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.

KEYWORDS:

• Hybrid nanoparticles (MoS₂+ GO)
• chemical reaction of mth order
• natural convection
• magnetic field
• multiple slip mechanisms
• porous medium

1. 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 three-dimensional 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-H₂O 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 H₂O-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 H₂O-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 H₂O-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 H₂O-based molybdenum disulphide (MoS₂) and silica hybrid nanofluid over the curved stretching surface. They reported that hybrid nanofluid has better low rates as compared to silica-H₂O nanofluid. Notifying the better thermal conductivity and very high boiling point the flow of propylene glycol-based MoS₂ and silica hybrid nanofluid was deliberated by Shaiq et al. [16]. An experimental study on the sesame oil-based carbon nanotube (CNT)/ MoS₂ nanofluid was accomplished by Gugulothu and Pasam [17]. Khashi’ie et al. [18] probed the 3D flow of Cu-alumina-H₂O 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, CNT-water 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 (Al₂O₃)–water (H₂O) hybrid nanofluid was explored by Taherialekouhi et al. [27]. Bhatti et al. [28] opted Darcy model for magnetic MgO–Ni/H₂O 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–MoS₂) 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 MoS₂ 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 [30–37]. 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 water-based 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 [41–47].

As per the author’s information, the natural convection of H₂O-based GO/MoS₂ hybrid nanofluid with higher-order CR has received a little attention. Therefore, the present study is to scrutinize the HMT performance of magnetized MoS₂–GO/ H₂O flow due to porous surface under the condition of natural convection, multiple slip mechanisms (velocity, thermal and solutal), suction/blowing and mth-order CR. The solution of the flow modelled equations of the MoS₂–GO/H₂O hybrid nanofluid is achieved by two different techniques, i.e. (i) shooting and (ii) bvp4c.

2. Mathematical model and formulation

2.1. Problem description

The present problem deals with the steady, viscous, incompressible, and three-dimensional flow of nano-fluid (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 $\left(u,v,w\right)$. The ambient temperature and concentration of working fluid have changed to $T_{\mathrm{\infty }}$ and $C_{\mathrm{\infty }}$, 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 MoS₂ and GO dispersed in base fluid (H₂O). The thermophysical features of MoS₂, GO and H₂O 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).

Figure 1. Flow geometry of the model.

Table 1. Thermophysical properties of solid particles (MoS₂ and GO) and base fluid (H₂O) [29].

	Thermophysical properties
Solid particles	$\rho \text{(kg/}{\text{m}}^{\text{3}}\text{)}$	$C_P\left(\text{J/kg}\text{K}\right)$	$\kappa \left(\text{W/mK}\right)$	$\sigma \left(\text{S/m}\right)$	$\text{β}\times {10}^{-5}\left({\text{K}}^{-\text{1}}\right)$	$Pr$
MoS2	5060	397.21	904.4	2.09×104	2.8424	–
GO	1800	717	5000	6.3×107	28.40	–
Base fluid
H2O	997.1	4179	0.613	0.05	21	6.2

Table 2. Correlation for calculation of effective properties of hybrid nanofluid [29].

