Dynamics of Ethylene Glycol-based Graphene and Molybdenum Disulphide Hybrid Nano uid Over a Stretchable Surface With Velocity and Thermal Slip Conditions

In this research study, numerical and statistical explorations are accomplished to capture the flow features of the dynamics of ethylene glycol-based hybrid nanofluid flow over an exponentially stretchable sheet with velocity and thermal slip conditions. Physical insight of viscous dissipation, heat absorption and thermal radiation on the flow-field is scrutinized by dissolving the nanoparticles of Molybdenum disulphide (MoS2) and graphene into ethylene glycol. The governing mathematical model is transformed into the system of similarity equations by utilizing the apt similarity variables. The numerical solution of resulting similarity equations with associated conditions are obtained employing three-stages Lobatto-IIIa-bvp4csolver based on a finite difference scheme in MATLAB. The effects of emerging flow parameters on the flow-field are enumerated through various graphical and tabulated results. Additionally, to comprehend the connection between heat transport rate and emerging flow parameters, a quadratic regression approximation analysis on the numerical entities of local Nusselt numbers and skin friction coefficients is accomplished. The findings disclose that the suction and thermal radiation have an adverse influence on the skin friction coefficients and heat transport rate. Further, a slight augmentation in the thermal slip factor causes a considerable variation in the heat transport rate in comparison to the radiation effect.


