Heat Transfer of Buoyancy and Radiation on the Free Convection Boundary Layer MHD Flow across a Stretchable Porous Sheet

Abstract

Theoretical influence of the buoyancy and thermal radiation effects on the MHD (magnetohydrodynamics) flow across a stretchable porous sheet were analyzed in the present study. The Darcy–Forchheimer model and laminar flow were considered for the flow problem that was investigated. The flow was taken to incorporate a temperature-dependent heat source or sink. The study also incorporated the influences of Brownian motion and thermophoresis. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived in this study. A favorable comparison with earlier published studies was achieved. Graphs were used to investigate and explain how different physical parameters affect the velocity, the temperature, and the concentration field. Additionally, tables are included in order to discuss the outcomes of the Sherwood number, the Nusselt number, and skin friction. The fundamental governing partial differential equations (PDEs), which are used in the modeling and analysis of the MHD flow problem, were transformed into a collection of ordinary differential equations (ODEs) by utilizing the similarity transformation. A semi-analytical approach homotopy analysis method (HAM) was applied for approximating the solutions of the modeled equations. The model finds several important applications, such as steel rolling, nuclear explosions, cooling of transmission lines, heating of the room by the use of a radiator, cooling the reactor core in nuclear power plants, design of fins, solar power technology, combustion chambers, astrophysical flow, electric transformers, and rectifiers. Among the various outcomes of the study, it was discovered that skin friction surges for 0.3 $\le F_1\le$ 0.6, 0.1 $\le k_1\le$ 0.4 and 0.3 $\le M\le$ 1.0, snf declines for 1.0 $\le G_r\le$ 4.0. Moreover, the Nusselt number augments for 0.5 $\le R\le$ 1.5, 0.2 $\le N_t\le$ 0.8 and 0.3 $\le N_b\le$ 0.9, and declines for 2.5 $\le Pr\le$ 5.5. The Sherwood number increases for 0.2 $\le N_t\le$ 0.8 and 0.3 $\le Sc\le$ 0.9, and decreases for 0.1 $\le N_b\le$ 0.7.

1. Introduction and Motivation

Magnetohydrodynamics (MHD) is the study of the magnetic impact of an electrically conducted incompressible fluid; it has received attention because of its major applications in different areas, such as X-ray radiation, crystal growth, asthma treatment, tumor therapy, plasma, fusing metal in electric heaters, nuclear reactors, gastric medication, and electrolytic biomedicine. MHD fluid can also be used to manage the cooling rates in certain industrial processes. With homogeneous magnetic fields, Chaudhary and Kumar [1] explored an unsteady 2D flow of an electrically conducted fluid toward a stretchable sheet. Naramgari and Sulochana [2] analyzed the chemical reaction and thermal radiation effects on a two-dimensional MHD steady flow toward a stretchable sheet using injection/suction. By using a chemical reaction, Reddy et al. [3] analyzed the impacts of MHD and buoyancy forces through rotating vertically porous plates. Mishra and Jena [4] explored steady two-dimensional incompressible, laminar, and electrically conducted fluids toward a stretchable sheet using magnetic fields and the viscous dissipation effect. Babu et al. [5] found numerical solutions of a steady two-dimensional MHD boundary layer flow toward a shrinking sheet by using the wall mass suction effect. Using the homotopy analysis method (HAM) technique, Mabood et al. [6] found the impact of radiation on the MHD flow toward an exponential stretchable sheet. Pal and Mondal [7] investigated heat transfer characteristics and the MHD non-Darcy boundary layer flow toward an electrically conducted fluid in the presence of the viscosity impact. The researchers in [8] demonstrated the MHD natural convection-based nanofluid flow toward a porous surface. Using the radiation effect, Mahmoud [9] investigated the heat transport flow of electrically conducted fluids across continuously moving vertical infinite plates with heat flux and suction. With variable heat, radiation effect, and mass diffusion, Kishore et al. [10] described the viscous incompressible MHD flow in porous media. Majeed et al. [11] evaluated the impact of energy activation and chemical reactions on the two-dimensional flow of heat transfer toward an exponential stretchable surface. With radiation influence and the boundary layer flow of viscous fluid, Sharma et al. [12] solved the problem of heat transfer due to an exponentially stretching/shrinking sheet. Poornima and Reddy [13] found the mathematical solutions of the convective flow of radiating boundary layer-combined nanofluids to a non-linear movable sheet using induced magnetic fields. Some studies involving MHD flow and thermal radiation have also been published in [14,15,16,17,18,19,20]. The researchers in [21] discussed the heat generation and Cattaneo–Christov heat flux effect on a hybrid nanofluid flow over two distinct geometries. Yaseen et al. [22] explored the Darcy–Forchheimer model and Cattaneo–Christov heat flux of MoS${}_2$-SiO${}_2$/kerosene oil between shrinking and rotating disks. Khan et al. [23] investigated the magnetohydrodynamic thin film flow through a porous stretching sheet by focusing on the impact of thermal radiation and viscous dissipation. Jawad et al. [24] discussed the analysis of the hybrid nanofluid stagnation point flow over a stretching surface by focusing on melting heat transfer.

