Effectiveness of Radiation on Magneto-Combined Convective Boundary Layer Flow in Polar Nanofluid around a Spherical Shape

Abstract

Many physical aspects emerging from the local structure and micromotions of liquid particles can be studied by utilizing the governing model of micropolar liquid. It has the ability to explain the behavior of a wide range of real fluids, including polymeric solutions, liquid crystals, lubricants, and animal blood. This earned it a major role in the treatment of many industrial and engineering applications. Radiative heat transmission induced by a combined convection flow of micropolar fluid over a solid sphere, and its enhancement via nanoparticle oxides, are investigated in this study. An applied magnetic field and a constant wall temperature are also considered. The Tiwari–Das model is used to construct the mathematical model. An approximate numerical solution is included using the Keller box method, in which its numerical calculations are performed via MATLAB software, to obtain numerical results and graphic outputs reflecting the effects of critical parameters on the physical quantities associated with heat transfer. The investigation results point out that a weakness in the intensity of the magnetic field, or an increment in the nanoparticle volume fraction, causes an increment in velocity. Raising the radiation parameter promotes energy transport, angular velocity, and velocity.

1. Introduction

Researchers lately focused on the phenomena of heat transfer enhancement based on the utilization of nanoparticles, revealing that the viscosity of the base fluid and the effectiveness of the thermal conductivity can be improved by the employment of metallic and non-metallic nanoparticles [1,2,3,4,5]. These progressive features made the topic of nanosolids a new technology of interest, which is reflected in the application in various functions of manufacturing procedures, mainly groundwater pollution, heat exchangers, thermal engineering, enhanced oil recovery, cooling phenomena, geothermal systems, magnetic cell separation, crude oil extraction, energy production, and nuclear waste storage, etc. [6,7,8,9,10,11]. In this regard, many studies were conducted that simulate the extent of the improvement in energy transport caused by these nanoparticles, and the most important factors influencing it. Khanafer et al. [12] investigate the enhancement of energy transmission utilizing nanofluids, while taking solid particle dispersion into consideration. Buongiorno [13] reports the influence of Brownian motion and thermophoresis on energy transport. Tiwari and Das [14] elucidate the critical effect of the volume fraction of nanosolids in the heat transport process. Ho et al. [15] provide an intriguing numerical examination of the influences due to uncertainties in viscosity and thermal conductivity on free convection. Nield and Kuznetsov [16] extend the Cheng–Minkowycz problem to the case of nanofluids, with the assistance of the Buongiorno model. Then, Kuznetsov and Nield [17] present a study in which the previous model is improved to include the case in which the size of the particles is passively controlled. Das et al. [18] model the entropy generation for magnetized nanoliquid flowing around an accelerating stretching sheet. Shahid et al. [19] study MHD nanoliquid flow, including gyrotactic microorganisms spreading on a stretching surface. Alwawi et al. [20] investigate the MHD mixed and combined convection of sodium alginate and carboxymethyl cellulose-water based Casson nanofluids flowing over spheres. More noteworthy reports are found in these refs. [21,22,23,24].

Lately, micropolar flow was an important subject of research because of its important role in many engineering and industrial applications problems, such as cervical flows, thrust-bearing technologies, radial diffusion paint rheology, and clean and contaminated engine lubricants, among others [25]. The dynamics of micropolar fluid was first mentioned in the theory of Eringen et al. [26,27], which considers the effect of local rotary inertia and couple stresses associated with practical micro-rotation. The micropolar fluids are popular non-Newtonian fluids with microstructures that consist of rigid particles and colloidal fluid elements, and are used in making apparatuses for chemical procedures, as well as various heat exchangers [28]. This, and many other applications, including, but not limited to, polymer preparing, and wire and fiber coating of the fluids, require utter understanding of the motion procedure. However, studies on micropolar fluids received a lot of attention recently, due to their vast applications. Khader and Sharma [29] examine the effect of a non-uniform heat source and thermal radiation on unsteady MHD micropolar fluid flow over a shirking/stretching sheet. Kumbinarasaiah and Raghunatha [30] investigate the two-dimensional flow of a rotating micropolar fluid in a porous channel, using the Laguerre wavelet exact Parseval frame method (LWPM). Bhat and Katagi [31] use the Keller box method to examine the steady and two-dimensional micropolar fluid flow between a porous and nonporous disk. Bilal et al. [32] examine the effect of thermal radiation of micropolar fluid across a permeable stretching surface. Abas et al. [33] study magnetohydrodynamic (MHD) dissipative micropolar fluid flow over a stretching sheet, with heat generation and slip effects. Mahabaleshwar et al. [33] examine the flow of an inclined MHD micropolar fluid in the presence of mass transpiration and thermal radiation. More recent studies on this subject are referred to in [34,35,36].

Magnetohydrodynamic (MHD) has the ability to optimize the transfer of heat from electrically conductive fluid, manipulate the fluid flow, and control separation flow. Therefore, in terms of industrial and engineering applications, the MHD flow is an important research topic that needs to be considered. Currently, more considerable attention was given to the study of MHD mixed convention flow around geometric surfaces [37]. For instance, Ibrahim and Zemedu [38] examine the MHD mixed convection boundary layer over a non-isothermal sphere with Soret and magnetic field effects. On the other hand, a study on the magnetic micropolar nanofluid oblique flow is presented by Tabassum et al. [39], in the presence of mixed convection. Moreover, Mkhatshwa et al. [40] consider the hall effect, chemical reaction, thermo-diffusion, and diffusion–thermal effects to investigate the MHD mixed conventional flow around a vertical cylinder, and Rahman et al. [41] investigate the MHD mixed convection over a horizontal circular cylinder that can conduct heat, and is placed at the center of a rectangular cavity. More information on this topic is found in [42,43,44,45]. On the other hand, thermal radiation effects recently received significant attention in the field of fluid dynamics. Physically, it is formed by the presence of surface gravity waves, which are propagated along the mean of the boundary layer flow. In addition, thermal radiation plays a wide role in the numerical solution investigations for heat transfer and temperature distributions in the existence of the boundary layer flow. Moreover, Mansour and Gorla [46] and Mohammadein and El-Amin [47] study the influence of thermal radiation on convection boundary layer flow with non-Newtonian fluids over a non-isothermal wedge and horizontal plate, respectively. Furthermore, [48,49,50,51,52] consider the radiation heat and mass transfer in mixed convection boundary layer flow on a solid sphere. Additionally, the mixed convection in micropolar fluid and radiation characteristics gained more attention in the literature, e.g., [52,53,54,55].

