Significance of Multi-Hybrid Morphology Nanoparticles on the Dynamics of Water Fluid Subject to Thermal and Viscous Joule Performance

Abstract

Three-dimensional flow via swirling porous disks and an annular sector is carried out using fully developed hybrid nanofluids. Here, a single-phase simulation based on thermophysical characteristics using various nanoparticle sizes and shapes is taken into account. A regression function connected with the permeable Reynolds number for injection and suction was created. We used the well-known and accurate “shooting approach” to apply to the governing, nonlinear, ordinary differential equation systems to obtain numerical results. Additionally, parametric research was employed to control the impact of embedded flow factors on concentration, velocity, and temperature. While the physical features of the bottom and upper disks, such as the skin friction coefficient and Nusselt number, are provided in a table, their characterization of the flow of several regulatory flow parameters, such as fluid velocity and temperature, is depicted graphically. The experimental range of nanoparticle fractions of 1% to 4% is considered with the Nusselt number having notable effects at $\phi$ = 4%. Both walls demonstrate the effects of an increase in injection factor, shear stress, and tensile stress. As the Eckert number rises at the lower wall, the rate of heat transfer dramatically increases, and the opposite is true for the upper wall. The rate of heat transmission is significantly impacted by the addition of different base fluids containing various kinds of nanoparticles. The aforementioned research created a solid foundation for the development of electronic computers with an emphasis on nanotechnology and biomedical devices.

1. Introduction

Nanofluids offer a more effective means of improving heat transfer. The capacity of common base liquids, similar to oil, ethylene glycol, and water, to improve the heat transport rate is insufficient. This perplexing situation was appropriately resolved by introducing extremely tiny solid particles into the basic liquids. Nanofluids are created by combining nano-measured pieces in suspension (e.g., 1–100 nm) in base liquids (oil, ethylene glycol, and water) and diffusing these tiny particles in the basic fluids. Metals (Fe, Al, Cu, Ag, Au) and oxides are common components in nanoparticles (Al₂O₃, TiO₂). Despite the numerous challenges associated with heat transfer mechanisms using a variety of fluids, researchers are looking for new types of fluids with heat transport properties. As a result of this venture, a new form of nanofluid known as a “hybrid nanofluid” has been investigated, which has a greater thermal conductivity influence than regular nanofluids. This is a new form of heat transport fluid that has been utilized to increase the pace of heat transmission. Engine cooling, electronic cooling, drug reduction, cooling generators, lubrication, and nuclear cooling are just a few of the applications for hybrid nanofluids. Applications of hybrid nanofluids in the fields of refrigeration, heating, ventilation, and air conditioning, heat exchangers, heat pipes, coolants in machining and manufacturing, electronic cooling, automotive industry, generator cooling, transformer cooling, nuclear system cooling, biomedical, space, ships, and defense are among the main areas of research. The properties of hybrid nanofluids have been the subject of several scientific investigations during the last few decades. Ahmed et al. [1] analyzed the time-dependent squeezed flow of magneto-nanofluid between two parallel disks. Nanoparticles of various shapes are suspended in the base. In the base fluid, three distinct-shaped nanoparticles are suspended. Abbas et al. [2] analyzed the three distinct types of nanoparticles (copper, aluminum oxide, and titanium dioxide) with varied forms (spherical, cylindrical, and brick) were employed, using water as the basic nanofluid. With the base fluid, they evaluated copper–alumina nano-ingredients. Motlagh et al. [3] utilized aluminum oxide to refine heat transfer in an inclined cavity. The thermal characteristics of nanofluids are strongly influenced by their size, kind, production technique, dispersibility of their nanoparticles, compatibility, and purity of the base fluid and nanoparticles. Metal oxides (${\mathrm{Fe}}_3{\mathrm{O}}_4$, MgO, ${\mathrm{Al}}_2{\mathrm{O}}_3$, ${\mathrm{TiO}}_2$, CuO), metal nitride (AIN), CNT, and metals (Au, Ag, Ni, Cu) are the most often utilized nanoparticles in base fluids. A spinning disk was the focus of an investigation by Acharya et al. [4] into the hybrid nanofluid flowing in the presence of Hall current. They regarded ${\mathrm{TiO}}_2$ and $\mathrm{Cu}$ nanoparticles to be a novel type of nano-liquid. With multi-walled carbon nanotubes over a movable wedge in a permeable surface, Akbar et al. [5] calculated the magnetic influence on flow. Using an impermeable spinning disk with set radial freedom of speech and expression, Dinarvand et al. [6] studied and verified the incompressible, continuous, steady 3D limit surface fluid motion of a water ZnO-Au hybrid nanofluid. Ramesh [7] investigated how three distinct hybrid nanoparticles move across a spinning disk in a permeable plate explained by the Forchheimer surface. Convective conditions are used to evaluate heat transfer performance. Xu et al. [8] investigated the unsteady mixed convection of a hybrid nanofluid due to moving disks. A basic, relatively uniform modeling approach that explains hybrid nanofluids containing several types of nanoparticles was generated to model the difficulty. An MHD hybrid simulation for a heat production system with seven various kinds of nanoparticles was built by Abdelmalek et al. [9]. Hybridized nanocomponents are modeled based on their form and size factor to determine their thermal conductivity and fluid viscosity. The similarity transform for the constant three-dimensional Navier-Stokes equations of flow between two flexible disk was examined mathematically by Dinarvand et al. [10]. Hybrid nanofluids flow of heat transfer a rotating disk, according to Acharya et al. [11]. Over a spinning disc in a magnetic field, the steady flow of an incompressible viscous electrically conducted hybrid nanofluid is taken into consideration. Yin et al. [12] examined nanofluid heat transfer on a radially spaced rotating disk. The micropolar dusty fluid, Coriolis force effects on dynamics of MHD rotating fluid when lorentz force is significant considered by Lou et al. [13]. Ashraf et al. [14] studied the significance of bioconvection on MHD tangent hyperbolic nanofluid flow of irregular thickness across a slender elastic surface. Behnam et al. [15] aimed to semi-analytically examine the boundary layer flow of a SiC-TiO₂/DO hybrid nanofluid under a steady vertical magnetic field over a porous spinning disk. The effect of contracting the spinning disk on nanofluids was studied by Hatami et al. [16]. For solution formation, they used the least square form. The 3D nanofluid flow over a stationary disk was analyzed by Mustafa et al. [17]. Under the influence of thermal energy and temperature difference effects, Dogonchi et al. [18] investigated the unstable squeezing flow and heat transfer of nanofluid between two simultaneous disks, some of which were penetrable but the others, stretchable/shrinkable. Researchers, Sajjan et al. [19], developed a Nonlinear Boussinesq and Rosseland approximations on 3D flow in an interruption of Ternary nanoparticles with various shapes of densities and conductivity properties. The incompressible hybrid nanofluid flow across an endless impermeable spinning disk was investigated in this paper. Hasnain et al. [20] conducted a theoretical analysis of heat transport enhancement in time-independent, three-dimensional, ternary, hybrid nanofluid flow, with base fluid water and motor oil under the impacts of thermal radiation, over a linear and nonlinearly stretched sheet. According to Khashi’ie et al. [21], magnetic nanofluids have been widely employed in both biological and environmental sectors, with a significant increase in numerical and experimental studies. As a result of the unique features of magnetic nanofluids, the goal of this research is to quantitatively investigate the three-dimensional flow of magnetic nanofluids (Fe₃O₄-H₂O, CoFe₂O₄-H₂O, Mn-ZnFe₂O₄-H₂O) over a shrinking surface with suction and heat radiation effects. A numerical solutions of the partial differential equations for investigating the significance of partial slip due to lateral velocity and viscous dissipation, the case of blood-gold Carreau nanofluid and dusty fluid by Koriko et al. [22]. Shah et al. [23] conducted a numerical simulation of a thermally enhanced EMHD flow of a heterogeneous micropolar mixture comprising (60%)-ethylene glycol (EG), (40%)-water (W), and copper oxide nanomaterials (CuO).