Properties	Applied models
Density	${\rho }_{hnf}=\left(1-{\varphi }_{Mo{S}_2}\right)\left[\left(1-{\varphi }_{GO}\right){\rho }_{bf}+{\varphi }_{GO}{\rho }_{GO}\right]+{\varphi }_{Mo{S}_2}{\rho }_{Mo{S}_2}$
Dynamic viscosity	${\mu }_{hnf}={\displaystyle \frac{{\mu }_{bf}}{{\left(1-{\varphi }_{GO}\right)}^{2.5}{\left(1-{\varphi }_{Mo{S}_2}\right)}^{2.5}}}$
Heat capacity	$(\rho C_p{)}_{hnf}=\left(1-{\varphi }_{Mo{S}_2}\right)\left[\left(1-{\varphi }_{GO}\right){\left(\rho C_p\right)}_{bf}+{\varphi }_{GO}{\left(\rho C_p\right)}_{GO}\right]+{\varphi }_{Mo{S}_2}(\rho C_p{)}_{Mo{S}_2}$
Thermal expansion	$(\rho \text{β}{)}_{hnf}=\left(1-{\varphi }_{Mo{S}_2}\right)\left[\left(1-{\varphi }_{GO}\right){\left(\rho \text{β}\right)}_{bf}+{\varphi }_{GO}{\left(\rho \text{β}\right)}_{GO}\right]+{\varphi }_{Mo{S}_2}(\rho \text{β}{)}_{Mo{S}_2}$
Thermal conductivity	${\displaystyle \frac{{\kappa }_{hnf}}{{\kappa }_{bf}}}={\displaystyle \frac{\frac{{\varphi }_{GO}{\kappa }_{GO}+{\varphi }_{Mo{S}_2}{\kappa }_{Mo{S}_2}}{{\varphi }_S}+2{\kappa }_{bf}+2\left({\varphi }_{GO}{\kappa }_{GO}+{\varphi }_{Mo{S}_2}{\kappa }_{Mo{S}_2}\right)-2{\varphi }_S{\kappa }_{bf}}{\frac{{\varphi }_{GO}{\kappa }_{GO}+{\varphi }_{Mo{S}_2}{\kappa }_{Mo{S}_2}}{{\varphi }_S}+2{\kappa }_{bf}-\left({\varphi }_{GO}{\kappa }_{GO}+{\varphi }_{Mo{S}_2}{\kappa }_{Mo{S}_2}\right)+{\varphi }_S{\kappa }_{bf}}}$
Electrical conductivity	${\displaystyle \frac{{\sigma }_{hnf}}{{\sigma }_{bf}}}=\left[1+{\displaystyle \frac{3{\sigma }_{GO}{\varphi }_{GO}+{\sigma }_{Mo{S}_2}{\varphi }_{Mo{S}_2}-3{\sigma }_{bf}{\varphi }_s}{{\sigma }_{GO}\left(1-{\varphi }_{GO}\right)+{\sigma }_{Mo{S}_2}\left(1-{\varphi }_{Mo{S}_2}\right)+{\sigma }_{bf}\left(2+{\varphi }_s\right)}}\right]$
	where ${\varphi }_s={\varphi }_{GO}+{\varphi }_{Mo{S}_2}$

2.2. Governing equations

Assuming the stated conditions, the flow model for hybrid nanofluid is [29] (1) $\begin{array}{rl}{u}_x+v_y+w_z& =0,\end{array}$(1) (2) $\begin{array}{rl}u{u}_x+v{u}_y+w{u}_z& ={\upsilon }_{hnf}{u}_{zz}+\frac{g{\left(\rho \text{β}\right)}_{hnf}}{{\rho }_{hnf}}\left(T-T_{\mathrm{\infty }}\right)\\ & -\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^{2}u-\frac{{\upsilon }_{hnf}}{k}u,\end{array}$(2) (3) $\begin{array}{rl}u{v}_x+v{v}_y+w{v}_z& ={\upsilon }_{hnf}{v}_{zz}+\frac{g{\left(\rho \text{β}\right)}_{hnf}}{{\rho }_{hnf}}\left(T-T_{\mathrm{\infty }}\right)\\ & -\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^{2}v-\frac{{\upsilon }_{hnf}}{k}v,\end{array}$(3) (4) $\begin{array}{rl}u{T}_x+v{T}_y+w{T}_z& =\frac{{\alpha }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}{T}_{zz}+\frac{Q_0}{{\left(\rho C_p\right)}_{hnf}}\left(T-T_{\mathrm{\infty }}\right)\\ & +\frac{{\sigma }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}{B}_0^2\left(u^2+v^2\right),\end{array}$(4) (5) $\begin{array}{rl}u{C}_x+v{C}_y+w{C}_z& =D_B{C}_{zz}-k_0(C-C_{\mathrm{\infty }}{)}^m.\end{array}$(5)