Keeping in mind the above numerical studies of the convection boundary layer flow, the existing work is concerned with mixed convection heat transfer characteristics for thermal radiation in micropolar nanofluid flowing under a magnetic field. To this end, a model was constructed for analyzing the nanoparticles, comprising of alumina and copper oxide, suspended in base fluids, considering the solid sphere geometry. In addition, the famous Tiwari–Das model was also employed in this study. After transforming the dimensional equations into partial differential equations, we then utilized the Keller box method to solve the resultant equations. The results obtained were compared with the previous literature to show their good agreement.

2. Basic Equations

Mixed convection with a uniform free stream $U_{\infty }$, on a solid sphere of radius a with constant wall temperature $T_w$, in the presence of micropolar nanofluid with magnetic field strength $B_0$, was investigated numerically in this paper. Two types of metals, namely, alumina oxide and copper oxide, were suspended in two different types of fluids, namely, water and methanol. Figure 1 shows the physical coordinate model system, the free stream temperature $T_{\infty }$, and $\widehat{\aleph }-\widehat{\gamma }$ are the perpendicular coordinates on the surface of the solid sphere. A heated and cooled sphere ($T_w$ > $T_{\infty }$ and $T_w$ < $T_{\infty }$, respectively) were also taken into account.

Under Boussinesq and boundary layer approximations, as well as utilizing the Tiwari–Das nanofluid model, considering magnetic and thermal radiation effects, the continuity, momentum, and energy micropolar equations are [14,47,54]:

$$\widehat{\nabla }.\widehat{V}=0,$$

(1)

$$\begin{array}{l}\left(\widehat{V}.\widehat{\nabla }\right)\widehat{V}=-\frac{1}{{\rho }_{nf}}\widehat{\nabla }\widehat{P}+\left(\frac{{\mu }_{nf}+\kappa }{{\rho }_{nf}}\right){\widehat{\nabla }}^2\widehat{V}+\frac{\kappa }{{\rho }_{nf}}\left(\frac{\partial \widehat{N}}{\partial \widehat{\gamma }}-\frac{\partial \widehat{N}}{\partial \widehat{\aleph }}\right)+\\ \frac{\left(\chi {\rho }_s{\beta }_s+\left(1-\chi \right){\rho }_f{\beta }_f\right)}{{\rho }_{nf}}\widehat{g}\left(T-T_{\infty }\right)-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{B}_0^2\widehat{u},\end{array}$$

(2)

$$\left(\widehat{V}.\widehat{\nabla }\right)\widehat{T}={\alpha }_{nf}{\widehat{\nabla }}^2\widehat{T}-\frac{1}{{(\rho c_p)}_{nf}}\frac{\partial Q_R}{\partial \widehat{\gamma }},$$

(3)

$${\rho }_{nf}J\left(\widehat{V}.\widehat{\nabla }\right)\widehat{N}=-\kappa \left(2\widehat{N}+\frac{\partial \widehat{u}}{\partial \widehat{\gamma }}-\frac{\partial \widehat{v}}{\partial \widehat{\aleph }}\right)+{\varphi }_{nf}{\widehat{\nabla }}^2\widehat{N}.$$

(4)

where the components $\widehat{g}$, $\widehat{N}$, $\widehat{P}$, and $\widehat{V}$ symbolize the gravitational, micro-rotation, pressure, and velocity vectors, respectively, while ${\widehat{\nabla }}^2$, J are called the Laplacian operator, and micro-inertia density, respectively. The momentum equation is divided into two equations in $\widehat{\aleph }-\widehat{\gamma }$ directions, depending on $\overline{\mathrm{g}}$ that can be written as $g_{\aleph }=g\sin (\aleph /a),g_{\gamma }=g\cos (\aleph /a)$, respectively. Also, by applying the boundary approaches of Reynolds number $\mathrm{Re}\to \infty$, which is equivalent to $(1/\mathrm{Re})\to 0$, and $-(\partial P/\partial \aleph )=u_e(\partial u_e/\partial \aleph )$ (combined convection flow case) [56,57], we can neglect the terms that contain (1/Re). So, the governing non-dimensional equations, in the Cartesian coordinate system are deduced, as follows:

$$\frac{\partial \widehat{r}\widehat{u}}{\partial \widehat{\aleph }}+\frac{\partial \widehat{r}\widehat{v}}{\partial \widehat{\gamma }}=0,$$

(5)

$$\begin{array}{l}\widehat{u}\frac{\partial \widehat{u}}{\partial \widehat{\aleph }}+\widehat{v}\frac{\partial \widehat{u}}{\partial \widehat{\gamma }}={\widehat{u}}_e\frac{d{\widehat{u}}_e}{\partial \widehat{\aleph }}+\left(\frac{{\mu }_{nf}+\kappa }{{\rho }_{nf}}\right)\frac{{\partial }^2\widehat{u}}{\partial {\widehat{\gamma }}^2}+\frac{\kappa }{{\rho }_{nf}}\frac{\partial \widehat{N}}{\partial \widehat{\gamma }}\\ +\frac{\left(\chi {\rho }_s{\beta }_s+\left(1-\chi \right){\rho }_f{\beta }_f\right)}{{\rho }_{nf}}g\left(T-T_{\infty }\right)\sin \left(\frac{\widehat{\aleph }}{a}\right)-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{B}_0^2\widehat{u},\end{array}$$

(6)

$$\widehat{u}\frac{\partial T}{\partial \widehat{\aleph }}+\widehat{v}\frac{\partial T}{\partial \widehat{\gamma }}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial {\widehat{\gamma }}^2}-\frac{1}{{(\rho c_p)}_{nf}}\frac{\partial Q_R}{\partial \widehat{\gamma }},$$

(7)

$${\rho }_{nf}J\left(\widehat{u}\frac{\partial \widehat{N}}{\partial \widehat{\aleph }}+\widehat{v}\frac{\partial \widehat{N}}{\partial \widehat{\gamma }}\right)=-\kappa \left(2\widehat{N}+\frac{\partial \widehat{u}}{\partial \widehat{\gamma }}\right)+{\varphi }_{nf}\frac{{\partial }^2\widehat{N}}{\partial {\widehat{\gamma }}^2},$$

