Morphology of Hybrid MHD Nanofluid Flow through Orthogonal Coaxial Porous Disks

Abstract

In this article, we study the novel features of morphological effects for hybrid nanofluid flow subject to expanding/contracting geometry. The nanoparticles are incorporated due to their extraordinary thermal conductivity and innovative work for hybrid nanofluids, which are assembled of aluminum oxides, Al₂O₃ metallic oxides, and metallic copper Cu. Cu nanoparticles demonstrate very strong catalytic activity, while Al₂O₃ nanoparticles perform well as an electrical insulator. The governing partial differential equations of the elaborated model are transformed into a system of nonlinear ordinary differential equations with the use of similarity variables, and these equations are numerically solved through a shooting technique based on the Runge–Kutta method. We develop a hybrid correlation for thermophysical properties based on a single-phase approach. A favorable comparison between shape and size factors for metallic and metallic-oxide nanoparticles is discussed via tables and figures. Moreover, the effect of embedding flow factors on concentration, velocity, and temperature is shaped in line with parametric studies, such as the permeable Reynolds number, nanoparticle volume fractions, and expansion/contraction parameters. The fluid velocity, temperature, and concentration are demonstrated in the presence of hybrid nanoparticles and are discussed in detail, while physical parameters such as the shear stress, flow of heat, and mass transfer at the lower and upper disks are demonstrated in a table. The hybrid nanoparticles show significant results as compared to the nanofluids. If we increase the nanoparticle volume fraction, this increases the thermal performance for an injection/suction case as well. The above collaborative research provides a strong foundation in the field of biomedical equipment and for the development of nanotechnology-oriented computers.

1. Introduction

The morphology of nanomaterials has proved effective in influencing the actions of nanomaterials under optical waves and photons. Similarly, the particle form influences nanoparticle (NP) movement in the circulatory system, immunological responses, and activity within internalized cells. Furthermore, the asymmetry of NPs induces particle chipping and rounding along the underflow walls of blood vessels. It also causes NP penetration and distribution in tumors and tissues. Furthermore, due to a diversity of reactions with the cell surface, NPs of diverse sizes and shapes might accumulate differently inside the same cell. Non-spherical NPs also prefer to stay in the blood as compared to spherical ones, as indicated by their higher blood circulation duration. The absorption efficiency of rod-shaped NPs is also superior to that of cubical, cylindrical, and spherical ones. Spherical NPs can be processed quickly after internalization, have acute margination effects, and have quicker blood circulation than other shapes. However, rod-shaped NPs show a comparatively slower processing time after internalization and can enter solid-tissue tumors with comparative ease, as discussed by Qureshi et al. [1]. Zeeshan et al. [2] investigated the various forms of NPs floating across a spinning disk on combined convection past steady flow. For a nanofluid (Nfd), the base fluid is known to be copper NPs of disk, cylindrical, and spherical types of varying sizes and water. Haq et al. [3] studied the completely formed Nfd pressing flux between parallel disks. They found three separate forms of condensed NPs in the base fluid. These NPs were Magnetite $\left({\mathrm{Fe}}_3{\mathrm{O}}_4\right)$, Mn–Zn ferrite (Mn–Zn ${\mathrm{Fe}}_2{\mathrm{O}}_4$), and cobalt ferrite (Co${\mathrm{Fe}}_3{\mathrm{O}}_4$). The movement and heat transfer of copper NPs embedded in the base fluid were studied by Khan et al. [4]. They included water as a liquid basis. They found a template for the thermal conductivity of Nfds by Hamilton and Crosser. The research is known to be done in three types of NPs, namely, platelet-, cylinder-, and brick-shaped particles. By including a mathematical decrease at the mainline level, Ashraf et al. [5] investigated computer modeling of continuous natural convection flow in the presence of a strong magnetic field and a weak magnetic quantity of Prandtl. The findings regarding the mathematical diffusion coefficient, related magnetic field, and buoyant body energy were used in the momentum equation to generate a series of boundary layer equations before moving on to the computer simulation.

It is evident from the literature that unified transfer of heat and mass, from diverse geometries, has massive engineering and geophysical implications, including in geothermal reservoirs, drying of porous solids, enhanced oil recovery, thermal insulation, and underground species transport. Heat and mass transfer are relevant in the many varieties of thermodynamic devices, such as pumps, compressors, turbines, nozzles, diffusers, refrigeration systems, air conditioning systems, microwave ovens, conventional ovens, and coffee makers, as discussed by Qureshi et al. [6]. This drew the attention of researchers towards such a combined process. Devi et al. [7] analyzed the motion of an electrically conductive fluid that is affected by thermal radiation and is surrounded by an infinite rotating porous disk with heat and mass transfer. On the contrary, unsteady mixed convection of a hybrid nanofluid (HNfd) due to a rotating disk was also observed. Accordingly, Hang Xu et al. [8] presented a generalized homogenous flow model, explaining HNfd as incorporating diversified versions of NPs, as a construct to model these issues. Moreover, Mabood et al. [9] investigated the effect of the fraction volume on metallic or metallic-oxide NPs + H₂O, radiation, viscous dissipation, and chemical reaction, including magnetohydrodynamic (MHD) heat and mass transfer stagnation point flow. The flow over a revolving disk on HNfd reveals the character of the Hall current, which was represented in another study by Acharya et al. [10]. Acharya et al. [11] studied the properties of heat and fluid movement in the form of thermophoretic transport; a numerical simulation was developed based on nonlinear paired partial differential equations respecting the conservation rules. A thermophoretic transport term is incorporated in the mass equation; hence, the influence of thermophoretic transport may be measured at various sphere locations. Hussain et al. [12] developed this by keeping a porous disk on metallic NPs, which addressed many issues due to the convection of heat and mass transfer of an incompressible viscous Nfd in a mathematical model. Das et al. [13] mathematically investigated a couple of connecting disks with a magnetic field and slip effect squeezed by the flow of Nfd. The equivalent disk on squeezed MHD flow of a Nfd is affected by both velocity and temperature slip, as discussed by Mohyud-Din et al. [14]. On a floating plane surface, Erickson et al. [15] observed the permeable expansion ratio and the collective effect of heat and mass transfer. On the other hand, the regular thermal conduction boundary layer flow and the heat and mass exchange of Nfd, which are influenced by several factors such as the size, shape, and kind of Nfd, as well as the type and operating temperature of the base fluid, were observed by Zakari et al. [16]. Turkyilmazoglu et al. [17] developed fluid flow and heat transfer as the by-product of a disk, rotating both upward and downward, during uneven motion. It was also found that this upward and downward movement of the disk exerts a similar effect to permeability across the wall, albeit with identifiable differences.

