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

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 MWCNTnanoﬂuids than SWCNTnanoﬂuids. 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.


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 modernday 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]). ese 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 magnetonanofluid 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. ey 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]. ey obtained essential progress of the convective heat transfer which mainly depends on Reynolds number and solid volume fraction of CNTs. e synthetic engine oil and ethylene glycol thermal conductivity improvement in the presence of MWCNTs was designated by Liu et al. [21]. ey 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. ey 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]. eir 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]. e 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]). e 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. en, 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). e 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 (C 2 H 6 O 2 ) 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.

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 2 Mathematical Problems in Engineering components (v r , v ϑ , v z ) represent in the directions of increasing (r, ϑ, z). e disk is rotated along z − axis with angular velocity Ω, and also, it is stretched in the radial direction with stretching rate s. e nanofluid flow is exposed to the magnetic field of uniform strength B 0 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 B 0 . e surface temperature of the disk and the ambient temperature are, respectively, denoted by T w 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.
Assuming (zp/zr) � (zp/zz) � 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]): with boundary conditions v r � rs where L 0 and N 0 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. erefore, the heat flux vector q satisfying the Cattaneo-Christov diffusion model can be adapted for nanofluid and given as (Akbar et al. [36], and Gangadhar et al. [42]) where λ t 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 λ t � 0. For steady flow, equation (7) is simplified into Substituting equations (8) into (5) and simplifying, we get the following energy equation: is the rate of strain tensor, π � k ij k ij is the i th and j th product of the components of the deformation rate tensor with itself, π c is the critical value of π, μ c is the Casson fluid dynamic viscosity coefficient, T y is the yield stress of Casson fluid, and v i and v j are the velocity components. e thermophysical properties of the nanoparticles (SWCNTs and MWCNTs) and base fluid (ethylene glycol) are given in Table 1 (Khalid et al. [15] and Yunus and Ghajar [47]).
Base fluid and the nanoparticles must be in thermal equilibrium and no slip should take place between them. e nanofluid effective density ρ nf , specific heat capacity (ρc p ) nf , dynamic viscosity μ nf , and electrical conductivity σ nf are given as (Alsagri et al. [18] and Khalid et al. [15]) where ϕ v is the nanoparticle volume fraction; ρ CNTs , (ρc p ) CNTs , and σ CNTs are, respectively, density, specific heat capacity, and electrical conductivity of CNTs, whereas ρ f , (ρc p ) f , μ f , and σ f are, respectively, density, specific heat capacity, dynamic viscosity, and electrical conductivity of the base fluid (ethylene glycol). e nanoparticle volume fraction ϕ 1 , ϕ 2 , and ϕ 3 are defined as For effective thermal conductivity of CNT nanofluid k nf , Xue's model which is effective for spherical and elliptical shape is utilized as follows (Alsagri et al. [18]):

Numerical Method of Solutions
We introduce the nondimensional variables based on [4] and [36]: Hence, the continuity equation (1) is satisfied, and the transformed equations for momentums and energy are found as follows: with transformed boundary conditions: where M � σ f B 2 0 /ρ f Ω is the magnetic parameter, Pr � ] f /α f is a Prandtl number, α � 2λ t Ω is the thermal relaxation parameter, A � s/Ω is the scaled stretching parameter, δ � L 0 2Ω/] f is the velocity slip parameter, and β � N 0 2Ω/] f is the thermal slip parameter. e radial and tangential directions shear stress at the surface of the disc for Casson nanofluid are defined as e total dimensionless form of surface shear stress at the surface of the disc is e surface heat flux of the disc for Casson nanofluid is given as e practical problem in engineering, the skin friction coefficient C f and the Nusselt number Nu (local heat transfer rate) of the disc, are defined as us, the dimensionless normalized skin friction coefficient and local heat transfer rate are given by where Re r is the local Reynolds number defined by Re r � (Ωr)r/] f . e 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. is 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).

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. e spectral quasilinearization method (SQLM) is used for the numerical computation with the number of collocation points N � 40 in space ξ and the scaled parameter L � 10. e physical properties of ethylene glycol (C 2 H 6 O 2 ), SWCNTs, and MWCNTs are employed from Table 1. e 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 5 th order of approximations for all radial wall stress − f″(0), tangential wall stress −h′(0), normalized skin friction coefficient f ″ (0) 2 + h ′ (0) 2 , and local Nusselt number −θ′(0). Table 4 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 (ϕ v ) 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  [29]. e study of important embedded parameters on radial velocity f′(ξ), tangential velocity h(ξ), and temperature θ(ξ) profiles is plotted in Figures 2-11 for ϕ v � 0.05, A � 0.5, M � 5, δ � 0.3, c � 0.5, Pr � 7.3, β � 0.2, and α � 0.3. Figures 2  and 3 represent the effects of nanoparticle volume fraction (0 < ϕ v < 0.1) on the radial velocity and temperature profiles, respectively, for the ethylene glycol-(C 2 H 6 O 2 -) based nanofluid with SWCNTs and MWCNTs. It is shown that, for both SWCNTs and MWCNTs, as the value of ϕ v 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.
is 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.
e 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. erefore, 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]. e effect of magnetic parameter M 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. is 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. is 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 c on radial velocity profile. Near the boundary surface, the velocity profile shows an increasing tendency as c increases.
is is anticipated since increasing the Casson fluid parameter c 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. Figure 9 illustrates the effect of velocity slip parameter δ on tangential velocity profile for both SWCNTand MWCNT nanofluids.
e 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. us, 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. e important physical quantities, the normalized skin friction coefficient C f , and the local heat transfer rate Nu are presented in Figures 12-15 for embedded parameters ϕ v � 0.05, A � 0.5, M � 5, δ � 0.3, c � 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 ϕ v . e normalized skin friction coefficient is improved with increasing values of both M and ϕ v . 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 c. 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. e variation of local heat transfer rate with M and ϕ v is demonstrated in Figure 14 for both SWCNT and MWCNT nanofluids. e 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 ϕ v . Further,

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. e effects of normally applied magnetic field, wall velocity, and thermal slip conditions are considered. e 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 distribution e normalized skin friction coefficient enhances for both SWCNT and MWCNT nanofluids as scaled stretching parameter, magnetic parameter, and Casson fluid parameter grow Local heat transfer rate enhances for both SWCNT and MWCNT nanofluids for higher values of scaled stretching parameter, Prandtl number, and thermal relaxation parameter e higher incidence of magnetic field raises the temperature filed, and it improves the thermal boundary layer thickness e momentum boundary layer thickness slows down as the velocity slip factor grows Increases 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 fluid Increase in the scaled stretching parameter thickens the momentum boundary layer while it thins down the thermal boundary layer Radial stretching of the disk is helpful to improve the cooling process of the rotating disk in practical applications    e 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. erefore, 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.  where D � 2 D/L ∞ , and the higher-order derivatives are found as powers of D as follows: where n is the order of derivative and D is the matrix of size (N + 1) × (N + 1). Applying the spectral method to the system of equations Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.