The Effects of Reduced Gravity and Radiative Heat Transfer on the Magnetohydrodynamic Flow Past a Non-Rotating Stationary Sphere Surrounded by a Porous Medium

Abstract

In the present study, the effects of reduced gravity and solar radiation on the magnetohydrodynamics (MHD) fluid flow and heat transfer past a solid and stationary sphere embedded in a porous medium are investigated. A model describing the considered configuration is put in dimensionless form using appropriate dimensionless variables and then transformed to primitive form for a smooth algorithm on a computing tool. A primitive form of the model is solved by employing the finite difference method. Solutions for variables of interest, such as velocity distribution and temperature field, along with their gradients, are depicted in graphs and tables. The main goal of the paper is to study the physical impact of reduced gravity on heat transfer and fluid flow around a sphere surface inserted in a porous medium in the presence of an applied magnetic field and solar radiation. The effects of the governing parameters, which are the reduced gravity parameter, magnetic field parameter, radiation parameter, porous medium parameter, and the Prandtl number, are discussed and physically interpreted. The displayed solutions indicate that velocity rises with the reduced gravity and solar radiation parameters but decreases with augmenting the Prandtl number, magnetic field parameter, and porous medium parameter. It is deduced from the presented results that the temperature becomes lower by increasing the values of the reduced gravity parameter and the Prandtl number, but, on the other hand, it becomes higher by increasing the values of the magnetic field, the porous medium, and the radiation parameters at all the considered positions of the surface of the sphere. A comparison between the present and already published results is performed to check the validity of the proposed numerical model.

1. Introduction

Researchers were interested in natural convection because of its importance in several natural phenomena and engineering applications. It occurs in the plume rising from fire air, as well as in ocean currents and other phenomena. Its primary use in the industrial sector is in free air cooling without the use of fans. Spherical-shaped components are important in both engineering and industrial applications. Kuiken and Merkin [1] investigated the heat transfer process caused by heated bars while considering the influence of reduced gravity. Ostrach [2] studied the free convection along a vertical plate by focusing on the effect of the Grashof number on the temperature and velocity fields. Merkin [3] studied the free convection on a vertical flat plate at small Prandtl numbers. Lin [4] analyzed the natural convective heat transfer in a squared cavity filled with water at maximum density. Ivey and Hemblin [5] studied the free convection for large Rayleigh number values and small aspect ratios in 2D cavities. Potter and Riley [6] used a cloud-free model to study the natural convection across a stationary sphere at high Grashof numbers. Riley [7] investigated the free convection past a sphere under imposed conditions and parameters included in the model. Ashraf et al. [8] carried out a numerical study on the oscillatory heat transfer past a stationary sphere. By considering the fluid dissipation effects, Ashraf and Fatima [9] numerically solved the unsteady fluid flow and heat transfer along a sphere. A numerical investigation was presented by Ashraf et al. [10] to study the free convection of a nanofluid. Researchers in [11,12,13,14,15,16] explored the thermophoretic effects on convective heat transfer around a sphere by considering various fluid properties. Ahmad et al. [17] performed a computational study using the finite difference method to investigate the convective, chemically reacting heat transfer along a curved surface.

From a technological point of view, magnetohydrodynamic effects are crucial due to their diverse applications, including geophysics, electrical power generation, etc. The effect of an externally applied magnetic field on the boundary layer flow is getting a lot of attention from researchers. Several geometries with various boundary conditions have been considered to study MHD convective heat transfer. MHD Sakiadis flow on an inclined surface was quantitatively examined by Abbas et al. [18], with a focus on the influence of varying density. Tamoor et al. [19] examined the magnetohydrodynamic Casson fluid flow over a stretching cylinder. Pattnaik et al. [20] evaluated the effects of the magnetic field and chemical reaction on convective heat and mass transfers over an exponentially extending sheet. The Homotopy Analysis Method was used by Mabood et al. [21] to numerically solve the magnetohydrodynamic flow with thermal radiation effects over an exponentially stretched sheet. Khan et al. [22] presented a numerical study on the bioconvection flow along a paraboloid surface under MHD effects. Bulinda et al. [23] discussed the MHD free convective flow along a corrugated vibrating bottom surface with considering the effects of Hall currents. Alwawi et al. [24] performed a study on the MHD natural convection flow of a Casson nanofluid around a solid sphere. Chamkha et al. [25] studied the double diffusive magnetoconvection past a sphere.

