Analysis of MHD tangent hyperbolic hybrid nanofluid flow with different base fluids over a porous stretched sheet

Abstract

This study investigates the flow of a two-dimensional magnetohydrodynamic(MHD) tangential hyperbolic hybrid nano fluid flow towards a porous stretched sheet with an inclined magnetic field under velocity and thermal slip conditions. Two distinct base fluids, (i) water and (ii) a mixture of $50\mathrm{\%}$ ethylene glycol($\mathrm{E}\mathrm{G}\mathrm{o}\mathrm{r}({\mathrm{C}\mathrm{H}}_2\mathrm{O}\mathrm{H}{)}_2$) and $50\mathrm{\%}$ water are employed as base liquids with Copper($\mathrm{C}\mathrm{u}$) and aluminium oxide(${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$) nanoparticles, to generate two different hybrid nanofluids as ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$. Scaling group transformations with a homotopy algorithm is used to study the effects of distinct physical parameters like the Weissenberg number, an inclined magnetic field, thermal radiation, velocity, and thermal slip conditions with a heat generation/absorption parameter and a permeable parameter on the velocity and temperature distributions. The outcomes of ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$ hybrid nano fluid are compared with ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ hybrid nano fluid in significant critical physical circumstances using graphs and tables. It is observed that the parameters $W_e$ and $H_a$, ${\theta }_M$, and λ increase the fluid temperature and decelerate the motion of fluid. The obtained results are validated by comparing them with the existing literature and found that the present results are consistent with the given data. Additionally, it is also noticed that the ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ hybrid nanofluid has a higher heat transmission rate as compared to ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$ hybrid nanofluid. Finally, the impact of nano particle volume fractions of copper and aluminium oxide on the velocity and temperature profiles is discussed to study their efficiency.

Keywords:

• Tangent hyperbolic hybrid nanofluid flow
• slip conditions
• porous medium
• stretchable sheet
• HAM approach

1. Introduction

Nanofluids are colloidal suspensions containing nanometer-sized particles in a base fluid. They have gained considerable interest recently due to their potential to enhance heat transfer in various engineering applications. By increasing thermal conductivity and improving stability and rheological properties, nanofluids can potentially boost the efficiency of systems in fields such as transportation, microelectronics, solar thermal, and more. The progress made in this field is promising, and nanofluids may represent an essential advancement in thermal engineering. The enhancement in heat transfer is due to the contributions made by a variety of parameters, some of which are the mass flow rate of the fluid, the volume fraction, and the thermal conductivity of the nano-particles themselves. The thermal conductivity of nanoparticles is extremely important because it influences how efficiently heat is transported from a hot surface to the nanoparticles and from the nanoparticles to the fluid in the surrounding area. The efficiency of heat transfer increases as the thermal conductivity of the nanoparticles increases. The mass flow rate of the fluid is also important because it determines how quickly the fluid can transfer heat away from the hot surface. The performance of the nanofluid as a heat transmission medium is also dependent on the nanoparticle volume fraction. Choi and Eastman first introduced the concept of using nanoparticles to enhance the thermal properties of fluids in 1995 [1]. A fluid containing nanoparticles is referred to as a nanofluid. Typically, the nanoparticles are made of materials like alumina(${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$), copper($\mathrm{C}\mathrm{u}$), silver($\mathrm{A}\mathrm{g}$), etc. There are many liquids such as water(${\mathrm{H}}_2\mathrm{O}$), oil, kerosene, or ethylene glycol($\mathrm{E}\mathrm{G}$ or $({\mathrm{C}\mathrm{H}}_2\mathrm{O}\mathrm{H}{)}_2$) are all acceptable base fluids. Nanofluids are heat transfer fluids. They are a form of high convective element composed of a base fluid and tiny-sized nanoparticles with high thermal conductivity. Recently, most of the research has been conducted regarding the nanofluid, which enhances heat transmission [2,3].

Hybrid nanofluids (HNFs), which are composed of multiple types of nanoparticles dispersed in a base fluid, have been getting more attention in recent years because they could help nanofluids conduct heat even better. Previous studies [4–7] have provided evidence to support the notion that hybrid nanofluids exhibit significantly higher thermal conductivity in comparison to their individual nanoparticle counterparts. This is due to the synergistic effect between the different types of nanoparticles, which can lead to improved heat transfer properties. As a result, hybrid nanofluids have become a popular choice as working fluids in convection devices such as heat exchangers, radiators, and cooling systems. An interesting research area is the computational analysis of magnetohydrodynamics(MHD) hybrid flow above a stretched sheet controlled by heat radiation [8–10]. MHD studies the dynamics of fluids interacting with magnetic fields while possessing non-negligible electrical conductivity. Applications for MHD include fusion research, MHD accelerators, and power generators. A fluid that conducts electricity in a consistent magnetic field was a concept first introduced by Hartmann [11]. Processing magnetic equipment, regulating heat transfer, cleaning crude oil of pollutants, Hall generators, and other applications are among the different uses for MHD viscous incompressible electrically conducting fluid flow. Such fluxes give rise to a perpendicular magnetic Lorentzian body force to the direction of the applied uniform magnetic field. This force helps regulate energy flow, reduce oscillations, and quell high-temperature plasmas. Much of the early research on nanofluids examined what happened when different types of nanoparticles were mixed with different base fluids and how effectively they transferred heat, such as the study of heat transfer on Jeffery-Hamel nanofluid containing $\mathrm{C}\mathrm{u}$ and $\mathrm{A}\mathrm{g}$ with base fluid as water [12], $\mathrm{C}\mathrm{u}$ and ferrosoferric oxide(${\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$) nanoparticles mixed with $\mathrm{E}\mathrm{G}$ [8], $\mathrm{C}\mathrm{u}$ and ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nanoparticles mixed with water [9], $\mathrm{C}\mathrm{u}$ and $\mathrm{A}\mathrm{g}$ nanoparticles with mixed water and $\mathrm{E}\mathrm{G}$ as the base fluid [10], magnesium oxide($\mathrm{M}\mathrm{g}\mathrm{O}$) and $\mathrm{A}\mathrm{g}$ nanoparticles mixed with water as the base fluid [13], etc.

Researchers have taken a significant interest in studying non-Newtonian type fluids due to their prevalence in nature and the wide range of fields in which they are used. These fluids are frequently found as slurries and waste and are critical for mining, biological treatments, and lubricating activities. Similarly, the rheological behaviours vary depending on the fluid substances. Numerous rheological models have been proposed as a result of the variety of rheological behaviours of non-Newtonian fluids, such as the Casson model [14], tangent hyperbolic fluid(THF) model [15], Williamson fluid model [16] with Carreau model [17] and are frequently used for shear rate-dependent viscosity. It is very interesting and important to study how tangent hyperbolic hybrid nanofluid(TH-HNF) flow and transfer heat over-stretching surfaces in fluid mechanics [18,19]. THFs are extensively utilized in high-speed engineering applications such as modelling flow in supersonic wind tunnels, predicting rocket engine performance, and building high-speed aircraft [20]. It is also used in biomedical engineering to estimate the risk of cardiovascular disease by modelling blood flow in arteries and veins [21,22]. Recently, the TH-HNF flow combining $\mathrm{C}\mathrm{u}$ and silicon dioxide(${\mathrm{S}\mathrm{i}\mathrm{O}}_2$) with $\mathrm{E}\mathrm{G}$ was examined [23]. In addition, the influence of a uniform magnetic field on the flow of an $\mathrm{E}\mathrm{G}$-water-based composite nanofluid, including graphene oxide(GO) and zirconium dioxide(${\mathrm{Z}\mathrm{r}\mathrm{O}}_2$), over an extended area has been investigated in [24]. The heat transfer in the MHD TH-HNF flow, including $\mathrm{G}\mathrm{O}$ and ${\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_3$ nanoparticles with water as a base fluid, have been investigated in [25]. Recently, the effect of iron-oxide nanoparticles with base fluid as a mixer of water and $\mathrm{E}\mathrm{G}$ has been studied [26]. The kerosene-based nanofluid with the nanoparticles ${\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_3$ and titanium dioxide(${\mathrm{T}\mathrm{i}\mathrm{O}}_2$) are of a spherical shape taken to observe the electroosmotic flow of immiscible fluids through a porous media [27].

Based on all this literature, most researchers studied the comparison of nanofluid with hybrid nanofluid. Still, the significance of base fluid's effect in a hybrid nanofluid is yet to be studied. In this study, two different base fluids, including water and a mixture of water and $\mathrm{E}\mathrm{G}$, are considered for the making of two distinct HNFs containing $\mathrm{C}\mathrm{u}$ and ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nanoparticles. The water's boiling point and freezing point are 100 degrees Celsius and 0 degrees Celsius, respectively. As we mix $50\mathrm{\%}$ $\mathrm{E}\mathrm{G}$ and $50\mathrm{\%}$ water, that mixture's boiling point and freezing point are maintained at 106–107 degrees Celsius and −37 to −40 degrees Celsius, respectively. So, to prevent freezing in cold weather and overheating in heat, the antifreeze solutions used in automotive and industrial applications can benefit significantly by using this mixture.

The proposed work examines the two-dimensional TH-HNF flow towards a porous stretchable sheet with an inclined uniform magnetic field under the velocity and thermal slip conditions with distinct base fluids. Copper and aluminium oxide are employed as solid nanoparticles. At the same time, water and a mixture of water(50$\mathrm{\%}$) and $\mathrm{E}\mathrm{G}\left(50\mathrm{\%}\right)$ are used as the base fluids. While the velocity and thermal slip conditions are implemented in the momentum and energy equations, respectively. Scaling group transformations [28,29] with a homotopy analysis method(HAM) [30–34] are used to study the impact of distinct physical parameters such as Weissenberg number, inclined magnetic field, thermal radiation, velocity and thermal slip conditions with heat absorption/generation parameter and permeable parameter on the velocity and temperature distributions. Finally, the significance of distinct base fluid with the hybrid nanofluid flow behaviour has been discussed concerning various physical parameters with the effects of slippery conditions on the velocity and temperature profiles of the hybrid nanofluid flow. The stretching surface phenomenon also has applications in the research of boundary layer flows, which are crucial for many engineering applications, including the construction of wind turbines and airplane wings. The TH-HNF flow past a stretching sheet has several potential applications in different fields, such as polymer and textile manufacturing, solar energy, cooling systems, and biomedicine.

