Slip Flow of Kerosene Oil Based SWCNT Nanofluid over Stretching Sheet with Radiation and Suction/Injection Effects

Present study numerically investigates a two dimensional steady laminar boundary layer nanofluid flow of single-wall carbon nanotubes (SWCNTs) immersed into kerosene oil, due to a linearly stretched sheet. Flow is subjected to the slip boundary condition and suction/injection effects. Employing suitable similarity transformations, governing PDEs of the arising problem are converted into coupled nonlinear non-dimensional ordinary differential equations. A set of obtained ODEs with assisting boundary conditions is solved numerically by applying finite element method (FEM). Effect of pertinent factors, velocity slip parameter, suction/injection parameter and solid volume fraction parameter on non-dimensional velocity and temperature profiles are characterized graphically. In addition, physical emerging parameters, local Nusselt’s number and local skin friction coefficient are computed and presented via table. Furthermore, derived numerical values of shear stress and heat flux at the surface are compared with previously published results.


Introduction
Slip effect takes place when cohesive forces between fluid particles dominate over adhesive forces between fluid particles and solid surface and normal velocity becomes non-zero. Usually, most of the analyses are performed with no-slip boundary conditions by considering Knudsen number as zero   0  Kn , but for some physical situations involving emulsions, suspensions, foams and polymer solutions, the no-slip boundary condition is no longer justifiable. Therefore, velocity slip and temperature jump can be materialized for the ranges of Knudsen number, 1 . 0 10 3    Kn (slip flow). Slip flow is of considerable significance for its numerous applications in field of technology and industry like polishing valves of artificial heart and internal cavities. Jang and Wereley (2004) proposed an analytical expression for pressure distribution of gas in a uniform and straight rectangular micro-channel in the slip flow regime. A numerical investigation was carried out by Mukhopadhyay and Andersson (2009) on heat transfer and unsteady viscous flow towards a stretching surface with impact of slip condition. In recent In past few decades, investigations on boundary layer flow due to a stretching surface have acquired abundant interest of researchers for its promising applications in engineering, metallurgical, industrial and manufacturing processes. For instance, crystal growing, aerodynamic extrusions of plastic and plastic sheets, wire drawing, glass blowing, hot rolling and paper production. Crane (1970) pioneered the analysis of a two-dimensional boundary layer flow over a stretching surface and later on, this work is extended by many researchers with MHD, heat and mass transfer, sole effect of rotation, non-Newtonian fluid and many more feasible combinations of various effects. Cortell (2005) established boundary layer flow and heat transfer in porous medium caused by a stretching surface. Mukhopadhyay and Layek (2008) considered a free convective flow of a fluid over a stretching surface under the impact of thermal radiation and irregular fluid viscosity. Akbar et al. (2013) studied a steady two-dimensional magnetohydrodynamic stagnation point flow of nanofluid past a stretching surface along thermal radiation effect and convective boundary condition. Gireesha et al. (2016) carried out a theoretical investigation on heat transfer in dusty fluid flow in porous medium over a nonisothermal stretching surface with radiation and Hall effects. Recently, Mishra and Kumar (2020) discussed thermal and velocity slip on MHD flow of a nanofluid towards a stretching cylinder.
Thermal radiation is a mode of heat transfer mechanism through which emission of thermal energy takes place in form of electromagnetic waves at the surface of matter with nonzero absolute temperature. In other words, it is a transformation of thermal energy into electromagnetic energy which is completely independent of any material medium. An analysis on radiation effects on magneto-hydrodynamic mixed free convective flow over a moving semiinfinite plate was carried out by Azzam (2002). Rad and Aghanajafi (2009) studied the thermal radiation impact on nanofluid flow in a rectangular micro-channel. Subsequently, several researchers, such as Hady et al. (2012), Sheikholeslami and Ganji (2015), Dogonchi and Ganji (2017), Eid and Makinde (2018), Krishna et al. (2019) and recently, Kanika et al. (2020) introduced problems concerning radiation effect on nanofluid flow.
Boundary layer suction in amongst boundary layer controlling techniques proposed by L. Prandtl in 1904, in an effort to slow down the boundary layer separation by reducing drag on the bodies in exterior flow. Suction and injection are of common concern in practical problems relating to aerodynamics and space sciences. El-Arabawy (2003) analysed suction and injection effects on micro-polar fluid flow over a continuously moving plate in presence of thermal radiation. Attia (2007) used uniform suction and injection impact on MHD flow of a fluid through rotating disk with heat transfer, Joule heating and Hall effects. Kandasamy et al. (2011) considered suction/injection effect on MHD nanofluid flow due to a stretching surface and observed that higher suction effect leads to decrease velocity and temperature profiles. Makinde and Chinyoka (2013), Khan et al. (2016), Mohammadein et al. (2018) and Chaudhary and Kanika (2020) also used suction and injection influence on various boundary layer flows.
In view of the above literatures, boundary layer flows of nanofluids, induced by stretching surface have been frequently addressed in recent studies. However, radiation and suction/injection impact on SWCNT-kerosene oil nanofluid over stretching sheet with slip boundary condition has still not been investigated. In this study, the numerical solution has been attained for the momentum and energy equations by implementing finite element method and results are portrayed graphically to interpret the behaviour of pertinent parameters. Moreover, the results of this study can lead to further improvements in the efficiency of heat exchangers and solar collectors systems (Ghalandari et al., 2020).

