Dynamics of ethylene glycol-based graphene and molybdenum disulfide hybrid nanofluid over a stretchable surface with 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 disulfide (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-bvp4c-solver 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.

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 (Al 2 O 3 , ZnO, TiO 2 , SiO 2 , MoS 2 , 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 disulfide (MoS 2 ) are disseminated into the ethylene glycol to prepare the hybrid nanofluid. The nanoparticles of graphene and molybdenum disulfide (MoS 2 ) 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 disulfide (MoS 2 ) 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-Al 2 O 3 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-Al 2 O 3 nanoparticles hybrid nanofluid flow through a stretchable surface

Mathematical foundation of the problem
In this numerical exploration, the dynamics of two-dimensional, steady-state, incompressible and electrically conducted flow of ethylene glycol-based hybrid nanofluid over an exponentially stretchable sheet under the stimulus of partial slip and thermal jump conditions are examined. The base fluid i.e., ethylene glycol is hybridized by immersing very fine nanoparticles of molybdenum disulfide (MoS 2 ) and graphene into the fluid. For the development of the mathematical model, the Cartesian coordinate system is considered where the x-axis is allied with the surface of the stretchable sheet, the y-axis is taken in a perpendicular direction and the flow-field is constrained to y > 0 . A physical diagram of the problem is exhibited in Fig. 1. The following assumptions are made to obtain the governing equations for the dynamics of hybrid nanofluid flow: • An unvarying oblique magnetic field ξ 0 is imposed in a direction that makes an angle α with the stretchable sheet, which is adequately weak to ignore the induced magnetic field Cramer and Pai 54 . • The hybrid nanofluid is imbued over the stretchable surface owing to the ambient velocity 1 (x).
• The base fluid ethylene glycol and the nanoparticles of MoS 2 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 θ ε while those of hybrid nanofluid is θ ∞ .
Based on the aforesaid assumptions, the constitutive flow equations of momentum and energy for the ethylene glycol-based hybrid nanofluid are obtained as 55 : The continuity equation: The momentum equation: The energy equation: The accompanying boundary conditions for the model are given as:  www.nature.com/scientificreports/ In aforesaid equations, ω 1 and ω 2 describe the hybrid nanofluid velocities along x and y axes respectively, θ indicates the hybrid nanofluid temperature, 1 (x) = 0 e x/l denotes stretchable sheet velocity, θ ε (x) = θ ∞ + (θ 0 − θ ∞ )e ax/2l reflects the hybrid nanofluid exponential temperature distribution over the sheet. a , θ 0 and 0 are respectively the temperature distribution parameter in the stretchable sheet, temperature and velocity references, C = C 1 /e x/2l signifies the Navier's velocity slip factor in which C 1 shows the primary slip velocity and δ = δ 1 /e x/2l indicates thermal slip factor wherein δ 1 is preliminary thermal slip value. Both the thermal and velocity slip factors got changed owing to the coordinate variable x and no-slip takes place when δ = C = 0 . H 0 represents the coefficient of heat absorption. The blowing/suction velocity is mentioned by where v 0 is the primary blowing/suction velocity strength.
For the developed hybrid nanofluid, the heat capacity, dynamic viscosity, density, thermal conductivity and electrical conductivity are denoted by ρC p hnf ,μ hnf ,ρ hnf ,K hnf and σ hnf respectively and are defined by the relations as mentioned in Table 1. Further, in the Table 1, the volume fraction of graphene and MoS 2 nanoparticles are respectively signified by φ 1 and φ 2 . μ f represents the dynamic viscosity of ethylene glycol, ρ g1 andρ m2 signify the density of graphene and MoS 2 nanoparticles, σ f is used to represent the electrical conductivity of ethylene glycol, σ g1 divulges the electrical conductivity of graphene, σ m2 embodies the electrical conductivity of MoS 2 .
ρC p f , ρC p g1 and ρC p m2 indicate the heat capacitance whereas k f ,k g1 andk m2 represents the thermal conductivity of ethylene glycol, graphene and MoS 2 respectively. The physical properties of ethylene glycol, graphene and MoS 2 are reported in Table 2.
Following the works of Hussanan et al. 58 and Brewster 59 , an optically thick fluid has been considered here, the radiation heat flux q r with the help of Rosseland approximation can be expressed as: In above expression (5), σ * signifies the Stefan-Boltzmann constant while k * specifies the heat absorption constant. Further, to linearize the term θ 4 the Taylor's series is implemented to expand it about θ ∞ (free stream temperature) and is expressed after ignoring higher degree terms as under: The energy Eq. (3) with the help of Eqs. (5) and (6) is reduced to (4a) at y = 0 : ,  56,57 .