2. Formulation of mathematical model

This work considers a steady and 2-D incompressible TH-HNF flow moving towards a stretching sheet. In addition, it is believed that the linear stretching creates the fluid flow, which coincides with the surface at y = 0 and is covered by the domain y>0. Considering these presumptions, the following additional stress tensor is created for the model under consideration [35–37]: (1) $\overline{\tau }=[{\delta }_{\mathrm{\infty }}+({\delta }_0+{\delta }_{\mathrm{\infty }})\tanh (\mathrm{\Lambda }\overline{{\mathrm{{\rm Y}}}^{\cdot }}{)}^n]\overline{{\mathrm{{\rm Y}}}^{\cdot }},$(1) where $\overline{{\mathrm{{\rm Y}}}^{\cdot }}$ is given by (2) $\overline{{\mathrm{{\rm Y}}}^{\cdot }}=\sqrt{\frac{1}{2}\sum _i\sum _j\overline{{\mathrm{{\rm Y}}}_{ij}^{\cdot }}\overline{{\mathrm{{\rm Y}}}_{ji}^{\cdot }}}=\sqrt{\frac{\mathrm{\Pi }}{2}}.$(2) In the above Equation (1(1) $\overline{\tau }=[{\delta }_{\mathrm{\infty }}+({\delta }_0+{\delta }_{\mathrm{\infty }})\tanh (\mathrm{\Lambda }\overline{{\mathrm{{\rm Y}}}^{\cdot }}{)}^n]\overline{{\mathrm{{\rm Y}}}^{\cdot }},$(1) ), $\overline{\tau }$ represents the additional stress tensor, ${\delta }_0$ and ${\delta }_{\mathrm{\infty }}$ are the zero shear rate and infinite shear rate viscosity, and Λ is the material constant(time-dependent). Here n stands for the power law index. The term Π in Equation (2(2) $\overline{{\mathrm{{\rm Y}}}^{\cdot }}=\sqrt{\frac{1}{2}\sum _i\sum _j\overline{{\mathrm{{\rm Y}}}_{ij}^{\cdot }}\overline{{\mathrm{{\rm Y}}}_{ji}^{\cdot }}}=\sqrt{\frac{\mathrm{\Pi }}{2}}.$(2) ) is given by $\mathrm{\Pi }=\frac{1}{2}tr\left(grad\right(V)+(grad\left(V\right){)}^T{)}^2$. We can neglect the ${\delta }_{\mathrm{\infty }}$ because we cannot discuss the phenomena for infinite shear viscosity. In this phenomenon, the flow of fluid is considered as tangential hyperbolic, which forms the thinning stress effects so $\mathrm{\Lambda }\overline{{\mathrm{{\rm Y}}}^{\cdot }}<1$. Therefore, Equation (1(1) $\overline{\tau }=[{\delta }_{\mathrm{\infty }}+({\delta }_0+{\delta }_{\mathrm{\infty }})\tanh (\mathrm{\Lambda }\overline{{\mathrm{{\rm Y}}}^{\cdot }}{)}^n]\overline{{\mathrm{{\rm Y}}}^{\cdot }},$(1) ) converts into the following form : (3) $\overline{\tau }={\delta }_0[1+n(\mathrm{\Lambda }\overline{{\mathrm{{\rm Y}}}^{\cdot }}-1\left)\right]\overline{{\mathrm{{\rm Y}}}^{\cdot }}.$(3) Where the y-axis is perpendicular to the stretched surface, the direction of the motion is measured along it in the x-axis. The sheet is stretched at a velocity ($U_w$) that varies directly with both the surface temperature ($T_w$) and the distance from the slit while keeping the origin fixed. It is assumed that the mass flux of the nano-sized particles at the wall is zero to avoid the effects of gravity. Now, a magnetic field of intensity $B_0$ is applied uniformly with an angle of inclination ${\theta }_M$ to the flow direction.

The main governing equations [21,23] for a laminar, steady, and incompressible model that considers the influence of an inclined magnetic field, a porous media, a thermal radiation, and a heat absorption or generation sources are given as follows and the geometric interpretation of the proposed model is shown in Figure 1: (4) $\begin{array}{rl}& \frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }y}=0,\end{array}$(4) (5) $\begin{array}{rl}& u_1\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}+u_2\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y}\\ & =(1-n){\nu }_{hnf}\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{y}^2}+n\sqrt{2}\mathrm{\Lambda }{\nu }_{hnf}\left(\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y}\right)\left(\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{y}^2}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}}{u}_1-\frac{{\nu }_{hnf}}{K_p}{u}_1,\end{array}$(5) (6) $\begin{array}{rl}& u_1\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+u_2\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\\ & ={\alpha }_{hnf}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+\frac{Q_0}{(\rho C_p{)}_{hnf}}(T-T_{\mathrm{\infty }})-\frac{1}{(\rho C_p{)}_{hnf}}\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y},\end{array}$(6) and the pertinent boundary conditions(BCs) given as follows [37]: (7) $\begin{array}{rl}& u_1(x,y=0)=U_w+L\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y},u_2(x,y=0)=0,\\ & T(x,y=0)=T_w+N\frac{\mathrm{\partial }T}{\mathrm{\partial }y},\end{array}$(7) (8) $\begin{array}{rl}& u_1(x,y)\to 0,T(x,y)\to T_{\mathrm{\infty }}\mathrm{a}\mathrm{s}y\to \mathrm{\infty },\end{array}$(8) where ${\nu }_{hnf}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\mathrm{a}\mathrm{n}\mathrm{d}{\alpha }_{hnf}=\frac{k_{hnf}}{(\rho C_p{)}_{hnf}}.$The components of the velocity is defined by $u_1$ and $u_2$ in the horizontal axis x and the vertical axis y directions, respectively, while ${\nu }_{hnf}$ is an effective kinematic viscosity of the HNF, ${\mu }_{hnf}$ represents an effective dynamic viscosity of the HNF, ${\rho }_{hnf}$ is an effective density of the HNF, Λ is the time-dependent material, $B_0$ is the magnetic field of uniform strength, ${\theta }_M$ is the angle of inclined magnetic field, ${\sigma }_{hnf}$ is an effective electrical conductivity of the HNF, $K_p$ indicates the permeability of the porous media, T is the temperature of the fluid, $(\rho C_p{)}_{hnf}$ is an effective heat capacity of the HNF, ${\alpha }_{hnf}$ is the thermal diffusivity of the HNF, $k_{hnf}$ is the thermal conductivity of the HNF, $Q_0$ is the heat generation/absorption source, $U_w=bx$ is the fluid velocity at surface, L and N are the velocity and the thermal slip parameters, $T_{\mathrm{\infty }}$ is the free stream temperature of the fluid and $T_w$ is the fluid temperature at the surface. Here the thermo-physical properties of base fluids and nano-particles given in Table 1. Here the term $q_r$ is known as Rosseland radiative heat flux which is defined as follows: (9) $q_r=-\frac{4}{3}\frac{{\sigma }^{\ast }}{k^{\ast }}\frac{\mathrm{\partial }{T}^4}{\mathrm{\partial }y},$(9) where ${\sigma }^{\ast }$ stands for Stefan–Boltzmann's constant and $k^{\ast }$ is the coefficient of the mean absorption. Now by using Taylor's series for $T^4$ centred about $T_{\mathrm{\infty }}$ and ignoring the higher order terms, one can get (10) $T^4\simeq T_{\mathrm{\infty }}^4+4T_{\mathrm{\infty }}^3(T-T_{\mathrm{\infty }}).$(10) Therefore, $q_r$ is the radiative heat flux and it can be calculated as: (11) $q_r=-\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{\mathrm{\partial }T}{\mathrm{\partial }y}.$(11)

Figure 1. Stretching sheet fluid flow model.

Table 1. Thermo-physical attributes of base fluids and nano-particles [24,38,39].

Physical properties	Units	$\mathrm{C}\mathrm{u}$	${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$	${\mathrm{H}}_2\mathrm{O}$	$\mathrm{E}\mathrm{G}\left(50\mathrm{\%}\right)+{\mathrm{H}}_2\mathrm{O}\left(50\mathrm{\%}\right)$
Density$\left(\rho \right)$	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	8933	3970	997.1	1056
Electric conductivity$\left(\sigma \right)$	$\mathrm{s}/\mathrm{m}$	59.6 $\times {10}^6$	35 $\times {10}^6$	0.005	0.00509
Thermal conductivity (k)	$\mathrm{W}/\mathrm{m}\mathrm{K}$	400	40	0.613	0.425
Specific heat $\left(C_p\right)$	$\mathrm{J}/\mathrm{k}\mathrm{g}\mathrm{K}$	385	765	4179	3288
Prandtl number $\left(P_r\right)$	–	–	–	6.2	29.86

•$\mathrm{C}\mathrm{u}$ and ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ are nanoparticles.•${\mathrm{H}}_2\mathrm{O}$ and $\mathrm{E}\mathrm{G}\left(50\mathrm{\%}\right)+{\mathrm{H}}_2\mathrm{O}\left(50\mathrm{\%}\right)$ are base fluids.

