Magneto Mixed Convection of Williamson Nanofluid Flow through a Double Stratified Porous Medium in Attendance of Activation Energy

Abstract

This article aims to develop a mathematical simulation of the steady mixed convective Darcy–Forchheimer flow of Williamson nanofluid over a linear stretchable surface. In addition, the effects of Cattaneo–Christov heat and mass flux, Brownian motion, activation energy, and thermophoresis are also studied. The novel aspect of this study is that it incorporates thermal radiation to investigate the physical effects of thermal and solutal stratification on mixed convection flow and heat transfer. First, the profiles of velocity and energy equations were transformed toward the ordinary differential equation using the appropriate similarity transformation. Then, the system of equations was modified by first-order ODEs in MATLAB and solved using the bvp4c approach. Graphs and tables imply the impact of physical parameters on concentration, temperature, velocity, skin friction coefficient, mass, and heat transfer rate. The outcomes show that the nanofluid temperature and concentration are reduced with the more significant thermal and mass stratification parameters estimation.

1. Introduction

Nanotechnology is the technique of analyzing and separating or adding an object’s atoms and molecules that need to be made very small. Over the last thirty years, nanotechnology has significantly impacted vast applications in the petroleum industry, food production, medicine, nuclear energy, cooling of the reactor, and the polymer industry. Primarily, in 1995, Choi and Eastman [1] “coined the term nanofluid by incorporating the substance of nanoparticles into base fluids and theoretically demonstrated their efficiency”. Based on their findings, they noticed a considerable increase in the thermal conduction of the base liquid. Buongiorno [2] demonstrated the role of Brownian motion and thermophoresis in a nanofluid. The seven slip mechanisms were studied by Buongiorno and Buongiorno’s concepts: inertia, thermophoresis, gravity, Magnus effects, fluid drainage, Brownian diffusion, and diffusionphoresis.

Williamson [3] developed “the flow of Pseudo-plastic equation” and analyzed the properties of the pseudo-plastic flow holding three constants. These are the viscous constant, plasticity constant, and the ratio between the viscous constant and the plasticity constant. Nadeem et al. [4] tested the Williamson fluid model for 2D flows through a stretching sheet. The role of radiation and heat absorption on an incompressible pseudo-plastic Williamson fluid over the unsteady flow of the boundary layer via a porous stretched surface was explored by Hayat et al. [5] and Karthikeyan et al. [6], and they discovered that increasing the Weissenberg number decreases the skin friction coefficient. According to Zeeshan et al. [7], water- and engine oil-based CNTs flowed through a porous medium. On the other hand, the MHD Williamson fluid flow through a nonlinear curved surface in convective homogeneous and heterogeneous reactions was implemented by K. Ahmed et al. [8]. H. Waqas et al. [9] presented a numerical result for the Carreau–Yasuda nanofluid in a porous medium with bioconvective microorganisms. Nasir Shehzad et al. [10] determined that the suction/injection parameter was generated due to a constant and porous medium in the presence of a heat source, and a chemical reaction was observed. The cutting-edge reports in Williamson nanofluid flow with thermal radiation and heat generation are seen in [11,12,13,14,15,16,17,18,19].

Cattaneo [20] suggested a modified Fourier’s law that included a relaxation time element to overcome the paradox of Fourier’s law and heat conduction. Christov [21] extended Cattaneo’s theory by including Oldroyd derivatives, and it was named the Cattaneo–Christov model theory. Eswaramoorthi et al. [22] expressed the impact of a Williamson fluid flow of two-dimensional Darcy–Forchheimer on a Riga plate. Dual stratification and a double Catteneo–Christov flux were established for the energy equations. The contribution of Jeffery fluid flow to the non-Fourier heat flux model on a nonlinear stretched surface with double stratification was studied by Hayat et al. [23] and Shankar Goud [24]. Ali et al. [25] used the Cattaneo–Christov dual diffusion model to discuss the 3D incompressible unsteady effect of magneto-hydrodynamics on the transient rotating flow of Maxwell viscous nanofluid. The relation between thermal boundary layer and thermal relaxation time was observed in Abu-hamdeh et al. [26]. Rashid et al. [27] examined the thermal radiation effects of Darcy–Forchheimer Maxwell fluid flow along an exponentially stretching surface with activation energy. Shafiq et al. [28] “reported the influence of convective boundary conditions, thermal radiation and chemical reaction on the three-dimensional flow of Darcy Forchheimer nanofluid across a rotating surface with Arrhenius activation energy”. “Entropy formation, activation energy, and binary chemical reaction effects on the Darcy Forchheimer flow of Williamson nanofluid through a nonlinear stretchable flat surface were deliberated” by Ghulam Rasool et al. [29] and Hayat et al. [30].

