Effect of natural convection on 3D MHD flow of MoS2–GO/H2O via porous surface due to multiple slip mechanisms

ABSTRACT The theme of this work is to examine the magnetized flow behaviour of hybrid nanofluid containing molybdenum disulphide (MoS2) and graphene oxide (GO) nanoparticles subject to multiple slip mechanisms, natural convection and chemical reaction of mth order. The working hybrid nanofluid is prepared by an amalgamation of nanoparticles, i.e. MoS2 and GO and conventional fluid (H2O). The existing equations of the model were solved by the techniques, i.e. shooting and bvp4c. To validate our codes, the comparison was presented by previous research. Here, we observed that, with an increase in thermal slip parameter values, only the thermal field profiles of MoS2–GO/H2O regularly depreciated. Moreover, when there was augmentation in velocity slip parameter, temperature and concentration functions rose, while both velocity outlines of MoS2–GO/H2O fluid declined. Furthermore, the velocity functions of MoS2–GO/H2O augmented and the concentration function declined with an increase in natural convection parameter values.


Introduction
In the present era of industry 4.0, several industrial applications require the instantaneous control of heat and mass transport phenomena for well productivity of the machines. Cooling becomes an important part in this aspect. For example, condensed solar cell in the solar panel must be cooled properly for better efficiency and long life. In fuel cells, the chemical energy of the controlled combustion of fuel is transformed into other forms of energy. This further assists in operating the industrial machinery, and simultaneously cooling the machine during the operation. Apart from these, the controlled heat and mass transfer (HMT) flows are also encountered in the use of natural energy resources like biomass, wind, hydro and solar power, etc. The study of enhanced HMT also acting a dynamic role in improved health and a pollution-free environment. These types of studies help in the prediction of the risk of environmental contamination by chemical or biological agents. In advanced drug delivery devices, the cooling or heating must be controlled and optimized to prevent the body cell damage during the drug delivery.
Considering the vast significance of simultaneous HMT, in the past few decades, many researchers have studied these flows under various circumstances. Siddheshwar and Mahabaleswar [1] studied the magnetohydrodynamic HMT flow of visco-elastic fluid past a stretching sheet due to some heat source. The analytical results of flow due to stretching sheet were found by Vajravelu and Canon [2]. Cortell [3] deliberated the heat and mass features of flow over a nonlinearly stretching sheet considering the nonlinear radiation effect. Rashidi et al. [4] probed the free convective flow of working fluid due to sheet with temperature and concentration differences and established the semi-analytic solution of the formulated problem by using homotopy analysis method (HAM). Muhammad et al. [5] analysed the threedimensional HMT flow over an exponentially stretching sheet, considering the dissipation effect.
With the development of machinery in the industry 4.0 era, the claim for better cooling systems with enhanced accuracy and consistency has augmented. The designs of the cooling systems are becoming compact day by day. To augment heat transfer rate (HTR), the designs of the devices have to be relooked, but due to several scientific constraints this is not actually very feasible. Therefore, the alternate way out for this problem is to use a coolant with high thermal properties. It has been studied that the insertion of a small percentage of nano-sized particles of metal or their oxides into the pure fluid enhances the thermal conductivity of these pure fluids by significant amounts. These fluids are called nanofluids. Due to the enhanced thermophysical possessions of nanofluids, researchers in the field of HMT included nanofluids in their studies. Khan and Pop [6] inspected the HMT flow of a nanofluid past a stretching sheet. Hayat et al. [7] discovered the three-dimensional flow of alumina-H 2 O nanofluid over a stretching sheet and investigated the nonlinear radiation effect and computed the solution for the formulated problem. A finite element study of natural convection of H 2 O-based alumina nanofluid was done by Sheikholeslami et al. [8]. They investigated the flow in a two-dimensional square cavity with a heat source placed centrally at one of the walls. Hayat et al. [9,10] found the semi-analytic solution of the 3D flow of working fluid due to exponentially stretching and bidirectional nonlinear stretched sheets by HAM. The flow was analysed under fixed mass and heat fluxes. Mahanthesh et al. [11] performed a comparative study of three different H 2 O-based nanofluids with respect to the 3D flow. This study comprised both linearly and nonlinearly stretching sheets. Furthermore, Reddy and Chamkha [12] found the impact of chemical reaction (CR) in the HMT flow of H 2 O-alumina nanofluid. The nonlinear radiation approximation was considered in the study. Mousa et al. [13] completed the solution of the heat transfer problem by the combined method MOL-PACT. Bhatti et al. [14] implemented a perturbation technique for the results of visco-elastic fluid flow in a channel.