Heat transfer analysis toward a porous stretchable sheet has received a lot of attention from architects, geologists, engineers, and mathematicians due to its occurrence in many practical and scientific situations, such as electronic cooling, thermal insulation, catalytic reactor, nuclear waste repository, oil production, geothermal system, petroleum industries, and energy storage units (see [25,26,27,28]). In 1856, Henry Darcy worked with homogeneous fluid flow passing through a porous medium. However, the classical Darcy law did not work for higher permeability flows and large velocities. In order to overcome this demerit, the researchers in [29] revised the Darcy velocity formula by including the velocity square into the equation of momentum to calculate the boundary layer flow and inertial effects. Pal and Mondal [30] described the Darcy–Forchheimer model to examine the mixed convection flow of a stretchable sheet using thermal radiation and variable viscosity. Ganesh et al. [31] examined the Darcy–Forchheimer model and hydromagnetic nanofluids on stretchable surfaces. The Darcy–Forchheimer–Casson fluid flow was investigated by Seth and Mandal [32], who discovered the influences of rotational and Casson parameters on the primary velocity. Under the saturated porous medium of the Darcy–Forchheimer flow, Seddeek [33] investigated the combined effects of viscous dissipation and the thermophoresis parameter. Hayat et al. [34] evaluated the 2D Darcy–Forchheimer flow of Maxwell fluids, induced by a stretchable sheet. Rajesh et al. [35] examined the heat transfer rate and the hybrid nanoparticle impact on the MHD flow through a vertical plate. The researchers in [36] studied the heat transfer analysis of nanoparticles (MoS${}_2$/H${}_2$O) and hybrid nanoparticles (MoS${}_2$-SiO${}_2$/H${}_2$O-C${}_2$H${}_6$O${}_2$) between two distinct plates by using heat absorption/generation. Raza et al. [37] explored the influence of Joules heating and the Brownian motion radiative nanofluid toward a stretchable cylinder. Faizan et al. [38] described the behaviors of heat and mass transfer for the Cattaneo–Christov nanofluid toward a Riga sheet.

The injection and suction have remarkable effects on boundary layer flow surfaces. The suction is used to upsurge the skin friction, whereas the injection works in the opposite direction. The injection of fluids through a porous sheet has practical applications in the boundary layer problem, such as wire coating, polymer fiber coating, and film cooling. Furthermore, they are important in the field of engineering, e.g., regarding radial diffusion and the construction of thrust bearings (see [39]). Suction is utilized in chemical progression to remove the reactant, whereas blowing is used to introduce a reactant, cool the surfaces, minimize the drag, and prevent corrosion (see [40,41,42]).

Many phenomena in applied sciences and engineering fields are governed by non-linear BVPs. As a result, BVPs have received attention from engineers, mathematicians, and physicists. Generally speaking, compared to linear ODEs and PDEs, it is significantly more complicated to find the analytical solutions of non-linear ODEs and PDEs, particularly when using analytical methods. Therefore, in 1992, Liao [43] proposed the homotopy analysis method (HAM) [44,45,46,47,48], relying on homotopy in topology. This approach is unaffected by significant large/small physical variations. The HAM approach provides great versatility in terms of changing and choosing the convergence areas and estimating rates. The method has an advantage over typical computational methodologies in that it avoids rounding off errors induced by the discretization procedure.

Judging by the aforementioned and other related literature studies, numerous studies have been presented for buoyancy and radiation influence on the MHD flow toward a stretchable sheet; however, limited studies are available for the Darcy–Forchheimer model with the Brownian motion, heat source, sink, or the thermophoresis effect on the MHD flow across a stretchable porous sheet. Therefore, the present study fills this gap. The novelty of this study is that it enhances the work of Daniel [49] by introducing the species conservation equation and the Darcy–Forchheimer model with the influence of the Brownian motion, heat source, heat sink, or thermophoresis effect on the MHD flow across a stretchable porous sheet. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived in this study. The fluid flow is considered laminar, incompressible, and electrically conducted. Mathematica software and homotopy analysis techniques were utilized to solve the modeled equations. The variations of several physical parameters on the flow, as well as thermal and concentration characteristics, are discussed briefly. The present study provides several applications, such as film cooling, wire coating, polymer fiber coating, construction of thrust bearings, radial diffusion, oil production, thermal insulation, X-ray radiation, fusing metal in electric heaters, and gastric medication.

The present study provides the following research questions:

• What is the general form of the buoyancy term in the momentum equation for a free convection boundary layer? How may it be approximated if the flow is due to temperature variations? What is the name of the approximation?
• What physical processes are represented by the terms of the momentum Equation (19)? By the energy Equation (20)? By the species conservation Equation (21)?
• What is the definition of the Prandtl number? How does its value affect the relative growth of the thermal boundary layer for the laminar flow toward a porous stretching sheet?

2. Preliminaries

It is commonly recognized that non-linear PDEs and ODEs for BVPs are significantly harder to solve than linear PDEs and ODEs, particularly when we attempt to use analytical techniques. Traditionally, asymptotic and perturbation approaches are frequently used to obtain analytic approximations of non-linear problems in engineering. Unfortunately, asymptotic and perturbation methods only work for weak nonlinear systems because they generally rely too heavily on small/large physical parameters. Therefore, certain analytical approximation techniques must be developed to be applicable to extremely non-linear problems, independently of any large/small physical parameters. Thus, in 1992, Liao [43] developed one of these semi-analytic approximation techniques.

We begin by presenting the following nonlinear differential equation:

$$\begin{array}{c}{\mathcal{N}}_1\left[u_1(x_1,t_1)\right]=0,\hfill \end{array}$$

(1)

where ${\mathcal{N}}_1$ and $u_1(x_1,t_1)$ represent the nonlinear operator and the unknown function, respectively. Liao [43] introduced a two-parameter set of equations in the embedded parameter $p\in$ [0,1], which is known as the $0\mathrm{th}$-order deformation equation given by

$$\begin{array}{c}(1-p){\mathcal{L}}_1[\varphi (x_1,t_1;p)-u_0(x_1,t_1)]=\hslash p{\mathcal{N}}_1\left[\varphi (x_1,t_1;p)\right],\hfill \end{array}$$

(2)

where ${\mathcal{L}}_1$, $\varphi (x_1,t_1;p)$ and $u_0\left(x_1\right)$ represent an auxiliary linear-operator, unknown function, and initial guess, respectively. We find at p = 0 and p = 1 that

$$\begin{array}{c}\varphi (x_1,t_1;0)=u_0(x_1,t_1),\hfill \end{array}$$

(3)

and

$$\begin{array}{c}\varphi (x_1,t_1;1)=u_1(x_1,t_1).\hfill \end{array}$$

(4)