Magnetic hydrodynamics (MHD), with its many significant uses, is also really relevant to be studied. The effect of a magnetic field on the motion of fluids is important to examine. Chemical reaction characteristics and boundary restriction inflow across a vertical surface were investigated by Rout et al. [24]. The chemical reaction in nano-liquid MHD flow was studied by Hayat et al. [25]. Rotation and chemical reactions were studied by Raddya et al. [26] in the MHD flow of nano-liquid. Hayat et al. [27] investigated chemical reactions in nanofluids flowing in MHD flow. Vajravelu et al. [28] analyzed the impact on the time-dependent MHD 3D heat and mass transfer flow of nanofluids squeezed among two simultaneous transpiration disks with velocity slip and heat reduction. Meanwhile, Das et al. [29] investigated that a pair of connected disks with magnetic fields and slip effects are squeezed by a mathematically modeled flow of nanofluid. Mohyud-Din et al. [30] explored the effects of an analogous disk on the compressed MHD flow of (N) that is impacted by both momentum and thermal slip. The mass and energy change in a convective heat transfer flow over a rotating disk with a uniform thin layer was investigated by Qayyum et al. [31]. Aziz et al. [32] found that the heat and mass transport of MHD through a spinning disk with viscous dissipation/emission and slip impact is approached by three-dimensional viscous nano-liquid flow. Elnaqeeb et al. [33] analyzed the natural convection flows of carbon nanotubes nanofluids with Prabhakar-like thermal transport. Uddin et al. [34] discussed heat generation and slip effects due to a revolving disk for the MHD 3D flow of nanofluids. The observed effects of MHD, viscous dissipation, and Joule heating were also reported by Khan et al. [35]. Nonlinear differential structures were calculated by the application of proper transformations using the built-in shooting process. Khan et al. [36] investigated the thermal transmission across two extending disks using MHD flow in the absence of viscous dissipation and Joule heating. MHD slip flow was calculated by Imtiaz et al. [37] through the spinning disk. Doh et al. [38] measured the flow of MHD fluid into a spinning disk. The MHD thin-film stream of nanofluid across a spinning disk was discussed by Shah et al. [39]. Rasool et al. [40] examined a numerical investigation of EMHD nanofluid flows over a convectively heated riga pattern positioned horizontally in a Darcy-Forchheimer porous medium. Rostami et al. [41] constructed a (SiO₂-Al₂O₃/water) hybrid nanofluid boundary layer flow on a vertical, porous, flat disk with a suspended step, employing continuous, convective, MHD-coupled heat transfer. In the work of Ghadikolaeia et al. [42], Cu–Al₂O₃ hybrid nanoparticles were evaluated in the presence of nonlinear thermal radiation, variable thermal conductivity, and varied morphologies of NPs (bricks, rods, platelets, and blades). Many researchers have been worked on analyzing the MHD hybrid nanofluid fluid flow of heat transfer [43,44,45,46,47].

It seems from the literature that no studies have been conducted to date to assess the effects of shape and size parameters on 3D MHD heat transfer in the hybrid nanofluids, Ag-Al₂O₃/H₂O, involving two orthogonal moving porous disks. Additionally, the effects of the applied magnetic field, shape and size factors, porosity parameter, Reynolds number, Prandtl number, Forchheimer number, and chemical reaction parameter should all be assessed. The main goals of the current research work are to develop the thermal characteristics of a carrier fluid, such as water, for practical and commercial reasons, as well as to assess the effects of hybridizing nanofluid flow over the surface of two orthogonal moving porous disks under magnetic influences. We also constructed and expressed this form of mathematics using the paradigm found in [48]. PDEs are regulated such that they may be transformed into dimensionless ordinary differential equations (ODEs). The fourth-order Runge–Kutta integration method and the “shooting method” were used to analyze the ODEs model numerically. The physical parameter data are described using tables and figures.

2. Mathematical Formulation

Consider the flow of incompressible, laminar, time-dependent MHD hybrid nanoparticles on the 3-dimensional fluid flow rate of heat transfer across a two orthogonally moving, porous disk. The disk is moving up and downward with a boundary 2r with a normalized velocity s’ (t) and the size of the diameters varies according to 2s(t). The impacts of magnetic fields and heat sources are specifically taken into consideration. Due to the low Reynolds number assumption, the induced magnetic field is disregarded. The practical model has a cylinder-shaped coordinate system $\left(r,\theta ,z\right)$. Here, v = (u (r, 0, and z), v (r, 0, and z), w (r, 0, and z)) is the fluid flow velocity. $T_1$ > $T_2$ are the set temperatures for the lower and upper walls of the permeable channel, respectively, as shown in the physical model (Figure 1). The following methodology can be used to represent the governing equation from Kashif et al. [49] which was extended to a 3-dimensional hybrid nanofluid:

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

(1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}-\frac{v^2}{r}=-\frac{1}{{\rho }_{hnf}}\frac{\partial p}{\partial r}+{\upsilon }_{hnf}(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2}+\frac{{\partial }^{2}u}{\partial z^2})-\frac{{\sigma }_e{B}_0^2}{{\rho }_{hnf}}u-\frac{{\upsilon }_{hnf}}{K_p}u-F{u}^2$$

(2)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}+\frac{uv}{r}={\upsilon }_{hnf}(\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}-\frac{v}{r^2}+\frac{{\partial }^{2}v}{\partial z^2})-\frac{{\sigma }_e{B}_0^2}{{\rho }_{hnf}}v-\frac{{\upsilon }_{hnf}}{K_p}v-F{v}^2$$

(3)

$$\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=-\frac{1}{{\rho }_{hnf}}\frac{\partial p}{\partial z}+{\upsilon }_{hnf}\left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2}\right)-\frac{{\upsilon }_{hnf}}{K_p}w-F{w}^2$$

(4)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}={\alpha }_{hnf}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2}\right)+\frac{{\mu }_{hnf}}{{\left(\rho c\right)}_p{}_{hnf}}{\left(\frac{\partial u}{\partial z}\right)}^2+\frac{{\sigma }_e{B}_0^2 u^2}{ {(\rho c_p)}_{hnf}}$$

(5)

where $B_0^2$ represents the strong magnetic field, $p$ stands for pressure, ${\sigma }_e$ demonstrates the electrical conductivity, $T$ represents the temperatures, ${\mu }_{hnf}$ illustrates the viscosity of the hybrid nanofluid, ${\rho }_{hnf}$ is the density of the hybrid nanofluid, and the kinematic viscosity of the hybrid nanofluid is represented by ${\upsilon }_{hnf}$, the specific heat capacitance of the hybrid nanofluid is ${(\rho c_p)}_{hnf}$, $k_{hnf}$ is the thermal conductivity of the hybrid nanofluid, the Brownian diffusion constant is shown, $K_p$ denotes the permeability of the porous medium, and $F$ is the non-uniform inertia coefficient.

