Numerical Simulation of Homogeneous–Heterogeneous Reactions through a Hybrid Nanofluid Flowing over a Rotating Disc for Solar Heating Applications

Abstract

Several materials, such as aluminum and copper, exhibit non-Newtonian rheological behaviors. Aluminum and copper nanoparticles are ideal for wiring power grids, including overhead power transmission lines and local power distribution lines, because they provide a better conductivity-to-weight ratio than bulk copper; they are also some of the most common materials used in electrical applications. Therefore, the current investigation inspected the flow characteristics of homogeneous–heterogeneous reactions in a hybrid nanofluid flowing over a rotating disc. The velocity slip condition was examined. The energy equation was developed by employing the first law of thermodynamics. Mixed convection thermal radiation and the convective condition effect were addressed. The dimensionless governing models were solved to give the best possible investigative solution using the fourth- and fifth-order Runge–Kutta–Felhberg numerical method. The effects of different influential variables on the velocity and temperature were scrutinized graphically. The effects of the variation of various sundry parameters on the friction factor and Nusselt numbers were also analyzed. The results revealed that the velocity gradient increased significantly for augmented values of the mixed convection parameter. Here, the velocity gradient increased more rapidly for a hybrid nanoliquid than for a nanofluid. The thermal distribution was enhanced due to a significantly increased radiation parameter.

1. Introduction

The recent technical application of rotating disks has made it one of the foremost domains of study in fluid dynamics. This has been an interest growing subject since the honorable study made by Von Karman in 1921, which considered an infinitely extended rotating disk. Cochran found the exact numerical solution to the equations and, later, it was generalized by Batchelor. The flow between coaxial rotating disks was reported by Brady and Durlofsky [1] for both open-ended and unshrouded disks. Turkyilmazoglu [2] analyzed the flow of a fluid that was instigated by a vertically moving rotating disk. Gowda et al. [3] scrutinized a hybrid nanoliquid flow in the presence of thermophoretic particle deposition over a rotating disk. Abbas et al. [4] examined the transfer of heat and mass by considering an Oldroyd-B fluid with mass flux and temperature gradients. Naganthran et al. [5] produced an important study that examined flows over a radially stretching or shrinking gyrating disk. Khan et al. [6] investigated the flow of Maxwell fluid under a rotating disk with magnetic flux. Jayadevamurthy et al. [7] investigated the bio-convective flow of a hybrid nanoliquid over a rotating disk by taking activation energy into account.

Chemical reactions affect a wide range of applications in the fields of chemical engineering and related industries. It is necessary to concentrate the flow of heat or mass, which involves components that are in the same or different phases of chemical reactions. By including the Hall effect in the mathematical model, Hayat et al.’s study [8] exemplified the investigation of H–H reactions in a Carreau liquid in which the heat transfer coefficient and temperature were investigated for different parameters. Hayat et al. [9] improvised the concept and adopted it for silver nanoparticles under a rotating disk by using the homotopy method to analyze the solution. Doh et al. [10] investigated the MHD flow of a nanofluid by employing suction parameters over a gyrating system and considering homogeneous–heterogeneous reactions. The result revealed that the thickness of the disk and the radial, axial, and azimuthal velocities were interdependent. Khan et al. [11] investigated TiO₂−GO nanoparticles between a stretching rotating system. Hafeez et al. [12] studied an flow of Oldroyd-B fluid with the influence of thermophoresis on the deposition of particles by a rotating disk. Khan et al. [13] considered a Maxwell nanofluid to investigate homogeneous–heterogeneous reactions that were subjected to a rotating disk. Christopher et al. [14] explored a 2D stream of a hybrid nanofluid through a stretchy cylinder by including the effect of homogeneous–heterogeneous reactions into the model.

Turkyilmazoglu [15] explained the convective flow of a micropolar liquid over a stretching surface. Raees et al. [16] produced a mathematical model to analyze the convective flow of a nanofluid in the presence of homogeneous and heterogeneous reactions to observe the stability of chemical reactions. Alghamdi [17] investigated a 3D nanofluid flow with mixed convection and activation energy when subjected to a rotating disk. Ahmed et al. [18] used a Boussinesq approximation to investigate convection that occurred due to buoyancy force in Maxwell 3D flow with nanoparticles subjected to a rotating cylinder. Abuzaid and Ullah [19] considered an Oldroyd-B fluid with nanoparticles to investigate mixed convection with activation energy and chemical reactions. Khan et al. [20] exemplified the flow of a nanofluid that was created through a rotating system with mixed convection energy activation and chemical reactions.