After the finding of nanofluids and their superior thermal properties, nanofluid has received an important consideration over the earlier two decades. As mentioned above, many theoretical and experimental studies have shown nanofluids to be a good resource for improved heat transfer rates. After these successful studies, the researchers have also tried to further enhance the stability, thermal and physical properties of fluids and came up with the idea of synthesizing the fluids by inserting the two or more nanoparticles of different nature in the base fluid directly or in the form of composites and named this new category of fluids as hybrid nanofluids. In the last decade, many studies have shown that the properly hybridized nanofluids may show promising results with respect to the improved HTR. Iqbal et al. [15] explored the nanoparticle shape effect on the flow of H 2 O-based molybdenum disulphide (MoS 2 ) and silica hybrid nanofluid over the curved stretching surface. They reported that hybrid nanofluid has better low rates as compared to silica-H 2 O nanofluid. Notifying the better thermal conductivity and very high boiling point the flow of propylene glycol-based MoS 2 and silica hybrid nanofluid was deliberated by Shaiq et al. [16]. An experimental study on the sesame oil-based carbon nanotube (CNT)/ MoS 2 nanofluid was accomplished by Gugulothu and Pasam [17]. Khashi'ie et al. [18] probed the 3D flow of Cu-alumina-H 2 O hybrid nanofluid due to its permeable surface. They reported the existence of a dual solution for the considered configuration and found one stable and another unstable solution. Very recently, a comparative study for the mixed convection of hybrid nanofluid with water and multi-walled CNTs-Cu, CNTwater and pure water was performed by Muhammad et al. [19].
After the ground breaking discovery of graphene by the Nobel laureates Geim and Novoselov [20], a new fertile ground has opened up for tremendous industrial applications. Researchers have proved that graphene is the lightest material with the highest thermal transport properties even at room temperature [21,22]. Apart from these properties of graphene, this material is also known to be a strong material with a very low density and high brittle property. The structure of graphene makes it available for good absorption and high CR rates. Scientists have also proved that graphene has a significantly large electron mobility rate at room temperature, which results in the high electrical conductivity of graphene. As graphene is such a magical material, therefore graphene and its oxides and graphene-nanofluids have received noteworthy attention in the past decade. Many theoretical and experimental studies have been accomplished towards the use of graphene (in any form) in lightweight manufacturing and nanoscale thermal engineering. Lee et al. [23] fabricated the reduced graphene oxide (GO) paper sheet and investigated the electrical, thermal and mechanical possessions of the material. They reported high thermal and electrical conductivities and improved tensile strength. Shiu and Tsai [24] studied the mechanical and thermal possessions of graphene/epoxy nanocomposites using the molecular dynamics simulation approach and found the enhanced thermal and mechanical properties in nanocomposites as compared to the graphene flakes. Aradhana et al. [25] synthesized the GO and condensed graphene-epoxy nanocomposite adhesives and explored the comparison in terms of their electrical, thermal and mechanical properties. Esfahani et al. [26] experimentally premeditated the thermal conductivity of water-based GO nanofluid and found the direct relationship between thermal conductivity on particle size and the viscosity of synthesized nanofluid. A contrast of the thermal conductivity of mono nanofluid and the GO-alumina (Al 2 O 3 )-water (H 2 O) hybrid nanofluid was explored by Taherialekouhi et al. [27]. Bhatti et al. [28] opted Darcy model for magnetic MgO-Ni/H 2 O flow through an elastic stretching surface. Ghadikolaei et al. [29] proposed the mathematical model for the 3D flow of working fluid with hybrid nanocomposite (GO-MoS 2 ) and hybrid base fluid (water-ethylene glycol) over a stretching surface and presented a relative study of hybrid nanocomposite nanofluid through hybrid conventional fluid and MoS 2 with hybrid base fluid. In this study brick, cylinder and platelet shaped nanoparticles were considered.
In nature and the industries like polymer industries, the flow induced by the existence of foreign particles and the CR between the different species is very common. During the chemical reactions, heat may also be generated which impacts the overall heat distribution in the system. The numerical or analytic solution for the CR effect on micropolar fluid flows under different circumstances over different types of obstacles is studied by various authors [30][31][32][33][34][35][36][37]. Furthermore, the higher-order CR effect on the flow of nanofluid is premeditated by Gopal et al. [38]. Very recently, Joshi et al. [39] studied the flow, heat and mass transport characteristics for single-walled CNTs and silver nanocomposite waterbased nanofluid. They probed the impact of nonlinear CR rates and heat generation on mixed convection flow and reported the fast mass transfer rate for higher-order CR. Abdelsalam et al. [40] offered a model for electromagnetic hybrid nanofluid flow through a sinusoidal channel due to the impact of laser radiation along with CR. Furthermore, the heat transfer performance of several fluids due to CR was presented in [41][42][43][44][45][46][47].
As per the author's information, the natural convection of H 2 O-based GO/MoS 2 hybrid nanofluid with higher-order CR has received a little attention. Therefore, the present study is to scrutinize the HMT performance of magnetized

