

(6)

where $\pi$ is the second invariant strain tensor.

For the above stated problem, the mathematical equations are^7,45

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

(7)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=\nu \frac{{\partial }^{2}u}{\partial z^2}+\sqrt{2}\nu \mathrm{\Gamma }\frac{\partial u}{\partial z}\frac{{\partial }^{2}u}{\partial z^2}-(\frac{\nu }{k}+\mathrm{Fu})u$$

(8)

$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=\nu \frac{{\partial }^{2}v}{\partial z^2}+\sqrt{2}\nu \mathrm{\Gamma }\frac{\partial v}{\partial z}\frac{{\partial }^{2}v}{\partial z^2}-(\frac{\nu }{k}+\mathrm{Fv})v$$

(9)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\frac{k}{\rho c_p}\frac{{\partial }^{2}T}{\partial z^2}-\frac{\partial q_r}{\partial z}+{\tau }_1[D_B\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2]$$

(10)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}=D_B\frac{{\partial }^{2}C}{\partial z^2}+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial z^2}\right)$$

(11)

where $q_r$ is the radiative heat flux which is defined as

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial z}$$

(12)

As $T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4$ , substituting equation (12) in equation (10) which reduces to

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\frac{k}{\rho c_p}(1+\frac{4{\sigma }^{*}{T}_{\infty }^3}{k^{*}})\frac{{\partial }^{2}T}{\partial z^2}+{\tau }_1[D_B\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2]$$

(13)

With boundary conditions

$$\begin{array}{c}\begin{array}{c}u=U_w=\mathrm{ax},v=V_w=\mathrm{by}\hfill \\ \multicolumn{1}{c}{-k\frac{\partial T}{\partial z}=h_f(T_f-T),-D_B\frac{\partial C}{\partial z}=h_s(C_s-C)\mathrm{at}\eta =0}\end{array}\}\hfill \\ u\to 0,v\to 0,w\to 0,T\to T_{\infty },C\to C_{\infty }\mathrm{as}\eta \to \infty \hfill \end{array}$$

(14)

In equations (6)–(14), $u,v,\mathrm{and}w$ are the components of velocity in their respective directions; $\mathrm{\Gamma }$ indicates the Williamson fluid parameter; $k$ represents the thermal conductivity; $\nu$ and $\rho$ specify the kinematic and dynamic viscosities, respectively; $F=C_b/\sqrt{B}x$ is the variable inertia coefficient of permeable medium; $T$ represents the temperature; $c_p$ and ${\tau }_1$ specify the heat and ratio of the heat capacities, respectively; $D_B$ and $D_T$ indicate the Brownian motion and thermophoretic coefficients, respectively; ${\sigma }^{*}$ represents the absorption coefficient; $k$ indicates the Stefan Boltzmann constant; c = a/b is the stretching parameter where $a\mathrm{and}b$ are positive constants, respectively; $h_f\mathrm{and}{h}_s$ indicate the convective heat and mass transmission coefficients, respectively; $T_f$ represents the convective fluid temperature; and $C$ is the concentration below the moving sheet.

The similarity variables is defined as

$$\begin{array}{l}u=ax{f}^{\prime }(\xi ),v=by{g}^{\prime }(\xi ),w=-{\left(a\nu \right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(f(\xi )+cg(\xi )\right)\\ \theta (\xi )=T-T_{\infty }/T_f-T_{\infty },\varphi (\xi )=C-C_{\infty }/C_f-C_{\infty },\xi =z{\left(\frac{a}{\nu }\right)}^{1/2}\end{array}$$

(15)

In view of the above equation, the continuity equation satisfied and equations (8)–(10) and (13) are reduced to

$$f″\prime -{(f\prime )}^2+(f+\mathrm{cg})f″+\mathrm{Wef}″f″\prime +\mathrm{gf}″-\gamma f\prime -\mathrm{Fr}{(f\prime )}^2=0$$

(16)

$$g″\prime -{(g\prime )}^2+(f+\mathrm{cg})g″+\mathrm{Weg}″g″\prime +\mathrm{fg}″-\gamma g\prime -\mathrm{Fr}{(g\prime )}^2=0$$

(17)

$$\frac{1}{\mathrm{Pr}}(1+\frac{4}{3}\mathrm{Rd})\theta ″+(f+\mathrm{cg})\theta \prime +\mathrm{Nb}\varphi \prime \theta \prime +\mathrm{Nt}{(\theta \prime )}^2=0$$

(18)

$$\varphi ″(\eta )+\mathrm{Le}(f+\mathrm{cg})\varphi \prime (\eta )+\left(\frac{\mathrm{Nt}}{\mathrm{Nb}}\right)\theta ″(\eta )=0$$