(8)

The nanofluid properties and the radiative heat flux term are defined as follows [42,45]:

$$\begin{array}{c}{\alpha }_{nf}=\frac{k_{nf}}{{(\rho C_p)}_{nf}},{\rho }_{nf}=(1-{\chi }_s){\rho }_f+{\chi }_s{\rho }_s,{\mu }_{nf}=\frac{{\mu }_f}{{(1-{\chi }_s)}^{2.5}},\\ {(\rho C_p)}_{nf}=(1-{\chi }_s){(\rho C_p)}_f+{\chi }_s{(\rho C_p)}_s,{\varphi }_{nf}=\left({\mu }_{nf}+\kappa /2\right)j,\\ \frac{k_{nf}}{k_f}=\frac{(k_s+2k_f)-2{\chi }_s(k_f-k_s)}{(k_s+2k_f)+{\chi }_s(k_f-k_s)},{\sigma }_{nf}=\left[1+\frac{3\left(\left({\sigma }_s/{\sigma }_f\right)-1\right){\chi }_s}{\left(\left({\sigma }_s/{\sigma }_f\right)+2\right)-\left(\left({\sigma }_s/{\sigma }_f\right)-1\right){\chi }_s}\right]{\sigma }_f,\\ Q_R=-\frac{4\tau }{3\omega }\frac{\partial T^4}{\partial \widehat{\gamma }}=\frac{16\tau }{3\omega }{T}^3\frac{\partial T}{\partial \gamma },\end{array}$$

(9)

where $\widehat{u}$ and $\widehat{v}$ are components of velocity, ${\chi }_s$, ${\alpha }_{nf}$, ${\mu }_{nf}$, ${\varphi }_{nf}=({\mu }_{nf}+\kappa /2)J$, ${(\rho C_p)}_{nf}$, $B_0$ are nanoparticle volume fraction, thermal diffusivity, the viscosity, spin gradient viscosity, heat capacity of the nanofluid, and magnetic field strength, respectively. Also, $\widehat{N}$ denotes the microrotation component normal to the $\widehat{\aleph }-\widehat{\gamma }$ plane. On the other hand, $Q_R$ is called the Rosseland diffusion approximation for radiation, which was offered by Howell et al. [58], such that $\tau$ and $\omega$ are Stefan–Boltzmann and mean absorption coefficients, respectively. The used boundary conditions for the above system of Equations (5)–(9) are (see [59]):

$$\begin{array}{c}\widehat{u}=\widehat{v}=0,T=T_w,\widehat{N}=-\frac{1}{2}\frac{\partial \widehat{u}}{\partial \widehat{\gamma }}\mathrm{as}\widehat{\gamma }=0,\\ \widehat{u}\to {\widehat{u}}_e(\widehat{\aleph }),T\to T_{\infty } \widehat{N}\to 0\mathrm{as}\widehat{\gamma }\to \infty \end{array}$$

(10)

we see that ${\widehat{u}}_e(\widehat{\aleph })=\frac{3}{2}{U}_{\infty }\sin \left(\widehat{\aleph }/a\right)$ and $\widehat{r}(\widehat{\aleph })=a\sin \left(\widehat{\aleph }/a\right)$ are the local free-stream velocity and the radial distance from the symmetrical axis to the surface of the sphere, respectively, and $\widehat{u}$ and $\widehat{v}$ are called the velocity components along the $\widehat{\aleph }-\widehat{\gamma }$ plane. The non-dimensional variables are recognized as follows (see [54]):

$$\begin{array}{c}\aleph =\frac{\widehat{\aleph }}{a},\gamma ={\mathrm{Re}}^{1/4}\left(\frac{\widehat{\gamma }}{a}\right),r(\aleph )=\frac{\widehat{r}(\widehat{\aleph })}{a},\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},\\ u=\frac{\widehat{u}}{U_{\infty }},v={\mathrm{Re}}^{1/2}\left(\frac{\widehat{v}}{U_{\infty }}\right),N=\left(\frac{a}{U_{\infty }}\right){\mathrm{Re}}^{-1/2}\widehat{N},\end{array}$$

(11)

where $\mathrm{Re}=U_{\infty }a/v_f$ is the Reynolds number, and $v_f$ is the kinematic viscosity of the fluid.

By substituting the variables included in Equations (9) and (11) in the system (5)–(9), we obtain the following dimensionless system for this problem:

$$\frac{\partial ru}{\partial \aleph }+\frac{\partial ru}{\partial \gamma }=0,$$

(12)

$$\begin{array}{l}u\frac{\partial u}{\partial \aleph }+v\frac{\partial u}{\partial \gamma }=u_e\frac{\partial u_e}{\partial \aleph }+\frac{{\rho }_f}{{\rho }_{nf}}\left(D\left({\chi }_s\right)+K\right)\frac{{\partial }^{2}u}{\partial {\gamma }^2}\\ +\frac{1}{{\rho }_{nf}}\left({\chi }_s{\rho }_s\left(\frac{{\beta }_s}{{\beta }_f}\right)+\left(1-{\chi }_s\right){\rho }_f\right)\lambda \theta \sin \aleph +\frac{{\rho }_f}{{\rho }_{nf}}K\frac{\partial N}{\partial \gamma }-\frac{{\rho }_f}{{\rho }_{nf}}\frac{{\sigma }_{nf}}{{\sigma }_f}\mathrm{M}u,\end{array}$$

(13)

$$\frac{\mathrm{Pr}}{1+(3/4)\mathrm{L}}\left(u\frac{\partial \theta }{\partial \aleph }+v\frac{\partial \theta }{\partial \gamma }\right)=\left[\frac{k_{nf}/k_f}{\left(1-{\chi }_s\right)+{\chi }_s{\left(\rho c_p\right)}_s/{\left(\rho c_p\right)}_f}\right]\frac{{\partial }^2\theta }{\partial {\gamma }^2},$$

