Energy Transfer through a Magnetized Williamson Hybrid Nanofluid Flowing around a Spherical Surface: Numerical Simulation

Abstract

A computational simulation of Williamson fluid flowing around a spherical shape in the case of natural convection is carried out. The Lorentz force and constant wall temperature are taken into consideration. In addition, upgrader heat transfer catalysts consisting of multi-walled carbon tubes, molybdenum disulfide, graphene oxide, and molybdenum disulfide are employed. The Keller box approach is used to solve the mathematical model governing the flow of hybrid Williamson fluid. To validate our findings, the key parameters in the constructed model are set to zero. Next, the extent of the agreement between our results and published results is observed. Numerical and graphical results that simulate the impressions of key parameters on physical quantities related to energy transmission are obtained, discussed, and analyzed. According to the results of this study, increasing the value of the Weissenberg number causes an increase in both the fluid temperature and drag force, while it also leads to a decrease in both the velocity of the fluid and the rate of energy transmission. Increasing the magnetic field intensity leads to a reduction in the rate of heat transfer, drag force, and fluid velocity while it has an appositive effect on temperature profiles.

1. Introduction

Recently, researchers in the field of hydrodynamics have focused their studies on non-Newtonian fluids because of the latter’s important engineering applications. The result of this focus is the emergence of many mathematical models that attempt to simulate the behavior of these fluids. One of the most important non-Newtonian fluid models considered in several papers is the Williamson fluid model proposed by Williamson [1]. He presented a governing equation explaining pseudoplastic fluid flow, thereby addressing the problems of mass transfer of the pseudoplastic, in which the shear-thinning features of non-Newtonian fluids were highlighted, and laboratory experiments were presented to examine the proposed approach. Later on, several studies extended the Williamson model in different directions. Lyubimov and Perminov [2] investigated the tinny film movement of a Williamson fluid via an inclined surface in the gravitational field. The authors analyzed the effect of the tangential and latter vibrations of the solid surface on layer flow and presented the results of the experiments, showing that a pronounceable average fluid flow was generated by the vibrations despite the weak gravitational field where the film is at rest. Nadeem and Akram [3] presented a study on the peristaltic transit of Williamson liquid in an asymmetric channel. Their study concluded that the curves of the pressure increase for large values of the Williamson factor are not linear, but for small values of the Williamson factor, they tend to behave as Newtonian liquids. Under the effects of nanoparticles, Nadeem and Hussain [4] investigated Williamson fluids’ two-dimensional flow over a stretching sheet. Nanomaterial chemically reactive flow-outcomes were modelled and analyzed using the rheological expressions of Williamson fluid by Hayat et al. [5]. The work considered a nonlinear bidirectional stretching sheet for the simulation. The outcome of the study shows that for higher values of magnetic parameters, the velocity always decays in the direction of travel. The Williamson fluid assessment of time-dependent flow was studied by Subbarayudu et al. [6]. The authors considered the radiative blood flow of the Williamson fluid against a wedge. Waqas et al. [7] and Hayat et al. [8] looked at incompressible, steady 2D nonlinear forced or mixed convective Williamson fluid flow on stretchable or flat surfaces, as well as the heat generation interaction of Williamson fluid flow in nonlinear forced or mixed convection subjects.

In 1995, Choi and Eastman [9] incorporated nanotechnology into the field of energy transmission for the first time, and this led to a quantum leap in improving the ability of the base fluids to transfer energy. Eastman et al. [10] affirmed that copper ultrafine particles can boost the heat transport properties of ethylene glycol. Chon et al. [11] reported the effect of temperature and nanoparticle size on the nanofluid’s thermal characteristics. Xuan and Li [12] identified the most critical factors influencing the rate of heat transfer, which may also be influenced by the factors addressed by the authors of [13,14,15]. Recently, Tiwari and Das [16] constructed a mathematical model that included highlighting the effect of the ultrafine particle-volume fraction on the physical properties relevant to heat transfer. Subsequently, many of the mathematical models that govern the issues associated with laminar boundary layers were extensions of the Tiwari and Das model. Tham et al. [17,18] utilized the Tiwari and Das model to report the combined convection flow past a sphere and cylinder. Dinarvand et al. [19] applied the Tiwari and Das model to simulate the magneto-combined convection of nanofluid about a vertical permeable cylinder. Swalmeh et al. [20,21,22] examined the natural convection flow of micropolar nanoliquid around spherical and cylindrical surfaces. Alwawi et al. [23,24,25,26,27] reported the effect of the magnetic field on the heat transmission rate of Casson nanofluid employing the Tiwari–Das model. See also [28,29,30,31,32].