Correspondingly, the role of magnetic orientation is monumental in affecting heat transport in HNfd. Likewise, emerging technologies including biotechnology, electrical engineering, and electromechanics have a more positive role in the case of magnetic field influence. Transformers, generators, and electric motors are useful applications of magnetic fields. For thermally conductive fluid, certain investigators researched the issue of magnetohydrodynamic (MHD) boundary layer flow, heat, and mass transfer among different layer geometries. Temperature profile contraction was observed with increased merits of heat energy deposition under layer constraint when Mutuku et al. [18] noticed an unsteady MHD boundary layer, temperature, and concentration profile of Nf over a surface with dual deposition in the layers. Sudarsana Reddy et al. [19,20] analyzed the MHD physical thermal conduction boundary layer flow, heat, and mass transfer features of a pair of separated Nfds, across a rotation disk and vertical cone, along with the chemical reaction. The free heat thermal conduction and mass transfer were measured in nanoliquid MHD flow through an irregular thickness across a slender elastic surface by Ashraf et al. [21]. Lou et al. [22] considered the micropolar dusty fluid with the dynamic effect of MHD rotating fluid. Several researchers have discussed heat and mass transfer and MHD boundary layer flow attributes across multiple geometries [23,24].

Choi [25] coined the idea of Nfds (the mixing of nanosized solid particles in base fluids) with higher thermal conductivity in comparison to the routine fluids available. Furthermore, the size of a Nfd plays a pivotal role in bringing stability to it, with a size of less than 100 nm considered a stable size for Nfd. Resultantly, Nfd has a handful of practical utilities as well, including in industrial cooling, paper, biomedicine, chemical reactors, and many more. Recent research has started considering HNfds (containing two or more NPs) in place of ordinary Nfds because of their similar yet comparatively superior performance outcomes. Nimmagadda and Priyadharshini et al. [26,27] measured an emerging concept in technology-related work. The viscosity and heat energy conductivity of an Al₂O₃-Cu/H₂O HNfd was calculated. Suresh et al. [28] showed that with a solid volume fraction of Al₂O₃-Cu NPs, all parameters were increased. In the same way, Sarkar et al. [29] brought the attention of researchers to the area of HNfds. Moghadassi et al. [30] studied the effects of Al₂O₃-H₂O Nfd, along with the effects of Al₂O₃-Cu/H₂O HNfd, showing that Al₂O₃-Cu/H₂O HNfd had greater heat transfer. This was followed by the analytical and numerical investigation of HNfd conducted by Huminic and Huminic et al. [31]. Ashraf et al. [32] investigated the effect of radiation on the continuous, viscous, and electrically conductive mixed-convection boundary layer movement of fluid over a semi-infinite porous longitudinally magnetized magnet with uniform sweat and a variable cross-cut magnetic field on the top.

In porous media, the presence of extensive fluid interaction surfaces can enhance the heat transmission effect. The porous medium alters the flow field conditions, thins the frontal layer, and typically has a higher conduction heat transfer coefficient than the fluid under study. Numerous technical applications, including polymeric, ceramic, and metallic foams, have made heat transmission in porous media a hot research area. For the past few decades, flow in porous medium has been the focus of active research. The effects of capillary forces and the kind of wettability of the medium on the displacement process were addressed in [33,34], which evaluated the process of imbibition in a porous medium under the influence of capillary forces in microgravity. The authors of [35,36,37] proposed a non-stationary mathematical model of multiphase fluid flow and computationally modeled the removal of a viscous fluid from a porous material while accounting for capillary effects in a sample of a porous medium. The temperature and velocity profiles above condensed material were theoretically determined within the framework of boundary layer approximation under the assumption of fuel gasification and gas-phase chemicals reacting in a diffusion flame. Logvinov et al. [38] investigated the removal of a viscous fluid from a rectangular Hele-Shaw cell. Tyurenkova et al. [39] studied the rate of material surface regression in turbulent and laminar flow regimes.

A review of the literature revealed to the authors that no research has been done on heat and mass transfer analysis of MHD incompressible hybrid nanofluid flow subject to two porous coaxial discs that are moving orthogonally. Thermal conductivity is most commonly described by form factors, and the viscosity model is developed as a hybrid model of thermophysical properties that depend on the size factors. These facts served as the impetus for the research herein, which was done to account for two distinct categories of nanomaterials and to depict a fully formed hybrid nanofluid that moves across porous disks in a strong magnetic field. The system was simplified to multiple related nonlinear ordinary differential equations (ODEs) that regulate the flow of an electromagnetic hydrodynamic hybrid nanofluid between orthogonal porous disks. Numerical calculation was performed to solve the system of ordinary differential equations via Runge–Kutta shooting techniques with the help of MATLAB software.