(14)

$$u\frac{\partial N}{\partial \aleph }+v\frac{\partial N}{\partial \gamma }=-\frac{{\rho }_f}{{\rho }_{nf}}K\left(2N+\frac{\partial u}{\partial \gamma }\right)+\frac{{\rho }_f}{{\rho }_{nf}}\left(D\left({\chi }_s\right)+\frac{K}{2}\right)\frac{{\partial }^{2}N}{\partial {\gamma }^2},$$

(15)

here, $D({\chi }_s)=1/{(1-{\chi }_s)}^{2.5}$, $\mathrm{Pr}=v_f/{\alpha }_f$ is the Prandtl number, $K=\kappa /{\mu }_f$, $\mathrm{M}=({\delta }_f{B}_0^{2}a/{\rho }_f{\nu }_f)$, $\lambda =Gr/{\mathrm{Re}}^2$, and $\mathrm{L}=\frac{\alpha \tau {(\rho c_p)}_{nf}}{4\omega T^3}$ are the micropolar, the magnetic, the mixed convection, and radiation parameters, respectively. In the constant wall temperature boundary conditions case, the Grashof number is given by $Gr=g{\beta }_f\left(T_w-T_{\infty }\right)\left(a^3/{\nu }_f^2\right)$. It is important to remember that $\lambda >0$ for an assisting flow ($T_w>T_{\infty }$) (heated flow), $\lambda <0$ for an opposing flow $(T_w<T_{\infty })$ (cooled flow), and $\lambda =0$ the forced convection flow.

The non-dimensional boundary conditions are:

$$\begin{array}{c}u=v=0,\theta =1,N=-\frac{1}{2}\frac{\partial u}{\partial \gamma }\mathrm{as}\gamma =0\\ u\to u_e(\aleph )=\left(3/2\right)\sin \aleph u\to 0,\theta \to 0,N\to 0\mathrm{as}\gamma \to \infty .\end{array}$$

(16)

To solve Equations (12)–(15), with the boundary conditions (16), suppose we have variables as below [20]:

$$\psi =\aleph r(\aleph )z(\aleph .\gamma ),\theta =\theta (\aleph ,\gamma ),N=\aleph n(\aleph ,\gamma ),$$

(17)

while $\psi$ is the stream function realized as $u=(1/r)(\partial \psi /\partial \gamma )$ and $v=-(1/r)(\partial \psi /\partial \aleph )$, which satisfies the continuity Equation (12). Substituting the Equation (17) into (13) to (15), we acquire the following partial differential equations:

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(D({\chi }_s)+K\right)\frac{{\partial }^{3}z}{\partial {\gamma }^3}+(1+\aleph \cot \aleph )z\frac{{\partial }^{2}z}{\partial {\gamma }^2}-{\left(\frac{\partial z}{\partial \gamma }\right)}^2+\frac{1}{{\rho }_{nf}}\left(\chi {\rho }_s\left(\frac{{\beta }_s}{{\beta }_f}\right)+(1-{\chi }_s){\rho }_f\right)\lambda \frac{\sin \aleph }{\aleph }\theta \\ +\frac{9\sin \aleph \cos \aleph }{4\aleph }+\frac{{\rho }_f}{{\rho }_{nf}}K\frac{\partial n}{\partial \gamma }-\frac{{\rho }_f}{{\rho }_{nf}}\frac{{\sigma }_{nf}}{{\sigma }_f}{\mathrm{Mz}}^{\prime }=\aleph \left(\frac{\partial z}{\partial \gamma }\frac{{\partial }^{2}z}{\partial \aleph \partial \gamma }-\frac{\partial z}{\partial \aleph }\frac{{\partial }^{2}z}{\partial {\gamma }^2}\right),\end{array}$$

(18)

$$\left[\frac{k_{nf}/k_f}{(1-{\chi }_s)+{\chi }_s{(\rho c_p)}_s/{(\rho c_p)}_f}\right]\frac{{\partial }^2\theta }{\partial {\gamma }^2}+\left(\frac{\mathrm{Pr}}{1+(3/4)\mathrm{L}}\right)(1+\aleph \cot \aleph )z\frac{\partial \theta }{\partial \gamma }=\aleph \left(\frac{\partial z}{\partial \gamma }\frac{\partial \theta }{\partial \aleph }-\frac{\partial z}{\partial \aleph }\frac{\partial \theta }{\partial \gamma }\right),$$

(19)

$$\frac{{\rho }_f}{{\rho }_{nf}}\left(D({\chi }_s)+\frac{K}{2}\right)\frac{{\partial }^{2}n}{\partial {\gamma }^2}+(1+\aleph \cot \aleph )z\frac{\partial n}{\partial \gamma }-\frac{\partial z}{\partial \gamma }n-\frac{{\rho }_f}{{\rho }_{nf}}K\left(2n+\frac{{\partial }^{2}z}{\partial {\gamma }^2}\right)=\aleph \left(\frac{\partial z}{\partial \gamma }\frac{\partial n}{\partial \aleph }-\frac{\partial z}{\partial \aleph }\frac{\partial n}{\partial \gamma }\right),$$

(20)

The boundary conditions become:

$$\begin{array}{c}z=\frac{\partial z}{\partial \gamma }=0,\theta =1,n=-\frac{1}{2}\frac{{\partial }^{2}z}{\partial {\gamma }^2}\mathrm{as}\gamma =0\\ \frac{\partial f}{\partial \gamma }\to \left(3/2\right)(\sin \aleph /\aleph )\theta \to 0,n\to 0\mathrm{as}\gamma \to \infty \end{array}$$

(21)

It can be noticed that at the lower stagnation point of the sphere (ℵ ≈ 0), equations reduce to the following ordinary differential equations

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(D\left({\chi }_s\right)+K\right)z^{‴}+2z{z}^{″}-{\left(z^{\prime }\right)}^2+\\ \frac{1}{{\rho }_{nf}}\left({\chi }_s{\rho }_s\left(\frac{{\beta }_s}{{\beta }_f}\right)+\left(1-{\chi }_s\right){\rho }_f\right)\lambda \theta +\frac{9}{4}+\frac{{\rho }_f}{{\rho }_{nf}}K\frac{\partial n}{\partial y}-\frac{{\rho }_f}{{\rho }_{nf}}\frac{{\sigma }_{nf}}{{\sigma }_f}\mathrm{M}{z}^{\prime }=0,\end{array}$$

