# Radiative Flow of Copper and Aluminum Nanoparticles with Heat Source Phenomenon: Dual Numerical Simulations and Stability Analysis

Authors: Sumera Dero; Liaquat Ali Lund; Al-Khaled Kamel; Asjad Muhammad Imran; Taseer Muhammad; Sami Ullah Khan; A Abbasi
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

The thermally developed flow of nanoparticles due to bidirectional moving space has been presented numerically. The additional impact of heat source and sink are also incorporated for enhancing the heat transfer rate. The copper and aluminum nanoparticles are selected for enhancing the thermal diffusion system. The thermal stability of nanoparticles is carefully addressed. The moving surface allows the shrinking and stretching phenomenon. The system of resulting equations with nonlinear nature is numerically swapped via shooting technique with help of MATLAB software. The dual numerical simulations for the thermally radiative problem with external heating phenomenon are being performed. It is noted from results that thermal stability of nanoparticles is more stable in the upper branch of shrinking/stretching surface. The presence of heat source provides extra heat and improves the thermal transportation phenomenon which is more progressive in upper branch. The enhancement in temperature profile due to copper nanoparticles is more stable as compared to aluminum nanoparticles. Moreover, the increasing change in the velocity is resulted upon increasing the rotation constant. The obtained results reflect applications in thermal sciences, heating systems, energy production, solar applications, nuclear reactions, biomedical applications, etc.


---

## Body

## 1. Introduction

Recently, motivations of researchers are noticed towards the nanofluid which reports leading contributions in the thermal engineering and industrial processes. The growing interest in nanomaterials is justified due to highly attractive applications in solar energy, thermal transportation systems, engineering devices, cooling systems, heating objectives, nuclear reactions, medical sciences, emission processes, etc. With nanostructured (1–100 nm), the nanofluids report excellent thermal activities. The interaction between nanoparticles and some base fluids reports results enhanced thermal mechanism of base fluids. Choi [1] focused first time on the thermal aspect of nanofluids via experimental approach. Later on, diverse research is being performed by investigators to highlight the thermal mechanism and applications of nanomaterials. Turkyilmazoglu [2] developed a slip nanofluid model for the annuli flow by utilizing the dual nanofluid and single relations of the nanofluid model. Nayak et al. [3] exploited the viscous dissipation consequences for illustrating the nanofluids flow with Oldroyd-B base material via bidirectional space. Basir et al. [4] performed the novel aspect of slip flow of nanofluid with sensitivity insurance and bioconvection phenomenon. The Maxwell nanofluid thermal determinations subject to the interaction of multiple types of slip consequences have been identified by Ahmad et al. [5]. The Buongiorno nanofluid model implementation for inspecting the viscoelastic fluid thermal capacity was reported in the work of Nadeem et al. [6]. The analytical simulation for the nanofluid Riga surface problem was investigated by Vaidya et al. [7]. Wang et al. [8] focused on the migration of nanofluid material minichannels having parallel plates. Abdelaziz et al. [9] discussed the characteristics of nanofluid pattern via horizontal tubes. The continuation of