The second leap related to heat transfer has come from upgrading the ultrafine particles by synthesizing more than one compound to create a hybrid nanomaterial with optimized thermal features. Turcu et al. [33] are recognized as being among the first to synthesize hybrid nanomaterials from polypyrrole-CNTs and Fe₃O₄–MWCNTs. Suresh et al. [34] produced a hybrid material with improved thermal conductivity, composed of copper and aluminum oxides. Baghbanzadeh et al. [35] described a technique for combining SiO₂ with MWCNTs to create hybrid ultrafine particles, as well as showing the best SiO₂ to MWCNTs ratio to generate the maximum thermal conductivity for the hybrid synthesis material. Zhou et al. [36] fabricated a polymer hybrid nanomaterial with excellent thermal conductivity. Leong et al. [37] presented an interesting review of hybrid nanocomposites, their synthesis methods, and the most important aspects to be taken into consideration, such as thermal conductivity, stability of the host fluid, and others. More comprehensive studies are provided by the authors of [38,39,40,41]. Accordingly, several numerical investigations have been carried out in an attempt to predict the behavior of hybrid nanofluids. Devi and Devi [42] confirmed in their numerical study of the hybrid nanoliquid flowing around a permeable sheet that hybrid nanoparticles of Al₂O₃-Cu can give water a higher rate of energy transmission than Cu mono nanoparticles. Hayat and Nadeem [43] highlighted 3D-hybrid nanofluid flow over a stretching surface, taking into consideration radiation effects, heat creation, and chemical reaction. Subhani and Nadeem [44] carried out a computational analysis comparing the heat transfer rates of a mono and hybrid micropolar nanofluid flowing in a porous medium on a stretching surface. Khashi’ie et al. [45] simulated the magneto-mixed convection flow of hybrid nonliquid around a shrinking cylinder with the assistance of the Tiwari–Das model. Alwawi et al. [46] discussed heat transfer boosters generated by a transient magnetic field through a hybrid fluid moving around a cylinder. It is worth mentioning here that these comprehensive studies [47,48,49,50] used the second grade of the ultrafine particles and combined the models of Tiwari–Das and Williamson to simulate the behavior of these fluids; which behavior is the closest to this study and one of the reasons that motivated us to proceed with this investigation. As a result of these massive experimental and numerical studies, we observed the wide-range inclusion of hybrid particles to enhance heat transfer in conceivable primary applications such as electronic and manufacturing cooling, solar energy, heat exchangers, etc. [51,52,53]. In the case of laminar-boundary layer flow, there are numerous effective approximation methods for dealing with the problems of heat transfer through fluids [54,55,56,57,58]. The Keller Box approximation was used in this paper because it is one of the best approximations for dealing with laminar-boundary layer problems, has many distinguishing characteristics, and has been widely used for more than three decades.

Based on the above literature, the fabricated molybdenum disulfide nanosolid (MoS₂) on the one hand, and multi-walled carbon nanotubes (MWCNTs) and graphene oxide (GO) on the other hand, are used to support the thermal feature of the host Williamson fluid, which flows over a spherical surface. An applied magnetic field is also included because of its substantial impact on the energy transmission characteristics of hybrid nanofluids and its unlimited use in many engineering and industrial applications. In addition, the state of convection produced by natural means is considered. To the best of our knowledge, there is no study available that investigates this problem. Accordingly, in this report, the relevant equations are solved by converting PDEs into the dimensionless form using an appropriate conversion. The implicit Keller box finite difference method is used for the local similar solution of dimensionless governing equations and the flow and energy transport characteristics of the Williamson hybrid nanofluid past a sphere. Computational outcomes are computed, addressed, and analyzed in the form of tables and figures in order to simulate the effects of key parameters on the Nusselt number, velocity, skin friction coefficient, and temperature.

2. Problem Description

A convection produced by natural means in an electro-conductive Williamson hybrid nanofluid flowing over a spherical body under the influence of a magnetic field B₀ was assumed, as shown in Figure 1. The constant wall temperature T_w > T_∞ as well as an applied magnetic field are considered. T_w stands for the wall temperature, T_∞ is the ambient temperature. Additionally, $\overline{\eta }$ is the curvilinear coordinate over the sphere border, measuring its surface’s circumference, and $\overline{\gamma }$ is the distance normal to the sphere’s surface.

According to the preceding considerations, governing equations could be formed as follows (see [23,59,60]):

$$\frac{\partial r\overline{u}}{\partial \overline{\eta }}+\frac{\partial r\overline{v}}{\partial \overline{\gamma }}=0,$$

(1)

$$\begin{array}{l}{\rho }_{Hnf}\left(\hat{u}\frac{\partial \overline{u}}{\partial \overline{\eta }}+\widehat{v}\frac{\partial \overline{u}}{\partial \overline{\gamma }}\right)=-\frac{\partial \overline{P}}{\partial \overline{\eta }}+\sqrt{2}v\Gamma \left(\frac{{\partial }^2\overline{u}}{\partial {\overline{\gamma }}^2}\frac{\partial \overline{u}}{\partial \overline{\gamma }}\right)+{\mu }_{Hnf}\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{\gamma }}^2}+\frac{{\partial }^2\overline{u}}{\partial {\overline{\eta }}^2}\right)+\\ {\rho }_{Hnf}{\beta }_{Hnf}g(T-T_{\infty })\sin \left(\frac{\overline{\eta }}{a}\right)-{\sigma }_{Hnf}{B}_0^2\hat{u},\end{array}$$

(2)

$$\begin{array}{l}{\rho }_{Hnf}\left(\overline{u}\frac{\partial \overline{u}}{\partial \overline{\eta }}+\overline{v}\frac{\partial \overline{u}}{\partial \overline{\gamma }}\right)=-\frac{\partial \overline{P}}{\partial \overline{\gamma }}+\sqrt{2}v\Gamma \left(\frac{{\partial }^2\overline{u}}{\partial {\overline{\gamma }}^2}\frac{\partial \overline{u}}{\partial \overline{\gamma }}\right)+{\mu }_{Hnf}\left(\frac{{\partial }^2\overline{u}}{\partial {\overline{\gamma }}^2}+\frac{{\partial }^2\overline{u}}{\partial {\overline{\eta }}^2}\right)+\\ {\rho }_{Hnf}{\beta }_{Hnf}g(T-T_{\infty })\cos \left(\frac{\overline{\eta }}{a}\right)-{\sigma }_{Hnf}{B}_0^2\overline{u},\end{array}$$

