MHD Mixed Convection Flow of Hybrid Ferrofluid through Stagnation-Point over the Nonlinearly Moving Surface with Convective Boundary Condition, Viscous Dissipation, and Joule Heating Effects

Abstract

This paper discusses a numerical study performed in analysing the performance regarding the magnetic effect on the mixed convection stagnation-point flow of hybrid ferrofluid, examining the influence of viscous dissipation, convective boundary condition as well as Joule heating across a nonlinearly moving surface. Additionally, the hybrid ferrofluid exhibits an asymmetric flow pattern due to the buoyancy force affecting the flow. Water $\left({\mathrm{H}}_2\mathrm{O}\right)$ is employed as the base fluid collectively with the mixtures of nanoparticles containing magnetite $\left(\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4\right)$ and cobalt ferrite $\left(\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4\right)$, forming a hybrid ferrofluid. The partial differential equation’s complexity is reduced by similarity transformation into a system of ordinary differential equations, which are then numerically solved by applying the MATLAB function bvp4c for a specific range of values regarding the governing parameters. Dual solutions were identified under both opposing and assisting flow conditions, and the stability analysis identified that the first solution was stable. Furthermore, it was also revealed that the addition of 1% $\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$ in hybrid ferrofluid led to a higher skin friction coefficient between 3.35% and 7.18% for both assisting and opposing flow regions. Additionally, the growth of magnetic fields results in a reduced heat transfer rate between 8.75% to 10.65%, whilst the presence of the suction parameter expands the range of solutions, which then delays the boundary layer separation. With the Eckert number included, the heat transfer rate continuously declined between 7.27% to 10.24%. However, it increased by about 280.64% until 280.98% as the Biot number increased.

1. Introduction

Convective transport has played a vital role in various real-world applications, for instance, energy-distribution networks, energy-generating plants, as well as environmental concerns. Convective transport notions and theories have been incorporated in several fluid models that contain both Newtonian and non-Newtonian fluid types owing to the breadth and depth of their theory as well as concepts. Here, the findings of the optimal thermal fluid result in further advancements in fluid flow modelling. Due to its economic significance, nanotechnology is now more prevalent across a broad range of industries. The application of nanotechnology, which involves the combination of base fluid and nanoparticles, has brought about a modification to heat transfer fluids. Beyond that, Choi and Eastman [1] used the term “nanofluid” to describe a brand-new kind of fluid. Magnetic nanoparticles are disseminated in a non-magnetic base fluid in a magnetic nanofluid, commonly known as ferrofluid [2]. In the nanotechnology sector, ferrofluid is employed in various applications, for example, coolants in thermal management systems, heat exchangers, and biomedical fields, for the treatment and diagnosis concerning diseases, such as cancer, rheumatoid arthritis, as well as angiocardiopathy [3]. Ferrofluid research in areas with severe heat transfer has thus received a lot of interest [4,5]. It may also encourage cross-disciplinary research in the fields of physics, chemistry, biology, as well as material science. Due to these concerns, various researchers and academicians have performed numerical and experimental studies on nano/ferrofluid behaviour concerning heat transfer issues [6,7,8,9,10,11].