The upper and lower porous surface boundary conditions for the above model are

$$\begin{array}{c}{z}_1=-s\left(t\right) u=0 v=-\frac{r{A}_1{s}^{\prime }\left(t\right)}{2s} w=-A_1{s}^{\prime }\left(t\right) T=T_1,\\ z_1=s\left(t\right) u=0 v=\frac{r{A}_1{s}^{\prime }\left(t\right)}{2s} w=A_1{s}^{\prime }\left(t\right) T=T_2.\end{array}\}$$

(6)

where A is the wall permeability factor and the dash denotes the derivative with respect to time $t$. After eliminating the pressure term from the governing equations, we add the following transformation of similarity [12]:

$$\eta =\frac{z_1}{s} u=-\frac{r{\upsilon }_f}{s^2}{F}_{\eta }\left(\eta ,t\right) v=\frac{r{\upsilon }_f}{s^2}G\left(\eta ,t\right) w=\frac{2{\upsilon }_f}{s}F\left(\eta ,t\right) \theta =\frac{T-T_2}{T_1-T_2}\}$$

(7)

where the similarity variable is represented by $\eta$.

First of all, the continuity equation is satisfied by the similarity variables stated in Equation (1). Furthermore, the similarity variables are used in the governing equations to acquire Equations (2) and (5).

$$\frac{{\upsilon }_{hnf}}{{\upsilon }_f}{F}_{\eta \eta \eta \eta }+\alpha (3F_{\eta \eta }+\eta F_{\eta \eta \eta })-2F{F}_{\eta \eta \eta }-\frac{s^2}{{\upsilon }_f}{F}_{\eta \eta t}+2G{G}_{\eta }-\frac{{\rho }_f}{{\rho }_{hnf}}M{F}_{\eta \eta }-\frac{{\upsilon }_{hnf}}{{\upsilon }_f}\frac{1}{\beta }{F}_{\eta \eta }-2Fr{F}_{\eta }{F}_{\eta \eta } = 0$$

(8)

$$\frac{{\upsilon }_{hnf}}{{\upsilon }_f}{G}_{\eta \eta }+\alpha (2G+\eta F_{\eta })+2G{F}_{\eta }-\frac{s^2}{{\upsilon }_f}{G}_t-2F{G}_{\eta }-\frac{{\rho }_f}{{\rho }_{hnf}}MG-\frac{{\upsilon }_{hnf}}{{\upsilon }_f}\frac{1}{\beta }G-Fr{G}^2 = 0$$

(9)

$$\begin{array}{c}{\theta }_{\eta \eta }+\frac{{\upsilon }_f}{{\alpha }_{hnf}}\left(\alpha \eta -2F\right){\theta }_{\eta }+[(1+0.1008 {\left(({\phi }_1\right)}^{0.69574}{\left(d{p}_1\right)}^{0.44708}+ {({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\left)\right)\hfill \\ F_{\eta \eta }{}^2+M{F}_{\eta }{}^2]E_c{P}_r\frac{{\kappa }_f}{{\kappa }_{hnf}}-\frac{k^2}{{\alpha }_{hnf}}{\theta }_t = 0\hfill \end{array}$$

(10)

with the boundary conditions

$$\begin{array}{c}\eta =-1, F=-Re,F_{\eta }=0,G=-Re,\theta =1,\\ \eta =1, F=Re,F_{\eta }=0,G=Re,\theta =0\end{array}\}$$

(11)

where α = $\frac{s{s}^{\prime }\left(t\right)}{{\upsilon }_f}$ is the wall expansion ratio, $Re=\frac{A_{1}s{s}^{\prime }\left(t\right)}{2{\upsilon }_f}$ is the permeable Reynolds number, $P_r=\frac{{\left(\mu c_p\right)}_f}{k_f}$ is the Prandtl number, $\beta =\frac{s^2}{K_p}$ is the porosity parameter, $E_c=\frac{U^2}{(T_1-T_2){\left(\rho c_p\right)}_f}$ is the Eckert number, $Fr=\frac{c_p}{\sqrt{K_p}}$ is the Forchheimer number, $M=\frac{{\sigma }_e{B}_0^2{s}^2}{{\mu }_f}$ is the magnetic parameter, ${\upsilon }_{hnf}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}$ is the viscosity of the hybrid nanofluid, and ${\alpha }_{hnf}=\frac{k_{hnf}}{{(\rho c_p)}_{hnf} }$ represents the thermal diffusivity of the hybrid nanofluid.

Finally, we set $F=f Re$, $G$ = $g Re$ and consider the case, following the work of Majdalani et al. [50], that when $\alpha$ is a constant, $f$ = $f$($\eta$) and $\theta =\theta \left(\eta \right)$, which leads to ${\theta }_t$ = 0,$g_{\eta t}$ = 0 and $f_{\eta \eta t}$ = 0. Thus, we have the following equations:

$$\frac{{\upsilon }_{hnf}}{{\upsilon }_f}{f}_{\eta \eta \eta \eta }+\alpha (3f_{\eta \eta }+\eta f_{\eta \eta \eta })-2Ref{f}_{\eta \eta \eta }+2Reg{g}_{\eta }-\frac{{\rho }_f}{{\rho }_{hnf}}M{f}_{\eta \eta }-\frac{{\upsilon }_{hnf}}{{\upsilon }_f}\frac{1}{\beta }{f}_{\eta \eta }-2FrRe{f}_{\eta }{f}_{\eta \eta } = 0$$

(12)

$$\frac{{\upsilon }_{hnf}}{{\upsilon }_f}{g}_{\eta \eta }+\alpha (2g+\eta g_{\eta })+2Re\left(g{f}_{\eta }-f{g}_{\eta }\right)-\frac{{\rho }_f}{{\rho }_{hnf}}Mg-\frac{{\upsilon }_{hnf}}{{\upsilon }_f}\frac{1}{\beta }g-FrRe{g}^2 = 0$$

(13)

$${\theta }_{\eta \eta }+\frac{{\upsilon }_f}{{\alpha }_{hnf}}Pr\left(\alpha \eta -2Ref\right){\theta }_{\eta }+[(1+0.1008 {\left(({\phi }_1\right)}^{0.69574}{\left(d{p}_1\right)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\left)\right)F_{\eta \eta }{}^2+M{F}_{\eta }{}^2]E_c{P}_r\frac{{\kappa }_f}{{\kappa }_{hnf}} = 0$$

(14)

$$\begin{array}{c}\begin{array}{c}f=-1,f_{\eta }=0, g=-1, \theta =1, at \eta =-1\end{array}\\ f=1, f_{\eta }=0, g=1, \theta =0, at \eta =1\end{array}\}$$

(15)

2.1. Practical and Engineering Interests

Skin friction and Nusselt number are physical characteristics that are essential for engineering applications in constructing nanoscale equipment. On both porous surfaces, all of these factors have been calculated.

