A Comparative Study of Williamson Hybrid Nanofluid Flow Consisting of Cu, GaN, and Al₂O₃ Nanoparticles in Ethylene Glycol over a Stretching Sheet with Suction/Injection and Heat Source/Sink

Mamidala Jyotshna^1*, Vadlakonda Dhanalaxmi²

¹Department of Mathematics, Maturi Venkata Subba Rao Engineering College (Affiliated to Osmania University), Hyderabad, India.
²Department of Mathematics, University College of Technology, Osmania University, Hyderabad, India.

DOI: 10.4236/jamp.2022.1012256                

Abstract

Several new techniques in the field of heat transfer in fluids have opened new avenues for studying the heat transfer effects in nanofluids and thermodynamic flow parameters, leading to novel applications. There have been studies on nanofluids, including metal, ceramic and magnetic nanoparticles mixed with base fluids such as Water, Kerosene, and Ethylene glycol. However, research on fluids employing semiconductor nanoparticles as supplements to base fluids to generate nanofluids and hybrid nanofluids is limited. For the investigation, Gallium nitrite, a binary semiconductor with excellent heat convection, is together with Cu metal nanoparticles and Al₂O₃ ceramic nanoparticles separately in the base fluid Ethylene glycol (EG) to form hybrid nanofluids. The effects of convective boundary conditions, thermal radiation, heat source/sink, suction/injection, and activation energy on three-dimensional Williamson MHD hybrid nanofluid flow of Cu + GaN + EG, Al₂O₃ + GaN + EG, and Cu + Al₂O₃ + EG are investigated on a stretched sheet with porosity. A similarity transformation is performed on the governing equations to transform them into dimensionless ordinary differential equations ODEs. Numerical analysis is carried out in MATLAB utilizing bvp5c and the shooting technique. The variations of velocity, temperature, and concentration profiles as a function of different physical effects are presented graphically with dimensionless parameters and explained the variations scientifically. As varied with different parameters, the values of the Skin-friction coefficient, Nusselt number, and Sherwood number are mentioned in the table.

Keywords

Williamson Hybrid Nanofluid, Gallium Nitride, Heat Transfer, Heat Source/Sink, Suction/Injection, Solid Volume Fraction

1. Introduction

For the last two decades, scientists have investigated fluid flow and heat transfer processes aiming at higher heat transfer efficiency. Therefore, the focus has been on improving the heat exchange process by adopting new heat transfer methods and fluids with different additives. In the process, the attention shifted toward Power law, Casson, Williamson, and Maxwell types of fluids. Subsequently, the fluids prepared by dispersing nanoparticles of size less than 100 nm, known as nanofluids, are recognized as better fluids than base fluids like Water, Kerosene, Ethylene glycol, etc. In modern technology, the performance of nanofluids paved the way for many applications, such as cooling processes, lubrication in automobiles, food processing, micro-electronic cooling refrigeration, and power generation. Sakiadis [1] and Crane [2] were the first to introduce the fluid flow past a stretching surface. Masuda et al. [3], for the first time, studied on change in thermal conductivity and viscosity of liquid by dispersing ultrafine particles. Choi [4] created the term “nanofluids”, used to improve the heat transfer mechanism. Williamson fluid is the only one of these nanofluids to have a reduced viscosity at high shear stress rates, demonstrating that its effective density should decrease independently as the shear rate rises. These fluids have several industrial uses, including lubrication, spinning machinery, and viscometry. Because of these unique features, nanofluids are used in multiple industrial areas, such as coolants in nuclear reactors, heat transfer agents, thermal detectors, solar thermal collectors, radiators, and microelectronic devices. Hayat [5] discussed different physical parameters (profiles) variations through graphical representation. Malik et al. [6] used the Williamson fluid model to calculate the numerical solution for the flow of a fluid with variable thermal conductivity in variable mode and heat transfer effect along a stretched cylinder. Kebede [7] described the heat and mass transport properties of Williamson nanofluid flow. Later, several researchers and scientists built on this concept to produce astounding achievements in flow and heat transfer. [8] [9] [10] [11]. Kuttan et al. [12] investigated the influence of four nanoparticles on the flow of a boundary layer via a stretched surface. Vinita and Poply [13] and Mondal et al. [14] have studied the effect of heat production on nanofluid flow via a stretched cylinder.

All the investigations on nanofluids failed in identifying suitable nanofluid(s) with an elevated heat transfer rate necessary for bulk production companies. To address this deficiency, hybrid nanofluids are developed by combining more than one variety of nanoparticles in a base fluid [15]. The thermal performance of hybrid nanofluids is characterized by various factors like thermal conduction, thickness and size of nanomaterials, heat and concentration of the molecules, and many more. It was also experimentally verified that hybrid nanofluids give better thermal flow in base fluids. These hybrid nanofluids have better thermal properties compared to base fluids and nanofluids. Recently, Khan et al. [16] considered a fluid flow consisting of silicon dioxide and molybdenum disulfide to know the effect of heat source or sink on thermal properties. Hossiany and Eid [17] investigated propylene glycol-water-based fluid’s heat transfer mechanism and fluid flow with hybrid nanoparticle suspension [15]. Muhammad Ramzan et al. [18] examined the flow of nanofluids, including EG and NiZnFe₂O₄ nanoparticles, across a curved surface and found that the Schmidt and Prandtl numbers decreased the fluid concentration.

Aluminum oxide (Al₂O₃) nanoparticles have received much interest due to their considerable increase in burning rate, coolant, ignition time, and temperature. In addition, Al₂O₃ has a unique far-infrared emission, and its nanoparticles have remarkable uses in aerospace due to high energy release during the oxidation process and heat shielding coatings in aircraft, among other things. Furthermore, Al₂O₃ nanoparticles are used in liquid form to increase ceramic density, fracture toughness, creep resistance, wear resistance, smoothness, and other properties in polymers, ceramics, rubber, and refractory goods. Sriharan et al. [19] investigated the heat transfer performance of metal oxide base nanofluids experimentally and discovered that Al₂O₃-deionized water nanofluids had a higher heat transfer coefficient. Umair Rashid and Adnan Ibrahim [20] have studied the impacts of nanoparticle shape on Al₂O₃-water nanofluid flows and report that the heat transfer augmentation in laminar shapes nanoparticles is more than in other shapes of nanoparticles.

