Evaluation of Magnetohydrodynamics of Natural Convective Heat Flow over Circular Cylinder Saturated by Nanofluid with Thermal Radiation and Heat Generation Effects

Abstract

The current study focuses on the natural-convection flow of nanofluids with boundary layer over a circular cylinder of uniform thermal wall with varying magnetic force from 0 to 1.5, radiative effects from 0 to 1, heat generation effects from 0 to 1, and Joule heating effects from 0 to 1. The problem is represented in the form of partial differential equations. The dimensional form of the equations is converted into a dimensionless form with the help of suitable stream functions. Then, the resultant equations are further reduced into the system of first-ordered differential equations, and the Keller box scheme is applied to obtain a solution numerically with the help of MATLAB code. The numerical solutions for Nusselt number, skin friction coefficient, Sherwood number, velocity profile, temperature profile, and concentration profile are represented with the help of graphs. The most interesting fact of the analysis is the flow of the fluid; the heat-mass and energy transfer rates could be managed in a controlled way through slight variations in the Brownian motion parameter from 0.1 to 0.7, in the Lewis number from 1 to 40, in the Eckert number from 0.1 to 0.4, in the thermophoresis parameter from 0.1 to 0.7, in the Prandtl number from 0.1 to 0.7, and in the buoyancy ratio from 0.1 to 0.7, as it is here analyzed and discussed.

1. Introduction

The configuration of viscous and incompressible fluids flowing with natural convection over a cylinder of circular form lying parallel to the x-axis is important for many engineering processes. The applications of these types of processes are geophysical heat elimination, food storage and processing, metallurgy sciences, geothermal removal of nuclear and non-nuclear reactors, and various branches of engineering, such as mechanical, chemical, and civil engineering. The pioneering studies by Blasius et al. [1] discussed the boundary-layer flow over the horizontally placed circular form of a cylinder in 1908. They found the velocity profile and evaluated the momentum transport equation of the forced-convection boundary-layer flow. Since then, these studies have found a new space in the cabinet of many researchers. Later on, Markin et al. [2] were the first who examined the exact solution of a free-convection flow over the isothermal cylinder. Later on, they found the exact solution of the mixed convective flow over an isothermal cylinder with the wall of the cylinder having a fixed temperature with the help of the Gortler–Blasius series expansion along with the finite difference scheme [3]. With the help of Markin et al.’s work, Ingham and Pop [4] made a great contribution in this era. They determined the solution of a free convective flow over an isothermal horizontal cylinder numerically. Recently, Mohamed et al. [5] investigated the viscous-dissipation flow over a horizontal circular cylinder of a nanofluid with free-convection boundary layer.

Although water, oil, and ethylene mixtures are usually employed in many flows, the heat transfer rate is very poor in such fluids. Since thermal conductivity plays a vital role in the heat transfer from a surface to a fluid, for a better result, it is necessary to invent a fluid that gives a better result. For this purpose, nano-sized metallic particles have been added to fluids in recent years. The term “Nanofluid”, used for this kind of fluid, was first introduced by Choi and Eastman [6] in 1995. Nanofluids are a mixture of nano-sized particles such as oxides and carbides suspended in the base fluid. Nanofluids enable us to develop clean-energy-source devices such as smartphones and laptops. Some interesting and meaningful experiments and numerical research studies on nanofluids were conducted by [7,8,9,10,11]. They predicted some thermo-physical properties of nanofluids and showed the impact of these properties on the convective heat flow using different models. Eastman et al. [12] found that an increment of up to 40% in heat conductivity could be achieved using ethylene–glycol mixed with 0.3% volumetric copper nanoparticles of 10 nm in size. Das et al. [13] obtained an increment from 10% to 30% in thermal conduction with an aluminum–water mixture with 4% of alumina.

In radiation, waves are emitted or transferred in the form of energy from surface particles or through space. So, radiation also impacts the rate of thermal diffusion and other thermo-physical properties of a fluid. Therefore, the thermal radiation effects on various geometrical surfaces were considered by different researchers. Hossain and Alim [14] examined the impact of free convective heat transfer with radiation on a flow over a thin, vertically placed cylinder. Hossain et al. [15] extended this idea to mixed convective thermal transport over a cylinder with circular diameter. The mystery of viscous fluids being temperature-dependent on a cylinder with circular diameter having fixed wall temperature was revealed by him. A few years ago, Elbashbeshy et al. [16] found out the effects of the radiative transfer of a fluid with a heat source and sink on the natural-convection flow over a truncated cone. A letter by Zeeshan et al. [17] briefly explains the effect of radiative MHD and of diminishing the internal energy of flows on titanium dioxide nanofluid water.

The internal generation of heat in a fluid is an important factor for consideration. Some beneficial work has been presented by many researchers on free-convection flows over the horizontal circular form of a cylinder having heat generation effects. Heat-generating phenomena have a significant impact on the problem involving internal chemical reactions in a fluid. The impact of heat generation on a fluid alters the temperature distribution. Vajravelu and Hadjinolaou [18] determined the characteristics of heat generation of a viscous-fluid flow over a linear stretching surface. They considered heat generation in terms of the volumetric rate. Molla et al. [19] examined the influence of heat generation along with the natural convective flow over the isothermal horizontally placed circular form of a cylinder. After that, their work was further extended by Molla et al. [20], whose extension was a uniform heat flux. Meanwhile, their study was verified by Cheng [21] and further extended to a surface with a free convective boundary-layer flow over the horizontal and elliptical form of a cylinder with heat-generating effects and constant heat flux of the surface. Recently, Mohammed et al. [22] briefly studied the effects of viscous dissipation over a horizontal cylinder with a natural convective flow.