2.2. Skin Friction Coefficients

The following parameter $C_{f1}$ or $C_{f-1}$ indicate the skin friction coefficients for the top and bottom porous disks respectively

$$\begin{array}{c}{C}_{f-1}=\frac{{\xi }_w{|}_{\mathsf{\eta } =-1}}{{\rho }_{bf}{\left(s^{\prime }{A}_1\right)}^2}=\frac{(1+0.1008({({\phi }_1)}^{0.69574}(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708})}{R{e}_r}\sqrt{{\left(f^{″}\left(-1\right)\right)}^2+{\left(g^{\prime }\left(-1\right)\right)}^2}\hfill \\ C_{f1}=\frac{{\xi }_w{|}_{\mathsf{\eta } =1}}{{\rho }_{bf}{\left(s^{\prime }{A}_1\right)}^2}=\frac{(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)}{R{e}_r}\sqrt{{\left(f^{″}\left(1\right)\right)}^2+{\left(g^{\prime }\left(1\right)\right)}^2}\hfill \end{array}$$

(16)

Here, $R{e}_r=4\left(\frac{s}{r}\right)\left(\frac{1}{{\left(Re\right)}^2}\right)$ represents the local Reynolds number and ${\xi }_w$ is the shear stress. The expression ${\xi }_w$ is defined as

$${\xi }_w=\sqrt{{\xi }_{zr}^2+{\xi }_{\theta z}^2}$$

When the upper and lower disks have radial and tangential shear stresses, they are denoted by ${\xi }_{rz}$ and ${\xi }_{\theta z}$:

$$\begin{gathered} {\xi }_{zr}={\mu }_{hnf}\left(\frac{\partial u}{\partial z}\right){|}_{\mathsf{\eta } =-1}={\mu }_{bf}(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)\left(\frac{r{\upsilon }_f}{s^3}\right)f^{″}\left(-1\right) \\ {\xi }_{\theta z}={\mu }_{hnf}\left(\frac{\partial v}{\partial z}\right){|}_{\mathsf{\eta } =-1}={\mu }_{bf}(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)\left(\frac{r{\upsilon }_f}{s^3}\right)g^{\prime }\left(-1\right) \end{gathered}$$

2.3. Nusselt Numbers

The top and bottom of a porous disk are the flow of heat transfer (Nusselt numbers), represented by $N{u}_{z-1}$ and $N{u}_{z1}$, and given as

$$\begin{gathered} N{u}_{z-1}=\frac{s{e}_z}{{\kappa }_f\left(T_1-T_2\right)}{|}_{\mathsf{\eta }=-1}=-\frac{k_{hnf}}{k_f }{\theta }^{\prime }\left(-1\right) \\ N{u}_{z1}=\frac{s{e}_z}{{\kappa }_f\left(T_1-T_2\right)}{|}_{\mathsf{\eta } =1}=-\frac{k_{hnf}}{k_f }{\theta }^{\prime }\left(1\right) \end{gathered}$$

(17)

Here, the heat flux is denoted as $s_z$, which follows that

$$\begin{gathered} e_z{|}_{\mathsf{\eta } =-1}=-k_{hnf}\left(\frac{\partial T}{\partial z}\right){|}_{\mathsf{\eta } =-1}=-\frac{\left(T_1-T_2\right)}{s}{k}_{hnf}{\theta }^{\prime }\left(-1\right) \\ {e}_z{|}_{\mathsf{\eta } =1}=-k_{hnf}\left(\frac{\partial T}{\partial z}\right){|}_{\mathsf{\eta } =1}=-\frac{\left(T_1-T_2\right)}{s}{k}_{hnf}{\theta }^{\prime }\left(1\right) \end{gathered}$$

where $Re=\frac{A_{1}s{s}^{\prime }\left(t\right)}{2{\upsilon }_f}$.

2.4. Numerical Solution

By using the thermophysical properties of hybrid nanofluids in the nonlinear ODEs Equations (12)–(14):

$$\begin{array}{c}\left(\frac{(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)}{\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+({\phi }_1)\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+\left({\phi }_2\right)\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)f^{″″}\left[\eta \right]-\alpha (3f^{″}\left[\eta \right]+\eta f^{‴}\left[\eta \right])-\hfill \\ 2Ref\left[\eta \right]f^{‴}\left[\eta \right]-\hfill \\ \left(\frac{1}{\left(\left(1-{\phi }_1-{\phi }_2\right)+{\phi }_1\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+{\phi }_2\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)M f^{″}[\eta ]\hfill \\ -\left(\frac{(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)}{\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+({\phi }_1)\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+\left({\phi }_2\right)\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)\frac{1}{\beta }{f}^{″}\left[\eta \right]-2FrRe{f}^{\prime }\left[\eta \right]f^{″}\left[\eta \right] = 0\hfill \end{array}$$

(18)

$$\begin{array}{c}\left(\frac{(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)}{\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+({\phi }_1)\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+\left({\phi }_2\right)\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)g^{″}\left[\eta \right]+\alpha (2g\left[\eta \right]+\eta g^{\prime }\left[\eta \right])+\hfill \\ 2Re\left(g\right[\eta \left]f^{\prime }\right[\eta ]-f\left[\eta \right]g^{\prime }\left[\eta \right])-\left(\frac{1}{\left(\left(1-{\phi }_1-{\phi }_2\right)+{\phi }_1\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+{\phi }_2\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)Mg[\eta ]-\hfill \\ \left(\frac{(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)}{\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+({\phi }_1)\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+\left({\phi }_2\right)\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)\frac{1}{\beta }g\left[\eta \right]-FrRe{g}^2\left[\eta \right] = 0\hfill \end{array}$$

(19)

$$\begin{array}{c}{\theta }^{″}\left[\eta \right]+(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+({\phi }_1)\left(\frac{{\rho }_{cp{s}_1}}{{\rho }_{cpbf}}\right)+\hfill \\ ({\phi }_2)\left(\frac{{\rho }_{cp{s}_2}}{{\rho }_{cpbf}}\right))\left(\frac{k_{s2}+(N-1)k_{mbf}+{\phi }_2(k_{mbf}-k_{s2})}{k_{s2}+(N-1)k_{mbf}-(N-1){\phi }_2(k_{mbf}-k_{s2})}\right)\left(\frac{k_{s1}+(N-1)k_{bf}+{\phi }_1(k_{bf}-k_{s1})}{k_{s1}+(N-1)k_{bf}-(N-1){\phi }_1(k_{bf}-k_{s1})}\right)\hfill \\ Pr\left(\alpha \eta -2Ref\left[\eta \right]\right){\theta }^{\prime }\left[\eta \right]+[(1+0.1008 {\left(({\phi }_1\right)}^{0.69574}{\left(d{p}_1\right)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\left)\right)\hfill \\ f^{″}{\left[\eta \right]}^2+M{f}^{\prime }{\left[\eta \right]}^2]E_c{P}_r\left(\frac{k_{s2}+\left(N-1\right)k_{mbf}+{\phi }_2\left(k_{mbf}-k_{s2}\right)}{k_{s2}+\left(N-1\right)k_{mbf}-\left(N-1\right){\phi }_2\left(k_{mbf}-k_{s2}\right)}\right)\left(\frac{k_{s1}+\left(N-1\right)k_{bf}+{\phi }_1\left(k_{bf}-k_{s1}\right)}{k_{s1}+\left(N-1\right)k_{bf}-\left(N-1\right){\phi }_1\left(k_{bf}-k_{s1}\right)}\right)=0\hfill \end{array}$$

