Joule Heating Effect of Buongiorno Model of Hybrid Nanofluid Flow With Homogeneous-Heterogeneous Reaction and Heat Generation/Absorption

1. Introduction

A nanofluid is a fluid that has been mixed with a base fluid, like water, ethylene glycol, or oil, and contains nanoparticles, which are particles smaller than 100 nanometers. Researcher are highly interested in studying nanofluids due to their enhance thermal and physical properties compared to base fluid. Nanoparticles in nanofluids can significantly increase thermal conductivity, improve heat transfer rates, and other properties like viscosity and stability. Nanofluid have a numerous of potential uses like cooling and heat transfer, energy sector, biomedical applications, manufacturing and materials, Automotive and Transportation, renewable energy, nanofluid based sensors and devices. The new class of heat transmission fluid, known as a nanofluid, was created by suspending nanoparticles in a base fluid and was considered by Choi and Eastman [1]. Xuan et al. [2] scrutinized performance of heat transmission of NF and predicting convective heat transmission of nanofluid by single phase fluids and multiphase feature of the nanofluid. Buongiorno [3] scrutinized the impacts of thermophoresis and Brownian diffusion on the properties of NF for boundary layer flow. Dharmalingam et al. [4] described experimently and mathematically flow of NF with heat transmission. Tielke et al. [5] investigated water based nanofluid for their thermal conductivity measurements and potential applications. A novel scaled correlation to forecast the measurement of NF thermal conductivity was presented by Coccia et al. [6]. Alami et al. [7] explored the implications and limitations of conventional heat transmission coefficient calculation methods. Eswara et al. [8] studied bioconvection nanofluid flow on stretching porous media permeable sheet with MHD. The 2-D micropolar NF laminar incompressible flow in a steady channel with MHD was studied by Alahmadi et al. [9]. Ali et al. [10] examined 2-D NF flow with EMHD, variable heat flux, thermal radiation. The research work related to nanofluid flow has seen in [11,12,13].

Hybrid nanfluid are a type of nanofluid that combines nanoparticles with different types, shapes, or materials to form a synergistic mixture within a base fluid. These combinations include a mix of metallic nonmetallic, or polymeric nanoparticles. The goal is to enhance specific properties by leveraging the unique characteristics of each type of nanoparticle. Research on hybrid nanofluids is ongoing, exploring different nanoparticle combinations and base fluid to achieve enhanced heat transfer, rheological behavior, and stability. Hybrid nanofluids have a significance application in various fields like heat exchangers and cooling system, Solar thermal system, electronics and thermal management, nuclear reactors, biomedical applications, manufacturing and materials processing, automotive, aerospace, renewable energy etc. Rosca et al. [14] studied Buongiorno model of HNF flow on a permeable stretch/shrink sheet. Hayat et al. [15] investigated 3-D HNF rotating flow on stretch sheet with radiation and heat generation impacts. Boundary layer flow of HNF with heat transmission on stretch/shrink sheets was discussed by Waini et al. [16]. El-Zahar et al. [17] described thermal convective boundary condition of HNF flow through a circular cylinder with MHD impact. The uses and advantages of HNF in photovoltaic systems and solar energy were covered by Rasheed et al. [18]. waini et al. [19] explored radiated HNF flow on steady nonlinear stretch/shrink sheet. Nadeem et al. [20] scrutinized HNF flow on a convective heat permeable shrink/stretch sheet at a stagnation point with MHD impacts. MHD HNF flow on an exponential stretch/shrink surface with heat generation/apsorption was studied by Sarfaraz et al. [21] and bvp4c is used for numerical solution. The related work has been seen in [22,23,24].