The magnetic hydrodynamic (MHD) effects on different surfaces represent an important area of study due to the large range of applications. The free-convection flow with MHD effects over the surface of different shapes has been studied by some researchers, and important results have been found. Hossain et al. [23] examined the heat flow rate in the presence of an MHD natural-convection flow over an oscillating and heated vertical flat plate. They obtained a solution with natural frequencies using the finite implicit scheme known as the Keller box method. Wilks [24] integrated the MHD natural convective flow with the vertical semi-infinite plate. Meanwhile, Azim and chowdhury [25] investigated the MHD effects along with the Joule heating effect and the heat generation effect over a horizontal circular cylinder and produced well-known results. The very next year, Azim [26] extended the same idea and produced a result with MHD and viscous dissipation effects. Later, Ellahi et al. [27] studied free convective and MHD nanofluids as well as flows over a single wall as well as multiple walls made of carbonated nanotubes and produced remarkable results. Furthermore, Nazeer et al. [28] extended that idea to produce results with the help of numerical simulations of MHD effects on a forced convective flow over a porous triangular cavity.

The above-cited contributions were restricted to a limited number of parameters variations applied numerically on selected surfaces at one time. None of them identified the natural convective flow of nanofluids including the viscous dissipation, magnetohydrodynamic, radiative, Joule heating, and heat generation effects over a horizontal circular cylinder. The governing partial differential equations are here solved and discussed numerically using the Keller box method and are presented graphically. To the best of our knowledge, until now, the presented problem has not been published by others yet.

2. Fundamental Governing Equations

Consider a circular cylinder with a horizontal axis having radius a normal to the axis of that cylinder. The fluid flowing over the horizontal cylinder is assumed to be a nanofluid, and it is immersed in an incompressible and viscous fluid with ambient temperature $T_{\infty }$. Furthermore, the cylinder is heated with constant temperature known as $T_w$, and the flow is steady and two-dimensional laminar natural convective with uniform heat flux. As magnetic effect M is not constant, constant magnetic field B₀ is also applied. The cylindrical surface is measured according to the orthogonal coordinate system of $\overline{x}$ and $\overline{y}$, where $\overline{x}=0$is considered to be a starting point as a lower stagnation point in the counter-clock-wise direction with the angle of $\raisebox{1ex}{$\overline{x}$}\!\left/ \!\raisebox{-1ex}{$a$}\right.$, and $\overline{y}$ is the coordinate axis normal to the coordinate $\overline{x}$. The physical model of the problem is shown in Figure 1.

Under the main assumptions of the law of conservation of mass, momentum and energy, the boundary-layer dimensional equations according to Boussinesq boundary-layer approximation are valid as shown below.

Continuity equation.

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

(1)

Momentum equation.

$$\overline{u}\frac{\partial \overline{u}}{\partial \overline{x}}+\overline{v}\frac{\partial \overline{u}}{\partial \overline{y}}=\upsilon \left[\frac{{\partial }^2\overline{u}}{{\partial \overline{y}}^2}\right]+g\beta \left(T-T_{\infty }\right)\left(\frac{\sin \overline{x}}{a}\right)+g{\beta }_c\left(C-C_{\infty }\right)\left(\frac{\sin \overline{x}}{a}\right)-\frac{{\sigma }_0 {\beta }_0^2\overline{u}}{\rho }$$

(2)

Energy equation.

$$\overline{u}\frac{\partial T}{\partial \overline{x}}+\overline{v}\frac{\partial T}{\partial \overline{y}}=\frac{K}{\rho C_p}\left[\frac{{\partial }^{2}T}{{\partial \overline{y}}^2}\right]+\tau \left[D_B\frac{\partial C}{\partial \overline{y}}\frac{\partial T}{\partial \overline{y}}+\frac{D_T}{T_{\infty }}{\left[\frac{\partial T}{\partial \overline{y}}\right]}^2\right]+\frac{\mu }{\rho C_p}{\left(\frac{\partial \overline{u}}{\partial \overline{y}}\right)}^2+\left(\frac{\sigma {\beta }_0^2}{\rho C_p}\right){\overline{u}}^2+\frac{Q_o\left(T-T_{\infty }\right)}{\rho C_p}-\frac{1}{\rho C_p}\nabla q_r$$

(3)

where the following apply.

$$q_r=-\frac{4\sigma \nabla T^4}{3\left({\alpha }_r+{\sigma }_s\right)};\tau =\frac{{\rho }_{cf}}{\mathrm{ϵ}\rho C_p}$$

$$\mathrm{Let}, T^4=4T_{\infty }^{3}T-3T_{\infty }^4\mathrm{implies}{q}_r=-\frac{16{\sigma T}_{\infty }^3\nabla T}{3\left({\alpha }_r+{\sigma }_s\right)}$$

Concentration equation.

$$\overline{u}\frac{\partial C}{\partial \overline{x}}+\overline{v}\frac{\partial C}{\partial \overline{y}}=D_B[\frac{{\partial }^{2}C}{{\partial \overline{x}}^2}+\frac{{\partial }^{2}C}{{\partial \overline{y}}^2}]+\frac{D_T}{T_{\infty }}[\frac{{\partial }^{2}T}{{\partial \overline{x}}^2}+\frac{{\partial }^{2}T}{{\partial \overline{y}}^2}]$$

(4)