Copper, in its nanomaterial form, has many uses in mechanical and thermal aspects. The exceptional physical properties of copper are due to its excellent heat and electrical conductivities, good machinability, non-magnetic, practically insoluble in water, and retention of mechanical and electrical properties at cryogenic temperatures. Among the other metals, copper has almost the second highest value of specific heat capacity. Gupta et al. [21] have enumerated magnetohydrodynamic three-dimensional boundary layer flow and heat transfer of water-driven copper and alumina nanoparticles induced by convective conditions. Later, Eastman et al. [22] showed that copper nanoparticles are more efficient than their oxide nanoparticles immersed with base fluids to enhance heat transfer. Elcio Nogueira [23] has sought an analytical solution for obtaining the outlet temperature of the hot and cold fluids in shell and tube heat exchangers consisting of Ethylene glycol and water as base fluids and CuO nanoparticles. He has observed the flow laminarization effect.

GaN, a III - V compound, is a very hard, mechanically stable wide band gap semiconductor material with high heat capacity and thermal conductivity. GaN semiconductor is recognized as one of the most promising materials for fabricating optical devices in the visible short wavelength and UV region. It has a high melting temperature of >1600˚C and a density as high as 6100 kg/m³. The very high breakdown voltages, electron mobility, and saturation velocity of GaN have also made it an ideal candidate for high-power and high-temperature microwave applications. They are also utilized in military electronic radars [24]. Due to their exceptional chemical, physical, and mechanical capabilities arising from size impact, GaN semiconductor nanoparticles are widely employed in the fabrication of solar cells, photo sensors, and significant electronic industries [25] [26] [27] of its suitable electronic applications, good thermal conductivity, and high heat capacity, one can predict its good prominence and participation in studying heat flow transfer in fluids.

Researchers have examined the flow of nanofluids containing magnetic (Fe₂O₃, ZnFe₂O₃, CoFe₂O₃), nonmetallic (MgO, TiO₂, Al₂O₃), and metallic (Cu, Ag, Au) nanoparticles under a variety of conditions, including stretching sheets, inclined magnetic fields, nonlinearly stretched porous sheets, thermal flux, etc. [28] - [33]. Likewise, very few papers have been published on ZnO [34] [35] and CdTe [36] semiconductor nanofluids, but the studies on hybrid nanofluids consisting of semiconductor compound nanoparticles (SCNPs) as one of the constituents in nanofluids are not found in the literature. Therefore, it is perceived that the results of heat transfer of hybrid nanofluids consisting of SCNPs under different external forces and boundary conditions are exciting and may acquire industrial applications. The envisaged industries are those producing hydraulic brake fluids and pharmaceutical vehicles. Hence, in the present analysis, attempts at the study of Williamson hybrid nanofluid, consisting of Cu + GaN, Al₂O₃ + GaN nanoparticles in the base fluid Ethylene glycol, running over a linearly stretched porous sheet under the influence of the magnetic field, heat source/sink, suction/injection, radiation, and Arrhenius activation energy. The results are compared with those of hybrid nanofluids consisting of Cu + Al₂O₃ nanoparticles in order to understand the changes in temperature, velocity, and concentration profiles caused with the inclusion of GaN semiconductor nanoparticles as a part of hybrid nanofluids.

2. Geometrical Model of the Problem

A steady, incompressible three-dimensional boundary layer Williamson hybrid nanofluid with convective boundary condition is assumed to run over a stretched porous sheet in the xy-plane with the velocity components u, v, and w along the x, y, and z directions, respectively. The surface is assumed to be stretching at a rate proportionate to its distance from the origin along the x and y directions, as shown in Figure 1. A uniform magnetic field of strength B_o is applied along the positive z-direction.

3. Mathematical Model

1) The governing equations:

To construct a mathematical model for the flow of a viscous nanofluid across

Figure 1. Geometrical diagram representing the problem.

a stretched sheet in three-dimension, with an incompressible steady-state laminar boundary layer flow, the conditions assumed are.

• The sheet is linearly stretched along the x- and y-axes with the velocities $u_w\left(x\right)=ax$ and $v_w\left(y\right)=by$, where a and b are stretching constants which are displayed in Figure 1, representing the problem.

• The ambient fluid’s temperature and concentration were indicated, respectively, by the letters T_∞ and C_∞ (at its free surface).

• Similarly, the uniform surface temperature (T_w) and concentration (C_w) of the sheet corresponding to the plane z = 0 are represented by T_w and C_w respectively. Here, the z-axis is normal to the flow which is constrained to z > 0.

• Cu, Al₂O₃, and GaN nanoparticles of͘ a homogeneous size and shape are combined with ethylene glycol (EG) as the base fluid to create the Williamson nanofluid flow.

• The stretched surface is subjected transversely, or along the z-direction, to the magnetic field of uniform field strength B_o. Since the fluid’s Reynolds number is regarded as low, the generated magnetic field in the liquid is insignificant.

• The linear stretched sheet has a small pressure gradient which can be neglected ( $\partial p/\partial x=0$ ).

When suction/injection, heat source or sink, thermal radiation, and activation energy are present, the Williamson hybrid nanofluid’s heat transport mechanism is examined using convection boundary conditions on a linearly stretched permeable sheet applied with a uniform magnetic field. The thermophysical characteristics of͘ the nanoparticles and base fluids employed in the current problem are displayed in Table 1. Table 2 provides the mathematical equations for several nanofluid physical parameters.