Homogenous and heterogeneous reactions refer to different types of chemical reactions on based on the phases of the reactants involved. In homogeneous reaction, all the products and reactants are in the same phase (e.g., all liquid, all solid, or all gaseous). In a heterogeneous reaction, reactant and products are in different phases (e.g., gas reaction with a solid, liquid reacting with a gas, etc). Homogenous and heterogeneous reactions have many applications in chemistry and other related fields, some of them are catalysis, environmental science, chemical engineering, material science, biological system, etc. HNF flow at the stagnation point on a stretch/shrink sheet with heterogeneous- homogeneous reactions was described by Waini et al. [25]. Xu et al. [26] explored Buongiorno Model of NF with homogeneous - heterogeneous reaction. Alarabi et al. [27] discussed HNF flow for Darcy-Forchheimer in stretch/shrink cylinder with MHD, Joule heating, and hetrogeneous-homogeneous reaction. Ramzan et al [28] studied NF flow with MHD impacts and homogeneous and heterogeneous reaction and utilized bvp4c for numerical computation. Anuar et al. [29] explored MHD flow of HNF at a stagnation point with heterogeneous - homogeneous reaction. Bala et al [30] scrutinized Casson NF flow on an unsteady porous media stretch/shrink sheet with heat source/sink and heterogeneous-homogeneous reaction.

The branch of physics deals with the study of the behavior of electrically conducting fluids in the occurrence of magnetic fields, including salt water, liquid metals, and plasma is known as MHD. Some uses of MHD are astrophysics, fusion energy research, electric power generation, MHD propulsion, geophysics, material processing, plasma and space research, environmental science, etc. Khashi’ie et al. [31] examined HNF flow with heat transmission and Joule heating effects on a moving plate. Reddy et al. [32] scrutinized MHD effects of viscous fluid flow with heat transmission in a porous media cylinder. Abbas et al. [33] explored incompressible MHD fluid flow on steady cylinder with variable thermal conductivity. MHD 2-D Williamson HNF incompressible flow with Joule heating influence was deliberated by Rashad et al. [34]. MHD Casson NF flow on stretched surface was scrutinized by Suresh et al. [35] along with the impacts of activation energy and thermal radiation. Hybrid nanofluid obtained by adding $A{l}_2{O}_3$ and $Cu$ in Base fluid as shown in Figure 1.

The following points highlight the novelty, goal, and aspects that this research communication and paper attempt to investigate:

• The mathematical 2-D flow model of HNF over porous stretch/shrink sheet using water as the base fluid and $Cu,A{l}_2{O}_3$ as nanoparticles.
• This analysis investigates an appropriate interpretation of Darcy-Forchhiemer and Buongiorno models of HNF flow on stretch/shrink sheet.
• The use of significant mechanisms such as heat generation/absorption, viscous dissipation, and Joule heating effect, homogeneous and heterogeneous reaction enhances the originality of this work.
• A system of PDEs is converted into a set of ODEs using similarity variables, and then HAM is applied for the solution of the obtained ODEs.
• Tables and graphs are used to explain the results of the numerical analysis. The percentage % comparison of NF and HNF of velocity and energy are shown through graph.

2. Problem Formulation

In this study, we examine Darcy-Forchheimer MHD flow of HNF with heterogeneous-homogeneous reaction on steady stretch/shrink sheet. Velocity components in $x-$ and $y-$ directions are signified as $\left(u,v\right)$. The plate normal coordinate is in $y$-direction, and stream wise flow is in $x-$ direction. It is supposed that the mass flux velocity is with$v_0$, $v_0<0$ for suction and $v_0>0$ for injection or withdraw of the fluid and the surface velocity is $u_w\left(x\right)$. Surface and ambient temperatures and concentrations are given as $T_{\infty },T_w$, and $C_{\infty },C_w$.

Homogeneous-heterogeneous reactions can write as

$$A+2B\to 3B,\hspace{0.17em}\hspace{0.17em}\mathrm{rates}=k_{1}a{b}^2$$

(1)

$$\hspace{0.17em}A\to B,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{rates}=k_{s}a$$

(2)

Where the chemical concentrations $a$ and $b$ for species, $A,B$, with rate constant $k_1$ and $k_s$. The governing equations of the flow problem are

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

(3)

$$\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)u=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}u}{\partial y^2}\right)-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}\left(B_0^{2}u\right)-\frac{{\nu }_{hnf}}{K}u-\frac{C_b}{\sqrt{K}{\rho }_{hnf}}{u}^2,$$

