Numerical Simulation of the Effects of Reduced Gravity, Radiation and Magnetic Field on Heat Transfer Past a Solid Sphere Using Finite Difference Method

Abstract

The current study deals with the reduced gravity and radiation effects on the magnetohydrodynamic natural convection past a solid sphere. The studied configuration is modeled using coupled and nonlinear partial differential equations. The obtained model is transformed to dimensionless form using suitable scaling variables. The finite difference method is adopted to solve the governing equation and determine the velocity and temperature profiles in addition to the skin friction coefficient and Nusselt number. Furthermore, graphic and tabular presentations of the results are made. The verification of the numerical model is performed by comparing with results presented in the literature and a good concordance is encountered. The main objective of this investigation is to study the effect of the buoyancy force caused by the density variation on natural convective heat transfer past a solid sphere. The results show that the velocity increases with the reduced gravity parameter and solar radiation but decreases with Prandtl number and magnetic field parameter. It is also found that the temperature increases the with solar radiation and magnetic field but decreases with the reduced gravity parameter and Prandtl number.

1. Introduction

Natural convection has become the interest of researchers and engineering due to its manifestation in several natural phenomena and engineering applications. This can be seen in the plume rising from the hot air from fire, oceanic currents, etc. Its principal applications in the industrial field are natural cooling and ventilation without the assistance of fans. The physics of reduced gravity is of great interest and the spherical-shaped components are significant in industry and engineering applications. Potter and Rily [1] investigated the free convection model for high Grashof number values. For fixed values of Prandtl and Grashof numbers, Riley [2] investigated natural convection around a stationary sphere. Heat Transfer process due to heated bar under the consideration of reduced gravity influence has been carried out by Kay [3]. An oscillatory heat transport process past a stationary sphere was performed by Ashraf et al. [4]. Ashraf and Fatima [5] numerically solved the unsteady fluid flow and heat transfer along a sphere by considering the fluid dissipation effect. Ashraf et al. [6] considered the nanofluid free convection in the plume region above a stationary sphere. The effects of thermophoretic transportation on convective heat transfer by taking different fluid characteristics along a sphere have been explored in [7,8,9,10,11,12]. Ahmad et al. [13] used the finite difference method to numerically study the combined effect of chemical reaction and convective heat transfer along a curved surface. Abbas et al. [14] numerically studied the MHD Sakiadis flow using variable density influence on an inclined surface. Rashad et al. [15] analyzed the energy transport in a nanofluid due to convection along cylindrical geometry inserted in porous media. Khan et al. [16] studied the MHD free convection in a vapor plume formed as a result of a fluid eruption while considering the effect of heat generation.

Bulinda et al. [17] investigated the magnetohydrodynamic free convective flow process along a corrugated vibrating bottom surface with a focus on the effects of Hall currents. Molla et al. [18] considered the impact of varying conductivity on the natural convection over a sphere immersed in quiescent fluid. Alwawi et al. [19] presented a study on the MHD natural convection flow of a Casson nanofluid around a solid sphere. Jenifer et al. [20] proposed a model of unsteady mixed convection past a sphere with the consideration of magnetic field and variable fluid property effects. Ahmad et al. [21] considered the chemically reacting natural convection past a curved surface with varying of the thermal conductivity and dynamic viscosity. Salleh et al. [22] considered the mixed convective flow around a sphere by considering the Newtonian heating effect. The effects of varying viscosity and chemical reaction on the natural convection process were investigated by Molla and Hossain [23]. Chamkha et al. [24] numerically modeled a non-Darcy heat transfer of a nanofluid past a porous vertical cone inserted in a permeable medium. Chamkha [25] investigated the MHD double diffusion natural convection past a sphere. Sparrow and Gregg [26] studied the free and forced convection flows over a flat plate for low Prandtl number values. Zhang et al. [27] studied the thermal-mechanical coupling propagation and transient thermal fracture in multilayer coatings. Liu et al. [28] considered a 3D solid-air model to simulate heat and mass transfer in vertical double tube heat exchangers. Some other relevant studies related to the current subject can be found in the literature [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45].

Based on the above described literature review, it can be concluded that combined effects of magnetohydrodynamic, reduced gravity, and solar radiation on natural convection past a sphere have not been the subject of any study before the present attempt. The developed model and solution procedure are presented in the next sections. The results are displayed in graphs and tables with a detailed discussion based on physical interpretations.