(19)

For the modeled problem, the following boundary conditions are defined

$$\begin{array}{c}f(0)=0,f\prime (0)=1,g(0)=1\hfill \\ g\prime (0)=c,f\prime (\infty )=0,g\prime (\infty )=0\hfill \\ \theta \prime (0)=-B{i}_1(1-\theta (0)),\hfill \\ \multicolumn{1}{c}{\varphi \prime (0)=-B{i}_2(1-\varphi (0))}\\ \theta (\infty )=\varphi (\infty )=0\hfill \end{array}$$

(20)

In the above equations, $\gamma =\nu /\mathrm{ak}$ signifies the porosity parameter, $\mathrm{We}=\mathrm{\Gamma }x\sqrt{2a^3/\nu }$ denotes the Williamson parameter, $\mathrm{Fr}=\nu C_b(B{)}^{-0.5}$ is the coefficient of inertia, $\mathrm{Pr}=\rho \nu c_p/k$ signifies the Prandtl number, $\mathrm{Rd}=4\sigma T_{\infty }^3/k{k}^{*}$ signifies the radiation parameter, $\mathrm{Nt}={\tau }_1{D}_T(T_f-T_{\infty })/\nu T_{\infty }$ denotes the thermophoresis parameter, $\mathrm{Nb}={\tau }_1{D}_T(C_f-C_{\infty })/\nu$ signifies the Brownian motion parameter, and $B{i}_1=h_f/k/h_f/k\sqrt{\nu /\nu aa}\sqrt{\nu /\nu aa},B{i}_2=h_s/D_B/h_s/D_B\sqrt{\nu /\nu aa}\sqrt{\nu /\nu aa}$ are the Biot numbers.

The skin fraction, Nusselt number, and Sherwood number are defined as

$$\begin{array}{c}\frac{1}{\sqrt{2}}{C}_f\sqrt{\mathrm{R}{\mathrm{e}}_x}=(f″(0)+\mathrm{We}{(f″)}^2(0))\hfill \\ \frac{1}{\sqrt{2}}{C}_f\sqrt{\mathrm{R}{\mathrm{e}}_x}=(g″(0)+\mathrm{We}{(g″)}^2(0))\hfill \end{array}$$

(21)

$$\mathrm{Nu}=-(1+\frac{4}{3}\mathrm{Rd})\theta \prime (0)$$

(22)

$$\mathrm{Sh}=-\varphi \prime (0)$$

(23)

Solution by HAM

To solve equations (16)–(19) with equation (20), the HAM is applied. The initial assumptions are taken as follows

$$f_0(\xi )=1-e^{-\xi },g_0(\xi )=c(1-e^{-\xi }),{\theta }_0(\xi )=\left(\frac{B{i}_1}{1+B{i}_1}\right)e^{-\xi },{\varphi }_0(\xi )=\left(\frac{B{i}_2}{1+B{i}_2}\right)e^{-\xi }$$

(24)

The $L_f,L_g,L_{\theta },\mathrm{and}{L}_{\varphi }$ are linear operators, which are taken as

$$L_f(f)=f″\prime -f\prime ,L_g(g)=g″\prime -g\prime ,L_{\theta }(\theta )=\theta ″-\theta ,L_{\varphi }(\varphi )=\varphi ″-\varphi$$

(25)

which have the succeeding properties

$$\begin{array}{c}{L}_f(l_1+l_2{e}^{-\xi }+l_3{e}^{\xi })=0,L_g(l_4+l_5{e}^{-\xi }+l_6{e}^{\xi })=0\hfill \\ L_{\theta }(l_7{e}^{-\xi }+l_8{e}^{\xi })=0,L_{\varphi }(l_9{e}^{-\xi }+l_{10}{e}^{\xi })=0\hfill \end{array}$$

(26)

where $l_1,l_2,l_3,l_4,l_5,l_6,l_7,l_8,l_9,l_{10}$ are constant coefficients.

The non-linear operators $N_f,N_g,N_{\theta }$ , and $N_{\varphi }$ are indicated as