(4)

$$\begin{array}{l}\left(v\frac{\partial }{\partial y}+u\frac{\partial }{\partial x}\right)T=\frac{k_{hnf}}{{\left(\rho c_p\right)}_{hnf}}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+{\left(\frac{\partial u}{\partial y}\right)}^2\frac{{\sigma }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}\\ +\tau \left(D_B\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+{\left(\frac{\partial T}{\partial y}\right)}^2\frac{D_T}{T_{\infty }}\right)+\frac{{\mu }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}{\beta }_0^2{u}^2+\frac{Q_0}{{\left(\rho c_p\right)}_{hnf}}\left(T-T_{\infty }\right)-\frac{1}{{\left(\rho c_p\right)}_{hnf}}\frac{\partial q_r}{\partial y},\end{array}$$

(5)

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

(6)

$$\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)a=D_A\left(\frac{{\partial }^{2}a}{\partial y^2}\right)-k_1\left(a{b}^2\right),$$

(7)

$$\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)b=D_B\left(\frac{{\partial }^{2}a}{\partial y^2}\right)+k_1\left(a{b}^2\right),$$

(8)

$$\begin{array}{l}u=u_w\left(x\right)=U_w\left(x\right)\lambda ,v=v_0,T=T_w,D_B\left(\frac{\partial C}{\partial y}\right)+\frac{D_T}{T_{\infty }}\left(\frac{\partial T}{\partial y}\right)=0,\\ D_A\left(\frac{\partial a}{\partial y}\right)=k_s\left(a\right),D_B\left(\frac{\partial b}{\partial y}\right)=-k_s\left(a\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}at\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y=0,\end{array}$$

(9)

$$u=u_e\left(x\right)\to 0,C\to C_{\infty },T\to T_{\infty },a\to a_0,b\to 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}at\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y\to \infty .$$

(10)

Where the temperature of HNF is $T$ and $C$ is the concentration. While, $D_A,D_B$ are the diffusion coefficients of species $A,B$ and $a_0>0$. Thermophoretic diffusion coefficient is $D_T$,$F=\frac{C_b}{x\sqrt{K}}$ is the inertia coefficient. Table 1 and Table 2 signify the thermophysical possessions of HNF.

Specific heat capacity of HNF is ${\left(\rho c_p\right)}_{hnf}$, density of HNF is ${\rho }_{hnf}$, density of base fluid is ${\rho }_f$, dynamic viscosity of HNF is ${\mu }_{hnf}$, electrical conductivity of base fluid is $\sigma$, ${\upsilon }_f$ is the kinematic viscosity of base fluid and electrical conductivity of HNF is ${\sigma }_{hnf}$. $k_{hnf},k_f$ is the thermal conductivity of HNF and base fluid. Nanoparticle’s volume frictions are ${\varphi }_1\left(A{l}_2{O}_3\right),{\varphi }_2\left(Cu\right)$.

Similarity transformations are;

$$\begin{array}{l}\eta =y{\left(\frac{a}{{\nu }_f}\right)}^{1/2},v=-{\left({\nu }_{f}a\right)}^{1/2}f\left(\eta \right),u=ax{f}^{\prime }\left(\eta \right),\Phi =\frac{C-C_{\infty }}{C_w-C_{\infty }},\\ \theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},a=a_{0}g\left(\eta \right),b=a_{0}h\left(\eta \right),v_0=-{\left({\nu }_{f}a\right)}^{1/2}S\end{array}$$

(11)

By using the Equation (11) in Equations (3)–(10), Equation (3) is identically proved, other equations are gotten as;

$$f^{‴}\left(\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}\right)-{f^{\prime }}^2+f{f}^{″}-M{f}^{\prime }\left(\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}\right)-\left(\frac{{\nu }_{hnf}}{{\nu }_f}\right)k_1{f}^{\prime }-\left(\frac{{\rho }_f}{{\rho }_{hnf}}{F}_1\right){\left(f^{\prime }\right)}^2=0,$$