(3)

$$\overline{u}\frac{\partial T}{\partial \overline{\eta }}+\overline{v}\frac{\partial T}{\partial \overline{\gamma }}={\alpha }_{Hnf}\left(\frac{{\partial }^{2}T}{\partial {\overline{\gamma }}^2}+\frac{{\partial }^{2}T}{\partial {\overline{\eta }}^2}\right),$$

(4)

The constant wall temperature boundary condition of the above governing equations will be defined as:

$$\overline{u}=\overline{v}=0, T=T_w at \overline{\gamma }=0,$$

$$\overline{u}\to 0, T\to T_{\infty }, p\to p_{\infty } at \overline{\gamma }\to \infty ,$$

(5)

where $\overline{u}$ and $\overline{v}$ are referents for the velocity components along $\overline{\eta }$, $\overline{\gamma }$ axes, respectively. $\overline{r}(\overline{\eta })$ is called the radial distance from the symmetrical axis to the surface of the sphere. P is pressure and $\Gamma$ is fluid relaxation time.

To convert the dimensional governing equations to the non-dimensional, we utilize the following non-dimensional variables, as follows:

$$\begin{array}{l}\omega =\left(\frac{\overline{\eta }}{a}\right)\gamma ={\mathrm{Gr}}^{1/4}\left(\frac{\overline{\gamma }}{a}\right), u=\left(\frac{a}{{\nu }_f}\right){\mathrm{Gr}}^{-1/2}\overline{u} ,v=\left(\frac{a}{{\nu }_f}\right){\mathrm{Gr}}^{-1/4}\overline{v}\\ \theta =\frac{T- T_{\infty }}{T_w- T_{\infty }},p=\frac{\widehat{p}- p_{\infty }}{{\rho }_f(v_f^2/a^2)},\end{array}$$

(6)

where Gr is determined physically as a non-dimensional number and symbolizes the ratio of the buoyancy forces to the viscous forces, and it can also be defined as $\mathrm{Gr}=g{\beta }_f(T_w-T_{\infty })\frac{a^3}{{\nu }_f^2}$ (see [61]). Pr is non-dimensional quantity, and it is a particular physical feature of fluids only. It is examined by considering the proportion of the velocity against the thermal of boundary layer thickness. On the other hand, the Prandtl number law is equal to momentum diffusivity over thermal diffusivity ($\mathrm{Pr}=\frac{v_f}{{\alpha }_f}$) (see [62]).

Table 1. Thermo-physical characteristics [49].

Properties of the Mono Nanofluid	Properties of the Hybrid Nanofluid
${\rho }_{nf}=\left(1-\varpi \right){\rho }_f+\varpi {\rho }_s,$	${\rho }_{Hnf}=\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1{\rho }_{s1}]+{\varpi }_2{\rho }_{s2},$
${\left(\rho c_p\right)}_{nf}=\left(1-\varpi \right){\left(\rho c_p\right)}_f+\varpi {\left(\rho c_p\right)}_s,$	${\left(\rho c_p\right)}_{Hnf}=\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){(\rho Cp)}_f+{\varpi }_1{(\rho Cp)}_{s1}]+{\varpi }_2{(\rho Cp)}_{s2},$
${\beta }_{nf}=\left(1-\varpi \right){\beta }_f+\varpi {\beta }_s$	${\beta }_{Hnf}=\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\beta }_f+{\varpi }_1{\beta }_{s1}]+{\varpi }_2{\beta }_{s2}.$
${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varpi \right)}^{2.5}},$	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\varpi }_1\right)}^{2.5}{\left(1-{\varpi }_2\right)}^{2.5}},$
$\frac{k_{nf}}{k_f}=\frac{\left(k_s+2k_f\right)-2\varpi \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+\varpi \left(k_f-k_s\right)},$	$\frac{k_{Hnf}}{k_{bf}}=\frac{k_{s2}+2k_{bf}-2{\varpi }_2\left(k_{bf}-k_{s2}\right)}{k_{s2}+2k_{bf}+{\varpi }_2\left(k_{bf}-k_{s2}\right)},$$\frac{k_{bf}}{k_f}=\frac{k_{s1}+2k_f-2{\varpi }_1\left(k_f-k_{s1}\right)}{k_{s1}+2k_f+{\varpi }_1\left(k_f-k_2\right)},$
${\alpha }_{nf}=\frac{k_{nf}}{{\left(\rho c_p\right)}_{nf}},$	${\alpha }_{Hnf}=\frac{k_{Hnf}}{{\left(\rho cp\right)}_{Hnf}},$
$\frac{{\sigma }_{nf}}{{\sigma }_f}=1 +\frac{ 3(\sigma -1)\varpi }{(\sigma +2)-(\sigma -1)\varpi }, \sigma = \frac{{\sigma }_s}{{\sigma }_f}$	$\frac{{\sigma }_{Hnf}}{{\sigma }_{bf}}=[\frac{ {\sigma }_{s2}+2{\sigma }_{bf}-2{\varpi }_2({\sigma }_{bf}-{\sigma }_{s2})}{{\sigma }_{s2}+2{\sigma }_{bf}+{\varpi }_2({\sigma }_{bf}-{\sigma }_{s2})}], \frac{{\sigma }_{bf}}{{\sigma }_f}=[ \frac{ {\sigma }_{s1}+2{\sigma }_f-2{\varpi }_1({\sigma }_f-{\sigma }_{s1})}{{\sigma }_{s1}+2{\sigma }_f+{\varpi }_1({\sigma }_f-{\sigma }_{s1})}]$

presents the thermo-physical characteristics of the hybrid nanofluids and mono nanofluids on which this study is based.