The model of Williamson fluid is taken [40] [41] [42] as

$\stackrel{\dot{}}{S}=-\stackrel{\dot{}}{p}I+\tau$, (1)

$\tau =\left[{\mu }_{\infty }+\frac{{\mu }_0-{\mu }_{\infty }}{1-\Gamma \stackrel{\dot{}}{\gamma }}\right]A_1$, (2)

Table 1. Thermo-physical properties of the fluid and nanoparticles.

Table 2. Mathematical equations of thermophysical properties of nanofluids and hybrid nanofluids [28] [37].

where $\tau$ —extra stress tensor; $\stackrel{\dot{}}{S}$ —Cauchy stress tensor; I—identity vector; $\stackrel{\dot{}}{p}$ —Pressure; ${\mu }_0$ —limiting viscosity at zero shear rate; $A_1$ —first Rivlin-Erickson tensor; ${\mu }_{\infty }$ —limiting viscosity at infinity shear rate; $\Gamma >0$ —time constant;

$\stackrel{\dot{}}{\gamma }=\sqrt{\frac{1}{2}trace\left(A_1^2\right)}$, (3)

Here in this case, they are all restrained by ${\mu }_{\infty }=0$ and $\Gamma \stackrel{\dot{}}{\gamma }<1$.

Thus, $\tau$ equal to

$\tau =\left[\frac{{\mu }_0}{1-\Gamma \stackrel{\dot{}}{\gamma }}\right]A_1$, (4)

Using binomial expansion to Equation (4), given as

$\tau ={\mu }_0\left[1+\Gamma \stackrel{\dot{}}{\gamma }\right]A_1.$ (5)

The governing equations [42] [43] [44] [45] are as follows

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$ (6)

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial z^2}+\sqrt{2}\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\Gamma \frac{\partial u}{\partial z}\frac{{\partial }^{2}u}{\partial z^2}-\frac{\sigma B_o^2}{{\rho }_{hnf}}u-\frac{{\mu }_{hnf}}{{\rho }_{hnf}{K}_p}u$ (7)

$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}v}{\partial z^2}+\sqrt{2}\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\Gamma \frac{\partial v}{\partial z}\frac{{\partial }^{2}v}{\partial z^2}-\frac{\sigma B_o^2}{{\rho }_{hnf}}v-\frac{{\mu }_{hnf}}{{\rho }_{hnf}{K}_p}v$ (8)

$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}={\alpha }_{hnf}\frac{{\partial }^{2}T}{\partial z^2}-\frac{1}{{\left(\rho c_p\right)}_{hn{f}_{}}}\frac{\partial q_r}{\partial z}+\frac{Q_o\left(T-T_{\infty }\right)}{{\left(\rho c_p\right)}_{hn{f}_{}}}$ (9)

$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}=D_B\frac{{\partial }^{2}C}{\partial z^2}-K_o^2\left(c-c_{\infty }\right){\left(\frac{T}{T_{\infty }}\right)}^m\exp \left(\frac{-E_a}{kT}\right)$ (10)

By Rosseland approximation, the radiative heat flux $\frac{\partial q_r}{\partial z}$ [46] is given by $\frac{\partial q_r}{\partial z}=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\frac{{\partial }^{2}T}{\partial z^2}$.

2) Boundary conditions

According to Geethan Kumar [42], the boundary conditions for the velocity, temperature, and concentration fields are:

$\begin{array}{l}z=0:u=u_w=ax,v=v_w=by,w=-W_w,-K\frac{\partial T}{\partial z}=h_f\left(T_w-T\right),C\to C_w\\ z\to \infty :u\to 0,v\to 0,T\to T_{\infty },C\to C_{\infty }\end{array}$ (11)

where K—the thermal conductivity and $h_f$ —the heat transfer coefficient.

3) Similarity equations

The similarity transformation equations are

$\begin{array}{l}u=ax{f}^{\prime }\left(\eta \right),v=ay{g}^{\prime }\left(\eta \right),w=-\sqrt{a{\nu }_f}\left(f\left(\eta \right)+g\left(\eta \right)\right)\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\eta =z\sqrt{\frac{a}{{\nu }_f}},\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }}\end{array}$ (12)

After using the above transformations, the continuity Equation (6) is identically satisfied, whereas the Equations (7)-(10) are transformed into 2^nd and 3^rd-order ODE’s given below.

$\left(1+L_1{L}_2{\lambda }_1{f}^{″}\right)f^{‴}=L_1{L}_2\left({f^{\prime }}^2-\left(f+g\right)f^{″}\right)+M{f}^{\prime }{L}_2+K{f}^{\prime }$ (13)

$\left(1+L_1{L}_2{\lambda }_2{g}^{″}\right)g^{‴}=L_1{L}_2\left({g^{\prime }}^2-\left(f+g\right)g^{″}\right)+M{g}^{\prime }{L}_2+K{g}^{\prime }$ (14)

$\left(\frac{k_{hnf}}{k_f}+\frac{4}{3R}\right){\theta }^{″}=-Pr\left(\left(f+g\right){\theta }^{\prime }{L}_3+Q\theta \right)$ (15)

$L_2{\varphi }^{″}=-\left(Sc\right)\left(\sigma \varphi {\left(\theta \delta +1\right)}^m\exp \left(\frac{-E}{1+\delta \theta }\right)-{\varphi }^{\prime }\left(f+g\right)\right)$ (16)

Here, $f,g,\varphi ,\theta$ are the functions of $\eta$,

$L_1=\left(\left(1-{\Phi }_2\right)\left(1-{\Phi }_1+{\Phi }_1\frac{{\rho }_1}{{\rho }_f}\right)+{\Phi }_2\frac{{\rho }_2}{{\rho }_f}\right)$,

$L_2={\left(1-{\Phi }_1\right)}^{2.5}{\left(1-{\Phi }_2\right)}^{2.5}$ _,

$L_3=\left(\left(1-\Phi {\varphi }_2\right)\left(1-{\Phi }_1+{\Phi }_1\frac{{\left(\rho C_p\right)}_1}{{\left(\rho C_p\right)}_f}\right)+{\Phi }_2\frac{{\left(\rho C_p\right)}_2}{{\left(\rho C_p\right)}_f}\right)$.

