Rheological Model for Generalized Energy and Mass Transfer through Hybrid Nanofluid Flow Comprised of Magnetized Cobalt Ferrite Nanoparticles

Al-Mubaddel, Fahad S.; Allehiany, F. M.; Nofal, Taher A.; Alam, Mohammad Mahtab; Ali, Aatif; Asamoah, Joshua Kiddy K.

doi:https://doi.org/10.1155/2022/7120982

Journal of Nanomaterials

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Abstract Introduction Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Classical and Fractional-Order Derivative Approaches in Hybrid Nanofluids

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Research Article | Open Access

Volume 2022 | Article ID 7120982 | https://doi.org/10.1155/2022/7120982

Rheological Model for Generalized Energy and Mass Transfer through Hybrid Nanofluid Flow Comprised of Magnetized Cobalt Ferrite Nanoparticles

Fahad S. Al-Mubaddel,^1,2F. M. Allehiany,³Taher A. Nofal,⁴Mohammad Mahtab Alam,⁵Aatif Ali,⁶and Joshua Kiddy K. Asamoah⁷

Academic Editor: Taza Gul

Received12 Jan 2022

Revised18 Feb 2022

Accepted05 Apr 2022

Published22 Apr 2022

Abstract

The goal of the current research is to evaluate a 3D stagnation point flow of Darcy Forchheimer’s hybrid nanofluid (NF) through a heated wavy flexible cylinder under the influence of slip conditions and varying thickness. A numerical model is developed for the purpose to magnify the energy and mass transmission rate and maximize the efficiency and performance of thermal energy conduction for a variety of commercial and biological purposes through methanol-based hybrid NF flow consisting of cobalt ferrite and copper nanoparticles. Due to their inclusive range of applications, copper and cobalt iron oxide nanoparticles are gaining a lot of attention in medical and technical research. The model has been articulated in the form of a set of PDEs, which are reduced by the resemblance substitutions to the system of ODEs. The obtained 1^st-order differential equations are further processed by the computational strategy PCM. For the sake of accuracy and credibility, the values are verified with the bvp4c package. The findings are physically exhibited and analyzed. It has been observed that the induced magnetic field lessens with the upshot of the magnetic term and enhances under the action of magnetic Prandtl number . The energy profile declines due to the variation of thermal jump constraint and boosts with the absorption and generation term.

1. Introduction

The flow around convex and concave bodies have been studied extensively in order to ensure the safety of the buildings by minimizing vortex-flaking, which causes a substantial amount of drag, noise, and vibration. Shape alteration is used as a flow control strategy as geometric interruptions [1]. Flow within a circular cylinder is used in many engineering mechanisms, but far less study has been conducted on flow over a cylinder in a constrained domain, such as flow in a horizontal channel or pipe flow. Many circumstances, such as blood flow via surgical supplies in veins and flow through cylindrical items near walls, necessitate consideration of wall effects while scaling a problem. Furthermore, whereas unstructured and random forms of external roughness, such as those seen in nature, have been studied, other types of organized roughness have not. A 3D printed solid with regular sinusoidal ridges may take curliness on its exterior [2]. When heat generation is created, Salahuddin et al. [3] investigated the differently designed nanomaterials that influenced the thermodynamic effectiveness and flow performance of nanoliquid flow owing to rigid and sinusoidal barriers. To assess the aerodynamic workloads of a 5 : 1 rectangular sinusoidal radius cylinder, Wu et al. [4] used a wind tunnel with numerous active mechanisms. Changing the amplitude and frequency results in a streamwise sequence that is completely coherent, Bilal et al. [5] investigated a nonuniform Maxwell nanoliquid flow across a stretched cylinder accompanied by a nonfluctuating suction/injection. It has been shown that the angular momentum of mass propagation grows considerably when the thermophoresis ratio is increased, but radial and angular velocity declines as the viscosity element is improved. Seo et al. [6] demonstrated a numerical estimation of a 3D flow through a rectangular enclosure. In comparison to a circular cylinder, the sinusoidal cylinder was tested to see if it might enhance total heat conduction efficiency. The influence of the cylinder shape on heat transition was noticeable, with performance improving by up to 27%. Bilal et al. [7] use up an angled extendable tube to explore the iron oxide Fe₃O₄ and carbon nanotube (CNT) hybrid nanofluid (HNF). The conclusions reveal that hybrid NF is the best heat enhancer and may be used for both heat transmission and cooling purposes. Some further applications, uses, and flow models can be found in [8–10].