Here $\varpi ,\rho ,(\rho c_p),\beta$ are called nanoparticle volume fraction, density, heat capacity, and thermal expansion, respectively. In addition, $\mu ,k,\alpha ,\sigma$ refer to dynamic viscosity, thermal conductivity, thermal diffusivity, and electrical conductivity, respectively. Furthermore, the subscripts s, f, nf, hnf symbolize solid, host fluid, nanoliquid, and hybrid nanoliquid, respectively.

Using Equation (6), the described hybrid nanofluid properties in Table 1, and the useful boundary layer approximations technique (Gr → ∞), we obtain (∂p/∂η) = 0 and (∂p/∂γ) = 0. In other words, this provides the next equations, those containing Williamson hybrid nanofluid effects and the magnetic field at the momentum equation:

$$\frac{\partial ru}{\partial \eta }+\frac{\partial rv}{\partial \gamma }=0$$

(7)

$$\begin{array}{l}u\frac{\partial u}{\partial \eta }+v\frac{\partial u}{\partial \gamma }=\frac{{\rho }_f}{{\rho }_{Hnf}}\left(\frac{1}{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\right)\frac{{\partial }^{2}u}{\partial {\gamma }^2}+\mathrm{We}\left(\frac{{\partial }^{2}u}{\partial {\gamma }^2}\frac{\partial u}{\partial \gamma }\right)\\ +\frac{1}{{\rho }_{Hnf}}\left(\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\varpi }_2\frac{{\rho }_{s2}{\beta }_{s2}}{{\beta }_f}\right)\theta \sin \eta - \frac{{\rho }_f}{{\rho }_{Hnf}} \frac{{\sigma }_{Hnf}}{{\sigma }_f}\mathrm{M}u,\end{array}$$

(8)

$$\begin{array}{c}u\frac{\partial \theta }{\partial \eta }+v\frac{\partial \theta }{\partial \gamma }\\ =\frac{1}{\mathrm{Pr}}\left[\frac{k_{Hnf}/k_f}{\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right)+{\varpi }_1{(\rho Cp)}_1/{(\rho Cp)}_f]+{\varpi }_2{(\rho Cp)}_{s2}/{(\rho Cp)}_f}\right]\frac{{\partial }^2\theta }{\partial {\gamma }^2},\end{array}$$

(9)

In the previous system of equations, We is a non-dimensional parameter measuring the Fluid relaxation time $\Gamma$, and it is described as $\mathrm{We}=\frac{\Gamma \eta G{r}^{3/4}}{a^3}$. $\mathrm{M}=\left(\frac{{\sigma }_f{B}_0^2{a}^2{\mathrm{Gr}}^{-1/2}}{{\rho }_f{v}_f}\right)$ is the magnetic parameter. Substituting the properties in Table 1 and Equation (6) yields the following non-dimensional form of boundary condition:

$$\begin{array}{c}u=v=0, \theta =1, \mathrm{at} \gamma =0,\\ u\to 0, \theta \to 0,p\to 0, \mathrm{as} \gamma \to \infty .\end{array}$$

(10)

The following transformation variables are effective in solving Equations (7)–(10), which are described as: (see [47])

$$\psi =\eta f(\eta ,\gamma ), \theta =\theta (\eta ,\gamma ),$$

(11)

this is consistent with the following formula:

$$u=\frac{\partial \psi }{\partial \gamma } \mathrm{and} v=-\frac{\partial \psi }{\partial \eta }$$

(12)

that satisfies the continuity equation, where $\psi$ is the stream function.

Thus, the reduction in the governing partial differential equations by substituting the transformation variables (11) and (12) is completed as shown below:

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{Hnf}}\left(\frac{1}{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\right)\frac{{\partial }^{3}f}{\partial {\gamma }^3}+\mathrm{We}\frac{{\partial }^{3}f}{\partial {\gamma }^3}\frac{{\partial }^{2}f}{\partial {\gamma }^2}+(1+\eta \cot \eta )f\frac{{\partial }^{2}f}{\partial {\gamma }^2}-{\left(\frac{\partial f}{\partial \gamma }\right)}^2+\\ \frac{1}{{\rho }_{Hnf}}\left(\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\varpi }_2\frac{{\rho }_{s1}{\beta }_{s2}}{{\beta }_f}.\right)\frac{\sin \eta }{\eta } \theta -\frac{{\rho }_f}{{\rho }_{Hnf}} \frac{{\sigma }_{Hnf}}{{\sigma }_f}\mathrm{M} \frac{\partial f}{\partial \gamma }\\ =\eta \left(\frac{\partial f}{\partial \gamma }\frac{{\partial }^{2}f}{{\partial }^2}-\frac{\partial f}{\partial \eta }\frac{{\partial }^{2}f}{\partial {\gamma }^2}\right)\end{array}$$

(13)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}\left[\frac{k_{Hnf}/k_f}{\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right)+{\varpi }_1{(\rho Cp)}_1/{(\rho Cp)}_f]+{\varpi }_2{(\rho Cp)}_{s2}/{(\rho Cp)}_f}\right]\frac{{\partial }^2\theta }{\partial {\gamma }^2}\\ +(1+\eta \cot \eta )f\frac{\partial \theta }{\partial \gamma }=\eta \left(\frac{\partial f}{\partial \gamma }\frac{\partial \theta }{\partial \eta }-\frac{\partial f}{\partial \eta }\frac{\partial \theta }{\partial \gamma }\right),\end{array}$$