2. Mathematical Formulation

In this paper, the single-phase approach was used to model the HNfd because the single-phase model is more suitable for Newtonian fluid and its thermophysical properties based on the NP volume of the fraction. It is also called the experimental model [40]. The detailed HNfd thermophysical properties used in this paper are shown in

Table 1. Thermophysical properties of $H{N}_{fd}$ [41].

Properties	$\mathbf{Hybrid} \mathbf{Nanofluid} (\mathbf{Cu}–\mathbf{A}{\mathbf{l}}_2{\mathbf{O}}_3–\mathbf{Water})$
$\mathbf{Density} \mathbf{\left(}\mathit{\rho }\mathbf{\right)}$	${\rho }_{hnf}={\phi }_1{\rho }_{s_1}+{\phi }_2{\rho }_{s_2}-\left(1-{\phi }_1-{\phi }_2\right) {\rho }_{bf}$
	${\mu }_{hnf}={\mu }_{bf}(1+0.1008({\left({\phi }_1\right)}^{0.69574}(d{p}_1{)}^{0.44708}+{\left({\phi }_2\right)}^{0.69574}{(d{p}_2)}^{0.44708}))$
$\mathbf{Viscosity} \mathbf{\left(}\mathit{\mu }\mathbf{\right)}$
$\mathbf{Heat} \mathbf{Capacity} \mathbf{\left(}\mathit{\rho }{\mathit{C}}_{\mathit{P}}\mathbf{\right)}$	${\left(\rho c_p\right)}_{hnf}={\phi }_1{\left(\rho c_p\right)}_{s_1}+{\phi }_2{\left(\rho c_p\right)}_{s_2}+\left(1-{\phi }_1-{\phi }_2\right) {\left(\rho c_p\right)}_{bf}$
	$k_{hnf}$ = $\frac{k_{s2}+\left(N-1\right)k_{mbf}-\left(N-1\right){\phi }_2\left(k_{mbf}-k_{s2}\right)}{k_{s2}+\left(N-1\right)k_{mbf}+{\phi }_2\left(k_{mbf}-k_{s2}\right)}{k}_{mbf}$
Thermal Conductivity (K)	Where
	$k_{mbf}$ = $\frac{k_{s1}+\left(N-1\right)k_{bf}-\left(N-1\right){\phi }_1\left(k_{bf}-k_{s1}\right)}{k_{s1}+\left(N-1\right)k_{bf}+{\phi }_1\left(k_{bf}-k_{s1}\right)}{k}_{bf}$

, and

Table 2. Thermophysical properties of base fluid and NPs [42].

Title	${\mathbf{H}}_2\mathbf{O} \left(\mathbf{f}\right)$	$\mathbf{A}{\mathbf{l}}_2{\mathbf{O}}_3\left({\mathit{\phi }}_{\mathbf{1}}\right)$	$\mathbf{Cu}\left({\mathit{\phi }}_{\mathbf{2}}\right)$
$\mathit{\rho } ($$\mathbf{kg}{m}^{-\mathbf{3}}$)	997.0	3970	8933
${\mathit{C}}_{\mathit{p}}$$(\mathbf{J} \mathbf{k}{g}^{-\mathbf{1}}{k}^{-\mathbf{1}})$	4180	765	385
$\mathit{\kappa } \left(w{m}^{-1}{k}^{-\mathbf{1}}\right)$	0.6071	40	400

presents the thermophysical properties of the basic fluid and NPs. Due to their significant amount of essential porous structure, the Cu NPs exhibit very robust catalytic efficiency. ${\mathrm{Al}}_2{\mathrm{O}}_3$ NPs are an insulating material, but ceramic material has rather good heat transparency. In a comparison of both NP types, the thermal conductivity of Cu NPs is relatively higher than that of ${\mathrm{Al}}_2{\mathrm{O}}_3$.

A two-dimensional Newtonian hybrid MHD nanofluid flow (${\mathrm{Al}}_2{\mathrm{O}}_3$–Cu/water) was considered between the deformable porous disks with cylindrical coordinates (r, θ, z); this was chosen as it can be proved as a most appropriate prototype for morphology analysis of heat and mass transfer. The fluid is incompressible, unsteady, isothermal, and laminar. The porous walls expand and contract up- and downward. The geometry was developed with significant modification as shown in Figure 1. The channel walls can move up and down at time-dependent rate $k^{\prime }\left(t\right)$ with range 2$k\left(t\right)$. The resulting magnetic field $B_0$ is disregarded under the minimal Reynolds number supposition. Terms $T_1$, $T_2$, $C_1$, and $C_2$ denote the temperature and concentration at the lower and upper permeable disks, correspondingly, with $T_1$ > $T_2$ and $C_1>C_2$. The laws of conservation of mass, momentum, energy, and concentration are expressed as follows [43]:

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

(1)

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

(2)

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

(3)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}={\alpha }_{hnf}\frac{{\partial }^{2}T}{\partial z^2},$$

(4)

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D_{hnf}\frac{{\partial }^{2}C}{\partial z^2},$$

(5)

where $B_0^2$ is the intensity of the magnetic field,$p$ denotes pressure, ${\sigma }_e$ denotes electrical conductivity, $T$ denotes the temperature, ${\upsilon }_{hnf}$ is the kinematic viscosity of $H{N}_{fd}$, ${\rho }_{hnf}$ is the density of $H{N}_{fd}$, ${\alpha }_{hnf}$ is the thermal diffusivity of $H{N}_{fd}$, and $D_{hnf}$ represents the diffusion coefficient of $H{N}_{fd}$. The initial boundary conditions for the lower and upper boundaries are:

$$\begin{array}{c}{z}_1=-k\left(t\right), u=0, w=-A_1{k}^{\prime }\left(t\right), T=T_1, C=C_1, \\ z_1=k\left(t\right), u=0, w=A_1{k}^{\prime }\left(t\right), T=T_2 C=C_2. \end{array}\}$$

(6)

Here, $A_1$ is the measure of partition penetrability, and the prime symbol denotes the derivative with respect to time t.