Assumptions for the Problem

The velocity components in the direction along $\overline{x}$and $\overline{y}$ are $\overline{u}$ and $\overline{v}$, respectively. Constant values such as $\mu$ and $\upsilon$, and dynamic and kinematic viscosities have more important characteristic properties in the considered fluid. Moreover, g is the gravitational constant of the acceleration field acting on the fluid. The thermal expansion is $\beta$, which controls the thermal behavior of the fluid, while ${\beta }_c$ is the coefficient of concentration expansion. As the fluid is composed of nanofluid particles, the attention diverts to magnetic-field strength ${\beta }_0$ and thermal conductivity ${\sigma }_0$. $T$ and $\rho$ are the local temperature and local density of the fluid, which determine the thickness of the fluid. $\tau$ is the ratio of the heat capacity of the nanoparticles to the heat capacity of the fluid. $C_p$ is the specific heat capacity at constant pressure. The thermal conductivity is represented by k. As radiation effects are also assumed to be applied to the fluid, $q_r$ is the radiative heat flux. In the last dimensional concentration equation, C is the nanofluid particle’s volume friction having different constant values at the surface and ambient point, which are $C_w$ and $C_{\infty }$, respectively.

$$\begin{array}{cccc}\mathrm{For}\overline{y}=0\hfill & \overline{u}={\overline{u}}_w\left(\overline{x}\right)=\in U_{\infty }\hspace{1em}\overline{v}=0& T=T_w& C=C_w\\ \mathrm{For}\overline{y}\to \infty \hfill & \overline{u}=U_{\infty },& T=T_{\infty }& C=C_{\infty }\end{array}$$

(5)

Now, we need to perform the conversion of the dimensional equations into the non-dimensional forms of the equations by introducing the variables shown below.

$$x=\frac{\overline{x}}{a}, y=\frac{\overline{y}}{a}G{r}^{1/4}, u=\frac{a}{\upsilon }G{r}^{-1/2}\overline{u}, v=\frac{a}{\upsilon }G{r}^{-1/4}\overline{v}, \theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}, \varphi =\frac{C-C_{\infty }}{C_w-C_{\infty }}$$

(6)

The above are further transformed into the non-dimensional forms as shown below.

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

(7)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\partial }^{2}u}{\partial y^2}+\theta \sin x+g{\beta }_c\frac{\left(C_w-C_{\infty }\right)a^3}{{\upsilon }^{2}Gr}\phi \sin x-Mu$$

(8)

$$u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}=\frac{1}{Pr}\left[1+\frac{4}{3}{R}_d\right]\frac{{\partial }^2\theta }{\partial y^2}+N_b\frac{\partial \theta }{\partial y}\frac{\partial \varphi }{\partial y}+N_t{\left[\frac{\partial \theta }{\partial y}\right]}^2+\frac{{\upsilon }^{2}Gr}{C_p{a}^2\left(T_w-T_{\infty }\right)}{\left[\frac{\partial u}{\partial y}\right]}^2+Q\theta$$

(9)

$$u\frac{\partial \varphi }{\partial x}+v\frac{\partial \varphi }{\partial y}=\frac{D_B}{\upsilon }\frac{{\partial }^2\varphi }{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{\mathsf{\Delta }T}{\mathsf{\Delta }\varphi }\frac{{\partial }^2\theta }{\partial y^2}$$

(10)

where the below apply.

$$Gr=\frac{g\beta \left(T_w-T_{\infty }\right)a^3}{{\upsilon }^2},M=\frac{{\sigma }_0 {\beta }_0^2{a}^2}{\mu {\left(Gr\right)}^{1/2}}$$

$$Pr=\frac{\nu }{\alpha },\alpha =\frac{K}{{\rho C}_p},\nu =\frac{\mu }{\rho },R_d=\frac{4{\sigma T}_{\infty }^3}{K\left({\alpha }_r+{\sigma }_s\right)}, Q=\frac{Q_o{a}^2}{{\rho C}_p\upsilon {\left(Gr\right)}^{1/2}}$$

$$N_t=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{{\upsilon T}_{\infty }};N_b=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{\upsilon };$$

Now, we transform all the equations into their dimensionless forms by introducing stream functions.

$$\psi =xf(x,\eta ) ,\eta =y,\varphi =\varphi \left(x,\eta \right), \theta =\theta \left(x,\eta \right), u=\frac{\partial \psi }{\partial x} , v=-\frac{\partial \psi }{\partial x}$$

(11)

Leading boundary-layer Equations (8)–(10) are further transformed into this form; furthermore, it is obvious that these streamline transformations satisfy continuity Equation (7).

$$x\left[f^{\prime }\frac{\partial f^{\prime }}{\partial x}-f^{″}\frac{\partial f}{\partial x}\right]=f^{‴}+f{f}^{″}-{\left[f^{\prime }\right]}^2+\left(\theta +Nr\varphi \right)\frac{\sin x}{x}-Mx{f}^{\prime }$$

(12)

$$x\left[f^{\prime }\frac{\partial \theta }{\partial x}-{\theta }^{\prime }\frac{\partial f}{\partial x}-xEc{\left(f^{″}\right)}^2\right]=\frac{1}{Pr}\left[1+\frac{4}{3}{R}_d\right]{\theta }^{″}+N_b{\theta }^{\prime }{\varphi }^{\prime }+N_t{\left({\theta }^{\prime }\right)}^2+f{\theta }^{\prime }+Q\theta +J{x}^2{f^{\prime }}^2$$

(13)

$$Lex\left[f^{\prime }\frac{\partial \varphi }{\partial x}-{\theta }^{\prime }\frac{\partial f}{\partial x}\right]=\left[{\varphi }^{″}+\frac{N_t}{N_b}{\theta }^{″}\right]+Lef{\varphi }^{\prime }$$

(14)

where the below apply.

$$Nr=\frac{G{r}_c}{G{r}_x},f^{\prime }\mathrm{is}\mathrm{differential}\mathrm{w.r.t}𝜂,Ec=\frac{{\upsilon }^{2}Gr}{a^2{C}_p\left(T_w-T_{\infty }\right)}, Le=\frac{\upsilon }{D_B}$$

The boundary condition becomes as shown below.

$$\begin{array}{cccc}\mathrm{For}y=0\hfill & f\left(x,0\right)=0\frac{\partial f}{\partial y}\left(x,0\right)=0& \theta \left(x,0\right)=1& \varphi \left(x,0\right)=1\\ \mathrm{For}y⟶\infty \hfill & \frac{\partial f}{\partial y}\left(x,\infty \right)⟶0& \theta \left(x,\infty \right)\to 0& \varphi \left(x,\infty \right)\to 0\end{array}$$

(15)

Now, we need to perform the conversion of all equations into ordinary differential equations.