(14)

subject to:

$$\begin{array}{c}f=\frac{\partial f}{\partial \gamma }=0,\theta =1 \mathrm{at} \gamma =0,\\ \frac{\partial f}{\partial \gamma }\to 0, \theta \to 0, \mathrm{as} \gamma \to \infty .\end{array}$$

(15)

Assuming that ϖ is approximately equal to 0 (at the stagnation point), Equations (13)–(15) are modified as follows:

$$\begin{array}{c}\frac{{\rho }_f}{{\rho }_{Hnf}}\left(\frac{1}{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\right)\frac{{\partial }^{3}f}{\partial {\gamma }^3}+\mathrm{We}\frac{{\partial }^{3}f}{\partial {\gamma }^3}\frac{{\partial }^{2}f}{\partial {\gamma }^2}+2f\frac{{\partial }^{2}f}{\partial {\gamma }^2}-{\left(\frac{\partial f}{\partial \gamma }\right)}^2+\\ \frac{1}{{\rho }_{Hnf}}\left(\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\varpi }_2\frac{{\rho }_{s1}{\beta }_{s2}}{{\beta }_f}.\right)\theta -\frac{{\rho }_f}{{\rho }_{Hnf}} \frac{{\sigma }_{Hnf}}{{\sigma }_f}\mathrm{M} \frac{\partial f}{\partial \gamma }=0,\end{array}$$

(16)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}\left[\frac{k_{hnf}/k_f}{\left(1-{\vartheta }_2\right)[\left(1-{\vartheta }_1\right)+{\vartheta }_1{(\rho Cp)}_1/{(\rho Cp)}_f]+{\vartheta }_2{(\rho Cp)}_{s2}/{(\rho Cp)}_f}\right]\frac{{\partial }^2\theta }{\partial {\gamma }^2}\\ +2f\frac{\partial \theta }{\partial \gamma }=0\end{array}$$

(17)

with the modified boundary conditions:

$$\begin{array}{c}f\left(0,\gamma \right)=f^{\prime }\left(0,\gamma \right)=0, \theta \left(0,\gamma \right)=1 \mathrm{as} \gamma =0,\\ f^{\prime }\left(0,\gamma \right)\to 0, \theta \left(0,\gamma \right)\to 0 \mathrm{as} \gamma \to \infty ,\end{array}$$

(18)

Our attention focuses on the skin friction C_f and Nusselt number Nu, which are closely related to energy transfer C_f and Nu given by Swalmeh [59] as:

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

(19)

where

$${\tau }_w={\mu }_{Hnf}{\left(\frac{\partial \overline{u}}{\partial \overline{\gamma }}+\left[\frac{\Gamma }{\sqrt{2}}{\left(\frac{\partial \overline{u}}{\partial \overline{\gamma }}\right)}^2\right]\right)}_{\overline{\gamma }= 0} ,q_w=- k_{Hnf}{\left(\frac{\partial T}{\partial \overline{\gamma }}\right)}_{\overline{\gamma }= 0}$$

(20)

C_f and Nu are reformulated into the following forms, using Equations (6) and (10):

$$C_f={\mathrm{Gr}}^{-1/4}\frac{1 }{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\omega \left(\frac{{\partial }^{2}f}{\partial {\gamma }^2} (\eta ,0)+\frac{\mathrm{We}}{2}{\left(\frac{\partial f}{\partial \gamma }(\eta ,0)\right)}^2\right),Nu= -{\mathrm{Gr}}^{1/4}\frac{k_{Hnf}}{k_f}\frac{\partial \theta }{\partial \gamma } (\omega ,0),$$

(21)

All parameters and symbols are shown in the nomenclature list.

3. Numerical Techniques

In this section, an efficient numerical procedure called the Keller box method is used for obtaining numerical solutions to Equations (13)–(15). Firstly, the Keller box method involves a finite difference scheme that reduces the order of PDEs to the system of first-order equations. Now, we begin by introducing the independent functions:

$$u\left(\eta ,\gamma \right)=f^{\prime }\left(\eta ,\gamma \right), v\left(\eta ,\gamma \right)=f^{″}\left(\eta ,\gamma \right),s\left(\eta ,\gamma \right)=\theta \left(\eta ,\gamma \right),$$

(22)

$$f^{\prime }=u,$$

(23)

$$u^{\prime }=v=f^{″},$$

(24)

$${\theta }^{\prime }=t,$$

(25)

Using the above transformation, Equations (13)–(15) can be represented in the following form:

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{Hnf}}\left(\frac{1}{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\right)v^{\prime }+\mathrm{We} v^{\prime }v+(1+\eta \cot \eta )fv-{\left(u\right)}^2+\\ \frac{1}{{\rho }_{Hnf}}\left(\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\varpi }_2\frac{{\rho }_{s1}{\beta }_{s2}}{{\beta }_f}.\right)\frac{\sin \eta }{\eta } s-\frac{{\rho }_f}{{\rho }_{Hnf}} \frac{{\sigma }_{Hnf}}{{\sigma }_f}\mathrm{M} u\\ =\eta \left(u\frac{\partial u}{\partial \eta }-\frac{\partial f}{\partial \eta }v\right),\end{array}$$