(12)

$$\begin{array}{l}\frac{1}{{\left(\rho c_p\right)}_{hnf}/{\left(\rho c_p\right)}_f}\left(k_{hnf}/k_f+\frac{4}{3}Rd\right){\theta }^{″}+\frac{\frac{{\mu }_{hnf}}{{\mu }_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{\left(\rho c_p\right)}}Ec{f^{″}}^2+\left(N_b{\theta }^{\prime }{\Phi }^{\prime }+N_t{{\theta }^{\prime }}^2\right)\\ +\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{\left(\rho c_p\right)}}EcM{f^{\prime }}^2+\mathrm{Pr}f{\theta }^{\prime }+\frac{1}{{\left(\rho c_p\right)}_{hnf}/{\left(\rho c_p\right)}_f}{Q}_1\theta =0,\end{array}$$

(13)

$${\Phi }^{″}+Scf{\Phi }^{\prime }+\frac{N_t}{N_b}{\Theta }^{″}=0,$$

(14)

$$\frac{1}{Sc}\left(g^{″}\right)+g^{\prime }f-K{h}^{2}g=0,$$

(15)

$$\frac{1}{Sc}\left(h^{″}\right)+h^{\prime }f+K{h}^{2}g=0,$$

(16)

$$\begin{array}{l}{f}^{\prime }\left(0\right)=\lambda ,\theta \left(0\right)=1,f\left(0\right)=S,N_t\left({\theta }^{\prime }\left(0\right)\right)+N_b\left({\Phi }^{\prime }\left(0\right)\right)=0,\\ \delta h^{\prime }\left(0\right)=K_s\left(g\left(0\right)\right),g^{\prime }\left(0\right)=K_s\left(g\left(0\right)\right),\\ f^{″}\left(\infty \right)\to 1,\theta \left(\infty \right)\to 0,\Phi \left(\infty \right)\to 0,h\left(\infty \right)\to 0,g\left(\infty \right)\to 1\end{array}$$

(17)

.

Where Schmidt number $Sc=\frac{{\nu }_f}{D_B}$, permeability parameter $k_1=\frac{{\mu }_f}{{\rho }_{f}Ka}$, magnetic field parameter $M=\frac{\sigma {\beta }_0^2}{{\rho }_{f}a}$,$F_1=\frac{C_b}{{\rho }_f\sqrt{K}}$ is interial parameter, $N_T=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{T_{\infty }{\upsilon }_f}$ is thermophoresis parameter, $Rd=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k^{*}{k}_f}$ is the thermal radiation, $N_b=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{{\upsilon }_f}$ is Brownian motion parameter, $\delta =\frac{D_B}{D_A}$ is the ratio of diffusion coefficient, $K=\frac{k_1{a}_0^2}{c}$ is the strength of homogeneous, $Ec=\frac{u_w^2}{c_p\left(T_w-T_{\infty }\right)}$ is Eckert number, $\mathrm{Pr}=\frac{\mu c_p}{k_f}$ is the Prandtl number, $K_s=\frac{k_s}{D_A\sqrt{\frac{c}{{\upsilon }_f}}}$ is the strength of heterogeneous. Skin friction, Sherwood number, and Nusselt number are

$$C_f=\frac{t_w}{{\mu }_f{u}_e^2},Sh=\frac{x{q}_m}{D_B\left(C_w-C_{\infty }\right)},Nu=\frac{x{q}_w}{k_f\left(T_w-T_{\infty }\right)}$$

(18)

Where

$$t_w=-{\mu }_{hnf}\left({\left(u_y\right)}_{y=0}\right),q_m=-D_B{\left(C_y\right)}_{y=0},q_w=-k_{hnf}\left({\left(T_y\right)}_{y=0}\right)+{\left(q_r\right)}_{y=0}$$

(19)

In Non-dimensional form, $C_f$, $Nu$, and $Sh$ are written as

$$\begin{array}{l}{C}_{f}R{e}_x^{0.5}=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}NuR{e}_x^{-0.5}=-\left(\frac{k_{nf}}{k_f}+\frac{4}{3}Rd\right){\theta }^{\prime }\left(0\right),\\ R{e}_x^{-0.5}Sh=-{\Phi }^{\prime }\left(0\right)\end{array}$$

(20)

