# Numerical Study of the Flow of Two Radiative Nanofluids with Marangoni Convection Embedded in Porous Medium

Authors: Ravi Gupta; Manish Gaur; Qasem Al-Mdallal; Sunil Dutt Purohit; Daya Lal Suthar
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

The aim of this study is to investigate the flow of two distinct nanofluids over a stretching surface in a porous medium with Marangoni convection. This investigation is studied under the effect of thermal radiation. Here, we have consideredFe3O4 and ZrO2 nanosized particles suspended in engine oil (EO) base fluid. For the numerical simulation of the flow, the fourth-order Runge-Kutta method and suitable similarity solutions were used. Numerical solutions with graphical representation are presented. Fe3O4/EO nanofluid is more significant in the cooling process in comparison to ZrO2/EO nanofluid. With increased radiation and temperature ratio parameters, a decrement in the temperature field has been noticed for both nanofluids. For increased values of volume friction parameter, a decrement is noticed for velocity profile and increment is noted for temperature profiles for both nanofluids. Also, a reduced velocity profile can be obtained with increased porosity parameter.


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## Body

## 1. Introduction

In the past two decades, nanofluids got the vast attention of researchers due to their proven efficiency in thermal conductivity. A base fluid with suspended nanometer-sized particles with different shapes and sizes is called nanofluids. These nanoparticles can be metallic or nonmetallic or oxidic, for example,Al, Cu, Ag, SiO2, Al2O3, Fe3O4, and ZrO2, and base fluids can be H2O, C2H6O2, and engine oil. Initially, Choi [1] investigated this phenomenon and named these fluids as “nanofluids.” Afterward, Xuan and Roetzel [2] studied the correlation between heat transfer and nanofluids and presented a numerical model. The heat transfer capacity of nanofluids is much higher than that of conventional fluid because the thermal conductivity of a fluid can be optimized by adding nanosized particles; a number of studies have been presented by researchers proving this concept for the nanofluids using different sizes, shapes, and concentrations of nanoparticles with different base liquids, on different geometries [3–13]. Some key applications of nanofluids are in the field of nuclear plants, micropolymer films, heat exchangers, electronic devices, space technology, and production of heat pipes.The convection characterized by the surface tension differences at the interface is defined as Marangoni convection. This interface dissipative flow can be relevant where different surface tension exists at interfaces, and by changing the temperature or concentration, this difference can be created. Marangoni convective boundary layer flows of nanoliquids have their applications in various fields like thin film, melting of crystals, welding, semiconductors, vapor bubbles, material sciences, soap films, silicon wafers, and microgravity conditions. Napolitano [14] was the first who gave this phenomenon and named it. Christopher and Wang [15] studied the Prandtl number effects for Marangoni convective flow through a flat surface. Furthermore, Aly and Ebaid [16] investigated radiative Marangoni convective flow of three different hybrid nanofluids with porosity and MHD effects. The thermal performance of glycerol/water mixture in the microchannel with Marangoni convection is studied by Yang et al. [17]. Afterward, this phenomenon is studied with different parameters, nanofluids, and geometries by some researchers [18–20].Although nanofluids have been studied by many researchers with different nanoparticles suspended in different base fluids, in this study, engine oil base fluid is considered which is rare in the literature. Marangoni convection is an important aspect of determining fluid motion and material processing, particularly in low-gravity hydrodynamics, for analyzing heat transfer interactions, and nanoparticles can improve the flow’s thermal performance. As per the best knowledge of the authors, the present study of Marangoni convective radiative flow of nanosized particles suspended in engine oil (EO) base fluid is novel. A comparative analysis for both nanofluids is also presented by the graphs and discussed and obtained a strong agreement with the existing literature.