(20)

where

$$H_1=\left(\frac{(1+0.1008({({\phi }_1)}^{0.69574}\left(d{p}_1{)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}\right)}{\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+({\phi }_1)\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+\left({\phi }_2\right)\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)$$

$$H_2=\left(\frac{1}{\left(\left(1-{\phi }_1-{\phi }_2\right)+{\phi }_1\left(\frac{{\rho }_{s_1}}{{\rho }_{bf}}\right)+{\phi }_2\left(\frac{{\rho }_{s_2}}{{\rho }_{bf}}\right)\right)}\right)$$

$$H_3=\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+\left({\phi }_1\right)\left(\frac{{\rho }_{cp{s}_1}}{{\rho }_{cpbf}}\right)+({\phi }_2)\left(\frac{{\rho }_{cp{s}_2}}{{\rho }_{cpbf}}\right)\right)$$

$$H_4=(1+0.1008 {\left(({\phi }_1\right)}^{0.69574}{\left(d{p}_1\right)}^{0.44708}+{({\phi }_2)}^{0.69574}{\left(d{p}_2\right)}^{0.44708}$$

$$D_1=\left(\frac{k_{s2}+\left(N-1\right)k_{mbf}+{\phi }_2\left(k_{mbf}-k_{s2}\right)}{k_{s2}+\left(N-1\right)k_{mbf}-\left(N-1\right){\phi }_2\left(k_{mbf}-k_{s2}\right)}\right)$$

$$D_2=\left(\frac{k_{s1}+\left(N-1\right)k_{bf}+{\phi }_1\left(k_{bf}-k_{s1}\right)}{k_{s1}+\left(N-1\right)k_{bf}-\left(N-1\right){\phi }_1\left(k_{bf}-k_{s1}\right)}\right)$$

$$\omega =D_1{D}_2$$

By inputting the values of $H_1$, $H_2$, $H_3$, $H_4$, $D_1,$ $D_1$, and $\omega$ in Equations (18)–(20), the final results are

$$H_1{f}^{\prime ‴}\left[\eta \right]-\alpha (3f^{″}\left[\eta \right]+\eta f^{‴}\left[\eta \right])-2Ref\left[\eta \right]f^{″}\left[\eta \right]-H_2\mathrm{M} f^{″}[\eta ]-H_1\frac{1}{\beta }{f}_{\eta \eta }-FrRe{f}_{\eta }{f}_{\eta \eta }=0$$

(21)

$$H_1{g}^{″}\left[\eta \right]+\alpha (2g\left[\eta \right]+\eta g^{\prime }\left[\eta \right])+2Re\left(g\left[\eta \right]f^{\prime }\left[\eta \right]-f\left[\eta \right]g^{\prime }\left[\eta \right]\right)-H_{2}Mg\left[\eta \right]-H_1\frac{1}{\beta }g\left[\eta \right]-FrRe{g}^2\left[\eta \right]=0$$

(22)

$${\theta }^{″}\left[\eta \right]+H_3\omega Pr\left(\alpha \eta -2Ref\left[\eta \right]\right){\theta }^{\prime }\left[\eta \right]+[H_4{F}_{\eta \eta }{}^2+M{F}_{\eta }{}^2]E_c{P}_r\omega =0$$

(23)

2.5. Solution of the Problem

We used the RK methodology with shooting methods to determine strategies to solve the current flow model (see Figure 2). To start the procedure, the change must be as follows:

$$q_1^{*}= f\left[\eta \right], q_2^{*}= f^{\prime }\left[\eta \right], q_3^{*}= f^{″}\left[\eta \right], q_4^{*}=f^{‴}\left[\eta \right], q_5^{*}= g\left[\eta \right], q_6^{*}= g^{\prime }\left[\eta \right], q_7^{*}=\theta \left[\eta \right], q_8^{*}={\theta }^{\prime }\left[\eta \right]$$

(24)

Finally, modify the model in the manner shown below from Equations (21)–(23):

$$f^{″″}\left[\eta \right]=\frac{1}{H_1}(\alpha (3f^{″}\left[\eta \right]+\eta f^{‴}\left[\eta \right])+2Ref\left[\eta \right]f^{″}\left[\eta \right]+H_2\mathrm{M} f^{″}[\eta ]+H_1\frac{1}{\beta }{f}_{\eta \eta }\left[\eta \right]+FrRe{f}_{\eta }\left[\eta \right]f_{\eta \eta }\left[\eta \right])$$

(25)

$$g^{″}\left[\eta \right]=\frac{1}{H_1}(-\alpha (2g\left[\eta \right]+\eta g^{\prime }\left[\eta \right])-2Re\left(g\left[\eta \right]f^{\prime }\left[\eta \right]+f\left[\eta \right]g^{\prime }\left[\eta \right]\right)+H_{2}Mg\left[\eta \right]+H_1\frac{1}{\beta }g\left[\eta \right]+FrRe{g}^2\left[\eta \right])$$

(26)

$${\theta }^{″}\left[\eta \right]=H_3\omega Pr\left(-\alpha \eta +2Ref\left[\eta \right]\right){\theta }^{\prime }\left[\eta \right]-[H_{4}f″{\left[\eta \right]}^2+M{f}^{\prime }{\left[\eta \right]}^2]E_c{P}_r\omega$$

(27)

By using substitution, embedded in Equation (24), the following system is attained:

$$\left[\begin{array}{c}{q_1^{*}}^{\prime }\\ {q_2^{*}}^{\prime }\\ {q_3^{*}}^{\prime }\\ {q_4^{*}}^{\prime }\\ {q_5^{*}}^{\prime }\\ {q_6^{*}}^{\prime }\\ {q_7^{*}}^{\prime }\\ {q_8^{*}}^{\prime }\end{array}\right]=\left[\begin{array}{c}{q}_2^{*}\\ q_3^{*}\\ q_4^{*}\\ \frac{1}{H_1}\left(\alpha \right(3q_3^{*}+\eta q_4^{*})+2Re{q}_1^{*}{q}_3^{*}+H_{2}M{q}_3^{*}]+H_1\frac{1}{\beta }{q}_3^{*}+2FrRe{q}_2^{*}{q}_3^{*}\left)\right)\\ q_6^{*}\\ \frac{1}{H_1}(-\alpha (2q_5^{*}+\eta q_6^{*})-2Re(q_5^{*}{q}_2^{*}+q_1^{*}{q}_6^{*})+H_{2}M{q}_5^{*}]+H_1\frac{1}{\beta }{q}_5^{*}+FrRe{q2}_5^{*})\\ q_8^{*}\\ H_3\omega Pr(-\alpha \eta +2Ref\left[\eta \right])q_8^{*}-[H_4{q_3^{*}}^2+M{q_2^{*}}^2]E_{\mathrm{c}}{P}_{\mathrm{r}}\omega \end{array}\right]$$

(28)

Consequently, the initial condition is

$$\left[\begin{array}{c}{q}_1^{*}\\ q_2^{*}\\ q_3^{*}\\ q_4^{*}\\ q_5^{*}\\ q_6^{*}\\ q_7^{*}\\ q_8^{*}\end{array}\right]=\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\\ -1\\ 1\\ 1\\ 0\end{array}\right]$$

Now, utilizing mathematics and a proper beginning condition, the aforementioned system is solved. The RK method, along with using the numerical “shooting method”, have both been considered in this case. This method makes handling the requisite dimensionless ODEs simple. First, we use the shooting approach to create the starting condition, ensuring that boundary criteria are met and the necessary level of precision is attained.