$$\begin{array}{c}\multicolumn{1}{c}{N_f[f(\xi ;\Re ),g(\xi ;\Re )]=\frac{{\partial }^{3}f(\xi ;\Re )}{\partial {\xi }^3}-{\left(\frac{\partial f(\xi ;\Re )}{\partial \xi }\right)}^2}\\ \multicolumn{1}{c}{+(f(\xi ;\Re )+g(\xi ;\Re ))\frac{{\partial }^{2}f(\xi ;\Re )}{\partial {\xi }^2}}\\ \multicolumn{1}{c}{+\left(\mathrm{We}\right)\frac{{\partial }^{2}f(\xi ;\Re )}{\partial {\xi }^2}\frac{{\partial }^{3}f(\xi ;\Re )}{\partial {\xi }^3}+g(\xi ;\Re )\frac{{\partial }^{2}f(\xi ;\Re )}{\partial {\xi }^2}}\\ \multicolumn{1}{c}{-\left(\gamma \right)\frac{\partial f(\xi ;\Re )}{\partial \xi }-\left(\mathrm{Fr}\right){\left(\frac{\partial f(\xi ;\Re )}{\partial \xi }\right)}^2}\end{array}$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{N_g[f(\xi ;\Re ),g(\xi ;\Re )]=\frac{{\partial }^{3}g(\xi ;\Re )}{\partial {\xi }^3}-{\left(\frac{\partial g(\xi ;\Re )}{\partial \xi }\right)}^2}\\ \multicolumn{1}{c}{+(f(\xi ;\Re )+g(\xi ;\Re ))\frac{{\partial }^{2}g(\xi ;\Re )}{\partial {\xi }^2}}\\ \multicolumn{1}{c}{+\left(\mathrm{We}\right)\frac{{\partial }^{2}g(\xi ;\Re )}{\partial {\xi }^2}\frac{{\partial }^{3}g(\xi ;\Re )}{\partial {\xi }^3}+f(\xi ;\Re )\frac{{\partial }^{2}g(\xi ;\Re )}{\partial {\xi }^2}}\\ \multicolumn{1}{c}{-\left(\gamma \right)\frac{\partial g(\xi ;\Re )}{\partial \xi }-\left(\mathrm{Fr}\right){\left(\frac{\partial g(\xi ;\Re )}{\partial \xi }\right)}^2}\end{array}$$

(28)

$$\begin{array}{c}\multicolumn{1}{c}{N_{\theta }[\theta (\xi ;\Re ),\varphi (\xi ;\Re )]=\frac{1}{\mathrm{Pr}}(1+\frac{4}{3}\mathrm{Rd})\frac{{\partial }^2\theta (\xi ;\Re )}{\partial {\xi }^2}+(f+\mathrm{cg})}\\ \multicolumn{1}{c}{\frac{\partial \theta (\xi ;\Re )}{\partial \xi }+\left(\mathrm{Nb}\right)\frac{\partial \varphi (\xi ;\Re )}{\partial \xi }\frac{\partial \theta (\xi ;\Re )}{\partial \xi }+\left(\mathrm{Nt}\right){\left(\frac{\partial \theta (\xi ;\Re )}{\partial \xi }\right)}^2}\end{array}$$

(29)

$$N_{\varphi }[\theta (\xi ;\Re ),\varphi (\xi ;\Re )]=\frac{{\partial }^2\varphi (\xi ;\Re )}{\partial {\xi }^2}+\mathrm{Le}(f+\mathrm{cg})\left(\frac{\partial \varphi (\xi ;\Re )}{\partial \xi }\right)+\frac{\mathrm{Nt}}{\mathrm{Nb}}\left(\frac{{\partial }^2\varphi (\xi ;\Re )}{\partial {\xi }^2}\right)$$

(30)

The zeroth order problems from equations (16)–(19) are

$$(1-\Re )L_f[f(\xi ;\Re )-f_0(\xi )]=\Re {\hslash }_f{N}_f[f(\xi ;\Re ),g(\xi ;\Re )]$$

(31)

$$(1-\Re )L_g[g(\xi ;\Re )-g_0(\xi )]=\Re {\hslash }_g{N}_g[f(\xi ;\Re ),g(\xi ;\Re )]$$

(32)

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

(33)

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

(34)

The equivalent boundary conditions are

$$\begin{array}{c}{f(\xi ;\Re )|}_{\xi =0}=0,{\frac{\partial f(\xi ;\Re )}{\partial \xi }|}_{\xi =0}=0,{\frac{\partial f(\xi ;\Re )}{\partial \xi }|}_{\xi \to \infty }=0\hfill \\ {g(\xi ;\Re )|}_{\xi =0}=c,{\frac{\partial g(\xi ;\Re )}{\partial \xi }|}_{\xi =0}=0,{g(\xi ;\Re )|}_{\xi \to \infty }=0\hfill \\ {\frac{\partial \theta (\xi ;\Re )}{\partial \xi }|}_{\xi =0}=-B{i}_1(1-\theta ),{\theta (\xi ;\Re )|}_{\xi \to \infty }=0\hfill \\ {\varphi (\xi ;\Re )|}_{\xi =0}=-B{i}_2(1-\varphi ),{\varphi (\xi ;\Re )|}_{\xi \to \infty }=0\hfill \end{array}$$