Sabu et al. [10] determined the heat transfer improvement for the alumina nanoparticles with slip constraints due to heated rotatory disk. Ojjela et al. [11] addressed the viscoelastic behavior and thermal capability for the nanoparticles with non-Newtonian base fluid. The stability determination of nanofluids with experimental approach was determined by Dong et al. [12]. Sivasankaran and Bhuvaneswari [13]investigated the heat transfer mechanism with constraints of pressure loss and porous space. Acharya [14] numerically focused on the natural convective flow of nanofluid due to heated wall surface in square enclosure. Mondal et al. [15] addressed the optimized thermal mechanism due to the couple stress nanofluid with radiative phenomenon. Acharya [16] discussed the finite element numerical simulations for the spinning flow of nanoparticles with active and passive control approach. Shafiq et al. [17] addressed the bioconvection aspect of nanofluid with zero mass constraints. Haya et al. [18] addressed the Joule heating effects while inspecting the enhanced aspect of heat transfer in third grade nanofluid. Some more recent work on nanofluid flow is presented in [19–24].The hybrid nanofluid is the modified form of nanofluids with more improved thermal activities. In fact, the hybrid nanofluid model is based on the interaction of more than one different types of nanoparticles along with base material. The preferences of the hybrid nanofluid model over simple nanomaterials are due to extra high activities and more thermal stability. Recently, different investigations are performed on hybrid nanofluid with different nanoparticles. Sowmya et al. [25] reported the inclusion of iron nanoparticles with existence of Lorentz force in cavity with two heated fins. Sundar et al. [26] claimed the hybrid nanofluid thermal properties with collector applications. Akhtar et al. [27] modeled a hybrid nanofluid problem in elliptic duct following the peristaltic mechanism. Eid and Mabood [28] noticed the ethylene glycol base fluid thermal inspection with hybrid nanofluid numerically. Eid and Mabood [28] presented the thermal research on the carbon nanotubes flow with heat generation features. Shahid et al. [29] implemented the Darcy porous law for studying the hybrid nanofluid prospective in parabolic surface. Madhukesh et al. [30] explored the thermal efficiencies of AA7072-AA7075 nanoparticles with Newtonian heating enrollment. Acharya [31] discussed the shape characteristics for the hybrid nanofluid in heated obstacles. Devi and Devi [32] visualized the copper and aluminum oxide nanoparticles thermal prospective accounted via stretching surface. Devi and Devi [33] observed the 3D thermal flow problem with Newtonian heating. Lund et al. [34] intended the rotating flow of hybrid nanofluid. Teh and Ashgar [35] determined the Joule heating for the rotating flow of hybrid nanofluid. Zaimi et al. [36] discussed the multiple solution of rotating viscoelastic fluid with hybrid nanofluid.This analysis deals with the impact of heat source and thermal radiation effects in flow of hybrid nanofluid due to rotating space. The main motivation for performing this investigation is to highlight the enhanced thermal aspect of heat transfer phenomenon subject to the magnetized copper and aluminum nanoparticles. The moving surface attained the stretching and shrinking phenomenon. The thermal stability of hybrid nanoparticles is focused. The shooting numerical algorithm is followed for the simulations process. The dual simulations are performed in each branch of moving space.