(26)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}\left[\frac{k_{Hnf}/k_f}{\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right)+{\varpi }_1{(\rho Cp)}_1/{(\rho Cp)}_f]+{\varpi }_2{(\rho Cp)}_{s2}/{(\rho Cp)}_f}\right]{\theta }^{″}\\ +(1+\eta \cot \eta )f\frac{\partial \theta }{\partial \gamma }=\eta \left(f^{\prime }\frac{\partial \theta }{\partial \eta }-\frac{\partial f}{\partial \eta }{\theta }^{\prime }\right),\end{array}$$

(27)

where the primes symbol indicates differentiation of the variable γ. In addition, the boundary conditions (16) are transformed into:

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

(28)

To make the steps mesh points in a two-dimensional η-γ plane, define kⁿ and h_n of the related step distances in η and γ orientations, respectively, as shown in Figure 2.

The mesh points are indicated below:

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

(29)

For any independent quantities, midpoint and first derivative in the η-direction and γ-direction, placed by finite difference, is employed as follows:

$${\left(\right)}_{j-1/2}^{n-1/2}=\frac{1}{4}\left({\left(\right)}_j^n+{\left(\right)}_{j-1}^n+{\left(\right)}_j^{n-1}+{\left(\right)}_{j-1}^{n-1}\right)$$

$${\left(\frac{\partial \left(\right)}{\partial \gamma }\right)}_{j-1/2}^{n-1/2}=\frac{1}{2h_j}\left({\left(\right)}_j^n-{\left(\right)}_{j-1}^n+{\left(\right)}_j^{n-1}-{\left(\right)}_{j-1}^{n-1}\right)$$

(30)

$${\left(\frac{\partial \left(\right)}{\partial \omega }\right)}_{j-1/2}^{n-1/2}=\frac{1}{2k^n}\left({\left(\right)}_j^n-{\left(\right)}_{j-1}^n+{\left(\right)}_j^{n-1}-{\left(\right)}_{j-1}^{n-1}\right)$$

The following are finite difference approximations of Equations (23)–(25) and (26)–(27) about the midpoint (η_n, γ_j−1/2):

$$f_j^n-f_{j-1}^n=h_j\left(u_{j-1/2}^n\right),$$

(31)

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

(32)

$$s_j^n-s_{j-1}^n=h_j\left(t_{j-1/2}^n\right),$$

(33)

$$\begin{array}{l}\frac{{\rho }_f}{{\rho }_{Hnf}}\left(\frac{1}{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\right) \left(v_j^{}-v_{j-1}^{}\right)+\mathrm{We}\left(v_j^{}+v_{j-1}^{}\right)\left(v_j^{}-v_{j-1}^{}\right)+\\ (1+\xi +\eta \cot \eta )\frac{\xi }{4}{h}_j(f_j^{}+f_{j-1}^{})(v_j^{}+v_{j-1}^{})- \left(\frac{1+\xi }{4}\right)h_j {(u_j^{}+u_{j-1}^{})}^2-\\ \frac{1}{2}\frac{{\rho }_f{\sigma }_{nf}}{{\rho }_{nf}{\sigma }_f} \mathrm{M}{h}_j (u_j^{}+u_{j-1}^{})+ \left(\frac{1+\xi }{2}\right)h_j{v}_{j-1/2}^{n-1}(f_j^{}+f_{j-1}^{}) -\left(\frac{1+\xi }{2}\right)h_j{f}_{j-1/2}^{n-1}(v_j^{}+z_{j-1}^{})f_{j-1/2}^{n-1}+\\ \frac{1}{2}\frac{1}{{\rho }_{Hnf}}\left(\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\varpi }_2\frac{{\rho }_{s1}{\beta }_{s2}}{{\beta }_f}\right)\frac{\sin {\eta }^{n-1l2}}{{\eta }^{n-1l2}}{h}_j(s_{{}_j}^{}+s_{{}_{j-1}}^{}) ={\left(L_1\right)}_{j-1/2}^{n-1}\end{array}$$

(34)

$$\begin{array}{l}\frac{1}{\mathrm{Pr}}\left[\frac{k_{Hnf}/k_f}{\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right)+{\varpi }_1{(\rho Cp)}_1/{(\rho Cp)}_f]+{\varpi }_2{(\rho Cp)}_{s2}/{(\rho Cp)}_f}\right]\left(t_j^{}-t_{j-1}^{}\right) -\\ \frac{\xi }{4}{h}_j(u_j^{}+u_{j-1}^n)(s_j^{}+s_{j-1}^{})+(1+\xi +\eta \cot \eta )\frac{1}{4}{h}_j(f_j^{}+f_{j-1}^{})(t_j^{}+t_{j-1}^{})+\frac{\xi }{2}{h}_j(u_j^{}+u_{j-1}^{})s_{j-1/2}^{n-1}-\\ \frac{\xi }{2}{h}_j{u}_{j-1/2}^{n-1}(s_j^{}+s_{j-1}^{})-\frac{\xi }{2}{h}_j(t_j^{}-t_{j-1}^{})f_{j-1/2}^{n-1}+\frac{\xi }{2}{h}_j{t}_{j-1/2}^{n-1}(f_j^{}+f_{j-1}^{})={\left(L_2\right)}_{j-1/2}^{n-1}\end{array}$$

