# MHD Slip Flow of CNT-Ethylene Glycol Nanofluid due to a Stretchable Rotating Disk with Cattaneo–Christov Heat Flux Model

Authors: Ayele Tulu; Wubshet Ibrahim
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This article deals with carbon nanoliquid flow due to stretchable rotating disk with the effect of Cattaneo–Christov heat flux model. Both SWCNTs and MWCNTs are considered with ethylene glycol as the base fluid. The effects of nanoparticle volume friction, normally applied magnetic field, stretching factor, velocity, and thermal slip factors are examined. The fundamental flow governing equations are transformed into dimensionless system of coupled nonlinear ordinary differential equations, and they are solved numerically using spectral quasi-linearization method (SQLM). Employing graphs and tables, the results of velocity and temperature fields as well as skin friction coefficient and local heat transfer rate are analyzed and presented via embedded parameters. The results reveal that higher velocity fields and lower temperature fields are noticed in the MWCNT nanofluids than SWCNT nanofluids. The higher incidence of magnetic field improves the thermal boundary layer thickness. A growth in velocity slip factor reduces the momentum boundary layer thickness of the nanoliquid flow. Generally, radial stretching of the disk is helpful in improving the cooling process of the rotating disk in practical applications.


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## Body

## 1. Introduction

Nanofluids have gained remarkable attention of researchers due to their inspiring heat transfer in various industrial and engineering applications. Common working fluids such as water, engine oils, and ethylene glycol have restricted thermal performances which limit their usage in modern-day cooling applications. Nanofluids consist of nanoscale particles such as copper, alumina, carbides, nitrides, metal oxides, graphite, and carbon nanotubes which enhance the thermal conductivity of base fluids (Mahanthesh et al. [1] and Ahmad et al. [2]). These nanofluids have widespread applications in modern systems of heating and cooling, solar cells, generation of new fuels, hybrid-powered engines, cancer therapy, drug delivery, and medicine (Hsiao [3] and Aziz et al. [4]). Due to these various applications of nanofluids, many studies associated with the flow of nanofluids have been conducted. For instance, Prasannakumara et al. [5] examined the boundary layer flow and heat transfer of fluid particles suspension with nanoparticles over a nonlinear stretching sheet. Khan et al. [6] analyzed three-dimensional steady MHD flow of Powell–Eyring nanofluid with convective and nanoparticles mass flux conditions. Also, Khan et al. [7] investigated unsteady three-dimensional Sisko magneto-nanofluid flow with heat absorption and temperature-dependent thermal conductivity. Recently, three-dimensional bioconvection nanofluid flow from a biaxial stretching sheet was presented by Amirsom et al. [8]. Besides, convinced works in this direction are explained by Shashikumar et al. [9], Khan et al. [10], Uddin et al. [11], and Khan et al. [12].In the last two decades, heat transfer of carbon nanofluids has received considerable attention as a result of their extensive applications in the fields of nanotechnology and medicine. Nanoparticles of carbon nanotubes (CNTs) were first discovered in 1991 by Lijima [13]. Carbon nanotubes are allotropes of carbon with a tube-shaped nanostructure such as frames of carbon atoms with diameter ranges from 1 to 100 nm. They have a remarkable conductivity which supports them to form a network of conductive tubes. Because of their configuration, some CNTs are good conductor of heat and electricity whereas others act as semiconductors. CNTs have 15 times thermal conductivity and 1000 times capability of copper and 200 times power and 5 times resistance of steel (Prajapati et al. [14] and Khalid et al. [15]). Carbon nanotubes have significance applications in nanotechnology, hardware, optics, energy storage, biomedical, ceramic, thermal defense, and various fields of material sciences and engineering (Hayat et al. [16]). Carbon nanotubes used in nanofluids are generally divided into single-wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs) contingent on their number of concentric layers of rolled grapheme sheets. SWCNTs contain catalyst for their synthesis, and they are prepared from covering layer of grapheme into a unified cylinder whereas MWCNTs can be formed without catalyst and comprise of numerous rolled layers of graphite with complex structure (Khan et al. [17]).Because of their unique morphology, electronic structural, mechanical properties, and innovative physicochemical