The solution $\varphi (x_1,t_1;p)$ of the $0\mathrm{th}$-order deformation equation distorts from the initial estimate $u_0(x_1,t_1)$ to the actual solution $u_1(x_1,t_1)$ of the given equation ${\mathcal{N}}_1\left[u_1(x_1,t_1)\right]=0$ as the embedded parameter $p\in$ [0,1] increases from 0 to 1. We can extend $\varphi (x_1,t_1;p)$ into a Maclaurin expansion with respect to p because it also depends on the embedded parameter $p\in$ [0,1], as follows:

$$\begin{array}{c}\varphi (x_1,t_1;p)=u_0(x_1,t_1)+{\displaystyle \sum _{m=1}^{\infty }}{u}_m(x_1,t_1)p^m,\hfill \end{array}$$

(5)

where

$$\begin{array}{c}{u}_m(x_1,t_1)={\left.\frac{1}{m!}\frac{{\partial }^m\varphi (x_1,t_1;p)}{\partial p^m}\right|}_{p=0}.\hfill \end{array}$$

(6)

Equation (5) is known as the homotopy Maclaurin expansion of $\varphi (x_1,t_1;p)$. Particularly, at p = 1, we have

$$\begin{array}{c}\varphi (x_1,t_1;1)=u_0(x_1,t_1)+{\displaystyle \sum _{m=1}^{\infty }}{u}_m(x_1,t_1).\hfill \end{array}$$

(7)

We obtain the following homotopy series solution if the aforementioned series converges to $\varphi (x_1,t_1;1)$, as indicated by Equation (4):

$$\begin{array}{c}{u}_1(x_1,t_1)=u_0(x_1,t_1)+{\displaystyle \sum _{m=1}^{\infty }}{u}_m(x_1,t_1),\hfill \end{array}$$

(8)

which satisfies the given equation ${\mathcal{N}}_1\left[u_1(x_1,t_1)\right]=0$ as proven by Liao [43]. Equation $u_m(x_1,t_1)$ is entirely determined by means of Equation (2). Taking the derivative of Equation (2) m times with respect to the embedded parameter p, if we divide by $m!$ and finally put p = 0, then we have

$$\begin{array}{c}{\mathcal{L}}_1[u_m(x_1,t_1)-{\chi }_m{u}_{m-1}(x_1,t_1)]=\hslash {\mathcal{D}}_{m-1}{\mathcal{N}}_1\left[\varphi (x_1,t_1;p)\right],\hfill \end{array}$$

(9)

where

$$\begin{array}{c}{\mathcal{D}}_{m-1}={\left.\frac{1}{(m-1)!}\frac{{\partial }^{m-1}}{\partial p^{m-1}}\right|}_{p=0},\hfill \end{array}$$

(10)

and

$${\chi }_m=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}m\le 1\hfill \\ 1,\hfill & \mathrm{if}m>1.\hfill \end{array}\right.$$

(11)

3. Mathematical Formulation

Here, we assume a steady, incompressible, and electrically conducted MHD flow toward a stretching porous sheet that is driven by the buoyancy forces. We also assume that the fluid flow is a laminar, two-dimensional, constant property condition in which the gravity force acts in the negative x-direction. A uniform magnetic field is applied normally to the sheet. The exception is for the influence of the variable density in the buoyancy force since it is this variation that induces fluid motion. The x-axis is taken along the stretching sheet in the direction of the motion while the y-axis is perpendicular to the sheet. Geometric representation of the physical flow model is shown in Figure 1. The transverse applied magnetic field and magnetic Reynolds number are assumed to be very small. Viscous dissipation, induced magnetic field, and the pressure gradient are considered negligible.

Under the above-stated assumption, the continuity equation is written as:

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

(12)

here, the term represents the net outflow (outflow minus inflow) of the mass in the x and y directions, the sum of which must be zero for the steady flow.

The appropriate form of the momentum equation can be represented as:

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^2\left(x\right)}{\rho }u-\frac{d{p}_{\infty }}{\rho dx}-g_1-u\frac{\upsilon }{k_0}-F_0{u}^2,$$

(13)

where $\frac{d{p}_{\infty }}{\rho dx}$ is the free stream pressure gradient in the region outside the boundary layer. In this region, u = 0, then we have

$$\frac{d{p}_{\infty }}{\rho dx}=-{\rho }_{\infty }{g}_1,$$

(14)

substituting Equation (14) into Equation (13), we obtain the following expression:

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^2\left(x\right)}{\rho }u+g_1(\Delta \rho /\rho )-u\frac{\upsilon }{k_0}-F_0{u}^2,$$

(15)

where $\Delta \rho ={\rho }_{\infty }-\rho$.

The third component on the right side of Equation (15) is the buoyancy force; the fluid flow originates because the density $\rho$ is a parameter. If the $\rho$ variation is only due to the temperature variation, then the component may be linked to a fluid property known as the volumetric thermal expansion coefficient.

$${\beta }_1=-\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial T}\right)}_p.$$

(16)

This thermodynamic property of the fluid provides a measure of the amount by which the density change responds to a change in the temperature at constant pressure. It is expressed in the following approximate form

$${\beta }_1\approx \frac{\Delta \rho }{\rho \Delta T}=-\frac{1}{\rho }\left(\frac{{\rho }_{\infty }-\rho }{T_{\infty }-T}\right),$$

(17)

it follows that

$$({\rho }_{\infty }-\rho )\approx \rho {\beta }_1(T-T_{\infty }).$$

(18)

This simplification is known as the Boussinesq approximation, and substituting into Equation (15), we obtain the following constituted boundary layers governing the equation (see Daniel [49] and Chamkha [50]):