The transformed boundary conditions given as

$\begin{array}{l}\text{at}\eta =0:f^{\prime }\left(\eta \right)=1,f\left(\eta \right)=S-g\left(\eta \right),g\left(\eta \right)=0,g^{\prime }\left(\eta \right)=\alpha ,\\ {\theta }^{\prime }\left(\eta \right)=-Bi\left(1-\theta \left(\eta \right)\right),\varphi \left(\eta \right)=1\\ \text{at}\eta \to \infty :f^{\prime }\left(\eta \right)\to 0,g^{\prime }\left(\eta \right)\to 0,\theta \left(\eta \right)\to 0,\varphi \left(\eta \right)\to 0\end{array}$ (17)

Here “prime” stands for the coefficient of differentiation w.r.t $\eta$.

The dimensionless parameters are listed as

$\begin{array}{l}{\lambda }_1=\Gamma x\sqrt{\frac{a^3}{2\nu }},{\lambda }_2=\Gamma y\sqrt{\frac{a^3}{2\nu }},M=\frac{\sigma B_o^2}{{\rho }_{f}a},P_r=\frac{{\left(\rho c_p\right)}_f{\nu }_f}{K_f},\\ S_c=\frac{{\nu }_f}{D_B},R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k^{*}{K}_f},\alpha =\frac{b}{a},\delta =\frac{T_w-T_{\infty }}{T_{\infty }},\sigma =\frac{K_o^2}{a},\\ E=\frac{E_a}{k{T}_{\infty }},Q=\frac{Q_o}{a{\left(\rho c_p\right)}_f},S=\frac{W_w}{\sqrt{a{\nu }_f}}\end{array}$ (18)

The heat transfer rates in this study, which are represented by the Skin-friction coefficient $C_f$, Nusselt number Nu, and Sherwood’s number $S_h$, are defined as (see, for example, [18] [44]) are defined as

$C_{fx}=\frac{{\tau }_{xz}}{{\rho }_f{v}_w^2},C_{fy}=\frac{{\tau }_{yz}}{{\rho }_f{v}_w^2},Nu=\frac{-x{q}_w}{T_w-T_{\infty }}$ and $S_h=\frac{-x{J}_w}{C_w-C_{\infty }}$ (19)

Here, shear stress at the surfaces is given as ${\tau }_{xz}={\mu }_{hnf}{\left[\frac{\partial u}{\partial z}+\frac{\Gamma }{\sqrt{2}}{\left(\frac{\partial u}{\partial z}\right)}^2\right]}_{z=0}$, ${\tau }_{yz}={\mu }_{hnf}{\left[\frac{\partial v}{\partial z}+\frac{\Gamma }{\sqrt{2}}{\left(\frac{\partial v}{\partial z}\right)}^2\right]}_{z=0}$, $q_w=-\left[k_{hnf}+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\right]{\left(\frac{\partial T}{\partial z}\right)}_{z=0}$ the wall heat, and

$J_w=-D_{hnf}{\left(\frac{\partial C}{\partial z}\right)}_{z=0}$ (20)

the mass transfers.

By using Equations (19) and (20), the dimensionless “Skin-friction coefficient, Nusselt number, and Sherwood number” [47] [48] [49] [50] are simplified as

$\begin{array}{l}2{\left(R{e}_x\right)}^{1/2}{C}_{fx}=\frac{1}{L_2}\left[f^{″}\left(0\right)+\frac{\lambda }{2}{\left(f^{″}\left(0\right)\right)}^2\right]\\ {\alpha }^{3/2}2{\left(R{e}_x\right)}^{1/2}{C}_{fy}=\frac{1}{L_2}\left[g^{″}\left(0\right)+\frac{\lambda }{2}{\left(g^{″}\left(0\right)\right)}^2\right]\\ {\left(R{e}_x\right)}^{-1/2}N{u}_x=-\left[\frac{k_{hnf}}{k_f}+\frac{4}{3}R\right]{\theta }^{\prime }\left(0\right)\\ {\left(R{e}_x\right)}^{-1/2}Sh=-\frac{D_{hnf}}{D_f}{\varphi }^{\prime }\left(0\right)\end{array}\}$ (21)

$R{e}_x=\frac{u_{w}x}{{\nu }_f},R{e}_y=\frac{v_{w}y}{{\nu }_f}$ are Reynold’s numbers.

4. Numerical Method and Validation

To validate the obtained results, the MATLAB built-in bvp5c program is employed. The RK-45 order approach is used to direct the shooting technique toward the boundary conditions while accounting for the lacking initial conditions. The numerical bvp5c was generated using the concept of a finite-difference method, and the results were compared in Table 3.

${f^{\prime }}_1=f_2,{f^{\prime }}_2=f_3\text{and}{f^{\prime }}_3=\frac{L_1{L}_2\left(f_2^2-\left(f_1+f_4\right)f_3\right)+M{f}_2{L}_2+K{f}_2}{1+L_1{L}_2{\lambda }_1{f}_3}$ (22)

${f^{\prime }}_4=f_5,{f^{\prime }}_5=f_6\text{and}{f^{\prime }}_6=\frac{L_1{L}_2\left(f_5^2-\left(f_1+f_4\right)f_6\right)+M{f}_5{L}_2+K{f}_5}{1+L_1{L}_2{\lambda }_2{f}_6}$ (23)

${f^{\prime }}_7=f_8,{f^{\prime }}_8=-\frac{Pr\left(\left(f_1+f_4\right)f_8{L}_3+Q{f}_7\right)}{\frac{k_f}{k_{hnf}}+\frac{4}{3R}}$ (24)

${f^{\prime }}_9=f_{10},{f^{\prime }}_{10}=-\frac{Sc\left(\sigma f_9{\left(f_7\delta +1\right)}^m\exp \left(\frac{-E}{1+\delta f_7}\right)-\left(f_1+f_4\right)f_{10}\right)}{L_2}$ (25)

Table 3. Attestation of Skin friction ( $C_{f}R{e}_x^{1/2}$ ) results for various values of the magnetic field fixed values K = λ₁ = λ₂ = Φ = Pr = R = Q = Sc = E = σ = δ = α = m = S = 0.