features, CNTs are the most resourceful material of this century. Moreover, the presence of carbon chains in CNTs does not convey any hazard to the atmosphere (Alsagri et al. [18]). Having all these implication of CNTs in mind, Kumaresan et al. [19] experimentally studied the heat transfer characteristics of nanofluids containing CNTs, and they disclosed nanofluids at very low nanoparticle volume fraction enhance high heat transfer rate. Heat transfer performance of CNT nanofluids flow through a horizontal tube was examined by Ding et al. [20]. They obtained essential progress of the convective heat transfer which mainly depends on Reynolds number and solid volume fraction of CNTs. The synthetic engine oil and ethylene glycol thermal conductivity improvement in the presence of MWCNTs was designated by Liu et al. [21]. They revealed that CNT-ethylene glycol nanofluid have better thermal conductivity as compared with ethylene glycol base fluid. Also, Mahanthesh et al. [22] investigated the Marangoni transport of dissipating SWCNT and MWCNT with water nanofluids under the influence of magnetic force and radiation. They reported that the thermal distribution of SWCNT nanoliquid is better than MWCNT nanoliquid. Some contributions on CNT-based nanofluids can be reviewed (Aman et al. [23], Khan et al. [23], Asadi et al. [24, 25], and Nasir et al. [26]).Rotating disk-induced flow of fluids have significance applications in numerous engineering and industrial sectors such as rotating machinery, electric power generating system, computer storage devices, air cleaning machines, crystal growth processes, gas turbine rotors, food processing, medical equipment, and others. Due to these, immense papers have been published regarding the flow of fluid due to rotating disk. For example, Hayat et al. [27] and Mustafa [28] analyzed MHD flow of nanofluid by a rotating disk with partial slip effects. Nanofluid flow near a stretchable rotating disk with axial magnetic field and convective conditions was analyzed by Mushtaq and Mustafa [29]. Also, Imtiaz et al. [30] studied the radiative flow of CNTs between stretchable rotating disks with convective conditions. Recently, the impacts of exponential space-dependent heat source on MHD slip flow of SWCNT and MWCNT nanoliquids past a stretchable rotating disk was investigated by Mahanthesh et al. [31]. Their result established that the thermal field for SWCNT-nanoliquid is higher than MWCNT-nanoliquid. Additionally, some efforts in this direction are published by Hayat et al. [32, 33], Khan et al. [34], and Mahanthesh et al. [35].The dynamics of heat transfer is very useful due to its abundant applications in industrial, engineering, and biomedical applications. Fourier was the first scholar who established the most successful classical heat flux model in continuum mechanics (Akbar et al. [36]). The main drawback of this model is the temperature field, and the whole system is instantly affected by initial disturbance. To avoid this unrealistic feature, first, Cattaneo [37] improved Fourier’s law by adding thermal relaxation time which tolerates the heat flux. Then, Christov [38] more modified the Cattaneo heat flux model from Maxwell–Cattaneo’s model. Also, the uniqueness of Cattaneo–Christov model for incompressible flow of fluids was tested by Tibllo and Zampoli [39]. Presently, much interest has been shown in the study of Cattaneo–Christov heat flux mode (Kundu et al. [40], Makinde et al. [41], Gangadhar et al. [42], and Hayat et al. [43]).Motivated by the above cited literatures, the purpose of this article is to study the flow of CNTs with ethylene glycol nanofluids due to stretchable rotating disk with Cattaneo–Christov heat flux model. Governing equations of the flow are highly nonlinear-coupled differential equations and solved numerically by employing spectral quasi-linearization method (SQLM). The spectral collocation method converges fast and provides more accurate approximations with less grid points (Motsa [44], Ibrahim and Tulu [45], and Uddin et al. [46]). To the best of the authors’ knowledge, no analysis has been published so far in this direction. Particularly, examining the effect of the Cattaneo–Christov heat flux model on SWCNTs and MWCNTs with ethylene glycol (C2H6O2) nanofluids flow and heat transfer using SQLM would be the contribution of this paper to the existing body of knowledge. Moreover, the effects of embedded parameters such as nanoparticle volume fraction, axially applied magnetic field, stretching factor, velocity, and thermal slip factors on velocity and temperature fields as well as skin friction coefficient and local heat transfer rate are examined, and the results are presented and discussed using graphs and tables.


