Effects of Second-Order Slip Flow and Variable Viscosity on Natural Convection Flow of 
                     
                        
                           
                              
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<jats:p>This study deals with natural convection unsteady flow of <jats:inline-formula>
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                  </jats:inline-formula> and <jats:inline-formula>
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                  </jats:inline-formula> 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 <jats:inline-formula>
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                  </jats:inline-formula> 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 <jats:inline-formula>
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                  </jats:inline-formula> nanofluid. Thus, the obtained results are promising for the application of hybrid nanofluids in the nanotechnology and biomedicine sectors.</jats:p>


Introduction
e 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. e 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. e 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 (Fe 3 O 4 ) (Rahmawati et al. [21]). is is due to the fact that Fe 3 O 4 exhibits more chemical inertness and stability, but it has lower thermal conductivity than CNTs. Also, CNTs-Fe 3 O 4 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-Fe 3 O 4 /water hybrid nanofluids were investigated by Sundar et al. [24]. ey 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/Fe 3 O 4 hybrid nanoparticles.
e 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 TiO 2 -SiO 2 /W-EG hybrid nanofluids properties. e free convection heat transfer and entropy generation analysis of water-Fe 3 O 4 /CNT hybrid nanofluid were reported by Shahsavar et al. [29]. Also, Sundar et al. [30] experimentally synthesized MWCNT/Fe 3 O 4 water-based hybrid nanofluids. ey 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-Fe 3 O 4 )/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. us, 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-Fe 3 O 4 /H 2 O hybrid nanofluids flow considering the effects of temperature-dependent viscosity.
ere 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]. ey revealed that temperature distribution and thermal boundary layer grow with bigger values of secondorder velocity slip conditions. Yet, there is limited information about the effect of second-order slip condition on CNTs − Fe 3 O 4 /H 2 O hybrid nanofluids flow over stretching surface. us, 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 analyze CNTs − Fe 3 O 4 /H 2 O hybrid nanofluids flow and heat transfer due to stretching surface. e impacts of temperature-dependent viscosity, free convection, and second-order slip condition are also considered. e governing equations are solved numerically employing the fast convergent and accurate technique, namely, spectral quasilinearization method (SQLM). e 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. e 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 − Fe 3 O 4 /H 2 O hybrid nanofluids flow and heat transfer and computing it by means of SQLM make this study novel and different from the former studies.

Mathematical Description of Problem
We consider an unsteady two-dimensional incompressible viscous flow of CNTs − Fe 3 O 4 with H 2 O hybrid nanofluids past semi-infinite linearly stretching surface embedded in a porous medium. e 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 − Fe 3 O 4 )/H 2 O, initially, CNTs are dispersed into base fluid and then Fe 3 O 4 is scattered in CNTs/H 2 O 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 − ζt))x, 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.
With foregoing assumptions and Boussinesq approximation, the transport equations of hybrid nanofluid boundary layer flow are established as follows (Manjunatha et al. [25]): e boundary conditions with second-order velocity slip and thermal slip are given as follows (Manjunatha et al. [25] and Ibrahim [39]): and initial condition is where u and v are the hybrid nanofluid velocity components in the x and y orientations, respectively; ρ hnf , μ hnf , k hnf , and (ρc p ) hnf are hybrid nanofluid effective density, dynamic viscosity, thermal conductivity, and specific heat capacity, respectively; g, β hnf , Q 0 , and N 0 are gravitational acceleration, hybrid nanofluid volume expansion coefficient, heat generation/absorption rate, and thermal slip factor, respectively. e temperature-dependent viscosity of the base fluid is given as follows (Manjunatha et al. [25]):

Mathematical Problems in Engineering
where μ 0 is reference viscosity of base fluid and R is variable viscosity parameter (note that R < 0 for gases and R > 0 for liquids). e slip velocity u 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 where a and b are constants given as where K n is Knudsen number, r � min[1/K n , 1], λ is the molecular mean free path, and α is the momentum accommodation coefficient with 0 ≤ α ≤ 1.
with transformed boundary conditions: where

