# Effects of Second-Order Slip Flow and Variable Viscosity on Natural Convection Flow ofCNTs−Fe3O4/Water Hybrid Nanofluids due to Stretching Surface

Authors: Ayele Tulu; Wubshet Ibrahim
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

This study deals with natural convection unsteady flow ofCNTs−Fe3O4/water hybrid nanofluids due to stretching surface embedded in a porous medium. Both hybrid nanoparticles of SWCNTs−Fe3O4 and MWCNTs−Fe3O4 are used with water as base fluid. Effects of hybrid nanoparticles volume friction, second-order velocity slip condition, and temperature-dependent viscosity are investigated. The governing problem of flow is solved numerically employing spectral quasilinearization method (SQLM). The results are presented and discussed via embedded parameters using graphs and tables. The results disclose that the thermal conductivity of CNTs−Fe3O4/H2O hybrid nanofluids is higher than that of CNTs−H2O nanofluids with higher value of hybrid nanoparticle volume fraction. Also, the results show that momentum boundary layer reduces while the thermal boundary layer gros with higher values of temperature-dependent viscosity and second-order velocity slip parameters. The skin friction coefficient improves, and the local heat transfer rate decreases with higher values of nanoparticle volume fraction, temperature-dependent viscosity, and second-order velocity slip parameters. Furthermore, more skin friction coefficients and lower local heat transfer rate are reported in the CNTs−Fe3O4/H2O hybrid nanofluid than in the CNTs−H2O nanofluid. Thus, the obtained results are promising for the application of hybrid nanofluids in the nanotechnology and biomedicine sectors.