## 2. Mathematical Description of Problem

We consider steady and incompressible flow of nanofluid due to stretchable rotating disk. We assume a nonrotating cylindrical coordinate frame (r, ϑ, z), and the velocity components (vr,vϑ,vz) represent in the directions of increasing (r, ϑ, z). The disk is rotated along z − axis with angular velocity Ω, and also, it is stretched in the radial direction with stretching rate s. The nanofluid flow is exposed to the magnetic field of uniform strength B0 acting parallel to the z-direction (Figure 1). Induced magnetic field caused by the motion of electrically conducting nanofluid is not considered because it is very small compared with B0. The surface temperature of the disk and the ambient temperature are, respectively, denoted by Tw and T∞. Moreover, effects of velocity and thermal slip boundary conditions are considered, and the Cattaneo–Christov heat flux model is used to analyze the heat transfer.Figure 1
Physical model of the flow problem.Assuming∂p/∂r=∂p/∂z=0, the governing equations for mass, momentum, and thermal energy transfer of Casson nanofluid flow past a stretchable rotating disk with axially applied magnetic field and slip boundary conditions are given as follows (Aziz et al. [4] and Hayat et al. [33]):(1)∂vr∂r+vrr+∂vz∂z=0,(2)ρnfvr∂vr∂r+vz∂vr∂z−vϑ2r=μnf1+1γ∂2vr∂r2+1r∂vr∂r+∂2vr∂z2−vrr2−σnfB02vr,(3)ρnfvr∂vϑ∂r+vz∂vϑ∂z+vrvϑr=μnf1+1γ∂2vϑ∂r2+1r∂vϑ∂r+∂2vϑ∂z2−vϑr2−σnfB02vϑ,(4)ρnfvr∂vz∂r+vz∂vz∂z=μnf1+1γ∂2vz∂r2+1r∂vz∂r+∂2vz∂z2,(5)ρcpnfvr∂T∂r+vz∂T∂z=−∇.q,with boundary conditions(6)vr=rs+L01+1γ∂vr∂z,vϑ=rΩ+L01+1γ∂vϑ∂z,vz=0,T=Tw+N0∂T∂z,atz=0,vr⟶0,vϑ⟶0,T⟶T∞asz⟶∞.where L0 and N0 are the velocity slip and thermal slip factors, respectively.In this study, even if nanofluids using a two-component model are considered, we assumed that there is no agglomeration of CNT nanoparticles within the nanofluid, and the base fluid and CNTs are assumed to be in thermal equilibrium and no slip occurs between them. Therefore, the heat flux vectorq satisfying the Cattaneo–Christov diffusion model can be adapted for nanofluid and given as (Akbar et al. [36], and Gangadhar et al. [42])(7)λt∂q∂t+v⋅∇q−q⋅∇v+∇⋅vq+q=−k∇T,where λt is the relaxation time of heat flux, k is the thermal conductivity, and v is the velocity vector. Equation (7) gives Fourier’s law when λt = 0.For steady flow, equation (7) is simplified into(8)λt∇⋅∇.qv+∇⋅q=−k∇2T,Substituting equations (8) into (5) and simplifying, we get the following energy equation:(9)vr∂T∂r+vz∂T∂z=αnf∂2T∂r2+1r∂T∂r+∂2T∂z2−λtvr2∂2T∂r2+vz2∂2T∂z2+2vrvz∂2T∂r∂z+vr∂vr∂r+vz∂vr∂z∂T∂r+vr∂vz∂r+vz∂vz∂z∂T∂z,The non-Newtonian Casson parameterγ is defined as γ=μγ2πc/Ty, whereas the rheological equation of state for the Cauchy stress tensor of Casson fluid flow can be expressed as (Khalid et al. [15] and Makinde et al. [41])(10)τij=2μγ+Ty2πkij,ifπ>πc,2μγ+Ty2πckij,ifπ<πc,where kij=1/2∂vi/∂xj+∂vj/∂xi is the rate of strain tensor, π = kijkij is the ith and jth product of the components of the deformation rate tensor with itself, πc is the critical value of π, μγ is the Casson fluid dynamic viscosity coefficient, Ty is the yield stress of Casson fluid, and vi and vj are the velocity components.The thermophysical properties of the nanoparticles (SWCNTs and MWCNTs) and base fluid (ethylene glycol) are given in Table1 (Khalid et al. [15] and Yunus and Ghajar [47]).Table 1
Thermo-physical properties of SWCNTs, MWCNTs and ethylene glycol at temperature 20°C.
Physical properties
Ethylene glycol(C2H6O2)
SWCNT
MWCNT
Density, ρ(kg/m3)
1109
2600
1600
Specific heat, cp(J/kg·K)
2382
425
796
Thermal conductivity, k(W/mK)
2.84
6600
3000
Electrical conductivity, σ(Sm−1)
0.01485
106–107
1.9 × 10−4Base fluid and the nanoparticles must be in thermal equilibrium and no slip should take place between them. The nanofluid effective densityρnf, specific heat capacity ρcpnf, dynamic viscosity μnf, and electrical conductivity σnf are given as (Alsagri et al. [18] and Khalid et al. [15])(11)ρnf=ϕvρCNTs+1−ϕvρf,ρcpnf=ϕvρcpCNTs+1−ϕvρcpf,μnf=μf1−ϕv2.5,σnfσf=1+3σCNTs/σf−1ϕvσCNTs/σf+2−σCNTs/σf−1ϕv,where ϕv is the nanoparticle volume fraction; ρCNTs, ρcpCNTs, and σCNTs are, respectively, density, specific heat capacity, and electrical conductivity of CNTs, whereas ρf, ρcpf, μf, and σf are, respectively, density, specific heat capacity, dynamic viscosity, and electrical conductivity of the base fluid (ethylene glycol). The nanoparticle volume fraction ϕ1, ϕ2, and ϕ3 are defined as(13)ϕ1=1−ϕv2.51−ϕv+ϕvρCNTsρf,ϕ2=1−ϕv+ϕvρcpCNTsρcpf,ϕ3=1+3σCNTs/σf−1ϕvσCNTs/σf+2−σCNTs/σf−1ϕvρf1−ϕvρf+ϕvρCNTs.For effective thermal conductivity of CNT nanofluidknf, Xue’s model which is effective for spherical and elliptical shape is utilized as follows (Alsagri et al. [18]):(13)knfkf=1−ϕv+2ϕvkCNTs/kCNTs−kflnkCNTs−kf/2kf1−ϕv+2ϕvkf/kCNTs−kflnkCNTs−kf/2kf.