3. HAM Solution

HAM is used for finding the solution of Equations (12)–(16) with BCs (17). Mathematica software is used for this purpose. The initial guesses are taken as:

$$f_0\left(\eta \right)=1-e^{-\eta },{\theta }_0\left(\eta \right)=e^{-\eta },{\Phi }_0\left(\eta \right)=e^{-\eta },g_0\left(\eta \right)=e^{-\eta },h_0\left(\eta \right)=e^{-\eta }$$

(21)

The linear operators are discoursed as:

$$\begin{array}{l}{L}_{\stackrel{⌢}{f}}(\stackrel{⌢}{f})=\stackrel{⌢}{f^{‴}}-{\stackrel{⌢}{f}}^{\prime },\hspace{0.17em}{L}_{\stackrel{⌢}{\theta }}(\stackrel{⌢}{\theta })={\stackrel{⌢}{\theta }}^{″}-\stackrel{⌢}{\theta },\hspace{0.17em}\hspace{0.17em}{L}_{\stackrel{⌢}{\Phi }}(\stackrel{⌢}{\Phi })={\stackrel{⌢}{\Phi }}^{″}-\stackrel{⌢}{\Phi },\\ L_{\stackrel{⌢}{h}}\left(\stackrel{⌢}{h}\right)=\stackrel{⌢}{h^{″}}-\stackrel{⌢}{h},L_{\stackrel{⌢}{g}}\left(\stackrel{⌢}{g}\right)={\stackrel{⌢}{g}}^{″}-\stackrel{⌢}{g},\end{array}$$

(22)

With properties

$$\begin{array}{l}{L}_{\stackrel{⌢}{f}}({\gamma }_1+{\gamma }_2{e}^{-\eta }+{\gamma }_3{e}^{\eta })=0,L_{\stackrel{⌢}{\Phi }}({\gamma }_6{e}^{-\eta }+{\gamma }_7{e}^{\eta })=0,L_{\stackrel{⌢}{\theta }}({\gamma }_4{e}^{-\eta }+{\gamma }_5{e}^{\eta })=0,\\ L_{\stackrel{⌢}{g}}({\gamma }_8{e}^{-\eta }+{\gamma }_9{e}^{\eta })=0,L_{\stackrel{⌢}{h}}({\gamma }_{10}{e}^{-\eta }+{\gamma }_{11}{e}^{\eta })=0,\end{array}$$

(23)

Where ${\gamma }_i\left(i=1-11\right)$ are constants.

The zero-order deformations are:

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

(24)

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

(25)

$$(1-\zeta ) L_{\stackrel{⌢}{\Phi }}\left[\stackrel{⌢}{\Phi }(\eta ;\zeta )-{\stackrel{⌢}{\Phi }}_0(\eta )\right]=p{\hslash }_{\stackrel{⌢}{\Phi }}{{\rm N}}_{\stackrel{⌢}{\Phi }}\left[\stackrel{⌢}{\Phi }(\eta ;\zeta ),\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{\theta }(\eta ;\zeta )\right]$$

(26)

$$(1-\zeta ) L_{\stackrel{⌢}{g}}\left[\stackrel{⌢}{g}(\eta ;\zeta )-{\stackrel{⌢}{g}}_0(\eta )\right]=p{\hslash }_{\stackrel{⌢}{g}}{{\rm N}}_{\stackrel{⌢}{g}}\left[\stackrel{⌢}{g}(\eta ;\zeta ),\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{h}(\eta ;\zeta )\right]$$

(27)

$$(1-\zeta ) L_{\stackrel{⌢}{h}}\left[\stackrel{⌢}{h}(\eta ;\zeta )-{\stackrel{⌢}{h}}_0(\eta )\right]=p{\hslash }_{\stackrel{⌢}{\theta }}{{\rm N}}_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{h}(\eta ;\zeta ),\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{g}(\eta ;\zeta )\right]$$

(28)