3. Results and Discussion

This section will use the built-in shooting technique to numerically calculate the acquired systems. The diverse variables physically focus on concentration, temperature, and velocity in this case. The relevant factors are discussed in-depth, i.e., the Reynolds number (Re), expansion ratio ($\alpha$), magnetic parameter (M), Prandtl number ($P_r$), volume fractions ${\phi }_1$ and ${\phi }_2$, diameter of the nanoparticles $dp1$ and $dp2$, Forchheimer number $\left(Fr\right)$, porosity parameter ($\beta )$, Eckert number ($E_c$), and size factor (n). Here, all the observation is described utilizing the numerical and graphical outcomes of $\mathrm{Ag}-{\mathrm{Al}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$, $\mathrm{Ag}-{\mathrm{Al}}_2{\mathrm{O}}_3/$ethylene glycol, $\mathrm{Ag}-{\mathrm{Al}}_2{\mathrm{O}}_3/$engine oil, $\mathrm{Ag}-{\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$, $\mathrm{Ag}-{\mathrm{TiO}}_2/$ethylene glycol, $\mathrm{Ag}-{\mathrm{TiO}}_2/$engine oil, $\mathrm{Cu}-{\mathrm{Al}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}, \mathrm{Cu}-{\mathrm{Al}}_2{\mathrm{O}}_3/$ethylene glycol, $\mathrm{Cu}-{\mathrm{Al}}_2{\mathrm{O}}_3/$engine oil, $\mathrm{Cu}-{\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$, $\mathrm{Cu}-{\mathrm{TiO}}_2/$ethylene glycol, and $\mathrm{Cu}-{\mathrm{TiO}}_2/$engine oil nanoparticles.

Table 1. Thermophysical properties of hybrid nanofluid [51].

Properties	(HNFDs)
Density ($\rho$)	${\rho }_{hnf}={\phi }_1{\rho }_{s_1}+{\phi }_2{\rho }_{s_{ 2}}-\left(1-{\phi }_1-{\phi }_2\right) {\rho }_{bf}$
Viscosity ($\mu$)	${\mu }_{hnf}={\mu }_{bf}(1+0.1008 {\left(({\phi }_1\right)}^{0.69574}{\left(d{p}_1\right)}^{0.44708}$+${({\phi }_2)}^{0.69574}{\left( d{p}_2\right)}^{0.44708}$))
$\mathrm{Heat} \mathrm{Capacity} (\rho C_P$)	${(\rho c_p)}_{hnf}={\phi }_1{\left(\rho c_p\right)}_{s_1}+{\phi }_2{(\rho c_p)}_{s_2}+\left(1-{\phi }_1-{\phi }_2\right) {\left(\rho c_p\right)}_{bf}$
Thermal Conductivity (K)	$k_{hnf}$ =$\left(\frac{k_{s2}+\left(n-1\right)k_{mbf}+{\phi }_2\left(k_{mbf}-k_{s2}\right)}{k_{s2}+\left(n-1\right)k_{mbf}-\left(n-1\right){\phi }_2\left(k_{mbf}-k_{s2}\right)}\right)k_{bf}$ where $k_{bf}$ = $\left(\frac{k_{s1}+\left(n-1\right)k_{bf}+{\phi }_1\left(k_{bf}-k_{s1}\right)}{k_{s1}+\left(n-1\right)k_{bf}-\left(N-1\right){\phi }_1\left(k_{bf}-k_{s1}\right)}\right)k_f$

focuses heavily on the thermophysical characteristics of hybrid nanofluid, while

Table 2. Different base fluids’ and nanoparticles’ thermophysical properties (Sheikholeslami and Ganji [52]).

Nanoparticles and Base Fluid	$\mathit{\rho } \left(\mathbf{k}\mathbf{g}{\mathbf{m}}^{-3}\right)$	${\mathit{C}}_{\mathit{p}}$$(\mathbf{J} \mathbf{k}{\mathbf{g}}^{-1}{\mathbf{k}}^{-1})$	$\mathit{\kappa } \left(\mathbf{w}{\mathbf{m}}^{-1}{\mathbf{k}}^{-1}\right)$
H2O	997.1	4179	0.613
Ethylene glycol	1115	2430	0.253
Engine oil	884	1910	0.144
Ag	10,500	235.0	429
Cu	8933	385	401
${\mathrm{Al}}_2{\mathrm{O}}_3$	3970	765	40
${\mathrm{TiO}}_2$	4250	686.2	8.9538

shows the thermophysical characteristics of the various base fluid and nanoparticle types.

Here, $k_{mbf}$ denotes the thermal conductivity for the shape base fluid, $k_{bf}$ is the thermal conductivity of the base fluid, $k_{s1}$, $k_{s2}$, and $k_f$ represent the thermal conductivity of the 1st and 2nd nanoparticles and base fluid, ${\mu }_{eff}$ denotes the effect viscosity, $n$ is the size of the factor, ${\mu }_{bf}$ denotes the viscosity of the base fluid, $d{p}_1$ and $d{p}_2$ are the first and second particle’s diameters, ${\phi }_1$ and ${\phi }_2$ are the volume fraction of the first and second particles, and ${\rho }_{s_1}$ and ${\rho }_{s_2}$ illustrate the 1st and 2nd solid particle densities show in Table 1.

Table 3. (a) Numerical size factor effect of heat transfer in different hybridized nanofluids. (b) Numerical size factor effect of heat transfer in different hybridized nanofluids.