The activation energy is the smallest quantity of energy forced to trigger a chemical reaction in a system. Energy exists in two types: kinetic and potential. A reaction among molecules could be incomplete due to kinetic energy loss or an inadequate collision. At this point, only the minimum amount of energy is required to initiate the chemical reaction. Bestman [31] was the first to investigate the impact of activation energy on natural action in a permeable boundary layer. Dawar et al. [32] addressed nonlinear stretching plates in magnetohydrodynamics pseudo-plastic nanofluid flow with activation energy. The method of homotopy analysis was performed by Alsaadi et al. [33] to examine the Arrhenius energy equation in the nanomaterial of magneto-Williamson flow. The influence of the activation energy, slip, porosity parameter, and entropy approach on the mixed convective flow of Darcy–Forchheimer along a stretched curved surface was noticed by Muhammad et al. [34]. Danook et al. [35] investigated the mixed convective heat transfer in a turbulent flow of nanofluid. Wasim Jamshed et al. [36] worked on the unsteady flow of a non-Newtonian Casson nanofluid with solar radiation using the Keller box method. The significance of MHD mixed convective flows Casson nanofluids over an elongating irregular surface immersed vertically in a Darcy–Brinkman porous medium was exploited by Alghamdi et al. [37]. Currently, investigators are analyzing the Arrhenius activation energy [38,39,40,41,42,43,44,45,46,47,48]. Other related studies have been conducted in [49,50,51,52,53,54,55].

For heat and mass transfer concepts, stratification is an essential component. Due to temperature differences, concentration variations, and differing fluid densities, it happens in inflow distribution. Heat and mass transport occur at the same time in the dual stratification process. Natural and mixed convection in a dual stratification medium are essential to study because of their applications. Groundwater reservoirs, industrial food, and regulating hydrogen and oxygen levels in the atmosphere are just a few examples of stratification. Sreelakshmi et al. [56] examined the steady flow of Maxwell fluid Darcy–Forchheimer over a stretching surface with thermal and solutal stratification. Darcy–Forchhemer MHD viscoelastic flow of nanofluid through a nonlinear stretching surface with dual stratification effects was deliberated by Hayat et al. [57]. Eswaramoorthi et al. [58] tested the impact of dual stratification and double non-Fourier heat flux model on the mathematical modeling of a Williamson fluid flow on a Darcy–Forchheimer over a Riga plate. Williamson fluid flow over a stretching in a linear surface was examined by Ahmed et al. [59] Some recent thermal and solutal stratification articles were found in [43,44,45,46,47,48,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70].

The vast majority of researchers collaborate on the mixed convective Darcy–Forchheimer flow with the non-Fourier heat flux model via the prescribed boundary layer but have not handled a dual stratified porous medium in Williamson nanofluid. Here, the gap was filled by the Williamson nanofluid on the double Cattaneo–Christov theory, radiation, dual stratification, and the impact of activation energy. The numerical findings were produced using the MATLAB bvp4c approach. Finally, in Williamson nanofluid flow, all the physical parameters were represented by a graphical process. This process is widely used in chemical and thermal engineering fields.

2. Development of the Flow Analysis

Assume a Williamson nanofluid’s steady flow through a linearly stretching surface in a Darcy–Forchheimer porous material. The Cattaneo–Christov theory, Arrhenius energy, thermal radiation, and magnetic field are studied. The flow process is revealed by thermal and solutal stratification. Throughout this work, the x and y directions represent velocity components of u and v, respectively (see Figure 1). The surface velocity is presumed to be $u_w$ = ax, where a > 0 denotes the stretching surface rate. The flow equation [69] is as follows:

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

(1)

$$\begin{gathered} u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=\vartheta \frac{{\partial }^{2}u}{\partial y^2}+\vartheta \sqrt{2}\wedge \frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\vartheta }{k_f}u+\frac{C_B}{x\sqrt{k_f}}{u}^2-\frac{\sigma B_0^2}{{\rho }_f}u \\ +\frac{1}{{\rho }_f}\left\{\begin{array}{c}\left(1-c_{\infty }\right)g{\rho }_{f\infty }{\mathsf{\Lambda }}_1\left(T-T_{\infty }\right)\\ -g\left({\rho }_p-{\rho }_{f\infty }\right)\left(C-C_{\infty }\right)\end{array}\right\} \end{gathered}$$