## 3. Numerical Method of Solutions

We introduce the nondimensional variables based on [4] and [36]:(14)vr=rΩf′ξ,vϑ=rΩhξ,vz=−2Ωνf1/2fξ,θξ=T−T∞Tw−T∞,ξ=z2Ωνf1/2,Hence, the continuity equation (1) is satisfied, and the transformed equations for momentums and energy are found as follows:(15)21+1γf″′+ϕ12ff″−f′2+h2−ϕ3Mf′=0,(16)21+1γh″+ϕ12fh′−2f′h−ϕ3Mh=0,(17)θ″+ϕ2Prkfknffθ′−αf2θ″+ff′θ′=0,with transformed boundary conditions:(18)f=0,f′0=A+1+1γδf″0,h0=1+1+1γδf′0,θ0=1+βθ′0,f′∞=h∞=θ∞⟶0,where M=σfB02/ρfΩ is the magnetic parameter, Pr=νf/αf is a Prandtl number, α = 2λtΩ is the thermal relaxation parameter, A=s/Ω is the scaled stretching parameter, δ=L02Ω/νf is the velocity slip parameter, and β=N02Ω/νf is the thermal slip parameter.The radial and tangential directions shear stress at the surface of the disc for Casson nanofluid are defined as(19)τzr=−μnf1+1γ∂vr∂rz=0,τzϑ=−μnf1+1γ∂vϑ∂rz=0.The total dimensionless form of surface shear stress at the surface of the disc is(20)τw=τzr2+τzϑ2=μf1−ϕv2.51+1γr2Ω3νff″02+h′021/2.The surface heat flux of the disc for Casson nanofluid is given as(21)qw=−knf∂T∂rz=0=−knf2ΩνfTw−T∞θ′0.The practical problem in engineering, the skin friction coefficientCf and the Nusselt number Nu (local heat transfer rate) of the disc, are defined as(22)Cf=τwρfrΩ2,Nur=rqwkfTw−T∞.Thus, the dimensionless normalized skin friction coefficient and local heat transfer rate are given by(23)RerCf=21−ϕv2.51+1γf″02+h′021/2,NurRer=−21−ϕv+2ϕvkCNTs/kCNTs−kflnkCNTs−kf/2kf1−ϕv+2ϕvkf/kCNTs−kflnkCNTs−kf/2kfθ′0,where Rer is the local Reynolds number defined by Rer=Ωrr/νf.The dimensionless system of nonlinear ODEs (15)–(17) are numerically solved using the spectral quasilinearization method (SQLM). SQLM converges fast and provides more accurate approximations with less grid points. This method is implemented by identifying nonlinear component of a differential equation, linearized the terms using the multivariable Taylor series expansion, and applied the Chebychev pseudospectral collocation method (for details, see Appendix A).