Due to their significance in engineering and industrial applications, such as oil reservoirs, resin transfer models, porous insulation, packed beds, geothermal energy, fossil fuel beds, and nuclear waste disposal, the flows saturating porous space have drawn researchers’ attention. Chitra and Kavitha [26] investigated the pulsatile flow in a circular pipe filled with a porous medium under the influence of a time-dependent pressure gradient. In [27,28,29,30,31], different processes of heat transfer and fluid flow on diverse geometries embedded in porous media have been carried out for various fluids. Hussain and Sheremet [32] carried out an analysis of the radiative nanofluid flow in porous media past a stretching surface under the influence of a magnetic field. Yan et al. [33] studied numerically a micro-combustor embedded in a porous medium. They concluded that the performance of such a device can be improved by increasing the thickness of the porous medium. In the presence of an anisotropic porous medium and stratified fluid, Jha et al. [34] presented semi-analytical results for transient natural convection fluid flow between two concentric vertical cylinders of infinite lengths. The laminar mixed convection in a porous channel with two distinct heat sources on the bottom wall was experimentally investigated by El-Kady [35]. The main goal was to demonstrate how the parameters of heat transfer are affected when a porous medium is used in the channel. For low values of the Prandtl number, Sparrow and Gregg [36] investigated free and forced convection flow across a flat plate.

Due to its diverse applications in industry, including boilers, rocket engines, nuclear reactor cooling, and thermal insulation, convective heat transfer, combined with thermal radiation effects, has attracted the attention of the research community. By taking into account radiation influence, Hossain and Thakkar [37] investigated the model of convective flow across an isothermal vertical plate. The MHD mixed convective flow on an isothermal surface submerged in a porous medium was the subject of a computational investigation performed by Damseh [38]. A study of the effects of thermal radiation on free convection through a permeable enclosure was considered by Zehamatkesh [39]. The radiative convective flow across a vertical plate submerged in a porous medium has been studied by Mondal et al. [40]. According to Muhammad et al. [41], micropolar fluid is impacted by heat generation, magnetic field, radiation, and chemical reaction as it flows through a porous moving surface. A study that focuses on the micropolar fluid flow with radiation action over a stretching sheet was considered by Bhattacharyya et al. [42]. The stagnation point flow of a micropolar fluid in the presence of radiation and a magnetic field was investigated by Pal et al. [43]. The MHD radiative mixed convection flow under an inclined surface was explored by Moradi et al. [44]. Sheikholeslami et al. [45] considered the effect of a magnetic field on the heat transfer and fluid flow in a nanofluid-filled semi-annulus. The effects of radiation and transpiration on the boundary layer flow of a micropolar fluid past a stretching sheet were discussed by Hussain et al. [46]. Mukhopadhyay et al. [47] investigated the boundary layer flow over an exponentially stretched sheet along with the effects of transpiration, radiation, and slip condition. Hayat et al. [48] investigated the thermally stratified radiative flow of third-grade fluid across a stretched surface. Parkesy et al. [49] studied the effect of radiation on transient magnetohydrodynamic flow between porous vertical channels of a micropolar fluid by applying a third-kind boundary condition. According to Uddin et al. [50], thermal radiation, heat generation, and absorption have an important impact on magnetohydrodynamic heat transfer in micropolar fluid past a wedge with Hall and ion-slip currents. Studies on the physics of thermal radiation on diverse fluid flow phenomena are discussed in [51,52,53].

Based on the above-described literature review, the combined effects of MHD, reduced gravity, and solar radiation on heat transfer and fluid flow along a stationary sphere embedded in a porous medium have not yet been considered. In the next sections, the mathematical formulation and solution process for this considered configuration are described. In addition, the results are presented in terms of temperature and velocity profiles.

2. Mathematical Formulation

Consider the viscous, steady, incompressible, two-dimensional, and natural convective flow past a non-rotating and stationary sphere. The reduced gravity, solar radiation, and applied magnetic field effects on optically dense thick fluid flow are considered in the current study. The magnetic field is normal to the flow direction. The coordinates along and normal to the flow are $\left(x,y\right)$, and the corresponding velocity components are ($u,v$). The temperature at the surface is $T_w$, and away from the surface, the free-stream temperature is $T_{\infty }$ with $T_w>T_{\infty }$. Following [1,2,3], the governing equations that are continuity, momentum, and energy equations are given as Figure 1:

$$\frac{\partial \left(\overline{r}u\right)}{\partial \overline{x}}+\frac{\partial \left(\overline{r}v\right)}{\partial \overline{y}}=0,$$