The conditions at the surface of the plate $\left(z=0\right)$ and far away from the plate $\left(z\to \mathrm{\infty }\right)$ are (6) $\begin{array}{l}\text{at}z=0\\ \left(u,v,w,T,C\right)=(ax+\ell u_z,by+\ell v_z,-W,T_w\\ +n{T}_z,C_w+p{C}_z)\\ \text{at}z\to \mathrm{\infty }\\ \left(u,v,T,C\right)=\left(0,0,T_{\mathrm{\infty }},C_{\mathrm{\infty }}\right).\end{array}\}$(6) Here, the symbols $\upsilon$, $g$, $\rho$, $\text{β}$, $\sigma$, $k$, $\alpha$, $C_P$, $Q_o$, $D_B$, $k_o$, $m$ and $\left(\ell ,n,p\right)$ 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. $\ell =n=p=0$, then the problem is turned into no-slip condition.

The dimensionless system of existing equations can be attained by introducing the following transformations [29]: (7) $\begin{array}{l}\eta =z(a/{\upsilon }_f{)}^{0.5},u=ax{f}^{\mathrm{\prime }}\left(\eta \right),v=ay{g}^{\mathrm{\prime }}\left(\eta \right),\\ w=-(a{\upsilon }_f{)}^{0.5}\left[f\left(\eta \right)+g\left(\eta \right)\right],\\ T=T_{\mathrm{\infty }}+\theta \left(\eta \right)\left(T_w-T_{\mathrm{\infty }}\right),\\ C=C_{\mathrm{\infty }}+\xi \left(\eta \right)\left(C_w-C_{\mathrm{\infty }}\right).\end{array}\}$(7) 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: (8) $\begin{array}{rl}& \frac{a_1}{a_2}{f}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+(f+g)f^{\mathrm{\prime }\mathrm{\prime }}-f{}^{\mathrm{\prime }2}+\frac{a_3}{a_2}Gr\theta \\ & -\left(K\frac{a_1}{a_2}+Mn\frac{a_4}{a_2}\right)f^{\mathrm{\prime }}=0,\end{array}$(8) (9) $\begin{array}{rl}& \frac{a_1}{a_2}{g}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+(f+g)g^{\mathrm{\prime }\mathrm{\prime }}-g{}^{\mathrm{\prime }2}+\frac{a_3}{a_2}Gr\theta \\ & -\left(K\frac{a_1}{a_2}+Mn\frac{a_4}{a_2}\right)g^{\mathrm{\prime }}=0,\end{array}$(9) (10) $\begin{array}{rl}& \frac{a_5}{a_6}{\theta }^{\mathrm{\prime }\mathrm{\prime }}\frac{1}{Pr}+(f+g){\theta }^{\mathrm{\prime }}+\frac{a_4}{a_6}MnEc(f{}^{\mathrm{\prime }2}+g{}^{\mathrm{\prime }2})+\frac{Q}{a_6}\theta =0,\end{array}$(10) (11) $\begin{array}{rl}& {\xi }^{\mathrm{\prime }\mathrm{\prime }}+(f+g)Sc{\xi }^{\mathrm{\prime }}-ScL{\xi }^m=0,\end{array}$(11)