## 2. Mathematical Model

Marangoni convective boundary layer flow of two different viscous, compressible nanofluids with thermal radiation, along the stretched surface, is studied in 2D. Engine oil is considered the base fluid with suspension ofFe3O4 and ZrO2 nanoparticles. Base fluid and nanosized particles of laminar flow are assumed to be thermally stable. The fluid flow is considered at y≥0; here, the x-axis is chosen parallel to the surface, and the y-axis is considered perpendicular to the stretching surface. Tw is the temperature at the surface, and T∞ is the ambient temperature. Likewise, as Marangoni convection is considered, a linear relation of surface tension with temperature is given by [19]
(1)γ=γ01−γ¯T−T∞.Here,γ0 is the surface tension and γ¯ is the rate of change of surface tension along T (temperature). Taking these assumptions, the equation of the governing convective flows of the nanofluid is described as follows [19]:
(2)∂u∂x+∂v∂y=0,(3)u∂u∂x+v∂v∂y=μnfρnf∂2u∂y2−μnfρnfku,(4)u∂T∂x+v∂T∂y=knfρCpnf∂2T∂y2−1ρCpnf∂qr∂y,with the boundary condition:(5)v=0,T=T∞+ax2,μnf∂u∂y=∂σ∂T∂T∂xaty=0,u⟶0,T⟶T∞asy⟶∞.Here, velocity notations inx and y directions are chosen as u and v accordingly. Radiative heat flux is represented by qr. The Rosseland approximation takes into account radiative heat flux which is reported by [20]
(6)qr=−4σ∗3k∗∂T4∂y=−16σ∗3k∗T3∂T∂y.In the above equation, the mean absorption coefficient is denoted byk∗ and σ∗ denotes the Stefan-Boltzmann constant. After substituting this qr into equation (4), the reduced equation is given as follows:
(7)u∂T∂x+v∂T∂y=knfρCpnf∂2T∂y2+16σ∗3ρCpnfk∗∂∂yT3∂T∂y.Here,μnf is the viscosity, ρnf is the density, κnf is the thermal conductivity, and ρCpnf is the heat capacity of nanofluids. Subscripts f and nf are used to denote base fluids and nanofluids accordingly. Also, the volume fraction is ϕ; also, m=3 is chosen for spherical-type nanoparticles. Physical constraints and thermophysical value of nanoliquids and base fluid are reported in Tables 1 and 2, respectively.Table 1
Table of physical constraints of nanofluids used.
μnf=μf/1−ϕ2.5Effective dynamic viscosityρnf=1−ϕρf+ϕρsEffective densityκnf=κs+m−1κf−m−1κf−κsϕ/κs+m−1κf+κf−κsϕκfThermal conductivityρCpnf=ρCpsϕ+ρCpf1−ϕHeat capacitanceTable 2
Thermophysical properties [19, 20].
ρkgm−3CpJkg‐1K‐1kWm‐1K‐1PrEO88419100.1446450Fe3O458106706—ZrO256805021.7—


## 3. Similarity Solutions

The following similarity transformation is used to solve our model [19]:
(8)η=ζ1y,φ=ζ2xfη,θη=T−T∞Tw−T∞,whereζ1=γ0γ¯aρf/μf21/3, ζ2=γ0γ¯aμf/ρf21/3, u=∂φ/∂y, and v=−∂φ/∂x.Following the foregoing transformations, equations (3) and (7) are turned into nonlinear ordinary differential equations as follows:
(9)f‴+1−ϕ+ϕρsρf1−ϕ2.5ff″−f′2−Kf′=0,(10)θ″1+RdAθθw−1+13+RdA3θ′2θw−1θθw−1+12+Bfθ′−2θf′=0.Here,A=κf/κnf=κs+m−1κf+κf−κsϕ/κs+m−1κf−m−1κf−κsϕ, B=Prf1−ϕ+ϕρs/ρf1−ϕ2.5, and C=1−ϕ2.5.The above equations are with the modified boundary conditions:(11)θ0=1,f0=0,f″0=−2C,θ∞=0,f′∞=0,whereRd=16σ∗T∞3/3k∗kf is the radiation parameter, Pr=ρCpfυf/κf is the Prandtl number, K=1/ψ12k is the permeability parameter, and θw=Tw/T∞ is the temperature ratio parameter.


## 4. Numerical Solution

With the help of the above-described similarity transformation, the governing equation of the flows has been converted into a set of the ordinary differential equations. Moreover, in order to tackle these equations using the Runge-Kutta method with a shooting technique, equations (9) and (10) together with the boundary conditions (11) turned into an initial value problem, as shown below.f=h1, f′=h2, f″=h3, θ=h4, and θ′=h5, where h3′=1−ϕ2.51−ϕ+ϕρs/ρfh22−h1h3+Kh2 and h5′=−RdA3h52θw−1h4θw−1+12−Bh1h5−2h4h2/1+RdAh4θw−1+13.The above equations are with the following transformed boundary conditions,h10=0, h30=−2C, and h40=1. The best approximated numerical results are obtained by some initial guesses with the shooting method. To get exact results up to 10−7 accuracy, an iterated approach is used in this method. The Prandtl number for EO is 6450, and the step size Δη=0.01 is considered.