(1)

$$\overline{u}\frac{\partial \overline{u}}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{u}}{\partial \overline{y}}=\nu \frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}+g\frac{\rho -{\rho }_{\infty }}{{\rho }_{\infty }}\mathit{sin}\overline{\frac{x}{a}}-\frac{\sigma B_0^2}{\rho }u-\frac{\nu }{K_o}u,$$

(2)

$$\overline{u}\frac{\partial T}{\partial \overline{x}}+\overline{v}\frac{\partial T}{\partial \overline{y}}={\alpha }_m\frac{{\partial }^{2}T}{\partial {\overline{y}}^2}-\frac{1}{\rho C_p}\frac{\partial q_r}{\partial y}.$$

(3)

The distance from the symmetric axis to the sphere surface (radial distance) is $\overline{r}=a sin\overline{\frac{x}{a}}$. $\left(\overline{u},\overline{v}\right)$ are the velocity components toward and normal to the flow directions. The symbols $a,g,\rho ,\nu ,\sigma ,B_0$, ${\alpha }_m=k/\rho C_P$, and $K_0$ designate the radius of the sphere, gravity acceleration, the density of the fluid, kinematic viscosity, thermal diffusivity, electrical conductivity, magnetic field, and permeability of the porous medium, respectively. In addition, $k$ is the thermal conductivity of the fluid, and $C_P$ is the specific heat at constant pressure.

Using the Rosseland approximation for radiation in an optically thick fluid, refs. [51,52,53], the radiative heat flux is simplified to:

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

(4)

$k^{*}$ stands for average absorption coefficient, and $\sigma$ is the Stefan-Boltzmann. The linearization of $T^4$ is given as follows:

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

Therefore, Equation $\left(4\right)$ becomes:

$$q_r=-\frac{16T_{\infty }^3\sigma }{3K_R}\frac{\partial T}{\partial y}.$$

(5)

Thus, Equation (3) becomes:

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{\infty }}{\rho C_p}\frac{{\partial }^{2}T}{\partial y^2}+\frac{16T_{\infty }^3\sigma }{\rho C_{P}3K_R}\frac{{\partial }^{2}T}{\partial y^2},$$

(6)

Further simplification of Equation (6) results in:

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k}{\rho C_p}\frac{\partial }{\partial y}\left[\frac{\partial T}{\partial y}+\frac{4.4T_{\infty }^3\sigma }{\rho C_{P}3k{K}_R}\frac{\partial T}{\partial y}\right],$$

(7)

The relationship between density and temperature, considering closeness of temperature to $T_m$, is as follows:

$$\frac{\rho -{\rho }_m}{{\rho }_m}=-\gamma {\left(T-T_m\right)}^2.$$

(8)

Furthermore, Equation (6) implies that for steady flow:

$$T\to T_m\pm \Delta Ty\to \pm \infty$$

(9)

For fixed $\Delta T$, consider the region $y\ge 0$ subject to the boundary conditions to obtain the symmetry conditions as:

$$\begin{array}{c}\overline{u}=0,\overline{v}=0,T=T_m \mathrm{at} y=0,\hfill \\ \overline{u}\to 0,T\to T_{\infty }\mathrm{as} y\to \infty .\hfill \end{array}$$

(10)

where $T_{\infty }=T+\Delta T$ and is related to ${\rho }_{\infty }$ by Equation (8). The reduced gravity is determined using the following expression:

$$g^{\prime }=g\frac{({\rho }_m-{\rho }_{\infty })}{{\rho }_{\infty }}$$

(11)

From Equation (8), the fluid particles’ acceleration having ${\rho }_m$ as density becomes:

$$g^{\prime }=g\gamma \frac{{\rho }_m}{{\rho }_{\infty }}{\left(T_{\infty }-T_m\right)}^2$$

(12)

Moreover, the skin friction coefficient and the Nusselt number at the surface are expressed as follows:

$$C_f=\frac{{\tau }_w}{\rho U^2},Nu=\frac{x{q}_w}{k\left(T_w-T_{\infty }\right)}$$

(13)

$$\mathrm{Where} {\tau }_w=\mu {\left(\frac{\partial \overline{u}}{\partial \overline{y}}\right)}_{\overline{y}=0},q_w=-k{\left(\frac{\partial T}{\partial \overline{y}}\right)}_{\overline{y}=0},$$

(14)

3. Solution Methodology

In this section, the resolution of the above-presented governing equations with considered boundary conditions are described in detail.