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^2\left(x\right)}{\rho }u+g_1{\beta }_1(T-T_{\infty })-u\frac{\upsilon }{k_0}-F_0{u}^2,$$

(19)

$$\begin{array}{c}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{\partial }^{2}T}{\partial y^2}+\frac{Q_1}{\rho c_p}(T-T_{\infty })+\frac{{\beta }_1^{*}u}{\rho c_p}(T_{\infty }-T)+\frac{1}{c_p\rho }\frac{\partial q_r}{\partial y}\hfill \\ \hfill +\tau \left[D_B\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right],\end{array}$$

(20)

and

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{{\partial }^{2}C}{\partial y^2}{D}_B+\frac{{\partial }^{2}T}{\partial y^2}\frac{D_T}{T_{\infty }}.$$

(21)

The appropriate boundary constraints are given by

$$\left\{\begin{array}{c}u=a_{1}x,v=v_w,T=T_w=T_{\infty }+A_{0}x\hfill \\ C=C_w=B_0{x}_{\infty }+C_{\infty }aty=0\hfill \\ u⟶0,C⟶C_{\infty },T⟶T_{\infty }asy⟶\infty .\hfill \end{array}\right.$$

(22)

Here, the velocity components are denoted by u and v along the x-axis and the y-axis, respectively, $c_p$ is the specific heat, T is the temperature, $B_0$ is the applied magnetic induction, $T_w$ is the wall temperature, ${\beta }_1$ is the coefficient of thermal expansion, $a_1$ is the stretching rate, $g_1$ is the acceleration due to gravity, $\upsilon$ is the fluid kinematic viscosity, $k_1^{*}$ is the mean absorption coefficient, $\rho$ is the fluid density, $v_w$ is the wall injection when $v_w>0$ and the wall suction when ($v_w<0$), $\sigma$ is the electrical conductivity, ${\beta }_1^{*}u(T_{\infty }-T)$ and $Q_1(T-T_{\infty })$ are the heat generation and absorption, where (${\beta }_1^{*}$ and $Q_1$ are constants) and ${\sigma }_1^{*}$ is the Stefan Boltzmann constant. The first term on the right-hand side of Equations (19)–(21) represents the net force due to the viscous shear stress, the net inflow of thermal energy due to the y-direction conduction, and the net inflow due to the y-direction diffusion, respectively. In terms of the free stream function $\psi (x,y)$, the components of velocity are described as follows:

$$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}.$$

(23)

For the similarity solutions, we use the following acceptable variable:

$$\eta =y\sqrt{\frac{a_1}{\upsilon }},\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},\psi (x,y)=\sqrt{\upsilon a_1}xf\left(\eta \right),\varphi =\frac{C-C_{\infty }}{C_w-C_{\infty }}.$$

(24)

The heat flux $q_r$ is calculated as follows:

$$q_r=-\frac{4{\sigma }_1^{*}}{3k_1^{*}}\frac{\partial T^4}{\partial y}.$$

(25)

Here, we have

$$T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4,$$

(26)

upon inserting Equations (24)–(26) into Equations (19)–(21), we have

$$f^{‴}+{f^{\prime }}^2+\theta G_r+f{f}^{″}-M^2{f}^{\prime }-{f^{\prime }}^2{F}_1+f^{\prime }{k}_1=0,$$

(27)

$$\frac{1}{Pr}(1+\frac{4R}{3}){\theta }^{″}+{{\varphi }^{\prime }}^2{N}_t+{\theta }^{\prime }f+{\theta }^{\prime }{\varphi }^{\prime }{N}_b+\Delta \theta -{\delta }_1\theta f^{\prime }=0,$$

(28)

and

$${\varphi }^{″}+{\varphi }^{\prime }fSc+{\theta }^{″}\frac{N_t}{N_b}=0.$$

(29)

The dimensionless BCs are given

$$\begin{array}{c}f\left(0\right)=-R_1,\theta \left(0\right)=1,\varphi \left(0\right)=1,f^{\prime }\left(0\right)=1,f^{\prime }(\infty )=1,\varphi (\infty )=0,\theta (\infty )=0.\hfill \end{array}$$

(30)

The converging parameters involved in the foregoing Equations (27)–(30) are given below:

$$\left\{\begin{array}{c}{F}_1=\frac{C_b}{\sqrt{k_0}}x,N_t=\frac{\tau D_T(T_w-T_{\infty })}{\upsilon T_{\infty }},k_1=\frac{\upsilon }{k_0{a}_1},N_b=\frac{\tau D_B(C_w-C_{\infty })}{\upsilon },\hfill \\ Sc=\frac{\upsilon }{D_B},M={\left(\frac{\sigma }{\rho a_1}\right)}^{\frac{1}{2}}{B}_0,{\delta }_1=\frac{{\beta }_1^{*}x}{\rho c_p},R=\frac{4{\sigma }_1^{*}{T}_{\infty }^3}{k_1^{*}{k}_0},Pr=\frac{\upsilon }{\alpha },\hfill \\ G_r=g_1{\beta }_1\frac{(T_w-T_{\infty })}{a_1^{2}x},R_1=\frac{v_w}{\sqrt{\upsilon a_1}},\Delta =\frac{Q_1}{\rho c_p{a}_1}.\hfill \end{array}\right.$$

(31)

We explained detail formulation of all governing equations in Appendix A.