Now, the first task is to convert the above system (4(4) $\begin{array}{rl}& \frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }y}=0,\end{array}$(4) )–(8(8) $\begin{array}{rl}& u_1(x,y)\to 0,T(x,y)\to T_{\mathrm{\infty }}\mathrm{a}\mathrm{s}y\to \mathrm{\infty },\end{array}$(8) ) into the non-dimensional form. For that, let us consider the following dimensionless parameters as follows: (12) $\begin{array}{rl}& \check{x}=\sqrt{\frac{b}{{\nu }_f}}x,\check{y}=\sqrt{\frac{b}{{\nu }_f}}y,{\check{u}}_1=\frac{u_1}{\sqrt{b{\nu }_f}},\\ & {\check{u}}_2=\frac{u_2}{\sqrt{b{\nu }_f}},\theta =\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}}.\end{array}$(12) By inserting the above parameters into the above system (4(4) $\begin{array}{rl}& \frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }y}=0,\end{array}$(4) )–(8(8) $\begin{array}{rl}& u_1(x,y)\to 0,T(x,y)\to T_{\mathrm{\infty }}\mathrm{a}\mathrm{s}y\to \mathrm{\infty },\end{array}$(8) ), it becomes (13) $\begin{array}{rl}& \frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{x}}+\frac{\mathrm{\partial }{\check{u}}_2}{\mathrm{\partial }\check{y}}=0,\end{array}$(13) (14) $\begin{array}{rl}& {\check{u}}_1\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{x}}+{\check{u}}_2\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{y}}\\ & =(1-n)\frac{{\nu }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2{\check{u}}_1}{\mathrm{\partial }{\check{y}}^2}+nb\sqrt{2}\mathrm{\Lambda }\frac{{\nu }_{hnf}}{{\nu }_f}\left(\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{y}}\right)\left(\frac{{\mathrm{\partial }}^2{\check{u}}_1}{\mathrm{\partial }{\check{y}}^2}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}{\check{u}}_1-\frac{{\nu }_{hnf}}{K_{p}b}{\check{u}}_1,\end{array}$(14) (15) $\begin{array}{rl}& {\check{u}}_1\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{x}}+{\check{u}}_2\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}}\\ & =\frac{{\alpha }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}+\frac{1}{(\rho C_p{)}_{hnf}{\nu }_f}\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}\\ & +\frac{Q_0}{(\rho C_p{)}_{hnf}b}\theta ,\end{array}$(15) with new BCs (16) $\begin{array}{rl}& {\check{u}}_1(\check{x},\check{y}=0)=\check{x}+L\sqrt{\frac{b}{{\nu }_f}}\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{y}},{\check{u}}_2(\check{x},\check{y}=0)=0,\\ & \theta (\check{x},\check{y}=0)=1+N\sqrt{\frac{b}{{\nu }_f}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}},\end{array}$(16) (17) $\begin{array}{rl}& {\check{u}}_1(\check{x},\check{y}\to \mathrm{\infty })\to 0,\theta (\check{x},\check{y}\to \mathrm{\infty })\to 0.\end{array}$(17) Now, we will utilize the stream function ψ such that ${\check{u}}_1=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}$ and ${\check{u}}_2=-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}$ to minimize the quantity of dependent variables. Equation (13(13) $\begin{array}{rl}& \frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{x}}+\frac{\mathrm{\partial }{\check{u}}_2}{\mathrm{\partial }\check{y}}=0,\end{array}$(13) ) is automatically satisfied by above stream function, and Equations (14(14) $\begin{array}{rl}& {\check{u}}_1\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{x}}+{\check{u}}_2\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{y}}\\ & =(1-n)\frac{{\nu }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2{\check{u}}_1}{\mathrm{\partial }{\check{y}}^2}+nb\sqrt{2}\mathrm{\Lambda }\frac{{\nu }_{hnf}}{{\nu }_f}\left(\frac{\mathrm{\partial }{\check{u}}_1}{\mathrm{\partial }\check{y}}\right)\left(\frac{{\mathrm{\partial }}^2{\check{u}}_1}{\mathrm{\partial }{\check{y}}^2}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}{\check{u}}_1-\frac{{\nu }_{hnf}}{K_{p}b}{\check{u}}_1,\end{array}$(14) ) and (15(15) $\begin{array}{rl}& {\check{u}}_1\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{x}}+{\check{u}}_2\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}}\\ & =\frac{{\alpha }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}+\frac{1}{(\rho C_p{)}_{hnf}{\nu }_f}\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}\\ & +\frac{Q_0}{(\rho C_p{)}_{hnf}b}\theta ,\end{array}$(15) ) will convert in terms of stream function as follows: (18) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }\check{y}\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\\ & =(1-n)\frac{{\nu }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}+nb\sqrt{2}\mathrm{\Lambda }\frac{{\nu }_{hnf}}{{\nu }_f}\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\left(\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}-\frac{{\nu }_{hnf}}{K_{p}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}},\end{array}$(18) (19) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}}\right)\\ & =\frac{{\alpha }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}+\frac{1}{(\rho C_p{)}_{hnf}{\nu }_f}\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}\\ & +\frac{Q_0}{(\rho C_p{)}_{hnf}b}\theta ,\end{array}$(19) with BCs (20) $\begin{array}{rl}& \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}(\check{x},\check{y}=0)=\check{x}+L\sqrt{\frac{b}{{\nu }_f}}\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2},\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}(\check{x},\check{y}=0)=0,\\ & \theta (\check{x},\check{y}=0)=1+N\sqrt{\frac{b}{{\nu }_f}}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}},\end{array}$(20) (21) $\begin{array}{rl}& \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}(\check{x},\check{y}\to \mathrm{\infty })\to 0,\theta (\check{x},\check{y}\to \mathrm{\infty })\to 0.\end{array}$(21)

3. Scaling group transformations

The scaling group transformations are Lie group transformations. Lie group transformations [28,29,42,43] refer to the transformations that preserve the group structure of a system, where a group is a set of transformations with a binary operation (such as a composition of transformations) that satisfies certain axioms. Scaling transformations are a type of transformation in which a quantity is multiplied by a constant factor and can form a Lie group under composition. In this sense, scaling group transformations can be considered a subgroup of Lie group transformations and can be used in various applications, such as finding similarity transforms to convert the system of PDEs into a system of ODEs. So here, we want to convert the above system of PDEs (18(18) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }\check{y}\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\\ & =(1-n)\frac{{\nu }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}+nb\sqrt{2}\mathrm{\Lambda }\frac{{\nu }_{hnf}}{{\nu }_f}\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\left(\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}-\frac{{\nu }_{hnf}}{K_{p}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}},\end{array}$(18) )–(19(19) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}}\right)\\ & =\frac{{\alpha }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}+\frac{1}{(\rho C_p{)}_{hnf}{\nu }_f}\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}\\ & +\frac{Q_0}{(\rho C_p{)}_{hnf}b}\theta ,\end{array}$(19) ) into the system of ODEs. For that, we need the similarity transforms(variables). The procedure to find similarity transforms as follows: (22) $\begin{array}{rl}\mathbb{G}:& {\psi }^{\ast }=\psi {\mathrm{e}}^{ϵ{\text{β}}_1},{\theta }^{\ast }=\theta {\mathrm{e}}^{ϵ{\text{β}}_2},{\mathrm{\Lambda }}^{\ast }=\mathrm{\Lambda }{\mathrm{e}}^{ϵ{\text{β}}_3},\\ & x^{\ast }=\check{x}{\mathrm{e}}^{ϵ{\text{β}}_4},y^{\ast }=\check{y}{\mathrm{e}}^{ϵ{\text{β}}_5},\end{array}$(22) here, $\mathbb{G}$ is group of scaling transformations which transforms the coordinates $(\psi ,\theta ,\mathrm{\Lambda },\check{x},\check{y})$ into new coordinates $({\psi }^{\ast },{\theta }^{\ast },{\mathrm{\Lambda }}^{\ast },x^{\ast },y^{\ast })$. With the help of group $\mathbb{G}$, the system (18(18) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }\check{y}\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\\ & =(1-n)\frac{{\nu }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}+nb\sqrt{2}\mathrm{\Lambda }\frac{{\nu }_{hnf}}{{\nu }_f}\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\left(\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}-\frac{{\nu }_{hnf}}{K_{p}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}},\end{array}$(18) )–(19(19) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }\check{y}}\right)\\ & =\frac{{\alpha }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}+\frac{1}{(\rho C_p{)}_{hnf}{\nu }_f}\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{\check{y}}^2}\\ & +\frac{Q_0}{(\rho C_p{)}_{hnf}b}\theta ,\end{array}$(19) ) can be written as: (23) $\begin{array}{rl}& (\frac{\mathrm{\partial }{\psi }^{\ast }}{\mathrm{\partial }{y}^{\ast }}\frac{{\mathrm{\partial }}^2{\psi }^{\ast }}{\mathrm{\partial }{x}^{\ast }\mathrm{\partial }{y}^{\ast }}-\frac{\mathrm{\partial }{\psi }^{\ast }}{\mathrm{\partial }{x}^{\ast }}\frac{{\mathrm{\partial }}^2{\psi }^{\ast }}{\mathrm{\partial }{y^{\ast }}^2}){\mathrm{e}}^{ϵ({\text{β}}_4+2{\text{β}}_5-2{\text{β}}_1)}\\ & =(1-n){\nu }_{hnf}\frac{{\mathrm{\partial }}^3{\psi }^{\ast }}{\mathrm{\partial }{y^{\ast }}^3}{\mathrm{e}}^{ϵ(3{\text{β}}_5-{\text{β}}_1)}\\ & +nb\sqrt{2}{\mathrm{\Lambda }}^{\ast }{\nu }_{hnf}\left(\frac{{\mathrm{\partial }}^2{\psi }^{\ast }}{\mathrm{\partial }{y^{\ast }}^2}\right)\left(\frac{{\mathrm{\partial }}^3{\psi }^{\ast }}{\mathrm{\partial }{y^{\ast }}^3}\right){\mathrm{e}}^{ϵ(5{\text{β}}_5-{\text{β}}_3-2{\text{β}}_1)}\\ & -(\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}+\frac{{\nu }_{hnf}}{K_{p}b})\frac{\mathrm{\partial }{\psi }^{\ast }}{\mathrm{\partial }{y}^{\ast }}{\mathrm{e}}^{ϵ({\text{β}}_5-{\text{β}}_1)},\end{array}$(23) (24) $\begin{array}{rl}& (\frac{\mathrm{\partial }{\psi }^{\ast }}{\mathrm{\partial }{y}^{\ast }}\frac{\mathrm{\partial }{\theta }^{\ast }}{\mathrm{\partial }{x}^{\ast }}-\frac{\mathrm{\partial }{\psi }^{\ast }}{\mathrm{\partial }{x}^{\ast }}\frac{\mathrm{\partial }{\theta }^{\ast }}{\mathrm{\partial }{y}^{\ast }}){\mathrm{e}}^{ϵ({\text{β}}_4+{\text{β}}_5-{\text{β}}_1-{\text{β}}_2)}\\ & =\frac{{\alpha }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^2{\theta }^{\ast }}{\mathrm{\partial }{y^{\ast }}^2}{\mathrm{e}}^{ϵ(2{\text{β}}_5-{\text{β}}_2)}\\ & +\frac{1}{(\rho C_p{)}_{hnf}{\nu }_f}\frac{16}{3}\frac{{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }}\frac{{\mathrm{\partial }}^2{\theta }^{\ast }}{\mathrm{\partial }{y^{\ast }}^2}{\mathrm{e}}^{ϵ(2{\text{β}}_5-{\text{β}}_2)}\\ & +\frac{Q_0}{(\rho C_p{)}_{hnf}b}{\mathrm{e}}^{-ϵ\left({\text{β}}_2\right)}{\theta }^{\ast }.\end{array}$(24) We know that the group $\mathbb{G}$ will not affect the invariance of the system. So, the invariance condition can be written as follows:

$\begin{array}{rl}& {\text{β}}_4+2{\text{β}}_5-2{\text{β}}_1=3{\text{β}}_5-{\text{β}}_1=5{\text{β}}_5-{\text{β}}_3-2{\text{β}}_1={\text{β}}_5-{\text{β}}_1,\\ & {\text{β}}_4+{\text{β}}_5-{\text{β}}_1-{\text{β}}_2=2{\text{β}}_5-{\text{β}}_2={\text{β}}_5-{\text{β}}_2=-{\text{β}}_2.\end{array}$By solving the above equations, we will get (25) ${\text{β}}_2={\text{β}}_5=0,{\text{β}}_1={\text{β}}_4=-{\text{β}}_3$(25) So, the set of scaling transformation $\mathbb{G}$ reduces to (26) $\begin{array}{rl}\mathbb{G}:& {\psi }^{\ast }=\psi {\mathrm{e}}^{ϵ{\text{β}}_1},{\theta }^{\ast }=\theta ,{\mathrm{\Lambda }}^{\ast }=\mathrm{\Lambda }{\mathrm{e}}^{-ϵ{\text{β}}_1},\\ & x^{\ast }=\check{x}{\mathrm{e}}^{ϵ{\text{β}}_1},y^{\ast }=\check{y}.\end{array}$(26) Now by expanding above the exponential function and neglecting the higher power of ϵ, one can get (27) $\begin{array}{rl}& {\psi }^{\ast }-\psi =\psi ϵ{\text{β}}_1,{\theta }^{\ast }-\theta =0,{\mathrm{\Lambda }}^{\ast }-\mathrm{\Lambda }=-\mathrm{\Lambda }ϵ{\text{β}}_1,\\ & x^{\ast }-x=xϵ{\text{β}}_1,y^{\ast }-\check{y}=0.\end{array}$(27) The characteristic equation for the above set of transformations can be written as follows: (28) $\frac{\mathrm{d}\check{x}}{\check{x}ϵ{\text{β}}_1}=\frac{\mathrm{d}\check{y}}{0}=\frac{\mathrm{d}\psi }{\psi ϵ{\text{β}}_1}=\frac{\mathrm{d}\theta }{0}=\frac{\mathrm{d}\mathrm{\Lambda }}{-\mathrm{\Lambda }ϵ{\text{β}}_1}$(28) From $\frac{\mathrm{d}\check{x}}{\check{x}ϵ{\text{β}}_1}=\frac{\mathrm{d}\check{y}}{0}$, we have $\check{y}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}=\gamma \left(\mathrm{s}\mathrm{a}\mathrm{y}\right)$. And the first and third terms $\frac{\mathrm{d}\check{x}}{\check{x}ϵ{\text{β}}_1}=\frac{\mathrm{d}\psi }{\psi ϵ{\text{β}}_1}$ give, $\frac{\psi }{\check{x}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}=F\left(\gamma \right)$. Similarly, from the first and fourth terms we have $\theta =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}=G\left(\gamma \right)$ and first and last terms give $\check{x}\mathrm{\Lambda }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}={\mathrm{\Lambda }}_0$. So the similarity transformations are obtained as: (29) $\check{y}=\gamma ,\psi =\check{x}F\left(\gamma \right),\mathrm{\Lambda }=\frac{{\mathrm{\Lambda }}_0}{\check{x}},\theta =G\left(\gamma \right).$(29) By inserting the above similarity variables into the system of PDEs (18(18) $\begin{array}{rl}& \left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }\check{y}\mathrm{\partial }\check{x}}\right)-\left(\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{x}}\right)\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\\ & =(1-n)\frac{{\nu }_{hnf}}{{\nu }_f}\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}+nb\sqrt{2}\mathrm{\Lambda }\frac{{\nu }_{hnf}}{{\nu }_f}\left(\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\check{y}}^2}\right)\left(\frac{{\mathrm{\partial }}^3\psi }{\mathrm{\partial }{\check{y}}^3}\right)\\ & -\frac{{\sigma }_{hnf}{B}_0^2{\sin }^2\left({\theta }_M\right)}{{\rho }_{hnf}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}-\frac{{\nu }_{hnf}}{K_{p}b}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}},\end{array}$(18) )–(21(21) $\begin{array}{rl}& \frac{\mathrm{\partial }\psi }{\mathrm{\partial }\check{y}}(\check{x},\check{y}\to \mathrm{\infty })\to 0,\theta (\check{x},\check{y}\to \mathrm{\infty })\to 0.\end{array}$(21) ), we will get the following system of ODEs (30) $\begin{array}{rl}& (1-n)F^{‴}+n{W}_e{F}^{″}{F}^{‴}+\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\rho }_{hnf}}{{\rho }_f}(F{F}^{″}-F^{\prime 2})\\ & -\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}{H}_a{\sin }^2\left({\theta }_M\right)F^{\prime }-\lambda F^{\prime }=0,\end{array}$(30) (31) $\begin{array}{rl}& (\frac{k_{hnf}}{k_f}+\frac{4}{3}{P}_r{R}_d)G^{″}+P_r\frac{(\rho C_p{)}_{hnf}}{(\rho C_p{)}_f}(F{G}^{\prime }-G{F}^{\prime })\\ & +P_{r}QG=0,\end{array}$(31) where F and G is a functions of γ and prime stands for derivative with respect to γ and BCs are as follows: (32) $\begin{array}{rl}& F\left(0\right)=0,F^{\prime }\left(0\right)=1+A{F}^{″}\left(0\right),G\left(0\right)=1+B{G}^{\prime }\left(0\right),\end{array}$(32) (33) $\begin{array}{rl}& F^{\prime }(\gamma \to \mathrm{\infty })\to 0,G(\gamma \to \mathrm{\infty })\to 0.\end{array}$(33) Here, $W_e=b\sqrt{2}{\mathrm{\Lambda }}_0$ is the Weissenberg number, $H_a=\frac{{\sigma }_f{B}_0^2}{b{\rho }_f}$ is the Hartmann number(magnetic field parameter), $\lambda =\frac{{\nu }_f}{K_{p}b}$ is a porous medium parameter, $P_r=\frac{{\nu }_f}{{\alpha }_f}$ is a Prandtl number, $R_d=\frac{4{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }{\nu }_f(\rho C_p{)}_f}$ is a thermal radiation parameter, $Q=\frac{Q_0}{(\rho C_p{)}_{f}b}$ is the coefficient of heat generation/absorption, $A=\sqrt{\frac{b}{{\nu }_f}}L$ is a velocity slip parameter, $B=\sqrt{\frac{b}{{\nu }_f}}N$ is a thermal slip parameter and from Table 2, we have $\begin{array}{rl}\frac{{\mu }_f}{{\mu }_{hnf}}& =\left[\right(1-{\varphi }_1\left)\right(1-{\varphi }_2){]}^{2.5},\\ \frac{{\rho }_{hnf}}{{\rho }_f}& =(1-{\varphi }_2)[{\varphi }_1\frac{{\rho }_{p_1}}{{\rho }_f}+(1-{\varphi }_1\left)\right]+{\varphi }_2\frac{{\rho }_{p_2}}{{\rho }_f},\\ \frac{{\sigma }_{hnf}}{{\sigma }_f}& =\left(\frac{{\sigma }_{p_2}-2{\varphi }_2({\sigma }_{bf}-{\sigma }_{p_2})+2{\sigma }_{bf}}{{\sigma }_{p_2}+{\varphi }_2({\sigma }_{bf}-{\sigma }_{p_2})+2{\sigma }_{bf}}\right)\\ & \times \left(\frac{{\sigma }_{p_1}-2{\varphi }_1({\sigma }_f-{\sigma }_{p_1})+2{\sigma }_f}{{\sigma }_{p_1}+{\varphi }_1({\sigma }_f-{\sigma }_{p_1})+2{\sigma }_f}\right),\\ & \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\frac{{\sigma }_{bf}}{{\sigma }_f}=\frac{{\sigma }_{p_1}-2{\varphi }_1({\sigma }_f-{\sigma }_{p_1})+2{\sigma }_f}{{\sigma }_{p_1}+{\varphi }_1({\sigma }_f-{\sigma }_{p_1})+2{\sigma }_f},\\ \frac{k_{hnf}}{k_f}& =\left(\frac{k_{p_2}-2{\varphi }_2(k_{bf}-k_{p_2})+2k_{bf}}{k_{p_2}+{\varphi }_2(k_{bf}-k_{p_2})+2k_{bf}}\right)\\ & \times \left(\frac{k_{p_1}-2{\varphi }_1(k_f-k_{p_1})+2k_f}{k_{p_1}+{\varphi }_1(k_f-k_{p_1})+2k_f}\right),\\ & \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\frac{k_{bf}}{k_f}=\frac{k_{p_1}-2{\varphi }_1(k_f-k_{p_1})+2k_f}{{\sigma }_{p_1}+{\varphi }_1(k_f-k_{p_1})+2k_f},\\ \frac{(\rho C_p{)}_{hnf}}{(\rho C_p{)}_f}& =(1-{\varphi }_2)[{\varphi }_1\frac{(\rho C_p{)}_{p_1}}{(\rho C_p{)}_f}+(1-{\varphi }_1\left)\right]\\ & +{\varphi }_2\frac{(\rho C_p{)}_{p_2}}{(\rho C_p{)}_f}.\end{array}$The coefficient of skin friction (${\mathcal{C}}_{fx}$) and the local Nusselt number $\mathcal{N}{u}_x$ [20] are calculated as: (34) $\begin{array}{rl}{\mathcal{C}}_{fx}& =\frac{{\mu }_{hnf}}{{\rho }_{hnf}(bx{)}^2}{\left[\right(1-n)\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }y}+\frac{n\mathrm{\Lambda }}{\sqrt{2}}\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{y}^2}]|}_{y=0},\end{array}$(34) (35) $\begin{array}{rl}\mathcal{N}{u}_x& =-\frac{x}{(T_w-T_{\mathrm{\infty }})}\frac{k_{hnf}}{k_f}[1+\frac{4}{3}\frac{4{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k^{\ast }{\nu }_f(\rho C_p{)}_f}]{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}_{y=0}.\end{array}$(35) The above physical quantities well be reduced into the dimensionless form as follows: (36) $\begin{array}{rl}{\mathcal{C}}_{fx}R{e}_x^{1/2}& =\frac{(1-n)F^{″}\left(0\right)+n{W}_e{F}^{″2}\left(0\right)}{\begin{array}{c}(1-{\varphi }_1{)}^{2.5}(1-{\varphi }_2{)}^{2.5}\\ \left[\right(1-{\varphi }_2)\left\{\right(1-{\varphi }_1)+{\varphi }_1\frac{{\rho }_{p_1}}{{\rho }_f}\}+{\varphi }_2\frac{{\rho }_{p_2}}{{\rho }_f}]\end{array}},\end{array}$(36) (37) $\begin{array}{rl}\mathcal{N}{u}_{x}R{e}_x^{-1/2}& =-\frac{k_{hnf}}{k_f}[1+\frac{4}{3}{R}_d]G^{\prime }\left(0\right).\end{array}$(37)

Table 2. The correlations for thermo-physical properties of the HNF [40,41].