3.1. Dimensionless Variables

The governing Equations (1)–(3) with boundary conditions (10) are put in their dimensionless forms using the following variables [13]:

$$x=\frac{\overline{x}}{a},y=\frac{\overline{y}G{r}^{\frac{1}{4}}}{a},\theta =\frac{T-T_{\infty }}{T_m-T_{\infty }},u=\frac{a\overline{u}G{r}^{\frac{1}{4}}}{\nu },v=\frac{a\overline{v}G{r}^{\frac{1}{4}}}{\nu },$$

(15)

When Equation (15) is used in Equations (1)–(3) with (10), they take the following form:

$$\frac{\partial \left(\mathit{sin}xu\right)}{\partial x}+\frac{\partial \left(\mathit{sin}xv\right)}{\partial y}=0,$$

(16)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\partial }^{2}u}{\partial y^2}+R_g\left(2\theta -{\theta }^2\right)sinx-Mu-Ku$$

(17)

$$u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}=\frac{1}{Pr}\left(1+\frac{4}{3}Rd\right)\frac{{\partial }^2\theta }{\partial y^2},$$

(18)

$$\begin{gathered} u=0,v=0,\theta =1, \mathrm{at} y=0, \\ u\to 0,\theta \to 0, \mathrm{as} y\to \infty . \end{gathered}$$

(19)

Here, $R_g=\frac{g^{\prime }}{g\beta \Delta T},Rd=kk\ast /4{\sigma }^{*}{T}_{\infty }^3$, Pr $=\frac{\nu }{\alpha }$, $K=\frac{a^{2}G{r}^{\frac{1}{4}}}{K_o}M=\frac{\sigma B_o^2{a}^{2}G{r}^{\frac{1}{4}}}{\rho \nu }$ are reduced gravity, the radiation parameter, the Prandtl number, the porosity parameter, and the magnetic field parameter, respectively. Here, $g^{\prime }$ is reduced gravity acceleration defined in Equation (12).

3.2. Primitive Variable Formulation

Before using the finite difference method (FDM), Equations (16)–(19) are transformed into a smooth form, and a numerical algorithm is written using FORTRAN programming language. The following variables are defined as [13]:

$$u\left(x,y\right)=x^{1/2}U\left(X,Y\right),v\left(x,y\right)=x^{-\frac{1}{4}}V\left(X,Y\right),Y=x^{-\frac{1}{4}}y,X=x,\theta \left(x,y\right)=\theta \left(X,Y\right).$$

(20)

By putting Equation (20) in Equations (16)–(19), the governing equations are expressed as:

$$XUcosX+(X\frac{\partial U}{\partial X}-\frac{Y}{2}\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial Y})sinX=0,$$

(21)

$$\begin{gathered} XU\frac{\partial U}{\partial X}+\frac{1}{2}{U}^2+\left(V-\frac{YU}{2}\right)\frac{\partial U}{\partial Y} \\ =\frac{{\partial }^{2}U}{\partial Y^2}+R_g\left(2\theta -{\theta }^2\right)sinX-MU-KU \end{gathered}$$

(22)

$$XU\frac{\partial \theta }{\partial X}+\left(V-\frac{YU}{2}\right)\frac{\partial \theta }{\partial Y}=\frac{1}{Pr}\left(1+\frac{4}{3}Rd\right)\frac{{\partial }^2\theta }{\partial Y^2}$$

(23)

The corresponding boundary conditions are:

$$\begin{array}{c}U=0,V=0,\theta =1,\mathrm{at} Y=0,\hfill \\ U\to 0,\theta \to 0,as Y\to \infty .\hfill \end{array}$$

(24)

3.3. Solution Scheme

The governing Equations (21)–(23) are discretized using the finite difference method. The $X$-axis is used to apply the backward difference, while the $Y$-axis is used to apply the central difference. After the flow equations are discretized, the unknown variables $(U_{i,j},V_{i,j},{\theta }_{i,j})$ are evaluated based on the boundary conditions specified in (24). The discretization is performed as follows:

$$\frac{\partial U}{\partial X}=\frac{U_{\left(i,j\right)}-U_{\left(i,j-1\right)}}{\Delta X},$$

(25)

$$\frac{\partial U}{\partial Y}=\frac{U_{\left(i+1,j\right)}-U_{\left(i,-1,j\right)}}{2\Delta Y},$$

(26)

$$\frac{{\partial }^{2}U}{\partial Y^2}=\frac{U_{\left(i+1,j\right)}-2U_{\left(i,j\right)}+U_{\left(i,-1,j\right)}}{\Delta Y^2}.$$