The process of the physical transition from solid to liquid caused by the increase in the internal energy of a substance is called melting. The heat and mass transfer of a moving surface system into a cooling medium plays a significant role in the behavior of interacting substances. The study of melting characters and their effects when moving over different mediums and materials is very important for keeping the material away from the molten state. Roberts [21] concentrated his research on a melting body of ice due to hot air. Epstein [22] illustrated the melting heat transfer over a laminar flow subjected to a plate to obtain the similarities between melting and diffusion. Kazmierczak et al. [23] emphasized the study of the convection flow of liquid in a porous medium that was subjected to a plate. Kumar [24] studied melting and Joule heating effects over a sheet. Krishnamurthy et al. [25] considered a Williamson fluid to study MHD boundary layer flow in a porous medium that was subjected to a chemical reaction over a stretching sheet. Radhika et al. [26] exemplified heat transfer and flow of a dusty hybrid nanoliquid moving past a surface and experiencing melting effects. Reddy et al. [27] explained the radiative flow of a nanoliquid moving past an extending sheet by considering the heat transfer due to melting. The heat transfer effects on Reiner–Philippoff nanofluid across a parabola were studied by Sajid et al. [28]. Ahmed and Mohamed [29] analyzed the heat and mass transfer impacts on hybrid nanofluid flow across a thin film. Mohamed and Hossainy [30] explored the flow of a hybrid nanofluid using computational methods. Mohamed et al. [31] studied the homogeneous–heterogeneous impacts on magnetodynamic Prandtl fluid flow.

Solar water heating systems use heat exchangers to transfer the solar energy that is absorbed in solar collectors to the liquid or air used to heat water or a space. Heat exchangers can be made of steel, copper, bronze, stainless steel, aluminum, or cast iron. Solar heating systems usually use copper because it is a good thermal conductor and has greater resistance to corrosion [32]. Recent numerical investigations by Hissouf et al. [33] and Jia et al. [34] showed that the thermal and electrical efficiency of solar heating is highly improved by using Cu and Al nanofluids. Hybrid nanofluids were utilized in several systems. Previous research showed that hybrid nanofluids exhibit promising results for solar heating applications, where the performance was greatly enhanced. Due to these extraordinary applications of hybrid nanofluids for solar heating, this article intends to fill the opening in the current literature.

Based on the available literature, the stimulation of solar heating on the hybrid nanoliquid flow over a rotating disc with the simulation of H–H reaction has not been studied to the best of the authors’ knowledge [35,36]. In this study, we investigated the influence of H–H reactions on a hybrid nanoliquid flowing over a rotating disc with solar heating. The focal point in the current study was to numerically examine the above-described flow. Furthermore, the findings of this research, which are explained in detail in this manuscript, are completely new and have never been published previously. Non-similar transformations were utilized to transform the governing equations of the fluid and entropy generation model into a dimensionless form. Maple software was used to numerically solve the transformed equations.

2. Mathematical Formulation

Consider a steady laminar flow of an incompressible hybrid nanofluid past a stretchable rotating disk. The Darcy–Forchheimer equation is used for porous medium analysis by neglecting viscous dissipation and magnetic effects. Homogeneous and heterogeneous reactions were considered at the surface. Under these conditions, the continuity, momentum, temperature, and concentration equations, which describe the flow and heat transfer, are stated as follows [3]:

$$\frac{\partial u}{\partial r}+\frac{\partial w}{\partial z}+\frac{u}{r}=0 ,$$

(1)

$$\left[u\frac{\partial u}{\partial r}+W\frac{\partial u}{\partial z}-\frac{v^2}{r}\right]=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left[\frac{{\partial }^{2}u}{\partial r^2}+\frac{{\partial }^{2}u}{\partial z^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2}\right]+\frac{g{\left(\beta \rho \right)}_{hnf}\left(T-T_{\infty }\right)}{{\rho }_{hnf}},$$

(2)

$$\left[u\frac{\partial v}{\partial r}+W\frac{\partial v}{\partial z}-\frac{uv}{r}\right]=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left[\frac{{\partial }^{2}v}{\partial r^2}+\frac{{\partial }^{2}v}{\partial z^2}+\frac{1}{r}\frac{\partial v}{\partial r}-\frac{v}{r^2}\right],$$