---

## Body

## 1. Introduction

The broad applications of heat transfer in various sectors of industry and biomedicine have required the accessibility of efficient thermal performance techniques. In the past few decades, several techniques of enhancing the thermal performance of working fluids have been realized by different researchers. Scattering of nanoparticles of metallic structures such as copper, carbides, alumina, nitrides, metal oxides, carbon nanotubes, and graphite in the working fluid is considered to be one of the innovative and efficient methods (Mahanthesh et al. [1]). At present, nanofluids are the noble options for the heat transfer fluids due to their remarkably higher thermal conductivity, and their use is common in heat exchangers, cooling systems, solar energy, biomedicine, and so forth (Aziz et al. [2]). Besides, the porous media have better dissipation area, which results in improved convective heat transfer. As a result, Reddy and Sreedevi [3] analyzed heat and mass transfer characteristics of nanofluid flow over porous stretching sheet. The non-Newtonian Casson nanofluid flow and heat transfer over stretching cylinder in a porous medium were investigated by Tulu and Ibrahim [4]. Furthermore, the effect of temperature and concentration on the thermal conductivity of ZnO-TiO/EG hybrid nanofluid using artificial neural network and curve fitting on experimental data was evaluated by Safaei et al. [5].At present, heat transfer of carbon nanofluids (CNTs) has received great attention due to their potential applications in the fields of nanotechnology and biomedicine. CNTs are allotropes of carbon prepared in cylindrical tubes of graphite with nanometer in diameter and a few millimeters in length (Hirlekar et al. [6]). CNTs are generally divided into single-wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs) depending on their number of concentric layers of rolled graphene sheets. SWCNTs contain catalyst for their synthesis and encompass a particular graphene sheet encircled around and form a cylinder. MWCNTs are concentrically nested cylinders of graphene sheets and they can be formed without catalyst. The inner and outer diameters of MWCNTs are determined by their number of layers (Khalid et al. [7]). CNTs are the resourceful material of modern technologies because of their special thermal, electrical, and mechanical characteristics, unique morphology, and innovative physicochemical features, and the presence of carbon chains in CNTs does not convey any hazard to the atmosphere (Tulu and Ibrahim [8] and Alsagri et al. [9]).Modern applications of carbon nanotubes in the biomedical field for drug delivery, cancer therapy, and other applications in medicines have attracted the attention of different researchers in studying the thermophysical properties and nanofluids flow of CNTs. For example, Alnaqi et al. [10] predicted the effect of functionalised MWCNTs on thermal performance factor of water under some Reynolds numbers using artificial neural network. Furthermore, CNTs have encouraging vigorous surface area that permits them to be functionalised and ornamented by numerous kinds of chemical and biological matters. Due to their needle-like shape, CNTs have potential to infiltrate more easily into cell walls and move among various bodies’ tissues (Cai et al. [11]). Also, they have the potential to carry drugs in the organism and have the ability to target and destroy specific cancer cells without harming the healthy cells (Srinivasan [12]). Moreover, in the long run, carbon nanotubes with friendly enzymes can be used as enzymatic biosensors that might concurrently sense and measure a multiplicity of biological molecules (Singh et al. [13]).Carbon nanotubes, however, do not have ample magnetic properties which limit their power for uses in biomedical applications. Furthermore, the weak solubility of CNTs in aqueous solutions and challenging of CNTs in added solvents limit their applications in the biomedical and other technological fields (Yang et al. [14]). Consequently, to overacome the above mentioned limitations of CNTs, researchers have considered different approaches of CNTs nanoparticles preparations (Asadi et al. [15], Asadi et al. [16], Bagherzadeh et al. [17], and Saba et al. [18] to mention few). Among these, magnetic delivery of CNTs is shown to be the best promising approach due to its ability to use a located magnet inside the body or an external magnet in several diagnostic and therapeutic agents at targeted tissues (Samadishadlou et al. [19]). Also, due to the nonhostile behavior of magnetic fields, employing magnetic CNTs in biomedical applications reduces their side effects to adjacent tissues (Mody et al. [20]). In view of these, the hybrid nanoparticles with CNTs that mostly received attention in the biotechnological and biomedical fields are CNTs with magnetite (Fe3O4) (Rahmawati et al. [21]). This is due to the fact that Fe3O4 exhibits more chemical inertness and stability, but it has lower thermal conductivity than CNTs. Also, CNTs-Fe3O4 nanoparticles advance targeting efficiency and enable magnetically engaged delivery. Besides, the magnetite nanocrystals inside the CNTs advance their drug loading ability, preserve them from agglomeration, and improve their chemical stability (Masotti and Caporali [22]).Recently, much effort has been given to developing new experimental work to obtain well-defined and highly stable hybrid nanoparticles which enhance the thermal conductivity of base fluid. For example, Tassaddiq et al. [23] studied the transfer of heat and mass over a rotating disk involving carbon nanotubes (CNTs) and magnetic ferrite nanoparticles together with carrier fluids. Also, the heat transfer and friction factor of MWCNT-Fe3O4/water hybrid nanofluids were investigated by Sundar et al. [24]. They reported that these hybrid nanofluids have significantly improved thermal and mechanical properties than regular fluids or mono nanofluids. Hybrid nanofluids have important applications in various engineering and biomedical fields such as in manufacturing, transportation, nuclear safety, military, modern electronic devices and supercomputers, and pharmaceutical and drug delivery (Manjunatha et al. [25]). Due to the mentioned applications, many scholars have been interested in hybrid nanofluids investigation. For example, using the polymer technique, Shi et al. [26] synthesized CNT/Fe3O4 hybrid nanoparticles. The impacts of mixing MgO-MWCNT hybrid nanoparticles in thermal oil were analyzed by Asadi et al. [27]. Recently, Zaresharif et al. [28] have undertaken hydrothermal analysis on natural convection and TiO2-SiO2/W-EG hybrid nanofluids properties. The free convection heat transfer and entropy generation analysis of water-Fe3O4/CNT hybrid nanofluid were reported by Shahsavar et al. [29]. Also, Sundar et al. [30] experimentally synthesized MWCNT/Fe3O4 water-based hybrid nanofluids. They reported that hybrid nanofluids create higher thermal conductivity and heat transfer compared to single nanoparticles-based nanofluids. Similarly, they observed that the viscosity of (MWCNT-Fe3O4)/water hybrid nanofluids significantly improved as compared to base fluid. Further, they suggested that more analyses and experiments are needed to fully realize the means of improving heat transfer of hybrid nanofluids.Various nanofluids flow studies have considered with constant physical properties of fluid. However, the viscosity of nanofluids may change significantly with temperature and play an indispensable role in nanofluids flow. For example, the heat produced via internal friction increases the temperature, which consecutively affects the stickiness of nanofluids. Thus, to precisely evaluate the flow nature of nanofluids, it is essential to consider this disparity of viscosity with temperature. Kuttan et al. [31] and Manjunatha and Gireesha [32] analyzed the flow of fluid over a flat surface with the effects of temperature-dependent viscosity. Also, Udawattha et al. [33] predicted the effective viscosity of nanofluids based on the rheology of suspensions of solid particles. So far, there is a lack of information on CNTs-Fe3O4/H2O hybrid nanofluids flow considering the effects of temperature-dependent viscosity.There are situations where no-slip boundary condition is not suitable. For example, for different non-Newtonian fluids, various polymer melts usually express small wall slip; and, normally, they are controlled by a monotonic relation and a nonlinear relationship between the slip velocity and the adhesive friction (Halim et al. [34]). Fluids offering slip boundary condition have important applications in some technological and biomedical fields, for instance, in expensive lubricating, optical coatings, refrigeration equipment, purifying of artificial heart valve, internal cavities, and other industrial processes. As a result, the effect of slip boundary condition has been considered by some researchers [35]. For example, Oyelakin et al. [36] and Tlili et al. [37] examined the effects of first-order slip boundary condition on the flow and heat transfer of nanofluid over stretching sheet and cylinder, respectively. Recently, the effects of second-order slip condition (Wu’s slip features) on nanofluids flow and heat transfer were reported by Khan et al. [38]. They revealed that temperature distribution and thermal boundary layer grow with bigger values of second-order velocity slip conditions. Yet, there is limited information about the effect of second-order slip condition on CNTs−Fe3O4/H2O hybrid nanofluids flow over stretching surface. Thus, one of the aims of this study is to fill the gaps in the above indicated knowledge.Motivated by the above cited literature survey, the purpose of this study is to analyzeCNTs−Fe3O4/H2O hybrid nanofluids flow and heat transfer due to stretching surface. The impacts of temperature-dependent viscosity, free convection, and second-order slip condition are also considered. The governing equations are solved numerically employing the fast convergent and accurate technique, namely, spectral quasilinearization method (SQLM). The effects of governing parameters on hybrid nanofluids flow velocity and temperature distributions are discussed and presented in tables and graphs as well. To the best of our knowledge, no analysis has been published in this direction yet. The developed model has potential applications in the biomedical fields such as cancer therapy, drug delivery, and enzymatic biosensors. Considering the effects of temperature-dependent viscosity and second-order slip condition on CNTs−Fe3O4/H2O hybrid nanofluids flow and heat transfer and computing it by means of SQLM make this study novel and different from the former studies.