(27)

Discretized Continuity equation:

$$\begin{gathered} V_{\left(i+1,j\right)}=V_{\left(i-1,j\right)}-2\frac{\Delta Y}{\Delta X}{X}_i\left(U_{\left(i,j\right)}-U_{\left(i,j-1\right)}\right)+\frac{Y_j}{2}\left(U_{\left(i+1,j\right)}-U_{\left(i-1,j\right)}\right) \\ -2\Delta Y{X}_i\frac{cos{X}_i}{\mathit{sin}{X}_i}{U}_{\left(i,j\right)}, \end{gathered}$$

(28)

Discretized Momentum equation:

$$(1+\frac{\Delta Y}{2}(V_{(i,j)}-\frac{Y_j}{2}{U}_{(i,j)})U_{(i-1,j)}+(-2-\frac{\Delta Y^2}{\Delta X}{X}_i{U}_{(i,j)}-M-K)U_{(i,j)}+(1-\frac{\Delta Y}{2}(V_{(i,j)}-\frac{Y_j}{2}{U}_{(i,j)}))U_{(i+1,j)}=-\Delta Y^{2}sin{X}_i{R}_g(2{\theta }_{(i,j)}-{\theta }^2{}_{(i,j)}),$$

(29)

Discretized Energy equation:

$$(\frac{1}{Pr}(1+\frac{4}{3}Rd)+\frac{\Delta Y}{2}(V_{(i,j)}-\frac{Y_j}{2}{U}_{(i,j)})){\theta }_{(i-1,j)}+(-\frac{2}{Pr}(1+\frac{4}{3}Rd)+\Delta Y^2{U}_{(i,j)}(1-\frac{X_i}{\Delta X})){\theta }_{(i,j)}+(\frac{1}{Pr}(1+\frac{4}{3}Rd)-\frac{\Delta Y}{2}(V_{(i,j)}-\frac{Y_j}{2}{U}_{(i,j)})){\theta }_{(i+1,j)}=-\frac{\Delta Y^2}{\Delta X}{X}_i{U}_{(i,j)}{\theta }_{(i,j-1)},$$

(30)

Discretized Boundary Conditon:

$$\begin{array}{c}{U}_{i,j}=0,V_{i,j}=0,{\theta }_{i,j}=1,\mathrm{at}{Y}_j=0,\hfill \\ U_{i,j}\to 0,{\theta }_{i,j}\to 0,as Y_j\to \infty .\hfill \end{array}$$

(31)

The approximate solutions determined with the finite difference method are discussed in detail in the forthcoming section. The convergence criterion used to achieve accurate numerical solutions for the variables $U,V$, $\theta$, and $\varphi$, respectively, is presented as follows:

$$\max \left|U_{ij}\right|+\max \left|V_{ij}\right|+\max \left|{\theta }_{ij}\right|\le ϵ,$$

where $ϵ={10}^{-5}$. The computation is started at $X=0$ and then goes downstream implicitly. The step sizes are taken as: $\Delta X=0.05$ and $\Delta Y=0.02$.

4. Results and Discussion

The numerical results of the studied configuration are presented and discussed in detail in the current section. Results are presented in term of the profiles of velocity $U$ and temperature $\theta$; skin friction $\partial U/\partial Y$ and heat transfer rate $\partial \theta /\partial Y$ for different values of reduced gravity number $R_g$, Prandtl number Pr, thermal radiation parameter $Rd$, magnetic field parameter $M$, and porous medium parameter $K$ are computed and presented in the form of graphs and tables.

The effects of the reduced gravity parameter $R_g$ on the velocity profile $U$ for Prandtl number $Pr$ = 7.0 at various points on the sphere are shown in Figure 2. The graph shows that when $R_g$ is improved, the velocity profile $U$ increases. The numerical results of fluid temperature for various values of $R_g$ are shown in Figure 3. The temperature curves show that the fluid temperature rapidly decreases as $R_g$ is strengthened, at all the considered points. The maximum temperature magnitude occurs at $X=\pi$. From a physical point of view, the increase of the reduced gravity parameter leads to a decrease in thermal expansion and thus a smaller temperature difference between the surface and the surrounding fluid, which results in a decrease in the fluid flow domain’s overall temperature. Figure 4 and Figure 5 present the effect of the porous medium parameter on the velocity and temperature profiles, respectively. Figure 4 highlights the influence of the porous medium parameter $K$ on the velocity field. It shows that as $K$ is increased, $U$ decreases at all the considered positions. Figure 5 depicts the behavior of $\theta$ for various values of $K$. It is noticed that as $K$ is raised, $\theta$ increases. The velocity and temperature profiles for various values of the solar radiation parameter $Rd$ are shown in Figure 6 and Figure 7 at various locations around the sphere. It should be mentioned that velocity and temperature increase when Rd is increased. Physically, these results are due to the enhancement of the overall heat transfer coefficient and absorption efficiency, which contribute to raising the temperature of the fluid flow domain. The effects of the magnetic field parameter $M$ on the temperature and velocity fields are shown in Figure 8 and Figure 9. With the intensification of $M$, it can be observed that the velocity is reduced, and the temperature is increasing. This fact is physically due to the generated Lorentz force caused by the interaction between the magnetic field and the flow motion. This force reduces the flow intensity, and because of the produced viscous resistance, the fluid’s temperature rises.