## 5. Results and Discussion

The comparative investigation of Marangoni convection ofFe3O4 and ZrO2 nanofluid flow is studied. This investigation is studied under the effect of thermal radiation over the stretching surface. Numerical simulation is performed with the above-described method, and results are presented graphically to illustrate the impacts of several nondimensional physical parameters for f′η (temperature profile) and θη (velocity profile).The impact ofK on f′η for Fe3O4/EO and ZrO2/EO nanofluids, respectively, is depicted in Figure 1. It is observed that with increased K, a decrement in f′η is seen for both nanofluids. With the fact K∝1/k, hence if we raise the coefficient of porosity, the permeability of the porous media decreases, resulting in a drop in fluid velocity.Figure 1
Velocity profile forK.The effect of volume frictionϕ on f′η is depicted in Figure 2. It is noticed that with increased ϕ, we get a decreased velocity profile. With rising ϕ, there is a significant influence of ϕ on the velocity profiles, with the speed of fluid being reduced at the vicinity of the surface and the reverse effect being visible further from the surface. It is also worth noting that the ZrO2/EO nanofluid has a larger velocity than the Fe3O4/EO nanofluid. This is due to the fact that density of ZrO2 is lower than that of Fe3O4, implying that ZrO2 is lighter in motion than Fe3O4.Figure 2
Velocity profile forϕ.The impact ofϕ on θη is depicted in Figure 3. It is observed that with increased ϕ, an increased temperature profile for both nanofluids is seen. With increased nanoparticles of both nanofluids, the temperature of fluid is increased; as a result, the temperature field may rise.Figure 3
Temperature profile forϕ.The impact ofRd on θη is depicted in Figure 4. It is concluded that with increased Rd along with θη, an increased temperature profile is observed. Radiations create additional heat within the fluids, which results in higher temperature fields. Furthermore, Rd has no substantial effect on the velocity fields. The radiation parameter, which is the inverse of the Stephan number, is used to determine the relative significance of thermal radiation versus conductive heat transmission. As a result of the thermal radiation’s dominance over conduction, it may be employed to regulate the thermal boundary layers rather well.Figure 4
Temperature profile forRd.The impact of the temperature ratio parameterθw on θη is depicted in Figure 5. It is concluded that with increased θw along with θη, an increased temperature profile is observed. As a recognizable outcome, it is likewise seen that the temperature profile of the Fe3O4/EO nanoliquid remains consistently greater than that of the ZrO2/EO nanoliquid.Figure 5
Temperature profile forθw.


## 6. Code of Verification

For the validation of this research, comparison with Das et al. [21] and Jamshed et al. [4] was made for local Nusselt numbers using several values of Pr as shown in Table 3. To do this, the remaining parameters in our simulation approach are set to zero (Rd=K=ϕ=θw=0). The calculated findings are found to be in good match with the previous results.Table 3
Comparison for different values of the Prandtl number for−θ′0 assumingRd=K=ϕ=θw=0.
PrJamshed et al. [4]Das et al. [21]Present study0.720.808761810.808761220.80876183111131.923574201.923574311.9235742473.073146513.073146793.07314652103.720554293.720554363.72055430


## 7. Conclusions

Marangoni convection ofFe3O4 and ZrO2 nanofluid flow past a stretching surface with thermal radiation embedded in porous media is considered. The flow’s governing model is encountered with the Runge-Kutta fourth-order technique and suitable similarity transformations. These results can be used to study the thermal performance with other boundary conditions.The following important outcomes are obtained.(i)
TheFe3O4/EO nanofluid is more significant in the cooling process in comparison to the ZrO2/EO nanofluid(ii)
With increased porosity parameterK, a decrement in the velocity field has been noticed for both nanofluids(iii)
With increasedRd and θw parameters, a decrement in the temperature field has been noticed for both nanofluids(iv)
For the increased values ofϕ, a decrement is noticed for the velocity profile and opposite behavior is observed for temperature profiles for Fe3O4/EO and ZrO2/EO nanofluids



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