Physical Quantities

Some of the most important physical quantities in the engineering processes are $S{h}_x$, $N{u}_x$, and $C_f$, which are indicated below:

$$S{h}_x=\frac{x{q}_m}{D_B\left(C_w-C_{\infty }\right)},N{u}_x=\frac{x{q}_w}{k_f\left(T_w-T_{\infty }\right)},C_f=\frac{{\tau }_w}{\rho u_w^2},$$

(32)

where

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

(33)

Using Equations (24) and (33) into Equation (32), we have

$${\left[Re\right]}^{-\frac{1}{2}}S{h}_x=-{\varphi }^{\prime }\left(0\right),{\left[Re\right]}^{-\frac{1}{2}}N{u}_x=-{\theta }^{\prime }\left(0\right),{\left[Re\right]}^{\frac{1}{2}}{C}_f=f^{″}\left(0\right).$$

(34)

Here $Re$ = $\frac{x{u}_w}{\upsilon }$ denotes the Reynolds number.

4. The HAM Solution

In order to determine an approximate solution for the dimensionless set of Equations (27)–(29) by incorporating the boundary condition as given in Equation (30), we shall apply the well-known homotopy analysis method (HAM) in the Mathematica software. To employ this method, we need initial guesses for the approximate solutions to Equations (27)–(29). These initial guesses are presented as follows:

$$\begin{array}{c}{\varphi }_0\left(\eta \right)=e^{-\eta },\hfill \\ {\theta }_0\left(\eta \right)=e^{-\eta },\hfill \\ f_0\left(\eta \right)=R_1-e^{-\eta }+\eta .\hfill \end{array}$$

(35)

The linear operators ${\mathcal{L}}_{\varphi }$, ${\mathcal{L}}_{\theta }$, and ${\mathcal{L}}_f$ are given below:

$$\begin{array}{c}{\mathcal{L}}_{\varphi }\left(\varphi \right)={\varphi }^{″},\hfill \\ {\mathcal{L}}_{\theta }\left(\theta \right)={\theta }^{″},\hfill \\ {\mathcal{L}}_f\left(f\right)=f^{‴}.\hfill \end{array}$$

(36)

The expanded forms of the linear operators in Equation (36) are given by

$$\begin{array}{c}{\mathcal{L}}_{\varphi }(c_1+c_2\eta )=0,\hfill \\ {\mathcal{L}}_{\theta }(c_3+c_4\eta )=0,\hfill \\ {\mathcal{L}}_f(c_5+c_6\eta +c_7{\eta }^2)=0.\hfill \end{array}$$

(37)

In this last Equation (37), $c_i$$(i=1,2,\dots ,7)$ are considered constant. Moreover, on the basis of Equation (35), the non-linear operators ${\mathcal{N}}_{\widehat{\varphi }}$, ${\mathcal{N}}_{\widehat{\theta }}$, and ${\mathcal{N}}_{\widehat{f}}$ are given as follows:

$$\begin{array}{c}{\mathcal{N}}_{\widehat{\varphi }}\left[\widehat{\varphi }(\eta ;p),\widehat{\theta }(\eta ;p),\widehat{f}(\eta ;p)\right]={\widehat{\varphi }}_{\eta \eta }+({\widehat{f}}_{\eta }\widehat{\varphi }+{\widehat{\varphi }}_{\eta }\widehat{f})Sc+{\widehat{\theta }}_{\eta \eta }\frac{N_t}{N_b},\hfill \end{array}$$

(38)

$$\begin{array}{c}{\mathcal{N}}_{\widehat{\theta }}\left[\widehat{\theta }(\eta ;p),\widehat{\varphi }(\eta ;p),\widehat{f}(\eta ;p)\right]\hfill \\ \hfill =\frac{1}{Pr}\left(1+\frac{4R}{3}\right){\widehat{\theta }}_{\eta \eta }+{\widehat{\varphi }}_{\eta }^2{N}_t+{\widehat{\theta }}_{\eta }\widehat{f}+{\widehat{\theta }}_{\eta }{\widehat{\varphi }}_{\eta }{N}_b+\Delta \widehat{\theta }-(1+{\delta }_1)\widehat{\theta }{\widehat{f}}_{\eta },\end{array}$$

(39)

and

$$\begin{array}{c}{\mathcal{N}}_{\widehat{f}}\left[\widehat{f}(\eta ;p)\right]={\widehat{f}}_{\eta \eta \eta }+{\widehat{f}}_{\eta }^2+\widehat{\theta }{G}_r+\widehat{f}{\widehat{f}}_{\eta \eta }-M^2{\widehat{f}}_{\eta }-{\widehat{f}}_{\eta }^2{F}_1+{\widehat{f}}_{\eta }{k}_1.\hfill \end{array}$$

(40)

The deformation equations that can be of the $0\mathrm{th}$ order are listed as follows:

$$\begin{array}{c}(1-p){\mathcal{L}}_{\widehat{\varphi }}\left[\widehat{\varphi }(\eta ;p)-{\widehat{\varphi }}_0\left(\eta \right)\right]=p{\hslash }_{\widehat{\varphi }}{\mathrm{N}}_{\widehat{\varphi }}\left[\widehat{\varphi }(\eta ;p),\widehat{\theta }(\eta ;p),\widehat{f}(\eta ;p)\right],\hfill \end{array}$$

(41)

$$\begin{array}{c}(1-p){\mathcal{L}}_{\widehat{\theta }}\left[\widehat{\theta }(\eta ;p)-{\widehat{\theta }}_0\left(\eta \right)\right]=p{\hslash }_{\widehat{\theta }}{\mathrm{N}}_{\widehat{\theta }}\left[\widehat{\varphi }(\eta ;p),\widehat{\theta }(\eta ;p),\widehat{f}(\eta ;p)\right],\hfill \end{array}$$

(42)

and

$$\begin{array}{c}(1-p){\mathcal{L}}_{\widehat{f}}\left[\widehat{f}(\eta ;p)-{\widehat{f}}_0\left(\eta \right)\right]=p{\hslash }_{\widehat{f}}{\mathrm{N}}_{\widehat{f}}\left[\widehat{f}(\eta ;p)\right].\hfill \end{array}$$

(43)