Properties Hybrid nanofluid
Heat capacity

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 ψ and similarity variables η as under 60 : where, Re, υ f and T(η) signify the Reynolds number, kinematic viscosity of water and hybrid nanofluid temperature in dimensionless form respectively. The Eqs. (8) and (9) result the following relations Here, prime shows the differentiation with regard to similarity variable η . Further, Eqs. (9) and (10) transform the mathematical model reported in "Mathematical foundation of the problem" section in below mentioned dimensionless forms: with allied boundary conditions The dimensionless parameters reported in the Eqs. (11) to (13) are M, X, Tr, Pr, Re, Ec, L, H a , S and D which represent the magnetic parameter, dimensionless coordinate, radiation parameter, Prandtl number, Reynolds number, Eckert number, velocity slip, heat absorption, injection/suction and thermal slip parameters respectively. These parameters are defined by below mentioned relations: 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 Nn x and skin friction coefficients Sf x are derived in both dimension and dimensionless forms. The dimension forms of Nn x and Sf x are defined as: 1 Pr where Re x = 1 (x)x/υ f signifies the local Reynolds number.

Numerical technique implementation.
To illustrate physically consistent and stable numerical solution corresponding to transformed similarity Eqs. (11) and (12) satisfying the allied conditions (13) the finite difference technique based bvp4c solver present in MATLAB is implemented. This solver was originally developed by Kierzenka and Shamoine 61 which works on three stages Lobatto IIIa algorithm. The algorithm results in 4th order accurate and uniform solution within the given range. The involved steps of the aforesaid method are furnished below: Step 1: Foremost, the new variables are introduced for the transformed similarity Eqs. (11) and (12) as under: Step 2: Then, by making use of relations (17), Eqs. (11) and (12) are changed into system of first order equations: Step 3: According to the assumed variables (18), the boundary conditions (13) are expressed as: Here, subscript a indicates the initial sheet position i.e., η = 0 while subscript b represents the boundary condition at infinity. The value of η is considered as η = 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.  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 Eqs. (11) and (12)  This phenomenon happens because a resisting force is persuaded due to the gradual improvement in the strength of the magnetic field. This resisting force termed as Lorentz force acts in the opposite direction of flow-field as well as hybrid nanoparticles and subsequently, the velocity of hybrid nanofluid is retarded whereas its temperature gets improved due to enrichment in frictional drag force. The hybrid nanofluid velocity profiles are declined while the temperature dispersal profiles are improved owing to the rise in the angle of aligned magnetic field under both velocity slip and without slip conditions, as revealed in Fig. 3. This tendency of profiles exhibits that the angle of the aligned magnetic field retards the velocity of hybrid nanofluid and the resistive force's strength is optimal when the exerted magnetic field is perpendicular to the stretchable sheet. The optimal resistive force diminishes the motion hybrid nanofluid and improves the temperature of the fluid. Figures 4 and 5 represent the diminishing effect of suction parameter (S > 0) on the scattering profiles of f ′ (η) and T(η) while these profiles are reversed in case of the injection parameter ( S < 0 ). The reason behind this behaviour of profiles is that the momentum boundary layer sticks to the surface of the stretchable sheet in the instance of suction, which breaks the flow momentum and as a result, both velocity and temperature of hybrid nanofluid are reduced. On the other hand, injection enhances fluid via lateral mass flux over the stretchable sheet and, in turn, appends the momentum of fluid flow. Consequently, both the velocity and temperature of hybrid nanofluid get improved. Figure 6 illustrates the influence of thermal radiation and heat absorption on the temperature of hybrid nanofluid. This figure shows that the temperature dispersal profiles upsurge due to the rise in the values of the radiation parameter (Tr). Generally, thermal radiation depends on the temperature of the surrounding, and it is emitted in an electromagnetic waveform. Therefore, thermal radiation can be assumed as a function of temperature. The