In comparison to common fluids like gasoline, freshwater, solo nanoparticle nanofluids, and acetylene, HNF is a revolutionary type of fluid that excels at energy conversions. HNFs can be used for a variety of thermal applications, as well as freezing in high-heat environments [11]. Hybrid NFs are used in solar energy, heat pumps, heat converters, air conditioners, automobile industry, electrical coolers, generators, radioactive systems, transmitters, ships, and bioscience. In this work, we are focusing on copper (Cu) and cobalt ferrite (CoFe₂O₄) NPs in the universal solvent water. Copper NPs in plant water extracts may be generated using a “green” chemical method called electrodeposition. Copper nanoparticles are being used as carriers for new antitubercular drugs [12]. Copper acts as an antifungal, antibiotic, and antimicrobial agent when it is added to freshwater for coatings, polymers, and textiles. Dietary supplements containing copper have a high absorptivity. Copper alloys and metals have high tensile strength [13]. Cobalt (Co) and iron (Fe) are metals. Fe lowers interstitial resistance, allowing for charge/ion mobility on the surface and a considerable increase in specific capacitance [14]. The use of imaging techniques like MRI, PET, and CT scan, among others, has proved crucial in detecting diseases efficiently. MRI is the most versatile of them all since it can provide both functional and morphological information while keeping excellent image quality. To make it more functional, bimagnetic particles are used. Bimagnetic core-shell cobalt ferrite NPs have emerged as a feasible option for generating new MRI contrast agents with improved magnetization. Bimagnetic NPs may also be used for drug transport and photothermal treatment, making them suitable entrants for the progress of novel nanotheragnostic drugs. Magnetic hydrotherapy is used to treat tumors because cancer cells are more sensitive to tiny temperature variations than healthy tissue. As a result, a rise in local temperature generated by the accumulation of magnetic NPs can kill cancer cells in the tumor while having little effect on normal tissues [15].

Several mathematicians and researchers address the mathematical approach to the abovementioned applications and challenges. Bilal et al. [16], for example, looked at the effects of electric and magnetic forces on the flow of water-based ferrous oxides and carbon nanotubes hybrid NFs over two revolving surfaces. The electric factor boosts the momentum boundary layer while lowering the thermal factor. Ramesh et al. [17] performed the covalent bonding reaction and activation energy characteristics in the flow of HNF through a stream-wise location using CoFe₂O₄ and Fe3O4 in EG+water. Wang et al. [18] employed an MWCNT-Fe3O4 hybrid nanoliquid to model the effects of metallic foam and nanomaterial on a typical solid heat sink’s thermal efficiency. Ibrahim et al. [19] assessed the effect of turbulators on enhancing energy efficiency, as well as the hydraulic efficiency of Cu water HNF in a solar accumulator, using numerical simulations and ANSYS software. The influence of concave and convex shape on the flow of a radiative hybrid NF (SiO2-MoS2/water) was investigated by Yaseen et al. [20]. The thermal efficiency boosts by 15.47 percent for flow over convex-shaped sheets and 14.28 percent for flow over beveled edge sheets when the volume percentage of SiO2 nanocrystals is raised from 1% to 5%. Wang et al. [21] experimentally and technically assessed the FeZn4Co/CNF electrocatalyst and discovered these nanomaterials. References [22–25] contain some relevant literature and applications of Cu and CoFe₂O₄ NPs in water for biomedical and engineering objectives.

Magnetization is among the most essential factors in manufacturing and engineering, with numerous uses. The interplay of fluid nanomaterials with magnetic fields affects the quality of various industrial items such as heat exchangers, gearboxes, and compressors. The impact of magnetic fields can regulate and make accessible the rate of cooling of numerous industrial devices. Magnetic fields are vital in interplanetary and astronomical magnetosphere applications, as well as aeronautic technologies and chemical science. The strength and distribution of the administered magnetics have a significant impact on the flow properties. Many academics submitted research articles in fluid mechanics that described the flow features under the influence of MHD. Hayat and Noreen [26] explored the role of thermal expansion and a generated MHD on the oscillatory transport of a 4th-order fluid across a vertical tunnel. Raju et al. [27] considered the cumulative implications of heat exchange and the exponential component on MHD flow across a semiplate. Some recent literature related to MHD hybrid nanofluid exists in [28–30].

