Heat Transmission Reinforcers Induced by MHD Hybrid Nanoparticles for Water/Water-EG Flowing over a Cylinder

Abstract

The assumptions that form our focus in this study are water or water-ethylene glycol flowing around a horizontal cylinder, containing hybrid nanoparticles, affected by a magnetic force, and under a constant wall temperature, in addition to considering free convection. The Tiwari–Das model is employed to highlight the influence of the nanoparticles volume fraction on the flow characteristics. A numerical approximate technique called the Keller box method is implemented to obtain a solution to the physical model. The effects of some critical parameters related to heat transmission are also graphically examined and analyzed. The increase in the nanoparticle volume fraction increases the heat transfer rate and liquid velocity; the strength of the magnetic field has an adverse effect on liquid velocity, heat transfer, and skin friction. We find that cobalt nanoparticles provide more efficient support for the heat transfer rate of aluminum oxide than aluminum nanoparticles.

1. Introduction

Flowing fluids are an important medium for transmitting energy, which cannot be neglected in several industrial, pharmaceutical, engineering, and food applications. This has led researchers to focusing on either searching for the best fluids that provide good heat transferability or trying to improve the efficiency of these media. Water was initially the best choice due to its abundance and ease of obtaining, and it possesses many critical thermophysical properties such as high heat capacity and thermal conductivity, so it met the needs of most industrial and engineering heat transfer applications. The mixture of water and ethylene glycol (EG) has also been used extensively in many energy systems, as ethylene glycol possesses distinct characteristics, such as its low freezing and high boiling points, high thermal conductivity, stability over a wide temperature range, and low viscosity, so it is suitable in operations requiring pumping. Accordingly, the combination of water and ethylene glycol is almost freeze-resistant, making it useful for charging operations and as an anti-freeze material in vehicles.

The physicist Feynman [1] was the first to address the nanoscale idea, as he suggested that the physical properties of materials differ according to the size of their particles. In the early 1990s, the scanning tunneling microscope was invented, which enabled researchers to study and monitor materials at the atomic level, which contributed to the wide spread of nanotechnology in several industrial, medicine, and engineering application [2,3,4,5,6,7]. Further experimental research and numerical simulation revealed that the lower the viscosity of the fluid and the greater its specific temperature, density, and thermal conductivity, the more efficient and able it is to transfer energy. In the field of heat transfer, Choi and Eastman [8] were amongst the first to improve the thermal features of flowing liquids through the adoption of nanoparticles. Eastman et al. [9] confirmed that combining copper nanoparticles with ethylene glycol promotes its thermal conductivity. Chon et al. [10] analyzed the impact of temperature and the size of ultrafine particles on the thermal conductivity of a nanoliquid. Xuan and Li [11] determined the most important factors affecting the rate of energy transmission, such as volume fraction, shapes, and dimensions of ultrafine particles; see also Refs. [12,13,14]. In 2007, Tiwari and Das [15] presented a mathematical model highlighting the effective role of the volume fraction of ultrafine particles in energy transmission. Since then, many mathematical models have been constructed that relied on this model. Nazar et al. [16] used Tiwari–Das’s nanofluid model to examine the combined convection flow of a nanofluid past a cylinder in a porous medium. Dinarvand et al. [17] constructed a mathematical model for investigating the mixed convection flow of a nanoliquid that relied on Tiwari–Das’s model. Swalmeh et al. [18] simulated the boundary layer flow of a micropolar nanofluid over a sphere using the Tiwari–Das model. Hamarsheh et al. [19] modeled the behavior of Casson nanofluid flow under the effect of a magnetic field with the aid of the Tiwari–Das model.