## 2. Problem Formulation

The stretching and shrinking phenomenon of 3D hybrid nanofluid is taken into account. The heat transfer analysis is considered in presence of external heating source and thermal radiation. The magnetic force impact is directed along thez-axis for 3D flow. The three-dimensional moving space attained the velocity uwx=λcxn. The stretching and shrinking results are attained for λ>0 and λ<0, respectively. The surface temperature of nanoparticles is notified with Tw while T∞ is for free stream temperature case as shown in Figure 1. The nanoparticles and surface present rotatory behavior with angular velocity Ω˜=Ω0x1−n along z-direction. Let u, v, and wbe representation of velocities along ,y, and z directions, respectively. The assumptions of low magnetic Reynolds number reduce the role of induced magnetic force. The governing relations in view of such considerations are as follows [26–28]:Figure 1
Flow illustration of problem.(1)ux+vy+wz=0,(2)uux+vuy+wuz−2Ω˜v=−Pxρhnf+μhnfρhnfuxx+uyy+uzz−σhnfρhnfB2u.(3)uvx+vvy+wvz+2Ω˜u=−Pyρhnf+μhnfρhnfvxx+vyy+vzzv−σhnfρhnfB2v,(4)uwx+vwy+wwz=−Pzρhnf+μhnfρhnfwxx+wyy+wzz,(5)ρcphnfuTx+vTy+wTz=khnfTxx+Tyy+TzzT−qrz+Q0T−T∞.The related boundary conditions are as follows:(6)w=wwx,v=0,u=uwx,T=Twatz=0u⟶0,v⟶0,T⟶T∞asz⟶∞.The relation for radiative fluxqr is given by qr=−4σ∗/3k∗∂T4/∂z. Table 1 and Table 2 provide the thermophysical properties.Table 1
Thermophysical features of hybrid nanofluid [32–34].
PropertiesHybrid nanofluidDynamic viscosityμhnf=μf/1−ϕCu2.51−ϕAl2O32.5Densityρhnf=1−ϕCu1−ϕAl2O3ρf+ϕAl2O3ρAl2O3+ϕCuρCuThermal conductivitykhnf=kCu+2knf−2ϕCuknf−kCu/kCu+2knf+ϕCuknf−kCu×knfwhere knf=kAl2O3+2kf−2ϕAl2O3kf−kAl2O3/kAl2O3+2kf+ϕAl2O3kf−kAl2O3×kfHeat capacityρcphnf=1−ϕCu1−ϕAl2O3ρcpf+ϕAl2O3ρcpAl2O3+ϕCuρcpCuElectrical conductivityσhnf=σCu+2σnf−2ϕCuσnf−σCu/σCu+2σnf+ϕCuσnf−σCu×σnfwhere σnf=σAl2O3+2σf−2ϕAl2O3σf−σAl2O3/σAl2O3+2σf+ϕAl2O3σf−σAl2O3×σfTable 2
The thermophysical properties [32–34].
PropertiesWaterH2OCopper (Cu)AluminaAl2O3ρkg/m3997.189333970cpJ/kgK4179385765kW/mK0.61340040σS/m0.055.96×1073.69×107Pr6.2The following similarity variables (7) are used to obtain similarity solutions for equations (1)–(5):(7)u=cxnf′η,v=cxngη,η=zcn+12ϑfxn−1/2w=−cϑfn+12xn−1/2fη+n−1n+1ηf′ηθη=T−T∞/Tw−T∞.The dimensionless set of equation is(8)f′′′+ξ1ξ2ff′′−2nn+1f′2+4Ωn+1g−2n+1σhnfσfξ2Mf′=0,(9)g′′+ξ1ξ2fg′−2nn+1f′g−4Ωn+1f′−2n+1σhnfσfξ2Mg=0,(10)ξ3Prkhnfkf+43Rdθ′′+θ′f+ξ3Qθ=0,subject to BCs,(11)f0=−b2n+1,f′0=λ,g0=0,θ0=1f′η⟶0,gη⟶0θη⟶0,asη⟶∞,ξ1=1−ϕCu1−ϕAl2O3+ϕAl2O3ρAl2O3ρf+ϕCuρCuρfξ2=1−ϕCu2.51−ϕAl2O32.5ξ3=11−ϕCu1−ϕAl2O3+ϕAl2O3ρcpAl2O3/ρcpf+ϕCuρcpCu/ρcpf,where prime “′” denotes the differentiation by η, M=σfB02/cρf is the magnetic number, Pr=ϑf/αf is Prandtl, Ω=Ω0/c is rotation parameter, Q shows the heat source/sink parameter, and Rd=4σ∗T∞3/k∗kf denotes the thermal radiation constant and b suction (b>0)/injection (b>0) constant.The dynamic of local Nusselt number and skin friction coefficient is(12)Cfx=μhnfρfuw2∂u∂z|z=0,Cfy=μhnfρfuw2∂v∂z|z=0,Nux=−khnf∂T/∂zz=0+qrz=0kfTw−T∞.Substituting (7) into (12) yields(13)ReCfx=1ξ2f′′0;ReCfy=1ξ2g′0Re2Nux=−khnfkf+43Rdθ′0,where Rex is the local Reynolds number.