with (12) $\begin{array}{r}\begin{array}{l}\text{at}\eta =0\\ \begin{array}{l}f\left(\eta \right)+g\left(\eta \right)=F,f^{\mathrm{\prime }}\left(\eta \right)=1+\lambda f^{\mathrm{\prime }\mathrm{\prime }}\left(\eta \right),g\left(\eta \right)=0,\\ g^{\mathrm{\prime }}\left(\eta \right)=\alpha +\lambda g^{\mathrm{\prime }\mathrm{\prime }}\left(\eta \right),\\ \theta \left(\eta \right)=1+St{\theta }^{\mathrm{\prime }}\left(\eta \right),\xi \left(\eta \right)=1+\delta {\xi }^{\mathrm{\prime }}\left(\eta \right)\end{array}\}\\ \text{as}\eta \to \mathrm{\infty }\\ f^{\mathrm{\prime }}\left(\eta \right)=g^{\mathrm{\prime }}\left(\eta \right)=\theta \left(\eta \right)=\xi \left(\eta \right)=0\}\end{array}\},\end{array}$(12) where $Gr=\frac{{\left(\rho \text{β}\right)}_{bf}g(T_w-T_{\mathrm{\infty }})}{a^{2}x{\rho }_{bf}}$ is the Grashof number, $K=\frac{{\upsilon }_{bf}}{ak}$ is the porosity parameter, $Ec=\frac{a^2{x}^2{\mu }_{bf}}{{\left(\rho C_p\right)}_{bf}{\upsilon }_{bf}(T_w-T_{\mathrm{\infty }})}$ is the Eckert number, $Mn=\frac{{\sigma }_{bf}{B}_0^2}{a{\rho }_{bf}}$ is the magnetic parameter, $Q=\frac{Q_0}{a{\left(\rho C_p\right)}_{bf}}$ is the heat generation coefficient, $Pr=\frac{{\left(\rho Cp\right)}_{bf}{\upsilon }_{bf}}{{\kappa }_{bf}}$ is the Prandtl number, $Sc=\frac{{\upsilon }_{bf}}{D_B}$ is the Schmidt number, $L=\frac{k_0{(C_w-C_{\mathrm{\infty }})}^{q-1}}{a}$ is the CR parameter, $F=\frac{W}{\sqrt{a{\upsilon }_{bf}}}$ is the suction or injection parameter as $\left(F>0,F<0\right)$, $\alpha =b{a}^{-1}$ is the stretching factor, $\left(\lambda ,St,\delta \right)$ are slip (velocity, thermal, concentration) parameters, i.e. $\lambda =\ell \sqrt{\frac{a}{{\upsilon }_{bf}}}$, $St=n\sqrt{\frac{a}{{\upsilon }_{bf}}}$, $\delta =p\sqrt{\frac{a}{{\upsilon }_{bf}}}$ and $(a_1=\frac{{\mu }_{hnf}}{{\mu }_{bf}},a_2=\frac{{\rho }_{hnf}}{{\rho }_{bf}},a_3=\frac{{\left(\rho \text{β}\right)}_{hnf}}{{\left(\rho \text{β}\right)}_{bf}},a_4=\frac{{\sigma }_{hnf}}{{\sigma }_{bf}},a_5=\frac{{\kappa }_{hnf}}{{\kappa }_{bf}},a_6=\frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_{bf}})$ are constants.

2.3. Quantities of engineering and physical importance

1. The skin-friction coefficients are well-defined as (13) $\left(C_{fx}=\frac{{\tau }_{wx}}{{\rho }_f{U}^2},C_{fy}=\frac{{\tau }_{wy}}{{\rho }_f{V}^2}\right),$(13) where ${\tau }_{wx}$ and ${\tau }_{wy}$ are wall shear stresses in x and y directions and are expressed as (14) $\begin{array}{r}\left({\tau }_{wx}={\mu }_{hnf}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }z}\right)}_{z=0},{\tau }_{wy}={\mu }_{hnf}{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }z}\right)}_{z=0}\right).\end{array}$(14)

   Substituting Equation (13) in Equation (12) and using Equation (7), we obtained the following dimension-free form of skin-friction coefficients (15) $\begin{array}{rl}\sqrt{R{e}_x}{C}_{fx}& =(a_1{a}_2{)}^{-1}{\left(\frac{d^{2}f}{d{\eta }^2}\right)}_{\eta =0}\text{and}\\ \sqrt{R{e}_y}{C}_{fy}& =\frac{{\alpha }^{-3/2}}{a_1{a}_2}{\left(\frac{d^{2}g}{d{\eta }^2}\right)}_{\eta =0}.\end{array}$(15)

2. The $Nu$ “Nusselt number” and $Sh$ “Sherwood number” are expressed as (16) $\begin{array}{rl}& (Nu=\frac{-{\kappa }_{hnf}x}{{\kappa }_{bf}\left(T_w-T_{\mathrm{\infty }}\right)}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right)}_{z=0},\\ & Sh=\frac{-x}{\left(T_w-T_{\mathrm{\infty }}\right)}{\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }z}\right)}_{z=0}).\end{array}$(16)