As an extension and development of the proposal presented by Choi and Eastman [8], hybrid nanoparticles have been introduced to the field of heat transmission through fluids. The idea of hybrid nanoparticles is based on the synthesis of two nanomaterials to produce a nanocomposite that possesses upgraded thermal characteristics. Suresh et al. [20] found that fabrication of Cu-Al₂O₃ nanocomposite including a small amount of copper, which has a high thermal conductivity with aluminum and has multiple features such as excellent stability and chemical inertness, shows properties integrated with those of alumina nanoparticles. Baghbanzadeh et al. [21] discussed a method of incorporating silica nanospheres (SiO₂) and multiwall carbon nanotubes (MWCNTs) to form hybrid nanoparticles; they also reported the optimal ratio of SiO₂ to MWCNTs in the nanocomposite that produces the highest thermal conductivity. Zhou et al. [22] produced a polymer nanocomposite with high efficiency in terms of thermal conductivity based on the hybrid nanoparticles idea. The reader is also referred to Refs. [23,24,25,26,27,28]. These improvements in thermal properties exhibited by the hybrid nanoparticles led them to being considered in several numerical studies related to heat transfer. Labib et al. [29] employed the mixture model to demonstrate the improvements in the heat transmission using CNTs–Al₂O₃ hybrid nanoparticles. Moghadassi et al. [30], in their numerical studies, found that adding small quantities of copper particles to a mono-nanofluid, which consisted of water and aluminum oxide, boosted the rate of heat transfer by 5%. Rostami et al. [31] numerically investigated the combined convection flow of SiO₂–Al₂O₃–water hybrid nanofluid past a vertical plate. Abdel-Nour et al. [32] employed the Galerkin finite element technique to address entropy generation in the flow of water hosting Al₂O₃–Cu hybrid nanocomposite in the presence of magneto-free convection. More numerical studies were conducted on hybrid nanofluids in Refs. [33,34,35,36,37,38].

Magneto-hydrodynamics or magneto-fluid dynamics (MHD) plays a vital role in many physical and engineering applications that depend on heat transmission mainly through electrically conductive fluids. Its applications are most evident with plasma propulsion, MHD molten pump jets or thrusters, flow control around hypersonic and re-entry vehicles, MHD power generation, biomedical applications, and others [39,40,41]. MHD is the science interested in studying the interaction between a magnetic field and electrically conducting moving fluids. The exposure of an electric-conducting fluid to an orthogonal magnetic field results in the creation of a type of force called Lorentz force, which affects all physical quantities related to heat transfer. Many research studies have examined the many aspects of this physical phenomenon in the heat transfer field. Sheikholeslami and Rokni [42] presented an article reviewing most of the previous studies that simulated the flow of nanofluids under a magnetic field. Jamaludin et al. [43] examined the boundary layer flow of nanofluids over a permeable sheet in the case of convection combined with slip conditions and the MHD effect. Dogonchi et al. [44] numerically studied the impacts of a magnetic field and joule heating on a nanofluid over a stretching plate considering Brownian motion. Alwawi et al. [45,46] modeled the magneto-free/mixed convection flow of Casson nanofluid over a solid sphere. The reader may also refer to Refs. [47,48,49].

The aforementioned literature produced our motivation to investigate hybrid particles promoting the heat transfer of water and ethylene glycol-water (EGW) flowing around a horizontal cylinder, which is widely used in heat transfer processes, especially in cooling. A variable orthogonal magnetic field was considered. The numerical approach of the solution to the mathematical model was computed using the Keller box method. In addition, we conducted comparisons between the physical quantities related to heat transfer for both host fluids.

2. Mathematical Model Formulation

As explained in Figure 1, we assume that the boundary layer flow of water/water-ethylene glycol, supported by hybrid nanocomposite over a circular cylinder, is affected by a transverse magnetic field. We also consider free convection and constant wall temperature. x and y indicate the axis that is measured on the circumference of the cylinder starting from the stagnation point and the axis that measures the vertical distance to its surface, respectively. $a$,$T_w$, $T_{\infty }$, g, and $B_0$ refer to the radius of circular cylinder, constant wall temperature, ambient temperature, gravity vector, and magnetic field strength, respectively.

Relying on the above consideration and under the usual Boussinesq incompressible hybrid nanofluid model, and applying the boundary layer approximations, the governing model of continuity, momentum, and energy equations can be described as [50,51,52,53]:

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

(1)

$$\begin{array}{l}{\rho }_{hnf}\left(\widehat{u}\frac{\partial \widehat{u}}{\partial \widehat{x}}+\widehat{v}\frac{\partial \widehat{u}}{\partial \widehat{y}}\right)=-\frac{\partial \widehat{p}}{\partial \widehat{x}}+{\mu }_{hnf}\left(\frac{{\partial }^2\widehat{u}}{\partial {\widehat{y}}^2}+\frac{{\partial }^2\widehat{u}}{\partial {\widehat{x}}^2}\right)+\hfill \\ {\rho }_{hnf}{\beta }_{hnf}g(T-T_{\infty })\sin \left(\frac{\widehat{x}}{a}\right)-{\sigma }_{hnf}{B}_0^2\widehat{u},\hfill \end{array}$$

