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

Authors: Fahad S. Al-Mubaddel; F. M. Allehiany; Taher A. Nofal; Mohammad Mahtab Alam; Aatif Ali; Joshua Kiddy K. Asamoah
Journal: Journal of Nanomaterials
 (2022)

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## 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 1st-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 M. The energy profile declines due to the variation of thermal jump constraint and boosts with the absorption and generation term.


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## Body

## 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 xy-surface where the fluid is considered at z>0. We suppose that the cylinder radius is extreme at point A called the noddle point through which fluid flow passes. Along the y-axis the wavy side of the cylinder is fixed, where the z-axis and x-axis are normal and upright to the wavy cylinder surface. Functions ue=ax and ve=bx epitomize the component of velocity at the stagnation point A. Here a and b are constants, in such a way b≤a,0<a (see Figure 1).Figure 1
The hybrid nanofluid flows in a wavy heat cylinder.Furthermore, we are analyzing the comportment of hybrid NF flow under the act of persistent magnetic field partaking uniform strengthM0. We suppose that M1, M2, and M3 are the magnetic field components in the directions of x, y, and z, respectively. At the cylinder surface, M1 and M2 approach to Mex and Mey, where M3 has vanished. Here, T1 and Tw are the surface and wall temperatures of the cylinder. The fundamental calculations that regulate the fluid flow are defined as follows [31]:
(1)∂u∂x+∂v∂y+∂w∂z=0,(2)∂M1∂x+∂M2∂y+∂M3∂z=0,(3)u∂u∂x+v∂u∂y+w∂u∂z−μe4πρhnfM1∂M1∂x+M2∂M1∂y+M3∂M1∂z=ρhnf∂2u∂z2+a2x−μe4πρhnfMexdMexdx−νk∗u−Fu2,(4)u∂v∂x+v∂v∂y+w∂v∂z−μe4πρhnfM1∂M2∂x+M2∂M2∂y+M3∂M2∂z=ρhnf∂2v∂z2+b2y−μe4πρhnfMeydMeydx−νk∗v−Fv2,(5)u∂M1∂x+v∂M1∂y+w∂M1∂z−M1∂u∂x+M2∂u∂y+M3∂u∂z=η0∂2M1∂z2,(6)u∂M2∂x+v∂M2∂y+w∂M2∂z−M1∂v∂x+M2∂v∂y+M3∂v∂z=η0∂2M2∂z2,(7)u∂T∂x+v∂T∂y+w∂T∂z=αhnf∂2T∂z2+Q0T−T∞ρCp,(8)u∂C∂x+v∂C∂y+w∂C∂z=Dhnf∂2C∂z2−KcC−C∞.Here,Kr is chemical reaction rate, R1 and R2 are the slip terms, Q0 is the heat source term, k∗ is the porosity term, Mex=xM0 and Mey=yM0 show the magnetic strength in x,y direction, and F=xCb/rk∗1/2 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:(9)u=uw+μhnfR1∂u∂z,v=vw+μhnfR1∂v∂z,w=0,M1=M2=M3=0,T=Tw+khnfR2∂T∂z,C=C0atz=0u⟶ue,v⟶ve,M1⟶Mex,M2⟶Mey,T⟶T∞,C⟶C∞atz⟶∞.The transformation variables are as follows:(10)u=axf′η,v=byg′η,w+fη+cgηav=0,M1=xM0h1η=0,M2=yM0h2η=0,M3+νfa1/2h1+h2M0=0,η=aνfz,T=T∞+Tw−T∞Θη,C=C∞+Cw−C∞ΦηBy incorporating Equation (10), we get the following:
(11)ℏ1ℏ21+cg+ff″−βℏ2h1h″12+h2h1″+1+1−Frf′2=0,(12)ℏ1ℏ21+cg+fg″−βℏ2h2h″22+h1h2+1+c−Frcg′2=0,(13)h1″+h1′f−f″h1−f″h2+ch1′gM=0,(14)h2″+h2′f−cg″h1−cg″h2+ch2′gM=0,(15)ℏ4ℏ3Θ″η+Θ′η+cΘ′ηg+Q0Θη=0,(16)ℏ6φ″η+φ′η+cφ′ηg+Krφη=0.The transform conditions are as follows:(17)f0=0,f′0=δ1ℏ1f″ηη=0,g0=0,g′0=ℏ1δ1g″ηη=0,h10=0,h20=0Θ0=δ2ℏ4,Φ0=1atη=0f′η⟶0,g′η⟶0,h1η⟶1,h2η⟶1,Θη⟶0,Φη⟶0asη=∞.Here,ℏ1=μhnf/μbf,ℏ2=ρhnf/ρbf,ℏ3=ρCphnf/ρCpbf,ℏ4=khnf/kbf,ℏ5=σhnf/σbf,ℏ6=Dhnf/Dbf.Here,h1 and h2 are magnetic field dimensionless terms. δ1 and δ2 are the velocity and thermal slip coefficient, where β is the magnetic parameters, Kr is the chemical reaction term, λ is the porosity term, Fr is the Forchheimer number, and Q1 is the heat absorption and generation term defined as follows:
(18)β=μe4πρfM0a2,M=νfη0,Kr=Kc1−λtc,λ=νk∗Ω,Fr=Cbk∗1/2,Q1=xQ0ρCp.Here,μe and η0 are the magnetic absorptivity and diffusivity.The interest physical quantities are as follows:(19)Cfx=τwxz=0uw2ρf,Cfy=τwyz=0vw2ρf,Nu=qwxTw−T∞kf,Sh=jwxCw−C∞Df,where(20)τwx=μhnf∂u∂zz=0,τwy=μhnf∂v∂zz=0,qw=−khnfdTdzz=0,jw=−DhnfdCdzz=0.The dimensionless form of Equation (19) is as follows:
(21)Re1/2Cfx=cyxℏ1f″0,Cfy=cyxℏ1g″0,Re1/2Nux=−khnfkfΘ′0,Re−1/2Sh=−Φ′0.