(35)

$${\left(l_1\right)}_{j-1/2}^{n-1} =-h_j{\left(\begin{array}{l}\frac{{\rho }_f}{{\rho }_{Hnf}}\left(\frac{1}{{(1-{\varpi }_1)}^{2.5}{(1-{\varpi }_2)}^{2.5}}\right) \frac{\left(v_j^{}-v_{j-1}^{}\right)}{h_j}+\\ \mathrm{We} v_{j-1}{v^{\prime }}_{j-1/2}(1-\xi +\eta \cot \eta )f_{j-1/2}^{}{v}_{j-1/2}^{}+(\xi - 1){\left( u_{j-1/2}^{}\right)}^2-\frac{{\rho }_f{\sigma }_{nf}}{{\rho }_{nf}{\sigma }_f} \mathrm{M}{u}_{j-1/2}^{} +\\ \left(\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right){\rho }_f+{\varpi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\varpi }_2\frac{{\rho }_{s1}{\beta }_{s2}}{{\beta }_f}\right)\frac{\sin {\eta }^{n-1l2}}{{\eta }^{n-1l2}}{s}_{j-1/2}^{}\end{array}\right)}^{n-1}$$

(36)

$${\left(l_2\right)}_{j-1/2}^{n-1}= -h_j{\left(\begin{array}{l}\frac{1}{\mathrm{Pr}}\left[\frac{k_{Hnf}/k_f}{\left(1-{\varpi }_2\right)[\left(1-{\varpi }_1\right)+{\varpi }_1{(\rho Cp)}_1/{(\rho Cp)}_f]+{\varpi }_2{(\rho Cp)}_{s2}/{(\rho Cp)}_f}\right]\frac{\left(t_j^{}-t_{j-1}^{}\right)}{h_j}\\ +\left(1+\eta \cot \eta -\xi \right)f_{j-1/2}^{}{t}_{j-1/2}^{}+\xi u_{j-1/2}^{}{s}_{j-1/2}^{}\end{array}\right)}^{n-1}$$

(37)

where $\xi =\frac{{\eta }^{n-1l2}}{k_n}$.

The boundary condition can be written as:

$$f_0^n = u_0^n=0, t_0^n=1,u_J^n=s_J^n=0,$$

(38)

Subsequently, the mathematical formula for the previous system (31)–(35) of non-linear algebraic equations will be linearized, by using Newton’s known method, and then solved by the block elimination technique. Furthermore, the numerical results are obtained by programming the algorithm of a linear system executed by MATLAB software. When we run the MATLAB program code, it needs to identify some specific computations: the boundary layer thickness $y_{\infty }$; proper step size $\Delta y$; and the step size $\Delta x$. $y_{\infty }$ is almost constant [63]. Furthermore, when $\mathrm{Pr}=6.2$, in this study, $y_{\infty }$ suitably lies between 3.5 and 8, to satisfy the boundary layer convergence. Once we obtain the suitable value of $y_{\infty }$, a sensible option of step size $\Delta y$ and step size $\Delta x$ should be determined. So, in the boundary layer flow, the step size $\Delta y=0.02$ and $\Delta x=0.005$ are appropriate to acquire accurate approximate numerical results. Moreover, these particular values successfully obtain outcomes which are almost compatible with previous findings, as displayed in

Table 2. Comparison of the outcomes for Nu at $\mathrm{Pr}=7,$ ${\varpi }_1={\varpi }_2=0$, $\mathrm{We}=0$, and $\mathrm{M}=0$.

$\mathit{\eta }$	[64]	[65]	Present
0	0.9581	0.9595	0.9593
$(1/18)\pi$	0.9559	0.9572	0.9568
$(1/9)\pi$	0.9496	0.9506	0.9499
$(1/6)\pi$	0.9389	0.9397	0.9397
$(2/9)\pi$	0.9239	0.9243	0.9242
$(5/18)\pi$	0.9045	0.9045	0.9046
$(1/3)\pi$	0.8858	0.8801	0.8833
$(7/18)\pi$	0.8518	0.8510	0.8526
$(4/9)\pi$	0.8182	0.8168	0.8178
$(1/2)\pi$	0.7792	0.7792	0.7792

and

Table 3. Comparison of the outcomes for C_f at $\mathrm{Pr}=7,$ ${\varpi }_1={\varpi }_2=0$, $\mathrm{We}=0$, and $\mathrm{M}=0$.

$\mathit{\eta }$	[64]	[65]	Present
0	0.0000	0.0000	0.0000
$(1/18)\pi$	0.0876	0.0875	0.0878
$(1/9)\pi$	0.1737	0.1735	0.1737
$(1/6)\pi$	0.2566	0.2563	0.2566
$(2/9)\pi$	0.3350	0.3345	0.3349
$(5/18)\pi$	0.4075	0.4068	0.4076
$(1/3)\pi$	0.4727	0.4715	0.4730
$(7/18)\pi$	0.5293	0.5380	0.5391
$(4/9)\pi$	0.5762	0.5745	0.5760
$(1/2)\pi$	0.6123	0.6103	0.6129

.

4. Results and Discussion

The current section provides an in-depth analysis of natural convection’s physical aspects produced by a magnetized Williamson hybrid nanofluid flowing around a spherical shape by highlighting the impressions of ultrafine particle fraction $\chi$, magnetic parameter $M$, and Weissenberg number $\mathrm{We}$ on physical quantities, as they relate to energy transmission. The pertinent parameters are chosen in the following ranges: $\chi =0.1, 0.15, 0.2$; $M=0.1, 0.5, 1$; and $\mathrm{We}= 0.1, 0.4, 0.5, 0.8$, in addition to fixing the Prandtl number at $\mathrm{Pr}=6.2$ (the Prandtl number for water) throughout the numerical calculations.