(35)

where $\Re \in [0,1]$ is the implanting parameter. When $\Re =0\mathrm{and}\Re =1$ , we have

$$\begin{array}{c}f(\xi ;0)=f_0(\xi ),f(\xi ;1)=f(\xi )\hfill \\ \mathrm{g}(\xi ;0)=g_0(\xi ),\mathrm{g}(\xi ;1)=g(\xi )\hfill \\ \theta (\xi ;0)={\theta }_0(\xi ),\theta (\xi ;1)=\theta (\xi )\hfill \\ \varphi (\xi ;0)={\varphi }_0(\xi ),\varphi (\xi ;1)=\varphi (\xi )\hfill \end{array}$$

(36)

Expanding $f(\xi ;\Re ),g(\xi ;\Re ),\theta (\xi ;\Re ),\mathrm{and}\varphi (\xi ;\Re )$ by Taylor’s series

$$\begin{array}{c}f(\xi ;\Re )=f_0(\xi )+\sum _{q=1}^{\infty }{f}_q(\xi ){\Re }^q\hfill \\ g(\xi ;\Re )=g_0(\xi )+\sum _{q=1}^{\infty }{g}_q(\xi ){\Re }^q\hfill \\ \theta (\xi ;\Re )={\theta }_0(\xi )+\sum _{q=1}^{\infty }{\theta }_q(\xi ){\Re }^q\hfill \\ \varphi (\xi ;\Re )={\varphi }_0(\xi )+\sum _{q=1}^{\infty }{\varphi }_q(\xi ){\Re }^q\hfill \end{array}$$

(37)

where

$$\begin{array}{c}{f}_q(\xi )={\frac{1}{q!}\frac{\partial f(\xi ;\Re )}{\partial \xi }|}_{\Re =0},{\mathrm{g}}_q(\xi )={\frac{1}{q!}\frac{\partial g(\xi ;\Re )}{\partial \xi }|}_{\Re =0}\hfill \\ {\theta }_q(\xi )={\frac{1}{q!}\frac{\partial \theta (\xi ;\Re )}{\partial \xi }|}_{\Re =0},{\varphi }_q(\xi )={\frac{1}{q!}\frac{\partial \varphi (\xi ;\Re )}{\partial \xi }|}_{\Re =0}\hfill \end{array}$$

(38)

The secondary constraints ${\hslash }_f,{\hslash }_g,{\hslash }_{\theta },\mathrm{and}{\hslash }_{\varphi }$ are nominated in such a way that the series (equation (36)) converges at $\Re =1$ . Changing $\Re =1$ in equation (36), we get

$$\begin{array}{c}f(\xi )=f_0(\xi )+\sum _{q=1}^{\infty }{f}_q(\xi )\hfill \\ g(\xi )=g_0(\xi )+\sum _{q=1}^{\infty }{g}_q(\xi )\hfill \\ \theta (\xi )={\theta }_0(\xi )+\sum _{q=1}^{\infty }{\theta }_q(\xi )\hfill \\ \varphi (\xi )={\varphi }_0(\xi )+\sum _{q=1}^{\infty }{\varphi }_q(\xi )\hfill \end{array}$$

(39)

The $q\mathrm{th}$ order problem satiates the following

$$\begin{array}{c}{L}_f[f_q(\xi )-{\chi }_q{f}_{q-1}(\xi )]={\hslash }_f{U}_q^f(\xi )\hfill \\ L_g[g_q(\xi )-{\chi }_q{g}_{q-1}(\xi )]={\hslash }_g{U}_q^g(\xi )\hfill \\ L_{\theta }[{\theta }_q(\xi )-{\chi }_q{\theta }_{q-1}(\xi )]={\hslash }_{\theta }{U}_q^{\theta }(\xi )\hfill \\ L_{\varphi }[{\varphi }_q(\xi )-{\chi }_q{\varphi }_{q-1}(\xi )]={\hslash }_{\varphi }{U}_q^{\varphi }(\xi )\hfill \end{array}$$

(40)

The boundary conditions are

$$\begin{array}{c}f(0)=f\prime (0)=f\prime (\infty )=0\hfill \\ g\prime (0)-c=g\prime (0)=g\prime (\infty )=0\hfill \\ \theta \prime (0)+B{i}_1(1-\theta (0))=\theta (\infty )=0\hfill \\ \varphi \prime (0)+B{i}_2(1-\varphi (0))=\varphi (\infty )=0\hfill \end{array}$$

(41)