Physical properties	HNF
Viscosity $\left(\mu \right)$	${\mu }_{hnf}=\frac{{\mu }_f}{(1-{\varphi }_1{)}^{2.5}(1-{\varphi }_2{)}^{2.5}}$
Density$\left(\rho \right)$	${\rho }_{hnf}=(1-{\varphi }_2)\{{\varphi }_1{\rho }_{p_1}+(1-{\varphi }_1\left){\rho }_f\right\}+{\varphi }_2{\rho }_{p_2}$
Heat capacity$\left(\rho C_p\right)$	$(\rho C_p{)}_{hnf}=(1-{\varphi }_2\left)\right\{{\varphi }_1{\rho }_{p_1}+(1-{\varphi }_1){\rho }_f\}+{\varphi }_2{\rho }_{p_2}$
Thermal conductivity(k)	$\frac{k_{hnf}}{k_{bf}}=\frac{k_{p_2}+2k_{bf}-2{\varphi }_2(k_{bf}-k_{p_2})}{k_{p_2}+2k_{bf}+{\varphi }_2(k_{bf}-k_{p_2})},$
	where $\frac{k_{bf}}{k_f}=\frac{k_{p_1}+2k_f-2{\varphi }_1(k_f-k_{p_1})}{k_{p_1}+2k_f+{\varphi }_1(k_f-k_{p_1})}$
Electric conductivity$\left(\sigma \right)$	$\frac{{\sigma }_{hnf}}{{\sigma }_{bf}}=\frac{{\sigma }_{p_2}+2{\sigma }_{bf}-2{\varphi }_2({\sigma }_{bf}-{\sigma }_{p_2})}{{\sigma }_{p_2}+2{\sigma }_{bf}+{\varphi }_2({\sigma }_{bf}-{\sigma }_{p_2})},$
	where $\frac{{\sigma }_{bf}}{{\sigma }_f}=\frac{{\sigma }_{p_1}+2{\sigma }_f-2{\varphi }_1({\sigma }_f-{\sigma }_{p_1})}{{\sigma }_{p_1}+2{\sigma }_f+{\varphi }_1({\sigma }_f-{\sigma }_{p_1})}$

•$p_1$ stands for first nanoparticle.•$p_2$ stands for second nanoparticle.•${\varphi }_1$ is volume fraction of $\mathrm{C}\mathrm{u}$ nano-particles.•${\varphi }_2$ is volume fraction of ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nano-particles.

4. Solution procedure with an analytical approach

This section describes the solution of the suggested model (30(30) $\begin{array}{rl}& (1-n)F^{‴}+n{W}_e{F}^{″}{F}^{‴}+\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\rho }_{hnf}}{{\rho }_f}(F{F}^{″}-F^{\prime 2})\\ & -\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}{H}_a{\sin }^2\left({\theta }_M\right)F^{\prime }-\lambda F^{\prime }=0,\end{array}$(30) )–(31(31) $\begin{array}{rl}& (\frac{k_{hnf}}{k_f}+\frac{4}{3}{P}_r{R}_d)G^{″}+P_r\frac{(\rho C_p{)}_{hnf}}{(\rho C_p{)}_f}(F{G}^{\prime }-G{F}^{\prime })\\ & +P_{r}QG=0,\end{array}$(31) ) along with the pertinent boundary conditions (32(32) $\begin{array}{rl}& F\left(0\right)=0,F^{\prime }\left(0\right)=1+A{F}^{″}\left(0\right),G\left(0\right)=1+B{G}^{\prime }\left(0\right),\end{array}$(32) )–(33(33) $\begin{array}{rl}& F^{\prime }(\gamma \to \mathrm{\infty })\to 0,G(\gamma \to \mathrm{\infty })\to 0.\end{array}$(33) ) utilizing homotopy algorithm(HAM) [30–34] to investigate outcomes such as skin friction coefficient, temperature and velocity profiles, and other parameters for magnetohydrodynamic (MHD) tangential hyperbolic fluid flow in the direction of a stretched sheet with slip circumstances and porous medium. Figure 2 depicts the implementation approach for determining the analytical solution using HAM. The dimensionless form of the energy and temperature equations have the initial guesses $(F_0,G_0)$ and auxiliary linear operators $({\mathfrak{L}}_F,{\mathfrak{L}}_G)$ as follow: (38) $\begin{array}{rl}& F_0\left(\gamma \right)=\frac{1}{1+A}(1-{\mathrm{e}}^{-\gamma })\mathrm{a}\mathrm{n}\mathrm{d}{G}_0\left(\gamma \right)=\frac{1}{1+B}{\mathrm{e}}^{-\gamma },\end{array}$(38) (39) $\begin{array}{rl}& {\mathfrak{L}}_F\left(F\right)=\frac{d^{3}F}{\mathrm{d}{\gamma }^3}-\frac{\mathrm{d}F}{\mathrm{d}\gamma }\mathrm{a}\mathrm{n}\mathrm{d}{\mathfrak{L}}_G\left(G\right)=\frac{d^{2}G}{\mathrm{d}{\gamma }^2}-G,\end{array}$(39) with satisfying conditions: (40) $\begin{array}{rl}& {\mathfrak{L}}_F(c_1+c_2{\mathrm{e}}^{-\gamma }+c_2{\mathrm{e}}^{\gamma })=0,\end{array}$(40) (41) $\begin{array}{rl}& {\mathfrak{L}}_G(c_4{\mathrm{e}}^{-\gamma }+c_5{\mathrm{e}}^{\gamma })=0,\end{array}$(41)

Figure 2. Flow chart of the HAM procedure.

where $c_i^{\prime }s(i=1,2,3,4,5)$ are arbitrary constants. Now to utilizing the HAM procedure, the zeroth order deformation equations for system (30(30) $\begin{array}{rl}& (1-n)F^{‴}+n{W}_e{F}^{″}{F}^{‴}+\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\rho }_{hnf}}{{\rho }_f}(F{F}^{″}-F^{\prime 2})\\ & -\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}{H}_a{\sin }^2\left({\theta }_M\right)F^{\prime }-\lambda F^{\prime }=0,\end{array}$(30) )–(33(33) $\begin{array}{rl}& F^{\prime }(\gamma \to \mathrm{\infty })\to 0,G(\gamma \to \mathrm{\infty })\to 0.\end{array}$(33) ), elaborated as (42) $\begin{array}{rl}& (1-q){\mathfrak{L}}_F\left[F\right(\gamma ,q)-F_0(\gamma \left)\right]=q{\hslash }_F{\mathcal{N}}_F\left[F\right(\gamma ,q\left)\right]\end{array}$(42) (43) $\begin{array}{rl}& (1-q){\mathfrak{L}}_G\left[G\right(\gamma ,q)-G_0(\gamma \left)\right]=q{\hslash }_G{\mathcal{N}}_G\left[F\right(\gamma ,q),G(\gamma ,q\left)\right]\end{array}$(43) (44) $\begin{array}{rl}& F(0,q)=0,{\frac{\mathrm{\partial }F(\gamma ,q)}{\mathrm{\partial }\gamma }|}_{\gamma =0}\\ & =1+A{\frac{{\mathrm{\partial }}^{2}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^2}|}_{\gamma =0},{\frac{\mathrm{\partial }F(\gamma ,q)}{\mathrm{\partial }\gamma }|}_{\gamma \to \mathrm{\infty }}\to 0,\end{array}$(44) (45) $\begin{array}{rl}& G(0,q)=1+B{\frac{\mathrm{\partial }G(\gamma ,q)}{\mathrm{\partial }\gamma }|}_{\gamma =0},G(\gamma \to \mathrm{\infty },q)\to 0.\end{array}$(45) Here, $q(\in [0,1\left]\right)$ is an embedding factor, ${\hslash }_F\ne 0$ and ${\hslash }_G\ne 0$ are convergence controlling parameters, ${\mathcal{N}}_F$ and ${\mathcal{N}}_G$ are non-linear operators defined as follows:

(46) $\begin{array}{rl}{\mathcal{N}}_F\left[F\right(\gamma ,q\left)\right]& =(1-n)\frac{{\mathrm{\partial }}^{3}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^3}+n{W}_e\frac{{\mathrm{\partial }}^{2}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^2}\frac{{\mathrm{\partial }}^{3}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^3}\\ & +\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\rho }_{hnf}}{{\rho }_f}\left(F\right(\gamma ,q)\frac{{\mathrm{\partial }}^{2}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^2}-{\left(\frac{{\mathrm{\partial }}^{3}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^3}\right)}^2)\\ & -(\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}{H}_a{\sin }^2({\theta }_M)+\lambda )\frac{\mathrm{\partial }F(\gamma ,q)}{\mathrm{\partial }\gamma },\end{array}$(46) (47) $\begin{array}{rl}& {\mathcal{N}}_G\left[F\right(\gamma ,q),G(\gamma ,q\left)\right]\\ & =(\frac{k_{hnf}}{k_f}+\frac{4}{3}{P}_r{R}_d)\frac{{\mathrm{\partial }}^{2}G(\gamma ,q)}{\mathrm{\partial }{\gamma }^2}\\ & +P_r\frac{(\rho C_p{)}_{hnf}}{(\rho C_p{)}_f}\left(F\right(\gamma ,q)\frac{\mathrm{\partial }G(\gamma ,q)}{\mathrm{\partial }\gamma }\\ & -G(\gamma ,q)\frac{\mathrm{\partial }F(\gamma ,q)}{\mathrm{\partial }\gamma })+P_{r}QG(\gamma ,q),\end{array}$(47) From Equations (42(42) $\begin{array}{rl}& (1-q){\mathfrak{L}}_F\left[F\right(\gamma ,q)-F_0(\gamma \left)\right]=q{\hslash }_F{\mathcal{N}}_F\left[F\right(\gamma ,q\left)\right]\end{array}$(42) ) and (43(43) $\begin{array}{rl}& (1-q){\mathfrak{L}}_G\left[G\right(\gamma ,q)-G_0(\gamma \left)\right]=q{\hslash }_G{\mathcal{N}}_G\left[F\right(\gamma ,q),G(\gamma ,q\left)\right]\end{array}$(43) ), we have (48) $\begin{array}{rl}& F(\gamma ,q=0)=F_0\mathrm{a}\mathrm{n}\mathrm{d}F(\gamma ,q=1)=F\left(\gamma \right),\end{array}$(48) (49) $\begin{array}{rl}& G(\gamma ,q=0)=G_0\mathrm{a}\mathrm{n}\mathrm{d}G(\gamma ,q=1)=G\left(\gamma \right),\end{array}$(49) From above, as the embedding factor q increases from 0 to 1, the homotopy solution approaches or converges to the original solution. By expanding Taylor's series around the embedding factor q, one can get (50) $\begin{array}{rl}& F(\gamma ,q)=F_0\left(\gamma \right)+\sum _{k=0}^{\mathrm{\infty }}{F}_k\left(\gamma \right)q^k\mathrm{a}\mathrm{n}\mathrm{d}\\ & G(\gamma ,q)=G_0\left(\gamma \right)+\sum _{k=0}^{\mathrm{\infty }}{G}_k\left(\gamma \right)q^k,\end{array}$(50) where (51) $\begin{array}{rl}& F_k\left(\gamma \right)={\frac{1}{k!}\left[\frac{{\mathrm{\partial }}^{k}F(\gamma ,q)}{\mathrm{\partial }{\gamma }^k}\right]|}_{q=0}\mathrm{a}\mathrm{n}\mathrm{d}\\ & G_k\left(\gamma \right)={\frac{1}{k!}\left[\frac{{\mathrm{\partial }}^{k}G(\gamma ,q)}{\mathrm{\partial }{\gamma }^k}\right]|}_{q=0}.\end{array}$(51) The choice of convergence control parameters is done to ensure the convergence of the series (50(50) $\begin{array}{rl}& F(\gamma ,q)=F_0\left(\gamma \right)+\sum _{k=0}^{\mathrm{\infty }}{F}_k\left(\gamma \right)q^k\mathrm{a}\mathrm{n}\mathrm{d}\\ & G(\gamma ,q)=G_0\left(\gamma \right)+\sum _{k=0}^{\mathrm{\infty }}{G}_k\left(\gamma \right)q^k,\end{array}$(50) ) at q = 1, i.e. (52) $\begin{array}{rl}& F(\gamma ,q)=F_0\left(\gamma \right)+\sum _{k=1}^{\mathrm{\infty }}{F}_k\left(\gamma \right)\mathrm{a}\mathrm{n}\mathrm{d}\\ & G(\gamma ,q)=G_0\left(\gamma \right)+\sum _{k=1}^{\mathrm{\infty }}{G}_k\left(\gamma \right).\end{array}$(52) Now differentiate Equations (42(42) $\begin{array}{rl}& (1-q){\mathfrak{L}}_F\left[F\right(\gamma ,q)-F_0(\gamma \left)\right]=q{\hslash }_F{\mathcal{N}}_F\left[F\right(\gamma ,q\left)\right]\end{array}$(42) ) and (43(43) $\begin{array}{rl}& (1-q){\mathfrak{L}}_G\left[G\right(\gamma ,q)-G_0(\gamma \left)\right]=q{\hslash }_G{\mathcal{N}}_G\left[F\right(\gamma ,q),G(\gamma ,q\left)\right]\end{array}$(43) ) $k(\ge 1)$ times with respect to q and substituting q = 0, we get the deformation equations of kth order as (53) $\begin{array}{rl}& {\mathfrak{L}}_F\left[F_k\right(\gamma )-{\mathrm{\aleph }}_k{F}_{k-1}(\gamma \left)\right]={\hslash }_F{\mathcal{R}}_k^F\left(\gamma \right)\end{array}$(53) (54) $\begin{array}{rl}& {\mathfrak{L}}_G\left[G_k\right(\gamma )-{\mathrm{\aleph }}_k{G}_{k-1}(\gamma \left)\right]={\hslash }_G{\mathcal{R}}_k^G\left(\gamma \right),\end{array}$(54) (55) $\begin{array}{rl}& F_k\left(0\right)=0,{\frac{\mathrm{\partial }{F}_k\left(\gamma \right)}{\mathrm{\partial }\gamma }|}_{\gamma =0}=0,{\frac{{\mathrm{\partial }}^2{F}_k\left(\gamma \right)}{\mathrm{\partial }{\gamma }^2}|}_{\gamma =0}=0,\\ & {\frac{\mathrm{\partial }{F}_k\left(\gamma \right)}{\mathrm{\partial }\gamma }|}_{\gamma \to \mathrm{\infty }}=0,\end{array}$(55) (56) $\begin{array}{rl}& G_k\left(0\right)=0,{\frac{\mathrm{\partial }{G}_k\left(\gamma \right)}{\mathrm{\partial }\gamma }|}_{\gamma \to \mathrm{\infty }}=0,G_k(\gamma \to \mathrm{\infty })=0,\end{array}$(56) where (57) $\begin{array}{rl}{\mathcal{R}}_k^F\left(\gamma \right)& =(1-n)F_{k-1}^{‴}+n{W}_e\sum _{l=0}^{k-1}{F}_l^{″}{F}_{k-1-l}^{‴}\\ & +\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\rho }_{hnf}}{{\rho }_f}(\sum _{l=0}^{k-1}{F}_l{F}_{k-1-l}^{″}-\sum _{l=0}^{k-1}{F}_l^{\prime }{F}_{k-1-l}^{\prime })\\ & -(\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}{H}_a{\sin }^2({\theta }_M)+\lambda )F_{k-1}^{\prime },\end{array}$(57) (58) $\begin{array}{rl}{\mathcal{R}}_k^G\left(\gamma \right)& =(\frac{k_{hnf}}{k_f}+\frac{4}{3}{P}_r{R}_d)G_{k-1}^{″}+P_r\frac{(\rho C_p{)}_{hnf}}{(\rho C_p{)}_f}\\ & \times (\sum _{l=0}^{k-1}{F}_l{G}_{k-1-l}^{\prime }-\sum _{l=0}^{k-1}{G}_l{F}_{k-1-l}^{\prime })+P_{r}Q{G}_{k-1},\end{array}$(58) and (59) $\begin{array}{r}{\mathrm{\aleph }}_k=\{\begin{array}{l}0,k\le 1.\\ 1,k>1.\end{array}\end{array}$(59) The general solution of system (53(53) $\begin{array}{rl}& {\mathfrak{L}}_F\left[F_k\right(\gamma )-{\mathrm{\aleph }}_k{F}_{k-1}(\gamma \left)\right]={\hslash }_F{\mathcal{R}}_k^F\left(\gamma \right)\end{array}$(53) )–(56(56) $\begin{array}{rl}& G_k\left(0\right)=0,{\frac{\mathrm{\partial }{G}_k\left(\gamma \right)}{\mathrm{\partial }\gamma }|}_{\gamma \to \mathrm{\infty }}=0,G_k(\gamma \to \mathrm{\infty })=0,\end{array}$(56) ) is (60) $\begin{array}{rl}{F}_k\left(\gamma \right)& =F_k^{\ast }\left(\gamma \right)+c_1+c_2{\mathrm{e}}^{-\gamma }+c_3{\mathrm{e}}^{\gamma },\end{array}$(60) (61) $\begin{array}{rl}{G}_k\left(\gamma \right)& =G_k^{\ast }\left(\gamma \right)+c_4{\mathrm{e}}^{-\gamma }+c_5{\mathrm{e}}^{\gamma },\end{array}$(61) where $F_k^{\ast }$ and $G_k^{\ast }$ are special solution of Equations (53(53) $\begin{array}{rl}& {\mathfrak{L}}_F\left[F_k\right(\gamma )-{\mathrm{\aleph }}_k{F}_{k-1}(\gamma \left)\right]={\hslash }_F{\mathcal{R}}_k^F\left(\gamma \right)\end{array}$(53) ) and (54(54) $\begin{array}{rl}& {\mathfrak{L}}_G\left[G_k\right(\gamma )-{\mathrm{\aleph }}_k{G}_{k-1}(\gamma \left)\right]={\hslash }_G{\mathcal{R}}_k^G\left(\gamma \right),\end{array}$(54) ) and $\begin{array}{rl}& c_3=c_5=0,c_1=-F_k^{\ast }\left(0\right)-{\frac{\mathrm{d}{F}_k^{\ast }}{\mathrm{d}\gamma }|}_{\gamma =0},\\ & c_2={\frac{\mathrm{d}{F}_k^{\ast }}{\mathrm{d}\gamma }|}_{\gamma =0},c_4=-G_k^{\ast }\left(0\right).\end{array}$

5. Convergence of HAM

To discover the complete solution of system (30(30) $\begin{array}{rl}& (1-n)F^{‴}+n{W}_e{F}^{″}{F}^{‴}+\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\rho }_{hnf}}{{\rho }_f}(F{F}^{″}-F^{\prime 2})\\ & -\frac{{\mu }_f}{{\mu }_{hnf}}\frac{{\sigma }_{hnf}}{{\sigma }_f}{H}_a{\sin }^2\left({\theta }_M\right)F^{\prime }-\lambda F^{\prime }=0,\end{array}$(30) )–(33(33) $\begin{array}{rl}& F^{\prime }(\gamma \to \mathrm{\infty })\to 0,G(\gamma \to \mathrm{\infty })\to 0.\end{array}$(33) ) with the help of the HAM, the interval for valid region must be determined. To accomplish this purpose, ℏ-curves are plotted in Figure 3, and then a region parallel to the horizontal axis is chosen as a viable region. From Figure 3(a), the valid interval of convergent solution of $F^{″}\left(0\right)$ for both HNFs ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ can be given as $-1.2\le {\hslash }_F\le 0.5\mathrm{a}\mathrm{n}\mathrm{d}-1.3\le {\hslash }_F\le 0.4$, respectively. Similarly, From Figure 3(b), the valid interval of convergent solution of $G^{\prime }\left(0\right)$ for same can be given as $-1.1\le {\hslash }_G\le 0.5\mathrm{a}\mathrm{n}\mathrm{d}-0.4\le {\hslash }_G\le 0.1$, respectively. Table 3 shows the $k-$ order approximate HAM solution of $F^{″}\left(0\right)$ and $G^{\prime }\left(0\right)$ for two distinct HNFs. From this table, we can conclude that the solution for $F^{″}\left(0\right)$ achieves in only 15th-order approximations, and for $G^{\prime }\left(0\right)$ it takes upto 30th-order approximations.