(3)

$$\left[u\frac{\partial w}{\partial r}+W\frac{\partial w}{\partial z}\right]=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left[\frac{{\partial }^{2}w}{\partial r^2}+\frac{{\partial }^{2}w}{\partial z^2}+\frac{1}{r}\frac{\partial w}{\partial r}\right],$$

(4)

$$\left[u\frac{\partial T}{\partial r}+W\frac{\partial T}{\partial z}\right]=\left[\frac{k_{hnf}}{{\left(\rho cp\right)}_{hnf}}+\frac{16\sigma *T_{\infty }^3}{3k*{\left(\rho cp\right)}_{hnf}}\right]\left[\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right],$$

(5)

$$\left[u\frac{\partial a}{\partial r}+W\frac{\partial a}{\partial z}\right]=D_A\left[\frac{{\partial }^{2}a}{\partial r^2}+\frac{{\partial }^{2}a}{\partial z^2}+\frac{1}{r}\frac{\partial a}{\partial r}\right]-k_{1}a{b}^2,$$

(6)

$$\left[u\frac{\partial b}{\partial r}+W\frac{\partial b}{\partial z}\right]=D_B\left[\frac{{\partial }^{2}b}{\partial r^2}+\frac{{\partial }^{2}b}{\partial z^2}+\frac{1}{r}\frac{\partial b}{\partial r}\right]+k_{1}a{b}^2.$$

(7)

The following boundary conditions were used:

At

$$z=0:u=a_{1}r,v=\Omega r,w=0,-k_f\frac{\partial T}{\partial z}={\Omega }_1\left(T_w-T\right),D_A\frac{\partial a}{\partial z}=k_{2}a,D_B\frac{\partial b}{\partial z}=-k_{2}a$$

$$z\to \infty :u\to 0,v\to 0,T\to T_{\infty },a\to a_0,b\to 0,$$

(8)

where the $\left(u,v,w\right)$ velocity components are in the increasing $\left(r,\varphi ,z\right)$ directions, ${\rho }_{hnf}$ is the effective density of the hybrid nanofluid, ${\mu }_{hnf}$ is the effective dynamic viscosity of the hybrid nanofluid, ${\left(\rho cp\right)}_{hnf}$ is the heat capacitance of the hybrid nanofluid, ${\nu }_{hnf}$ is the kinematic viscosity of the hybrid nanofluid, ${\nu }_f$ is the kinematic viscosity of the fluid, ${\mu }_f$ is the dynamic viscosity of the fluid, ${\rho }_f$ is the density of the fluid, $T$ is the fluid temperature, $k*$ is the mean absorption coefficient, $\sigma *$ is the Stefan–Boltzmann constant, $\left(D_B,D_A\right)$ are the diffusion coefficients, $\left(k_1,k_2\right)$ are the rate constants, $\left(a,b\right)$ are the concentrations of chemical species, $a_1$ is the stretching constant, ${\left(\beta \rho \right)}_{hnf}$ is the thermal expansion coefficient, and ${\Omega }_1$ is the heat transfer coefficient.

The Von Karman’s similarity variables are defined as follows [3]:

$$u=r\Omega f\prime \left(\eta \right),v=r\Omega F\left(\eta \right),w=-2\sqrt{v_f\Omega }f\left(\eta \right),$$

$$\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_{{}_w}-T_{\infty }},a=a_{0}g\left(\eta \right),b=a_{0}h\left(\eta \right),\eta =\sqrt{\frac{\Omega }{v_f}}z,$$

(9)

where $\Omega$ is a constant angular velocity, $\eta$ is the transformed coordinate, $f\prime \left(\eta \right)$ is the radial velocity,$f\left(\eta \right)$ is the axial velocity, $F\left(\eta \right)$ is the tangential velocity, $\theta \left(\eta \right)$ is the non-dimensional temperature, $T_{\infty }$ is the ambient temperature, and $T_w$ is the surface temperature.