Table 4. Thermophysical properties of H₂O and booster nanosolid [25,46,66,67].

Physical Properties	Water	MWCNT	GO	MoS2
ρ (kg/m3)	997.1	1600	1800	5060
k (W/mK)	0.613	3000	5000	904.4
cp (J/kgK)	4179	796	717	397.21
σ (Sm−1)	5.5 × 10−6	1.9 × 10−4	6.30 × 107	2.09 × 104
β × 10−5 (K−1)	21	44		2.8424

shows the thermophysical properties of H₂O and booster nanosolid as used in this analysis.

Figure 3 illustrates the extent to which magnetic parameter intensification affects skin friction of mono and hybrid nanofluids while maintaining the ultrafine particle fraction $\chi$ and Weissenberg number $\mathrm{We}$ at constant values. Skin friction values decline as magnetic parameter values increase. This decline is due to the restriction in fluid flow produced by a rise in the strength of the magnetic field, which restricts convection and therefore decreases skin friction. Figure 4 describes the influence of the Weissenberg number on skin friction. It is noted that skin friction reduces as the Weissenberg number is elevated, owing to fluid thickening and increased viscous force at high We values. Figure 5 traces the effect of the ultrafine particle volume fraction on the mono/hybrid nanofluid’s skin friction, considering that the magnetic parameter and Weissenberg number are fixed values. It has been observed that increasing the ultrafine particle volume fraction tends to reduce skin friction, whether for hybrid or mono-nanofluid. The influence of escalating magnetic parameter values on the Nusselt number is expounded in Figure 6. It indicates that the ascending value of the magnetic parameter reduces the Nusselt number. In reality, increasing the intensity of the magnetic field interrupts the fluid movement, which in turn limits heat transmission. This means a decrease in the Nusselt number. In Figure 7, a decline in the Nusselt number is associated with an augmentation in the Weissenberg number, which seems to indicate that heat is transported from the spherical surface to the boundary layer, and the temperature is observed to increase with the Weissenberg number. As a result, the spherical surface is efficiently cooled with higher Weissenberg number values. Figure 8 conveys the variations of Nusselt number for various values of the ultrafine particle volume fraction at fixed We = 0.5 and magnetic parameter M = 3. Obviously, increasing the volume fraction of nanoparticles assists in raising the curve of the Nusselt number. More specifically, growth in the volume fraction of ultrafine particles, whether MWCNT or GO, contributes to improving the thermal conductivity of the mono nanofluid MoS₂/water, and as a result, the heat transfer rate increases, and therefore the Nusselt number increases. Moreover, in terms of energy transfer rate, the examined hybrid/mono-nano liquids can be arranged in ascending order regardless of the influencing parameter: MoS₂/H₂O < GO-MoS₂/H₂O < MWCNTs-MoS₂/H₂O. This superiority of composition of the carbon nanotubes and water in terms of the heat transfer rate may be due to the superior thermal conductivity of this nanofluidic hybrid. The behavior of the hybrid nanofluid’s temperature under the effect of M is shown in Figure 9. The temperature of the hybrid fluid in the boundary laminar layers increases with the strength of the imposed magnetic field. The main reason behind this dramatic rise in temperature profiles is the Lorentz force generated by the crossing transverse magnetic field, which in turn increases the friction and consequently elevates the temperature of the hybrid nanofluid. The influence of the Weissenberg number We on temperature is seen in Figure 10. It is worth noting that the temperature of the nanofluid rises as its values escalate. This occurs due to the higher resistance caused by the increased viscosity. Figure 11 describes the impression of ultrafine particle volume fraction variation on the hybrid nanofluid’s temperature. An elevation in the volume fraction of ultrafine particles produces an increase in heat transmission from the sphere’s surface to the host liquid, which aids in increasing the thermal boundary layer thickness. Figure 12 contains graphical results for the critical impact of magnetic parameters on the hybrid nanofluid’s velocity. This is known as the Lorentz force phenomenon, something which occurs when the magnetic field crosses a flowing fluid, and where this force restrains the movement of the fluid, slowing it down. In Figure 13, the flow of the nanofluid is found to diminish when the Weissenberg number is raised, implying that the velocity of the nanofluid is inhibited. This restriction in fluid movement is attributed to increased viscosity effects. The variation of velocity curves together with the ultrafine particle volume fraction values escalated are reported in Figure 14. The rise in the volume fraction leads to enhanced energy transfer, which in turn raises the fluid’s velocity.

5. Conclusions

The energy transfer of Williamson fluid flowing around a spherical shape supported by upgraded promoter nanosolid in the case of MHD convection produced by natural means was simulated computationally. It yielded the following meaningful points:

1-

The use of hybrid nanosolids stimulates energy transfer through the host fluid;

2-

The Weissenberg number has a positive effect on temperature and friction drag, while it negatively affects energy transfer and fluid velocity;

3-

Growing the strength of the magnetic field decreases velocity, friction drag, and energy transfer rate but raises temperature;

4-

Increasing the volume fraction of catalyzed nanomaterials (whether for MWCNTs or GO) improves energy transfer, raises the fluid temperature, and reduces friction drag;

5-

In terms of energy transfer rate, the examined hybrid/mono-nano liquids can be arranged in ascending order regardless of the influencing parameter as:

MoS₂/H₂O < GO-MoS₂/H₂O < MWCNTs-MoS₂/H₂O.