Here

$$U_q^f(\xi )=f″{\prime }_{q-1}+(f{\prime }_{q-1}{)}^2+(f+\mathrm{cg}){f″}_{q-1}+\mathrm{We}\left(\sum _{k=0}^{q-1}{f″}_{q-1}{f″\prime }_k\right)+{gf″}_{q-1}-\left(\gamma \right)f{\prime }_{q-1}-\left(\mathrm{Fr}\right){\left({f\prime }_{q-1}\right)}^2$$

(42)

$$U_q^g(\xi )=g″{\prime }_{q-1}+(g{\prime }_{q-1}{)}^2+(f+\mathrm{cg}){g″}_{q-1}+\mathrm{We}\left(\sum _{k=0}^{q-1}{g″}_{q-1}{g″\prime }_k\right)+{fg″}_{q-1}-\left(\gamma \right)g{\prime }_{q-1}-\left(\mathrm{Fr}\right){\left({g\prime }_{q-1}\right)}^2$$

(43)

$$U_q^{\theta }(\xi )=\frac{1}{\mathrm{Pr}}(1+\frac{4}{3}\mathrm{Rd}){\theta ″}_{q-1}+(f+\mathrm{cg})\theta {\prime }_{q-1}+\mathrm{Nb}\left(\sum _{k=0}^{q-1}{\varphi \prime }_{q-1-k}{\theta \prime }_k\right)+\mathrm{Nt}{\left({\theta \prime }_{q-1}\right)}^2$$

(44)

$$U_q^{\varphi }(\xi )={\theta ″}_{q-1}+(f+\mathrm{cg})\varphi {\prime }_{q-1}+\mathrm{Nb}\left(\sum _{k=0}^{q-1}{\varphi \prime }_{q-1-k}{\theta \prime }_k\right)+\frac{\mathrm{Nt}}{\mathrm{Nb}}\left({\varphi ″}_{q-1}\right)$$

(45)

where

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

(46)

HAM convergence

To calculate the series solution of velocity profile in x-direction, velocity profile in y-direction, temperature profile, and concentration profile, the assisting parameters ${\hslash }_f,{\hslash }_g,{\hslash }_{\theta },\mathrm{and}{\hslash }_{\varphi }$ of HAM are in authority of convergence. At 15th order approximations, $\hslash -\mathrm{curves}$ for $f″(0),g″(0),\theta \prime (0),\mathrm{and}\varphi \prime (0)$ are schemed in Figures 2–5. The convergence regions for velocity profile in x-direction, velocity profile in y-direction, temperature profile, and concentration profile are displayed in their valid regions, respectively.

Figure 2.

$\hslash -\mathrm{curve}$ for $f″(0)$ .

Figure 3.

$\hslash -\mathrm{curve}$ graph of velocity profile $g″(0)$ .

Figure 4.

$\hslash -\mathrm{curve}$ for $\theta \prime (0)$ .

Figure 5.

$\hslash -\mathrm{curve}$ for $\varphi \prime (0)$ .