To continue the investigation, we provide the following set of transformation variables [43].

$$\eta =\frac{z_1}{k},u=-\frac{r{\upsilon }_f}{k^2}{F}_{\eta }\left(\eta ,t\right),w=\frac{2{\upsilon }_f}{k}F\left(\eta ,t\right), \theta =\frac{T-T_2}{T_1-T_2}, \chi \left(\eta \right)=\frac{C-C_2}{C_1-C_2}. \}$$

(7)

The continuity Equations (1) and (5) were fulfilled using the similarity transformation. Consequently, Equations (1)–(5) were transformed into the dimensionless forms:

$$\frac{{\upsilon }_{hnf}}{{\upsilon }_f}{F}_{\eta \eta \eta \eta }+\alpha (3F_{\eta \eta }+\eta F_{\eta \eta \eta })-2F{F}_{\eta \eta \eta }-\frac{k^2}{{\upsilon }_f}{F}_{\eta \eta t}-\frac{{\rho }_f}{{\rho }_{hnf}}\mathrm{M}{F}_{\eta \eta }=0,$$

(8)

$${\theta }_{\eta \eta }+\frac{{\upsilon }_f}{{\alpha }_{hnf}}(\alpha \eta -2F){\theta }_{\eta }-\frac{k^2}{{\alpha }_{hnf}}{\theta }_t=0,$$

(9)

$${\chi }_{\eta \eta }+Sc (\alpha \eta -2F){\chi }^{\prime }-k^2{\chi }_t=0.$$

(10)

where α = $\frac{k{k}^{\prime }\left(t\right)}{{\upsilon }_f}$ is the wall expansion ratio, Re = $\frac{A_{1}k{k}^{\prime }\left(t\right)}{2{\upsilon }_f}$ is the permeable Reynolds number, $Sc$ = $\frac{{\upsilon }_f}{D}$ is the Schmidt number, ${\upsilon }_{hnf}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}$ is the kinematic viscosity of hybrid nanofluid, ${\alpha }_{hnf}=\frac{k_{hnf}}{{\left(\rho c_p\right)}_{hnf} }$ is the thermal diffusivity of hybrid nanofluid, $Pr=\frac{{\left(\mu C_p\right)}_{bf}}{k_{bf}}$ is the Prandtl number, and M = $\frac{{\sigma }_e{B}_0^2{k}^2}{{\mu }_f}$ is the magnetic parameter.

The dimensionless physical limitations listed below were considered:

$$\begin{array}{c} \eta =-1, F=-Re,F_{\eta }=0,\theta =1, \chi =1,\\ \mathrm{and} \eta =1, F=Re,F_{\eta }=0,\theta =0, \chi =0.\end{array}\}$$

(11)

Finally, Majdalani et al. [44] set F = $f$ Re, α is a constant, $f$ = $f$ $\left(\eta \right)$, θ = $\theta \left(\eta \right),$ and $\chi =\chi \left(\eta \right)$, which leads to ${\theta }_t$ = 0, $f_{\eta \eta t}$ = 0, and ${\chi }_t=0$. Thus, we have the following equations:

$$\frac{{\upsilon }_{hnf}}{{\upsilon }_f}{f}_{\eta \eta \eta \eta }+\alpha (3f_{\eta \eta }+\eta f_{\eta \eta \eta })-2Ref{f}_{\eta \eta \eta }-\frac{{\rho }_f}{{\rho }_{hnf}}\mathrm{M}{f}_{\eta \eta }=0,$$

(12)

$${\theta }_{\eta \eta }+\left(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+\left({\phi }_1\right)\left(\frac{{\rho }_{cp{s}_1}}{{\rho }_{cpbf}}\right)+\left({\phi }_2\right)\left(\frac{{\rho }_{cp{s}_2}}{{\rho }_{cpbf}}\right)\right)\frac{ k_{mbf} }{ k_{hnf} }\frac{ k_{bf} }{ k_{mbf} }Pr(\alpha \eta -2Ref){\theta }_{\eta }=0,$$

(13)

$${\chi }^{″}+Sc(\alpha \eta -2fRe){\chi }^{\prime }=0.$$

(14)

$$\begin{array}{c} \eta =-1, f=-1, f_{\eta }=0,\mathsf{\theta }=1, \chi =1, \\ \eta =1, f=1, f_{\eta } =0,\mathsf{\theta }=0 \chi =0. \end{array}\}$$

(15)

Quantities of engineering interest: Skin friction, the Nusselt number, and the Sherwood number are physical parameters that are crucial to the technical goal of modeling equipment at the nanoscale. All of these variables were estimated on both porous surfaces.