(a)
${\mathit{\phi }}_1={\mathit{\phi }}_2$	$\mathrm{Ag}-{\mathrm{Al}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$					$\mathrm{Ag}-{\mathrm{Al}}_2{\mathrm{O}}_3/$Ethylene glycol
	${\mathit{n}}_3$	${\mathit{n}}_{3.7}$	${\mathit{n}}_{4.8}$	${\mathit{n}}_{5.7}$	${\mathit{n}}_{16.2}$	${\mathit{n}}_3$	${\mathit{n}}_{3.7}$	${\mathit{n}}_{4.8}$	${\mathit{n}}_{5.7}$	${\mathit{n}}_{16.2}$
1%	$0.0182$	$0.0192$	$0.0206$	$0.0218$	$0.0368$	$0.0178$	$0.0186$	$0.0201$	$0.0213$	0.0362
2%	$0.0231$	$0.0252$	$0.0286$	$0.0314$	$0.0694$	$0.0219$	$0.0239$	$0.02731$	$0.0301$	$0.0678$
3%	$0.0286$	$0.0322$	$0.0381$	$0.0431$	$0.1077$	$0.0267$	$0.0301$	$0.0357$	$0.0406$	$0.10461$
4%	$0.0349$	$0.0402$	$0.0489$	$0.0565$	$0.1474$	$0.0320$	$0.0371$	$0.0454$	$0.0527$	$0.1428$
${\mathit{\phi }}_1={\mathit{\phi }}_2$	$\mathrm{Ag}-Ti{O}_2/H_{2}O$					$Ag-Ti{O}_2/$Ethylene glycol
	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$
1%	$0.0176$	$0.0184$	$0.0195$	$0.02049$	$0.0311$	$0.0172$	$0.0176$	$0.0189$	$0.02044$	$0.0307$
2%	$0.0216$	$0.0233$	$0.026$	$0.0282$	$0.0541$	$0.0205$	$0.0228$	$0.025$	$0.0272$	$0.0535$
3%	$0.0261$	$0.0289$	$0.033$	$0.03731$	$0.0814$	$0.0242$	$0.0276$	$0.0320$	$0.0366$	$0.0802$
4%	$0.0311$	$0.0353$	$0.042$	$0.0477$	$0.1111$	$0.0279$	$0.0339$	$0.0411$	$0.0468$	$0.1099$
${\mathit{\phi }}_1={\mathit{\phi }}_2$	$Ag-A{l}_2{O}_3/$Engine oil					$Ag-Ti{O}_2/$Engine oil
	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$
1%	$0.0170$	$0.017$	$0.0192$	$0.0203$	$0.0351$	$0.0167$	$0.01751$	$0.0186$	$0.0195$	$0.0299$
2%	$0.0201$	$0.0221$	$0.0252$	$0.0279$	$0.0643$	$0.0196$	$0.0221$	$0.0238$	$0.0259$	$0.0511$
3%	$0.0236$	$0.0268$	$0.0321$	$0.0367$	$0.0984$	$0.0228$	$0.0254$	$0.0297$	$0.0332$	$0.0761$
4%	$0.0275$	$0.0321$	$0.0398$	$0.0466$	$0.1338$	$0.0263$	$0.0301$	$0.0363$	$0.0415$	$0.1028$
(b)
${\mathit{\phi }}_1={\mathit{\phi }}_2$	$Cu-A{l}_2{O}_3/H_{2}O$					$Cu-A{l}_2{O}_3/$Ethylene glycol
	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$
1%	$0.0174$	$0.0178$	$0.0191$	$0.0201$	$0.0305$	$0.0171$	$0.0174$	$0.0183$	$0.0202$	$0.0299$
2%	$0.0205$	$0.0227$	$0.02535$	$0.0275$	$0.053$	$0.0201$	$0.0220$	$0.0249$	$0.0268$	$0.0529$
3%	$0.0241$	$0.0279$	$0.0323$	$0.036$	$0.0795$	$0.02397$	$0.0279$	$0.0324$	$0.0361$	$0.0803$
4%	$0.0287$	$0.0337$	$0.0403$	$0.0458$	$0.1084$	$0.0276$	$0.0325$	$0.0404$	$0.0447$	$0.1090$
${\mathit{\phi }}_1={\mathit{\phi }}_2$	$Cu-Ti{O}_2/H_{2}O$					$Cu-Ti{O}_2/$Ethylene glycol
	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$
1%	$0.0175$	$0.0181$	$0.0193$	$0.0202$	$0.0308$	$0.0171$	$0.018$	$0.0193$	$0.0202$	$0.0302$
2%	$0.0211$	$0.0227$	$0.0254$	$0.0276$	$0.0532$	$0.0211$	$0.0227$	$0.0254$	$0.0276$	$0.0536$
3%	$0.0252$	$0.0286$	$0.0329$	$0.0362$	$0.0797$	$0.0245$	$0.0280$	$0.0325$	$0.0362$	$0.0362$
4%	$0.0298$	$0.034$	$0.0405$	$0.046$	$0.1101$	$0.0287$	$0.0329$	$0.0405$	$0.0461$	$0.1095$
${\mathit{\phi }}_1={\mathit{\phi }}_2$	$Cu-A{l}_2{O}_3/$Engine oil					$Cu-Ti{O}_2/$Engine oil
	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$	${\mathit{n}}_{\mathbf{3}}$	${\mathit{n}}_{\mathbf{3.7}}$	${\mathit{n}}_{\mathbf{4.8}}$	${\mathit{n}}_{\mathbf{5.7}}$	${\mathit{n}}_{\mathbf{16.2}}$
1%	$0.0163$	$0.017$	$0.0182$	$0.0191$	$0.0293$	$0.0164$	$0.0171$	$0.0182$	$0.01915$	$0.0296$
2%	$0.0187$	$0.0203$	$0.0228$	$0.0248$	$0.0495$	$0.0188$	$0.0204$	$0.0229$	$0.0249$	$0.0497$
3%	$0.0213$	$0.0239$	$0.0279$	$0.0314$	$0.0730$	$0.0215$	$0.024$0	$0.0281$	$0.0316$	$0.0734$
4%	$0.0242$	$0.0295$	$0.0337$	$0.0387$	$0.0984$	$0.0244$	$0.0280$	$0.0341$	$0.0390$	$0.0988$

a,b analyzed the effects of different size factors (spherical, bricks, cylinder, platelet, and laminar) and several types of nanoparticles and base fluid comparisons shown in heat transfer. With the several kinds of hybridizing nanoparticles (HNFD1, HNFD2, HNFD3, HNFD4, HNFD5, HNFD6, HNFD7, HNFD8, HNFD9, HNFD10, HNFD11, HNFD12), the flow of heat transfer (Nusselt number) is improved if we raise the values of both nanoparticle volumes fractions ${\phi }_1$ and ${\phi }_2$. Moreover, compared to other shape factors and hybridizing nanoparticles, the laminar shape factor (HNFD1) is superior for the flow of heat transmission (Nusselt number).

Table 4. Shear stress and tensional stress impact several non-dimensional parameters.

$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{R}\mathit{e}$	$\mathit{M}$	${\mathit{\phi }}_1$	${\mathit{\phi }}_2$	$\mathit{d}{\mathit{p}}_1$	$\mathit{d}{\mathit{p}}_2$	$\mathit{F}\mathit{r}$	${\mathit{f}}^{″}\left(\mathit{\eta }\right)$	${\mathit{g}}^{\prime }\left(\mathit{\eta }\right)$
−0.5	1	0.3	1	0.1	0.1	0.01	0.01	0.5	$5.247$	$3.255$
−1									$6.329$	$4.415$
−1.5									$7.422$	$5.565$
	3								$5.132$	$3.093$
	6								$5.102$	$3.051$
	9								$5.092$	$3.037$
		0.6							$6.503$	$4.013$
		0.9							$7.847$	$4.926$
		1.2							$9.73$	$5.664$
			2						$5.414$	$3.48$
			4						$5.725$	$3.883$
			6						$6.014$	$4.237$
				0.2					$6.221$	$4.068$
				0.3					$7.278$	$4.933$
				0.4					$8.401$	$5.84$
					0.2				$5.539$	$3.501$
					0.3				$5.84$0	$3.752$
					0.4				$6.149$	$4.009$
						0.05			$5.241$	$3.24$9
						0.1			$5.237$	$3.245$
						0.15			$5.233$	$3.2425$
							0.05		$5.241$	$3.24$9
							0.1		$5.237$	$3.245$
							0.15		$5.233$	$3.2425$
								1	$5.614$	$3.3004$
								1.5	$6.0123$	$3.344$
								2	$6.439$	$3.388$

shows the numerical effects of shear stress and tensional stress on non-dimensional parameters. Increases the values of the $Re$, $\alpha , {\phi }_1$, ${\phi }_2$, $M$, and $Fr$ then the flow of fluid gradually increases in shear and tensional stress. Moreover, to enhance the values of the porosity parameter and diameter of the first and second nanoparticles, the shear and tensional stress values were reduced in both porous disks.

Table 5. Eckert parameter E_c effects on numerical heat transfer for Re = 3, α = 2 and Pr = 6.2.

$\mathit{E}\mathit{c}$	$\left|{\mathit{\theta }}^{\prime }\left(-1\right)\right|$	$\left|{\mathit{\theta }}^{\prime }\left(1\right)\right|$
2	1.9051	0.2206
4	2.5239	0.0906
6	2.9857	0.0437
8	3.369	0.0232
10	3.7172	0.0129

represents the impact of the inflow of heat transfer ($E_c)$ for both porous disks. As a result, as we raise the values of ($E_c)$, the flow of heat transfer for the lower disk increases while strictly dropping for the upward disk. Physically, the Eckert number is the product of the specific enthalpy difference between the wall and the fluid and the kinetic energy. As a result, a rise in the Eckert number results in work being performed against the viscous fluid stresses, which converts kinetic energy into internal energy.

Table 6. Different nondimensional parameter effects on the Nusselt number.

