Numerical Investigation of Darcy–Forchheimer Hybrid Nanofluid Flow with Energy Transfer over a Spinning Fluctuating Disk under the Influence of Chemical Reaction and Heat Source

The present computational model is built to analyze the energy and mass transition rate through a copper and cobalt ferrite water-based hybrid nanofluid (hnf) flow caused by the fluctuating wavy spinning disk. Cobalt ferrite (CoFe2O4) and copper (Cu) nanoparticles (nps) are incredibly renowned in engineering and technological research due to their vast potential applications in nano/microscale structures, devices, materials, and systems related to micro- and nanotechnology. The flow mechanism has been formulated in the form of a nonlinear set of PDEs. That set of PDEs has been further reduced to the system of ODEs through resemblance replacements and computationally solved through the parametric continuation method. The outcomes are verified with the Matlab program bvp4c, for accuracy purposes. The statistical outputs and graphical evaluation of physical factors versus velocity, energy, and mass outlines are given through tables and figures. The configuration of a circulating disk affects the energy transformation and velocity distribution desirably. In comparison to a uniform interface, the uneven spinning surface augments energy communication by up to 15%. The addition of nanostructured materials (cobalt ferrite and copper) dramatically improves the solvent physiochemical characteristics. Furthermore, the upward and downward oscillation of the rotating disc also enhances the velocity and energy distribution.


Introduction
The study of the hybrid nanofluid (hnf) flow over a spinning disc with energy and mass transitions has a significant commitment to current innovations and advanced applications. Some of them are electric power generation systems, biomedical devices, aeronautical science, co-rotating apparatus, rotating devices, chemical reactions, the hydrothermal and bioconvection effect on the Maxwell hnf on an extending cylinder. It was found that the percentage of microbes decreases as the quantities of the Peclet number increase. Some related literature and applications of CoFe 2 O 4 and Cu nps in the water for biomedical and engineering purposes may be initiated in [34,35].
The purpose of this study is to expand an idea suggested by Mohebbi et al. [36], by studying the consequence of the different nanoparticles, Cu and CoFe 2 O 4 water-based hybrid NFs, on a wavy circling fluctuating disc. The second priority is to augment the productivity and implementation of thermal energy conveyance for a range of biological, industrial, and commercial uses. In order to maximize the thermal efficiency of water-based hybrid nanoliquid across a rotating surface, this paper investigates the effects of a nano composition and MHD on the hnf flow. The Darcy-Forchhemier, chemical reaction, and heat source terms all contributed to the study's uniqueness.

Governing Equations
We assumed a 3D unsteady hybrid NF flow comprised of Cu and CoFe 2 O 4 nano particulates over a fluctuating wavy moving gyrating disc. Initially, the disc is at a(0) = h. Then, with some movement ω = a(t) (angular velocity), the disc moves at Z = a(t) in the vertical direction. The disc moves with the velocity Ω(t) at the z-axis as shown in Figure 1. The buoyancy effect is presumed to be neglected. It is supposed that the Cu and CoFe 2 O 4 nanoparticulate nanomaterials are disseminated homogenously. The buoyant impacts are minimal, proving that they are insignificant when compared to the flow's inertia force. The magnetic effect is employed uniformly. On behalf of exceeding presumptions, the basic equations are expressed as [37][38][39]: Here, F r and F θ are the body forces along x and z directions defined as [37]: Here, Ha is the Hartmann number Ha = LB 0 σ µ and θ is the direction, whereas, in the above equations, Kr, k, and Q 0 are the chemical reaction rate, porosity term, and heat source, respectively.
The associated boundary conditions are: The transformation variables are: The dimensionless form of Equation (17) is: , (0), . 1 1

Numerical Solution
The basic procedure of the PCM approach applied to a set of ODEs (Equations (10 (14) and (15)) is functionalized as [40,41]: By incorporating Equation (9), we obtain: The transform conditions are: Here, S is the disk fluctuation term, Kr is the rate of chemical reaction, ω is the disk's rotation, λ is the porosity parameter, Fr is the Forchheimer factor, γ is the thermal energy ratio constraint, and is the heat source defined as: The physical quantities are: where The dimensionless form of Equation (17) is:

Results and Discussion
This section reveals the physical trend and explains the mechanism behind each result. The following observations have been made: Figures 2-6 explain the outlines of the velocity f (η) field against the variation in φ 1 = φ Cu , cobalt ferrite φ 2 = φ Fe 2 O 4 , disk fluctuation parameter S, porosity term λ, and Forchhemier number Fr, respectively. Because water's specific heat ability is more than those of the Cu and cobalt ferrite nanostructures, including them in the base fluid decreases their average heat absorption efficiency, leading to a rise in fluid acceleration as illustrated in Figures 2 and 3. The upward and downward oscillation of the turning disc encourages molecules of water to transfer instantly, raising the fluid's axial velocity as perceived in Figure 4. It is obvious that the porosity and Forchhemier number lessen the fluid velocity as reported in Figures 5 and 6. The variation in porosity term λ enhances the fluid kinetic viscosity while declining the disk rotation rate, so as a result, the flow speed diminishes.
Figures 10-13 demonstrate the behavior of the heat ( ) θ η profile via the copper 1 φ nanomaterial, cobalt ferrite 2 φ nanoparticles, thermal energy ratio term γ , and h source  , correspondingly. Because water's specific heat ability is more than those of t Cu and cobalt ferrite nanostructures, dispersing such nanostructures in a working flu decreases its heat flux absorbency, increasing the fluid temperature as seen in Figures  and 11. This property of nano particulates in the hybrid nanofluid makes it more valua for the biomedical and engineering field because their inclusion improves the thermal ficiency of base fluid, which is mostly used in medical and industrial apparatus. As show in Figures 12 and 13 the thermal energy conveyance rate decreases when the therm power ratio component γ improves, whereas it tends to increase as the heat abso tion/generation term rises.          Figures 7-9 particularize the radial velocity h(η) profile trend against the injection +β term, suction −β coefficient, and disk spinning constant ω influence, respectively. Both sucking and infusion effects on the edge of the revolving disc provide an impedance to the flow stream, resulting in a drop in the peripheral flow velocity, as seen in Figures 7 and 8. The increasing disc centrifugal acceleration also energizes the fluid particulates, causing an intensification in fluid radial velocity across an irregular surface as highlighted in Figure 9.        Figures 10-13 demonstrate the behavior of the heat θ(η) profile via the copper Cu φ 1 nanomaterial, cobalt ferrite φ 2 nanoparticles, thermal energy ratio term γ, and heat source , correspondingly. Because water's specific heat ability is more than those of the Cu and cobalt ferrite nanostructures, dispersing such nanostructures in a working fluid decreases its heat flux absorbency, increasing the fluid temperature as seen in Figures 10 and 11. This property of nano particulates in the hybrid nanofluid makes it more valuable for the biomedical and engineering field because their inclusion improves the thermal efficiency of base fluid, which is mostly used in medical and industrial apparatus. As shown in Figures 12 and 13 the thermal energy conveyance rate decreases when the thermal power ratio component γ improves, whereas it tends to increase as the heat absorption/generation term rises.      γ Figure 13. The nature of energy ( ) θ η versus heat absorption/generation term  .      Tables 1 and 2 represent the thermal properties and experimental values o and Cu nano particulates, respectively. Table 3 describes the numerical valuat bvp4c and published work with the PCM results, to ensure accuracy. The veloci and energy fields are associated with the determination. Tables 4 and 5 establis tive scrutiny for the Nusselt number and skin friction amid copper and copper brid nanoliquid.     [39].

Properties
Viscosity Thermal Conductivity

Conclusions
The computational estimation of hybrid nanoliquid comprised of CoFe 2 O 4 and Cu nanomaterial flows caused by the oscillation of a rotating wavy disc with energy dissemination is described in the proposed investigation. The goal of the suggested study is to advance the reliability of thermal energy transportation for a diversity of commercial and biological sectors. The observations are described as a system of PDEs that are graphically and statistically calculated using the PCM procedure. Below are the main discoveries from the aforesaid assessment:

•
The dispersion of copper Cu (φ 1 = φ Cu ) and cobalt ferrite φ 2 = φ CoFe 2 O 4 nanoparticles in the working fluid water significantly boosts the mass and energy transfer rate.

•
The upward and downward oscillation of the turning disc encourages molecules of water to transfer instantly, raising the fluid's axial velocity.

•
The variation in porosity term λ and Forchhemier number Fr reduces the fluid velocity. • Both sucking −β and infusion +β effects on the texture of the revolving disc provide an impedance to the flow stream, which drops the fluid velocity.

•
The effect of thermal energy ratio term γ reduces, while the heat source term improves the fluid temperature.