(2)

$$\begin{array}{l}{\rho }_{hnf}\left(\widehat{u}\frac{\partial \widehat{u}}{\partial \widehat{x}}+\widehat{v}\frac{\partial \widehat{u}}{\partial \widehat{y}}\right)=-\frac{\partial \widehat{p}}{\partial \widehat{y}}+{\mu }_{hnf}\left(\frac{{\partial }^2\widehat{u}}{\partial {\widehat{y}}^2}+\frac{{\partial }^2\widehat{u}}{\partial {\widehat{x}}^2}\right)+\hfill \\ {\rho }_{hnf}{\beta }_{hnf}g(T-T_{\infty })\cos \left(\frac{\widehat{x}}{a}\right)-{\sigma }_{hnf}{B}_0^2\widehat{v},\hfill \end{array}$$

(3)

$$\widehat{u}\frac{\partial T}{\partial \widehat{x}}+\widehat{v}\frac{\partial T}{\partial \widehat{y}}={\alpha }_{hnf}\left(\frac{{\partial }^{2}T}{\partial {\widehat{y}}^2}+\frac{{\partial }^{2}T}{\partial {\widehat{x}}^2}\right),$$

(4)

subject to:

$$\begin{gathered} \widehat{u}=\widehat{v}=0,T=T_{w}at\widehat{y}=0, \\ \widehat{u}\to 0,T\to T_{w}at\widehat{y}\to \infty , \end{gathered}$$

(5)

The non-dimensional variables are expressed as:

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

(6)

where $\mathrm{Gr}=g{\beta }_f(T_w-T_{\infty })\frac{a^3}{{\nu }_f^2}$ is the Grashof number.

Table 1. Thermo-physical properties. Data from [53].

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

displays the thermophysical properties of both the mono and hybrid nanofluids.

Using the non-dimensional variable in Equation (6), the properties listed in Table 1, and the boundary layer approximations (as Gr $\to \infty$), we have (𝜕p/𝜕y) = 0 and (𝜕p/𝜕y) = 0 [52]. Consequently, we ignore the associated terms in the previous equations as follows:

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

(7)

$$\begin{array}{l}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\rho }_f}{{\rho }_{hnf}}\left(\frac{1}{{(1-{\chi }_1)}^{2.5}{(1-{\chi }_2)}^{2.5}}\right)\frac{{\partial }^{2}u}{\partial y^2}\hfill \\ +\frac{1}{{\rho }_{hnf}}\left(\left(1-{\chi }_2\right)[\left(1-{\chi }_1\right){\rho }_f+{\chi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\chi }_2\frac{{\rho }_{s2}{\beta }_{s2}}{{\beta }_f}\right)\theta \sin x-\frac{{\rho }_f}{{\rho }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}\mathrm{M}u,\hfill \end{array}$$

(8)

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

(9)

where $\mathrm{M}=\left(\frac{{\sigma }_f{B}_0^2{a}^2{\mathrm{Gr}}^{-1/2}}{{\rho }_f{v}_f}\right)$ is the magnetic parameter, $\mathrm{Pr}=v_f/{\alpha }_f$ is the Prandtl number, and the boundary conditions are:

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

(10)

The reduction process of Equations (7)–(10) requires using the following transformation [51]:

$$\psi =xf(x.y),\theta =\theta (x,y),$$

(11)

where $\psi$ is the stream function given by:

$$u=\frac{\partial \psi }{\partial y}\mathrm{and}v=-\frac{\partial \psi }{\partial x},$$

(12)

Consequently, we obtain the following reductive equations:

$$\begin{gathered} \frac{{\rho }_f}{{\rho }_{hnf}}\left(\frac{1}{{(1-{\chi }_1)}^{2.5}{(1-{\chi }_2)}^{2.5}}\right)\frac{{\partial }^{3}f}{\partial y^3}+f\frac{{\partial }^{2}f}{\partial y^2}-{\left(\frac{\partial f}{\partial y}\right)}^2-\frac{{\rho }_f}{{\rho }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}\mathrm{M}\frac{\partial f}{\partial y} \\ \begin{array}{c}\hfill +\frac{1}{{\rho }_{hnf}}\left(\left(1-{\chi }_2\right)[\left(1-{\chi }_1\right){\rho }_f+{\chi }_1\frac{{\rho }_{s1}{\beta }_{s1}}{{\beta }_f}]+{\chi }_2\frac{{\rho }_{s2}{\beta }_{s2}}{{\beta }_f}.\right)\frac{\sin x}{x}\theta \hfill \\ \hfill =x\left(\frac{\partial f}{\partial y}\frac{{\partial }^{2}f}{\partial x\partial y}-\frac{\partial f}{\partial x}\frac{{\partial }^{2}f}{\partial y^2}\right),\hfill \end{array} \end{gathered}$$

(13)

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

(14)

subject to:

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

(15)

As x approaches zero (known as the stagnation point), Equations (13)–(15) transform to the following ODEs:

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

(16)

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

(17)

and the boundary conditions are:

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

(18)

where the primes refer to differentiation with respect to y.