## 3. Numerical Solution

The main phases, while employing the parametric methodology, are as follows [34–38]:Step 1.
Simplifying the modeled equations(22)ϑ1=fη,ϑ2=f′η,ϑ3=gη,ϑ4=g′η,ϑ5=h1η,ϑ6=h1′η,ϑ7=h2η,ϑ8=h2′η,ϑ9η=Θη,ϑ10=Θ′η,ϑ11=Φη,ϑ12=Φ′η.
By putting (22) in (11)–(16) and (17), we get the following:
(23)ℏ1ℏ21+cϑ3+ϑ1ϑ2′−βℏ2ϑ5ϑ′26+ϑ7ϑ1′+1+1−Frϑ22=0,(24)ℏ1ℏ21+cϑ3+ϑ1ϑ4′−βℏ2ϑ7ϑ′28+ϑ5ϑ7+1+c−Frcϑ42=0,(25)ϑ6′+ϑ6ϑ1−ϑ2′ϑ5−ϑ2′ϑ7+cϑ6ϑ3M=0,(26)ϑ8′+ϑ8ϑ1−cϑ4′ϑ5−cϑ4′ϑ7+chϑ8ϑ3M=0,(27)ℏ4ℏ3ϑ10′η+ϑ10η+cϑ10ηg+Q0ϑ9η=0,(28)ℏ6ϑ12′η+ϑ12η+cϑ12ηϑ3+Krϑ11η=0,and the transform conditions are as follows:
(29)ϑ10=0,ϑ20=δ1ℏ1ϑ2′ηη=0,ϑ30=0,ϑ40=δ1ℏ1ϑ4′ηη=0,ϑ50=0,ϑ70=0,ϑ90=δ2ℏ4,ϑ110=1atη=0,ϑ2η⟶0,ϑ4η⟶0,ϑ5η⟶1,ϑ7η⟶1,ϑ9η⟶0,ϑ11η⟶0asη=∞.Step 2.
Introducing parameterp(30)ℏ1ℏ21+cϑ3+ϑ1ϑ2′−βℏ2ϑ5ϑ′26+ϑ7ϑ1′+1+1−Frϑ22−1p=0,(31)ϑ6′+ϑ1ϑ6=1p−ϑ2′ϑ5−ϑ2′ϑ7+cϑ6ϑ3M=0,(32)ϑ8′+ϑ1ϑ8−1p−cϑ4′ϑ5−cϑ4′ϑ7+chϑ8ϑ3M=0,(33)ℏ4ℏ3ϑ10′η+ϑ10η−p+cϑ10ηg+Q0ϑ9η=0,(34)ℏ6ϑ12′η+ϑ12η−1p+cϑ12ηϑ3+Krϑ11η=0.Step 3.
Differentiating by parameter “p”
By differentiating Equations (30)–(34) w. r. t parameter p, we get the following:
(35)V′=AV+R,(36)dζidτ,where i=1,2,⋯⋯⋯11.Step 4.
Applying the superposition principle(37)V=aU+W.
For each element, resolve the two Cauchy problems listed below.(38)U=aU,(39)W′=AW+R.By putting Equation (39) in Equation (37), we get
(40)aU+W′=AaU+W+R.Step 5.
Solving the Cauchy problems
By utilizing implicit scheme,(41)Ui+1−UiΔη=AUi+1,Wi+1−WiΔη=AWi+1.
The final iterative form is as follows:(42)Ui+1=UiI−ΔηA,Wi+1=Wi+ΔηRI−ΔηA.