(22)

$$\left[\frac{k_{nf}/k_f}{\left(1-{\chi }_s\right)+{\chi }_s{\left(\rho c_p\right)}_s/{\left(\rho c_p\right)}_f}\right]{\theta }^{″}+2\left(\frac{\mathrm{Pr}}{1+(3/4)\mathrm{L}}\right)z{\theta }^{\prime }=0,$$

(23)

$$\frac{{\rho }_f}{{\rho }_{nf}}\left(D\left({\chi }_s\right)+\frac{K}{2}\right)n^{″}+2z{n}^{\prime }-z^{\prime }n-\frac{{\rho }_f}{{\rho }_{nf}}K\left(2n+z^{″}\right)=0.$$

(24)

The boundary conditions become:

$$\begin{array}{c}z\left(0\right)=z^{\prime }\left(0\right)=0,\theta \left(0\right)=1,n\left(0\right)=-\frac{1}{2}{z}^{″}\left(0\right)\mathrm{as}\gamma =0,\\ z^{\prime }\to \frac{3}{2},\theta \to 0,n\to 0\mathrm{as}\gamma \to \infty ,\end{array}$$

(25)

where primes denote differentiation with respect to $\gamma$. Other symbols are defined in the nomenclature section.

The interesting physical quantities, in dimensional form, are local skin friction C_f and Nusselt number Nu, and they can be written as:

$$C_f=\left(\frac{a}{{\mu }_f{U}_{\infty }}\right){\mathrm{Re}}^{-1/2}{\left(({\mu }_{nf}+\kappa )\frac{\partial \widehat{u}}{\partial \widehat{\gamma }}+\kappa \widehat{N}\right)}_{\widehat{\gamma }=0},Nu={\mathrm{Re}}^{-1/2}{\left(\frac{k_{nf}a}{k_f(T_{\infty }-T_f)}\frac{\partial T}{\partial \widehat{\gamma }}+Q_R\right)}_{\widehat{\gamma }=0}.$$

(26)

Using the non-dimensional variables (11) and boundary conditions (21), local skin friction C_f and Nusselt number Nu are:

$$C_f={\mathrm{Re}}^{-1/2}\left(D\left({\chi }_s\right)+\frac{K}{2}\right)\aleph \frac{{\partial }^{2}z}{\partial {\gamma }^2}\left(\gamma ,0\right),Nu=-{\mathrm{Re}}^{-1/2}(1+\frac{4}{3}L){\left(\frac{k_{nf}}{k_f}\frac{\partial \theta }{\partial \gamma }\right)}_{\gamma =0}.$$

(27)

3. Numerical Method

The Keller box numerical method is widely applied because it is flexible, much faster, elastically programmable, and efficient, as well as having high accuracy [60,61,62]. Moreover, given the remarkable nature of the Keller box scheme, some attention should be given to its characteristics and accuracy. The scheme is widely employed in order to be the most adaptable and fixable as compared to other methods. The Keller box method is a significant scheme for solving the convection flow equations, particularly the parabolic equations for the boundary layer models [63,64]. Furthermore, the implicit finite difference scheme (the Keller box method) includes three main techniques, namely, the finite difference method, central differences, and Newton’s method, to attain a linear system and solve it via the block tri-diagonal elimination technique [65,66]. The numerical outcomes are obtained by the MATLAB program with excellent results (tables), and smooth curves (figures), and excellent agreement with the previous literature.

Firstly, the Keller box scheme comprises a finite difference method for solving partial differential Equations (18)–(20), which supersedes each of the high derivatives in the differential equation with a first-order equation system, as follows:

$$\begin{array}{c}u\left(\aleph ,\gamma \right)=z^{\prime }\left(\aleph ,\gamma \right)\to (u=z^{\prime })\\ v\left(\aleph ,\gamma \right)=z^{″}\left(\aleph ,\gamma \right)\to (v=z^{″}=u^{\prime })\\ q\left(\aleph ,\gamma \right)=h^{\prime }\left(\aleph ,\gamma \right)\to (q=h^{\prime })\\ c(\aleph ,\gamma )={\theta }^{\prime }(\aleph ,\gamma )\to c={\theta }^{\prime },\end{array}$$

(28)

The differential Equations (18)–(20) to be approximated are:

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(\frac{1}{{(1-{\chi }_s)}^{2.5}}\right)v^{\prime }+(1+\aleph \cot \aleph )zv-{\left(u\right)}^2+\frac{1}{{\rho }_{nf}}\left(\left(1-{\chi }_s\right){\rho }_f+{\chi }_s\frac{{\rho }_s{\beta }_s}{{\beta }_f}\right)\frac{\sin \aleph }{\aleph }\theta \\ +\frac{{\rho }_f}{{\rho }_{nf}}K{n}^{\prime }-\frac{{\rho }_f}{{\rho }_{nf}}\frac{{\sigma }_{nf}}{{\sigma }_f}\mathrm{M}u=\aleph \left(u\frac{\partial u}{\partial \aleph }-\frac{\partial z}{\partial \aleph }v\right),\end{array}$$

(29)

$$\left(\frac{1+(3/4)\mathrm{L}}{\mathrm{Pr}}\right)\left[\frac{(k_{nf}/k_f)}{\left(1-{\chi }_s\right)+{\chi }_s{(\rho Cp)}_s/{(\rho Cp)}_f]}\right]c^{\prime }+(1+\aleph \cot \aleph )z\frac{\partial \theta }{\partial \gamma }=\aleph \left(z^{\prime }\frac{\partial \theta }{\partial l}-\frac{\partial z}{\partial l}c\right),$$

(30)

$$\frac{{\rho }_f}{{\rho }_{nf}}\left(\frac{1}{{(1+{\chi }_s)}^{2.5}}+\frac{K}{2}\right)q^{\prime }+(1+\aleph \cot \aleph )zq-un-\frac{{\rho }_f}{{\rho }_{nf}}K\left(2h+v\right)=\aleph \left(u\frac{\partial n}{\partial \aleph }-q\frac{\partial z}{\partial \aleph }\right),$$

(31)

Such that the prime symbol signalizes the differential of the variable γ. In the same procedure, the boundary conditions (21) will be obtained as:

$$\begin{array}{l}z(\aleph ,0)=0,z(\aleph ,0)=0,\theta =1,\\ z^{\prime }(\aleph ,\infty )=0,\theta (\aleph ,\infty )=0,\end{array}$$

(32)

To execute the mesh points in the two-dimensional $\aleph$-$\gamma$ plane, let kⁱ and g_j be the steps sizes of the relevant step distances in $\aleph$ and $\gamma$ directions, respectively, as clarified in Figure 2.