The BCs for Equations (41)–(43) are given by

$$\begin{array}{c}{\left.\widehat{f}(\eta ;p)\right|}_{\eta =0}=-R_1,{\left.\widehat{\theta }(\eta ;p)\right|}_{\eta =0}=1,{\left.\widehat{\varphi }(\eta ;p)\right|}_{\eta =0}=1,{\left.\frac{\partial \widehat{f}(\eta ;p)}{\partial \eta }\right|}_{\eta =0}=1,\hfill \\ {\left.\frac{\partial \widehat{f}(\eta ;p)}{\partial \eta }\right|}_{\eta =\infty }=1,{\left.\widehat{\theta }(\eta ;p)\right|}_{\eta =\infty }=0,{\left.\widehat{\varphi }(\eta ;p)\right|}_{\eta =\infty }=0.\hfill \end{array}$$

(44)

Moreover, when p = 0 and p = 1, we have

$$\begin{array}{c}\widehat{\varphi }(\eta ;0)=\widehat{{\varphi }_0}\left(\eta \right),\hfill \\ \widehat{\theta }(\eta ;0)=\widehat{{\theta }_0}\left(\eta \right),\hfill \\ \widehat{f}(\eta ;0)=\widehat{f_0}\left(\eta \right),\hfill \end{array}$$

(45)

and

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

(46)

Now, by using the Maclaurin expansion for $\widehat{\varphi }(\eta ;p)$, $\widehat{\theta }(\eta ;p)$, and $\widehat{f}(\eta ;p)$ at p = 0, we have

$$\begin{array}{c}\widehat{\varphi }(\eta ;0)={\widehat{\varphi }}_0\left(\eta \right)+{\displaystyle \sum _{m=1}^{\infty }}{\widehat{\varphi }}_m\left(\eta \right)p^m,\hfill \\ \widehat{\theta }(\eta ;0)={\widehat{\theta }}_0\left(\eta \right)+{\displaystyle \sum _{m=1}^{\infty }}{\widehat{\theta }}_m\left(\eta \right)p^m,\hfill \\ \widehat{f}(\eta ;0)={\widehat{f}}_0\left(\eta \right)+{\displaystyle \sum _{m=1}^{\infty }}{\widehat{f}}_m\left(\eta \right)p^m.\hfill \end{array}$$

(47)

Here, we have

$$\begin{array}{c}{\widehat{\varphi }}_m\left(\eta \right)={\left.\frac{1}{m!}\frac{{\partial }^n\widehat{\varphi }(\eta ;p)}{\partial p^n}\right|}_{p=0},\hfill \\ {\widehat{\theta }}_m\left(\eta \right)={\left.\frac{1}{m!}\frac{{\partial }^n\widehat{\theta }(\eta ;p)}{\partial p^n}\right|}_{p=0},\hfill \\ {\widehat{f}}_m\left(\eta \right)={\left.\frac{1}{m!}\frac{{\partial }^n\widehat{f}(\eta ;p)}{\partial p^n}\right|}_{p=0}.\hfill \end{array}$$

(48)

The deformation equations that can be of the mth-order are defined as follows:

$$\begin{array}{c}{\mathcal{L}}_{\widehat{\varphi }}\left[{\widehat{\varphi }}_m\left(\eta \right)-{\chi }_m{\widehat{\varphi }}_{m-1}\left(\eta \right)\right]={\hslash }_{\widehat{\varphi }}{\mathfrak{R}}_m^{\widehat{\varphi }}\left(\eta \right),\hfill \end{array}$$

(49)

$$\begin{array}{c}{\mathcal{L}}_{\widehat{\theta }}\left[{\widehat{\theta }}_m\left(\eta \right)-{\chi }_m{\widehat{\theta }}_{m-1}\left(\eta \right)\right]={\hslash }_{\widehat{\theta }}{\mathfrak{R}}_m^{\widehat{\theta }}\left(\eta \right),\hfill \end{array}$$

(50)

and

$$\begin{array}{c}{\mathcal{L}}_{\widehat{f}}\left[{\widehat{f}}_m\left(\eta \right)-{\chi }_m{\widehat{f}}_{m-1}\left(\eta \right)\right]={\hslash }_{\widehat{f}}{\mathfrak{R}}_m^{\widehat{f}}\left(\eta \right).\hfill \end{array}$$

(51)

together with the BCs given by

$$\begin{array}{c}\widehat{\varphi }\left(0\right)=1,\widehat{f}\left(0\right)=-R_1,\widehat{\theta }\left(0\right)=1,\widehat{f^{\prime }}\left(0\right)=1,\widehat{f^{\prime }}(\infty )=1,\widehat{\theta }(\infty )=0,\widehat{\varphi }(\infty )=0.\hfill \end{array}$$

(52)

5. Results and Discussion

Here, in this section, we shall discuss the convergence of the above-derived solution. The analytical solution includes the convergence control parameter ℏ, which can affect the rate of approximation and the converging regions for the HAM solutions. The constant ℏ curve is quite rational whenever convergence takes place at p = 1. The convergence of Equations (41)–(43) is completely particularized by the secondary restriction ${\hslash }_f$, ${\hslash }_{\theta }$, and ${\hslash }_{\varphi }$. There was a choice to control the series solutions and achieve their convergence. The probabilities of ℏ are represented by the ℏ curve for the fifth-order computed HAM solution. The efficient regions of ℏ are −2.0 $<{\hslash }_f<$ 1.0, −4.0 $<{\hslash }_{\theta }<$ 1.0 and −3.0 $<{\hslash }_{\varphi }<$ 1.0. The convergence of the homotopy analysis method by the ℏ curve was employed for $f^{″}\left(0\right)$, ${\theta }^{\prime }\left(0\right)$ and ${\varphi }^{\prime }\left(0\right)$, as demonstrated in Figure 2, Figure 3 and Figure 4.