(2)

$$\begin{gathered} u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+{\mathrm{\Gamma }}_T\left(u^2\frac{{\partial }^{2}T}{\partial x^2}+v^2\frac{{\partial }^{2}T}{\partial y^2}+\left(u\frac{\partial u}{\partial x}\frac{\partial T}{\partial x}+v\frac{\partial u}{\partial y}\frac{\partial T}{\partial x}\right)+2uv\frac{\partial T^2}{\partial x\partial y}\right) \\ +u\frac{\partial v}{\partial x}\frac{\partial T}{\partial y}+v\frac{\partial v}{\partial y}\frac{\partial T}{\partial y} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{k_f}{{\left(\rho c\right)}_f}\frac{{\partial }^{2}T}{\partial y^2}+\frac{1}{{\left(\rho c\right)}_f}\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k_p}\frac{{\partial }^{2}T}{\partial y^2}+\frac{Q_1}{{\rho }_f{C}_p}\left(T-T_{\infty }\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\tau \left(D_m\left(\frac{\partial T}{\partial y}\right)\left(\frac{\partial C}{\partial y}\right)+\frac{D_n}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right) \end{gathered}$$

(3)

$$\begin{gathered} u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+{\mathrm{\Gamma }}_C\left[u^2\frac{{\partial }^{2}C}{\partial x^2}+v^2\frac{{\partial }^{2}C}{\partial y^2}+\left(u\frac{\partial u}{\partial x}\frac{\partial C}{\partial x}+v\frac{\partial u}{\partial y}\frac{\partial C}{\partial x}\right)+2uv\frac{{\partial }^{2}C}{\partial x\partial y}+u\frac{\partial v}{\partial x}\frac{\partial C}{\partial y} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+v\frac{\partial v}{\partial y}\frac{\partial C}{\partial y}\right]=D_m\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_n}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}-k_r^2\left(C-C_{\infty }\right){\left(\frac{T}{T_{\infty }}\right)}^{n}exp\left(-\frac{E_A}{kT}\right) \end{gathered}$$

(4)

The boundary conditions are

$$\begin{gathered} u=U_w\left(x\right)=ax,v=-V_w\left(x\right),T=T_w\left(x\right)=T_0+bx,C=C_w\left(x\right)=C_0+c_{1}x\mathrm{at}y=0, \\ u\to 0,\frac{\partial u}{\partial y}\to 0,T\to T_{\infty }=T_0+b_{1}x,C\to C_{\infty }=C_0+c_{2}x\mathrm{at}y\to \infty \end{gathered}$$

(5)

Consider

$$\eta =\sqrt{\frac{a}{\vartheta }}y,u=ax{f}^{\prime }\left(\eta \right),v=-\sqrt{a\vartheta }f\left(\eta \right)\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_0},\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_o}$$

(6)

Equations (2)–(4) can be modified as follows by using (6):

$$f^{‴}+f{f}^{″}-f^{{\prime }^2}+Wi{f}^{″}{f}^{‴}-K{f}^{\prime }+\lambda \left(\theta -B_N\varphi \right)-Fc{f}^{{\prime }^2}-M{f}^{\prime }=0$$

(7)

$$\begin{gathered} \frac{1}{Pr}\left(1+\frac{4}{3}Rd\right){\theta }^{″}-{\omega }_{\theta }\left(f^2{\theta }^{″}+\theta f^{{\prime }^2}+f^{{\prime }^2}{S}_{\theta }-f{f}^{\prime }{\theta }^{\prime }-f{f}^{″}\theta -f{f}^{″}{S}_{\theta }\right) \\ +N_B{\varphi }^{\prime }{\theta }^{\prime }+N_T{\theta }^{{\prime }^2}+H_A\theta -f^{\prime }\theta +f{\theta }^{\prime }-S_{\theta }{f}^{\prime }=0 \end{gathered}$$