In the model that we are investigating, the mesh points are referenced as below:

$$\begin{array}{l}{\gamma }_0=0,{\gamma }_i={\gamma }_{i-1}+g_i,i=0,1,2,3,\dots ,J.{\gamma }_{\infty }={\gamma }_J,\\ {\aleph }^0=0,{\aleph }^i={\aleph }^i={\aleph }^{i-1}+k^i,i=0,1,2,3,\dots ,N.\end{array}$$

(33)

For i = 1, 2, …, N, j = 1, 2, …, J, the midpoint and first derivative for these quantities, can be obtained at the inner mesh points ( ) as follows:

$$\begin{array}{c}{\left(\right)}_{j-1/2}^{n-1/2}=\frac{1}{4}\left({\left(\right)}_j^n+{\left(\right)}_{j-1}^n+{\left(\right)}_j^{n-1}+{\left(\right)}_{j-1}^{n-1}\right)\\ {\left(\frac{\partial \left(\right)}{\partial \gamma }\right)}_{j-1/2}^{n-1/2}=\frac{1}{2g_j}\left({\left(\right)}_j^n-{\left(\right)}_{j-1}^n+{\left(\right)}_j^{n-1}-{\left(\right)}_{j-1}^{n-1}\right)\\ {\left(\frac{\partial \left(\right)}{\partial \aleph }\right)}_{j-1/2}^{n-1/2}=\frac{1}{2k^n}\left({\left(\right)}_j^n-{\left(\right)}_{j-1}^n+{\left(\right)}_j^{n-1}-{\left(\right)}_{j-1}^{n-1}\right),\end{array}$$

(34)

Consequently, the systems of Equations (24) and (25)–(27) are extracted at the midpoint (ℵ, γ_j−1/2), which is called “centering”, as the form:

$$z_j^n-z_{j-1}^n=g_j\left(u_{j-1/2}^n\right),$$

(35)

$$u_j^n-u_{j-1}^n=g_j\left(v_{j-1/2}^n\right),$$

(36)

$${\theta }_j^n-{\theta }_{j-1}^n=g_j\left(q_{j-1/2}^n\right),$$

(37)

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(\frac{1}{{(1-{\chi }_s)}^{2.5}}\right)\left(v_j-v_{j-1}\right)+(1+\xi +\aleph \cot \aleph )\frac{\xi }{4}{g}_j(z_j+z_{j-1})(v_j+v_{j-1})-\left(\frac{1+\xi }{4}\right)g_j{(u_j+u_{j-1})}^2-\\ \frac{1}{2}\frac{{\rho }_f{\sigma }_{nf}}{{\rho }_{nf}{\sigma }_f}\mathrm{M}{g}_j(u_j+u_{j-1})+\left(\frac{1+\xi }{2}\right)g_j{v}_{j-1/2}^{n-1}(z_j+z_{j-1})-\left(\frac{1+\xi }{2}\right)g_j{z}_{j-1/2}^{n-1}(v_j+v_{j-1})+\\ \frac{1}{2}\frac{1}{{\rho }_{Hnf}}\left(\left(1-{\chi }_s\right){\rho }_f+{\chi }_s\frac{{\rho }_s{\beta }_s}{{\beta }_f}\right)\frac{\sin {\aleph }^{n-1l2}}{{\aleph }^{n-1l2}}{g}_j({\theta }_j+{\theta }_{j-1})+\frac{{\rho }_f}{{\rho }_{nf}}K\left(\frac{1+\xi }{2}\right)(n_j-n_{j-1)})={\left(L_1\right)}_{j-1/2}^{n-1}\end{array}$$

(38)

$$\begin{array}{l}\left(\frac{1+(3/4)\mathrm{L}}{\mathrm{Pr}}\right)\left[\frac{k_{nf}/k_f}{\left(1-{\chi }_s\right)+{\chi }_s{(\rho Cp)}_s/{(\rho Cp)}_f}\right]\left(q_j^{}-q_{j-1}^{}\right)-\\ \frac{\xi }{4}{g}_j(u_j^{}+u_{j-1}^n)({\theta }_j^{}+{\theta }_{j-1}^{})+(1+\xi +\aleph \cot \aleph )\frac{1}{4}{g}_j(z_j^{}+z_{j-1}^{})(q_j^{}+q_{j-1}^{})+\frac{\xi }{2}{g}_j(u_j^{}+u_{j-1}^{}){\theta }_{j-1/2}^{n-1}-\\ \frac{\xi }{2}{g}_j{u}_{j-1/2}^{n-1}({\theta }_j^{}+{\theta }_{j-1}^{})-\frac{\xi }{2}{g}_j(q_j^{}-q_{j-1}^{})z_{j-1/2}^{n-1}+\frac{\xi }{2}{g}_j{q}_{j-1/2}^{n-1}(z_j^{}+z_{j-1}^{})={\left(L_2\right)}_{j-1/2}^{n-1}\end{array}$$

(39)

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(D({\chi }_s)+\frac{K}{2}\right)(q_j-q_{j-1})+\frac{(1+\xi )}{4}{g}_j(z_j-z_{j-1})(q_j-q_{j-1})-\\ \frac{(1+\xi )}{4}(u_j-u_{j-1})(n_j-n_{j-1})-\frac{{\rho }_f}{{\rho }_{nf}}\frac{\xi }{2}K{g}_j(n_j+n_{j-1})-\frac{{\rho }_f}{{\rho }_{nf}}\frac{\xi }{2}K{g}_j(v_j+v_{j-1})\\ +\frac{\xi }{2}{g}_j{n}_{j-1/2}^{n-1}\left(u_j+u_{j-1}\right)-\frac{\xi }{2}{g}_j{u}_{j-1/2}^{n-1}\left(n_j+n_{j-1}\right)={\left(L\right)}_3^{n-1},\end{array}$$