2. Problem Analysis

The studied configuration (Figure 1) corresponds to a two-dimensional viscous, steady, and incompressible flow past a stationary sphere in the presence of an external horizontal magnetic field. Coordinates along and normal to the flow are $\left(x,y\right)$. Temperature of a surface is$T_w$ and free stream temperature is $T_{\infty }$with $T_w>T_{\infty }$. By following [1,2,3], the mathematical formulation is expressed as:

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

(1)

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

(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 is $\overline{r}=\mathrm{a}\sin \overline{\frac{x}{a}}$ and is known as radial distance. The notations $\left(\overline{u},\overline{v}\right)$ are velocity components toward and normal to the flow direction. The symbols $a, g, \rho , \nu , {\sigma }_e, B_0$ and ${\alpha }_m$ designate the radius of the sphere, gravity acceleration, fluid density, kinematic viscosity, electric conductivity, thermal diffusivity and magnetic field strength, respectively.

Below is a presentation of the radiant heat flow $q_r$:

𝑞𝑟 = −4𝜎3𝐾𝑅𝜕𝑇4𝜕𝑦.

(4)

$K_R$ stands for mean absorption coefficient. Stefan–Boltzmann constant is represented by the symbol $\sigma$. Equation (4)’s right side $T^4$ is given as follows.

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

So, Equation (4) becomes:

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

(5)

and Equation (3) is expressed as:

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

(6)

Further simplification of Equation (6) gives:

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

(7)

The relationship between density and temperature 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:

$$T\to T_m\pm \mathsf{\Delta }Ty\to \pm \infty$$

(9)

For fixed $\mathsf{\Delta }T$. Consider region $y\ge 0$ subject to the boundary conditions to obtain the symmetry in this situation.

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

(10)

where $T_{\infty }=T+\mathsf{\Delta }T$ and has the relation with ${\rho }_{\infty }$ shown by Equation (8). Determining the reduced gravity is simple.

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

(11)

Fluid particle acceleration has a density ${\rho }_m$. Thus, from Equation (8):

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

(12)

Moreover, skin friction coefficient and 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. Dimensionless Variables

The equations given in (1)–(3) subject to conditions given in (10) are made dimensionless by employing the following non-dimensional variables [9]:

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

(15)

By introducing Equation (10) into Equations (1)–(3) with (8) we have,

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

(16)

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

(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, $R_g=\frac{g^{\prime }}{g \beta \mathsf{\Delta }T}, Rd=k{K}_R/4{\sigma }^{*}{T}_{\infty }^3$, Pr$=\frac{\nu }{\alpha }$ and $M=\frac{\sigma B_o^2{a}^{2}G{r}^{\frac{1}{4}}}{\rho \nu }$ are reduced gravity, radiation parameter, Prandtl number, and magnetic field parameter, respectively. Here, $g’$ is reduced gravity acceleration defined in Equation (12).

4. Solution Methodology

The above Equations (16)–(19) are discretized using the finite difference method. Theses equations are firstly transformed to a smooth form, then a numerical algorithm is written using FORTRAN coding language. The dimensionless variables are defined as [9]:

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

(20)

By putting Equation (20) in (16)–(19) the set of governing equations becomes:

$$\mathrm{XU}\cos \mathrm{X}+\left(\mathrm{X}\frac{\partial \mathrm{U}}{\partial \mathrm{X}}-\frac{\mathrm{Y}}{2}\frac{\partial \mathrm{U}}{\partial \mathrm{Y}}+\frac{\partial \mathrm{V}}{\partial \mathrm{Y}}\right)\sin X=0,$$

(21)

$$XU\frac{\partial U}{\partial X}+\frac{1}{2}{U}^2+\left(V-\frac{YU}{2}\right)\frac{\partial U}{\partial Y}=\frac{{\partial }^{2}U}{\partial Y^2}+Rg\left(2\theta -{\theta }^2\right)\sin X-\mathrm{M}U$$

(22)

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

(23)

The corresponding boundary conditions are:

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

(24)

Method of Solution

The finite difference method is used to solve the set of governing Equations (21)–(23) with the boundary conditions (24). The $X$-axis is used to apply the backward difference, while the$Y$-axis is used to apply the central difference. After the discretization of the governing equations, the discretized variables are identified as $(U_{i,j},V_{i,j}, {\theta }_{i,j})$. The detailed discretization is described as follow:

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

(25)

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

(26)

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

(27)

The following system of algebraic equations is obtained by combining Equations (25)–(27), Equations (21)–(23), and the boundary conditions specified in Equation (24).