## 4. Results and Discussion

Numerical results for CNTs with ethylene glycol nanofluids flow due to a stretchable rotating disk with Cattaneo–Christov heat flux mode are stated here. The spectral quasilinearization method (SQLM) is used for the numerical computation with the number of collocation pointsN = 40 in space ξ and the scaled parameter L = 10. The physical properties of ethylene glycol (C2H6O2), SWCNTs, and MWCNTs are employed from Table 1. The convergence of the SQLM solution and the stability of the results are tested. Tables 2 and 3 reveal that the convergence of solutions for both SWCNTs and MWCNTs with ethylene glycol nanofluids is achieved at 5th order of approximations for all radial wall stress − f″(0), tangential wall stress −h′(0), normalized skin friction coefficient f″02+h′02, and local Nusselt number −θ′(0).Table 2
SQLM Solutions Convergence for SWCNTs − C2H6O2 when ϕv = 0.05, A = 0.5, M = 5, δ = 0.3, γ = 0.5, Pr = 7.3, β = 0.2, α = 0.3 are used.
Order
−f″(0)
−h′(0)
f
″
0
2
+
h
′
0
2
−θ′(0)
01
0.23689710
0.89154598
0.922482783
0.67685416
02
0.23359837
0.89107446
0.92519082
0.67565587
03
0.23359687
0.89107467
0.92118505
0.67567940
04
0.23359687
0.89107467
0.92118487
0.67567940
05
0.23359687
0.89107467
0.92118487
0.67567940Table 3
SQLM Solutions Convergence for MWCNTs − C2H6O2 when ϕv = 0.05, A = 0.5, M = 5, δ = 0.3, γ = 0.5, Pr = 7.3, β = 0.2, α = 0.3 are used.
Order
−f′′(0)
−h′(0)
f
″
0
2
+
h
′
0
2
−θ′(0)
01
0.22142446
0.80709704
0.83691960
0.72862155
02
0.21820148
0.80799670
0.83694119
0.74087932
03
0.21819376
0.80799811
0.83694054
0.74082968
04
0.21819374
0.80799812
0.83694054
0.74082974
05
0.21819374
0.80799812
0.83694054
0.74082974Table4 encompasses the computations of normalized skin friction coefficient and local Nusselt number for changing values of important involved parameters such as nanoparticle volume fraction, scaled stretching parameter, Casson fluid parameter, velocity slip parameter, and thermal slip parameter for both SWCNTs and MWCNTs with ethylene glycol nanofluids. Here, it is shown that the normalized skin friction coefficient grows as nanoparticle volume fraction (ϕv) increases from 0.01 to 0.1 for SWCNT nanofluid. For both SWCNTs and MWCNT nanofluids, normalized skin friction coefficient enhances as scaled stretching parameter and Casson fluid parameter grow. Further, a decreasing trend in normalized skin friction coefficient is perceived for increasing value of velocity slip parameter. Furthermore, from Table 4, we perceived that for both SWCNT and MWCNT nanofluids, the local Nusselt number shows an increasing tendency for higher values of scaled stretching parameter and Casson fluid parameter. However, it shows a decreasing tendency if the value of Casson fluid parameter is greater than 0.5. It is also noted that the local Nusselt number is reduced for greater values of nanoparticle volume fraction, velocity slip, and thermal slip parameters for both SWCNT and MWCNT nanofluids. For some parameters, a comparable result was established in the literatures of Aziz et al. [4] and Mushtaq and Mustafa [29].Table 4
Values of the normalized skin friction coefficientf″02+h′02 and local heat transfer −θ′(0) for SWCNTs − C2H6O2 and MWCNTs − C2H6O2 nanofluid, with M = 5, Pr = 7.3, α = 0.3 for various physical parameters.
ϕ
v
A
β
γ
δ
f
″
0
2
+
h
′
0
2
−θ′(0)
SWCNTs
MWCNTs
SWCNTs
MWCNTs
0.01
0.5
0.2
0.5
0.3
0.918047
0.900415
0.902380
0.920289
0.05
0.921185
0.836940
0.67568
0.740830
0.1