Problem description
The present problem deals with the steady, viscous, incompressible, and three-dimensional flow of nanofluid (hybrid) over a vertical porous plate. Figure 1 discloses that the plate is located in the xz plane, i.e. y = 0, the plate is considered towards the x-direction, the vertical direction of the plate is z-axis and the y-direction is termed as normal of xz plane. The velocity components in the respective directions are (u, v, w). The ambient temperature and concentration of working fluid have changed to T ∞ and C ∞ , correspondingly. Furthermore, B 0 (magnetic field strength) is used in the direction normal to the surface with thermal expansion due to temperature difference (second term on the RHS of Equations (2) and (3)). The flow model is fully advanced and open. Therefore, the mechanism of heat transfer is supported with heat source and Ohmic heating, while in the working fluid mth-order CR is accounted for along with the mass transfer. The effects of slips mechanism (velocity, thermal and solutal) and suction are also considered. In this work, the nanofluid which we have utilized is a mixture of two different nanoparticles, namely MoS 2 and GO dispersed in base fluid (H 2 O). The thermophysical features of MoS 2 , GO and H 2 O are listed in Table 1. Here, the shape of the dispersed nanoparticles is considered to be spherical, owing to this we considered the modified Maxwell model and Brinkman model for the calculation of effective thermal conductivity and dynamic viscosity, respectively (see Table 2).

Governing equations
Assuming the stated conditions, the flow model for hybrid nanofluid is [29] The conditions at the surface of the plate (z = 0) and far away from the plate (z → ∞) are Here, the symbols υ, g, ρ, β, σ , k, α, C P , Q o , D B , k o , m and ( , n, p) present in the above equations represent kinematic viscosity, gravity, density, coefficient of thermal expansion, electrical conductivity, porosity, thermal diffusivity, specific heat, heat source, diffusion coefficient, coefficient of CR, order of CR and slip factors (velocity, thermal and concentration), respectively. Furthermore, if the terms of Equation (6), i.e. = n = p = 0, then the problem is turned into no-slip condition.
The dimensionless system of existing equations can be attained by introducing the following transformations [29]: Using similarity transformation defined above and correlations defined in Table 2, Equation (1) is gratified  identically and Equations (2)-(5) take the resulting dimensionless form: a 5 a 6 θ 1 Pr ξ with where Gr = Thermal conductivity are constants.

Quantities of engineering and physical importance
(a) The skin-friction coefficients are well-defined as where τ wx and τ wy are wall shear stresses in x and y directions and are expressed as Substituting Equation (13) in Equation (12) and using Equation (7), we obtained the following dimension-free form of skin-friction coefficients and (b) The Nu "Nusselt number" and Sh "Sherwood number" are expressed as Again using Equation (7), the expression for Nu and Sh given in Equation (16) is reduced to the following form: where

Method 1
Equations (2)-(5) governing the fluid flow subject to Equation (6) are tackled by the numerical method, i.e. Runge-Kutta-Fehlberg (RKF) with shooting scheme. To implement the methodology, the following procedure is followed: (1) Reduce Equations (2)-(5) to the dimensionless form Equations (8)- (11) and use Equation (7) with associated initial and boundary conditions. (2) Equations (8)-(11) are highly nonlinear and are discretized to obtain a system of first-order differential equations (DEs). (3) The obtained system in step 2 is in the form of an initial value problem (IVP) with some missing initial conditions. (4) Now, putting values to missing initial conditions using the shooting method and solving by the RKF method. In this study, all the computations are carried out for η max = 6 with step size η = 0.001 and tolerance limits of 10 −5 . Also, for η max > 6 the change in variables, i.e. skin friction, Nusselt number and Sherwood number is almost negligible hence the applied numerical scheme is stable and consistent.

Method 2
The solution of Equations (8)-(11) along with Equation (12) is tackled by the bvp4c tool which is available in MATLAB 2020a. For the outcomes of each parameter, the different loop is applied using the bvp4c tool (see [48]). In this article, this tool is particularly used for contour plots.