Again using Equation (7), the expression for $Nu$ and $Sh$ given in Equation (16) is reduced to the following form: (17) $\begin{array}{rl}& (R{e}_x^{-1/2}Nu=\frac{-{\kappa }_{hnf}}{{\kappa }_{bf}}{\left(\frac{d\theta }{d\eta }\right)}_{\eta =0},\\ & R{e}_x^{-1/2}Sh=-{\left(\frac{d\xi }{d\eta }\right)}_{\eta =0}),\end{array}$(17) where $\left(R{e}_x=x\sqrt{\frac{a}{{\upsilon }_{bf}}},R{e}_y=y\sqrt{\frac{a}{{\upsilon }_{bf}}}\right)$ are termed as local Reynolds number.

3. Methods of solution

3.1. 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.

5. Calculating the residue for far-field conditions and repeat step (4), if residue > error tolerance.

6. If residue < error tolerance, then the solution is obtained $\begin{array}{rl}& \text{max}\{|f^{\mathrm{\prime }}\left({\eta }_{max}\right)-0|,|g^{\mathrm{\prime }}\left({\eta }_{max}\right)-0|,|\theta \left({\eta }_{max}\right)-0|,\\ & |\xi \left({\eta }_{max}\right)-|0\}\le 0.00001(\text{Error}\text{tolerance})\end{array}$

In this study, all the computations are carried out for ${\eta }_{max}=6$ with step size $\mathrm{\Delta }\eta =0.001$ and tolerance limits of 10⁻⁵. Also, for ${\eta }_{max}>6$ 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.

3.2. 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.

4. Code of validation

To validate the exactness of our code, values of $-\sqrt{R{e}_x}{C}_{fx}$ and $-\sqrt{R{e}_y}{C}_{fy}$ are calculated for the distinct value of $\alpha$ 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.

Table 3. Comparison of numerical values of surface drag forces for $Mn$ and $\alpha$ when $\phi =0,Gr=0$ and $Ec=F=Q=0$.

		$-\sqrt{R{e}_x}{C}_{fx}$		$-\sqrt{R{e}_y}{C}_{fy}$
		Hayat	Present	Hayat	Present
$\alpha$	$Mn$	et al. [10]	outcomes	et al. [10]	outcomes
0.2	0.0	1.1815	1.181495	2.6420	2.641982
	0.3	1.2192	1.219195	2.7261	2.726087
	0.5	1.2833	1.283287	2.8695	2.869498
0.1	0.1	1.1357	1.135687	3.5913	3.591389
0.3		1.2339	1.233898	2.2527	2.252698
0.5		1.3248	1.324788	1.8735	1.873498

5. Results and discussion

This portion of the research declared the influences of various active parameters, i.e. magnetic field $\left(Mn\right)$, Grashof number $\left(Gr\right)$, porosity $\left(K\right)$, slip variables ($\lambda$, $St$, $\delta$), suction $\left(F>0\right)$ or injection $\left(F<0\right)$, Schmidt number $\left(Sc\right)$, order of CR $\left(m\right)$ and CR parameter $\left(L\right)$ on physical variables, i.e. velocities $\left(f^{\mathrm{\prime }}\left(\eta \right),g^{\mathrm{\prime }}\left(\eta \right)\right)$, temperature $\left(\theta \left(\eta \right)\right)$ and concentration $\left(\xi \left(\eta \right)\right)$ fields are illustrated through graphs. Furthermore, the behaviour of varying $Mn,Ec\text{and}K$ 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 MoS₂ and GO as 3% each. To have a better understanding the response of active parameters is categorized in the following section:

Figure 2. Concentration outlines for $Sc$.

Figure 3. Concentration outlines for $L$.

Figure 4. Concentration outlines for $Mn$.

Figure 5. Concentration outlines for $K$.

Figure 6. Concentration outlines for $Gr$.

Figure 7. Concentration outlines for $F$.

Figure 8. Concentration outlines for $\delta$.

Figure 9. Concentration outlines due to $\lambda$.

Figure 10. Concentration outlines for $m$.