## 3. Stability Analysis

Two branches of solution against the involved parameters are observed. In order to inspect the branch of solution with more physical relevance, the stability analysis has been performed for both branches. The governing equations (2)-(5) with unsteady form are(14)ut+uux+vuy+wuz−2Ω˜v=−Pxρhnf+μhnfρhnfuxx+uyy+uzz−σhnfρhnfB2u,(15)vt+uvx+vvy+wvz+2Ω˜u=−Pyρhnf+μhnfρhnfvxx+vyy+vzzv−σhnfρhnfB2v,(16)wt+uwx+vwy+wwz=−Pzρhnf+μhnfρhnfwxx+wyy+wzz,(17)ρcphnfTt+uTx+vTy+wTz=khnfTxx+Tyy+TzzT−qrz+Q0T−T∞.The unsteady similarity solution can be considered as follows [33–35]:(18)u=cxnfηη,τ,v=cxngη,τ,w=−cϑfn+12xn−1/2fη,τ+n−1n+1ηfηη,τ−2τn+1ηfη,τθη,τ=T−T∞/Tw−T∞.,where the η is the same as in equation (7) and the dimensionless time is the τ=ctxn−1, where the subscription represents the derivative of the variable subscripted. Remember that continuity (1) is also satisfied by the values u,v, and w given above. Equation (18) is substituted into equations (14)-(17), and one can get(19)fηηη+ξ1ξ2ffηη−2nn+1fη2+4Ωn+1g+2n−1n+1τfτfηη−fηfτη−2n+1fτη−2n+1σhnfσfξ2Mfη=0,(20)gηη+ξ1ξ2fgη−2nn+1fηg−4Ωn+1fη+2n−1n+1τfτgη−gτfη−2n+1gτ−2n+1σhnfσfξ2Mg=0,(21)ξ3Prkhnfkf+43Rdθηη+θηf+ξ3Qθ+2τfτθη−θτfη−θτ=0,subject to BCs,(22)f0,τ=−b2n+1,f′0,τ=λ,g0,τ=0,θ0,τ=1f′η,τ⟶0,gη,τ⟶0θη,τ⟶0,asη⟶∞.It is assumed that [28](23)fη,τ=f0η+e−ετFη,τ,gη,τ.=g0η+e−ετGη,τ,θη,τ=θ0η+e−ετHη,τ.The solution associated to the formulated equations (8)–(10) with boundary conditions (11) is denoted with f0η,g0η, and θ0η. The decay and growth disturbance rate is defined with ε. It is assumed that η,τ,Gη,τ, and Hη,τ and derivatives of these expressions are smaller in contrast to the steady state situation. Such assumptions are followed due to the linear stability of flow pattern. Upon replacing equation (26) into set of expressions (19)–(21) with taking =0,Fη,τ, Gη,τ, and Hη,τ are associated to F0η, G0η, and H0η. On this end, the linear eigenvalue problem is formulated as(24)F0″′+ξ1ξ2f0F0″+F0f0″+2n+1ε−2nf0′F0′+4Ωn+1G0−2n+1σhnfσfξ2MF0′=0,(25)G0″+ξ1ξ2g0′F0+G0′f0−2nn+1f0′G0+F0′g0−4Ωn+1F0′+2n+1εG0−2n+1Mσhnfσfξ2G0=0,(26)ξ3Prkhnfkf+43RdH0″+θ0′F0+H0′f0+ξ3QH0+εH0=0,with(27)F00=0,F0′0=0,G00=0,H00=0F0′η⟶0,G0η⟶0H0η⟶0,asη⟶∞.We can solve equations (24)–(26) along with BCs (27) by using the numerical values of f0η,g0η, and θ0η which were obtained from the solutions of equations (8)–(10) with BCs (11). It is worth noting that the homogeneous linear equations (24)–(26) subject to constants (27) construct an eigenvalue system with eigenvalue ε. By solving equations (24)–(27), a set of eigenvalues ε1<ε2<ε3<... is obtained.