Results and discussion

To discuss the influence of stretching parameter $c$ , Williamson parameter $\mathrm{We}$ , Porosity parameter $\gamma$ , coefficient of inertia $\mathrm{Fr}$ , thermal radiation parameter $\mathrm{Rd}$ , thermophoresis parameter $\mathrm{Nt}$ , Brownian motion parameter $\mathrm{Nb}$ , Prandtl number Pr and Boit numbers $B{i}_1,B{i}_2$ on velocity profile $f\prime (\xi )$ in x-direction, velocity profile $g\prime (\xi )$ in y-direction, temperature profile $\theta (\xi )$ , and concentration $\varphi (\xi )$ profile, Figures 6–24 are plotted. Figures 6–9 are schemed to observe the effect of $c$ on velocities $f\prime (\xi ),g\prime (\xi )$ , temperature $\theta (\xi )$ , and concentration profile $\varphi (\xi )$ , respectively. The stretching parameter $c$ represents the ratio of velocity of the fluid in x-direction to the velocity of the fluid in y-direction. The stretching parameter $c$ declines the velocity profile $f\prime (\xi )$ in x-direction is displayed in Figure 6. This behavior is due to the indirect relation of the stretching parameter $c$ to velocity profile $f\prime (\xi )$ in x-direction which resists the fluid flow. The effect of stretching parameter $c$ on velocity profile $g\prime (\xi )$ in y-direction is displayed in Figure 7. The stretching parameter $c$ is associated with the velocity profile $g\prime (\xi )$ , that is why the velocity profile $g\prime (\xi )$ upsurges with the increasing stretching parameter $c$ . The impact of stretching parameter $c$ on temperature profile $\theta (\xi )$ is displayed in Figure 8. Physically, the stretching parameter $c$ thicker the thermal and momentum boundary layer which slower the heat dissipation causing the heating up the plate. Therefore, the temperature profile $\theta (\xi )$ rises with the increasing stretching parameter $c$ . Figure 9 displays the effect of stretching parameter $c$ on concentration profile $\varphi (\xi )$ . We have observed the reducing behavior in concentration profile $\varphi (\xi )$ with the rise in stretching parameter $c$ . The influence of Williamson parameter $\mathrm{We}$ on velocities profiles $f\prime (\xi )\mathrm{and}g\prime (\xi )$ is schemed in Figures 10 and 11. It is ratio of the relaxation time to retardation time. The escalating Williamson parameter $\mathrm{We}$ increases the relaxation time, because of which the fluid particles consume extra time in restoring their position and as a consequence, the increase in viscosity arises, and the particles’ velocity reduces. That is why the velocity profiles $f\prime (\xi )\mathrm{and}g\prime (\xi )$ reduce with the increasing Williamson parameter $\mathrm{We}$ . The influence of porosity parameter $\gamma$ on velocities profiles $f\prime (\xi )\mathrm{and}g\prime (\xi )$ is displayed in Figures 12 and 13. The porous medium plays an important role during flow phenomena. Physically, the porous medium affects the boundary layer of the fluid flow which as a result accelerates the fluid velocity. Due to this fact, the velocities profiles $f\prime (\xi )\mathrm{and}g\prime (\xi )$ decrease with the increasing porosity parameter $\gamma$ . The effect of coefficient of inertia $\mathrm{Fr}$ on velocities profiles $f\prime (\xi )\mathrm{and}g\prime (\xi )$ is displayed in Figures 14 and 15. The coefficient of inertia $\mathrm{Fr}$ decreases the velocity profiles $f\prime (\xi )\mathrm{and}g\prime (\xi )$ . The effect of thermal radiation parameter $\mathrm{Rd}$ on temperature profile $\theta (\xi )$ is displayed in Figure 16. Physically, the thermal radiation shows a significant role in the progression of heat transmission. The increase in thermal radiation the temperature of the boundary layer flow upsurges. That is why the velocity profile $\theta (\xi )$ rises with the increasing thermal radiation parameter $\mathrm{Rd}$ . The influence of Prandtl number Pr on temperature profile $\theta (\xi )$ is depicted in Figure 17. The increasing Prandtl number Pr reduces the thermal diffusion, which results the boundary layer thickness and diminishes the temperature of the fluid. Therefore, the temperature profile $\theta (\xi )$ diminishes with the increasing Prandtl number Pr. Figures 18 and 19 depict the impact of thermophoresis parameter $\mathrm{Nt}$ on temperature $\theta (\xi )$ and concentration $\varphi (\xi )$ profiles. Here, we witnessed that the increasing thermophoresis $\mathrm{Nt}$ rises the temperature $\theta (\xi )$ and concentration $\varphi (\xi )$ profiles. Figures 20 and 21 show the effect Brownian motion parameter $\mathrm{Nb}$ on temperature $\theta (\xi )$ and concentration $\varphi (\xi )$ profiles. From here, we observed the increasing behavior in temperature profile $\theta (\xi )$ while reducing behavior in concentration profile $\varphi (\xi )$ . Figures 22 and 23 show the influence of Biot number $B{i}_1$ on temperature $\theta (\xi )$ and concentration $\varphi (\xi )$ profiles. From here, we observed the increasing behavior in temperature $\theta (\xi )$ and concentration $\varphi (\xi )$ profiles with the increasing Biot number $B{i}_1$ . A similar impact is observed in Figure 24 with the increasing Biot number $B{i}_2$ .

Figure 6.

Impact of $c$ on $f\prime (\xi )$ when We = 0.1, γ = 0.2, Fr = 0.3.

Figure 7.

Impact of $c$ on $g\prime (\xi )$ when $\mathrm{We}=0.1,\gamma =0.2,\mathrm{Fr}=0.3$ .

Figure 8.

Impact of $c$ on $\theta (\xi )$ when $\mathrm{Nt}=0.5,B{i}_1=0.1,\mathrm{Nb}=0.4,\mathrm{Pr}=0.6,\mathrm{Le}=0.7,\mathrm{Rd}=0.8$ .

Figure 9.

Impact of $c$ on $\varphi (\xi )$ when $B{i}_1=0.1,\mathrm{Nb}=0.4,B{i}_2=0.2,\mathrm{Nt}=0.5$ .

Figure 10.

Impact of $\mathrm{We}$ on $f\prime (\xi )$ when $c=0.1,\gamma =0.2,\mathrm{Fr}=0.3$ .

Figure 11.

Impact of $\mathrm{We}$ on $g\prime (\xi )$ when $c=0.1,\gamma =0.2,\mathrm{Fr}=0.3$ .