Figure 11. Temperature outlines for $Mn$.

Figure 12. Temperature outlines for $F$.

Figure 13. Temperature outlines for $St$.

Figure 14. Temperature outlines due to $\lambda$.

Figure 15. Velocity outlines along x-axis for $Mn$.

Figure 16. Velocity outlines along x-axis due to $K$.

Figure 17. Velocity outlines along x-axis for $Gr$.

Figure 18. Velocity outlines along x-axis due to $\lambda$.

Figure 19. Velocity outlines along x-axis for $F$.

Figure 20. Velocity outlines along y-axis for $Mn$.

Figure 21. Velocity outlines along y-axis for $K$.

Figure 22. Velocity outlines along y-axis for $Gr$.

Figure 23. Velocity outlines along y-axis due to $\lambda$.

Figure 24. Velocity outlines along y-axis for $F$.

Figure 25. $\sqrt{R{e}_x}{C}_{fx}$ outlines due to $Mn$ and $Ec$.

Figure 26. $\sqrt{R{e}_y}{C}_{fy}$ outlines due to $Mn$ and $Ec$.

Figure 27. $R{e}_x^{-1/2}Nu$ outlines due to $Mn$ and $Ec$.

Figure 28. $R{e}_x^{-1/2}Sh$ outlines due to $Mn$ and $Ec$.

1. 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. $Sc=\frac{{\upsilon }_{bf}}{D_B}$; physically it relates the thickness of momentum and concentration boundary layer (BL). Thus a higher value of $Sc\left(>1\right)$ 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 $0.1\text{to}0.5$ 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 $\left(Gr\right)$. 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 $\left(\delta \right)$ and velocity $\left(\lambda \right)$ slips, respectively. Here, the concentration along with the width of associated BL decreases continuously with $\delta$ (see Figure 8) while it escalates with $\lambda$ (see Figure 9). Figure 10 displays how the order of CR $\left(m\right)$ affects the concentration profile $\xi \left(\eta \right)$, here $\xi \left(\eta \right)$ 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 the concentration profiles of $m=2,3,4,5$ is very less.

2. Temperature profile: Dissimilarity of temperature curves and the associated BL for active parameters $Mn,F,St\text{and}\lambda$ are displayed in Figures 11–14. It could be anticipated from Figures 15 and 20 that increasing the magnetic field lessens the flow rate 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 $\theta \left(\eta \right)$ correspond to negative $F$ is positive $\left(>0\right)$ for $\eta <0.25$ 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 $St\text{and}\lambda$, correspondingly. It is examined from these data that the temperature of fluid decreases (increases) for thermal (velocity) slip parameters.

3. Velocity profile: Figures 15–24 inspect the aftermath of $Mn,K,Gr,\lambda \text{and}F$ on the distribution of nondimensional velocity fields along $x$- and $y$- directions, i.e. $f^{\mathrm{\prime }}\left(\eta \right)\text{and}{g}^{\mathrm{\prime }}\left(\eta \right)$. Figures 15 and 20 present the velocity distribution for hybrid nano-fluid along both axes for varying $Mn$. It can be visualized from the plot that enhancing the strength 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 $f^{\mathrm{\prime }}\left(\eta \right)\text{and}{g}^{\mathrm{\prime }}\left(\eta \right)$ 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 $f^{\mathrm{\prime }}\left(\eta \right)$, and asymptotically satisfies the conditions $f^{\mathrm{\prime }}\left(\eta \to \mathrm{\infty }\right)=0$. Besides, the velocity BL becomes narrower with cumulative porosity parameters. The effect of Grashof number on $f^{\mathrm{\prime }}\left(\eta \right)\text{and}{g}^{\mathrm{\prime }}\left(\eta \right)$ 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 $\lambda$ 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.