Continuity equation:

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

(28)

Momentum equation:

$$\left(1+\frac{∆Y}{2}(V_{\left(i,j\right)}-\frac{Y_j}{2}{U}_{\left(i,j\right)}\right)U_{\left(i-1,j\right)}+\left(-2-\frac{∆Y^2}{∆X}{X}_i{U}_{\left(i,j\right)}-M\right)U_{\left(i,j\right)}+\left(1-\frac{∆Y}{2}\left(V_{\left(i,j\right)}-\frac{Y_j}{2}{U}_{\left(i,j\right)}\right)\right)U_{\left(i+1,j\right)}=-∆Y^{2}sin{X}_{i}Rg(2{\mathsf{\theta }}_{\left(i,j\right)}-{\mathsf{\theta }}^2{}_{\left(i,j\right)} )$$

(29)

Energy equation:

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

(30)

The convergence criterion is presented as follows to achieve accurate numerical solutions for the variables $U, V$ and $\mathsf{\theta }$, respectively.

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

where $ϵ={10}^{-5}$. The computation is started at $X=0$ and then marches downstream implicitly.

5. Grid Independency Test and Numerical Model Verification

To check the accuracy of the numerical model a grid independency test is performed by considering different grid numbers. The grid independency test is performed for $M=1.0$, $Rd=1.0, R_g=5.0$and $Pr=7.0$ at the circumferential position $X=\mathsf{\pi }/4.$ The results of this test are presented in

Table 1. Grid independency test for $M=1.0,Rd=1.0,R_g=10.0,\mathrm{and} \mathrm{Pr}=7.0$ at $X=\mathsf{\pi }/4$.

No. of Grid Points	${\left(\frac{\partial U}{\partial Y}\right)}_{Y=0}$	$-{\left(\frac{\partial \theta }{\partial Y}\right)}_{Y=0}$
25.0	2.01299	0.79793
50.0	2.12478	0.77268
100.0	2.15249	0.76919
200.0	2.15950	0.76891
250.0	2.16036	0.76892
500.0	2.16150	0.76894

. It is noticed that an excellent solution accuracy is observed as the number of grid points is increased. The deviation between 200 point and 500-point grids for the skin friction and heat transfer rate at position $X=\mathsf{\pi }/4$ are calculated. It is noticed that deviation percentages are 0.09% and 0.003% for skin friction and heat transfer rate, respectively. Thus, the 200-point grid can be considered as sufficient for the convergence and accuracy of the results and is chosen to perform all the calculations. This grid corresponds to step sizes $\mathsf{\Delta }X=0.05$ and $\mathsf{\Delta }Y=0.02$. The solutions determined using the finite difference method based on the selected grid are discussed in detail in the forthcoming section.

In order to verify the validity of the proposed numerical model, a comparison of the findings of ${\left(\frac{\partial U}{\partial Y}\right)}_{Y=0}$ with earlier published results for several Pr values are presented in

Table 2. Verification of numerical model; comparison of ${\left(\frac{\partial U}{\partial Y}\right)}_{Y=0}$ with the results of Sparrow & Gregg [26] in the absence of the reduced gravity term for $M=0\mathrm{and} Rd=0,$at $\mathsf{\pi }/2$.

$\mathit{P}\mathit{r}$	Sparrow & Gregg [26]	Present
0.03	0.93841	0.93740
0.02	0.95896	0.95870
0.008	0.99550	0.99400

. In can be noticed that a good concordance exists between the results.