The converted initial conditions are

$f_2\left(0\right)=1,f_1\left(0\right)=S-f_4\left(0\right),f_4\left(0\right)=0,f_5=\alpha ,f_8=-Bi\left(1-f_7\right),f_9=1$

$f_2\to 0,f_5\to 0,f_7\to 0,f_9\to 0$ (26)

Analyzing the governing Equations (7)-(10) with boundary conditions (11) is time-consuming. Instead, we used the bvp5c technique to solve the equations numerically in MATLAB. The computations begin with preset values of the physical parameters λ₁ = λ₂ = 0.2, σ = E = δ = 1.0, S = 0.2, Sc = 0.6, Pr = 6.2, R = K = M = Bi = α = 0.5. Unless otherwise specified, the same base values can be assumed throughout the investigation. Finally, we cross-checked our results with the solutions of [46] [51] (see Table 3) and found extremely strong agreement in the results. Tolerances on the order of 10⁻⁵ are used in the calculations, and all findings are correct within the stipulated tolerances.

5. Results and Discussion

The flow parameters shown in Figures 2-30 are estimates of species’ momentum, energy, and mass [27].

The Numerical results of skin friction drag along x and y directions ( $-C_{f}R{e}_x^{1/2}$ and $-{\alpha }^{3/2}{C}_{f}R{e}_y^{1/2}$ ), Nusselt ( $R{e}_x^{-1/2}Nu$ ), and Sherwood ( $R{e}_x^{-1/2}Sh$ ) numbers, the essential emerging parameters, are estimated using different dimensionless parameters for Cu + GaN + EG, Al₂O₃ + GaN + EG, and Cu + Al₂O₃ + EG hybrid nanofluids. Their numerical values are shown in Tables 4-6. Skin friction drags in the x and y-direction increase as M, K,α, Φ₂, and S increases but decreases as λ₁ andλ₂ increse for Cu + GaN + EG, Al₂O₃ + GaN + EG, Cu + Al₂O₃ + EG. The Nusselt numbers decrease as M, K, λ₁ andλ₂, values increase, whereas they increase with the increase in the values of α, Φ₂, and S for Cu + GaN + EG, Al₂O₃ + GaN + EG, Cu + Al₂O₃ + EG. Sherwood numbers rise with the increase in α and S and decrease with the increase in M, K, λ₁ andλ₂.

Figure 2. Variation of λ₁ with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 3. Variation of λ₂ with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 4. Variation of λ₁ with $\theta \left(\eta \right)$.

Figure 5. Variation of λ₂ with $\theta \left(\eta \right)$.

Figure 6. Variation of λ₁ with $\varphi \left(\eta \right)$.

Figure 7. Variation of λ₂ with $\varphi \left(\eta \right)$.

Figure 8. Variation of M with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 9. Variation of M with $\theta \left(\eta \right)$.

Figure 10. Variation of M on $\varphi \left(\eta \right)$.

Figure 11. Variation of S on $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 12. Variationof S on $\theta \left(\eta \right)$.

Figure 13. Variation of S on $\varphi \left(\eta \right)$.

Figure 14. Variation of Q with $\theta \left(\eta \right)$.

Figure 15. Variation of Q with $\varphi \left(\eta \right)$.

Figure 16. Variation of K with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 17. Variation of K with $\theta \left(\eta \right)$.

Figure 18. Variation of K with $\varphi \left(\eta \right)$.

Figure 19. Variation of λ₂ with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 20. Variation of Φ₂ with $\theta \left(\eta \right)$.

Figure 21. Variation of Φ₂ with $\varphi \left(\eta \right)$.

Figure 22. Variation of Pr with $\theta \left(\eta \right)$.

Figure 23. Variation of R with $\theta \left(\eta \right)$.

Figure 24. Variation of Bi with $\theta \left(\eta \right)$.

Figure 25. Variation of Sc with $\varphi \left(\eta \right)$.

Figure 26. Variation of α with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 27. Variation of α with $\theta \left(\eta \right)$.

Figure 28. Variation of α with $\varphi \left(\eta \right)$.

Figure 29. Variation of E with $\varphi \left(\eta \right)$.

Figure 30. Variation of σ with $\varphi \left(\eta \right)$.

Table 4. Numerical values of C_f , Nu, and Sh [47] (Cu + GaN + EG).

Table 5. Numerical values of C_f , Nu, and Sh [47] (Al₂O₃ + GaN + EG).

Table 6. Numerical values of C_f , Nu and Sh [47] (Cu + Al₂O₃ + EG).

Table 7 shows the numerical results of Nusselt numbers, which increase with the increase in Pr and Bi but decrease with the increasing values of R and Q. The Sherwood numbers increase with the increased values of R, Bi, Q, Sc, δ, σ, and m, whereas they decrease with the increasing values of Pr and E.

Three different hybrid nanofluids (hnfs) were formed by adding two nanoparticles from Cu, Al₂O₃, and GaN in Ethylene glycol (EG) base fluid, i.e., Al₂O₃ + GaN + EG, Cu + GaN + EG, and Cu + Al₂O₃ + EG abbreviated as AGE, CGE, and CAE respectively are formed. The three hnfs, AGE, CGE, and CAE, are considered in the present study. Figures 2-7 depict the changes in velocity, temperature, and concentration profiles of three nanofluids, AGE, CGE, and CAE nanofluids, for λ₁ = 0.01, 0.2 and λ₂ = 0.01, 0.3 (Williamson parameters) with the boundary layer thickness parameter η. Graphical variations of the results concerning CAE (Cu + Al₂O₃ + EG) hybrid nanofluid (hnf) are included in every figure to compare the results obtained on hnfs consisting of GaN (as one of the nanoparticles) in the nanofluids CGE and AGE. It is noticed from Figure 2 that the axial and transverse velocity profiles $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$, decrease and increase respectively with the rise in λ₁. Generally, the williamson parameter λ₁ directly proportional to relaxation time [42]; thus, the fluid velocity increases along the axial direction resulting in a decrease in transverse velocity with a rise in λ₁. Exactly the opposite trend prevails with $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$ profiles of AGE, CGE, and CAE hnfs have λ₂ as enhances (Figure 3). As the fluid slows down, the outflow of heat from the fluid reduces, resulting in a rise in the fluid temperature. Figure 4 presents temperature profiles of AGE, CAE, and CGE hnfs for λ₁ = 0.01 and 0.2. Similarly, Figure 5 for λ₂ = 0.01 and 0.3. Both plots indicate that the profiles grow with increasing CGE, CAE, and AGE, demonstrating increased heat energy with increasing Williamson parameters. A rise in λ increases the viscosity of a nanofluid; as a result, the flow of the liquid rises, as