The velocity profiles for various values of Pr are sketched in Figure 10. The increase of Pr leads to the decrease of velocity, due to the enhancement of the viscous dissipations. The highest value for $U$ is achieved at $X=\frac{\pi }{2}$. Figure 11 shows the effects of Pr on $\theta$. The results indicate that the temperature of the fluid reduces as Pr is increased. From a physical point of view, this is due to the reduction of the thermal conductivity of the fluid that causes the reduction of temperature. The kinematic viscosity is increased by increasing Pr, and, as a result, the viscous effects become stronger, and the velocity of the fluid decreases.

Table 1. Effect of $Pr$ on $\left(a\right){\left(\frac{\partial U}{\partial Y}\right)}_{Y=0}\left(b\right)-{\left(\frac{\partial \theta }{\partial Y}\right)}_{Y=0}$ for different values of $R_g$, $M=0.1$, K = 0.1, and $Rd=0.1,\mathit{Pr}=7.0$.

${\mathit{R}}_{\mathit{g}}$	${\left(\frac{\partial \mathit{U}}{\partial \mathit{Y}}\right)}_{\mathit{Y}=0}$	$-{\left(\frac{\partial \mathit{\theta }}{\partial \mathit{Y}}\right)}_{\mathit{Y}=0}$
0.1	0.07511	0.32437
0.5	0.25270	0.49175
0.7	0.32540	0.53556
0.9	0.39301	0.57069
1.0	0.42536	0.58607
2.0	0.71561	0.69778
5.0	1.42254	0.87804
10.0	2.39128	1.04443

is showing the numerical results of skin friction $\partial U/\partial Y$ and the rate of heat transfer $\partial \theta /\partial Y$ for different values of reduced gravity parameter $R_g$ when the remaining parameters are kept constant. It is deduced that as $R_g$ is increased, both skin friction and the rate of heat transfer become stronger at the leading edge $Y=0$. It is noted that all the results satisfy the given boundary conditions.

Table 2. Comparison of the results of ${\left(\frac{\partial U}{\partial Y}\right)}_{Y=0}$ with those of Sparrow & Gregg [36] when reduced gravity term is absent for $M=0,Rd=0,\mathrm{K}=0$ at $\pi /2$.

Pr	Sparrow & Gregg [36]	Present
0.03	0.93841	0.93740
0.02	0.95896	0.95870
0.008	0.99550	0.99400

presents the comparison of the present results concerning skin friction with already published results for special cases, and it is observed that there is excellent agreement. This confirms the validity of the current numerical model.

5. Conclusions

The effects of reduced gravity, solar radiation, and an external magnetic field on the heat transfer and fluid flow past a stationary and non-rotating sphere submerged in a fluid and embedded in a porous medium is discussed in the current work. The main findings are summarized as follows:

• When $R_g$ is increased, $U$ increases at all the points on the sphere, and the maximum value occurs at $X=\frac{\pi }{2}$. As $R_g$ is increased, the buoyancy force increases and accelerates the fluid motion. Also, the velocity increases with increasing the values of porous medium parameter $K$.
• The increase in $R_g$ results in the reduction of $\theta$, and the maximum value is obtained at $X=\pi$.
• The increase of Pr leads to a decrease of velocity and temperature, due to the enhancement of viscous effects and the reduction of thermal conductivity.
• The augmentation of $Rd,K$, and $M$ cause the raising of temperature and the reduction of velocity.
• Skin friction and heat transfer rate increase with increasing the values of $R_g$.
• The verification of the used numerical model is performed by comparing it to previously published results on skin friction, and good agreement is encountered.