thermal effect improves the conduction properties of nanofluid and hence, thicken the boundary layer and as a result, the temperature of fluid gets enhanced. Further, the temperature dispersal profiles of hybrid nanofluid are reduced due to the enhancement in heat absorption parameter (H a ). The reason behind this physical behaviour of the fluid is that a rise in the values of H a results from the augmentation in the heat-absorbing capacity of the hybrid nanofluid and consequently, the fluid temperature gets diminished. Figure 7 is depicted to capture the inspirations of viscous dissipation and thermal slip factor on the temperature of hybrid nanofluid. It is observed that the temperature dispersal profiles of hybrid nanofluid are improved owing to the enhancement in Eckert number (Ec) while it reduced due to thermal slip factor (D). Since, Eckert number ( Ec ) represents the relation between enthalpy and kinetic energy. It is involved in converting kinetic energy to the form of internal energy in opposing the viscous fluid stresses. As, Ec increases the internal energy also gets increased and consequently fluid temperature gets improved. This infers that the inspiration of viscous dissipation is responsible to raise the temperature of hybrid nanofluid. On the other hand, the thermal slip parameter is accountable to lessen the hybrid nanofluid temperature under both the velocity slip and no-slip situations. Physically, it is justified because from the surface of the stretchable sheet a reduced quantity of heat flows to the hybridized fluid in increasing the thermal slip parameter. Additionally, the graphical illustrations suggest that the thermal boundary layer thickness is improved by the magnetic field, angle of aligned magnetic www.nature.com/scientificreports/ field, viscous dissipation, injection and thermal radiation while it is declined due to augmentation in suction, heat absorption, and the thermal slip factor. The wall heat transport rate and shear stress function are analysed from the engineering outlooks. For this, the numerical form of local Nusselt number Nn x and skin friction coefficients Sf x are obtained under both the velocity slip (L = 0.5) and no-slip (L = 0) situations against emerging flow parameters. These values are provided in Tables 4 and 5. It is perceived from Table 4 that the numerical values of Sf x are increasing owing to rise in magnetic field, angle of aligned magnetic field and suction parameter (S > 0) whereas these values are decreased due to injection parameter (S < 0) . 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 Nn x are reduced by increasing angle of aligned magnetic field, Eckert number, magnetic, injection and thermal slip parameters whereas these numeric are enhanced due to upsurge in heat absorption, suction and radiation parameters. This quantifies that the heat transport rate can be improved by the inspirations of heat absorption, suction and thermal radiation while viscous dissipation, magnetic field, injection, angle of aligned magnetic field and thermal slip factor are significant to reduce the heat transport rate of hybrid nanofluid. Further, it is noticed that the shear stress function at the surface of the stretchable sheet is lower in the case of velocity slip condition as compared to the no-slip condition.

Quadratic regression analysis: approximations of skin friction coefficients and wall temperature gradients
In this section, for the quadratic regression approximation, a statistical method is accomplished to know the connection between two or more emerging flow parameters. Precisely, regression approximation is employed to analyse how the features of an emerging flow parameter change due to the variation of another flow parameter Tables 6 and 7 demonstrate the coefficients of quadratic regression approximated values of Sf x and Nn x respectively for different enduring parameters. The optimum relative error bounds ε and ε 1 for Sf x and Nn x have also been analysed by using the relations ε = Sf x (est) − Sf x /Sf x and ε 1 = Nn x (est) − Nn x /Nn x respectively. These   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 MoS 2 nanoparticles over an exponentially stretchable sheet with partial slip and thermal jump conditions. Some important conclusions of the study are highlighted below:  www.nature.com/scientificreports/ • 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.      Table 6. Quadratic regression approximated coefficients of Sf x owing to variations in S and L and optimum relative error bound ε are tabulated as mentioned below.