(8)

$$\begin{gathered} \frac{1}{SC}{\varphi }^{″}+f{\varphi }^{\prime }-{\omega }_{\varphi }\left(f^2{\varphi }^{″}-f{f}^{″}{S}_{\varphi }-f{f}^{″}\varphi +f^{{\prime }^2}\varphi +f^{{\prime }^2}{S}_{\varphi }-f{f}^{\prime }{\varphi }^{\prime }\right)-f^{\prime }\varphi +f^{\prime }{S}_{\varphi }+\frac{N_T}{SC{N}_B}{\theta }^{″} \\ -{\mathsf{\sigma }}_{\mathrm{o}}\varphi {\left(1+\theta {\mathsf{\delta }}_{\mathrm{o}}\right)}^n\exp \left(\frac{-E}{1+\theta {\mathsf{\delta }}_{\mathrm{o}}}\right)=0 \end{gathered}$$

(9)

The boundary conditions are

$$\begin{gathered} \eta \to 0,f\left(0\right)=fw;f^{\prime }\left(0\right)=1;\theta \left(0\right)=1-S_{\theta };\varphi \left(0\right)=1-S_{\varphi }, \\ \eta \to \infty ,f^{\prime }(\infty )=0;\theta (\infty )=0;\varphi (\infty )=0 \end{gathered}$$

(10)

$E=\frac{E_A}{k{T}_{\infty }}$, $Fc$= $\frac{C_B}{\sqrt{k_f}}$, $Gr=\left(\frac{\left(g\beta \left(1-C_{\infty }\right)\left(T_w-T_{\infty }\right)x^3\right)}{{\vartheta }^2}\right)$, $H_A=\frac{Q_1}{{\rho }_f{C}_{p}a}$, $fw$ = −$\frac{V_w}{\sqrt{a\vartheta }}$, $\mathrm{M}=\frac{\sigma B_0^2}{{\rho }_{f}a}$, $Sc=\frac{\vartheta }{D_B},N_B=\frac{\tau D_B{C}_{\infty }}{\vartheta }$, $B_N=\left(\frac{\left({\rho }_p-{\rho }_{f_{\infty }}\right)C_{\infty }}{{\rho }_{f_{\infty }}{\wedge }_1\left(1-C_{\infty }\right)\left(T_w-T_0\right))}\right)$, $Pr=\frac{k_f}{{\left(\rho c\right)}_f}$, $R{e}_x=\frac{U_{w}x}{\vartheta }$, $Wi=\wedge x\sqrt{\frac{2a^3}{\vartheta }}$, $\lambda =\left(\frac{Gr}{R{e}_x^2}\right)=\left(\frac{\left(g{\wedge }_1\left(1-C_{\infty }\right)\left(T_w-T_{\infty }\right)\right)}{a^{2}x}\right)$, $N_T=\left(\frac{\tau D_T\left(T_w-T_0\right)}{T_{\infty }\vartheta }\right)$, $R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k_p{k}_f}$, $\delta =\left(\frac{\left(T_w-T_0\right)}{T_{\infty }}\right)$.

Physical quantities for skin friction, Nusselt, and Sherwood number are obtained as follows:

$$C_f=\frac{2{\tau }_{\omega }}{\rho U_{\omega }^2},Nu=\frac{x{q}_{\omega }}{k_f({\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_0)},S{h}_x=\frac{x{j}_{\omega }}{D_m({\mathrm{C}}_{\mathrm{w}}-{\mathrm{C}}_0)}$$

(11)

where

$$\begin{array}{c}{\tau }_{\omega }=\mu \left(\frac{\partial u}{\partial y}\left[1+\bigwedge \sqrt{\frac{1}{2}}\frac{\partial u}{\partial y}\right]\right)\\ q_{\omega }=-\left(k_f\frac{\partial T}{\partial y}+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k_p}\frac{\partial T}{\partial y}\right)\\ j_{\omega }=-D_m\frac{\partial C}{\partial y}\end{array}$$

The following are the dimensionless parts of local skin friction, heat, and mass transfer rates.

$$\begin{array}{c}\frac{1}{2}{C}_{f}R{e}^{\frac{1}{2}}=f^{″}\left(0\right)+\frac{Wi}{2}{f}^{″}{\left(0\right)}^2,\\ R{e}^{-\frac{1}{2}}Nu=-\left(1+\frac{4}{3}Rd\right){\theta }^{\prime }\left(0\right),\\ R{e}^{-\frac{1}{2}}S{h}_x=-{\varphi }^{\prime }\left(0\right).\end{array}$$

(12)

3. The Solution Methodology

A system of nonlinear ODEs (7)–(9) with boundary conditions (10) is solved via the MATLAB bvp4c code. The problems are converted into first-order ODEs using the mathematical algorithm described below (Figure 2).