Figure 12.

Impact of $\gamma$ on $f\prime (\xi )$ when $c=0.1,\mathrm{We}=0.2,\mathrm{Fr}=0.3$ .

Figure 13.

Impact of $\gamma$ on $g\prime (\xi )$ when $c=0.1,\mathrm{We}=0.2,\mathrm{Fr}=0.3$ .

Figure 14.

Impact of $\mathrm{Fr}$ on $f\prime (\xi )$ when $c=0.1,\gamma =0.2,\mathrm{We}=0.3$ .

Figure 15.

Impact of $\mathrm{Fr}$ on $g\prime (\xi )$ when $c=0.1,\gamma =0.2,\mathrm{We}=0.3$ .

Figure 16.

Impact of $\mathrm{Rd}$ on $\theta (\xi )$ when $B{i}_1=0.1,\mathrm{Pr}=0.6,\mathrm{Le}=0.7,\mathrm{Nb}=0.4,\mathrm{Nt}=0.5,c=0.8$ .

Figure 17.

Impact of Pr on $\theta (\xi )$ when $\mathrm{Nt}=0.5,\mathrm{Rd}=0.6,B{i}_1=0.1,\mathrm{Nb}=0.4,\mathrm{Le}=0.7,c=0.8$ .

Figure 18.

Impact of $\mathrm{Nt}$ on $\theta (\xi )$ when $\mathrm{Rd}=0.5,\mathrm{Le}=0.7,\mathrm{Pr}=0.6,B{i}_1=0.1,\mathrm{Nb}=0.4,c=0.8$ .

Figure 19.

Impact of $\mathrm{Nt}$ on $\varphi (\xi )$ when $B{i}_1=0.1,\mathrm{Nb}=0.4,\mathrm{Le}=0.7,B{i}_2=0.2,c=0.8$ .

Figure 20.

Impact of $\mathrm{Nb}$ on $\theta (\xi )$ when $\mathrm{Nt}=0.5,B{i}_1=0.1,\mathrm{Rd}=0.4,\mathrm{Le}=0.7,\mathrm{Pr}=0.6,c=0.8$ .

Figure 21.

Impact of $\mathrm{Nb}$ on $\varphi (\xi )$ when $\mathrm{Nt}=0.4,B{i}_1=0.1,\mathrm{Le}=0.7,B{i}_2=0.2,c=0.8$ .

Figure 22.

Impact of $B{i}_1$ on $\theta (\xi )$ when $\mathrm{Nt}=0.5,\mathrm{Rd}=0.1,\mathrm{Le}=0.7,\mathrm{Nb}=0.4,\mathrm{Pr}=0.6,c=0.8$ .

Figure 23.

Impact of $B{i}_1$ on $\varphi (\xi )$ when $\mathrm{Nb}=0.1,\mathrm{Nt}=0.4,\mathrm{Le}=0.7,B{i}_2=0.2,c=0.8$ .

Figure 24.

Impact of $B{i}_2$ on $\varphi (\xi )$ when $\mathrm{Nb}=0.1,B{i}_1=0.2,\mathrm{Nt}=0.4,\mathrm{Le}=0.7,c=0.8$ .

Figures 25–28 are plotted to observe the residual error for velocity $f\prime (\xi )$ in x-direction, velocity $g\prime (\xi )$ in y-direction, $\theta (\xi )$ temperature, and concentration $\varphi (\xi )$ profiles, respectively.

Figure 25.

Residual error for $f(\xi )$ .

Figure 26.

Residual error for $g(\xi )$ .

Figure 27.

Residual error for $\theta (\xi )$ .

Figure 28.

Residual error for $\varphi (\xi )$ .

Figures 29–32 are plotted for the comparison of HAM and numerical method (ND-Solve technique) for velocity $f\prime (\xi )$ in x-direction, velocity $g\prime (\xi )$ in y-direction, $\theta (\xi )$ temperature, and concentration $\varphi (\xi )$ profiles, respectively.

Figure 29.

HAM and numerical comparison for $f\prime (\xi )$ .

Figure 30.

HAM and numerical comparison for $g\prime (\xi )$ .

Figure 31.

HAM and numerical comparison for $\theta (\xi )$ .

Figure 32.

HAM and numerical comparison for $\varphi (\xi )$ .