The qualities of nano/ferrofluid are further enhanced by introducing a hybrid nanofluid, which combines two different nanomaterials to enhance thermal and rheological characteristics. Here, the occurrence of two or more nanoparticle types in a hybrid nanofluid improves its heat-transfer efficiency, thermophysical properties, as well as stability in comparison to those single-nanoparticle nanofluids [12]. The study of hybrid nanofluid has become more important in the heat transmission field considering renewable energy, electromechanical engineering, as well as automotive cooling [13]. This results from the wide variety of applications that can be found for the technology. In contrast, hybrid nanofluid transmits heat more quickly than other materials. Multiple review articles [14,15,16,17] have provided in-depth descriptions of the hybrid nanofluid. Hybrid nanoparticles were deployed in the earlier investigations by Turcu et al. [18] and Jana et al. [19] in their analysis. Aminuddin et al. [20] conducted a study on the fluid flow and heat transfer problem on a hybrid nanofluid containing graphene oxide $\left(\mathrm{GO}\right)$ and iron dioxide $\left(\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_4\right)$ dispersed in ethylene glycol (EG) as the base fluid. Their study on hybrid nanofluid includes several factors, such as magnetic field, thermal radiation, and viscous dissipation along with the presence of velocity slip. Interestingly, they found out that the heat transfer performance of hybrid nanofluid was enhanced with the influence of thermal radiation and velocity slip. Beyond that, Kho et al. [21] investigated the fluid flow and heat transfer problem of $\mathrm{Ag}-\mathrm{Ti}{\mathrm{O}}_2$ hybrid nanofluid on different geometry, which is on a permeable wedge displaying several effects, such as a magnetic field, thermal radiation, and viscous dissipation. In their findings, the heat transfer rate for $\mathrm{Ag}-\mathrm{Ti}{\mathrm{O}}_2$ hybrid nanofluid was enhanced with a higher concentration of $\mathrm{Ti}{\mathrm{O}}_2$. Recently, Atashafrooz et al. [22] conducted a study on $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{CuO}/\mathrm{water}$ hybrid nanofluid for fluid flow and heat transfer problems inside an open trapezoidal enclosure with the impacts of magnetic field and thermal radiation. The fact that many researchers have evaluated numerical as well as experimental studies employing the hybrid nanofluid concept after that is worth noting. Scholars frequently employ Takabi and Salehi’s [23] as well as Devi and Devi’s [24,25] thermophysical correlations for hybrid nanofluids when undertaking numerical studies driven by a volume fraction-based model for hybrid nanofluids. The recent studies on thermophysical correlations by Devi and Devi [26,27,28,29] and [30,31] on thermophysical correlations by Takabi and Salehi.

Hybrid ferrofluid, which features a range of nanoparticles scattered throughout the ferrofluid, has been discovered by researchers to build a more effective fluid. Hybrid ferrofluids have caught the interest of numerous academicians in finding solutions to practical heat transmission issues because of several significant applications, which include heat dissipation, damping, dynamic sealing, etc. The utilisation of hybrid ferrofluid as a seed material to cure acid mine drainage (AMD) is an application that can be highlighted as especially significant. A long-lasting option for wastewater treatment, especially for AMD, will be provided using a hybrid ferrofluid. This is because AMD will affect animal life, aquatic life, human life, plant life, and other forms of life, as well as inflict serious environmental damage [32]. Mathematical models have proliferated rapidly alongside the expansion of theoretical work. However, a review of the existing research reveals that only a small number of studies have concentrated on heat and flow transfer regarding the hybrid ferrofluid. Kumar et al. [33] proposed further study of MHD hybrid ferrofluid under the influence of non-uniform heat sources and sank on the transport of radiative thin films. They found that a hybrid ferrofluid is significantly more efficient in heat conduction than ferrofluid itself. The subsequent study of an MHD hybrid ferrofluid subjected to asymmetrical heat fall/rise was conducted by Tlili et al. [34]. In other research, Manh et al. [35] investigated how a hybrid ferrofluid submerged in porous media reacted to magnetic and radiation fields in terms of heat transmission. Recent research on this topic found that Zainodin et al. [36] examined the heat source/sink effect as well as velocity slip on exponentially deformable sheets around stagnation-point in a mixed convection flow on a hybrid ferrofluid ($\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$). In addition to the works already mentioned, references [37,38,39] contain other noteworthy publications on hybrid ferrofluid.