0.926017
0.761583
0.505748
0.604996
0.05
0.1
0.859218
0.772487
0.301925
0.341617
0.5
0.921185
0.836940
0.675679
0.740830
0.8
0.994724
0.911512
0.882447
0.958942
0.05
0.5
0.2
0.921185
0.836940
0.675679
0.740830
0.5
0.921185
0.836940
0.656374
0.719663
1.0
0.921185
0.836940
0.626539
0.686951
0.05
0.5
0.2
0.1
0.386210
0.350375
0.636355
0.691414
0.5
0.921185
0.836940
0.675679
0.740830
5
1.559282
1.417469
0.635683
0.707745
0.05
0.5
0.2
0.5
0.1
1.017835
0.925215
0.824280
0.885015
0.3
0.921185
0.836940
0.675679
0.740830
0.5
0.855123
0.775726
0.584179
0.649354The study of important embedded parameters on radial velocityf′(ξ), tangential velocity h(ξ), and temperature θ(ξ) profiles is plotted in Figures 2–11 for ϕv = 0.05, A = 0.5, M = 5, δ = 0.3, γ = 0.5, Pr = 7.3, β = 0.2, and α = 0.3. Figures 2 and 3 represent the effects of nanoparticle volume fraction (0 < ϕv < 0.1) on the radial velocity and temperature profiles, respectively, for the ethylene glycol- (C2H6O2-) based nanofluid with SWCNTs and MWCNTs. It is shown that, for both SWCNTs and MWCNTs, as the value of ϕv increases, the radial velocity and temperature profiles of the nanofluid increase. It is also recognized that there is higher velocity distribution in the MWCNT nanofluid than SWCNT nanofluid, whereas the opposite trend is observed for temperature distribution. This is due to the fact that SWCNTs have higher density than MWCNTs. Figures 4 and 5, respectively, show the radial velocity and temperature profiles for changed value of scaled stretching parameter A. The parameter A is given as A = s/Ω, and it measures a radial stretching rate. Figure 4 reveals that, as parameter A grows, the radial velocity momentum boundary layer becomes thicker. Actually, the radial stretching rate raises as A enhances, and it accelerates radially outward flow. From Figure 5, it is also noticed that temperature profile declines and thermal boundary layer becomes thinner as the value of A enhances. Therefore, in practical applications, radial stretching of the disk is helpful to improve the cooling process of the rotating disk. A related result was established in the literature of Mushtaq and Mustafa [29].Figure 2
Radial Velocity profiles for various values ofϕv.Figure 3
Temperature profiles for various values ofϕv.Figure 4
Radial Velocity profiles for various values of A.Figure 5
Temperature profiles for various values of A.Figure 6
Tangential velocity profiles for various values of M.Figure 7
Temperature profiles for various values of M.Figure 8
Radial velocity profiles for various values ofγ.Figure 9
Tangential Velocity profiles for various values ofδ.Figure 10
Temperature profiles for various values ofβ.Figure 11
Temperature profiles for various values ofα.The effect of magnetic parameterM on tangential velocity and temperature profiles are demonstrated in Figures 6 and 7, respectively. For both SWCNT and MWCNT nanofluids, tangential velocity declines for increasing value of M. This is expected since magnetic field acting in the axial direction provides resistance force which accounts for dropping the radial and tangential fluid velocities. In Figure 7, an enhancing tendency of temperature profile is perceived as M increases. Hence, incidence of magnetic field raises the thermal boundary layer thickness as the temperature increases. It is also noticed that there are lower velocity and higher temperature distributions in the SWCNT nanofluid than in the MWCNT nanofluid. This is due