To examine and analyze the flow characteristics, two physical quantities are highlighted: the local skin friction coefficient C_f and the local Nusselt number Nu, which were described by Alwawi et al. [50] as follows:

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

(19)

where

$${\tau }_w={\mu }_{hnf}{\left(\frac{\partial \widehat{u}}{\partial \widehat{y}}\right)}_{\widehat{y}=0},q_w=-k_{hnf}{\left(\frac{\partial T}{\partial \widehat{y}}\right)}_{\widehat{y}=0}$$

(20)

By applying the non-dimensional variables in Equations (6) and (15), C_f and Nu transform to:

$$C_f={\mathrm{Gr}}^{-1/4}\frac{1}{{(1-{\chi }_1)}^{2.5}{(1-{\chi }_2)}^{2.5}}x\frac{{\partial }^{2}f}{\partial y^2}(x,0),\mathrm{Nu}=-{\mathrm{Gr}}^{1/4}\frac{k_{hnf}}{k_f}\frac{\partial \theta }{\partial y}(x,0),$$

(21)

3. Numerical Approach

The model that governs our context is treated numerically based on the Keller box method (KBM). Generally, the PDEs of the boundary layer flow problems of a hybrid fluid are solved numerically using this method. The Keller box is an implicit finite difference scheme method used for finding precise numerical results for the PDFs, which was constructed by Keller and Bramble [54]. Recently, it was used by Swalmeh et al. [55] and Hamarsheh et al. [19] to find the numerical results of the influences of convection boundary layer flow in a micropolar nanofluid.

Briefly, KBM has four steps: (1) converting the governing equations of nonlinear PDEs into a first-order system; (2) using the central differences method to obtain the finite difference equations; (3) linearizing these finite difference equations via Newton’s method, and then systemizing them to matrix-vector form; (4) applying the block tri-diagonal elimination method to solve the matrix-vector linear system. The numerical results of the physical quantities for the obtained linear system were tabulated and plotted using MATLAB version 7.0.0 (MathWorks, Natick, MA, USA).

The values of the parameters that were considered were $\chi =0.1,0.15,0.2$ and M > 0. The thermo-physical properties of the nanoparticles and the host fluid used in the mathematical calculation are reported in

Table 2. Different values of thermo-physical properties of nanoparticles and host fluid. Data from [56,57,58].

Physical Properties	Host Fluid		Nanoparticles
	H2O + EG 50%	Water	Co (Cobalt)	Al	Al2O3
k (W/mK)	0.425	0.613	100	237	40
ρ (kg/m3)	1056	997.1	8900	2701	3970
ρcp (J/kgK)	3288	4179	420	902	765
Β × 10−5 (K−1)	0.00341	21	1.3 × 10−5	2.31	0.85
σS m−1	0.00509	5.5 × 10−6	1.602 × 107	3.5 × 107	1 × 10−10
Pr	29.86	6.2	-	-	-

.

The verification of the accuracy of the current numerical results for C_f and Nu using the prior results published by Merkin [59] and Alwawi et al. [50] is demonstrated in

Table 3. Comparison of C_f with previous numerical results (${\chi }_1={\chi }_2=0,\mathrm{and}\mathrm{M}=0$) at Pr = 1.

x	Merkin [59]	Alwawi et al. [50]	Current Study
0	0.0000	0.0000	0.0000
π/6	0.4151	0.4158	0.4148
π/3	0.7558	0.7538	0.7544
π/2	0.9579	0.9563	0.9573
2π/3	0.9756	0.9735	0.9751
5π/6	0.7822	0.7480	0.7469
π	0.3391	0.3311	0.3330

and

Table 4. Comparison of Nu with previous numerical results (${\chi }_1={\chi }_2=0,\mathrm{and}\mathrm{M}=0$) at Pr = 1.

x	Merkin [59]	Alwawi et al. [50]	Current Study
0	0.4214	0.4214	0.4219
π/6	0.4161	0.4162	0.4166
π/3	0.4007	0.4008	0.4010
π/2	0.3745	0.3744	0.3744
2π/3	0.3364	0.3360	0.3355
5π/6	0.2825	0.2754	0.2714
π	0.1945	0.1926	0.1939

.