By substituting the similarity variables into Equations (1)–(9), we obtained:

$$f‴-{\epsilon }_1\left[{\left(f\prime \right)}^2-2ff″-{\left(F\right)}^2\right]+\gamma {\epsilon }_2=0,$$

(10)

$$F″-{\epsilon }_1\left[2f\prime F-2fF\prime \right]=0,$$

(11)

$$\left(\frac{k_{hnf}}{k_f}+R\right)\theta ″+2\mathrm{Pr}{\epsilon }_{3}f\theta \prime =0\prime$$

(12)

$$\frac{1}{Sc}g″+2fg\prime -\zeta *g{h}^2=0,$$

(13)

$$\frac{1}{Sc}h″+2fh\prime +\zeta *g{h}^2=0,$$

(14)

where

$${\epsilon }_1={\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}\left[\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+{\varphi }_1\frac{\rho s_1}{{\rho }_f}\right]+{\varphi }_2\frac{\rho s_2}{{\rho }_f}\right] ,$$

$${\epsilon }_2={\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}\left[\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+{\varphi }_1\frac{\left(\rho \beta \right)s_1}{{\left(\rho \beta \right)}_f}\right]+{\varphi }_2\frac{\left(\rho \beta \right)s_2}{{\left(\rho \beta \right)}_f}\right],$$

$${\epsilon }_3=\left[\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+{\varphi }_1\frac{\left(\rho c_p\right)s_1}{{\left(\rho c_p\right)}_f}\right]+{\varphi }_2\frac{\left(\rho c_p\right)s_2}{{\left(\rho c_p\right)}_f}\right],$$

Similarly, the transformed boundary conditions were as follows:

$$\begin{array}{l}f\prime \left(0\right)=m_1,f\left(0\right)=0,F\left(0\right)=1,\theta \prime \left(0\right)=-Bi\left(1-\theta \left(0\right)\right),\\ g\prime \left(0\right)=K_{s}g\left(0\right),\delta *h\prime \left(0\right)=-K_{s}g\left(0\right)\end{array}$$

(15)

$$f\prime \to 0,\theta \to 0,F\to 0,g\to 1,h\to 0 \mathrm{at} z\to \infty$$

(16)

The diffusion coefficients $D_A$ and $D_B$ were assumed to be comparable and hence the ratio of the diffusion coefficients reduced to 1, i.e., $\delta =1$; thus, $h\left(\eta \right)+g\left(\eta \right)=1$.

From Equations (13) and (14):

$$\frac{1}{Sc}g″+2fg\prime -\zeta *{\left(1-g\right)}^2=0 \mathrm{with} g\prime \left(0\right)=K_{s}g\left(0\right),g(\infty )\to 1,$$

(17)

where $m_1=\frac{a_1}{\Omega }$ is the stretching parameter, $Sc=\frac{{\nu }_f}{D_A}$ is the Schmidt number, $\delta *=\frac{D_B}{D_A}$ is the ratio of the diffusion coefficients, $\zeta *=\frac{k_1{a}_0^2}{\Omega }$ is the homogeneous reaction parameter, $K_s=\sqrt{\frac{{\upsilon }_f}{\Omega }}\frac{k_2}{D_A}$ is the heterogeneous reaction parameter, $\mathrm{Pr}=\frac{{\nu }_f{\left(\rho cp\right)}_f}{k_f}$ is the Prandtl number, $R=\frac{16\sigma *T_{\infty }^3}{3k*k_f}$ is the radiation parameter, $\frac{{\Omega }_1}{k_f}\sqrt{\frac{{\nu }_f}{\Omega }}$ is the Biot number, and $\gamma =\frac{g{\left(\beta \right)}_f\left(T_w-T_{\infty }\right)}{r{\Omega }^2}$ is the mixed convection variable.

The thermophysical characteristics of a hybrid nanofluid are as follows:

$${\nu }_{hnf}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}},{\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}},$$

$$\frac{{\rho }_{hnf}}{{\rho }_f}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+{\varphi }_1\frac{\rho s_1}{{\rho }_f}\right]+{\varphi }_2\frac{\rho s_2}{{\rho }_f},$$

$$\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+{\varphi }_1\frac{\left(\rho \beta \right)s_1}{{\left(\rho \beta \right)}_f}\right]+{\varphi }_2\frac{\left(\rho \beta \right)s_2}{{\left(\rho \beta \right)}_f},$$

$$\frac{{\left(\rho cp\right)}_{hnf}}{{\left(\rho cp\right)}_f}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+{\varphi }_1\frac{\left(\rho cp\right)s_1}{{\left(\rho cp\right)}_f}\right]+{\varphi }_2\frac{\left(\rho cp\right)s_2}{{\left(\rho cp\right)}_f},$$