(40)

$${\left(L_1\right)}_{j-1/2}^{n-1}=-g_j{\left(\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(\frac{1}{{(1-{\chi }_s)}^{2.5}}\right)\frac{\left(v_j^{}-v_{j-1}^{}\right)}{g_j}+(1-\xi )z_{j-1/2}{v}_{j-1/2}+(\xi -1){\left(u_{j-1/2}^{}\right)}^2\\ -\frac{{\rho }_f{\sigma }_{nf}}{{\rho }_{nf}{\sigma }_f}\mathrm{M}{u}_{j-1/2}^{}+\left(\left(1-{\chi }_s\right)+{\chi }_s\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}\right)\frac{\sin {\aleph }^{n-1l2}}{{\aleph }^{n-1l2}}{\theta }_{j-1/2}^{}+\frac{{\rho }_f}{{\rho }_{nf}}K{q}_{j-1/2}\end{array}\right)}^{n-1}$$

(41)

$${\left(L_2\right)}_{j-1/2}^{n-1}=-g_j{\left(\begin{array}{l}\left(\frac{1+(3/4)\mathrm{L}}{\mathrm{Pr}}\right)\left[\frac{k_{nf}/k_f}{\left(1-{\chi }_s\right)+{\chi }_s{(\rho Cp)}_s/{(\rho Cp)}_f}\right]\frac{\left(q_j^{}-q_{j-1}^{}\right)}{g_j}\\ +\left(1+\aleph \cot \aleph -\xi \right)z_{j-1/2}^{}{q}_{j-1/2}^{}+\xi u_{j-1/2}^{}{\theta }_{j-1/2}^{}\end{array}\right)}^{n-1}$$

(42)

$${\left(L_1\right)}_{j-1/2}^{n-1}=-g_j{\left(\begin{array}{l}\frac{{\rho }_f}{{\rho }_{nf}}\left(\frac{1}{{(1-{\chi }_s)}^{2.5}}+\frac{K}{2}\right)(q_j-q_{j-1})+(1-\xi )z_{j-1/2}{q}_{j-1/2}\\ +(\xi -1)\left(n_{j-1/2}^{}{u}_{j-1/2}^{}\right)-\frac{{\rho }_f}{{\rho }_{nf}}K(n_{j-1/2}^{}+v_{j-1/2}^{})\end{array}\right)}^{n-1}$$

(43)

where $\xi =\frac{{\aleph }^{n-1l2}}{k^i}$

The boundary condition can be expressed as:

$$z_0^n=u_0^n=0,{\theta }_0^n=1,u_J^n={\theta }_J^n=0,$$

(44)

4. Validation of Numerical Findings

The governing Equations (14)–(16), subject to the boundary conditions (17), were solved numerically using the Keller box method presented by [67]. Before determining the effects of micropolar nanofluid, the values of Cf should be compared to those given by Nazar and Pop [68] at K = 0, ${\chi }_s=0$, M = 0, and L = 0, as displayed in

Table 1. Values of $C_f$ for $\mathrm{K}=0$   ${\chi }_s=0$, M = 0, and L = 0 (Newtonian fluid), Pr = 0.7, and various values of $\lambda$.

ℵ	$\mathit{\lambda }$
	−4	−3	−2	−1	−0.5	0.0	0.74	0.75
0°	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)
	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)	(0.0000)
10°	0.07899	0.1794	0.2627	0.3401	0.3768	0.4146	0.46221	0.4631
	(0.0801)	(0.1806)	(0.2662)	(0.3438)	(0.3804)	(0.4160)	(0.4669)	(0.4675)
20°	0.11270	0.3240	0.4981	0.6522	0.7277	0.7967	0.8898	0.9004
	(0.1149)	(0.3261)	(0.5000)	(0.6564)	(0.7301)	(0.8014)	(0.9031)	(0.9045)
30°		0.3890	0.6669	0.8998	1.0009	1.1767	1.2754	1.2736
		(0.4024)	(0.6718)	(0.9098)	(1.0211)	(1.1284)	(1.2813)	(1.2833)
40°		0.3609	0.7509	1.0726	1.2003	1.3689	1.5740	1.5721
		(0.3704)	(0.7535)	(1.0790)	(1.2292)	(1.3733)	(1.5775)	(1.5802)
50°			0.7109	1.14780	1.3015	1.4782	1.7700	1.7715
			(0.7181)	(1.1434)	(1.3350)	(1.5172)	(1.7737)	(1.7771)
60°			0.5226	1.1047	1.2890	1.4518	1.8536	1.8561
			(0.5295)	(1.0866)	(1.3246)	(1.4577)	(1.8580)	(1.8621)
70°				0.8899	1.1870	1.4539	1.8222	1.8270
				(0.8929)	(1.1889)	(1.4583)	(1.8260)	(1.8307)
80°				0.5093	0.9113	1.2403	1.6741	1.6802
				(0.5280)	(0.9190)	(1.2480)	(1.6800)	(1.6855)
90°					0.4735	0.9100	1.4224	1.4317
					(0.4813)	(0.9154)	(1.4289)	(1.4352)
100°						0.4119	1.0745	1.0477
						(0.4308)	(1.0847)	(1.0922)
110°							0.6141	0.6000
							(0.6543)	(0.6637)
120°								0.0079
								(0.0380)

. It is found that the values of C_f are in perfect agreement.

5. Graphical Findings and Discussion

MATLAB software assisted in obtaining graphical outcomes for several values of micropolar nanofluid parameters for the physical quantities, such as $C_f$, $Nu$, $\theta (0,\gamma )$, $z^{\prime }(0,\gamma )$, and $h(0,\gamma )$ at various positions $\aleph$, for the assisting $(\lambda >0)$ and opposing $(\lambda <0)$ flow. The data regarding the thermo–physical characteristics of the used fluid and nanoparticles are listed in

Table 2. Thermo–physical properties of based fluids and nanoparticles [37,43,69].

Material	ρ(kg/m3)	ρCp(J/kgK)	k(W/mK)	B × 10−5K−1	σ (s/m)	Pr
Water	997.1	4179	0.613	21	5.5 × 10−6	6.2
Methanol	792	2545	0.2035	149	5 × 10−7	7.3
Al2O3	3970	765	40	0.85	3.5 × 107	…
CuO	6510	540	18	0.85	3.96 × 107	…

.