## 4. Result and Discussion

The preceding is some of the findings that have been noticed:Velocity profilefη,gηFigures2(a)–2(d) particularize the presentation of velocity fη,gη profile against the variation of magnetic parameter β and velocity slip term δ1, respectively. Figures 2(a) and 2(b) reveal that the fluid velocity profile reduces under the upshot of the magnetic term β. 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 δ1.Figure 2
The presentation of velocityfη,gη profile versus (a, b) magnetic parameter β and (c, d) velocity slip parameter δ1.
(a)(b)(c)(d)Figures3(a)–3(c) illustrate the performance of velocity fη profile against the variation of copper ϕ1 nanoparticles, cobalt ferrite ϕ2 nanoparticles, and Darcy Forchheimer Fr, respectively. Figures 3(a) and 3(b) expose that the velocity field substantially boosts with the action of copper ϕ1 and cobalt ferrite ϕ2 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).Figure 3
The performance of velocityfη profile versus (a) copper ϕ1 nanoparticles, (b) cobalt ferrite ϕ2 nanoparticles, and (c) Darcy Forchheimer Fr.
(a)(b)(c)Induced magnetic fieldh1′η,h2′ηFigures4(a)–4(d) highlight the presentation of h1′η,h2′η profiles versus β (magnetic constraint) and M. Figures 4(a) and 4(b) show that the induced magnetic field profile decreases with the effect of the magnetic parameter β. 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 M encourages the h1′η,h2′η profiles. The fundamental reason for this is that multiplying the ratios of M corresponds to a reduced magnetic diffusive, resulting in a loss of magnetic field strength. It improves the curve. Hence, the rising credit of M improves the induced magnetic field profile.Figure 4
The performance of induced magnetic fieldh1′η,h2′η profile versus (a, b) magnetic parameter β and (c, d) magnetic Prandtl number M.
(a)(b)(c)(d)Temperature profileΘηFigures5(a)–5(d) illustrate the performance of energy Θη profile against the variation of copper ϕ1 nanoparticles, cobalt ferrite ϕ2 nanoparticles, thermal jump parameter δ2, and heat source term Q1. Figures 5(a) and 5(b) explain that the energy Θη profile boosts with the positive variation of copper ϕ1 and cobalt ferrite ϕ2 nanoparticles. We have discussed before that the thermal conductivity of fluid enhances, while specific heat capacity is condensed under the action of copper ϕ1 and cobalt ferrite ϕ2 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 δ2 and heat source term Q1. The energy profile declines due to the variation of thermal jump constraint. To put it another way, the energy field is a diminishing function δ2. Logically, increasing δ2 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 Q1 boost the temperature field because its effect generates heat, which causes the rises in energy profile as shown in Figure 5(d).Figure 5
The performance of energyΘη profile versus (a) copper ϕ1 nanoparticles, (b) cobalt ferrite ϕ2 nanoparticles, (c) thermal jump parameter δ2, (d) heat source term Q1.