The impact of stretching parameter $c$ , Williamson parameter $\mathrm{We}$ , porosity parameter $\gamma$ , thermal radiation parameter $\mathrm{Rd}$ , thermophoresis parameter $\mathrm{Nt}$ and Brownian motion parameter $\mathrm{Nb}$ , and Prandtl number Pr on skin fraction $C_f$ , local Nusselt number $\mathrm{Nu}$ , and Sherwood number $\mathrm{Sh}$ are shown in Tables 1–3. The increasing stretching parameter $c$ reduces the skin fraction. This behavior is due to the indirect relation of the stretching parameter $c$ to skin fraction $C{f}_x$ in x-direction which resists the fluid flow, while in y-direction the stretching parameter has opposite behavior to the skin fraction $C{f}_y$ . The Williamson parameter $\mathrm{We}$ shows increasing behavior to the skin fraction $\mathrm{Cf}$ . This behavior is due to the particles of the fluid that take more time in restoring their position and as a result, the increase in viscosity arises, and the fraction of the fluid particles at the surface of the sheet increases. The porosity parameter $\gamma$ shows reducing behavior to the skin fraction $\mathrm{Cf}$ . Actually, the porous medium affects the boundary layer of the fluid which as a result accelerates the velocity of the fluid particles. This accelerating fluid particles increase the skin fraction $\mathrm{Cf}$ . The increasing stretching parameter $c$ increases is heat transfer rate $\mathrm{Nu}$ . This behavior is due to fact that the stretching parameter $c$ thicker the thermal and momentum boundary layer which slower the heat dissipation causing the heating up the plate. The Prandtl number Pr increases the heat transfer rate $\mathrm{Nu}$ . The thermal radiation $\mathrm{Rd}$ increases the heat transfer rate $\mathrm{Nu}$ . Physically, thermal radiation upsurges the boundary layer flow which results increase in heat transfer rate $\mathrm{Nu}$ . Thermophoresis $\mathrm{Nt}$ and Brownian motion $\mathrm{Nb}$ parameters reduce the heat transfer rate $\mathrm{Nu}$ . The increasing stretching parameter $c$ , thermophoresis $\mathrm{Nt}$ , and Brownian motion $\mathrm{Nb}$ parameters reduce the mass transfer rate $\mathrm{Sh}$ .

Table 1.

Numerical results of embedded parameters on skin fraction.

$c$	$\mathrm{We}$	$\gamma$	$C{f}_x$	$C{f}_y$
0.1	0.1	0.1	–0.969150	–0.454130
0.2			–0.970217	–0.455364
0.3			–0.971740	–0.581819
	0.2		–0.870760	–1.056940
	0.3		–0.768846	–0.825998
	0.4		–0.624216	–0.599542
		0.2	–0.617283	–0.620103
		0.3	–0.602723	–0.562401
		0.4	–0.579271	–0.444439

Table 2.

Numerical results of embedded parameters on local Nusselt number.

$c$	Pr	$\mathrm{Rd}$	$\mathrm{Nt}$	$\mathrm{Nb}$	$\mathrm{Nu}$
0.1	4.5	0.1	0.1	0.1	0.459589
0.2					0.459816
0.3					0.463456
	4.6				0.463902
	4.7				0.467118
	4.8				0.470111
		0.2			0.143878
		0.3			0.192457
		0.4			0.208134
			0.2		0.205091
			0.3		0.199132
			0.4		0.187810
				0.2	0.183980
				0.3	0.170990
				0.4	0.154704

Table 3.

Numerical results of embedded parameters on Sherwood number.

$c$	$\mathrm{Nt}$	$\mathrm{Nb}$	$\mathrm{Sh}$
0.1	0.1	0.1	0.459589
0.2			0.459816
0.3			0.465634
	0.2		0.443825
	0.3		0.433098
	0.4		0.423083
		0.2	0.431602
		0.3	0.412100
		0.4	0.397994

Conclusion

The three-dimensional Williamson fluid flow has been examined in this research article. The fluid flow is analyzed over porous stretched sheet with the influence of thermal radiation. The key findings are listed as follows:

• The velocity profile in x-direction is decreased with the augmented stretching, Williamson, porosity, and inertial coefficient parameters.

• The velocity profile in y-direction is increased with the enlarged stretching parameter, while reduced with the augmented Williamson, porosity, and inertial coefficient parameters.

• The temperature profile is increased with the enlarged stretching, thermophoresis, radiation, and Brownian motion parameters and Biot number while decreased with the increased Prandtl number.

• The concentration profile is increased with the increased thermophoresis parameter and Biot numbers while decreased with the enlarged stretching and Brownian motion parameters.

• The skin fraction is increased with the increased Williamson and porosity parameters while reduced with the increased stretching parameter.

• The Nusselt number is increased with the increased stretching and thermal radiation parameters, and Prandtl number while reduced with the enlarged Brownian motion and thermophoresis parameters.

• The Sherwood number is increased with the enlarged stretching parameter while reduced with the increased Brownian motion and thermophoresis parameters.