Let us, here at this point, introduce some variables.

$$f^{\prime }=u , u^{\prime }=v , s^{\prime }=t,g^{\prime }=p$$

(16)

$$\theta =s , \varphi =g , \alpha =\frac{x_{n-\frac{1}{2}}}{k_n} , B=\frac{\sin x}{x}$$

Momentum Equation (12) becomes as shown below.

$$x\left[u\frac{\partial u}{\partial x}-v\frac{\partial f}{\partial x}\right]=v^{\prime }+fv-u^2+\left(s+N_{r}g\right)B-Mxu$$

(17)

Energy Equation (13) becomes as shown below.

$$x\left[u\frac{\partial s}{\partial x}-t\frac{\partial f}{\partial x}-xEc{v}^2\right]=\frac{1}{Pr}\left[1+\frac{4}{3}{R}_d\right]t^{\prime }+N_{b}pt+N_t{t}^2+ft+Qs+J{x}^2{u}^2$$

(18)

Concentration Equation (14) becomes as shown below.

$$L_{e}x\left[u\frac{\partial g}{\partial x}-p\frac{\partial f}{\partial x}\right]=\left[p^{\prime }+\frac{N_t}{N_b}{t}^{\prime }\right]+L_e\left(fp\right)$$

(19)

The boundary equation becomes, with all these steps, as follows.

$$\begin{array}{ccccc}\mathrm{For}\eta =0\hfill & f\left(x,0\right)=0& u\left(x,0\right)=0& s\left(x,0\right)=1& g\left(x,0\right)=1\\ \mathrm{For}\eta ⟶\infty \hfill & & u\left(x,\infty \right)\to 0& s\left(x,\infty \right)\to 0& g\left(x,\infty \right)\to 0\end{array}$$

(20)

The net points on the rectangular form are defined below.

$$x^0=0,x^i=x^{i-1}+k_i, i=1,2,\dots ,I$$

(21)

$${\eta }^0=0,{\eta }_j={\eta }_{j-1}+h_j,j=1,2,\dots ,J,{\eta }_J={\eta }_{\infty },$$

Furthermore, the points in the formed rectangle are as shown in Figure 2. The application of the numerical scheme can be seen in ref. [29,30,31,32,33,34].

3. Solution Approach

Due to these transformations, the first derivative in the x-direction can be written as shown below.

$$\frac{\partial ( )}{\partial x}=\frac{{( )}^n-{( )}^{n-1}}{k_n}$$

Moreover, the points on the rectangle are defined as shown below.

$${\left(\eta \right)}_{j-\frac{1}{2}}^i=\frac{1}{2}\left[{\left(\eta \right)}_{j-1}^i+{\left(\eta \right)}_j^i\right]$$

(22)

$${\left(x\right)}_j^{i-\frac{1}{2}}=\frac{1}{2}\left[{\left(x\right)}_j^{i-1}+{\left(x\right)}_j^i\right]$$

Now, We ${\mathrm{apply}\mathrm{Newton}}^{\prime }s \mathrm{method}$ for linearization to Equations (16)–(20).

$$\frac{f_j^n-f_{j-1}^n}{h_j}=\frac{u_j^n-u_{j-1}^n}{2}=u_{j-1/2}^n,$$

(23)

$$\frac{u_j^n-u_{j-1}^n}{h_j}=\frac{v_j^n-v_{j-1}^n}{2}=v_{j-1/2}^n,$$

(24)

$$\frac{s_j^n-s_{j-1}^n}{h_j}=\frac{t_j^n-t_{j-1}^n}{2}=t_{j-1/2}^n,$$

(25)

$$\frac{g_j^n-g_{j-1}^n}{h_j}=\frac{p_j^n-p_{j-1}^n}{h_j}=p_{j-1/2}^n,$$

(26)

$$\begin{gathered} \left[\frac{v_j^n-v_{j-1}^n}{h_j}+\frac{v_j^{n-1}-v_{j-1}^{n-1}}{h_j}\right]+\left[{\left(fv\right)}_{j-\frac{1}{2}}^n+{\left(fv\right)}_{j-\frac{1}{2}}^{n-1}\right]-\left[{\left(u^2\right)}_{j-\frac{1}{2}}^n+{\left(u^2\right)}_{j-\frac{1}{2}}^{n-1}\right]+B\left[s_{j-\frac{1}{2}}^n+s_{j-\frac{1}{2}}^{n-1}\right]+B{N}_r\left[g_{j-\frac{1}{2}}^n+g_{j-\frac{1}{2}}^{n-1}\right]-M{x}_{n-\frac{1}{2}}\left[u_{j-\frac{1}{2}}^n+u_{j-\frac{1}{2}}^{n-1}\right]= \\ \frac{x_{n-\frac{1}{2}}}{k_n}\left[\left[u_{j-\frac{1}{2}}^n[u_{j-\frac{1}{2}}^n-u_{j-\frac{1}{2}}^{n-1}\right]+u_{j-\frac{1}{2}}^{n-1}\left[u_{j-\frac{1}{2}}^n-u_{j-\frac{1}{2}}^{n-1}\right]-v_{j-\frac{1}{2}}^n\left[f_{j-\frac{1}{2}}^n-f_{j-\frac{1}{2}}^{n-1}\right]-v_{j-\frac{1}{2}}^{n-1}\left[f_{j-\frac{1}{2}}^n-f_{j-\frac{1}{2}}^{n-1}\right]\right]. \end{gathered}$$