Let $f=y\left(1\right),$ $f^{\prime }=y\left(2\right),$ $f^{″}=y\left(3\right),\theta =y\left(4\right),$ ${\theta }^{\prime }=y\left(5\right),$ $\varphi =y\left(6\right)$, and ${\varphi }^{\prime }=y\left(7\right)$. The following is a list of first-order ODEs:

$$\begin{array}{c}{y}^{\prime }\left(1\right)=y\left(2\right),\hfill \\ y^{\prime }\left(2\right)=y\left(3\right),\hfill \\ y^{\prime }\left(3\right)=yy1=\left(\frac{1}{1+Wiy\left(3\right)}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}*(-y\left(1\right)y\left(3\right)+y{\left(2\right)}^2+Ky\left(2\right)+Fcy{\left(2\right)}^2+My\left(2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\lambda \left(y\left(4\right)-B_{N}y\left(6\right)\right)),\hfill \\ y^{\prime }\left(4\right)=y\left(5\right),\hfill \\ y^{\prime }\left(5\right)=yy2=\left(\frac{1}{1+\left(\frac{4}{3}\right)Rd}-Pr{\omega }_{\theta }y{\left(1\right)}^2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}*(-Pry\left(1\right)y\left(5\right)+Pry\left(2\right)y\left(4\right)+Pr{S}_{\theta }y\left(2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+Pr{\omega }_{\theta }(y\left(4\right)y{\left(2\right)}^2+y{\left(2\right)}^2{S}_{\theta }-y\left(1\right)y\left(2\right)y\left(5\right)-y\left(1\right)y\left(3\right)y\left(4\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-y\left(1\right)y\left(3\right)S_{\theta })-Pr{H}_{A}y\left(4\right)-Pr{N}_{B}y\left(7\right)y\left(5\right)-Pr{N}_{T}y{\left(5\right)}^2),\hfill \\ y^{\prime }\left(6\right)=y\left(7\right),\hfill \\ y^{\prime }\left(7\right)=yy3=\left(1/\left(1-Sc{\omega }_{\varphi }y{\left(1\right)}^2\right)\right)*(-Scy\left(1\right)y\left(7\right)+Scy\left(2\right)y\left(6\right)+Sc{S}_{\varphi }y\left(2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+Sc{\omega }_{\varphi }(y\left(6\right)y{\left(2\right)}^2+y{\left(2\right)}^2{S}_{\varphi }-y\left(1\right)y\left(2\right)y\left(7\right)-y\left(1\right)y\left(3\right)y\left(6\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-y\left(1\right)y\left(3\right)S_{\varphi })-\left(N_T/N_B\right)*yy2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+Sc{\sigma }_o{\left(1+{\delta }_{o}y\left(4\right)\right)}^{n}y\left(6\right)\exp \left(-E/\left(1+{\delta }_{o}y\left(4\right)\right)\right))\hfill \end{array}$$

with boundary condition

$$\begin{array}{c}y0\left(1\right)=fw,y0\left(2\right)=1,y0\left(4\right)=\left(1-S_{\theta }\right),y0\left(6\right)=\left(1-S_{\varphi }\right),\hfill \\ y\inf \left(2\right)=0,y\inf \left(4\right)=0,y\inf \left(6\right)=0\hfill \end{array}$$

4. Results and Discussion

Table 1. Correlation of Nusselt number $\left(NuR{e}^{-\frac{1}{2}}\right)$ when Wi = Fc = K = R =${\omega }_{\theta }$=$H_A={\omega }_{\varphi }=0$, M =$B_N=N_B=0.5$, Sc = 5, and ${\delta }_0=1$.