Introduction
Recently, an innovative class of nanotechnology is developed with better chemical and thermal features by hybrid nanofluids. A hybrid nanofluid is prepared by the dispersal of two or more nano-sized metal or metal oxides into a conventional (base) fluid. The thermophysical features and significance of nanoparticles of various metals (Cu, Au, Ag, etc.), metal oxides (Al2O3, ZnO, TiO2, SiO2, MoS2, etc.), metal nitrides (Boron nitride BN, AIN), metal carbides (SiC), carbon materials (CNTs, graphene, diamonds, etc.) and hybrid nanomaterials are available in the literature [1]. Each nanoparticle has inimitable thermal features and is utilized as per the need of the thermal systems. In this research exploration, the nanoparticles of graphene and molybdenum disulphide (MoS2) are disseminated into the ethylene glycol to prepare the hybrid nanofluid. The nanoparticles of graphene and molybdenum disulphide (MoS2) have tremendous thermal performance and are applicable in various thermal systems [2]. The graphene (allotrope of carbon) nanoparticles own the unique material, chemical, electrical and physical characteristics because it has a single layer of atoms, biocompatibility certainties, expanded surface area, cell growth capability, fast mobility of electrons, stability, and high thermal conductivity. On the other hand, molybdenum disulphide (MoS2) consists of a layered structure and has distinctive properties such as chemically inertness, photo corrosion resistance and anisotropy, etc. Experimental findings revealed that hybrid nanofluids have better efficiency as compared to nanomaterials. Novel characteristics of hybrid nanofluids are significant in thermal storage, solar heating, transformer cooling, biomedical industry, heat pumps, refrigeration, welding, aircraft, spacecraft, lubrication, generator and electronic cooling, etc. Suresh et al. [3] quantified the mechanism of thermal transport on the dynamics of water conveying Cu-Al2O3 hybrid nanofluid. They analysed the significant improvement in convective heat transfer owing to synthesized hybrid particles' addition as compared to water.
Devi and Devi [4] implemented the Runge-Kutta-Fehlberg algorithm to analyse the heat transport rate of three-dimensional water-driven Cu-Al2O3 nanoparticles hybrid nanofluid flow through a stretchable surface under the inspiration of suction and Lorentz force. The numerical findings disclosed that the heat transport rate of water conveying Cu-Al2O3 nanoparticles hybrid nanofluid is better than Cu-water nanofluid. Further, in the extended work, Devi and Devi [5] showed that the heat transport rate of water-driven Al2O3-Cu hybrid nanofluid can be improved to 17.3% as compared to base liquid water whereas 11.2% than the Cu-water nanofluid. Recently, Khashi'ie et al. [6] enumerated the significance of the magnetic field and suction strength on the time-dependent squeezing flow of water-based Al2O3-Cu hybrid nanofluid through a parallel channel. They observed that the heat transport rate of the hybrid nanofluid at the lower plate can be reduced due to the inclusion of injection, squeezing and magnetic parameters while it can be improved at the upper plate of the channel nearly by 30.35% owing to the augmentation of suction and magnetic field strength.
Because of the applications in industry, the novel features of thermal radiation cannot be overlooked exclusively in the designing of steadfast equipment, furnaces, electrical power generation, nuclear plants, gas turbines, satellites, missiles, and also in designing of various cutting-edge energy conversion systems. These days, owing to the diminution of traditional energy sources, researchers are giving much attention to renewable energy resources. Solar energy is the fundamental basis of renewable energy and the thermal effect performs as an important fragment to change solar energy to the apt form for different industrial applications. Keeping the significance of radiation effect into account, the influence of nonlinear radiation on the Cu-Al2O3 nanoparticles driven flow of micropolar dusty hybrid nanofluid via a stretchable surface was scrutinized by Ghadikolaei et al. [7]. The nonlinear radiation impact on the thermal conductive flow of hybrid nanofluid (water/Cu-Al2O3) through a three-dimensional stretchable surface in a rotating medium using the least square approach was surveyed by Usman et al. [8]. Further, Sheikoleslami et al. [9] inspected the non-Darcy model of hybrid nanofluid considering Hartmann effects and a radiation term. They analysed that the temperature gradient gets highly affected owing to augmenting values of buoyancy parameter. Shoaib et al. [10] studied the three-dimensional rotating flow of magneto-hybrid nanofluid past an extendable surface with thermal radiation. Some illustrious research articles highlighting the novel features of thermal radiation can be seen in references [11][12][13][14][15][16][17][18]. In addition, the inclusion of heat generation/absorption is important in controlling the heat transport rate in several industrial processes such as in glass fibre production, hot rolling, endothermic reactions, paper production, heat conversation due to nuclear fuel wastage, etc.
Following this, Hayat and Nadeem [19] elucidated the implication of heat absorption/generation on the heat transport features of three-dimensional chemically reactive and radiative hybrid nanofluid flow through a stretchable surface. They concluded from their numerical exploration that the inspirations heat generation, radiation and chemical reaction are significant to achieve the higher heat transport rate of hybrid nanofluid as compare to simple nanofluid. This numerical exploration was further extended by Hayat et al. [20] to capture the effects of heat generation/absorption, considering the hybrid nanofluid in a rotating medium.
Recently, Li et al. [21] assessed the mathematical model to illustrate the effect of heat generation/ absorption on the entropy optimised convective flow in a rotating cone. They concluded that the upsurge in viscosity parameter and buoyancy ratio variable result in a significant rise in the tangential velocity.
One point to note here is that, in all the research articles reported above, the majority of authors have overlooked viscous dissipation term because of the feeble result on the flow field but its bearing in polymer manufacturing, lubrication, instrumentations, food processing, etc., is substantial because it augments the temperature distribution features and subsequently upsurges the heat transport rate. Few novel research articles describing the implication of viscous dissipation on various flow problems persuaded by a stretchable surface/thin stirring needle are cited in references [22][23][24][25]. In addition, Joule dissipation demonstrates the features of volumetric heat source in magneto-fluid flows and combined Joule and viscous dissipations inspirations are noteworthy in various heat-treated materials. Owing to this reason, Seth et al. [26], Daniel et al. [27] and Seth and Singh [28] and taken both the terms of Joule and viscous dissipations together in their mathematical models. Flow features of propylene glycol conveying, entropy optimized, magneto and dissipative Darcy-Forchheime nanofluid via a stretchable surface was illustrated by Abbas et al. [29]. Further, Wang et al. [30] explored the irreversibility features of entropy optimized magneto-nanofluid flow via variable thick surface under inspirations of viscous dissipation and Joule heating. Numerical exploration was done by Ibrahim and Khan [31] to analyse the influence of viscous dissipation on SWCNT and MWCNT conveying mixed convection nanofluid flow. Slip conditions are being extensively inspected by numerous researchers owing to their importance in the heat transport process which occurs due to the dissimilar fluid velocity and the velocity of fluid near the boundary. In industry, slip is widely used in micro heat exchangers, microelectronics cooling devices, polishing, artificial heart valves, internal cavities and drug delivery system [32][33][34]. Lately, Hussain et al. [35] inspected the flow of graphene and ethylene glycol-based Maxwell nanofluid via a stretchable surface under the inspirations of thermal radiation, slip conditions, viscous and Joule dissipations. This problem was further extended by Sharma et al. [36] considering heat absorption into account. The numerical findings divulged that viscous dissipation, radiation and slip parameters are important in controlling the temperature of the flow field. Also, heat transport rate is more sensitive to radiation parameters in contrast with viscous dissipation. Very recently, Wahid et al. [37] investigated the dynamics of hybrid nanofluid flow in the presence of velocity slip and heat generation via an exponentially stretchable surface. They analysed that the shear stress and wall temperature gradient can be augmented by enhancing the volume fraction of copper nanoparticles.  An unvarying oblique magnetic field 0  is imposed in a direction that makes an angle  with the stretchable sheet, which is adequately weak to ignore the induced magnetic field Cramer and Pai [38].  The hybrid nanofluid is imbued over the stretchable surface owing to the ambient  The base fluid ethylene glycol and the nanoparticles of MoS2 and graphene are in a thermal equilibrium state and no-slip befalls between them.  The polarization impact is overlooked owing to the non-appearance of the externally exerted electric field.  Inspirations of heat absorption, viscous dissipation and optically thick radiation are unified to improve the heat transport rate.
 The temperature of hybridized fluid at the stretchable sheet is   while those of hybrid nanofluid is   .
Based on the aforesaid assumptions, the constitutive flow equations of momentum and energy for the ethylene glycol-based hybrid nanofluid are obtained as: The continuity equation: The momentum equation: The energy equation: The accompanying boundary conditions for the model are given as: In aforesaid equations, 12 Following the works of Hussanan et al. [41] and Brewster [42], an optically thick fluid has been considered here, the radiation heat flux r q with the help of Rosseland approximation can be expressed as: In above expression (5)  The energy equation (3) with the help of equations (5) and (6) is reduced to