(27)

$$\begin{gathered} \frac{1}{Pr}\left[1+\frac{4}{3}{R}_d\right]\left[\frac{t_j^n-t_{j-1}^n}{h_j}+\frac{t_j^{n-1}-t_{j-1}^{n-1}}{h_j}\right]+N_b\left[{\left(pt\right)}_{j-\frac{1}{2}}^n+{\left(pt\right)}_{j-\frac{1}{2}}^{n-1}\right]+N_t\left[{\left(t^2\right)}_{j-\frac{1}{2}}^n+{\left(t^2\right)}_{j-\frac{1}{2}}^{n-1}\right]+\left[{\left(ft\right)}_{j-\frac{1}{2}}^n+{\left(ft\right)}_{j-\frac{1}{2}}^{n-1}\right]+Q\left[{\left(s\right)}_{j-\frac{1}{2}}^n+{\left(s\right)}_{j-\frac{1}{2}}^{n-1}\right]+ \\ J{x}_{n-\frac{1}{2}}^2\left[{\left(u^2\right)}_{j-\frac{1}{2}}^n+{\left(u^2\right)}_{j-\frac{1}{2}}^{n-1}\right]=\frac{x_{n-\frac{1}{2}}}{k_n}\left[u_{j-\frac{1}{2}}^n\left[s_{j-\frac{1}{2}}^n-s_{j-\frac{1}{2}}^{n-1}\right]+u_{j-\frac{1}{2}}^{n-1}\left[s_{j-\frac{1}{2}}^n-s_{j-\frac{1}{2}}^{n-1}\right]-t_{j-\frac{1}{2}}^n\left[f_{j-\frac{1}{2}}^n-f_{j-\frac{1}{2}}^{n-1}\right]-t_{j-\frac{1}{2}}^{n-1}\left[f_{j-\frac{1}{2}}^n-f_{j-\frac{1}{2}}^{n-1}\right]- \\ {x}_{n-\frac{1}{2}}\left[k_{n}Ec\left[{\left(v^2\right)}_{j-\frac{1}{2}}^n+{\left(v^2\right)}_{j-\frac{1}{2}}^{n-1}\right]\right]\right]-\frac{x_{n-\frac{1}{2}}}{k_n}{x}_{n-\frac{1}{2}}{k}_{n}Ec{\left(v\right)}_{j-\frac{1}{2}}^{n-1}{\left(v\right)}_{j-\frac{1}{2}}^n. \end{gathered}$$

(28)

$$\begin{gathered} \left[\frac{p_j^n-p_{j-1}^n}{h_j}+\frac{p_j^{n-1}-p_{j-1}^{n-1}}{h_j}\right]+\frac{N_t}{N_b}\left[\frac{t_j^n-t_{j-1}^n}{h_j}+\frac{t_j^{n-1}-t_{j-1}^{n-1}}{h_j}\right]+L_e\left[{\left(fp\right)}_{j-\frac{1}{2}}^n+{\left(fp\right)}_{j-\frac{1}{2}}^{n-1}\right]= \\ {L}_e\frac{x_{n-\frac{1}{2}}}{k_n}\left[u_{j-\frac{1}{2}}^n\left[g_{j-\frac{1}{2}}^n-g_{j-\frac{1}{2}}^{n-1}\right]+u_{j-\frac{1}{2}}^{n-1}\left[g_{j-\frac{1}{2}}^n-g_{j-\frac{1}{2}}^{n-1}\right]-p_{j-\frac{1}{2}}^n\left[f_{j-\frac{1}{2}}^n-f_{j-\frac{1}{2}}^{n-1}\right]-p_{j-\frac{1}{2}}^{n-1}\left[f_{j-\frac{1}{2}}^n-f_{j-\frac{1}{2}}^{n-1}\right]\right]. \end{gathered}$$

(29)

Moreover, the boundary condition becomes as shown below.

$$f_0^n=0,u_0^n=0,s_0^n=1,g_0^n=1,u_J^n=0,s_J^n=0,s_J^n=0,g_J^n=0$$

(30)

Equations (23)–(29) can be reduced to what follows.

$$f_j-f_{j-1}-\frac{h_j}{2}\left(u_j+u_{j-1}\right)=0,$$

(31)

$$u_j-u_{j-1}-\frac{h_j}{2}\left(v_j+v_{j-1}\right)=0,$$

(32)

$$s_j-s_{j-1}-\frac{h_j}{2}\left(t_j+t_{j-1}\right)=0,$$

(33)

$$g_j-g_{j-1}-\frac{h_j}{2}\left(p_j+p_{j-1}\right)=0,$$

(34)

$$\begin{gathered} \left[v_j^n-v_{j-1}^n\right]+h_j\left[{\left(fv\right)}_{j-\frac{1}{2}}^n\right]-h_j\left[{\left(u^2\right)}_{j-\frac{1}{2}}^n\right]+B{h}_j\left[s_{j-\frac{1}{2}}^n\right]+B{N}_r{h}_j\left[g_{j-\frac{1}{2}}^n\right]-Mx{h}_j\left[u_{j-\frac{1}{2}}^n\right]-\alpha h_j\left[{\left(u^2\right)}_{j-\frac{1}{2}}^n\right]+\alpha h_j\left[{\left(fv\right)}_{j-\frac{1}{2}}^n\right]-\alpha h_j{f}_{j-\frac{1}{2}}^{n-1}{v}_{j-\frac{1}{2}}^n+ \\ \alpha h_j{v}_{j-\frac{1}{2}}^{n-1}{f}_{j-\frac{1}{2}}^n={\left(R_1\right)}_{j-\frac{1}{2}}^{n-1} \end{gathered}$$