Figure 3. $\hslash -$curves for velocity and temperature fields. (a) $\hslash -$curves of $F^{″}\left(0\right)$ for two HNFs. and (b) $\hslash -$curves of $G^{\prime }\left(0\right)$ for two HNFs.

Table 3. kth order approximate HAM solutions of $F^{″}\left(0\right)$ and $G^{\prime }\left(0\right)$ for two hybrid nanofluids.

	$F^{″}\left(0\right)$		$G^{\prime }\left(0\right)$
k	${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$/${\mathrm{H}}_2\mathrm{O}$	${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$/$\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$	${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$/${\mathrm{H}}_2\mathrm{O}$	${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$/$\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$
1	−1.168317	−1.151324	−1.155268	−1.353191
2	−1.153767	−1.147482	−1.018038	−1.340087
3	−1.164525	−1.155716	−1.041461	−1.354278
4	−1.161725	−1.154500	−1.027622	−1.350012
5	−1.163064	−1.155218	−1.029380	−1.349484
10	−1.162695	−1.155081	−1.024600	−1.348752
15	−1.162703	−1.155083	−1.023823	−1.349177
20	−1.162703	−1.155083	−1.023502	−1.324321
25	−1.162703	−1.155083	−1.023263	−1.324067
30	−1.162703	−1.155083	−1.023043	−1.323852
35	−1.162703	−1.155083	−1.022841	−1.323841
40	−1.162703	−1.155083	−1.022841	−1.323841

•k is the order of term approximation.•${\hslash }_F=-1.07$ and ${\hslash }_G=-0.43$ for base fluid ${\mathrm{H}}_2\mathrm{O}$.•${\hslash }_F=-1.04$ and ${\hslash }_G=-0.09$ for base fluid $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$.

6. Outcomes and discussion

This section discusses how the parameters affect the hydrothermal properties of the HNF flow towards a stretchable sheet. Thermal radiation, porous material, heat generation/absorption, and a magnetic field are used to exhibit physically realistic effects. Two different liquids ${\mathrm{H}}_2\mathrm{O}$ and $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ are used as base fluids in which $\mathrm{C}\mathrm{u}$ and ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nanoparticles are mixed to make hybrid nanofluids. In this study, we examine water-based and EG+water-based hybrid nanofluids separately. Here, we discuss the outcome of the given model numerically and graphically. Throughout the study, the default values are considered as ${\varphi }_1=0.04,{\varphi }_2=0.04,We=0.2,n=0.2,H_a=0.2,A=0.1,{\theta }_M=\pi /2,\lambda =0.1,R_d=0.2,Q=0.2\mathrm{a}\mathrm{n}\mathrm{d}B=0.1$, whenever it is not specified. To validate the solution obtained using HAM, the numerical results are compared with existing numerical data. Table 4 shows absolute values of skin friction coefficient ${\mathcal{C}}_{fx}R{e}_x^{1/2}$ without porous medium and without considering any nanofluid for various values of $W_e$, $H_a$, n and also compared with existing data. This table shows that the surface drag force goes down with increasing values of $W_e$ and n. While the value of $H_a$ increases, the skin friction rises. Table 5 shows the effect of $H_a$ in the absence of n and $W_e$.

Table 4. Absolute values of the surface drag force for various values of $n,W_e$ and $H_a$ and ${\varphi }_1={\varphi }_2=0$ without porous medium.

		Akbar et al. [35]			Zakir and Zaman [37]			HAM
n	$H_a$	$W_e=0.0$	$W_e=0.3$	$W_e=0.5$	$W_e=0.0$	$W_e=0.3$	$W_e=0.5$	$W_e=0.0$	$W_e=0.3$	$W_e=0.5$
0.0	0.0	1	1	1	1	1	1	1	1	1
0.1	0.0	0.94868	0.94252	0.93321	0.94868	0.94248	0.93826	0.94868	0.94248	0.93826
0.2	0.0	0.89442	0.88019	0.87023	0.89442	0.88023	0.87026	0.89443	0.88023	0.87026
0.3	0.5	1.09544	0.98806	0.96032	1.02472	0.98804	0.96001	1.02470	0.98803	0.95996
0.3	1.0	1.26491	1.13448	1.09609	1.18322	1.13454	1.09616	1.18321	1.13449	1.09575
0.3	1.5	1.41421	1.26189	1.21229	1.32288	1.26193	1.21235	1.32286	1.26165	1.21047

Table 5. The validation of $-F^{″}\left(0\right)$ for distinct values of $H_a$ and ${\varphi }_1={\varphi }_2=0$ without porous medium.

$H_a$	Akbar et al. [35]	Fathizadeh et al. [44]	HAM
0	1	1	1
0.25	1.1180	1.1180	1.1180
0.5	–	–	1.2247
1	1.4142	1.4142	1.4142
5	2.4495	2.4495	2.4491
10	3.3166	3.3166	3.3122

Impact of the Weissenberg number ($W_e$): $W_e$ is a non-dimensional quantity used in fluid mechanics to describe the behaviour of viscoelastic fluids. The Weissenberg number measures the relative importance of viscous and elastic effects in a viscoelastic fluid. In this study, the increased values of $W_e$ causes the fluid flow to be more difficult to move by improving its viscosity and friction. Figure 4 is drawn to depict the effect of $W_e$ on velocity and temperature distributions. A higher value of $W_e$ actually reflects an increased importance of the elastic aspects of the fluid and makes it more resistant to flow. Figure 4(a) clearly shows the reduction of the fluidity while the value of $W_e$ increases. From this figure, we can also observe that the flow of ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ HNF is a little faster than ${\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$ HNF. Figure 4(b) shows the results of the temperature profiles of two distinct hybrid nanofluids versus the Weissenberg number $W_e$. It demonstrates that the thermal boundary layer thickens as n increases. As a result, the heat rises in the fluid. The temperature distribution for water-base hybrid nanofluid stays ahead of $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ based hybrid nanofluid.

Figure 4. Velocity and temperature fields for distinct values of $W_e$. (a) Velocity fields for two HNFs. and (b) Temperature fields for two HNFs.

Impact of the Hartmann number ($H_a$) and angle of inclination (${\theta }_M$): In MHD, the Hartmann number is a measure that doesn't have any units. It describes how a fluid behaves when there is a uniformly inclined magnetic field around it. Figure 5 depicts the effect of $H_a$ on velocity and temperature distributions. And also compares the flow behaviours of a $H_2$-based HNF and an $(\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O})$-based HNF. From Figure 5(a), it is revealed that the velocity of the fluid declines as the $H_a$ value increases, but here the $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ based HNF remains little ahead of water-based HNF. This is because the magnetic field forms when the electrically conducting fluid is moving in the transverse direction, increasing flow resistance owing to the Lorentz force. As the value of the $H_a$ rises, the temperature field rises because heat is produced as a result of the Lorentz force's resistive character. We also analysed that the water-based HNF has a higher temperature than the other. Figure 6 illustrates the effect of an inclined magnetic field on the velocity and temperature of the fluid. From Figure 6(a,b), it has been shown that increasing the inclined magnetic field decreases velocity profiles while increasing temperature profiles. This could occur because an increasing angle-varying magnetic field strengthens the applied magnetic field.

Figure 5. Velocity and temperature fields for distinct values of $H_a$. (a) Velocity fields for two HNFs. and (b) Temperature fields for two HNFs.

Figure 6. Velocity and temperature fields for various values of ${\theta }_M$. (a) Velocity fields for two HNFs. and (b) Temperature fields for two HNFs.

Impact of the power-law index (n): The n-value is a dimensionless quantity that describes the shear-thinning or shear-thickening characteristics of a non-Newtonian fluid. Figure 7 shows how the values of n changes the flow of a fluid's velocity profile. Figure 7(a) shows that when n is increased, the velocity profile drops. As the power law index n increases, there is a corresponding increase in the fluid temperature. It can be observed in Figure 7(b). In both profiles, we can also compare two different HNFs.

Figure 7. Velocity and temperature fields for distinct values of n. (a) Velocity fields for two HNFs. and (b) Temperature fields for two HNFs.

Impact of the porous medium parameter (λ): Porosity is a measure of the void space in a material. The velocity and temperature fields are affected by the porous medium. The Figure 8 illustrates this observation. As the porosity of a porous medium increases, the available space for the fluid to flow through decreases, resulting in a reduction in flow velocity. Figure 8(a) shows the lagging in the fluid velocity for enhancing the value of λ. In contrast, Figure 8(b) illustrates that the temperature profile exhibits an upward trend with increasing values of λ. A comparison of two different HNFs is provided.

Figure 8. Velocity and temperature fields for various values of λ. (a) Velocity fields for two HNFs. and (b) Temperature fields for two HNFs.

Impact of the velocity slip variable (A): The slippery effect, which modifies the flow rates of both HNFs, is seen in Figure 9. Fluid particles are essential in controlling the slippery support that is given to the fluid. Due to this, the overall fluidity becomes slower than usual as a result. It can be seen in Figure 9(a). From Figure 9(b), we can observe that the thermal dispersion becomes higher for elevated values of slip variable λ, given the comparison of both HNFs for velocity and temperature fields.

Figure 9. Velocity and temperature fields for distinct values of A. (a) Velocity profiles for two HNFs. and (b) Temperature fields for two HNFs.