Figure 3 and Figure 4 visualize the influence of the magnetic field factor on both skin friction and Nusselt number. It is noticeable that the increment in the magnetic field parameter $M$ leads directly to a decrease in both skin friction and Nusselt number. This could be due to the restriction in nanofluid flow, as a result of the Lorentz force formation caused by an increase in the strength of the magnetic field, which, in turn, acts to curb energy transport and skin friction. Figure 5 and Figure 6 are plotted to describe the extent of the domination of the volume fraction of nanoparticles ${\chi }_s$ on skin friction and Nusselt number, respectively. It is well-known that the increasing value of nanoparticle volume fraction improves the thermal conductivity of the nanoliquid. This, of course, improves the transfer of energy between the spherical surface and the nanoliquid, in addition to reducing the friction force, resulting in an elevation of the curves of Nusselt number, and a lowering of the curves of skin friction. The effect of micropolar parameter $K$ on skin friction and Nusselt number is reported in Figure 7 and Figure 8, respectively. It is observed that $K$ has an inverse relationship with $C_f$ and $Nu.$ Actually, the gain in micropolar factor raises nanofluid viscosity, which decreases both the Nusselt number and skin friction. Figure 9 and Figure 10 show how the skin friction and Nusselt number are affected by the combined convection parameter $\lambda$, respectively. It is obvious that the gain in the mixed convection coefficient increases buoyant forces that enhance momentum transfer, thereby increasing the energy transport rate and friction drag. It is also worth noting that moving from a cooling ($\lambda$ < 0) state to a heating state ($\lambda >0$) contributes to changing the behavior of micropolar nanofluid, in which Al₂O₃–H₂O changes its behavior, as it possesses the lowest value for skin friction upon cooling, and has the highest value for skin friction upon heating. Figure 11 and Figure 12 focus on showing the effect of the radiation parameter L on the Nusselt number and skin friction, respectively. Releasing more thermal radiation means more heat is added to the micropolar liquid. In other words, the radiation parameter acts as an additional heat source, and this increases the effectiveness of energy transfer and enhances the forces of friction. Moreover, in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, it is found that the Nusselt number has its greatest value when methanol (CH₃OH) is used as a base fluid, due to its lower value of thermal conductivity (Table 1), which plays a critical role in raising the Nusselt number, as seen in Equation (18).

The impact of changing the parameters $M,{\chi }_s,\mathrm{K},\lambda ,$ and $\mathrm{L}$ on the temperature $\theta$, velocity $\partial z/\partial \gamma$, and angular velocity $h$ is investigated separately in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27. According to Figure 13, Figure 14 and Figure 15, increasing the magnetic field strength causes a rise in temperature, as well as a decline in velocity and angular velocity. The formation of the Lorentz force, which is created by the crossing of a magnetic field through a moving fluid, is the reason behind these behaviors. This force increases the fluid’s resistance to movement, and raises its temperature. Figure 16, Figure 17 and Figure 18 show the responses of temperature, velocity, and angular velocity to the effect of the volume fraction of nanoparticles ${\chi }_s$, respectively. It is well-known that the increase in the volume fraction of the nanoparticles contributes to an increase in the rate of energy transfer. Furthermore, the collision of ultrafine particles in the flow generates more thermal energy and, thus, raises the temperature of the nanofluid. Simultaneously, the increase in ${\chi }_s$ increases the density of the fluid, impeding its movement. According to Figure 19, Figure 20 and Figure 21, the volume fraction of the nanoparticles and the micropolar parameter have the same effect, in that the profiles of temperature, velocity, and angular velocity show the same impression. The temperature rises as it rises, while velocity and angular velocity decline. This actually occurs because raising the micropolar parameter raises the viscosity of the nanofluid. Figure 22, Figure 23 and Figure 24 indicate the effects of increasing the mixed convection parameter on the temperature, velocity, and angular velocity, respectively. As a result of the increase in mixed convection parameter, the buoyant forces are boosted, which makes the temperature profiles decrease, and the velocity and angular velocity of the nanofluid increase. Furthermore, the physical quantities of the micropolar nanofluid can completely change from cooling ($\lambda$ < 0) to heating ($\lambda >0$) states, with Al₂O₃–H₂O having the highest velocity and angular velocity at heating, but it has the lowest value of the same two physical quantities at cooling compared to the other nanofluids studied. Figure 25, Figure 26 and Figure 27 are plotted to examine the effect of the thermal radiation parameter L on the profiles of temperature, angular velocity, and velocity. It is clear that increasing the radiation parameter raises the temperature of the micropolar fluid, and increases both the angular velocity and the velocity. An explanation for this behavior may be that an increase in the rate of radiation causes more energy to be released into the micropolar liquid, and this, of course, improves the mentioned physical quantities. Through the study of the different parameters that act on the temperature profiles, it is seen that the CuO–H₂O micropolar nanofluid acquires the top value of the temperature profile when compared with the other micropolar nanofluids. The physical properties of the copper oxide used as a nanoparticle could be the main reason for the high temperatures, such as the lower thermal conductivity and higher density of CuO.

6. Conclusions

The existing work is concerned with the impression of thermal radiation on the heat transfer characteristics of a magnetized micropolar nanofluid flowing around a spherical surface. To this end, a governing model was constructed, based on the Tiwari–Das nanofluid model, and solved numerically through the Keller box method. The following points focus on the main results:

• The use of aluminum oxide (Al₂O₃) as a nanoparticle in nanofluids always shows the highest values of physical quantities, such as local skin friction, Nusselt number, velocity, and angular velocity compared to copper oxide (CuO) nanoparticles, but for the temperature profile, it tends to favor copper oxide, which occupies the top of the temperature values;
• Nanoparticles suspended in water as a base fluid obtain higher values for measurements of local skin friction, temperature, velocity, and angular velocity compared to an ethanol base fluid. However, ethanol had the highest values for the Nusselt number compared to water;
• The mixed convection and radiation parameters are the only two parameters that raise the quantities of local skin friction, velocity, and angular velocity by their increment;
• Raising the values of the $M,{\chi }_s,\mathrm{and}K,$ parameters is responsible for promoting the temperature, but the mixed convection parameter is not included in this behavior. Its increment causes a decrease in temperature;
• The Nusselt number is directly proportional to each volume parameter and mixed convection parameter, while it is inversely proportional to the magnetic parameter and micropolar parameter.