Skin friction coefficients: Skin friction coefficients for the lower and upper discs are represented by the variables $C_{f1}$ and $C_{f-1}$ and are stated as

$$C_{f-1}=\frac{{\zeta }_y{|}_{\mathsf{\eta } =-1}}{{\rho }_f{\left(s^{\prime }A\right)}^2}=\frac{(1+0.1008({\left({\phi }_1\right)}^{0.69574}(d{p}_1{)}^{0.44708}+{\left({\phi }_2\right)}^{0.69574}{(d{p}_2)}^{0.44708}))}{R{e}_r}{f}^{″}\left(-1\right),$$

(16)

$$C_{f1}=\frac{{\zeta }_y{|}_{\mathsf{\eta } =1}}{{\rho }_f{\left(s^{\prime }A\right)}^2}=\frac{(1+0.1008({\left({\phi }_1\right)}^{0.69574}(d{p}_1{)}^{0.44708}+{\left({\phi }_2\right)}^{0.69574}{(d{p}_2)}^{0.44708}))}{R{e}_r}{f}^{″}\left(1\right),$$

where ${\xi }_{zr}$ represents the shear stress at the lower and upper discs in the radial direction, and $R{e}_r=4\left(\frac{k}{r}\right){\left(\frac{1}{Re}\right)}^2$ stands for the local Reynolds number.

$$\begin{gathered} {\zeta }_y={\mu }_{hnf}(\frac{\partial u}{\partial y}){|}_{\mathsf{\eta } =-1}={\mu }_{bf}(1+0.1008({\left({\phi }_1\right)}^{0.69574}(d{p}_1{)}^{0.44708}+ \\ {\left({\phi }_2\right)}^{0.69574}{(d{p}_2)}^{0.44708}))\left(\frac{r{\upsilon }_f}{s^3}\right)f^{″}\left(-1\right), \end{gathered}$$

$$\begin{gathered} {\zeta }_y={\mu }_{hnf}(\frac{\partial u}{\partial y}){|}_{\mathsf{\eta } =1}={\mu }_{bf}(1+0.1008({\left({\phi }_1\right)}^{0.69574}(d{p}_1{)}^{0.44708}+ \\ {\left({\phi }_2\right)}^{0.69574}{(d{p}_2)}^{0.44708}))\left(\frac{r{\upsilon }_f}{s^3}\right)f^{″}\left(1\right). \end{gathered}$$

Nusselt numbers: The ratio of conductive to convective heat flow inside a fluid at a boundary is known as the Nusselt number ($Nu$). Fluid movement (advection) and diffusion are components of convection (conduction). A Nusselt number of 1 represents pure conduction as the mode of heat transfer. The analysis of the heat transfer rates (Nusselt numbers) $N{u}_{\mathsf{\eta } =-1}$ and $N{u}_{\mathsf{\eta } =1}$ at the lower and upper discs is provided as

$$Nu{|}_{\mathsf{\eta } =-1}=\frac{k{s}_z}{{\kappa }_f\left(T_1-T_2\right)}{|}_{\mathsf{\eta } =-1}=-\frac{k_{hnf}}{k_f }{\theta }^{\prime }\left(-1\right),$$

$$Nu{|}_{\mathsf{\eta } =1}=\frac{k{s}_z}{{\kappa }_f\left(T_1-T_2\right)}{|}_{\mathsf{\eta } =1}=-\frac{k_{hnf}}{k_f }{\theta }^{\prime }\left(1\right),$$

(17)

where heat flux is denoted by $s_z$, which follows as

$$s_z{|}_{\mathsf{\eta } =-1}=-k_{hnf}\left(\frac{\partial T}{\partial z}\right){|}_{\mathsf{\eta } =-1}=-\frac{\left(T_1-T_2\right)}{k}{k}_{hnf}{\theta }^{\prime }\left(-1\right),$$

$$s_z{|}_{\mathsf{\eta } =1}=-k_{hnf}\left(\frac{\partial T}{\partial z}\right){|}_{\mathsf{\eta } =1}=-\frac{\left(T_1-T_2\right)}{k}{k}_{hnf}{\theta }^{\prime }\left(1\right).$$

Sherwood numbers: The proportion of convectional mass transfer to diffusional mass transfer is known as the Sherwood number. The following mathematical equations describe the mass transfer rates (Sherwood numbers) $Sh{|}_{\mathsf{\eta } =-1}$ and $Sh{|}_{\mathsf{\eta } =1}$ at the lower and upper discs:

$$Sh{|}_{\mathsf{\eta } =-1}=\frac{k{q}_z}{D_{hnf}\left(C_1-C_2\right)}{|}_{\mathsf{\eta } =-1}=-{\chi }^{\prime }\left(-1\right),$$

$$Sh{|}_{\mathsf{\eta } =1}=\frac{k{q}_z}{D_{hnf}\left(C_1-C_2\right)}{|}_{\mathsf{\eta } =1}=-{\chi }^{\prime }\left(1\right),$$

(18)

where

$$q_z{|}_{\mathsf{\eta } =-1}=-D_{hnf}\left(\frac{\partial C}{\partial z}\right){|}_{\mathsf{\eta } =-1}=-D_{hnf}\frac{\left(C_1-C_2\right)}{k}{\chi }^{\prime }\left(-1\right),$$

$$q_z{|}_{\mathsf{\eta } =1}=-D_{hnf}\left(\frac{\partial C}{\partial z}\right){|}_{\mathsf{\eta } =1}=-D_{hnf}\frac{\left(C_1-C_2\right)}{k}{\chi }^{\prime }\left(1\right),$$

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

Numerical Solution Procedure: To obtain numerical solutions for the transformed ODEs (12)–(14) with suitable transformed boundary conditions (15), Runge–Kutta and the efficient “shooting method’’ approach were taken into consideration. With this approach, the necessary dimensionless ODEs were simply handled. The nonlinear particle differential problem was transformed into a highly nonlinear linked system of ordinary differential equations, forming Equations (12)–(14).