Pr	${\mathit{N}}_{\mathit{T}}$	$\mathit{E}$	${\mathit{\sigma }}_0$	$\mathit{n}$	$\mathit{\lambda }$	$\mathit{N}\mathit{u}\mathit{R}{\mathit{e}}^{-1/2}$
						Mustafa et al. [69]	Present
2	0.5	1	1	0.5	0.5	0.706605	0.706604
4	0.5	1	1	0.5	0.5	0.935952	0.935955
7	0.5	1	1	0.5	0.5	1.132787	1.132788
10	0.5	1	1	0.5	0.5	1.257476	1.257482
5	0.1	1	1	0.5	0.5	1.426267	1.426269
5	0.5	1	1	0.5	0.5	1.013939	1.013938
5	0.7	1	1	0.5	0.5	0.846943	0.846928
5	1.0	1	1	0.5	0.5	0.649940	0.649939
5	0.5	0	1	0.5	0.5	0.941201	0.941209
5	0.5	1	1	0.5	0.5	1.013939	1.013943
5	0.5	2	1	0.5	0.5	1.064551	1.064563
5	0.5	4	1	0.5	0.5	1.114549	1.114191
5	0.5	1	0	0.5	0.5	1.145304	1.145301
5	0.5	1	1	0.5	0.5	1.013939	1.013938
5	0.5	1	2	0.5	0.5	0.926282	0.926281
5	0.5	1	5	0.5	0.5	0.798671	0.798669
5	0.5	1	2	−1	0.5	1.030805	1.030804
5	0.5	1	2	−0.5	0.5	0.999470	0.999468
5	0.5	1	2	0	0.5	0.964286	0.964285
10	0.5	1	2	1	0.5	0.886830	0.886830
10	0.5	1	2	0.5	0	1.032281	1.032280
10	0.5	1	2	0.5	0.5	1.056704	1.056706
10	0.5	1	2	0.5	3	1.154539	1.154538
10	0.5	1	2	0.5	5	1.215937	1.215938

shows the association between Nusselt numbers taken from Mustafa et al.’s results and our results. We were able to match our results identically to Mustafa’s results. The quantitative data of the skin friction drag force ($1/2C_{f}R{e}^{1/2}$), heat transfer rate ($NuR{e}^{-1/2}$), and Sherwood number ($S{h}_{x}R{e}^{1/2}$) for the several values of Richardson number λ, Weissenberg number Wi, Forchheimer number Fc, Magnetic parameter M, and suction/injection parameter fw were presented in

Table 2. Numerical analysis of $1/2C_{f}R{e}^{1/2}$,$NuR{e}^{-1/2}$, and $S{h}_{x}R{e}^{1/2}$ for different parameters Wi, Fc, λ, M, and fw.

Wi	Fc	λ	M	fw	$1/2{\mathit{C}}_{\mathit{f}}\mathit{R}{\mathit{e}}^{1/2}$	$\mathit{N}\mathit{u}\mathit{R}{\mathit{e}}^{-1/2}$	$\mathit{S}{\mathit{h}}_{\mathit{x}}\mathit{R}{\mathit{e}}^{-1/2}$
0	0.4	0.5	0.5	0.3	−1.493123	1.667677	0.688683
0.1	0.4	0.5	0.5	0.3	−1.455877	1.661396	0.681731
0.2	0.4	0.5	0.5	0.3	−1.41351	1.653626	0.673835
0.3	0.4	0.5	0.5	0.3	−1.362763	1.643289	0.664469
0.2	0	0.5	0.5	0.3	−1.329383	1.662128	0.682787
0.2	0.2	0.5	0.5	0.3	−1.372209	1.657807	0.678189
0.2	0.4	0.5	0.5	0.3	−1.41351	1.653626	0.673835
0.2	0.6	0.5	0.5	0.3	−1.45342	1.649575	0.669706
0.2	0.4	0	0.5	0.3	−1.470747	1.646644	0.666783
0.2	0.4	0.2	0.5	0.3	−1.44786	1.64946	0.669584
0.2	0.4	0.4	0.5	0.3	−1.424968	1.652245	0.672411
0.2	0.4	0.6	0.5	0.3	−1.402068	1.655	0.675269
0.2	0.4	0.5	0	0.3	−1.167756	1.642785	0.789742
0.2	0.4	0.5	0.5	0.3	−1.41351	1.653626	0.673835
0.2	0.4	0.5	1	0.3	−1.56368	1.632112	0.646948
0.2	0.4	0.5	1.5	0.3	−1.696232	1.612796	0.626083
0.2	0.4	0.5	0.5	−0.3	−1.13794	1.366538	0.605637
0.2	0.4	0.5	0.5	−0.1	−1.22469	1.476979	0.612521
0.2	0.4	0.5	0.5	0.1	−1.316926	1.576889	0.632016
0.2	0.4	0.5	0.5	0.3	−1.41351	1.653626	0.673835

. Moreover, it was discovered that as Wi and λ values grow, the skin friction coefficient also increases, whereas it was significantly decreased when Fc, fw, and M increased.

Table 3. Numerical study of $NuR{e}^{-1/2}$ for various parameters R, ${\omega }_{\theta }$, and $S_{\theta }$.