The objective of this study is to build on a concept proposed by Salahuddin et al. [31] by investigating the effects of methanol-based hybrid NFs consisting of Cu and CoFe₂O₄ nanoparticles on heat and mass transmission. The fluid flow has been examined in a heated wavy flexible cylinder under the upshot of slip condition, variable thickness, Darcy Forchheimer, heat absorption/generation, and chemical reaction. The second intention is to improve thermal energy conduction productivity and performance for a variety of commercial and biological applications. The PCM approach is used to simulate the problem, and the results are compared to those obtained using the Matlab software bvp4c.

2. Mathematical Formulation

We supposed the steady 3D stagnation point flow of HNF flow over a heated stretchy wavy cylinder. The hybrid NF is a solution of copper Cu and cobalt ferrite CoFe₂O₄ nanomaterial in methanol fluid. The cylinder is located on -surface where the fluid is considered at . We suppose that the cylinder radius is extreme at point A called the noddle point through which fluid flow passes. Along the -axis the wavy side of the cylinder is fixed, where the -axis and -axis are normal and upright to the wavy cylinder surface. Functions and epitomize the component of velocity at the stagnation point A. Here and are constants, in such a way (see Figure 1).

Furthermore, we are analyzing the comportment of hybrid NF flow under the act of persistent magnetic field partaking uniform strength . We suppose that , , and are the magnetic field components in the directions of , , and , respectively. At the cylinder surface, and approach to and , where has vanished. Here, and are the surface and wall temperatures of the cylinder. The fundamental calculations that regulate the fluid flow are defined as follows [31]:

Here, is chemical reaction rate, and are the slip terms, is the heat source term, is the porosity term, and show the magnetic strength in direction, and is the nonuniform inertia factor constant.

Here Equation (1) describes the conservation of mass. Equation (2) shows the magnetic flux. Equations (3) and (4) are the momentum equations that pronounce the conduct of fluid flow. Equations (5) and (6) represent magnetic induction. Equations (7) and (8) are the energy and mass equations that describe the energy and mass transference around and near the wavy surface of the cylinder.

The initial and boundary conditions are as follows:

The transformation variables are as follows:

By incorporating Equation (10), we get the following:

The transform conditions are as follows:

Here, .

Here, and are magnetic field dimensionless terms. and are the velocity and thermal slip coefficient, where is the magnetic parameters, is the chemical reaction term, is the porosity term, is the Forchheimer number, and is the heat absorption and generation term defined as follows:

Here, and are the magnetic absorptivity and diffusivity.

The interest physical quantities are as follows:

where

The dimensionless form of Equation (19) is as follows:

3. Numerical Solution

The main phases, while employing the parametric methodology, are as follows [34–38]:

Step 1. Simplifying the modeled equations By putting (22) in (11)–(16) and (17), we get the following: and the transform conditions are as follows:

Step 2. Introducing parameter

Step 3. Differentiating by parameter “”
By differentiating Equations (30)–(34) w. r. t parameter , we get the following: where .

Step 4. Applying the superposition principle For each element, resolve the two Cauchy problems listed below.

By putting Equation (39) in Equation (37), we get

Step 5. Solving the Cauchy problems
By utilizing implicit scheme, The final iterative form is as follows:

4. Result and Discussion

The preceding is some of the findings that have been noticed:

Velocity profile

Figures 2(a)–2(d) particularize the presentation of velocity profile against the variation of magnetic parameter and velocity slip term , respectively. Figures 2(a) and 2(b) reveal that the fluid velocity profile reduces under the upshot of the magnetic term . Physically, it is clear that the magnetic field creates resistive force around its self, which provides hurdles (Lorentz force) to the flow field, and as a result, fluid flow declines. Figures 2(c) and 2(d) show that the fluid velocity diminishes with the varying effect of velocity slip term .

(a)

(b)

(c)

(d)

Figures 3(a)–3(c) illustrate the performance of velocity profile against the variation of copper nanoparticles, cobalt ferrite nanoparticles, and Darcy Forchheimer , respectively. Figures 3(a) and 3(b) expose that the velocity field substantially boosts with the action of copper and cobalt ferrite nanoparticles. The specific heat capacity of methanol is remarkably greater, while the thermal conductivity is less than the copper and cobalt ferrite nanomaterials, that is why the inclusion of hybrid nanoparticles, especially copper, reduces its average heat-absorbing efficiency, which results in the enhancement of velocity field. The upshot of Darcy Forchheimer’s number degenerates the velocity distribution as shown in Figure 3(c).