Table 7. Numerical values of Nu and Sh number [47] for Cu + GaN + EG, Al₂O₃ + GaN + EG, and Cu + Al₂O₃ + EG.

does particle concentration. Figure 6 and Figure 7 show the concentration profiles $\varphi \left(\eta \right)$ for all the hnfs for various values of λ₁ and λ₂, respectively. Further, the increasing order of concentration profiles of hnfs is CGE, CAE, and AGE.

Figure 8 shows the diminished axial and transverse velocities corresponding to AGE, CAE, and CGE hnfs with the rise in the magnetic parameter (M = 0.5, 1, 1.5). Owing the Lorentz force that emerges due to the application of M, which opposes the fluid flow, causes a decrease in the axial and transverse velocities. Also, the Lorentz force produces internal friction between fluid particles, causing a rise in the temperature of the fluid and this effect increases with the increase in the magnetic field [42]. The visual evidence is that the velocities of AGE, CAE, and CGE hnfs decrease, and the temperature profiles increase (Figure 9) in the same order. Hence, we understand that CGE hnfs are better cooling agents in cryogenic systems. Further, the increase in the magnetic field causes a temperature rise, implying that the magnetic field causes deterrence to the cooling effect. The same conclusion may be reached from the concentration profiles of all three hnfs (see Figure 10).

The effect of suction/injection on AGE, CAE, and CGE hnfs is analyzed for suction parameters S = −0.2, 0.0, 0.2. For these parameters, the velocity, temperature, and concentration profiles are represented in Figures 11-13. For obvious reasons, the suction causes a decrease, and injection increases the velocity profile, irrespective of the type of hnf. Likewise, the temperature profile increases with suction and decreases with injection. When the wall suction (S > 0) is considered, the results imply a decrease in the boundary layer thickness and an increase in the fluid velocity. S = 0 represents a non-porous plate. For injection (S < 0), a contrary behavior is observed. However, the overall effect on these profiles of AGE, CAE, and CGE hybrid nanoparticles remains the same when compared to the effect of the Williamson parameter. The results indicate that the temperature decreases with increasing suction, which increases due to an increase in injection. Further, suction or injection effects are more or less similar in all three types of hnfs, i.e., independent of the type of nanoparticles present in the fluid. We can see from Figure 13 that the concentration and its boundary layer thickness decrease with an increase in suction/injection parameter S. The tiny nature of different nanoparticles may receive a similar effect due to suction or injection. The decrease and increase in concentration profile with suction and injection, respectively, and in both cases, a decrease in concentration profile with the boundary layer thickness (Figure 1).

The influence of dimensional less parameter Q representing heat source (Q > 0)/sink (Q < 0) on the temperature and concentration profiles for AGE, CAE, and CGE hnfs are revealed in Figure 14 and Figure 15. These graphs indicate that the temperature is enhanced and the concentration profile is diminished with the increases in the values of Q. This is due to an increase in the thickness of the thermal boundary layer and a decrease in the heat transfer rate from the surface to the fluid. Mohamed R.Eid and Mohamed A. Nafe [38] observed similar results in MHD hnfs. Geethan Kumar et al. [42] have also observed an increase in the thickness of the thermal boundary layer of chemically reactive Williamson fluid.

It can be perceived from Figure 16 that the axial and transverse velocities of AGE, CAE, and CGE hnfs decrease as the porosity parameter K increases. The increase in porosity causes more permeability to the fluid along the z-direction, increasing the fluid velocity along the negative z-direction and causing the reduction of velocities. Contrarily, the porosity parameter manifests a rise in temperature and concentration profiles. Figure 17 and Figure 18 demonstrate that this effect is independent of the type of nanoparticles in the hnf mixture. AsK increases, the net amount of decrease in velocity reduces, which will cause less growth in the temperature profile.

Figures 19-21 show the effect of volume fraction Φ₂ on axial and transverse velocities ( $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$ ), $\theta \left(\eta \right)$, and $\varphi \left(\eta \right)$ profiles. As Φ₂ surges, the axial and transverse velocity profiles increase in the order of CGE, AGE, and CAE hnfs (Figure 19). Further, the increase of Φ₂ causes an increase in the resistance force within the fluids consisting of GaN nanoparticles in such a way that the temperature profiles exhibit the opposite trend (Figure 20), i.e., in the order of AGE and CGE. The temperature profile of AGE remains below the temperature profile of CGE because the thermal conductivity boundary layer thickness of AGE decreases more than that of CGE. Also, the change in boundary layer thickness in CAE is not as much as AGE and CGE since it does not contain GaN nanoparticles. Lie group analysis of a Powell-Eyring nanofluids flow over a stretching surface by Hammed Abiodun Ogunseye et al. [52] has also given a similar conclusion on Al₂O₃-water nanofluids. Figure 21 shows that the concentration profiles of CGE and AGE are more than that of CAE, and among the hnfs having GaN nanoparticles, CGE hnf has a higher concentration profile than that of AGE hnf. Kandasamy et al. [53] observed a similar result in Cu-water and SWCNT-water nanofluids. The decrease in concentration profile of SWCNT-water nanofluids is due to the higher diffusion boundary layer thickness possessed by SWCNT-water nanofluids when compared to Cu-water fluids since the CNTs have extraordinary [53]. Due to the nitride element present in GaN, CGE and AGE hnfs show higher concentration profiles than CAE hnf. Therefore, CGE and AGE are assigned lower diffusion boundary layer thicknesses than CAE hnf. Among CGE and AGE hnfs, the concentration profile of AGE hnf is less than CGE hnf, and AGE hnf may be assigned a higher diffusion boundary layer thickness than CGE. Further, CAE hnf is assigned with the highest diffusion boundary layer thickness than CGE and AGE as it shows the lower concentration profile among all the studied CGE, AGE, and CAE hnfs.