Mathematical Formation 2.1 Problem Statement
To develop the model, consider a steady two-dimensional viscous flow of an incompressible nanofluid induced by a stretching surface under slip boundary condition and uniform suction/injection effects, as shown in Figure 1. The nanofluid consists of SWCNTs with host fluid kerosene oil. It is assumed that host fluid and nano-tubes are in thermal equilibrium. Moreover, the thermal radiation effect is taken into consideration and viscous dissipation is considered as negligible in this study.
subjected to the following boundary conditions: where u and v are the velocity constituents in x and  y axes directions, subscript nf refers nanofluid, l is the slip The Rosseland approximation (Pantokratoras and Fang, 2012) is utilized to simplify the radiative heat flux as Taylor series expansion.

Nanofluid Properties
The physical parameters for SWCNT-kerosene oil nanofluid, namely, dynamic viscosity here f and CNT are subscripts representing base fluid and nano-sized particles and f is the volume fraction of nanotubes. Moreover, the thermo-physical properties of nanotubes and base fluid are presented in the

Similarity Conversion
The following dimensionless variable are to be utilized to transform the above mathematical model in non-dimensional form (Jafari and Freidoonimehr, 2015).
, that identically satisfies the continuity equation (1). Therefore, the resulting non-linear dimensionless ordinary differential equations, by plugging the above variables in equations (2)-(4), are obtained as follows: In the above equations, primes represent derivatives with respect to similarity variable  , f is the dimensionless stream function,  is the dimensionless temperature, is the velocity slip parameter. It is noteworthy that S is positive for suction and negative for injection process.

Parameters of Engineering Interest 3.1 Skin Friction Coefficient
The statement of local skin friction coefficient  (9) into equation (13), the parameter is evaluated as: is the local Reynolds number.

Nusselt Number
The local Nusselt number x Nu is expressed as follows: is the heat flux at surface, i.e., at 0  y . Substituting the transformations given in equation (9), the above equation is obtained as: Nr Nu (16)

Solution Approach
Intending to numerically solve the system of non-linear ordinary differential equations (10)-(11) with assisting boundary conditions equation (12) , therefore, the resulting coupled system of differential equations is obtained as follows:  (19) and the corresponding boundary conditions are: The variational form associated with the equations (17) here 1  , 2  and 3  are the weight functions corresponding to the functions f , h and  .
Substituting the following finite element approximations in equations (21)-(23), to obtain the finite element model Therefore, the finite element model of the equations (21)-(23) for c th element is formulated as:   The matrix constructed by assemblage of element equations, is of order . 3003 3003 It holds a system of linear equation, which is to be solved by employing a direct or indirect iterative process. Subsequently, implementing the boundary conditions, the system of remaining 2998 equations can be evaluated by Newton-Raphson technique. The process is terminated when the solution is converged with desired accuracy 7 10  , i.e., where,  refers either f , h or  and j is representing the iterative step.

Results and Discussion
In this portion, an attempt is made to analyze the impact of controlling parameters, namely, solid Impact of uniform mass flux parameters on dimensionless velocity profile is illustrated in Figure  2. The flow velocity is observed to be decreased due to increasing effect of suction while the reversal response is recorded for increasing injection effect. This is an outcome of the fact that boundary layer gets more close to the surface which in result destroys the momentum, which leads to fall in fluid velocity. Figure 3 shows the responses of dimensionless temperature profiles to mass flux parameter. It is analyzed that temperature as well as thermal boundary layer thickness are decreasing for larger values of suction/injection parameter.
The investigation of velocity slip parameter on velocity and temperature profiles in presence of suction effect is demonstrated in Figures 4 and 5    is detected in the fluid flow for boosting values of slip parameter. The slip boundary condition is implemented when the impact of fluid viscosity at the surface is negligible. Since slip effect causes weaker bond between the surface and fluid as more resistance is experienced in conveyance of velocity of stretching sheet to the fluid flow, therefore, it decreases the velocity. Further, Figure 5 reveals that higher values of velocity slip parameter correspond to rise in temperature throughout the boundary layer.
Figures 6 and 7 exhibit the effect of solid volume fraction parameter on dimensionless momentum and thermal profiles respectively. It is found that velocity distribution is an increasing function of solid volume fraction parameter for SWCNTs. From a physical perspective, enhancement in nanoparticle volume fraction gives rise to convective flow. Further, nanomaterial volume fraction has a significant impact on temperature field. From Figure 7 it is observed that temperature profile also shows the increasing behavior for higher solid volume fraction. It is apparent that thermal conductivity and diffusivity increases when volume of nanoparticles is increased. Hence, improvement in heat transfer leads to rise in temperature. Figure 8 portrays the outcome of thermal radiation parameter on temperature profiles and it shows that higher values of radiation parameter enhance the fluid temperature. This is evident due to the fact that increment in radiation parameter decreases the Rosseland radiative mean International Journal of Mathematical, Engineering andManagement Sciences Vol. 6, No. 3, 860-877, 2021 https://doi.org/10.33889/IJMEMS.2021.6.3

Conclusions
Flow of SWCNT-kerosene oil nanofluid over a linear stretching surface has been addressed in appearance of velocity slip and suction/injection effects. Finite element method is employed to solve the governing non-linear ordinary differential equations. The main outcomes of this problem can be summarized as follows:


Fluid velocity, surface shear stress, temperature and surface heat flux are all increasing functions of solid volume fraction parameter and decreasing functions of mass flux parameter.  Thickness of momentum boundary layer reduces in the range 8 . 2   , whereas thermal boundary layer thickness, surface heat flux as well surface shear stress increases for large values of velocity slip parameter.  Local Nusselt number and temperature both enhances with increment in radiation parameter.

Conflict of Interest
Authors declare that they have no conflict of interest.