## 2. Mathematical Description of Problem

We consider an unsteady two-dimensional incompressible viscous flow ofCNTs−Fe3O4 with H2O hybrid nanofluids past semi-infinite linearly stretching surface embedded in a porous medium. The flow is situated to y>0, where x and y are, respectively, in the direction of flow and normal to the surface, as shown in Figure 1. Also, we considered the base fluid and the hybrid nanoparticles to be in thermal equilibrium so that no slip occurs between them. It is worth mentioning that, to develop the targeted hybrid nanofluids CNTs−Fe3O4/H2O, initially, CNTs are dispersed into base fluid and then Fe3O4 is scattered in CNTs/H2O nanofluids. In the beginning, at time t=0, both the surface and hybrid nanofluids are at rest with uniform temperature T˜s. For time t>0, the surface begins stretching linearly with velocity u˜w=u˜0/1−ζtx, where u˜0 is constant (u˜0>0 for a stretching and u˜0<0 for a shrinking surface), ζ≥0 is constant, t is time, and ζt<1. At the same time, the surface temperature is upturned to T˜b, which is then kept constant.Figure 1
Physical model of the flow problem.With foregoing assumptions and Boussinesq approximation, the transport equations of hybrid nanofluid boundary layer flow are established as follows (Manjunatha et al. [25]):(1)∂u˜∂x+∂v˜∂y=0,(2)ρhnf∂u˜∂t+u˜∂u˜∂x+v˜∂u˜∂y=μhnf∂2u˜∂y2+∂u˜∂y∂∂yμhnf−μhnfKu˜+gρβthnfT˜−T˜s,(3)ρcphnf∂T˜∂t+u˜∂T˜∂x+v˜∂T˜∂y=khnf∂2T˜∂y2+Q0T˜−T˜s.The boundary conditions with second-order velocity slip and thermal slip are given as follows (Manjunatha et al. [25] and Ibrahim [39]):(4)t≥0:u˜x,t=u˜w+u˜slip,v˜=0,T˜=T˜b+N0∂T˜∂yaty=0,t≥0:u˜x,t⟶0,v˜⟶0,T˜⟶T˜sasy⟶∞,and initial condition is(5)t<0:u˜=0,v˜=0,T˜=T˜s,∀x,y,where u˜ and v˜ are the hybrid nanofluid velocity components in the x and y orientations, respectively; ρhnf, μhnf, khnf, and ρcphnf are hybrid nanofluid effective density, dynamic viscosity, thermal conductivity, and specific heat capacity, respectively; g, βhnf, Q0, and N0 are gravitational acceleration, hybrid nanofluid volume expansion coefficient, heat generation/absorption rate, and thermal slip factor, respectively.The temperature-dependent viscosity of the base fluid is given as follows (Manjunatha et al. [25]):(6)μf=μ0e−Rθξ,where μ0 is reference viscosity of base fluid and R is variable viscosity parameter (note that R<0 for gases and R>0 for liquids).The slip velocityu˜slip at the surface, where Wu’s [40] slip velocity equation is usable for random Knudsen number, Kn, and is employed by researcher Ibrahim [39], is specified as(7)u˜slip=a∂u˜∂y+b∂2u˜∂y2,where a and b are constants given as(8)a=233−αr2α−321−r2Knλ,b=−14r4+2Kn21−r2λ2,where Kn is Knudsen number, r=min1/Kn,1, λ is the molecular mean free path, and α is the momentum accommodation coefficient with 0≤α≤1.The values of thermophysical properties of nanoparticles CNTs,Fe3O4, and base fluid are given in Table 1 (Tulu and Ibrahim [8], Alsagri et al. [9], Sundar et al. [24], and Hayat et al. [41]).Table 1
Thermophysical properties values of base fluid, CNTs, andF3O4.
Physical propertiesH2OSWCNTMWCNTFe3O4ρkg/m3997.1260016005180cpJ/kgK4179425796670kW/mK0.613660030009.7σS/m0.051061.9×10−425000β10−5K−12127441.05The hybrid nanofluid effective thermophysical properties models are given in Table2 (Alsagri et al. [9] and Manjunatha et al. [25]). ϕCNTs is the CNTs nanoparticle volume fraction, and ϕFO is the Fe3O4 nanoparticle volume fraction. ρf, ρcpf, μf, and ρβtf are, respectively, density, specific heat capacity, dynamic viscosity, and volumetric thermal expansion coefficient of the base fluid. ρCNTs, ρcpCNTs, kCNTs, and ρβtCNTs; ρFO, ρcpFO, kFO, and ρβtFO, are, respectively, density, specific heat capacity, thermal conductivity, and volumetric thermal expansion coefficient of CNTs and F3O4.Table 2
Thermophysical models of hybrid nanofluid.
Thermophysical propertiesHybrid nanofluidCNTs−Fe3O4/H2ODensitykg/m3ρhnf=1−ϕFO1−ϕCNTsρf+ϕCNTsρCNTs+ϕFOρFOHeat capacityJ/kgKρcphnf=1−ϕFO1−ϕCNTsρcpf+ϕCNTsρcpCNTs+ϕFOρcpFOViscosityNs/m2μhnf=μf/1−ϕCNTs2.51−ϕFO2.5Thermal conductivityJ/Kkhnf/knf=1−ϕFO+2ϕFOkFO/kFO−knflnkFO+knf/2knf/1−ϕFO+2ϕFOknf/kFO−knflnkFO+knf/2knfwhereknf/kf=1−ϕCNTs+2ϕCNTskCNTs/kCNTs−kflnkCNTs+kf/2kf/1−ϕCNTs+2ϕCNTskf/kCNTs−kflnkCNTs+kf/2kfThermal expa. coeff.K−1βthnf=1−ϕFO1−ϕCNTsβtf+ϕCNTsβtCNTs+ϕFOβtFOThe continuity equation (1) is adequately fulfilled introducing the following dimensionless variables based on Oyelakin et al. [36] and Manjunatha et al. [25]:(9)ξ=u˜0νf1−ζty;u˜=u˜01−ζtxf′ξ;v˜=−u˜0νf1−ζtfξ;θξ=T˜−T˜sT˜b−T˜s,where f′ξ is a dimensionless velocity and θξ is the dimensionless temperature.Consequently, the transformed equations of momentum and energy are specified as follows.(10)f‴ξ+ϕ1ϕ2fξf″ξ−Af′ξ+ξ2f″ξ−f′2ξ+ϕ3τθξ−Rθ′ξf″ξ−κf′ξ=0,(11)khnfkfθ″ξ+ϕ4Prfξθ′ξ−A2θξ+ξ2θ′ξ+PrHθξ=0,with transformed boundary conditions:(12)f=0,f′0=1+γf″0+δf‴0;θ0=1+βθ′0,ξ≥0,f′∞⟶0;θ∞⟶0,ξ≥0.where(13)ϕ1=1−ϕFO1−ϕCNTs+ϕCNTsρCNTsρf+ϕFOρFOρf,ϕ2=1−ϕCNTs2.51−ϕFO2.5,ϕ3=1−ϕFO1−ϕCNTs+ϕCNTsβtCNTsβtf+ϕFOβtFOβtf,ϕ4=1−ϕFO1−ϕCNTs+ϕCNTsρcpCNTsρcpf+ϕFOρcpFOρcpf,where A=ζ/u0 is unsteady parameter, κ=νf1−ζt/Ku˜0 is permeability parameter, Pr=νfρcpf/kf is a Prandtl number, H=1−ζtQ0/u˜0ρcpf is heat generation/absorption rate parameter, τ=gβtT˜b−T˜s/xu˜02 is Grashof number, γ=au˜0/νf is the first-order velocity slip parameter, δ=bu˜0/νf is the second-order velocity slip parameter, and β=N0u˜0/νf is the thermal slip parameter.The shearing stress at the wall is defined for hybrid nanofluid as(14)τw=μhnf∂u˜∂yy=0=u˜0μf1−ϕCNTs2.51−ϕFO2.5u˜0νfxf″0.The surface heat flux at the wall is given for hybrid nanofluid as(15)qw=−khnf∂T˜∂yy=0=−khnfu˜0νfT˜b−T˜sθ′0.The surface drag and wall heat transfer rates are characterized by the skin friction coefficientCf and the local Nusselt number Nu, respectively, and they are obtained as(16)Cfx=τwρfu˜02,Nux=xqwkfT˜b−T˜s.Thus, the dimensionless skin friction coefficient and local Nusselt number are given by(17)Rex1/2Cfx=11−ϕCNTs2.51−ϕFO2.5f″0,(18)Rex−1/2Nux=−khnfkfθ′0,where Rex is the local Reynolds number defined by Rex=u˜wx/νf.