(35)

$$\begin{gathered} \frac{1}{Pr}\left[1+\frac{4}{3}{R}_d\right]\left[t_j^n-t_{j-1}^n\right]+N_b{h}_j\left[{\left(pt\right)}_{j-\frac{1}{2}}^n\right]+N_t{h}_j\left[{\left(t^2\right)}_{j-\frac{1}{2}}^n\right]+h_j\left[{\left(ft\right)}_{j-\frac{1}{2}}^n\right]+Q{h}_j\left[{\left(s\right)}_{j-\frac{1}{2}}^n\right]+J{x}_{n-\frac{1}{2}}^2{\left(u^2\right)}_{j-\frac{1}{2}}^n+\frac{h_j{\left(\alpha k_n\right)}^{2}Ec}{2}{v}^2{}_{j-\frac{1}{2}}^n+ \\ \frac{h_j{\left(\alpha k_n\right)}^{2}Ec}{2}{v}_{j-\frac{1}{2}}^{n-1}{v}_{j-\frac{1}{2}}^n-h_j\alpha {\left[us\right]}_{j-\frac{1}{2}}^n+h_j\alpha {\left[ft\right]}_{j-\frac{1}{2}}^n+\alpha h_j{s}_{j-\frac{1}{2}}^{n-1}{u}_{j-\frac{1}{2}}^n-\alpha h_j{u}_{j-\frac{1}{2}}^{n-1}{s}_{j-\frac{1}{2}}^n-\alpha h_j{f}_{j-\frac{1}{2}}^{n-1}{t}_{j-\frac{1}{2}}^n+\alpha h_j{t}_{j-\frac{1}{2}}^{n-1}{f}_{j-\frac{1}{2}}^n={\left(R_2\right)}_{j-\frac{1}{2}}^{n-1} \end{gathered}$$

(36)

$$\begin{gathered} \left[\frac{p_j^n-p_{j-1}^n}{h_j}\right]+\frac{N_t}{N_b}\left[\frac{t_j^n-t_{j-1}^n}{h_j}\right]+L_e\left[{\left(fp\right)}_{j-\frac{1}{2}}^n\right]-L_e\alpha \left[{\left(ug\right)}_{j-\frac{1}{2}}^n\right]+L_e\alpha \left[{\left(fp\right)}_{j-\frac{1}{2}}^n\right]+L_e\alpha g_{j-\frac{1}{2}}^{n-1}\left[u_{j-\frac{1}{2}}^n\right]-L_e\alpha u_{j-\frac{1}{2}}^{n-1}\left[g_{j-\frac{1}{2}}^n\right]-L_e\alpha f_{j-\frac{1}{2}}^{n-1}\left[p_{j-\frac{1}{2}}^n\right]+ \\ {L}_e\alpha p_{j-\frac{1}{2}}^{n-1}\left[u_{j-\frac{1}{2}}^n\right]={\left(R_3\right)}_{j-\frac{1}{2}}^{n-1} \end{gathered}$$

(37)

R_i is defined in the Appendix A. Furthermore, the steps required to solve Equations (31)–(37) are also explained in Appendix A; thus, we obtain the following equations.

$$\begin{gathered} \left(a1\right)\delta v_j+\left(a2\right)\delta v_{j-1}+\left(a3\right)\delta f_j+\left(a4\right)\delta f_{j-1}+\left(a5\right)\delta u_j+\left(a6\right)\delta u_{j-1}+\left(a7\right)\delta s_j+\left(a8\right)\delta s_{j-1}+\left(a9\right)\delta g_j+ \\ \left(a10\right)\delta g_{j-1}={\left(r_5\right)}_{j-\frac{1}{2}}^{n-1} \end{gathered}$$

(38)

$$\begin{gathered} \left(b1\right)\delta t_j+\left(b2\right)\delta t_{j-1}+\left(b3\right)\delta f_j+\left(b4\right)\delta f_{j-1}+\left(b5\right)\delta s_j+\left(b6\right){\delta }_{j-1}+\left(b7\right)\delta u_j+\left(b8\right)\delta u_{j-1}+\left(b9\right)\delta p_j+ \\ \left(b10\right)\delta p_{j-1}+\left(b11\right)\delta v_j+\left(b12\right)\delta v_{j-1}={\left(r_6\right)}_{j-\frac{1}{2}}^{n-1} \end{gathered}$$

(39)

$$\begin{gathered} \left(c1\right)\delta p_j+\left(c2\right)\delta p_{j-1}+\left(c3\right)\delta t_j+\left(c4\right)\delta t_{j-1}+\left(c5\right)\delta f_j+\left(c6\right)\delta f_{j-1}+\left(c7\right)\delta g_j+\left(c8\right)\delta g_{j-1}+\left(c9\right)\delta u_j+ \\ \left(c10\right)\delta u_{j-1}={\left(r_7\right)}_{j-\frac{1}{2}}^{n-1} \end{gathered}$$

(40)

Similarly, we apply the $\delta$ operator to the boundary equation, which becomes as shown below.

$$\delta f_0=0,\delta u_0=0, \delta s_0=0, \delta g_0=0, \delta u_J=0, \delta s_J=0, \delta g_J=0$$

(41)

The system of linearized Equations (38)–(41) is transformed into three diagonal block matrices, which can be written as shown.