to the fact that SWCNTs have higher electrical conductivity than MWCNTs. An analogous result was also found by Aziz et al. [4] and Mushtaq and Mustafa [29]. Figure 8 shows the influence of Casson fluid parameter γ on radial velocity profile. Near the boundary surface, the velocity profile shows an increasing tendency as γ increases. This is anticipated since increasing the Casson fluid parameter γ tends to reduce in yield stress, and the fluid flows easily. However, as far the radial velocity is away from the boundary surface, the reverse effect resulting in the thinning of the momentum boundary layer is due to the presence of the magnetic field effect.Figure9 illustrates the effect of velocity slip parameter δ on tangential velocity profile for both SWCNT and MWCNT nanofluids. The tangential velocity profile shows diminishing trend for increasing value of δ. In slip condition, since the fluid velocity near the surface of the sheet is no more equal to the velocity of stretchable rotating disk, the momentum boundary layer thickness slows down as δ grows. Figure 10 represents greater thermal slip parameter β leads to the drop of temperature profile for both SWCNT and MWCNT nanofluids with the highest effect recognized at the wall of the disk. Thus, an increases in the thermal slip factor β reduces the heat transfer from the surface of the disk to the fluid. Figure 11 shows the effect of the thermal relaxation parameter α on temperature distribution. It reveals that the temperature profile reduces with increasing value of α. Physically, thermal relaxation time is the time required by the fluid particles to transfer heat energy to its adjacent particles. Consequently, as α raises, the material particles need extra time to transfer heat to its adjacent particles and this leads to less transfer of heat from the disk to the fluid.The important physical quantities, the normalized skin friction coefficientCf, and the local heat transfer rate Nu are presented in Figures 12–15 for embedded parameters ϕv = 0.05, A = 0.5, M = 5, δ = 0.3, γ = 0.5, Pr = 7.3, β = 0.2 , and α = 0.3. Figure 12 depicts the variation of normalized skin friction coefficient with magnetic field parameter M and nanoparticle volume fraction ϕv. The normalized skin friction coefficient is improved with increasing values of both M and ϕv. From Figure 13, it is also observed that the normalized skin friction coefficient is increased with higher values of scaled stretching parameter A and Casson fluid parameter γ. Form both Figures, it is also recognized that there is a higher normalized skin friction coefficient in the SWCNT nanofluid than in the MWCNT nanofluid. The variation of local heat transfer rate with M and ϕv is demonstrated in Figure 14 for both SWCNT and MWCNT nanofluids. The local heat transfer rate is decreased with the greater value of M. On the other hand, the local heat transfer rate improves with rise values of M and ϕv. Further, Figure 15 clearly illustrates that the local heat transfer rate is enhanced with greater values of both Prandtl number Pr and thermal relaxation parameter α. In addition, lower local heat transfer rate in the SWCNT nanofluid than MWCNT nanofluid is reported.Figure 12
Skin friction coefficient for various values of M andϕv.Figure 13
Skin friction coefficient for various values of A andγ.Figure 14
Local heat transfer rate for various values of M andϕv.Figure 15
Local heat transfer rate for various values of Pr andα.