Porous medium Fe 3 O 4 -coated CNTs
Stretching sheet is the second-order velocity slip parameter, and β � N 0 ����� u 0 /] f is the thermal slip parameter. e shearing stress at the wall is defined for hybrid nanofluid as e surface heat flux at the wall is given for hybrid nanofluid as e surface drag and wall heat transfer rates are characterized by the skin friction coefficient C f and the local Nusselt number Nu, respectively, and they are obtained as us, the dimensionless skin friction coefficient and local Nusselt number are given by where Re x is the local Reynolds number defined by Re x � (u w x/] f ).

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: Mathematical Problems in Engineering with boundary conditions 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 f j+1 and θ j+1 .
First, it is essential to change the semi-infinite domain to a truncated domain [0, L ∞ ]. en, the interval [0, L ∞ ] is transformed to the interval [− 1, 1] using the linear transformation ξ � (1/2)L ∞ (χ + 1). Further, the Gauss-Lobatto points are selected to define the nodes in [− 1, 1] as reported in Trefethen [44]. at is, where N is the number of collocation points used. Using the Chebyshev spectral collocation method, equations (19) and (20) are discretized. e 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]. at is, where T is a vector function at the collocation points. Higher-order derivatives are found as powers of D as 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: with transformed boundary condition where Mathematical Problems in Engineering 7 are vectors of sizes (N + 1) × 1; I and [. . .] d , respectively, represent an identity and a diagonal matrix of size (N + 1) × (N + 1). e 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 (E d ) for velocity (f ′ ) and temperature (θ) at the (r + 1) th iteration as follows (Motsa [42]): If the iteration scheme converges, the error (E d ) 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.
us, the proper initial guesses that fulfill the governing equations of the considered flow problem are

Results and Discussion
Natural convection flow of (CNTs − e convergence analysis was done for the skin friction coefficient and local Nusselt number, and almost the 5 th -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.
us, 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.
Variation in fluids thermal conductivity with nanoparticles volume friction at room temperature is presented in Table 7.  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. e effect of variable viscosity parameter R 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 8 Mathematical Problems in Engineering 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]. e impacts of Grashof number (natural convection parameter) τ on velocity and temperature profiles are presented in Figures 7 and 8, respectively. e 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]. e 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. Figures 10 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, secondorder 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. erefore, 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. e 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     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 ϕ CNT and ϕ FO , 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.
Furthermore, under unsteady condition, the skin friction coefficient f ″ (0) and local heat transfer rate − θ ′ (0) are illustrated for some parameters in Figures 14-17. e skin           e 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 − Fe 3 O 4 /H 2 O hybrid nanofluid than in the MWCNTs − H 2 O 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.

Conclusion
In this work, unsteady natural convection flow of (CNTs − Fe 3 O 4 )/H 2 O hybrid nanofluids due to stretching surface embedded in a porous medium with the effects of secondorder slip flow and temperature-dependent viscosity is examined. e governing problems of flow are solved numerically using SQLM. e main results of the present study can be summarized as follows: e momentum boundary layer reduces, while the thermal boundary layer grows with bigger values of temperature-dependent viscosity and second-order velocity slip parameters. erefore, 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. e 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 the CNTs − Fe 3 O 4 /H 2 O hybrid nanofluids than in the CNTs − H 2 O nanofluids.
ese 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 − Fe 3 O 4 /H 2 O hybrid nanofluids have better flow distributions with good stability of thermal properties than CNTs − H 2 O nanofluids, and the former can be more recommended for the treatment of cancer therapy and drug delivery in the biomedical sectors.

A:
Unsteady parameter C fx : Local skin friction coefficient (C p ): Specific heat capacity of the material D: Chebyshev differentiation matrix