Numerical solution
The mathematical model reported in the above section includes highly nonlinear partial differential equations, therefore, to identify the numerical solution of these equations subject to the allied conditions, it is relevant to introduce a stream function  and similarity variables  as under: 12 ,, yx where,

 
Re, and f T  signify the Reynolds number, kinematic viscosity of water and hybrid nanofluid temperature in dimensionless form respectively.
The equations (8) and (9) result the following relations Here, prime shows the differentiation with regard to similarity variable  . Further, equations (9) and (10) transform the mathematical model reported in section 2 in below mentioned dimensionless forms: with allied boundary conditions The dimensionless parameters reported in the equations (11) to (13)

Wall temperature gradient and skin friction coefficients
To scrutinize the wall heat transport rate and shear stress function from the engineering outlooks, the expressions of local Nusselt number

Numerical technique implementation
To illustrate physically consistent and stable numerical solution corresponding to transformed similarity equations (11) and (12)  Step 1: Foremost, the new variables are introduced for the transformed similarity equations (11) and (12) as under: y y y Step 2: Then, by making use of relations (17), equations (11) and (12) are changed into system of first order equations: Step 3: According to the assumed variables (18) Here, subscript a indicates the initial sheet position i.e., 0   while subscript b represents the boundary condition at infinity. The value of  is considered as 5   for the infinite boundary condition.
Step 4: Finally, the initial guess was provided at the initial mesh points to obtain the solution.
These steps are repeated till the obtained numerical solutions satisfy the boundary conditions (19) asymptotically.