## 5. Conclusion

In this study, CNTs with ethylene glycol nanofluid flow and heat transfer due to stretchable rotating disk with Cattaneo–Christov heat flux model are examined. The effects of normally applied magnetic field, wall velocity, and thermal slip conditions are considered. The governing equations of the flow are solved numerically employing the spectral quasilinearization method (SQRM). From the main outcomes of the present observation, the following conclusions are drawn:Higher radial and tangential velocities distributions are observed in the MWCNT nanofluid than in the SWCNT nanofluid, whereas the trend is opposite for temperature distributionThe normalized skin friction coefficient enhances for both SWCNT and MWCNT nanofluids as scaled stretching parameter, magnetic parameter, and Casson fluid parameter growLocal heat transfer rate enhances for both SWCNT and MWCNT nanofluids for higher values of scaled stretching parameter, Prandtl number, and thermal relaxation parameterThe higher incidence of magnetic field raises the temperature filed, and it improves the thermal boundary layer thicknessThe momentum boundary layer thickness slows down as the velocity slip factor growsIncreases in the thermal slip factor tend to drop the temperature distribution at the wall of the disk, and it reduces the heat transfer from the surface of the disk to the fluidIncrease in the scaled stretching parameter thickens the momentum boundary layer while it thins down the thermal boundary layerRadial stretching of the disk is helpful to improve the cooling process of the rotating disk in practical applications


### 5.1. Future Directions

The present study results of CNTs with ethylene glycol nanofluid flow and heat transfer due to a stretchable rotating disk would have contributed valuable information to the uses of CNT nanofluids in various fields of nanotechnology. However, the week magnetic properties and week solubility of CNTs limit their applications in various technological and biomedical fields. Consequently, nowadays, the hybrid nanoparticles of magnetite metals with CNTs are typically receiving attention and experimentally investigating by researchers. Therefore, numerical analysis of hybrid nanofluid flow and heat transfer of magnetite metals with CNTs hybrid nanoparticles with various base fluids will be the future work of the researchers.


## 5.1. Future Directions

The present study results of CNTs with ethylene glycol nanofluid flow and heat transfer due to a stretchable rotating disk would have contributed valuable information to the uses of CNT nanofluids in various fields of nanotechnology. However, the week magnetic properties and week solubility of CNTs limit their applications in various technological and biomedical fields. Consequently, nowadays, the hybrid nanoparticles of magnetite metals with CNTs are typically receiving attention and experimentally investigating by researchers. Therefore, numerical analysis of hybrid nanofluid flow and heat transfer of magnetite metals with CNTs hybrid nanoparticles with various base fluids will be the future work of the researchers.



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*Source: 1374658-2020-03-19.xml*