5.1. Velocity Profile

The important characteristics of the flow profile are graphically displayed in this subsection for certain embedded parameters, such as the magnetic parameter, the inertial parameter, the porosity parameter, and the Grashof number. Figure 5, Figure 6, Figure 7 and Figure 8 represent the impact of $f^{\prime }\left(\eta \right)$, which is displayed, respectively, for varying rates of M, $F_1$, $k_1$, and $G_r$. Figure 5 characterizes the results of $f^{\prime }\left(\eta \right)$ for the dissimilar rate of M when $G_r$ = 2.0, $k_1$ = 0.4, and $F_1$ = 0.5. From this graph, we predict that $f^{\prime }\left(\eta \right)$ is dropped with the rising of M. Physically, the magnetic effects create drag forces, which produce resistance and can slow down the fluid motion, thereby reducing $f^{\prime }\left(\eta \right)$. Figure 6 characterizes the result of $f^{\prime }\left(\eta \right)$ for the dissimilar rate of $F_1$ when $G_r$ = 2.0, $N_t$ = 0.6, $k_1$ = 0.4, and M = 0.8. It can be observed from Figure 6 that the $f^{\prime }\left(\eta \right)$ curves are reduced against the rising of the values of $F_1$. The inertial factor is directly proportional to the drag coefficient and porous medium. Thus, for high values of $F_1$, both the drag coefficients and porosity of the medium rise. Consequently, a resisting force is boosted and, therefore, lower velocity is obtained, corresponding to bigger values of $F_1$. Figure 7 depicts the variations of $f^{\prime }\left(\eta \right)$ for various values of $k_1$ when $G_r$ = 2.0, $N_b$ = 0.4, $F_1$ = 0.5, and M = 0.8. This curve shows that $f^{\prime }\left(\eta \right)$ decreases for some rising values of $k_1$. Porous medium boosted the resistance to the fluid motion. Since the free convection Reynolds number is not an independent parameter and the heat transfer coefficients are small, a new dimensionless grouping of the Grashof number plays an important role, which incorporates the coefficients of the thermal expansion ${\beta }_1$ in the formulation. Figure 8 depicts the behavior of $f^{\prime }\left(\eta \right)$ for varying values of $G_r$ when $k_1$ = 0.4, $Sc$ = 0.3, $F_1$ = 0.5, and M = 0.8. This figure shows that $f^{\prime }\left(\eta \right)$ increases for larger rates of $G_r$. The Grashof number is a measurement of the ratio of the buoyancy force to the viscous force operating on the fluid.

5.2. Thermal and Concentration Profiles

The important characteristics of the thermal and concentration properties are graphically displayed in this subsection for certain embedded parameters, such as the Brownian motion parameter, the thermophoresis parameter, the radiation parameter, the heat generation/absorption parameter, the Prandtl number, and the Schmidt number. Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 show the impact of $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$, which are, respectively, plotted for certain values of $\Delta$, $N_b$, $N_t$, R, $Pr$, and $Sc$. The behavior of the heat source $(\Delta >0)$ or the heat sink parameter $(\Delta <0)$ on $\theta \left(\eta \right)$ has been analyzed in Figure 9 and Figure 10. From these figures, it is recognized that the heat source $(\Delta >0)$ enhanced the heat transfer variation. This enhancement in the temperature of the fluid causes the greatly-induced flow toward the plates through the buoyancy impact. Moreover, the figures show that the heat sink parameter $(\Delta <0)$ declines $\theta \left(\eta \right)$. This decline in $\theta \left(\eta \right)$ causes a reduction in the boundary layer flow, which is the result of the buoyancy impact on the thermal problem. Figure 11 and Figure 12 elaborate on the effects of $N_b$ on $\theta \left(\eta \right)$ when $N_t$ = 0.7, $Pr$ = 2.5, R = 1.3 and $\varphi \left(\eta \right)$ when $N_t$ = 0.7 and $Sc$ = 0.4. These results recognize that $\theta \left(\eta \right)$ is enhanced for bigger values of $N_b$, whereas the reverse trend is noted for $\varphi \left(\eta \right)$. Figure 13 and Figure 14 indicate the effects of $N_t$ on $\theta \left(\eta \right)$ whenever $N_b$ = 0.6, $Pr$ = 2.7, R = 1.6, and $\varphi \left(\eta \right)$ whenever $N_b$ = 0.6 and $Sc$ = 0.4. Due to the enhancement in the thermophoretic parameter, the thermophoresis force is enhanced, which helps the fluid to migrate from the heated location to the cooler one. Thus, clearly, $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ are enhanced for higher rates of $N_t$. Figure 15 indicates the behavior of R on $\theta \left(\eta \right)$. It is necessary to keep in mind that more of an estimation of R strengthens $\theta \left(\eta \right)$ because the radiation factor produces thermal energy in the flow regions. Therefore, increments have been observed in $\theta \left(\eta \right)$. The behavior of $Pr$ on $\theta \left(\eta \right)$ is exhibited in Figure 16. From this graph, we conclude that an increase in $Pr$ declines the $\theta \left(\eta \right)$ curves. The physical interpretations of the Prandtl number follow its definition as a ratio of the momentum diffusivity to the thermal diffusivity. It is common in the problem of heat transfer that $Pr$ is utilized to reduce the relative thickness of the momentum and the thermal boundary layers. Figure 17 indicates that there is a reduction in $\varphi \left(\eta \right)$ for $Sc$ = 0.5, 1.2, 1.8, and 2.4 when $N_t$ = 1.2 and $N_b$ = 1.4. Physically, the Schmidt number provides the measurement of the relative effectiveness of the mass and momentum transport by diffusion in the concentration and the velocity boundary layer.