4. The surface drag forces, Nusselt number and Sherwood number analysis: Impact of variation in the magnitude of magnetic field against Eckert number is outlined for $\left(\sqrt{R{e}_x}{C}_{fx},\sqrt{R{e}_y}{C}_{fy}\right)$, $R{e}_x^{-1/2}Nu$ and $R{e}_x^{-1/2}Sh$ in Figures 25–28, respectively. Physically, increasing the value magnetic field parameter creates the Lorentz force which resists the motion of a fluid (see Figures 15 and 16), as a result, the surface drag force along x- and y-axes increases as shown in Figures 25 and 26. Owing to this fact, the profiles of $R{e}_x^{-1/2}Nu$ and $R{e}_x^{-1/2}Sh$ increase with growing $Mn$ see Figures 27 and 28. Since increasing the Eckert number increases the energy of the flowing fluid by increasing the dissipative force working amidst the fluid layers. Therefore, profiles of $R{e}_x^{-1/2}Nu$ increase see Figure 27.

The surface plots of $\sqrt{R{e}_x}{C}_{fx},\sqrt{R{e}_y}{C}_{fy}$, $R{e}_x^{-1/2}Nu$ and $R{e}_x^{-1/2}Sh$ with regard to active parameters $Mn$ and $K$ are shown in Figures 29–32. In, Figures 29 and 30, the influence of change in the values of parameters such as $Mn$ and $K$ on surface drag forces is analysed. When there was augmentation in values of $Mn$ and $K$ the modulus of drag force coefficients increased, this is happen because of velocity functions of MoS₂–GO/H₂O hybrid nanofluid are frequently dropped with a rise in the values of $Mn$ and $K$. In the last of this section, Figures 31 and 32 reveal the contour plots of $R{e}_x^{-1/2}Nu$ and $R{e}_x^{-1/2}Sh$ due to the impact of parameters $Mn$ and $K$. It is very clear that both HMT rates raise due to a rise in $Mn$ and $K$. Since enhancing the strength of $Mn$ and $K$ both thermal and concentration fields of MoS₂–GO/H₂O hybrid nanofluid augmented.

Figure 29. $\sqrt{R{e}_x}{C}_{fx}$ versus $Mn\text{and}K$.

Figure 30. $\sqrt{R{e}_y}{C}_{fy}$ versus $Mn\text{and}K$.

Figure 31. $R{e}_x^{-1/2}Nu$ versus $Mn\text{and}K$.

Figure 32. $R{e}_x^{-1/2}Sh$ versus $Mn\text{and}K$.

6. Closing remarks

The magnetized flow of MoS₂–GO/H₂O fluid through a porous surface under the condition of multiple slip mechanisms (velocity, thermal and solutal), mth-order CR, suction/blowing and natural convection is presented in this article. The flow model equations are solved by the techniques, i.e. bvp4c and shooting. For the validation of our code, the comparison is made with Hayat et al. [10] for the values $-\sqrt{R{e}_x}{C}_{fx}$ and $-\sqrt{R{e}_y}{C}_{fy}$. The main summary of this research includes the following points:

• The heat and mass transfer performance of MoS₂-GO/H₂O fluid is enhanced with an increase in a magnetic field $Mn$ and Eckert number $Ec$.

• The velocities, thermal and concentration distribution outlines of MoS₂–GO/H₂O are depreciated when the working fluid moves from injection to suction region.

• When an increment in the values of $Gr$, $Sc$ and $\delta$, concentration curves of working fluid decrease.

• The MoS₂–GO/H₂O fluid concentration outlinesdepreciated due to a rise in the values of $m$.

• Mass transfer functions increase with parameters such as magnetic field $Mn$ and porosity parameter $K$.

• On increasing the strength of magnetic field, thermal field is enhanced while both x- and y-directions velocity is reduced.

• The velocity functions are reduced with a parameter $K$.

• The velocity functions of MoS₂–GO/H₂O are augmented with an increase in Grashof $Gr$ values.

• With the increase in thermal slip $St$ values, only the thermal field profiles of MoS₂–GO/H₂O regularly depreciated.

• On increasing velocity slip $\lambda$ values, both velocity outlines of MoS₂–GO/H₂O fluid declined while $\theta \left(\eta \right)$ and $\xi \left(\eta \right)$ outlines increased.

Disclosure statement

No potential conflict of interest was reported by the authors.