## 4. Results and Discussion

Following to the investigation of Devi and Devi [33, 34] and Lund et al. [35], here water is considered as base material with copper (Cu) and aluminum nanoparticles Al2O3. Moreover, volume fraction of aluminum is kept constant against the variation of copper volume oxide (0.01≤ϕCu≤0.1). The comparison validation for ensuring the validity task has been done in Table 3 with work of Zaimi et al. [37] with excellent comparison results.Table 3
Values ofReCfx and ReCfy for several values of Ω when λ=1,ϕAl2O3=ϕCu=0,b=0,n=1.
ReCfxReCfyΩ[35]Present results[28]Present results0–1.0000−1.00006250.00000.00000000.5–1.1384−1.1383806–0.5128−0.51276021–1.3250−1.3250287–0.8371−0.83709832–1.6523−1.6523520–1.2873−1.28725883–1.9289−1.9289315–1.6248−1.62473574–2.1716−2.1715931–1.9054−1.90539295–2.3901−2.3901398–2.1506−2.1505265Figures2–4 are sketched to observe the physical aspect of ϕCu on f′′0,g′0, and −θ′0. Results are visualized for suction case. The dual branches of solution are results for various values of suction case. The critical points for ϕCu=0.01 are bc1=−2.4009 while bc2=−2.3145 and bc3=−2.25401 are the critical points against ϕCu=0.05 and ϕCu=0.01, respectively. Two different categories of branches such as dual branch b≤bci and no branch b>bci for suction parameter b exist. Moreover, larger change in ϕCu results increment in bci which preserve a separation. The dominant values of suction do not allow tension in moving particles, and vorticity is not smothered. In upper branch, with larger suction, the decrement in wall shear force and heat transfer rate is noted. In lower branch, the heat transfer rate is lower but wall shear force enhanced. In upper branch, no significant change in g′0 due to suction parameter is inspected while in lower branch, g′0 declined.Figure 2
Impact ofb on f″0 for fixed values of ϕCu.Figure 3
Impact ofb on g′0 for fixed values of ϕCu.Figure 4
Impact ofb on −θ′0 for fixed values of ϕCu.Figures5–7 show the variation of ′′0,g′0 and −θ′0 magnetic parameter M against copper solid volume fraction ϕCu. As ϕCu grows, the critical point of Mc shifts to the left, assisting in the separation of the boundary layer. Mc1=0.4877,Mc2=0.34892, and Mc3=0.2225 referred to critical points of M for ϕCu=0.01,0.05, and 0.1. As a result, the dual-branch representation is possible for M≥Mci where i=1,2,3; however, no branch exists for M<Mci. The variation in f′′0 and −θ′0 get dominant change against ϕCu in the upper branch for a fixed value of M, while the reversal phenomenon can be seen in the bottom branch. Furthermore, as the magnetic field is increased, the skin friction coefficient g′0 decreases by maintaining constant values of ϕCu in the upper branch while slower trend is predicted in the lower branch.Figure 5
Impact ofM on f′′0 for fixed values of ϕCu.Figure 6
Impact ofM on g′0 for fixed values of ϕCu.Figure 7
Impact ofM on −θ′0 for fixed values of ϕCu.The influences of positive constantn on f′′0,g′0, and −θ′0 is reported in Figures 8 and 9The obtained results are prepared in view of stretching/shrinking parameter λ. A lower observation off′′0for stretching constant is noted. In upper zone, upon increasing λ, the change in g′0 is declining while g′0 increases. The thermal boundary layer separation is greater for linear stretching as compared to nonlinear case. Moreover, the heat transfer rate for stretching case is more progressive as compared to shrinking case. From Figure 10, a declining change in f′′0 is observed for greater λ.Figure 8
Impact ofλ on −θ′0 for fixed values of n.Figure 9
Impact ofλ on f″0 for fixed values of n.Figure 10