4. Results and Discussion

Figure 2 and Figure 3 illustrate the impact of growth magnetic parameter M on the Nusselt number and the skin friction of water and water-EG hybrid nanofluids at fixed nanoparticles volume fractions (${\chi }_1=0.1,{\chi }_2=0.05$). This magnetic current suppresses both the rate of energy transmission and the skin friction of both fluids, which occurs because the passage of a magnetic current across a flowing fluid formulates a type of force that curbs the energy of the particles of the fluid. The figures show the basic role of cobalt and aluminum in supporting the heat transfer of aluminum oxide, produced due to the efficiency of aluminum in raising the rate of heat transfer.

Figure 4 and Figure 5 show the opposite effect of the imposed magnetic field on the fluid velocity and the direct effect on the temperature, which were expected due to the creation of the Lorentz force, which works to slow the movement of the fluid and thus reduce its velocity, resulting in an increase in its temperature. Interestingly, the hybrid nanofluid, composed of water-ethylene glycol and Co-Al₂O₃ had a lower temperature compared with the hybrid nanofluid consisting of water and Co-Al₂O₃ with an increased magnetic parameter, which means less energy dissipation and increased stability. However, when we compared the hybrid nanoparticles fabricated from the synthesis of Co and Al₂O₃ with the hybrid nanoparticles formed from the synthesis of Al and Al₂O₃, the latter was found to be better as it showed the lowest temperature.

Figure 6 and Figure 7 highlight the behavior of the Nusselt number and skin friction with increasing volume fraction coefficient of cobalt (Co) or aluminum (Al); the volume fraction coefficient of aluminum oxide (${\chi }_1=0.1$) the and magnetic parameter M = 0.1 remained fixed. We found that the increase in the volume fraction coefficient promoted the rate of heat transfer and skin friction. The increase in ${\chi }_2$, which helps promote the density and thermal conductivity of the mono nanoliquid, resulted in increases in both the skin friction coefficient and the Nusselt number. Notably, the combination of Al-Al₂O₃/water-EG was more efficient in terms of energy transfer rate, and its skin friction coefficient was the highest.

The effect of the growth in the nanoparticle volume fraction (${\chi }_2$) for cobalt and aluminum on the velocity and temperature profiles with constant aluminum oxide (${\chi }_1=0.1$) and magnetic parameter (M = 0.1) is demonstrated in Figure 8 and Figure 9. As expected, the growth in the nanoparticles volume fraction, whether for cobalt or aluminum, contributed significantly to the acceleration of heat transfer from outside the surface of the cylinder to the mono nanoliquid, which aided in the expansion of the thermal boundary layers. As a result, the temperature of the mono nanoliquid rose. Conversely, the increase in nanoparticles volume fraction, whether for cobalt or aluminum, inhibited mono nanoliquid velocity.

5. Conclusions

In this study, we dealt with a hybrid nanofluid promoting heat transfer using water and ethylene glycol-water flowing around a horizontal cylinder, which are widely used in heat transfer processes, especially in cooling. A variable orthogonal magnetic field was considered and we compared the effects of the two fluids. The key findings are:

• The heat transmission rate of the hybrid-nanofluid is higher compared with that of the mono nanofluid.
• The heat transmission rate of ethylene glycol-water is higher than that of water, regardless of the incorporated hybrid nanoparticles.
• The magnetic parameter tends to raise the temperature profile, whereas it has a reverse effect on skin friction, velocity, and Nusselt number.
• Aluminum and cobalt nanoparticles are essential to support the thermal properties of aluminum oxide, which promote energy transfer.
• C_f, Nu, and the temperature of Al₂O₃/ethylene glycol-water nanofluid increase and by the addition of nanoparticles (Co and Al); they consequently formed a hybrid nanofluid with different values of M and ${\chi }_2$.
• MHD hybrid nanofluid flow is clearly affected by the nanoparticle volume fraction (${\chi }_2$) and magnetic (M) parameters. So, the findings of this study provide additional information to the field of fluid mechanics.