(a)

(b)

(c)

Induced magnetic field

Figures 4(a)–4(d) highlight the presentation of profiles versus (magnetic constraint) and . Figures 4(a) and 4(b) show that the induced magnetic field profile decreases with the effect of the magnetic parameter . Actually, the improving values of magnetic term suppressed the induced magnetic field which indicates deteriorating conduct of the induced magnetic field. Figures 4(c) and 4(d) report that the positive influence of encourages the profiles. The fundamental reason for this is that multiplying the ratios of corresponds to a reduced magnetic diffusive, resulting in a loss of magnetic field strength. It improves the curve. Hence, the rising credit of improves the induced magnetic field profile.

(a)

(b)

(c)

(d)

Temperature profile

Figures 5(a)–5(d) illustrate the performance of energy profile against the variation of copper nanoparticles, cobalt ferrite nanoparticles, thermal jump parameter , and heat source term . Figures 5(a) and 5(b) explain that the energy profile boosts with the positive variation of copper and cobalt ferrite nanoparticles. We have discussed before that the thermal conductivity of fluid enhances, while specific heat capacity is condensed under the action of copper and cobalt ferrite nanoparticles. That is why such a situation has been noticed in Figures 5(a) and 5(b). Figures 5(c) and 5(d) show an opposite behavior of energy profile versus thermal jump parameter and heat source term . The energy profile declines due to the variation of thermal jump constraint. To put it another way, the energy field is a diminishing function . Logically, increasing enables the wavy cylinder to expand. As a result of this, the thickness of the cylinder rises, reducing the energy field curvature. Heat is emitted as energy by nanosized particles in practice. The more the input of microparticles, the greater the heat production as energy. The heat absorption and generation term boost the temperature field because its effect generates heat, which causes the rises in energy profile as shown in Figure 5(d).

(a)

(b)

(c)

(d)

Concentration profile

Figures 6(a)–6(c) report the performance of concentration profile versus copper nanoparticles, cobalt ferrite nanoparticles, and chemical reaction term , respectively. Figures 6(a) and 6(b) describe that the mass transfer profile improves with the positive deviation of copper and cobalt ferrite nanoparticulate. We have reviewed earlier that the thermal conductivity of fluid enhances, while specific heat capacity is condensed under the action of copper and cobalt ferrite nanoparticles. That is why such a situation has been noticed in Figures 6(a) and 6(b). The chemical reaction coefficient positively affects the mass transfer, because it also encourages fluid particles to move fast, which results in the positive variation as elaborated in Figure 6(c).

(a)

(b)

(c)

Tables 1 and 2 exemplify the thermochemical possessions and model of base fluid, copper, and cobalt iron oxide individually. Tables 3 and 4 report the statistical assessment of PCM and bvp4c techniques, to confirm the legality of the current report. The energy field and mass transition profile are associated with the determination. Tables 3 and 4 also reveal the comparative assessments between simple and hybrid NF. It has been clearly perceived that the mass and heat transfer ratio of hybrid NF as compared to simple NF or ordinary fluid is greater.

5. Conclusion

The objective of this research is to build a computational model to investigate the effects of methanol-based hybrid NFs consisting of Cu and CoFe₂O₄ nanoparticles on heat and mass communication. The fluid flow has been examined in a heated wavy flexible cylinder under the impact of slip condition, variable thickness, Darcy Forchheimer, heat absorption/generation, and chemical reaction. The PCM approach is used to simulate the problem, and the results are compared to those obtained using the Matlab software bvp4c. The key observations are as follows: (i)The velocity profile reduces with the effect of the magnetic parameter , velocity slip constant , and Darcy Forchheimer’s number (ii)The velocity and energy field significantly boosts with the inclusion of copper and cobalt ferrite nanoparticulates in the base fluid methanol(iii)The profile decreases with the effect of the , while enhances under the action of parameter (iv)The energy profile declines due to the variation of thermal jump constraint and boosts with the absorption and generation term (v)The mass propagation rate can be significantly enhancing with the effect of chemical reaction parameter (vi)The hybrid NF has greater tendency to enhance the energy and velocity of base fluid as compared to the ordinary NF

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors have declared no conflict of interest.

Acknowledgments

The authors are thankful to the Deanship of Scientific Research, King Khalid University, Abha, Saudi Arabia, for financially supporting this work through the General Research Project under Grant no. R.G.P.2/160/43. Taif University Researchers Supporting Project number (TURSP-2020/31), Taif University, Taif, Saudi Arabia.

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Copyright © 2022 Fahad S. Al-Mubaddel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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