$$\left[\begin{array}{ccccccc}\left[A_1\right]& \left[C_1\right]& & & & & \\ \left[B_2\right]& \left[A_2\right]& \left[C_2\right]& & & & \\ & & & \ddots & & & \\ & & & \ddots & & & \\ & & & \ddots & & & \\ & & & & \left[B_{J-1}\right]& \left[A_{J-1}\right]& \left[C_{J-1}\right]\\ & & & & & \left[B_J\right]& \left[A_J\right]\end{array}\right]\left[\begin{array}{c}\left[{\delta }_1\right]\\ \left[{\delta }_2\right]\\ ⋮\\ ⋮\\ ⋮\\ \left[{\delta }_{J-1}\right]\\ \left[{\delta }_J\right]\end{array}\right]=\left[\begin{array}{c}\left[r_1\right]\\ \left[r_2\right]\\ ⋮\\ ⋮\\ ⋮\\ \left[r_{J-1}\right]\\ \left[r_J\right]\end{array}\right]$$

(42)

Generally, this is written as follows.

$$A\delta =r$$

(43)

where $\left[A_i\right]$, $\left[B_i\right]$, $\left[C_i\right]$, $\left[{\delta }_J\right]$, and $\left[r_J\right]$ are defined in Appendix A.

$$\left[{\delta }_1\right]=\left[\begin{array}{c}{\delta v}_0\\ {\delta t}_0\\ {\delta p}_0\\ {\delta f}_1\\ {\delta v}_1\\ {\delta t}_1\\ {\delta p}_1\end{array}\right], \left[{\delta }_j\right]=\left[\begin{array}{c}{\delta u}_{j-1}\\ {\delta g}_{j-1}\\ {\delta s}_{j-1}\\ {\delta f}_j\\ {\delta v}_j\\ {\delta t}_j\\ {\delta p}_j\end{array}\right], 2\le j\le J$$

(44)

$$\left[r_j\right]=\left[\begin{array}{c}{\left(r_1\right)}_{j-1/2}\\ {\left(r_2\right)}_{j-1/2}\\ {\left(r_3\right)}_{j-1/2}\\ {\left(r_4\right)}_{j-1/2}\\ {\left(r_5\right)}_{j-1/2}\\ {\left(r_6\right)}_{j-1/2}\\ {\left(r_7\right)}_{j-1/2}\end{array}\right], 1\le j\le J$$

(45)

System (43), by block elimination, can be written as shown.

$$A=LU$$

(46)

where $L \mathrm{and} U$ are defined as follows.

$$L=\left[\begin{array}{ccccc}\left[{\alpha }_1\right]& & & & \\ \left[{𝛣}_2\right]& \left[{\alpha }_2\right]& & & \\ & & \ddots & & \\ & & \ddots & \left[{\alpha }_{J-1}\right]& \\ & & & \left[{𝛣}_J\right]& \left[{\alpha }_J\right]\end{array}\right]\mathrm{and}U=\left[\begin{array}{cccccc}\left[I\right]& \left[{𝛤}_1\right]& & & & \\ & \left[I\right]& \left[{𝛤}_2\right]& & & \\ & & & \ddots & & \\ & & & \ddots & & \\ & & & & \left[I\right]& \left[{𝛤}_{J-1}\right]\\ & & & & & \left[I\right]\end{array}\right]$$

where [I] is of order 7, known as the identity matrix, and $\left[{\alpha }_i\right]$ and $\left[{𝛤}_i\right]$ are also of the order of 7 × 7 matrices.

The elements of L and U can be written as follows.

$$\left[{\alpha }_1\right]=\left[A_1\right]$$

(47)

$$\left[A_1\right]\left[{𝛤}_1\right]=\left[C_1\right]$$

(48)

$$\left[{\alpha }_j\right]=\left[A_j\right]-\left[B_j\right]\left[{𝛤}_{j-1}\right], j=2,3,\dots ,J$$

(49)

By Equation (46), Equation (43) becomes as shown.

$$LU\delta =r$$

(50)

If $U\delta =W$, then Equation (50) becomes as shown.

$$LW=r$$

(51)

where the following applies.

$$W=\left[\begin{array}{c}\left[W_1\right]\\ \left[W_2\right]\\ ⋮\\ \left[W_{J-1}\right]\\ \left[W_J\right]\end{array}\right] ,$$

(52)

where $\left[W_j\right]$ is a column matrix of order 7 and can be determined by forward sweep.

$$\left[{\alpha }_1\right]\left[W_1\right]=\left[r_1\right]$$

(53)

$$\left[{\alpha }_j\right]\left[W_j\right]=\left[r_j\right]-\left[B_j\right]\left[W_{j-1}\right], j=2,3,\dots ,J$$

(54)

Once W is determined, with the help of backward sweep, we find out the solution.

$$\left[{\delta }_J\right]=\left[W_J\right]$$

(55)

$$\left[{\delta }_j\right]=\left[W_j\right]-\left[{𝛤}_j\right]\left[{\delta }_{j+1}\right], j=1,2,3,\dots ,J-1$$

(56)

Since $\delta$ is found, we have the approximated solution of the required system (43). The problem is solved using MATLAB coded as given in Appendix B.

4. Outcomes and Discussions

In Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, the comparisons ofthe thermophoresis parameter and Brownian motion are shown with the help of graphs. It is clear from Figure 3 that with the increases in the thermophoresis parameter (Nt) and Brownian motion (Nb), skin friction increases. The terms of Brownian motion and thermophoresis parameter are inversely related to the viscosity of the fluid. Furthermore, skin friction is related to the friction between the surface and the fluid flow near the surface. So, with the increases in the Brownian motion and thermophoresis viscosity of the fluid and the decrease in the motion of the fluid, skin friction increases as a result. In Figure 4, the Nusselt number is discussed. The Nusselt number is related to the rate of heat transfer of the fluid; moreover, with the increase in thermophoresis, the temperature of the fluid increases, and the fluid easily transfers heat. Therefore, with the increase in the thermophoresis parameter, the Nusselt number increases. Moreover, the Brownian motion parameter (Nb) is related to the concentration of the mass particles. So, with the increase in Brownian motion, the Nusselt number decreases. In Figure 5, the Sherwood number results are reported. The Sherwood number decreases as the thermophoresis number increases; however, for Brownian motion, the Sherwood number increases, as it is related to the mass concentration of the fluid. In Figure 6, the concentration profile results are reported. As the thermophoresis parameter (Nt) and the Brownian motion parameter are enhanced, the concentration profile increases, but the trend becomes opposite when the graph approaches the lowest point.

In Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, the comparisons of the buoyancy ratio and the Prandtl number are reported with the help of graphs. In Figure 9, the increase in the buoyancy ratio makes the fluid become less dense and flow easily. Therefore, skin friction increases. However, the Prandtl number decreases, as the Prandtl number is directly related with the viscosity of the fluid. In Figure 11, the Sherwood number increases with the increase in the buoyancy ratio, and as the Prandtl number increases, the Sherwood number decreases. It is due to the increase in the mass flow rate of the concentration of the fluid. In Figure 13, the velocity profile shows that with the increases in the Prandtl number and the buoyancy ratio, the velocity profile becomes smaller. This effect is due to viscosity, which becomes high with the increases in the Prandtl number and the buoyancy ratio. In Figure 14, the temperature profile shows that by increasing the Prandtl number and the buoyancy ratio, we lower the temperature profile; therefore, with the increases in the Prandtl number and the buoyancy ratio, the density of the fluid diminishes.

In Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, the comparisons of the heat generation and radiation parameters are reported. In Figure 15, skin friction increases with the increases in the heat generation effect and the radiation parameter. This effect is due to the increase in the temperature of the fluid caused by the increases in these parameters. The fluid becomes less dense, and the friction between the fluid and surface diminishes, so skin friction increases. The same trend can be seen for the Nusselt number by increasing heat generation in Figure 16; however, with the increase in radiation, the Nusselt number decreases, as the radiation parameter is related to the ambient temperature, so when radiation increases, the ambient temperature also increases; therefore, the rate of heat transfer flow from the fluid to the ambient temperature decreases. Hence, the Nusselt number decreases. In Figure 17, the Sherwood number decreases with the increase in heat generation, as the heat generation parameter is inversely related to the concentration of nanoparticles. Hence, the Sherwood number decreases. In Figure 18, the same can be observed, whereby the concentration profile decreases as heat generation increases. In Figure 19, the velocity profile increases as heat generation and radiation increase. This effect is due to the increases in heat generation and radiation, whereby the fluid becomes less dense, and the fluid flows easily. In Figure 20, it is obvious that with the increases in heat generation and radiation, the temperature profile increases, as it rises the temperature of the fluid. So, the temperature profile increases.

In Figure 21, Figure 22 and Figure 23, the comparisons of the effects of Joule heating and the magnetic parameter on the fluid are reported. In Figure 21, with the increase in Joule heating, skin friction increases. This effect is because Joule heating is inverse to the density of the fluid, so the fluid becomes frictionless as Joule heating increases. Moreover, the magnetic parameter lowers skin friction, because the magnetic parameter is directly related to density. Therefore, the greater the magnetic parameter is, the denser the fluid is, and the friction between the surface and the fluid layer becomes high. In Figure 22, the Joule heating effect enhances the Nusselt number, as it is inversely related to the concentration of nanoparticles as well as the temperature difference between the fluid and the ambient temperature; therefore, the rate of heat transfer becomes dominant. The opposite case is verified for the magnetic parameter, whose increase decreases the Nusselt number.

In Figure 24, Figure 25 and Figure 26, the comparisons of the Lewis number and Eckert number are reported with the help of graphs. In Figure 24, skin friction decreases with the increase in the Lewis number, as it is directly proportional to fluid viscosity. Therefore, the friction between the surface and the fluid increases as a result, as skin friction is enhanced by the increase in the Lewis number. Furthermore, the Eckert number enhances skin friction, as the Eckert number is directly related to the Grashof number of nanoparticles. So, the fluid flow becomes dense. Therefore, skin friction increases. In Figure 25, as the Lewis number is directly related to viscosity, the Nusselt number increases as the Lewis number and the Eckert number both increase, so the rate of heat transfer becomes easier. The Eckert number increases the Grashof number; therefore, the Nusselt number also increases. In Figure 26, the Sherwood number decreases as the Eckert number and the Lewis number increase.

5. Conclusions

The effects of radiation, heat generation, and magnetic hydrodynamic properties imposed on a nanofluid with a steady, incompressible, natural-convection flow over the horizontally circular form of a cylinder are numerically determined by the very efficient finite element method known as the Keller box method. Buoyancy force and thermophoresis effects are embedded in the nanofluid, and the resultant highly non-linear partial differential equations, including continuity, momentum, energy, and concentration equations, for the mass of nanofluid particles present in the fluid are solved numerically. Then, the governing equations are transformed into their non-dimensional forms with the help of suitable stream functions. The numerical solution is obtained with the help of an implicit finite difference scheme in MATLAB, and the following results are obtained:

• It can be predicted from the graph that the Eckert number increases; then, the Nusselt number increases, but the Sherwood number decreases. This effect is due to the increases in the volume of the nanofluid particles and in the Nusselt number. Furthermore, it does not affect the velocity, concentration, or temperature profile;
• The increases in Brownian motion and buoyancy ratio increase the Nusselt number due to an increase in the volume ratio of the nanoparticles, and rate of heat transfer is enhanced by the increase in the mass ratio;
• The magnetic parameter reduces the Sherwood number and the Nusselt number due to the effects of the movement of the nanoparticles, which is reduced with the increase in the magnetic parameter;
• Heat generation also decreases the Nusselt number, as the rate of heat transfer is reduced due to the reduction in the temperature difference between the wall of the cylinder and the nanofluid.