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

(19)

$$\begin{array}{l}{\theta }^{″}\left[\eta \right]\\ +(\left(1-\left({\phi }_1+{\phi }_2\right)\right)+\left({\phi }_1\right)\left(\frac{{\rho }_{cp{s}_1}}{{\rho }_{cpbf}}\right)\\ +\left({\phi }_2\right)\left(\frac{{\rho }_{cp{s}_2}}{{\rho }_{cpbf}}\right))\left(\frac{k_{s2}+\left(N-1\right)k_{mbf}+{\phi }_2\left(k_{mbf}-k_{s2}\right)}{k_{s2}+\left(N-1\right)k_{mbf}-\left(N-1\right){\phi }_2\left(k_{mbf}-k_{s2}\right)}\right)\left(\frac{k_{s1}+\left(N-1\right)k_{bf}+{\phi }_1\left(k_{bf}-k_{s1}\right)}{k_{s1}+\left(N-1\right)k_{bf}-\left(N-1\right){\phi }_1\left(k_{bf}-k_{s1}\right)}\right)\end{array}$$

$$Pr(\alpha \eta -2Ref\left[\eta \right])\theta {}^{\prime }\left[\eta \right]=0,$$

(20)

$$\chi {}^{″}\left[\eta \right]+\mathrm{Sc}\left(\alpha \eta -2\mathrm{Re}f\left[\eta \right]\right){\chi }^{\prime }\left[\eta \right]=0,$$

(21)

Here, we assume that

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

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

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

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

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

$\omega =D_1{D}_2$.

Putting values of $H_1$, $H_2, H_3$, $D_1,$ $D_2$, and $\omega$ in Equations (19)–(21), we obtain the final result as follows:

$$H_1{f}^{⁗}\left[\eta \right]+\alpha (3f{}^{″}\left[\eta \right]+\eta f{}^{‴}\left[\eta \right])-2Ref\left[\eta \right]f{}^{″}\left[\eta \right]-H_2\mathrm{M} f{}^{″}[\eta ]=0,$$

(22)

$$\theta {}^{″}\left[\eta \right]+H_3\omega Pr(\alpha \eta -2Ref\left[\eta \right])\theta {}^{\prime }\left[\eta \right]=0,$$

(23)

$$\chi {}^{″}\left[\eta \right]+\mathrm{Sc}\left(\alpha \eta -2\mathrm{Re}f\left[\eta \right]\right){\chi }^{\prime }\left[\eta \right]=0.$$

(24)

To solve the existing flow model, we used the RK technique with the addition of shooting methods. The following substitution is required to begin the process:

$$w_1=f\left[\eta \right], w_2=f{}^{\prime }\left[\eta \right], w_3=f{}^{″}\left[\eta \right], w_4=f{}^{‴}\left[\eta \right],w_5=\theta \left[\eta \right], w_6=\theta {}^{\prime }\left[\eta \right], w_7=\chi \left[\eta \right], w_8={\chi }^{\prime }\left[\eta \right].$$

(25)

First, in Equations (22)–(24), we change the model in the following pattern:

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

(26)

$${\theta }^{″}\left[\eta \right]=-(H_3\omega Pr(\alpha \eta -2Ref\left[\eta \right]{\theta }^{\prime }\left[\eta \right]),$$

(27)

$$\chi {}^{″}\left[\eta \right]=\mathrm{Sc}\left(-\alpha \eta +2\mathrm{Re}f\left[\eta \right]\right){\chi }^{\prime }\left[\eta \right],$$

The following system is obtained by using the substitution contained in Equation (25):

$$\left[\begin{array}{c}{w}_1^{\prime }\\ w_2^{\prime }\\ w_3^{\prime }\\ w_4^{\prime }\\ w_5^{\prime }\\ w_6^{\prime }\\ w_7^{\prime }\\ w_8^{\prime }\end{array}\right]=\left[\begin{array}{c}{w}_2\\ w_3\\ w_4\\ \frac{1}{H_1}\left(-\alpha \left(3w_3+\eta w_4\right)+2Re{w}_1{w}_3+H_2\mathrm{M} w_3\right) \\ w_6\\ -\left(H_3\omega Pr\left(\alpha \eta -2Re{w}_1\right)w_6\right) \\ w_7\\ \mathrm{Sc}\left(2Re{w}_1-\alpha \eta \right)w_8 \end{array}\right],$$

(28)

Consequently, the initial condition is:

$$\left[\begin{array}{c}{w}_1^{\prime }\\ w_2^{\prime }\\ w_3^{\prime }\\ w_4^{\prime }\\ w_5^{\prime }\\ w_6^{\prime }\\ w_7^{\prime }\\ w_8^{\prime }\end{array}\right]=\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\end{array}\right].$$

(29)

The above system was then solved using mathematics and a suitable initial condition. Here, Runge–Kutta and the well-known accurate “shooting method” were taken into consideration. The required dimensionless ODEs can easily be tackled with this method. First of all, we obtained the initial condition by using the shooting method in such a way that boundary conditions were satisfied and achieved the desired level of accuracy.

3. Results and Discussion

To understand the problem, we must carry out a numerical evaluation of the nonlinear differential equations. The impact of relevant physical parameters like the expansion/contraction ratio parameter ($\alpha )$, permeable Reynolds parameter $(Re$), Prandtl number $\left(Pr\right)$, shape factor $\left(N\right)$, magnetic parameter $(M$), Schmidt number $(Sc$), diameter of NPs $(d{p}_1$ and $d{p}_2$), and volume friction parameter $({\phi }_1$ and ${\phi }_2$) on the velocity, temperature, and concentration profiles is shown graphically in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. We validated our findings against previously published research articles using

Table 3. Comparison of results regarding the heat transfer rate at the lower disk for $Re=0, {\phi }_2=0,M=0$.