Here $\zeta$ is the embedding parameter, and the non-zero auxiliary parameters are, ${\hslash }_{\stackrel{⌢}{\theta }}$, ${\hslash }_{\stackrel{⌢}{\Phi }}$ and ${\hslash }_{\stackrel{⌢}{g}},{\hslash }_{\stackrel{⌢}{h}}$.$N_{\stackrel{⌢}{f}}$, $N_{\stackrel{⌢}{\theta }}$, $N_{\stackrel{⌢}{g}}$,$N_{\stackrel{⌢}{h}}$ and $N_{\stackrel{⌢}{\Phi }}$ are the nonlinear operators which are defined as:

$$\begin{array}{l} {{\rm N}}_{\stackrel{⌢}{f}} \left[\stackrel{⌢}{f}(\eta ;\zeta )\right]=\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{\stackrel{⌢}{f}}_{\eta \eta \eta }-{\left({\stackrel{⌢}{f}}_{\eta }\right)}^2+\stackrel{⌢}{f}{\stackrel{⌢}{f}}_{\eta \eta }-\left(\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}\right)M{\stackrel{⌢}{f}}_{\eta }\\ -\frac{{\nu }_{hnf}}{{\nu }_f}{k}_1{\stackrel{⌢}{f}}_{\eta }-\frac{{\rho }_f}{{\rho }_{hnf}}{F}_1{\left({\stackrel{⌢}{f}}_{\eta }\right)}^2,\end{array}$$

(29)

$$\begin{array}{l}{{\rm N}}_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{\theta }(\eta ;\zeta ),\stackrel{⌢}{\Phi }(\eta ;\zeta )\right]=\frac{1}{{\left(\rho c_p\right)}_{hnf}/{\left(\rho c_p\right)}_f}\left(k_{hnf}/k_f+\frac{4}{3}Rd\right){\stackrel{⌢}{\theta }}_{\eta \eta }\\ +\frac{\frac{{\mu }_{hnf}}{{\mu }_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{\left(\rho c_p\right)}}Ec{\stackrel{⌢}{f}}_{\eta \eta }^2+\left(N_b{\stackrel{⌢}{\theta }}_{\eta }{\stackrel{⌢}{\Phi }}_{\eta }+N_t{\stackrel{⌢}{\theta }}_{\eta }^2\right)\\ +\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{\left(\rho c_p\right)}}EcM{f^{\prime }}^2+\mathrm{Pr}\stackrel{⌢}{f}{\stackrel{⌢}{\theta }}_{\eta }+\frac{1}{{\left(\rho c_p\right)}_{hnf}/{\left(\rho c_p\right)}_f}{Q}_1\stackrel{⌢}{\theta }\end{array}$$

(30)

$${{\rm N}}_{\stackrel{⌢}{\Phi }}\left[\stackrel{⌢}{\Phi }(\eta ;\zeta ),\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{\theta }(\eta ;\zeta )\right]={\stackrel{⌢}{\Phi }}_{\eta \eta }+Sc\stackrel{⌢}{f}\stackrel{⌢}{\Phi }+\frac{N_t}{N_b}Sc\left(n+1\right)\stackrel{⌢}{f}{\stackrel{⌢}{\theta }}_{\eta \eta },$$

(31)

$${{\rm N}}_{\stackrel{⌢}{g}} \left[\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{g}(\eta ;\zeta ),\stackrel{⌢}{h}(\eta ;\zeta )\right]=\left({\stackrel{⌢}{g}}_{\eta \eta }\right)\frac{1}{Sc}+\stackrel{⌢}{f}{\stackrel{⌢}{g}}_{\eta }-K{\stackrel{⌢}{h}}^2\stackrel{⌢}{g},$$

(32)

$${{\rm N}}_{\stackrel{⌢}{h}} \left[\stackrel{⌢}{f}(\eta ;\zeta ),\stackrel{⌢}{g}(\eta ;\zeta ),\stackrel{⌢}{h}(\eta ;\zeta )\right]=\frac{1}{Sc}\left({\stackrel{⌢}{h}}_{\eta \eta }\right)+{\stackrel{⌢}{h}}_{\eta }\stackrel{⌢}{f}+K{\stackrel{⌢}{h}}^2\stackrel{⌢}{g},$$