Impact ofλ on g′0 for fixed values of n.Figures11–13 show the effect of copper solid volume fraction ϕCu on velocity profiles f′η and gη in the x- and y-directions, as well as a temperature profile θη. Different profiles in both branches fulfill the far-field BCs η⟶∞ for various values of ϕCu and hence validate the obtained numerical results. In Figure 11, the field of velocity decreases as parameter ϕCu increases in the upper surface while contradictory results are noted in the lower branch. This is because the volume fraction slows the velocity of the fluid, which reduces the thickness of the momentum layer and, as a result, the velocities in the x-direction. On the other hand, when ϕCu rises, velocity rate increases in the y-direction for both branches (refer to Figure 12). Furthermore, as expected, heat transfer rate gets arisen for copper solid volume percentage ϕCu rises in both branches.Figure 11
Variation off′η against η for various values of ϕCu.Figure 12
Variation ofgη against η for various values of ϕCu.Figure 13
Variation ofθη against η for various values of ϕCu.The graphical outcomes for rotation constantΩ parameter on velocity along y-directions (gη) have been worked out in Figure 14. The solution has a symmetrical nature. When the value of Ω is positive, velocity increases in both branches. When the value of Ω is negative, the velocity of the hybrid nanofluid decreases. It is also discovered that when −0.06≤Ω≤0.06, two branches are possible. Figure 15 shows the effect of heat sink/source parameter on temperature profiles. It has been observed that temperature of hybrid nanofluid increases in both solutions when the parameter Q increases. This enhancement is due to external heat source. This aids in the formation of the thermal boundary layer thickness.Figure 14
Variation ofgη against η for various values of Ω.Figure 15
Variation ofθη against η for various values of Q.It is well justified fact that upon existing of different solution branches, the importance of stability analysis cannot be denied identifying the more realistically branch. Usually, the upper branch is referred to the physical branch as it meets with the boundary assumptions. However, the stability of lower branch is also importance and can be observed via stability procedure. After carefully noticing, it is claimed that upper branch is more stable in view ofε1. The positive sign of ε1 ensures that stability of branch and reflects the decay and growth results. The negative values of ε1 show that the branch is unstable. Both positive and negative values of ε1 upper and lower branches are presented via Table 4. Figure 16 shows the lower numerical values of eigenvalue for upper and lower surface branches.Table 4
The values ofε1 for various values of ϕCu and b when λ=−1,M=0.5,n=2.5,Pr=6.2,Rd=Q=0.5.
ϕCubε1Upper branchLower branch0.01−32.2281−2.2827−2.51.9924−1.90010.03−32.6295−2.8547−2.52.1824−2.43730.05−32.8626−3.0626−2.52.25463−2.4515Figure 16
The values ofε1 for various values of M.


## 5. Conclusions

In a rotating hybrid nanofluid, the laminar steady boundary flow on a stretching/shrinking sheet was investigated. The governing PDEs were numerically solved inbvp4c using the well-known collocation approach known as the 3-stage-Lobatto-3a method. Numerical results for various values of physical application parameters have been graphically presented and thoroughly analyzed. The values of ReCfx and ReCfy are compared to published publications in order to validate the obtained numerical results, which show a positive agreement. In the precise range of the magnetic, suction, and shrinking/stretching parameters, dual branches have been discovered. The following are a few key points from the current study:(i)
For suction case, the heat transfer rate is larger as compared to injection phenomenon in the upper branch.(ii)
In upper branch, no significance change in y-directional velocity is observed while declining change in velocity is noticed for lower branch.(iii)
The presence of external heat source enhanced the heat transfer rate in both branches. However, the increment in heat transfer is relatively larger in the upper branch.(iv)
No dual branches exist outside the magnetic parameter and stretching/shrinking constant.(v)
Upon increasing the copper nanoparticle volume fraction, the enhancement in temperature rate is measured.(vi)
The fluctuation in thermal boundary layer is more progressive for linear stretching instead of nonlinear case.



---

*Source: 1736516-2022-07-01.xml*