(a)(b)(c)(d)Concentration profileΦηFigures6(a)–6(c) report the performance of concentration Φη profile versus copper ϕ1 nanoparticles, cobalt ferrite ϕ2 nanoparticles, and chemical reaction term Kr, respectively. Figures 6(a) and 6(b) describe that the mass transfer Φη profile improves with the positive deviation of copper ϕ1 and cobalt ferrite ϕ2 nanoparticulate. We have reviewed earlier that the thermal conductivity of fluid enhances, while specific heat capacity is condensed under the action of copper ϕ1 and cobalt ferrite ϕ2 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).Figure 6
The performance of concentrationΦη profile versus (a) copper ϕ1 nanoparticles, (b) cobalt ferrite ϕ2 nanoparticles, and (c) chemical reaction term Kr.
(a)(b)(c)Tables1 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.Table 1
The statistical properties of copper, cobalt iron oxide, and methanol [32, 33].
ρ (kg/m3)Cp (j/kgK)k (W/mK)σ (S/m)Methanol79225450.20350.5×10−6Copper (Cu)89333854015.96×107CoFe2O449077003.75.51×109Table 2
The thermal properties of the hybrid nanofluidϕ1=ϕCu,ϕ2=ϕCoFe2O4 [32, 33].
PropertiesViscosityμhnf/μbf=1/1−ϕCu−ϕCoFe2O42Densityρhnf/ρbf=ϕCuρCu/ρbf+ϕCoFe2O4ρCoFe2O4/ρbf+1−ϕCu−ϕCoFe2O4Thermal capacityρCphnf/ρCpbf=ϕCuρCpCu/ρCpbf+ϕCoFe2O4ρCpCoFe2O4/ρCpbf+1−ϕCu−ϕCoFe2O4Thermal conductivitykhnf/kbf=ϕCukCu+ϕCoFe2O4kCoFe2O4/ϕCu+ϕCoFe2O4+2kbf+2ϕCukCu+ϕCoFe2O4kCoFe2O4−2ϕCu+ϕCoFe2O4kbf/ϕCukCu+ϕCoFe2O4kCoFe2O4/ϕCu+ϕCoFe2O4+2kbf−2kCuϕCu+kCoFe2O4ϕCoFe2O4+ϕCu+ϕCoFe2O42kbfElectrical conductivityσhnf/σbf=ϕCuσCu+σCoFe2O4ϕCoFe2O4/ϕCoFe2O4+ϕCu+2σbf+2ϕCuσCu+ϕCoFe2O4σCoFe2O4−2ϕCu+ϕCoFe2O4σbf/ϕCuσCu+ϕCoFe2O4σCoFe2O4/ϕCu+ϕCoFe2O4+2σbf−ϕCuσCu+ϕCoFe2O4σCoFe2O4+ϕCu+ϕCoFe2O4σbfTable 3
Statistical results for Nusselt number.
PCMbvp4cPCMbvp4cδ2Q1ϕ1,ϕ2knf/kfΘ′0knf/kfΘ′0khnf/kfΘ′0khnf/kfΘ′00.20.04755350.04744350.04845310.04844310.40.03551230.03540420.03661220.03660230.60.03658520.03647430.03698530.03697520.80.02921070.02910030.02714070.02713230.00.05655880.05644760.05545550.05544560.40.05757600.05746520.05759610.05758540.80.05799620.05788610.05899650.05898461.20.010.06744200.067343100.06834600.06833500.020.06842410.06832200.06932710.06931600.030.06913240.06903130.07131420.07130410.040.07224190.07234070.07433190.0743217Table 4
Numerical outcomes for Sherwood number.
PCMbvp4cPCMbvp4cKrϕ1,ϕ2Dnf/DfΦ′0Dnf/DfΦ′0Dhnf/DfΦ′0Dhnf/DfΦ′00.20.06324280.06322280.06424210.06422200.40.06294220.06292100.06394370.06392340.60.06159440.06157420.06149460.06147430.80.59303620.59301600.59103410.59101410.010.06277130.06275110.06777360.06776340.020.06388330.06386310.07288550.07286510.030.66875510.66873400.77746050.77744020.040.70266190.70266170.79068150.7906613


## 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 parameterβ, velocity slip constant δ1, and Darcy Forchheimer’s number Fr(ii)
The velocity and energy field significantly boosts with the inclusion of copperϕ1 and cobalt ferrite ϕ2 nanoparticulates in the base fluid methanol(iii)
Theh1′η,h2′η profile decreases with the effect of the β, while enhances under the action of parameter M(iv)
The energy profile declines due to the variation of thermal jump constraint and boosts with the absorption and generation termQ1(v)
The mass propagation rate can be significantly enhancing with the effect of chemical reaction parameterKr(vi)
The hybrid NF has greater tendency to enhance the energy and velocity of base fluid as compared to the ordinary NF



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