(33)

$$\begin{array}{l}{\stackrel{⌢}{f}}_{\eta }\left(0,\zeta \right)=\lambda ,\stackrel{⌢}{f}\left(0,\zeta \right)=S,N_t\stackrel{⌢}{\theta }\left(0,\zeta \right)+N_b\stackrel{⌢}{\Phi }\left(0,\zeta \right)=0,\stackrel{⌢}{\theta }\left(0,\zeta \right)=1,\\ \delta {\stackrel{⌢}{h}}_{\eta }\left(0,\zeta \right)=K_s\stackrel{⌢}{g}\left(0,\zeta \right),{\stackrel{⌢}{g}}_{\eta }\left(0,\zeta \right)=K_s\stackrel{⌢}{g}\left(0,\zeta \right),\\ {\stackrel{⌢}{f}}_{\eta \eta }\left(\infty ,\zeta \right)=1,\stackrel{⌢}{\Phi }\left(\infty ,\zeta \right)=0,\stackrel{⌢}{\theta }\left(\infty ,\zeta \right)=0,\stackrel{⌢}{h}\left(\infty ,\zeta \right)=0,\stackrel{⌢}{g}\left(\infty ,\zeta \right)=1\end{array}$$

(34)

.

4. Result and Discussion

In this section we explain in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24 the impacts of different parameters like, $M, F_1, k_1, \mathrm{Pr}, Rd, N_b, Sc, N_t, K_s, K,$ on $f^{\prime },\theta ,\Phi ,g,C_f,Nu\hspace{0.17em}\hspace{0.17em}\&\hspace{0.17em}Sh$. The influence of growing the $M$ on $f^{\prime }$ is shown in Figure 7 It has now been extensively established that $M$ reduces velocity by creating drag force, which opposes the fluid’s motion. Transfer phenomena are resisted by the$M$. This is because, due to the $M$, changes in the $M$ also cause changes in the Lorentz force, which increases resistance to transportation events. Under all conditions, $f^{\prime }$ decline at a significant distance from the Stretch/shrink surface. Figure 8 shows the effect of the $M$ on $\theta$. When perceived in Figure 8, the thickness of the thermal BL raises as $M$ is improved.

The effects of $k_1$ on $f^{\prime }$ are shown in Figure 9. It is evident that the presence of porous medium consequences is augmented fluid flow restriction, instigation it to move more and more slightly, which therefore decline $f^{\prime }$. So, the impedance to fluid motion enlargements as $k_1$ augments. Then, the temperature rises as a consequence of a decline in $f^{\prime }$. Thus, it can be discussed that upsurge in $k_1$ lessens the thickness of the BL, subsequent in an upsurge in the rate of heat transmission. The influence $F_1$ on $f^{\prime }$ is shown in Figure 10. When $F_1$ is increased, the velocity of HNF is increased, this is because the increment in $F_1$ produced inertial drag force, which is a barrier to $f^{\prime }$.

The impact of $\mathrm{Pr}$ on $\theta$ is scrutinized in Figure 11. Augmentation in $\mathrm{Pr}$ in declined in the thermal BL, which outcomes in reduction in $\theta$. When $\mathrm{Pr}$ is augmented, ${\upsilon }_{nf}$ becomes larger than density, creating an opposing force against the fluid flow. For dissimilar values of $Ec$ for stretch /Shrink surface the feature of $\theta$ is show in Figure 12. With the augmentation of $Ec$, the $\theta$ is upsurges as shown in Figure 7. Here it is noted that $Ec$ confines the fluid motion and $Ec=0$ signifies no viscous dissipation. The cause behind is $Ec$ means it is the ratio of the square of the velocity of fluid far away from the surface of the boundary to the product of the specific heat at constant temperature. In case of $Ec>0$, so that the thermal BL upsurges.

The increment in $\theta$ for growing in $Rd$ can be detected in Figure 13. The Rd describes the rate at which conduction heat transmission to thermal radiation transmission occurs. Rd augments heat transmission because the augmentation in $Rd$ will rise the thickness of the boundary layer. The thickness of the thermal boundary layer and $\theta$ increased when $Rd$ is increased, which supplying more heat to HNF flow.

The behavior of the $Sc$ in mass species is depicted in Figure 14. When $Sc$ was inclined, the particle concentration dropped. Physically,$Sc$ is the ratio of mass diffusivity to kinematic viscosity. The kinematic viscosity of fluid particles is skewed in relation to the influence of Schmidt number due to their physical characteristics. On the other hand, mass diffusivity decreases with $Sc$ variation. In comparison to the Schmidt number role, concentration layers have a decreasing function.