Code of validation
To validate the exactness of our code, values of − √ Re x C fx and − Re y C fy are calculated for the distinct value of α and Mn and are compared with those of Hayat et al. [10] (see Table 3). An excellent agreement between them leads to approval of the accurateness of our codes.

Results and discussion
This portion of the research declared the influences of various active parameters, i.e. magnetic field (Mn), Grashof number (Gr), porosity (K), slip variables (λ, St, δ), suction (F > 0) or injection (F < 0), Schmidt number (Sc), order of CR (m) and CR parameter (L) on physical variables, i.e. velocities (f (η), g (η)), temperature (θ (η)) and concentration (ξ(η)) fields are illustrated through graphs. Furthermore, the behaviour of varying Mn, Ec and K on physical variables, i.e. skin friction, Nusselt number and Sherwood number is illustrated using 2D plots and contour plots and discussed in detail. Moreover, all the computations are conducted by considering the value of Pr = 6.2, Q = 1 and the concentration of nanoparticles MoS 2 and GO as 3% each. To have a better understanding the response of active parameters is categorized in the following section: (i) Concentration profile: The impact of Sc on the concentration profiles is revealed in Figure 2. The Schmidt number Sc defined as the quotient of kinematic diffusivity and mass diffusivity, i.e. Sc = υ bf D B ; physically it relates the thickness of momentum and concentration boundary layer (BL). Thus a higher value of Sc(> 1) indicates reduction on the mass diffusivity and hence the concentration profile decreases (see Figure 2). Figure 3 plots the influence of L on concentration profile, it is seen that as L rises from 0.1 to 0.5 the concentration increases. Figure 4 displays the influence of Mn on concentration profile. It appears that concentration profile      is an increasing function of Mn and so does the BL. The impact of increasing K on concentration is publicized in Figure 5, the fluid concentration rises with K. Figure 6 displays the concentration profile for numerous data of Grashof number (Gr). It is spotted that the concentration profile declines with growing Gr. Figure 7 presents the impact of F on concentration profile, increasing the value of   F from −3 (injection) to +3 (suction) results in the decline of concentration, this behaviour continues till the profile reaches its ambient conditions. Figures 8 and 9 present the characteristics of concentration profile for growing values of concentration (δ) and velocity (λ) slips, respectively. Here,   the concentration along with the width of associated BL decreases continuously with δ (see Figure  8) while it escalates with λ (see Figure 9). Figure 10 displays how the order of CR (m) affects the concentration profile ξ(η), here ξ(η) decreases with increasing m. The width of concentration BL for m = 1 is wider compared to boundary layers correspond to m = 2, 3, 4, 5. Also, the variation between      of fluid and hence accelerates the temperature profiles see Figure 11. Figure 12 depicts that thickness of thermal BL decreases with growing values of F, the slope of the curve θ(η) correspond to negative F is positive (> 0) for η < 0.25 which means nearby the wall of the extending sheet the fluid possess higher temperature for injection region   and thereafter the temperature of the fluid declines continuously and reaches its ambient conditions. Moreover, the temperature of the fluid decreases throughout the domain for suction. Figures 13 and  14 probed the response of growing St and λ, correspondingly. It is examined from these data that      of Mn outcomes in the lessening of the velocity along with associated BL thickness. This is happening because of the resistive force experienced by the fluid particles engendered by Mn. The aspect of the porosity parameter K on f (η) and g (η) for F > 0 is exhibited in Figures 16 and 21. These plots illustrate that growth in the amount of K tops to the decline of f (η), and asymptotically satisfies the conditions f (η → ∞) = 0. Besides, the velocity BL becomes narrower with cumulative porosity parameters. The effect of Grashof number on f (η) and g (η) is delineated in Figures 17 and  22. Grashof number is a dimension-free number that represents the fraction of buoyancy force and viscous force. Growing Grashof number refers to either accelerating the buoyancy force or decreasing the viscous force, in either case, the fluid velocity increases. Thus, the fluid velocity increases in both directions as shown in Figures 17 and 22.
The upshots of the parameter λ are drawn in Figures 18 and 23. It is witnessed from the plots that escalation of the slip parameter (velocity) resists the motion of the fluid, hence the constituent of velocity in the x and y directions decreases. Figures  19 and 24 illustrate that switching from injection to suction regions results in the decline of velocity in both directions. It is worth mentioning that in either orientation, i.e. x-or y-axis the alteration in the dimensionless field is showing a similar pattern with deviation in the width of the corresponding BL. (iv) The surface drag forces, Nusselt number and Sherwood number analysis: Impact of variation in the magnitude of magnetic field against Eckert

Disclosure statement
No potential conflict of interest was reported by the authors.