$$\frac{k_{hnf}}{k_{bf}}=\frac{k_{s2}+2k_{bf}-2{\phi }_2\left(k_{bf}-k_{s2}\right)}{k_{s2}+2k_{bf}+{\phi }_2\left(k_{bf}-k_{s2}\right)}, \frac{k_{bf}}{k_f}=\frac{k_{s1}+2k_f-2{\phi }_1\left(k_f-k_{s1}\right)}{k_{s1}+2k_f+{\phi }_1\left(k_f-k_{s1}\right)},$$

$$\frac{k_{hnf}}{k_f}=\frac{\left(\frac{{\varphi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\varphi }_{cu}{k}_{cu}}{\varphi }+2k_f+2\left({\varphi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\varphi }_{cu}{k}_{cu}\right)-2\varphi k_f\right)}{\left(\frac{{\varphi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\varphi }_{cu}{k}_{cu}}{\varphi }+2k_f-\left({\varphi }_{A{l}_2{O}_3}{k}_{A{l}_2{O}_3}+{\varphi }_{cu}{k}_{cu}\right)+\varphi k_f\right)},$$

where ${\rho }_{f,}{\rho }_{s1,}{\rho }_{s2,}{\left(Cp\right)}_{hnf},{\left(Cp\right)}_{s1},{\left(Cp\right)}_{s2}$ represent the densities of the fluid, solid nanoparticles of Al₂O₃, and solid nanoparticles of Cu, as well as the specific heat capacitances of the hybrid nanofluid, solid nanoparticles of Al₂O₃, and solid nanoparticles of Cu, respectively. $k_{hnf}$ is the thermal conductivity of the hybrid nanofluid, $k_f,k_{s1},k_{s2}$ are the thermal conductivities of the base fluid, solid nanoparticles of Al₂O₃, and solid nanoparticles of Cu. $\left(\rho c{p}_f\right)$ is the heat capacitance of base fluid and $\left(\rho c{p}_s\right)$ is the heat capacitance of the solid nanoparticles. ${\rho }_s$ is the density of the solid nanoparticles, ${\mu }_f$ is the effective dynamic viscosity of the base fluid, and $h_{nf}$ represents the hybrid nanofluid.

$Nu$ and $C_f$ are given as follows:

$$Nu=\frac{r{q}_w}{k_f\left(T_w-T_{\infty }\right)},C_f=\frac{\sqrt{{\tau }_{W{T}^2}+{\tau }_{W{\varphi }^2}}}{\rho f{\left(\Omega r\right)}^2}$$

Here, ${\tau }_{WT}$ is the radial stress, ${\tau }_{W\varphi }$ is the transverse shear stress, $q_w$ is the heat flux at the disk surface, and $j_w$ is the mass flux.

The non-dimensional forms of $Nu$ and $C_f$ are as follows:

$${\mathrm{Re}}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}Nu=-\frac{k_{hnf}}{k_f}\theta \prime \left(0\right),{\mathrm{Re}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{C}_f=\frac{\sqrt{f\prime {’}^2\left(0\right)}+g{’}^2\left(0\right)}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}}$$

where $\mathrm{Re}=\frac{\Omega r^2}{v_f}$ is the local Reynolds number.

3. Solution Method

The final transformed system of ODEs, as shown in Equations (10)–(12) and Equation (17), with the reduced boundary conditions in Equation (16), was solved numerically by employing the RKF technique using Maple software. A comparative analysis is reported in

Table 1. Comparison of the numerical solutions for $F\prime \left(0\right)$, $-G\prime \left(0\right)$, $-H(\infty )$, and $-\theta ’\left(0\right)$, when ${\varphi }_1={\varphi }_2=0$, $m_1=0$, and $\mathrm{Pr}=6.2$.

	Ref. [33]	Ref. [34]	Present
$F\prime \left(0\right)$	0.510186	0.51023262	0.51022941
$-G\prime \left(0\right)$	0.61589	0.61592201	0.61591990
$-H(\infty )$		0.88447411	0.88446912
$-\theta \prime \left(0\right)$		0.93387794	0.93387285

to assess the validity of our employed numerical method with earlier works, which shows a good agreement with the current results.

4. Results and Discussion

We considered the flow of a hybrid nanofluid with homogeneous–heterogeneous reactions over a rotating disc. The convective condition with mixed convection thermal radiation effect was considered. The thermophysical properties of water, alumina, and copper are tabulated in

Table 2. Thermophysical properties of water, alumina, and copper [28].