Due to the high level of practical interest, research has been performed on the flow properties and heat transmission concerning mixed convection. There are a wide variety of technological and industrial applications for mixed convection flow, including electronic gadgets, pipeline transportation, and nuclear reactors. The term “mixed convection” refers to a process in which both forced and free convection occurs. Convection, in general, is a method of heat transmission in which fluid moves from a hotter substance to a colder substance. It happened when both forced and natural convection mechanisms were operating at once to support flow and heat transmission. Sparrow et al. [40] conducted an earlier investigation involving mixed convection flow on the boundary layer by adding the assisting and opposing flow in their analysis. Concurrently, Jamaludin et al. [41] consider the suction effect along with heat source/sink for mixed convection flow in a nanofluid. Following then, interest in mixed convection among scholars increased, leading to some noteworthy mention [42,43,44]. Since then, scholars have taken MHD into account in mixed convective flow to significantly improve boundary layer and heat transmission features. Since it depends on the heat observing variable, the MHD term is especially crucial for verifying heat transfer while producing significant items (MHD power generation, nuclear reactor, MHD generator) [45]. Khan and Rasheed [46] investigated the heat transfer performance of nanofluid in MHD mixed convection flow with thermal radiation by accounting for the magnetic field impact in the mixed convection flow. They identified that the Lorentz force reduced fluid flow velocity, as well as reduced skin friction following the resistance it caused in flows through MHD. The effect of suction and thermal radiation on MHD mixed convection flow for ferrofluid over the nonlinearly moving surface was then addressed by Jamaludin et al. [8]. It is appealing that the existence of MHD and suction expanded the solution range and delayed boundary layer separation. In addition, Wahid et al.’s [47] examination of the heat transfer effectiveness of a hybrid nanofluid established that the efficiency rose as the magnetic values rose. After decades of study, researchers have been fascinated by the analysis of MHD mixed convection flow, which has led to several important mentions in studies of boundary layers and heat transfer rate [27,48,49,50].

In several fields of engineering and business, the Joule heating mechanism is being used more frequently, including the wiring of electrical and electronic devices. In theory, heat is generated because of resistive losses during the Joule heating process, which transforms electrical energy into thermal energy. The Joule heating characteristics are often described by the Eckert number and magnetic parameter in combination, while the viscous dissipation effects are usually described by the Eckert number alone. According to Reddy and Reddy [51], this control parameter is employed to raise the nanofluid temperature from the boundary layer flow perspective. Here, Khashi’ie et al. [52] examined the influence of Joule heating in a $\mathrm{Cu}$–$\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{water}$hybrid nanofluid by taking into consideration the effect of MHD and suction over a moving plate. Their finding was that heat transfer performance is reduced by the Eckert number caused by Joule heating, while it has zero impact on the boundary layer separation. Concurrently, Yashkun et al. [53] assessed the MHD mixed convection hybrid nanofluid flow regarding Joule heating passing through the exponentially moving surface. Their research revealed the presence of a dual solution for the shrinking region within a particular range, and the impact of Ec widened the range of solutions. Recently, Zainal et al. [54] sought to conduct research on hybrid nanofluids toward stagnation-point flow for the unsteady flow problem while considering the influence of Joule heating together with viscous dissipation. The examination of boundary layers and heat transport in Joule heating has been a fascinating topic addressed by researchers over the years, and it has produced several noteworthy mentions [55,56,57,58].

Additionally, the attention in boundary layer flow exploration also includes the convective boundary condition. Convective boundary conditions are presumed to exist when heat convection and conduction at a material’s surface both flow in the same direction. Physically speaking, it is possible to conceptualise it as a state in which the heat conduction rate at the surface equals the heat convection rate. This kind of scenario offers broad and accurate forecasting, especially in a variety of engineering and industrial tasks like the conjugate heat transfer around fins, material drying, heat exchangers, the transpiration cooling process, and so forth. Similarly, high-temperature processes are required for convective heat transfers. Such circumstances occur in nuclear power plants, material drying, laser therapy, haemodialysis, oxygenation, sanitary fluid transmission, and gas turbines [59,60]. The effect of convective boundary conditions on MHD flow regarding hybrid nanofluid over the permeable moving surface is discussed by Aly and Pop [61] in their paper. They found that the temperature profile changed because of the rise in the Biot number, which caused an increase in the thickness of the thermal boundary layer in both instances of stretching and shrinking. Jusoh et al. [62] analyse the heat transfer performance for Ag–Cu hybrid nanofluid flow over the stretching/shrinking surface with the influence of convective boundary conditions along with viscous dissipation. They determined that as the Biot number rose, the temperature profile steadily decreased because the thermal reversal was caused by viscous dissipation. Recently, Khashi’ie et al. [63] evaluated the heat transfer performance regarding the MHD stagnation-point of hybrid nanofluid flow on a shrinking disk, including the influence of convective boundary conditions, Joule heating as well as viscous dissipation. Notably, a few intriguing articles [64,65,66,67] investigated convective boundary conditions.