		Kashif et al. [45]	Present Results	Kashif et al. [45]	Present Results
$\phi$	${\phi }_1$	$\alpha <0$	$\alpha <0$	$\alpha >0$	$\alpha >0$
0%	0%	3.16640	3.16650	1.67941	1.67956
5%	5%	3.61121	3.61130	1.91744	1.91749
10%	10%	4.16060	4.16071	2.21351	2.21378
15%	15%	4.84304	4.84315	2.58392	2.58399
20%	20%	5.69882	5.69893	3.05193	3.051989

before visualizing the results. An excellent coincidence was obtained, confirming the validity of the method. The current analysis was produced by entering the following values for the parameters: $Re=1, {\phi }_2=0.01, d{p}_1=0.01, d{p}_2=0.021, \mathrm{and} M=1$. The competitive contrast gives support to the recorded numerical performance.

3.1. Effect of Variable Shear Stress (f′(η)), Heat Transfer (θ′(η)), Mass Transfer (χ(η)), and Thermal Conductivity of Hybrid Nanofluid (k_hnf)

Table 4. Calculations of the effect of shear stress at the lower and upper disks for Al₂O₃ + Cu/H₂O $H{N}_{fd}$ for different nondimensional parameters.

Al2O3 + Cu/H2O
$\mathit{\alpha }$	${\mathit{\phi }}_1$	${\mathit{\phi }}_2$	$\mathit{R}\mathit{e}$	$\mathit{M}$	$\mathit{d}{\mathit{p}}_1$	$\mathit{d}{\mathit{p}}_2$	$|{\mathit{f}}^{\prime }\left(-1\right)|$	$|{\mathit{f}}^{\prime }\left(1\right)|$
1				1	0.01	0.01	2.86822	2.86822
2			1				1.82187	1.82187
3	0.01 = 1%	0.01 = 1%					0.77265	0.77265
4							0.29907	0.29907
	0.02 = 2%	0.01 = 1%	1	1			2.86178	2.86178
1	0.03 = 3%						2.85581	2.85581
	0.04 = 4%						2.85028	2.85028
1	0.01 = 1%	0.02 = 2%	1	1			2.84421	2.84421
		0.03 = 3%					2.82103	2.82103
		0.04 = 4%					2.79874	2.79874
1	0.01 = 1%	0.01 = 1%	2				3.72387	3.72387
			3				5.00722	5.00722
			4				6.67346	6.67346
				2			3.09975	3.09975
1	0.01 = 1%	0.01 = 1%	1	3			3.31576	3.31576
				4			3.51844	3.51844
					0.05		5.241	5.241
					0.1		5.237	5.237
					0.15		5.233	5.233
					0.05	0.05	5.241	5.241
1	0.01 = 1%	0.01 = 1%	1	1		0.1	5.237	5.237
						0.15	5.233	5.233

presents the impact of different parameters like $\alpha$, $Re$, ${\phi }_1$, ${\phi }_2$, $M$, $d{p}_1$, and $d{p}_2$ on the shear stress for both porous disks with hybrid nanofluid (Al₂O₃ + Cu/H₂O). Increasing the values of $\alpha$, ${\phi }_1$, ${\phi }_2$ $d{p}_1$, and $d{p}_2$ decreased the shear stress, but an opposite effect was observed for $Re, M$ for the upper and lower disks.

Table 5. Heat transfer rates at the lower and upper disks for Al₂O₃ + Cu/H₂O $H{N}_{fd}$.

			Al2O3 + Cu/H2O
$\mathit{P}\mathit{r}$	N	$\mathit{\alpha }$	$| {\mathit{\theta }}^{\prime }\left(-1\right)|$	$| {\mathit{\theta }}^{\prime }\left(1\right)|$
5	3	1	2.20736	2.20736
5.7			2.52049	2.52049
6.2			2.74784	2.74784
7			3.11511	3.11511
6.2	3	1	3.00884	3.00884
	3.7		2.9688	2.9688
	4.9		2.90299	2.90299
	5.7		2.86822	2.86822
6.2	3	1	3.00884	3.00884
		2	1.03645	1.03645
		3	0.20387	0.20387
		4	0.02797	0.02797

indicates the effect of parameters $Pr, N,$ and $\alpha$ on heat transfer flow at the upper and lower disks. By enhancing the values of $N$ and $\alpha$, the heat transfer flow rate was reduced, but $Pr$ had the opposite impact for both porous disks. The mass transfer flow effect with Sc, Re, and $\alpha$ for both porous disks with $H{N}_{fd}$ is shown in

Table 6. Simulation results of the numerical impact of flow mass transfer at the lower and upper disks for Al₂O₃ + Cu/H₂O $H{N}_{fd}$.

			Al2O3 + Cu/H2O
$\mathit{S}\mathit{c}$	$\mathit{R}\mathit{e}$	$\mathit{\alpha }$	$|\mathit{\chi }\left(-1\right)|$	$|\mathit{\chi }\left(1\right)|$
1			0.77628	0.77628
2	1	1	1.13597	1.13597
3			1.56842	1.56842
4			2.05396	2.05396
1	1		0.77628	0.77628
	2	1	1.43009	1.43009
	3		2.26351	2.26351
	4		3.1994	3.1994
	1	1	0.77628	0.77628
1		2	0.58169	0.58169
		3	0.43021	0.43021
		4	0.31574	0.31574

. Table 6 shows that with increasing Sc and Re values, the flow behavior of the mass transfer rate also increased for both porous disks, but the reverse was observed for α for the upper and lower disks.