6. Results and Discussion

In current section the numerical solutions of the set of the equations governing the considered configuration are discussed in detail. The effects of the parameters governing the studied configuration on the velocity$U$, and temperature $\mathsf{\theta }$, profiles in addition to the coefficient of skin friction $C_f$and the Nusselt number $Nu$ are presented in form of graphs and tables. Figure 2 depicts the effect of the reduced gravity parameter $Rg$ on the velocity profile $U$ for Prandtl number Pr = 7.0 at different locations of the sphere. It is noticed that at diverse positions of the sphere when $Rg$is increased, the fluid velocity increases. In Figure 3 the numerical outcomes of temperature $\theta$ for different values of $Rg$are displayed. The temperature curves show that at higher $Rg$values, the fluid temperature drops quickly at all the considered positions. From physical point of view the trend of temperature is logical because the intensification in reduced gravity parameter leads to a reduction in thermal expansion and temperature difference between the surface and ambient temperature, and hence the overall temperature of the fluid flow domain decreases. However, the maximum magnitude for temperature is attained at $X=\mathsf{\pi }$. The velocity profiles for several Pr values are sketched in Figure 4. It is remarked that the increase of Pr leads to a reduction of the fluid velocity at all the considered positions. This decrease is due to increase of the fluid viscosity that causes the increase of the viscous effect and hence the decrease of flow intensity. It is also to be mentioned that the highest value for $U$ is achieved at $X=\frac{\pi }{2}$. Figure 5 shows the effects of Pr on the temperature profiles. Graphical outcomes indicate that the temperature of the fluid reduces as Pr is enhanced. From physical point of this is due to the decrease of the thermal conductivity when Pr is increased. In fact, the capability of the fluid to conduct the heat is reduced and thus the temperature of the fluid decreases. Figure 6 and Figure 7 show the behaviors of velocity and temperature profiles for several values of radiation parameter $Rd$ at several positions around the sphere. It is noticed that temperature and fluid velocity rise with the augmenting numerical of $Rd$ values. An intensification in $Rd$ leads to a rise the thermal conductance of the fluid and the mean absorption coefficient that helps to boost the temperature of the fluid flow as shown in Figure 6 and Figure 7. Figure 8 and Figure 9 illustrate the impact of magnetic field parameter $M$ on the temperature and velocity profiles. It is to be mentioned that the velocity is decreasing, and the temperature is increasing with the intensification of the magnitude of the magnetic field. From a physical point of view, the application of the external applied magnetic field perpendicular to the flow direction generates a Lorentz force that opposes the flow and reduce the fluid velocity. In addition, due to the resistance to the flow, the viscous effect causes an augmentation of fluid temperature.

Table 3. Consequences of reduced gravity parameter $Rg$ on kin friction coefficient $C_f$ and Nusselt number $Nu$.

		${\mathit{C}}_{\mathit{f}}$					$\mathit{N}\mathit{u}$
$\mathit{X}$	$\mathit{R}\mathit{g}$ = 0.1	$\mathit{R}\mathit{g}$ = 1.0	$\mathit{R}\mathit{g}$ = 5.0	$\mathit{R}\mathit{g}$ = 10.0	$\mathit{R}\mathit{g}$ = 0.1	$\mathit{R}\mathit{g}$ = 1.0	$\mathit{R}\mathit{g}$ = 5.0	$\mathit{R}\mathit{g}$ = 10.0
$\pi /6$	0.10860	0.64508	2.16368	3.63897	0.16210	0.28475	0.42706	0.50809
$\pi /4$	0.14868	0.86768	2.90589	4.886676	0.17741	0.31446	0.47131	0.56063
$\pi /2$	0.19412	1.12016	3.74840	6.30307	0.19241	0.34257	0.51314	0.61033
$\pi$	0.00013	0.00135	0.00701	0.01477	0.10090	0.09991	0.09515	0.08816

presents the results of the of skin friction coefficient and Nusselt number versus the reduced gravity parameter $Rg$. An increasing in the values of $Rg$ enhances the skin friction coefficient and Nusselt number at all the considered circumferential positions.

7. Conclusions

In the current study, the effects of reduced gravity, solar radiation, and external magnetic field on the natural convection past a stationary sphere immersed in a fluid are investigated. The main findings related to effects of reduced gravity and Prandtl number on the velocity $U$, temperature $\theta$, skin friction coefficient $C_f$, and Nusselt number $Nu$ are summarized as:

• When $Rg$is increased, the velocity increases, and the temperature decreases due the enhancement of the buoyancy force.
• The increase of Pr leads to the decrease of the velocity and temperature of the fluid, due to the increase of the viscosity.
• The application of an external magnetic field causes the reduction of the flow intensity and an augmentation of the temperature.
• The increase in $Rd$ cause an increase of the temperature and a reduction of the velocity.