Physical Properties		Water	Alumina	Copper
Density	$\rho \left(kg{m}^{-3}\right)$	$997.1$	$3970$	$8933$
Specific heat	$C_p \left(Jk{g}^{-1}{K}^{-1}\right)$	$4179$	$765$	$385$
Heat conductivity	$k_f \left(W{m}^{-1}{K}^{-1}\right)$	$0.613$	$40$	$401$
Thermal expansion coefficient	$\beta \left(K^{-1}\right)$	$21\times {10}^{-5}$	$0.85\times {10}^{-5}$	$1.67\times {10}^{-5}$
Electrical conductivity	$\sigma {\left(\Omega m\right)}^{-1}$	$0.05$	$1\times {10}^{-5}$	$5.96\times {10}^{-7}$

. To investigate the influence of physical parameters, with the assistance of the relevant similarity analysis, the proposed model was converted to a set of ordinary differential equations (ODEs) using the involved variables. For a convergent solution, we employed the RKF-45 shooting method. In this current section, the major effects of several parameters on the radial velocity $f\prime$, the tangential velocity $F$, and the temperature profile $\theta$ are given via graphical illustrations.

Figure 1 displays the impact of $\gamma$ on the fluid velocity profile $f\prime$ for both fluid cases: the hybrid nanoliquid and nanoliquid. It reveals that $f\prime$ increased greatly for increased values of $\gamma$. Here, $f\prime$ increased more rapidly for the hybrid nanoliquid than for the nanofluid. The effects on $F$ due to various values of $\gamma$ are depicted in Figure 2. This figure shows that $F$ increased with an increase of $\gamma$ for both the hybrid nanofluid and the nanofluid. It was found that the velocity of the nano liquid increased gradually compared to the velocity of the hybrid nanofluid. Figure 3 shows the consequence of ${\varphi }_1$ on $f\prime$ for both the hybrid nanofluid and the nanofluid. This graph shows that an increase in the ${\varphi }_1$ values increased the velocity. Moreover, the velocity of the nanoliquid increased slower than that of the hybrid nanofluid. Figure 4 illustrates the influences on $f\prime$ for increased values of $m_1$ for increased values of. Here, the velocity increased with an increase of $m_1$ for both fluid cases and the velocity of the hybrid nanoliquid increased quickly compared to that of the nanofluid. The effect of $m_1$ on $F$ for both fluid cases is given in Figure 5. It denotes that an improvement in $m_1$ greatly increased $F$. However, $F$ increased rapidly for the hybrid nanoliquid compared with that of the nanoliquid. Figure 6 is plotted to demonstrate the effect of $\mathrm{Pr}$ on $\theta$ for both liquid cases. This figure shows that $\theta$ decreased for increasing values of $\mathrm{Pr}$. One can notice from this figure that the decrease in the thermal distribution in the hybrid nanofluid was slower than that of the nanofluid. Physically, an increase in $\mathrm{Pr}$ corresponds to a thermal boundary layer thickness decrease. Hence, the rate of heat transfer decreased and, as a result, the temperature of the liquid was reduced. The effect of $R$ on $\theta$ for both fluid cases is shown in Figure 7. The thermal distribution increased with an increase in $R$. However, the thermal distribution increased remarkably in the hybrid nanoliquid compared with that of the nanoliquid. This happened because the radiation parameter afforded easier convection for the thermal distribution. Figure 8 shows the impact of $Bi$ on $\theta$ for both liquid cases. This figure reveals that $\theta$ increased with an increase in $Bi$ for both fluid cases. Here, the thermal distribution increased slower in the nanoliquid compared to the hybrid nanoliquid. Physically, the Biot number is interconnected to the convective boundary condition on the sheet. An increase in the Biot number rapidly increased the temperature near the boundary of the surface. Due to this convective heating, the thermal boundary layer thickness also increased. The effect of $Sc$ on $g$ is given in Figure 9. The increase in $g$ for various values of $Sc$ for both liquid cases is observed here, where $g$ increased faster for the hybrid nanoliquid compared to the nanoliquid. Physically, $Sc$ is defined by the ratio of the viscous diffusion rate to the molecular rate. For a larger mass diffusivity, the decrease is shown using the Schmidt number.