R	${\mathit{\omega }}_{\mathit{\theta }}$	${\mathit{S}}_{\mathit{\theta }}$	$\mathit{N}\mathit{u}\mathit{R}{\mathit{e}}^{-1/2}$
0	0.1	0.2	1.292138
0.5	0.1	0.2	1.653626
1	0.1	0.2	1.877266
1.5	0.1	0.2	1.926091
0.5	−0.1	0.2	1.64429
0.5	0	0.2	1.662016
0.5	0.1	0.2	1.653626
0.5	0.2	0.2	1.561214
0.5	0.1	0	1.849146
0.5	0.1	0.1	1.753203
0.5	0.1	0.2	1.653626
0.5	0.1	0.3	1.550338

and

Table 4. Numerical results of $S{h}_{x}R{e}^{1/2}$ for different parameters Sc, $N_T$, ${\omega }_{\varphi }$, and $S_{\varphi }$.

Sc	${\mathit{N}}_{\mathit{T}}$	${\mathit{\omega }}_{\mathit{\varphi }}$	${\mathit{S}}_{\mathit{\varphi }}$	$\mathit{S}{\mathit{h}}_{\mathit{x}}\mathit{R}{\mathit{e}}^{-1/2}$
0.5	0.5	0.1	0.2	0.015548
1	0.5	0.1	0.2	0.673835
1.5	0.5	0.1	0.2	1.186802
2	0.5	0.1	0.2	1.628294
1	0.2	0.1	0.2	1.038076
1	0.3	0.1	0.2	0.911577
1	0.4	0.1	0.2	0.790267
1	0.5	0.1	0.2	0.673835
1	0.5	0	0.2	0.591323
1	0.5	0.1	0.2	0.673835
1	0.5	0.2	0.2	0.759523
1	0.5	0.3	0.2	0.848478
1	0.5	0.1	0	0.899109
1	0.5	0.1	0.1	0.786382
1	0.5	0.1	0.2	0.673835
1	0.5	0.1	0.3	0.561474

incorporated the effects of the embedded parameters Radiation R, Thermal relaxation time parameter ${\omega }_{\theta }$, Thermal stratification $S_{\theta }$, Thermophoresis parameter $N_T$, Mass relaxation parameter ${\omega }_{\varphi },$ Mass stratification $S_{\varphi }$, and Schmidt number Sc on heat and mass diffusion rates. The higher variation of ${\omega }_{\theta }$, $S_{\theta }$, $N_T$, and $S_{\varphi }$ was related to the reduced mass and heat transfer rates. It was also accelerated as the thermal radiation, Schmidt number, and mass relaxation time were increased.

The embedded parameters with fixed values $Wi=0.2,{\mathsf{\delta }}_{\mathrm{o}}={\omega }_{\theta }=0.1,{\omega }_{\varphi }=0.1,\mathrm{Pr}=2,Fc=0.4,\mathrm{K}=0.2,{\mathsf{\sigma }}_{\mathrm{o}}=1,B_N=0.5,\mathrm{R}=0.5,H_A=-0.5,N_B=0.5,N_T=0.5,\mathrm{n}=0.5,fw=0.3,S_{\theta }=0.2,\mathrm{Sc}=1.0,M=0.5,\lambda =0.5,E=1\mathrm{and}{S}_{\varphi }=0.2$ are existence in velocity $f^{\prime }$, temperature $\theta$ and concentration $\varphi$ profiles. Specifications of the suction/injection parameter $fw$ on velocity field $f$ with the range of $0\le \eta \le 4$ the fluid velocity profile of the nanofluid while comparing the values of Fc and K were noted in Figure 3. In the graph flow between Fc and K, then values of $fw$ varied from $fw$= −0.3 to +0.3 increase, and the velocity profile $f^{\prime }$ of the graph automatically decreases. The influence of mixed convection parameters $\lambda$ on the velocity field for various values of the parameters is shown in Figure 4. As shown in Figure 4, the velocity field decays with raising values of $\lambda$ = 0 to 0.6. Supporting flow is represented on a warm surface by $\lambda$ values greater than zero, whereas resisting flow is shown on a cold surface by $\lambda$ values less than zero. The influence of the Forchheimer number Fc on the velocity profile is seen in Figure 5 for varied $Wi$ = 0.0 and 0.4 values. It is determined that the velocity field diminishes when the Forchheimer number rises. The effect of the Weissenberg number $Wi$ on velocity flow is seen in Figure 6 for both magnetic field parameter scenarios. Figure 6 shows that increasing $Wi$ depreciates the velocity field. The relaxation time is prolonged, which restricts fluid motion. The Weissenberg number is used in physical studies of viscoelastic flows to test the impact of elastic to viscous forces. As a result of lowering the velocity of the boundary layer, the Forchheimer number Fc and the porous medium K have different parameter values. In Figure 7 and Figure 8, we present the variation of the Weissenberg number $Wi$ and the magnetic field M of the base flow of the velocity profile. Figure 7 and Figure 8 show that increasing $Wi$ and M lowers the velocity distribution. The Lorentz force is formed when the magnetic field is strengthened. This force aids in reducing the velocity distribution as well as $Wi$.