## 3. Method of Solutions

In this section, the nonlinear differential equations (10) and (11) are solved numerically by SQLM. For complete explanation of the method, the interested person is referred to Motsa [42] and Ibrahim and Tulu [43]. Using this method, the system of nonlinear differential equations in two unknowns f and θ gives the following iterative scheme of linear differential equations:(19)fj+1″′+ϕ1ϕ2fj−A2ξ−Rθj′fj+1″−ϕ1ϕ22fj′+A+κfj+1′+ϕ1ϕ2fj″fj+1+ϕ1ϕ2ϕ3τθj+1−Rfj″θj+1′=ϕ1ϕ2fjfj″−fj′2−Rθj′fj″,(20)khnfkfθj+1″+ϕ4Prfj−A2ξθj+1′+PrH−2Aϕ4θj+1+ϕ4Prθj′fj+1=ϕ4Prfjθj′,with boundary conditions(21)fj+1=0,fj+1′0=1+γfj+1″0+δfj+1″′0θj+10=1+βθj+1′0,fj+1′∞⟶0;θj+1∞⟶0,where the terms j and j+1 are at the previous and current iteration levels, respectively.Equations (19) and (20) establish the iterative scheme for the SQLM. A numerical solution is then found employing the Chebyshev spectral collocation method. Beginning with appropriate initial approximations, the iteration schemes are used to obtain fj+1 and θj+1.First, it is essential to change the semi-infinite domain to a truncated domain0,L∞. Then, the interval 0,L∞ is transformed to the interval −1,1 using the linear transformation ξ=1/2L∞χ+1. Further, the Gauss-Lobatto points are selected to define the nodes in [−1, 1] as reported in Trefethen [44]. That is,(22)χi=cosπiN,i=0,1,2,…,N;−1≤χ≤1,where N is the number of collocation points used.Using the Chebyshev spectral collocation method, equations (19) and (20) are discretized. The derivatives of the unknown functions f and θ at the collocation points are defined using the Chebyshev differentiation matrix D as given in Trefethen’s work [44]. That is,(23)dfdξ=∑p=0NDpkfχp=DϜ,p=1,2,3,…,N,where D=2D/L∞ and Ϝ=f0,f1,−−−,fNT is a vector function at the collocation points. Higher-order derivatives are found as powers of D as(24)Ϝnξ=DnϜ,where n is the order of derivative and D is matrix of size N+1×N+1.Applying the spectral method to the system of equations (19) and (20), it can be solved as a coupled matrix:(25)Ψ11Ψ12Ψ21Ψ22Ϝj+1Θj+1=b1b2,with transformed boundary condition(26)Ϝj+1χN=0,Ϝj+1χN−1=1+γfj+1″0+δfj+1″′0,Ϝj+1χ0=0,Θj+1χN=1+βθr+1′0,Θj+1χ0=0,where(27)Ψ11=D3+ϕ1ϕ2fj−A2ξ−Rθj′dD2−ϕ1ϕ22fj′+A+κdD+ϕ1ϕ2fj′′d,Ψ12=ϕ1ϕ2ϕ3τI−Rfj′′dD,Ψ21=ϕ4Prθj′d,Ψ22=khnfkfD2+ϕ4Prfj−A2ξdD+PrH−2Aϕ41I,b1=ϕ1ϕ2fjfj′′−fj′2−Rθj′fj′′,b2=ϕ4Prfjθj′,Ϝj+1=fj+1,0,fj+1,1,−−−,fj+1,NT,Θj+1=θj+1,0,θj+1,1,−−−,θj+1,NT,are vectors of sizes N+1×1; I and …d, respectively, represent an identity and a diagonal matrix of size N+1×N+1.The stability and convergence of the iteration schemes for velocity and temperature distributions can be evaluated by considering the norm of their differences between two successive iterations. Hence, for each iteration scheme, we can define the maximum error (Ed) for velocity (f′) and temperature (θ) at the r+1th iteration as follows (Motsa [42]):(28)Ed=Maxfr+1′−fr′∞,Ed=Maxθr+1−θr∞.If the iteration scheme converges, the error (Ed) is expected to reduce with an increase in the number of iterations. In this study, the maximum error of the unknown functions f′ and θ can be calculated for a given number of collocation points N until the criteria for the convergence tolerance set are fulfilled at iteration r.When SQLM is employed, the choice of initial guesses is crucial. Thus, the proper initial guesses that fulfill the governing equations of the considered flow problem are(29)f0ξ=1−e−ξ1+γ−δ,θ0ξ=e−ξ1+β.