Table 7. Numerical results on the thermal conductivity $k_{hnf}$ of $H{N}_{fd}$ given different shape and size factors.

(Al2O3–Cu/H2O)
$\mathit{N}=\mathbf{Shape} \mathbf{and} \mathbf{Size} \mathbf{Factor}$		$\mathit{N}=3$	$\mathit{N}=3.7$	$\mathit{N}=4.8$	$\mathit{N}=5.7$
${\mathit{\phi }}_1$	${\mathit{\phi }}_2$	${\mathit{k}}_{\mathit{h}\mathit{n}\mathit{f}}$	${\mathit{k}}_{\mathit{h}\mathit{n}\mathit{f}}$	${\mathit{k}}_{\mathit{h}\mathit{n}\mathit{f}}$	${\mathit{k}}_{\mathit{h}\mathit{n}\mathit{f}}$
0.01 = 1%	0.01 = 1%	$0.236201$	$0.23873$	0.24260	0.24566
0.02 = 2%		$0.020056$	$0.021461$	0.02372	0.025623
0.03 = 3%		$0.000103$	$0.000129$	0.00017	0.0002306
0.04 = 4%		0.000009	$0.0000128$	0.000021	0.000031
0.01 = 1%	0.02 = 2%	0.020096	0.021544	0.02412	0.025898
	0.03 = 3%	0.000101	0.000128	0.000187	0.000237
	0.04 = 4%	0.000008	0.000013	0.000022	0.000032

shows the effect of the volume fractions ${\phi }_1$ and ${\phi }_2$ for similar shapes (spherical, plates, bricks, cylindrical) and size factors of NPs. If we raise the values of volume fractions ${\phi }_1$ and ${\phi }_2$, the thermal conductivity of the hybrid nanofluid (Al₂O₃ + Cu/H₂O) is increased in several types of shape factor but is much better in terms of thermal conductivity rate for the platelet shape factor.

3.2. Velocity and Temperature Profile Effects of α, M, Re, Pr, and φ

Figure 2 shows the effect of the expansion/contraction parameter α varying from negative to positive; the radial velocity fluid flow rose from the center of the wall but was reduced by the momentum boundary layer thickness for both porous disks. Furthermore, in all expanding/contracting cases, the fluid flow decelerated and, in effect, induced a decrease in velocity near the profile. The line graph flows symmetrically. Figure 6 shows that increasing the values of both volume fractions ($\phi ={\phi }_1$ and ${\phi }_2$) enhanced the thermal boundary layer thickness flow for the upper disk but with the opposite flow of fluid in the lower disk. The NPs physically disperse heat. Applying many NPs at the same time expends more energy, increasing the levels and thickness of the temperature field. Figure 7 and Figure 8 present the increasing behavior of the middle of the wall of the axial and radial velocity profiles through the impact of magnetic parameter M when it is greater than 0. We may conclude from these results that the transversal absorption magnetic field normalizes the fluid velocity. The magnetic effect causes the vibration of the particles inside the fluid, which is governed by the Lorentz force. Therefore, increasing the value of M decreases the pattern of the internal wall but increases the momentum boundary layer thickness in the radial velocity profile. Figure 9 demonstrates the $Pr$ number’s impact on the temperature profile. The ratio of thermal diffusivity to momentum diffusivity is defined as the Prandtl number. With increasing values of $Pr$, the flow of fluid thermal boundary layer thickness was reduced from the upper disk, and the opposite flow of fluid occurred in the lower disk, as shown in Figure 9.

3.3. Concentration Profile Effects of α,Sc and Re

Figure 3 shows that the impact of α values varied from negative to positive on the concentration profile. At large values of α, the impact on the flow of mass transfer was opposite in the upper and lower disks. The evolution of the concentration field Sc is illustrated in Figure 4. It emphasizes that enlarged Sc indicated a reduced diffusion rate, which resulted in a decay in the concentration distribution. Figure 5 presents the influence of the permeability parameter $Re$ on the concentration profiles for the fixed values ${\phi }_1={\phi }_2=$ 0.01, $Sc$ = 1, $Pr=$ 6.2, and $M=$1. Physically, the Reynolds number demonstrates the ratio of inertial force to viscous force. Figure 5 illustrates that increasing the value of the permeability parameter Re in the center of the wall increased the thickness of the concentration boundary layer.

4. Conclusions

In this article, laminar, MHD, unsteady, and incompressible Newtonian hybrid nanofluid flow combined with the effect of morphology through a deformable porous disk was observed. We used two types of nanoparticles for this purpose: metallic and metallic oxides, with water as the base fluid. By using the software Mathematica with the help of a shooting method based on the fourth-order Runge–Kutta technique, we drew the following results from this research:

• Increasing the numerical values of the magnetic parameter (M) and Reynolds number (Re) enhanced the values of shear stress in the upper and lower disks;
• Increasing the value of the permeable Reynolds number resulted in opposite flow rates of mass transfer in both porous disks, as shown in the concentration profile;
• The heat transfer rate was enhanced with increased Prandtl number in the presence of hybrid nanoparticles;
• With increasing values of Sc and Re, the flow of mass transfer grew in both porous disks;
• The thermal conductivity of Al₂O₃ + Cu/H₂O hybrid nanofluid flow showed good results with a platelet shape factor at a volume fraction of 4%.