Figure 22 exhibits the effect of Prandtl number Pr on temperature profile $\theta \left(\eta \right)$ of CGE, AGE, and CAE hnfs. “Prandtl number Pr is defined as the ratio of momentum diffusivity to the thermal diffusivity.” When the thermal diffusivity increases, Pr and temperature field depreciate [54]. This behavior is expected in most nanofluids and has also been observed by several researchers in nanofluids with different combinations of nanoparticles and base fluids [34] [55] [56]. Further, the graphical presentation given in Figure 22 shows that the thermal diffusivities of CGE, AGE, and CAE hnfs, vary according to the relation CGE < CAE < AGE. The explanation for this relationship is that the sum of specific heat capacities and thermal conductivities of the nanoparticles mixed with the base fluid Ethylene glycol vary according to the descending order CGE > CAE > AGE.

Thermal radiation, the strength identified by a radiation parameter R, enhances the nanofluid temperature by thermal boundary layer thickness. Thermal radiation adds heat to the nanofluid, causing a rise in the fluid temperature. As a result of thermal radiation, the temperature field of a nanofluid improves with the increase in the parameter R. The change in temperature profile as a function of R for CGE, AGE, and CAE hnfs is presented in Figure 23. As expected, the variations in this figure showed improvement in the temperature profile with an increase in the thermal radiation parameter R. This means that all the hnfs absorb heat from the thermal radiation and increase the thermal boundary layer thicknesses and temperature. It is worth mentioning that heat energy absorption occurs among the fluid molecules by enhancing their vibrational energies proportional to the quantum of heat energy absorbed. With the increase in heat radiation, the content of heat absorbed by all CGE, AGE, and CAE hybrid nanofluids remains almost the same because the volume fraction of nanoparticles in each hnf is much less, and mostly the water molecules take part in the process of absorption.

Figure 24 depicts temperature profiles of CGE, AGE, and CAE hnfs, demonstrating the effect of Bi on heat dispersion. The temperature profile rises as the Biot number increases because the Biot number relates to internal conductive resistance, and the surface provides convective resistance. As the Biot number increases, so does the internal conductive resistance, increasing the cross-boundary layer thickness. And thus, the fluid temperature increases. Without abnormalities, the temperature profiles of the three hnfs increased with an increase in Biot number Bi. The variation of concentration field of CGE, AGE, and CAE with Schmidt number Sc is depicted the Figure 25. Schmidt number is “the ratio of momentum diffusivity to the Brownian diffusivity.” The Brownian diffusivity parameter increases due to the random collision of molecules in nanofluid, which intern lowers the Sc. As a result, the concentration boundary layer thickness grows as Sc improves, decreasing the mass fraction field. From Figure 25, the nanoparticle concentration declines with the enhancement in Schmidt number in all the hnfs (CGE, CAE, AGE) because Sc is inverse proportion to the diffusion rate of mass [42]. Therefore, mass diffusion decreases as the Schmidt number increases in all the hybrid nanofluids under study.

“The stretching rate parameter α is the ratio of the transverse velocity to the axial velocity. In general, the porosity parameter reduces fluid velocity.” It signifies the increase in the stretching parameter of the porous sheet, and the transverse velocity exceeds the axial velocity [42]. Henceforth, the rise in the value of α causes a rise in the transverse velocity and, simultaneously, a decrease in the axial velocity. This trend is shown graphically in Figure 26 for CGE, AGE, and CAE hybrid nanofluids. The stretching rate parameter α also causes a decrease in the temperature and concentration profiles (Figure 27 and Figure 28).

The influence of activation parameter E on the $\varphi \left(\eta \right)$ of CGE, AGE, and CAE hybrid nanofluids is presented graphically in Figure 29. in fact, E is the energy that is sufficient to continue the chemical reaction and is known as “the least amount of energy necessary to initiate and sustain a chemical reaction.” In general, the rise in the activation energy causes an increase in the nanoparticle concentration. Aamir Hamid et al. [57] have studied the “Impacts of chemical reaction with activation energy on the unsteady flow of magneto-Williamson nanofluids and concluded that the increase in the destructive chemical reaction parameter σ > 0 tends to reduce the nanoparticle concentration profile” [57]. An increase in activation energy parameter E causes an impact on increasing the concentration profiles of all three hnfs.

According to Figure 30, the nanoparticle concentration in the hybrid nanofluid falls as the boundary layer thickness of the solute decreases due to an increase in the chemical reaction parameter σ. Diffusivity of the fluid varies due to changes in the intensity of chemical reaction, and thus the concentration declines. Kandasamy et al. [53] discovered that when the chemical reaction and buoyancy ratio increase, the concentration of water-based SWCNTs, Cu, and Al₂O₃ decreases, but the rate of mass transfer increases due to the combined impact of diffusion conductivity and kinematic viscosity of the nanoparticles. They also discovered that the diffusion boundary layer thickness of water-base Cu and SWCNTs increases faster than that of Al₂O₃-water as the chemical reaction progresses [53]. Figure 30 indicates that 1) the concentration profiles of CGE, AGE, and CAE hnfs decrease with the increase in the chemical reaction parameter σ and 2) the deviation in the profile geometry between two consecutive profiles decreases, gradually, as σ progresses. It could be due to a gradual decrease in the amount of fluid for the reaction or an increase in the chemical reaction boundary layer thickness.