## 4. Results and Discussion

Natural convection flow ofCNTs−Fe3O4/H2O hybrid nanofluids due to stretching surface in non-Darcy porous medium with the effects of second-order slip and variable viscosity is considered. For the study, it is assumed that 1% (0.01 volume) of CNTs, 4% (0.04 volume) of Fe3O4, and 95% (0.95 volume) of H2O are in use for the preparation of CNTs−Fe3O4/H2O hybrid nanofluid. The thermophysical properties of Fe3O4, SWCNTs, MWCNTs, and water in use are from Table 1. To obtain convergent, stable, and accurate results, the suitable intervals for the parameters used in this study are fixed to (30)0<ϕCNTs<0.05,0<ϕFO<0.1,0<A<5,0.0<δ<0.2,0<γ<3,0<κ<15,0<R<3,0<τ<0.1,0<β<2,−3<H<1.5.The convergence analysis was done for the skin friction coefficient and local Nusselt number, and almost the5th-order iteration is enough up to eight digits of approximations for both f″0 and −θ′0, as is shown in Table 3. Also, the accuracy of the employed method is checked using the grid-invariance test choosing mesh with nodes N = 10, 20, 50, and 100 for skin friction coefficients and local Nusselt number, as shown in Table 4. Once increasing the number of nodes to more than 50, the accuracy is not affected up to five decimal-point but only to increase the compilation time. Thus, all the results of this study are obtained with number of nodal points N=50. Besides, the accuracy of the current results is also confirmed comparing the values of f″0 and −θ′0 with formerly available literatures for comparable parameters, as shown in Tables 5 and 6, respectively, and the results are in a very sound agreement.Table 3
SQLM Convergence forf″0 and −θ′0 of SWCNTs−Fe3O4/H2O and MWCNTs−Fe3O4/H2O hybrid nanofluids for N=50,ξ=3,ϕCNTs=0.01,ϕFO=0.04,A=0.8,δ=0.02,γ=0.2,τ=0.02,κ=5,Pr=6.2,β=0.3,R=1&H=−0.5.
f″0−θ′0OrderSWCNTsMWCNTsSWCNTsMWCNTs11.595309023.514992843.042446623.1036304922.078584724.439676603.058244433.1357451232.078693964.439509003.058244433.1357582742.078693954.439508713.058244433.1357582352.078693954.439508713.058244433.1357582362.078693954.439508713.058244433.13575823Table 4
Grid-independence test forMWCNT−Fe3O4/H2O hybrid nanofluid when ξ=3 and the values of parameters ϕCNTs=0.01,ϕFO=0.04,A=0.8,δ=0.02,γ=0.2,τ=0.02,κ=5,Pr=6.2,β=0.3,R=1&H=−0.5 are used.
f″0−θ′0OrderN = 10N = 20N = 50100N = 10N = 20N = 50N = 10013.255463.514993.514993.514993.103073.103633.103633.1036322.959394.431264.439684.439683.138653.135993.135743.1357432.963324.432294.439514.439512.621213.135973.135763.1357642.963324.432294.439514.439512.621213.135973.135763.1357652.963324.432294.439514.439512.621213.135973.135763.13576Table 5
Comparison of the present result off″0 with γ when ϕCNTs=ϕFO=A=δ=k=R=τ=0.
γSahoo and Do [45]Noghrehabadi et al. [46]Oyelakin et al. [36]Present result0.01.0011541.0000001.0000001.0000000.10.8714470.8720820.8720830.8720830.20.7749330.7763770.7763770.7763770.30.6997380.7015480.7015480.7015480.50.5891950.5911960.5911950.5911961.00.4284500.4301600.4301600.4301602.00.2828930.2839800.2839790.2839815.00.1444300.1448410.1447140.14484210.00.0810910.0812430.0809320.08124520.00.0437480.0437900.0435690.043792Table 6
Comparison of the present result of−θ′0 with Pr when ϕCNTs=ϕFO=A=H=R=β=δ=γ=κ=τ=0.
PrWang [47]Khan and Pop [48]Manjunatha et al. [25]Present result0.700.45390.4539—0.4539322.000.91140.91130.911350.911367.001.89541.89541.895401.8954020.003.35393.35393.353903.3539070.006.46226.4621—6.46220Variation in fluids thermal conductivity with nanoparticles volume friction at room temperature is presented in Table7. The thermal conductivity of both SWCNT−H2O and MWCNT−H2O nanofluids improves for bigger values of nanoparticle volume fraction; and it is almost doubled when 5 percent of nanoparticles is added to the base fluid. Also, the thermal conductivity of both SWCNT−Fe3O4/H2O and MWCNT−Fe3O4/H2O hybrid nanofluids is better for increasing values of hybrid nanoparticle volume fraction; and it improves nearly by 100 percent when 1 percent of CNTs nanoparticles and 3 percent of Fe3O4 nanoparticles are added to the base fluid. These observations are comparable with experimental results of Sundar et al. [30].Table 7
Thermal conductivityknf of CNTs−H2O nanofluid and khnf of CNTs−Fe3O4/H2O hybrid nanofluid.