Validation of numerical findings and scheme
The obtained results and the employed numerical technique have been validated by comparing the numerical entities of  Table 3, which disclose venerable agreement between the numerical results and executed numerical scheme. Thus, it reveals that the developed numerical scheme and the results of this paper are acceptable and valid.

Results and discussion
Due to the enormously nonlinear nature and intricacy, the solution of leading equations (11) and (12)      whereas these values are decreased due to injection parameter   0 S  . This unveils that at the surface of the stretchable sheet under both velocity slip and no-slip situations the shear stress function is improved owing to upsurge in the angle of aligned magnetic field, suction and magnetic field while it gets reduced due to injection. Table 5 illustrates that under both the situations of velocity slip and no-slip, the numerical findings of The quadratic regression approximation model for the estimated x Sf owing to change in suction parameter S and velocity slip parameter L is given by whereas, quadratic regression approximation formula for the estimated   Tables 6 and 7.
It is worthy to remark that as the strength of the magnetic field or viscous dissipation effect improves, the coefficients of S or Tr becomes negative as observed from Tables 6 and 7, respectively. This finding reveals that the suction and thermal radiation have an adverse influence on the approximated skin friction coefficients and wall temperature gradients respectively. Moreover, from the tabulated values it is found that the coefficients of the velocity slip parameter are greater than those of the suction parameter in magnitude; which reveals the findings that a small variation in the velocity slip parameter L results in a significant change in the shear stress function in comparison to the suction parameter S. Likewise, a small augmentation in the thermal slip factor causes a considerable variation in the heat transport rate in comparison to the thermal effect. In addition, it is observed that the optimum relative error of quadratic regression approximation for the reduced wall temperature gradient is approximately zero, and the approaching rate towards this admirable accuracy level is faster than that of the quadratic regression approximation for the reduced skin friction coefficients.

Conclusions
In this research study, due to the remarkable significance in the advancement of robust apparatus used in the energy sector, nuclear power station, medical sciences, satellites, sensing outlets, gas turbines and supercapacitors, etc., numerical and statistical explorations have been accomplished to capture the flow features of the dynamics of ethylene glycol-based hybrid nanofluid containing graphene and MoS2 nanoparticles over an exponentially stretchable sheet with partial slip and thermal jump conditions. Some important conclusions of the study are highlighted below:  The fluid motion of hybrid nanofluid is strongly retarded by increasing the strength of the magnetic field and angle of the aligned magnetic field however leave a reversal impact on the temperature of the hybrid nanofluid  Both the temperature and velocity dispersal profiles are enhanced for improving the injection parameter while the suction parameter has reversed impact on the profiles  The temperature of hybrid nanofluid can be augmented by improving viscous dissipation and thermal effects while it can be reduced owing to a rise in the values of heat absorption parameter and thermal slip factor  The shear stress functional values can be enhanced due to the improvement in the suction, angle of aligned magnetic field and magnetic field effects while it can be reduced owing to rise in injection parameter at the surface of the stretchable sheet under both the situation of velocity slip and no-slip  The heat transport rate can be improved by the inspirations of heat absorption, suction and thermal radiation while the angle of an aligned magnetic field, viscous dissipation, magnetic field, injection and thermal slip factor are significant to reduce the heat transport rate of hybrid nanofluid  The quadratic regression approximation analysis discloses that the suction and thermal radiation have an adverse influence on the approximated skin friction coefficients and wall temperature gradients. Further, a small variation in the velocity slip parameter results in a significant change in the shear stress function in comparison to the suction parameter. Likewise, a small augmentation in the thermal slip factor causes a considerable variation in the heat transport rate in comparison to the thermal effect.  The optimum relative error of quadratic regression approximation for the reduced wall temperature gradient is approximately zero, and the approaching rate towards this admirable accuracy level is faster than that of the quadratic regression approximation for the reduced skin friction coefficients.