$\mathit{R}\mathit{e}$	$\mathit{\alpha }$	$\mathit{P}\mathit{r}$	$\left|\mathit{N}{\mathit{u}}_{\mathit{z}-1}\right|$
−1.5	1	1	$0.19442$
−1			$0.30838$
1			$1.30681$
1.5			$1.69612$
−1	−2		$0.62786$
	−1		$0.30838$
	1		$0.23170$
	2		$0.05776$
	−1	2	$2.58310$
		4	$5.6498$
		6	$8.7222$
		8	$11.754$

demonstrates the impact of Re, $\alpha ,$ and $P_r$ on the Nusselt number. The Nusselt number grows significantly when the Re and $P_r$ values are higher than the rate of heat transfer. Physically, a dimensionless number called the Prandtl number connects a fluid’s viscosity to its thermal conductivity. As a result, it evaluates the relationship between the transmission of momentum and a fluid’s capability for heat transport. Additionally, the flow of heat transfer is decreased since the expansion ratio $\alpha$ is greater than working. Before visualizing the findings, we used

Table 7. Comparison of the upper disk’s flow of heat transfer rate for fixed values of Re = 0, Pr = 2.92, M = 0.

	Hussain and Akbar [53]	Present Results	Hussain and Akbar [53]	Present Results
Ec	α < 0	α < 0	α > 0	α > 0
0	1.2461	1.2462	1.5675	1.5676
0.3	4.7629	4.7630	5.99363	5.99364
0.5	7.9383	7.9384	9.98939	9.98940
0.8	12.701	12.702	15.9834	15.9835
1.1	17.464	17.465	21.9767	21.9768

to confirm our findings against previously published research publications. An outstanding coincidence was observed, supporting the accuracy of the Mathematica code for the “shooting method”. By providing the values for the following parameters, the present analysis is estimated: $Re=0$, $Pr=2.92$, $M=0$.

Figure 3 displays the thermal conductivity of hybrid nanofluid $k_{hnf}$ with several types of nanoparticles and only one base fluid water (HNDF1, HNDF4, HNDF7, HNDF10) or different types of size factors used (spherical, bricks, cylinder, platelet, and laminar). Figure 3 shows four sections: for section, no. 1 we used HNDF1; no. 2, HNDF4; no. 3, HNDF7; and no. 4, HNDF10. The sections were used with increasing values of the volume fractions and several types of shape factors (spherical, bricks, cylinder, platelet, and laminar), but we noticed that when the NP volume fraction level was 0.04 in HNFDs1, better thermal conductivity efficiency was attained at 16.2. This implies that if we progressively raise the quantitative value of ${\phi }_1$ and ${\phi }_2$, the thermal conductivity will rise along with the form factor. Figure 4 demonstrates the effect of viscosity-hybrid nanofluid ${\mu }_{hnf}$ with volume fractions of ${\phi }_1$ and ${\phi }_2$ and diameters of $d{p}_1$ and $d{p}_2$ for both NP under the various form and size factor value diameters used (spherical, bricks, cylindrical, platelet, laminar). Additionally, with a size factor of 16.2, the influence of viscosity is particularly strong as well as diameters of 25 when the NP’s volume fraction ${\phi }_1$ and ${\phi }_2$ is 0.04, as shown in Figure 4. We found that increasing the volume fraction and shape or size parameter numbers increased the thermal conductivity and viscosity of the hybrid nanofluid in Figure 3 and Figure 4. In this instance, laminar-shaped nanomaterials outperform other NP forms in terms of heat and mass transferability.

Figure 5 reveals the Forchheimer number $\left(Fr\right)$ impact of the radial velocity profile influenced by the fixed values $\alpha =2, Re=1, {\phi }_1={\phi }_2=0.01, dp1=dp2=10, M=2, \beta =1$. A commonly analyzed that the Forchheimer number $\left(Fr\right)$ is greater than zero impact of the normal boundary layer increases for an upper and lower porous disk.

Figure 6 presents the impact of the Forchheimer number $\left(Fr\right)$ on the tangential velocity profile. The momentum boundary layer of the tangential velocity profile is reduced in the upper and lower disk if we raise the values of the Forchheimer number.

Figure 7 demonstrates the porosity parameter effect in the radial velocity profile. If we raise the values of the porosity parameter ($\beta$), then the radial velocity profile values rise to the middle of the wall and are reduced for both porous disks. Figure 8 shows that the impact of the porosity parameter ($\beta$) is analyzed by the tangential velocity profile. Raising the values of $\beta$ then gradually increases the flow of fluid to the center of the wall and the same behavior of the upper disk is observed but the opposite behavior is shown in the lower disk.

The expansion ratio impact on the radial velocity profile is shown in Figure 9. If the expansion ratio $\alpha$ is greater or less than zero, the flow of the momentum boundary layer thickness rises to the middle of the wall but the flow of the hybrid nanofluid is reduced in both porous disks. Figure 10 represents the effect of the temperature profile for the fixed values of $Re=1, {\phi }_1={\phi }_2=0.02,d{p}_1=d{p}_2=20, M=4$. In this case, a rise in the values $\alpha$ of the flow of thermal transfer rates acts oppositely in both porous disks.

Figure 11 illustrates the effects of magnetic parameters on the radial velocity profiles with constant values of $\alpha =1, Re=1, Pr=6.2$. When the magnetic parameter $M$ values are greater than zero, the flow of the momentum boundary layer thickness is enhanced for the upper and lower disk. This outcome is consistent with the idea that the applied magnetic field acts as a resistance force that is essential for slowing down and controlling fluid movement. Furthermore, Figure 12 shows the tangential velocity profile flow of the momentum boundary layer acting oppositely in the upper and lower porous disk.

The permeable Reynolds number impact of the radial velocity profile and temperature profile is displayed in Figure 13 and Figure 14. Physically, the Reynolds number is equal to the ratio of the inertia force over the viscous forces. Geometrically, we used the Reynolds numbers less than 2000 so that the flow was laminar or the shapes were graphically symmetrical. Furthermore, if the permeable Reynolds number values grow, then the flow of momentum and the thermal boundary is decreasing the middle of the wall but is enhanced for both boundary layer porous disks.

4. Conclusions

The hybrid nanofluid flow of heat transfer under the impact of shape and size factors, unsteady, three-dimensional, Newtonian, MHD fluid flow across two orthogonally moving porous coaxial disks. We use several NP types for this purpose. The basic fluids utilized were water, engine oil, and ethylene glycol. Metallic and metallic-oxide NP were also employed. Using velocity and temperature profiles, the impacts of nanoparticle shape and size variables were investigated. It was found that the form of the particle has a significant impact on the velocity and temperature profiles. The following are the key findings:

• Laminar nanoparticles 16.2 within the volume fraction of 4 percent in HNFD1 have the maximum thermal conductivity and viscosity intensity, as compared to other shapes spherical, brick, cylindrical, and platelet nanoparticles, respectively.
• The flow rate of heat transfer is enhanced on both porous disks by increasing the expansion of the value ratio $\alpha$ and permeable Reynolds number $Re$.
• Enhance the values of permeability Reynolds number $Re$, Forchheimer number $Fr$, and magnetic parameter $M$, the momentum boundary layer increases in the upper and lower porous surface.
• Shear stress and tensional stress flow rates are slowed down if the porosity parameter, as well as the diameters of the nanoparticles, $d{p}_1$, and $d{p}_2$, are larger than zero.
• Heat transfer rates are raised on both porous disks by increasing the values of Prandtl number $Pr$ and Reynolds number $Re$.
• Eckert number parameter values are raised and the flow of heat transfer rates in both porous disks is reduced.