6. Conclusions

A numerical analysis of a 3-D Williamson MHD fluid consisting of Ethylene glycol embedded with Cu-GaN, Al₂O₃-GaN, and Cu-GaN hybrid nanoparticles passed over a linearly stretching porous sheet with Suction/Injection, and Heat Source/Sink, Radiation, and Arrhenius activation energy addressed, and the results presented in the article. Various hydrodynamic profiles, numerical data of Skin friction coefficient, Nusselt number, and Sherwood numbers are estimated. The shooting technique algorithm based on the Runge-Kutta method was employed to secure the results. The results presented are:

• As the local Williamson parameter λ₁ increases, the axial and the transverse velocities decrease and increase, respectively, in the three CGE, AGE, and CAE hybrid nanofluids. The temperature profiles increase in the order of CGE, CAE, and AGE hybrid nanofluids indicating higher heat flow and a more significant concentration flow profile in CGE nanofluids than in the other two fluids.

• Axial and transverse velocities corresponding to AGE, CAE, and CGE hybrid nanofluids diminish with an increase in the magnetic field parameter. However, the changes observed in the variation of velocity profile with the Williamson parameter and magnetic field are the same and indicate that CGE hybrid nanofluid is a better cooling agent among all three.

• The suction parameter (S > 0) causes a decrease in the boundary layer thickness and an increase in the fluid velocity. A contrary behavior occurs for injection (S < 0). The effect of suction/injection on AGE, CAE, and CGE is the same, i.e., independent of the type of nanoparticles present in the fluid.

• As the parameter of heat source Q rises from the sink) to +Q (source), the temperature profile is enhanced, and the concentration profile is diminished due to an increase in the thermal boundary layer thickness. A similar increase in temperature profile due to an increase in the thermal boundary layer thickness occurs in CGE, CAE, and AGE hybrid nanofluids with the increase in Radiation parameter.

• The x- and y-axis velocities and the related momentum boundary layer thickness in AGE, CAE, and CGE hybrid nanoparticles decrease as the porosity parameter K increases.

• Diffusion boundary layer thickness of CGE, CAE, and AGE hybrid nanofluids are in the order of CAE < CGE < AGE, implying that GaN nanoparticles enhance the diffusion boundary layer of a nanofluid.

• The increase in Prandtl number causes a decrease in the temperature profiles of CGE, CAE, and AGE hybrid nanofluids in the order CGE < CAE < AGE based on the order of thermal diffusivity variation CGE < CAE < AGE.

• The temperature profile of the three hybrid nanofluids increases with an increase in Biot number Bi, and mass diffusion decreases with an increased Schmidt number in all the hybrid nanofluids.

• The value of stretching rate parameter α causes a rise in the transverse velocity and lowers the axial velocity, the temperature, and concentration profiles of CGE, CAE, and AGE hnfs.

• An increase in activation energy parameter E shows an impact of increasing the concentration profiles of all the three hybrid nanoparticles

• As the “σ” progresses, 1) the concentration profiles of CGE, AGE, and CAE hybrid nanofluids decrease, and 2) the deviation in the profile geometry between two consecutive profiles decreases gradually, and it could be due to a gradual decrease in the amount of fluid available for reaction or an increase in the chemical reaction boundary layer thickness.

Acknowledgements

The authors would like to thank the Head, Department of Mathematics, University College of Science, Osmania University, Hyderabad, India, for his participation in the discussions and help. One of the authors, M. Jyotshna, thanks the Principal and Head, Department of Applied Sciences and Humanities, Maturi Venkata Subba Rao Engineering College, Nadergul, Hyderabad, for their constant encouragement and support in conducting the research work.

Authors’ Information

M. Jyotshna is an Assistant Professor at the Department of Mathematics, Maturi Venkata Subba Rao Engineering College (Affiliated to Osmania University) Nadergul, Hyderabad-501510, India. She is also a Researcher at the Department of Mathematics, University College of Science, Osmania University, Hyderabad-500007, India. Her research interests are Applied Mathematics and Nanofluids. ORCID Number: https://orcid.org/0000-0002-3657-6322.

V. Dhanalaxmi is a Professor of Mathematics at the University College of Technology, Osmania University. Hyderabad-500007, India. Her research interests are fluid mechanics and Applied Mathematics.

Nomenclature

$E_a$ Activation energy parameter

$S_c$ Schmidt number

$A_1$ First Rivlin-Erickson tensor

$B_o$ Magnetic field strength

$c_p$ Specific heat at constant Pressure

${\sigma }^{*}$ Stefan Boltzmann constant

$D_B$ Diffusion coefficient

$k^{*}$ Mean absorption coefficient

$K_o$ Chemical reaction constant

$K_p$ Permeability of the porous medium

$P_r$ Prandtl number

$Q_o$ Heat source/sink

${\alpha }_{nf}$ Thermal diffusivity

${\alpha }_{nf}$ Thermal diffusivity

${\lambda }_1$ & ${\lambda }_2$ Williamson parameters

${\mu }_0$ Limiting viscosity at zero shear rate

${\mu }_{\infty }$ Limiting viscosity at infinity shear rate

${\mu }_{nf}$ Dynamic viscosity of the nanofluid

${\rho }_{nf}$ Nanofluid density

α Stretching ratio parameter

C Concentration of the fluid

E Activation energy parameter

I Identity vector

k Boltzmann constant

m Fitted rate constant

M Magnetic parameter

$\stackrel{\dot{}}{p}$ Pressure

Q Heat source/sink parameter

R Radiation parameter

S Suction/injection

$\stackrel{\dot{}}{S}$ Cauchy stress tensor

T Temperature of the fluid

Γ Time constant

δ Temperature difference parameter

ρ Density

σ Chemical reaction parameter

σ Electric conductivity

τ Extra stress tensor

Г Williamson fluid parameter

Conflicts of Interest

The authors declare no conflicts of interest.

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