knfCNTs−H2OkhnfCNTs−Fe3O4/H2OϕCNTsSWCNTsMWCNTsϕCNTsϕFOSWCNTsMWCNTs0.000.6130000.6130000.010.000.7193900.7193900.010.7193900.7096260.010.011.0398701.0401270.020.8279470.808217—0.021.0802911.0808120.030.9387390.908834—0.031.1212771.1220660.041.0518361.011541—0.041.1628371.1639020.051.1673091.116402—0.051.2049861.206332The impacts of important physical parameters on velocityf′ξ and temperature θξ profiles are plotted for both SWCNT−Fe3O4/H2O and MWCNT−Fe3O4/H2O hybrid nanofluids when the values of the parameters in use for the entire results to be fixed are ϕCNTs=0.01,ϕFO=0.04,A=0.8,δ=0.02,γ=0.2,τ=0.02,κ=5,Pr=6.2,β=0.3,R=1, and H=−0.5, unless it is specified. Figures 2 and 3, respectively, represent that the greater value of nanoparticle volume fraction (ϕFO) leads to increasing both the velocity and temperature profiles of hybrid nanofluids. Due to higher density of SWCNTs compared to MWCNTs, also, it is recognized that there is higher velocity distribution in the MWCNT−Fe3O4/H2O than in the SWCNT−Fe3O4/H2O hybrid nanofluids, whereas the the opposite trend is perceived for temperature distribution. Thus, physically, SWCNT−Fe3O4/H2O hybrid nanofluid is more efficient than MWCNT−Fe3O4/H2O hybrid nanofluid for practical applications. The influences of unsteady parameter A on both velocity and temperature profiles of hybrid nanofluids flow are, respectively, reported in Figures 4 and 5. As A increases, near the boundary surface, the velocity profiles rise to its highest value; then it changes downward and finally it reduces to zero. Also, it is revealed that an increase in A leads to dropping the temperature distribution of both hybrid nanofluids, as illustrated in Figure 5.Figure 2
Velocity profiles for various values ofϕFO.Figure 3
Temperature profiles for some values ofϕFO.Figure 4
Velocity profiles for some values ofA.Figure 5
Temperature profiles for some values ofA.The effect of variable viscosity parameterR on velocity profiles of hybrid nanofluids flow is shown in Figure 6. Except adjacent to the boundary surface, the velocity profiles show a decreasing tendency as R increases. Also, the thickness of the thermal boundary layer enhances for bigger values of R, which increases the temperature profile. As well, the same pattern of velocity and temperature distribution in hybrid nanofluids flow was observed by Tulu and Ibrahim [8] and Manjunatha et al. [25]. The impacts of Grashof number (natural convection parameter) τ on velocity and temperature profiles are presented in Figures 7 and 8, respectively. The increment in τ tends to enhance the fluid velocity profiles and reaches a maximum; then it gradually falls to zero as it is far from the surface. Also, it is observed that the thermal boundary layer thickness gradually reduces with bigger value of τ. Physically, free convection flows are steadily transferred from the stretching surface to the free stream, and increase in τ indicates the progress of free convection currents. Similar result of flow distribution was found by Manjunatha et al. [25]. The effect of permeability parameter κ on velocity field of hybrid nanofluids flow is demonstrated in Figure 9. Here, except near the wall surface, the greater value of κ tends to diminish the velocity profile and the thickness of momentum boundary layer while for the temperature field the opposite behavior is also reported.Figure 6
Velocity profiles for some values ofR.Figure 7
Velocity profiles for some values ofτ.Figure 8
Temperature profiles for some values ofτ.Figure 9
Velocity profiles for some values ofκ.Figures10 and 11 show the variation of second-order velocity slip parameter δ on velocity and temperature profiles of both SWCNTs and MWCNTs hybrid nanofluids. Adjacent to the wall of the surface, the velocity profiles show an increasing tendency. However, at a distance from the wall, it reduces with bigger value of δ. Physically, second-order slip parameter δ increases the resistance of fluid motion, which decreases the fluid flow field and momentum boundary layer thickness. Furthermore, improved value of δ leads to improving the temperature profile and thermal boundary layer thickness. Therefore, the occurrence of second-order slip condition in the hybrid nanofluids flow plays an important role in the nanotechnology. Figures 12 and 13, respectively, illustrate the variation of temperature field with respect to thermal slip parameter β and Prandtl number Pr. The temperature distributions of both SWCNTs and MWCNTs hybrid nanofluids fall with added value of β, and more influence is noticed at the surface, as demonstrated in Figure 12. Also, it is illustrated that the temperature profile and thermal boundary layer thickness decrease significantly as the value of Prandtl number increases. Physically, the thermal diffusivity of the fluid tends to decline and results in weak diffusion of heat inside the fluid with bigger values of Prandtl number.Figure 10
Velocity profiles for some values ofδ.Figure 11
Temperature profiles for some values ofδ.Figure 12