5.3. Table Discussions

Our discussions on the numerical outcomes of $S{h}_x$, $C_f$, and $N{u}_x$ for different convergence factors, such as $F_1$, $k_1$, M, $G_r$, R, $Pr$, $N_t$, $N_b$, and $Sc$ are exhibited in

Table 1. Numerical outcomes of $S{h}_x$ for $N_t$, $N_b$ and $Sc$.

${\mathit{N}}_{\mathit{t}}$	${\mathit{N}}_{\mathit{b}}$	$\mathit{Sc}$	${\mathit{Sh}}_{\mathit{x}}$
0.2	0.1	0.3	2.45038307
0.4			2.36012785
0.6			2.34056085
0.8			2.31059187
	0.1		2.26205129
	0.3		2.27140738
	0.5		2.29025809
	0.7		2.29506702
		0.3	2.46230876
		0.5	2.45037182
		0.7	2.44570887
		0.9	2.42071925

,

Table 2. Numerical outcomes of $C_f$ for $k_1$, M, $G_r$, and $F_1$.

${\mathit{F}}_1$	${\mathit{k}}_1$	M	${\mathit{G}}_{\mathit{r}}$	${\mathit{C}}_{\mathit{f}}$
0.3	0.1	0.3	1.0	1.63205892
0.4				1.64218038
0.5				1.64575398
0.6				1.66094124
	0.1			1.23056802
	0.2			1.24059735
	0.3			1.25192114
	0.4			1.25602565
		0.3		1.43271729
		0.5		1.45952172
		0.8		1.47062175
		1.0		1.48073426
			1.0	2.03968538
			2.0	1.90250917
			3.0	1.86620846
			4.0	1.80911127

and

Table 3. Numerical outcomes of $N{u}_x$ for R, $Pr$, $N_t$, and $N_b$.

R	$\mathit{Pr}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{N}}_{\mathit{b}}$	${\mathit{Nu}}_{\mathit{x}}$
0.5	2.5	0.1	0.2	2.53802156
1.0				2.52508295
1.2				2.51862063
1.5				2.50752047
	2.5			1.54531398
	3.5			1.59370589
	4.5			1.64950842
	5.5			1.66097014
		0.2		1.10459009
		0.4		1.02054179
		0.6		0.85628206
		0.8		0.80049531
			0.3	1.27353338
			0.5	1.25904107
			0.7	1.23718056
			0.9	1.21562169

. Table 1 indicates the numerical outcomes of $S{h}_x$ for $N_t$ = 0.2, 0.4, 0.6, 0.8, $N_b$ = 0.1, 0.3, 0.5, 0.7, and $Sc$ = 0.3, 0.5, 0.7, 0.9. The result of $S{h}_x$ is enhanced for a greater estimation of $N_t$ and $Sc$ while $S{h}_x$ diminishes for the rising credit of $N_b$. It is clearly depicted from Table 2 that $C_f$ boots up for higher rates of $F_1$, $k_1$ and M, whereas the opposite behavior is seen for higher values of $G_r$. The variations of $Pr$, $N_t$, $N_b$, and R upon $N{u}_x$ are discussed in Table 3. It is clearly observed from Table 3 that $N{u}_x$ increases for more increments of R, $N_b$, and $N_t$, while it reduces for higher values of $Pr$. Furthermore,

Table 4. Comparison of $-{\theta }^{\prime }\left(0\right)$ for $R_1$ and ${\delta }_1$.

${\mathit{R}}_1$	${\mathit{\delta }}_1$	Daniel [49]	Chamkha [50]	Present Study
0	0.5	0.94765	0.94769	0.94661
0	1.0	1.07895	1.07996	1.07713
0.45	0.5	0.82396	0.82397	0.82238
0.45	1.0	0.96190	0.96191	0.96052
−1.5	0.5	1.57077	1.57077	1.57061
−1.5	1.0	1.66182	1.66184	1.66047

represents the comparison of $-{\theta }^{\prime }\left(0\right)$ for several values of $R_1$ and ${\delta }_1$ by using HAM with the results by Daniel [49] and Chamkha [50]. It can also be observed from Table 4 that the obtained results are in close agreement with those published in the earlier investigations.

6. Concluding Remarks

In this study, we proposed a mathematical model in order to investigate how radiation and buoyancy affect the MHD 2D boundary layer’s fluid flow across a stretchable porous sheet. The flow of fluid is considered laminar, incompressible, and electrically conducted. The behavior of thermophoresis, heat source, sink, and the Brownian motion were systematically investigated in this work. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived if the flow is due to temperature variations. The results, which we presented in this work, are shown to match with those in the earlier published studies and are found to be in remarkable agreement. Before solving the modeled problem, similarity variables were utilized to convert the fundamental governing PDEs into the corresponding ODEs. A semi-analytical HAM approach was then applied for solving Equations (27)–(29) with the boundary conditions in Equation (30). The analysis was carried out up to the fifth-order HAM approximation solution.

The core findings of this work are summarized as follows:

• $f^{\prime }\left(\eta \right)$ increased for high values of $G_r$ and decreased for the increment values of $k_1$, $F_1$, and M.
• $\theta \left(\eta \right)$ enhances with the rising rates of $(\Delta >0)$, $N_b$, R, and $N_t$, and diminishes with higher $Pr$ and negative values of $\Delta$.
• For more estimations of $N_t$, there is an increment in the $\varphi \left(\eta \right)$ curves.
• $\varphi \left(\eta \right)$ gradually diminishes against higher values of $Sc$ and $N_b$.
• Higher estimations of M, $F_1$, and $k_1$ upsurge $C_f$, whereas the reverse is seen for $G_r$.
• $N{u}_x$ enhances due to increments of $N_t$, R and $N_b$, while it reduces due to higher $Pr$ values.
• $S{h}_x$ reduces due to greater estimation of $N_b$ and decreases for higher values of $N_t$ and $Sc$.
• The role of the Grashof number is the same in free convection as that of the Reynolds number in the forced convection.