Impact of the radiation parameter $R_d$, the heat source parameter Q, and the thermal slip parameter B on the temperature field: The significance of thermal radiation in the energy balance of a fluid flow in MHD is of considerable importance. For the increasing levels of thermal-based radiation $R_d$, Figure 10(a) displays a state of the thermal diffusion. In other words, when radiative heat transfer increases, the temperature of the HNF rises. Physically, increasing radiation releases more heat energy within the flow, which aids in increasing the temperature of the TH-HNF. Figure 10(b) depicts the temperature distribution for a varying heat source or sink parameter Q. In this figure, we can see that as the values of Q go up, the fluid temperature goes down. From Figure 10(c), we can observe that the thickness of the thermal boundary layer decreases as the thermal slip parameter B increases. Hence, the temperature of the HNF drops. That is because fluid transfers a small quantity of heat away from the stretching sheet. In all these three cases, we can see that water-based HNF is much ahead of the other one.

Figure 10. Temperature profiles for distinct values of $R_d,Q,B$. (a) Temperature fields for two HNFs at various values of $R_d$. (b) Temperature fields for two HNFs at various values of Q and (c) Temperature profiles for two HNFs at various values of B.

Influence of nanoparticle volume fraction variables ${\varphi }_1$ and ${\varphi }_2$: The proportion of the volume of nanoparticles present in the fluid can influence a fluid's velocity and temperature profiles. When nanoparticles are introduced into a fluid, they can interact with the fluid in several ways, such as altering its viscosity, density, and thermal conductivity. Consequently, nanoparticle volume fraction affects fluid velocity and temperature. It can be seen in Figure 11. The velocity profile falls as the volume fraction of both nanoparticles increases, as can be seen in Figure 11(a). The more excellent thermal attraction time is found in fluids with constrained velocity. Therefore, the thickness of the thermal boundary layer rises as increasing volume fractions. It can be observed in Figure 11(b).

Figure 11. Velocity and temperature profiles for various values of ${\varphi }_1$ and ${\varphi }_2$. (a) Velocity profiles for two HNFs at various values of ${\varphi }_1$ and ${\varphi }_2$. and (b) Temperature profiles for two HNFs at various values of ${\varphi }_1$ and ${\varphi }_2$.

Parametric impression on the skin friction and the local Nusselt number: The skin friction and the local Nusselt number are essential in fluid dynamics, especially when studying how heat moves through a fluid by convection. Skin friction is the force per unit area that acts parallel to a solid surface on a fluid in contact. The fluid's viscosity induces it and is frequently described as the shear stress at the solid's surface. The rate of heat transfer or the local Nusselt number, on the other hand, is a dimensionless number that is used to describe the convective heat transfer between a fluid and a solid surface. In Tables 6 and 7, the parametrical viewpoints are displayed. It can be observed that the surface drag force is reduced by increasing the volume fraction of $\mathrm{C}\mathrm{u}$. In contrast, the skin friction coefficient is raised by increasing the volume fraction of ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ in both base fluids. However, it can be seen that $\mathrm{E}\mathrm{G}$+water-based HNF have slightly more skin friction and also a high Nusselt number. A similar effect on the rate of heat transfer as the $\mathrm{C}\mathrm{u}$ and ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nanoparticles volume fractions rise. The surface drag force and the heat transfer rate decrease when the values of the $W_e$, the n, and the velocity slip variable A are increased. On the other hand, when the values of the $H_a$, an angle of the inclined magnetic field ${\theta }_M$, and the porosity parameter λ are increased, the skin friction coefficient is increased and reduces the rate of heat transfer. Also, the more significant thermal radiation parameter $R_d$, heat source parameter Q, and thermal slip parameter B drop the impact on the Nusselt number.

Table 6. Impact of distinct physical parameters on skin friction for two HNFs.

${\varphi }_1$	${\varphi }_2$	$W_e$	n	$H_a$	A	${\theta }_M$	λ	${\mathcal{C}}_{fx}R{e}_x^{1/2}$ $\mathrm{C}\mathrm{u}+{\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/$ ${\mathrm{H}}_2\mathrm{O}$	${\mathcal{C}}_{fx}R{e}_x^{1/2}$ $\mathrm{C}\mathrm{u}+{\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/$ $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$
0.04	0.04	0.2	0.2	0.2	0.1	$\pi /2$	0.1	0.777340	0.787920
0.01								0.830638	0.837197
0.02								0.810402	0.818529
0.03								0.792759	0.802208
	0.01							0.766732	0.775355
	0.02							0.769942	0.779229
	0.03							0.773478	0.783417
		0.1						0.782632	0.793237
		0.2						0.777340	0.787920
		0.3						0.771937	0.782490
			0.1					0.838296	0.849595
			0.2					0.777340	0.787920
			0.3					0.711195	0.721004
				0.0				0.724626	0.733585
				0.1				0.751613	0.761412
				0.2				0.777340	0.787920
					0.0			0.921208	0.932466
					0.1			0.777340	0.787920
					0.2			0.676618	0.686454
						$\pi /6$		0.738287	0.747673
						$\pi /4$		0.751613	0.761412
						$\pi /2$		0.777340	0.787920
							0.0	0.752428	0.762251
							0.1	0.777340	0.787920
							0.2	0.801194	0.812483

Table 7. Impact of various physical parameters on Nusselt number for two HNFs.

${\varphi }_1$	${\varphi }_2$	$W_e$	n	$H_a$	A	${\theta }_M$	λ	$R_d$	Q	B	$\mathcal{N}{u}_{x}R{e}_x^{-1/2}$ $\mathrm{C}\mathrm{u}+{\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/$ ${\mathrm{H}}_2\mathrm{O}$	$\mathcal{N}{u}_{x}R{e}_x^{-1/2}$ $\mathrm{C}\mathrm{u}+{\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3/$ $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$
0.04	0.04	0.2	0.2	0.2	0.1	$\pi /2$	0.1	0.2	0.2	0.1	1.63527	2.15278
0.01											1.57282	2.00957
0.02											1.59354	2.05615
0.03											1.61464	2.10387
	0.01										1.54604	1.99001
	0.02										1.57574	2.04324
	0.03										1.60546	2.09748
		0.1									1.64075	2.15289
		0.2									1.63527	2.15278
		0.3									1.62971	2.14754
			0.1								1.69028	2.20063
			0.2								1.63527	2.15278
			0.3								1.56407	2.09136
				0.0							1.70475	2.21368
				0.1							1.66983	2.18269
				0.2							1.63527	2.15278
					0.0						1.77922	2.31151
					0.1						1.63527	2.15278
					0.2						1.51570	2.02155
						$\pi /6$					1.68711	2.19802
						$\pi /4$					1.66983	2.18269
						$\pi /2$					1.63527	2.15278
							0.0				1.66877	2.18175
							0.1				1.63527	2.15278
							0.2				1.60173	2.12437
								0.1			1.80856	2.58112
								0.2			1.63527	2.15278
								0.3			1.51557	1.91191
									0.1		1.79554	2.29309
									0.2		1.63527	2.15278
									0.3		1.40489	1.97823
										0.0	1.82215	2.48834
										0.1	1.63527	2.15278
										0.2	1.48313	1.89693

7. Final outcomes with concluding remarks

The current research has attempted to examine the two-dimensional MHD TH-HNF flow towards a porous stretchable sheet under the velocity and thermal slip conditions upon considering the porous medium on the sheet. The figures and tables show the significant influence of external physical factors on the TH-HNF flow over a stretched sheet, including inclined magnetic field, thermal radiation, heat absorption/generation parameter, and permeability parameter. Two distinct tangent hyperbolic working fluids are considered in this study, where the first is the $\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}{\mathrm{O}}_3/{\mathrm{H}}_2\mathrm{O}$ HNF, and the second is the $\mathrm{C}\mathrm{u}-\mathrm{A}\mathrm{l}{\mathrm{O}}_3/\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$ HNF. The computational outcomes of this research are presented graphically and numerically. Here are some of the most essential takeaways from this study:

• The velocities of the ${\mathrm{H}}_2\mathrm{O}$-based and $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$-based HNF decrease due to the increasing Wiessenberg number $W_e$ values. Still, the velocity of the EG+water-based HNFs stays slightly ahead of the other one. Similarly, other parameters like an inclined magnetic field, velocity slip parameter, and permeable parameter are also obstacles to the fluid flow.

• Conversely, with high values of parameters like the Weissenberg number, inclined magnetic field, velocity slip condition, porosity parameter, thermal radiation, and heat generation parameter, the temperature distributions of both ${\mathrm{H}}_2\mathrm{O}$-based and $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$-based HNFs are boosted. In contrast, the temperature distribution is reduced by the thermal slip condition. But, here, the thickness of the thermal boundary layer of ${\mathrm{H}}_2\mathrm{O}$-based HNF is remain higher than $\mathrm{E}\mathrm{G}+{\mathrm{H}}_2\mathrm{O}$-based HNF.

• The velocity profiles reduce as the volume fractions of copper and aluminium oxide nanoparticles grow significantly. In contrast, the temperature profiles increase considerably due to these higher volume fractions.

• The surface drag force can be enhanced by increasing the volume fractions of ${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ nanoparticles in both base fluids. Still, increasing the amount of $\mathrm{C}\mathrm{u}$ nanoparticles in both HNFs makes the surface drag force go down. Similar effects can be observed on the Nusselt number. But here, ethylene glycol and water-based HNF have a more significant skin friction coefficient and Nusselt number.

• Finally, the skin friction and the local Nusselt number decrease with the increasing the values of $W_e$, n, and velocity slip parameter A values. On the other hand, greater values of $H_a$, magnetic field inclination ${\theta }_M$, and the porosity parameter values result in higher skin friction coefficients and slower Nusselt numbers. Also, the influence on the Nusselt number can be reduced by increasing the values of the thermal radiation parameter $R_d$, the heat sink/source parameter Q, and the thermal slip parameter B.

In this study, we have considered two distinct base fluids, (i) water and (ii) a proportion of a mixture of $50\mathrm{\%}$ $\mathrm{E}\mathrm{G}$ and $50\mathrm{\%}$ ${\mathrm{H}}_2\mathrm{O}$, to form two different HNFs. In future, the base fluid can be considered as a mixture of different proportions of volume ratios of $\mathrm{E}\mathrm{G}$ to ${\mathrm{H}}_2\mathrm{O}$, such as $20\mathrm{\%}:80\mathrm{\%},40\mathrm{\%}:60\mathrm{\%},60\mathrm{\%}:40\mathrm{\%},80\mathrm{\%}:20\mathrm{\%}$ with the consideration of different shapes of nanoparticle such as bricks, cylinders, platelets, and blades.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

No data is associated with the manuscript.