The effects $K$ and $K_s$ on $g\left(\eta \right)$ are shown on Figure 15 and Figure 16. The declining pattern of $g\left(\eta \right)$ of HNF is perceived. It is reliable with the fact that the reaction rate rises as the values of $K$ and $K_s$ rise, which reason the lessening in diffusion rate.

Furthermore, the effects are shown in Figure 17 and Figure 18, which displays that both $f^{″}\left(0\right)$ and ${\theta }^{\prime }\left(0\right)$ are the growing functions of $M$. Physically, rise in $M$ as a result in an increasing in Lorentz force. Lorentz force creates retardation force in fluid particles. So this decelerate force slows down flow of HNF and rises in both $f^{″}\left(0\right)$ and ${\theta }^{\prime }\left(0\right)$.

Figure 13 and Figure 14 displays the influences of $k_1$ and $F_1$ on $f^{″}\left(0\right)\hspace{0.17em}$. When growing occur in $k_1$ and $F_1$ $f^{″}\left(0\right)\hspace{0.17em}$ of HNF is augmented.

Figure 21 and Figure 22 are drawn to scrutinize the incentive of $N_t$ and $N_b$ on ${\theta }^{\prime }\left(0\right)$. The reverse effect is detected for upsurge in $N_t$ and $N_b$. Base fluids with higher nanoparticle concentrations exhibit improved thermal physical properties. The concentration of additional nanoparticles within the base fluid rises with their addition, increasing the likelihood of an intermolecular collision and raising K.E raised the temperature. As a result, HNF experiences an increase in ${\theta }^{\prime }\left(0\right)$. Also, an upsurge in $N_b$ caused the movements of nanoparticles to shift from lower to higher concentrated regions. Figure 23 and Figure 24 shows the reverse impact of $N_t$ and $N_b$ on $\Phi \left(0\right)$. Figure 25a,b shows the percentage comparison of NF and HNF of velocity and energy.

5. Table Discussion

The impacts of $F_1, k_1, M, \mathrm{Pr}, N_b, N_t, Rd, Ec, {\varphi }_1, {\varphi }_2$ on $C_f\hspace{0.17em}\&\hspace{0.17em}Nu$ are presented in Table 3. Growing in $F_1, k_1, M$,$C_f$ is increased. The impacts of $\mathrm{Pr}, Ec, Rd\hspace{0.17em}, M, {\varphi }_1, {\varphi }_2, N_b, N_t$ on $Nu$ are presented in Table 3. Nusselt number is augmented when the values of $Ec, N_t\hspace{0.17em}, M, {\varphi }_1, {\varphi }_2$ is enlarged, while decline for increasing in $\mathrm{Pr}, N_b$. The impacts of $Sc, N_b, N_t\hspace{0.17em}, {\varphi }_1, {\varphi }_2$ on $Sh$ are presented in Table 4. Sherwood number is increased for increasing in $Sc, N_t, {\varphi }_1, {\varphi }_2$ and decline for increasing in $N_b\hspace{0.17em}$.

6. Conclusions

In this research work we examine homogenous and heterogen $2.11695$ ous reaction of Bongiorno model Darcy-Forchheimer MhD HNF flow through stretching /shrinking sheet with the impact heat viscous dissipation, thermal radiation, generation/absorption, Joule heating effect. The system of ODEs is derived from the set of PDEs by applying the proper transformations for similarity variables. The obtained ODEs is then solved by using HAM on mathematica software. The concluding remarks are obtained from the study are follows

• The $f^{\prime }$ is a declining function of $k_1, M, F_1$.
• Increasing in $M, Rd, Ec,$$\theta \left(\eta \right)$ is decreased.
• Augmenting in $Sc$ decline the $\Phi \left(\eta \right)$.
• The decreasing pattern of $g\left(\eta \right)$ is observed for increasing in $K, K_s$.
• $C_f$ is the growing function of $k_1, F_1.$ Nu and $C_f$ is the growing function of$M$.
• The reverse impact is observed for $Nu$ and $Sh$, while increasing in $N_b, N_t$.