Figure 10 shows the influence of $K_s$ on $g$ for both fluid cases. It reveals that $g$ decreased with an increase in $K_s$ values. $g$ decreased quicker for the hybrid nanoliquid than for the nanoliquid. Since the heterogeneous reaction parameter $K_s$ is inversely related with mass diffusivity, $g$ decreased. Figure 11 indicates the effect of ${\delta }^{*}$ on $g$ for both liquid cases. An increase in $g$ was perceived for increased values of ${\delta }^{*}$. Moreover, $g$ increased more rapidly for the hybrid nanoliquid than for the nanoliquid. The effect of ${\varsigma }^{*}$ on $h$ is illustrated in Figure 12. It indicates that $h$ increased rapidly with an increase in ${\varsigma }^{*}$ values for both fluid cases. However, $h$ increased more rapidly for the hybrid nanoliquid than for the nanoliquid. Figure 13a,b shows the streamline pattern for both the nanofluid and hybrid nanofluid, respectively, when setting $\gamma =0.5$. Figure 14a,b shows the streamline pattern for the nanofluid when setting $m_1=0.5$ and $m_1=0.5$, respectively.

Table 3. Impact of the pertinent parameters on $C_f$ for both fluid cases.

$\mathit{\gamma }$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	${\mathit{m}}_1$	${\mathit{C}}_{\mathit{f}}$
				Nanofluid	Hybrid Nanofluid
0.1				−1.083206	0.977902
0.2				−1.138167	−1.054503
0.3				−1.198512	−1.133617
	0.1			−0.184883	−0.137800
	0.2			−0.164147	−0.122279
	0.3			−0.142617	−0.106240
		0.1		−0.144872	−0.157821
		0.2		−0.125478	−0.148756
		0.3		−0.104887	−0.95471
			0.1	−0.772552	−0.922382
			0.2	−0.874816	−1.080466
			0.3	−0.969263	−1.214500

displays the impacts of the pertinent parameters on the friction factor for both fluid cases.

Table 4. Impacts of the pertinent parameters on $N{u}_x$ for both fluid cases.

$\mathbf{Pr}$	$\mathit{R}$	$\mathit{\gamma }$	$\mathit{B}\mathit{i}$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	$\mathit{N}{\mathit{u}}_{\mathit{x}}$
						Nano	Hybrid
0.1						0.177526	0.176076
0.3						0.191266	0.192070
0.5						0.191602	0.208061
	0.1					0.252315	0.191266
	0.3					0.228968	0.186031
	0.5					0.208061	0.181655
		0.1				0.158576	0.177080
		0.2				0.162353	0.179877
		0.3				0.166506	0.183226
			0.1			0.170182	0.186840
			0.2			0.061450	0.078860
			0.3			0.045218	0.521440
				0.1		2.452295	2.457524
				0.2		2.349007	2.359885
				0.3		2.241073	2.258100
					0.1	2.652480	2.557524
					0.2	2.435200	2.459885
					0.3	2.341073	2.358100

displays the impacts of the pertinent parameters on the heat transference rate for both fluid cases.

5. Final Remarks

The current study explored the influence of homogeneous–heterogeneous reactions on a hybrid nanofluid flowing over a rotating disc. The dimensionless governing models were solved to find the best possible investigative solution using the fourth- and fifth-order Runge–Kutta–Fehlberg numerical method. The effects of different influential variables on the velocity and temperature were investigated graphically. The variations in various sundry parameters on the friction factor and Nusselt numbers were also analyzed. The main conclusions of the current study are as follows:

• The velocity gradient increased greatly for increased values of $\gamma$. Here, $f\prime$ increased more rapidly for the hybrid nanoliquid than for the nanofluid.
• The velocity increased with an increase in $m_1$ for both fluid cases.
• The thermal distribution increased significantly with an increase in $R$.
• An increase in $g$ was perceived for increased values of ${\delta }^{*}$.
• $h$ increased greatly with an increase in ${\varsigma }^{*}$ values for both fluid cases.
• The combination of copper and aluminum was shown to provide high heat conductivity, resistance to atmospheric and aqueous corrosion, and longevity, which means they offer strong advantages over any other material in solar heating applications.
• Nanofluids of copper and aluminum enhanced the performance of the solar collectors, such as increasing the efficiency of the solar collectors.
• The thermal storage and absorber temperatures reached up to 153 °C and 178 °C, respectively, while using our hybrid nanofluids.