Specifications of the heat absorption $H_A$ on temperature profile $\theta$ within the range of $0\le \eta \le 8$ were established in Figure 9. In the graph, the flow depends on thermal radiation values. When the values of $H_A$ increase, the temperature field also increases. Figure 10 demonstrates the impact of the suction/injection parameter on temperature for different parameters of the Forchheimer number Fc. The temperature field reduces for higher values of fw. Figure 11 reveals the influence of the radiation parameter R on the temperature profile. In these cases, the temperature profile was increased for various values of the thermal radiation grown by varying the thermal relaxation time parameters ${\omega }_{\theta }$. For various Brownian motion and chemical reaction parameters, Figure 12 and Figure 13 indicate the performance of growing levels of thermal and solutal stratification parameters. The stratification parameter is the ratio of free stream temperature to fluid surface temperature. A significantly larger stratification parameter causes a rise in free stream temperature or a decrease in a nanofluid stream, whereas the concentration profile exhibits the inverse correlation. Temperature and concentration distribution within the boundary layer and the ambient fluid were reduced as the $S_{\theta }$ and $S_{\varphi }$ values increased. The profiles of non-dimensional temperature and concentration against thermal relaxation time ${\omega }_{\theta }$ and mass relaxation time parameter ${\omega }_{\varphi }$ were plotted in Figure 14 and Figure 15. The temperature field shrinks for $N_T$ = 0.0 and 0.5 as ${\omega }_{\theta }$ increases. For $E=0.0,0.5,$ an upsurge in ${\omega }_{\varphi }$ decreases the concentration gradient. Figure 16 shows the effect of Sc on the concentration profile with the range of $0\le \eta \le 8$ when $N_T$= 0.0. It was discovered that concentration decreased as Sc increased. Because of this, the Schmidt number has an opposite relation with mass diffusivity. The characteristics of the thermophoresis parameter $N_T$ on the profile of concentration were observed in Figure 17 for $S_{\varphi }$ = 0.0 and 0.2. The concentration here rises as a function of $S_{\varphi }$. The presence of a high value of $N_T$ helps reduce the concentration boundary layer.

Figure 18, Figure 19 and Figure 20 show the role of numerous parameters on skin friction coefficients and heat and mass transfer rates. We concluded from Figure 18a,b that the skin friction coefficient rises at fw and Wi while also increasing in Ri and Wi. Figure 19a,b display a lower Nusselt number due to a lower R and ${\omega }_{\theta }$ as well as diminished $N_T$ and $S_{\theta }$. Figure 20a,b depict the mass diffusion rate for various estimates of fw and $S_{\varphi }$. The mass diffusion rate increased in this case as the values of fw and $S_{\varphi }$ increased, and Sc and ${\omega }_{\varphi }$ also increased.

5. Conclusions

The numerical and analytical results of the Darcy–Forchheimer Williamson nanofluid flow through a linear stretched surface were observed in this paper. In addition, the effects of thermal and solutal stratification, activation energy, and the Cattaneo–Christov dual flux were all considered. The following are the outcomes of this work.

• The velocity profile was reduced by the Weissenberg number and Forchheimer number, while the mixed convective parameter shows the increasing tendency in velocity profile.
• The temperature distribution was raised with a high thermal relaxation time and radiation values.
• For higher estimations of Schmidt number and mass relaxation time, the concentration profile diminished.
• Increases in the thermal and mass stratification parameters reduce the temperature and concentration profile.
• Heat and mass transfer rates were declined for large values of thermal radiation, thermal relaxation time, mass stratification, and suction parameter.

The findings discussed in this work should benefit scientists and engineers in various chemical and thermal engineering applications such as nuclear reactors, cooling systems, and hybrid power systems.