Temperature profiles for some values ofβ.Figure 13
Temperature profiles for some ofPr.The computations of the physical quantities of interest such as the skin friction coefficientf″0 and local heat transfer rate −θ′0 are, respectively, given in Tables 8 and 9 with some embedded parameters for SWCNTs−Fe3O4/H2O and MWCNTs−Fe3O4/H2O hybrid nanofluids. For both hybrid nanofluids, the skin friction coefficient tends to enhance with bigger values of nanoparticle volume fractions ϕCNT and ϕFO, Grashof number, and heat absorption parameters. However, the reverse effect occurs for bigger values of Prandtl number, permeability, first-order velocity slip, and thermal slip parameters, as demonstrated in Table 8. Also, for both hybrid nanofluids, the local heat transfer rate shows an increasing tendency for bigger values of Grashof number, first-order velocity slip parameter, and Prandtl number, as shown in Table 9. Furthermore, the local heat transfer rate tends to reduce for greater values of nanoparticle volume fractions ϕCNT and ϕFO, permeability, thermal slip, and heat absorption parameters. Also, a similar result was found in the work of Manjunatha et al. [25]. Physically, the obtained results show that the presence of nanoparticle volume fractions and heat absorption parameters plays an important role in the industrial application of hybrid nanofluids, since the skin friction coefficient tends to enhance, while the local heat transfer rate tends to reduce with bigger values of these parameters.Table 8
Values of the skin friction coefficientf″0 of SWCNTs−Fe3O4/H2O and MWCNTs−Fe3O4/H2O hybrid nanofluids with N=50,ξ=3,A=0.8,R=1&δ=0.02 for some embedded physical parameters.
ϕFOϕCNTτγβκHPrSWCNTsMWCNTs0.000.010.020.20.35−0.56.21.302659623.643390120.052.262933934.619011420.103.113818605.398801580.040.00−2.01625523−2.016255230.012.078693954.439508710.0516.4160524525.840495980.010.01−0.226997850.996822660.022.078693954.439508710.034.317480047.770848800.020.12.962554056.368854520.22.078693954.439508710.51.084315392.273750840.20.12.401437354.956560160.32.078693954.439508710.51.802080583.996936310.31.03.352212625.889283745.02.078693954.4395087110.01.063040333.268391865.0−5.02.078693954.439508710.02.194757024.544611870.52.327796454.66293515−0.51.03.297830625.907028285.02.233945574.6151373510.01.730923034.06568197Table 9
Values of the local heat transfer−θ′0 of SWCNTs−Fe3O4/H2O and MWCNTs−Fe3O4/H2O hybrid nanofluids with N=50,ξ=3,A=0.8,R=1&δ=0.02 for some embedded physical parameters.
ϕFOϕCNTτγβκHPrSWCNTsMWCNTs0.000.010.020.20.35−0.56.23.313275943.393841150.053.000742093.077257010.102.741967062.812574380.040.03.182473023.182473020.013.058249433.135758230.052.713197442.897389050.010.012.994548733.049474030.023.058249433.135758230.033.113005523.206593550.020.13.017990243.052699830.23.058249433.135758230.53.102479143.224973200.20.13.261772643.346202960.33.058249433.135758230.52.878396522.949816290.31.03.129059323.215424685.03.058249433.1357582310.03.009841893.077646675.0−5.03.058249433.135758230.02.753301362.843378840.52.416756892.52484894−0.51.01.283036991.341859235.02.760680352.8382867610.03.840903153.91355315Furthermore, under unsteady condition, the skin friction coefficientf″0 and local heat transfer rate −θ′0 are illustrated for some parameters in Figures 14–17. The skin friction coefficient reduces, while the local heat transfer rate increases with bigger values of unsteadiness parameter A. The skin friction coefficient improves with bigger values of both thermal dependent viscosity R and second-order velocity slip parameter δ, as demonstrated in Figures 14 and 15, respectively. Also, the local heat transfer rate reduces gradually with greater value of both thermal dependent viscosity R and second-order velocity slip parameter δ, as reported in Figures 16 and 17, respectively. Besides, there are superior skin friction coefficient and lower local heat transfer rate in the MWCNTs−Fe3O4/H2O hybrid nanofluid than in the MWCNTs−H2O mono nanofluid. In general, these findings show us that the presence of both thermal dependent viscosity and second-order velocity slip condition in the nanofluids flow adds a significant role in the nanotechnology and biomedical uses as the skin friction coefficient tends to improve, while the local heat transfer rate tends to decline with bigger values of these parameters.Figure 14
Skin friction coefficientCf for some values of A and R.Figure 15
Skin friction coefficientCf for some values of A and δ.Figure 16
Local heat transfer rateNu for some values of A and R.Figure 17
Local heat transfer rateNu for some values of A and δ.


## 5. Conclusion

In this work, unsteady natural convection flow ofCNTs−Fe3O4/H2O hybrid nanofluids due to stretching surface embedded in a porous medium with the effects of second-order slip flow and temperature-dependent viscosity is examined. The governing problems of flow are solved numerically using SQLM. The main results of the present study can be summarized as follows:The thermal conductivity ofCNTs−Fe3O4/H2O hybrid nanofluids improves more than CNTs−H2O nanofluids with increasing values of hybrid nanoparticle volume fraction; and the progress is nearly by 100 percent when 1 percent of CNTs nanoparticles and 3 percent of Fe3O4 nanoparticles are added to the base fluid.Both velocity and temperature profiles ofCNTs−Fe3O4/H2O hybrid nanofluids flow enhance with bigger value of nanoparticle volume fraction. Moreover, there is lower velocity distribution in the SWCNT−Fe3O4/H2O than in the MWCNT−Fe3O4/H2O hybrid nanofluids, while the opposite trend is reported in the temperature distribution. Thus, physically, SWCNT−Fe3O4/H2O hybrid nanofluid is more efficient than MWCNT−Fe3O4/H2O hybrid nanofluid in engineering applications.The momentum boundary layer reduces, while the thermal boundary layer grows with bigger values of temperature-dependent viscosity and second-order velocity slip parameters. Therefore, the occurrence of second-order slip condition and temperature-dependent viscosity in the hybrid nanofluids flow plays an important role in the nanotechnology and biomedicine sectors.The skin friction coefficient improves with bigger values of nanoparticle volume fraction, Grashof number, temperature-dependent viscosity, and second-order velocity slip parameters. Besides, there are superior skin friction coefficient and lower local heat transfer rate in theCNTs−Fe3O4/H2O hybrid nanofluids than in the CNTs−H2O nanofluids. These results show that the hybrid nanofluids have a more important role than working fluids or mono nanofluids in industrial uses.In general, the obtained results confirm thatCNTs−Fe3O4/H2O hybrid nanofluids have better flow distributions with good stability of thermal properties than CNTs−H2O nanofluids, and the former can be more recommended for the treatment of cancer therapy and drug delivery in the biomedical sectors.



---

*Source: 8407194-2021-03-19.xml*