As far as the authors are concerned, based on the literature reviewed and experience, no prior study has examined the boundary layer flow as well as heat transfer analysis, including the identification of the dual solutions and the stability of the system on the effects of Joule heating, viscous dissipation, along with the convective boundary condition by integrating mixed convection hybrid ferrofluid stagnation-point flow across a nonlinearly permeable moving surface. Moreover, a review of the existing research reveals that a limited number of studies have concentrated on heat and flow transfer regarding the similarity variable proposed by Shen et al. [68]. Hence, the current work is motivated by the aforementioned literature and aims to fill the void by broadening the work of Jamaludin et al. [8] as well as Shen et al. [68] in developing new mathematical models following the Tiwari and Das’s model [69]. This is performed via the nonlinearly permeable moving surface by including new control parameters, such as viscous dissipation, Joule heating, including convective boundary conditions. Beyond that, this current work also contributed with a distinct type of fluid that incorporates an advanced type of nanofluid known as $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid into the MHD mixed convection stagnation-point flow compared to the previous studies. Noteworthy to mention, the novel aspect of this work is the unforeseen complexity of the flow and heat transfer behaviour of $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid comes from the interaction of the relevant regulating parameters, such as convective boundary conditions, Joule heating, viscous dissipation, as well as effects, such as a magnetic field and suction across a nonlinearly permeable moving surface. Via the similarity transformations, one can acquire the fluctuation in a mathematical model that may subsequently be solved numerically via the use of the MATLAB program bvp4c. An additional interesting aspect that will be explored is the possibility of dual solutions (first as well as second solutions). As proposed by Merkin [70] as well as Weidman and Turner [71], the dual solutions will go through the stability analysis procedure. The stability between the two different solutions is an important consideration that must be considered. It provides a technique for determining which solution is stable, as other scholars [27,42,54] have already done in their outstanding study on stability analysis.

2. Mathematical Formulations

This paper covered the steady, two-dimensional mixed convection flow of a viscous and incompressible electrically conducting fluid and heat transfer through a penetrable nonlinearly stretching/shrinking sheet. The coordinate frame was described as illustrated in Figure 1, where the stretched $x$-axis by the surface of the sheet and the measured orthogonal of the $y$-axis to the surface. Additionally, the sheet was positioned vertically, and the flow was along the $x$-axis. Moreover, the surface velocity $u_w\left(x\right)=b{x}^n$ caused a stretching $(b>0)$ or a shrinking $(b<0)$ of the wall surface in the $x$-direction. The flow was affected by Joule heating and a transverse magnetic field of intensity $B_y\left(x\right)=B_0{x}^{\left(n-1\right)/2}$ that was employed along the positive $y$-axis orthogonal to the surface being shrunk or stretched. It was hypothesised that the produced magnetic field was inconsequential regarding the results of the current research. Additionally, the flow near the stagnation points free-stream velocity is expressed by the symbol $u_e\left(x\right)$, whereas the velocity of the wall’s mass flux is defined by $v_w\left(x\right)=-\frac{n+1}{2}\sqrt{a{v}_f}{x}^{\left(n-1\right)/2}S$, in which $v_w<0$ in the case of injection as well as $v_w>0$ in the case of suction.

Next, convection from the hot fluid is used to maintain the sheet’s temperature $T_w\left(x\right)=T_{\infty }+T_0{x}^{2n-1}$, where $h_f=h_0{x}^{\left(n-1\right)/2}$ (see Fatunmbi et al. [66]) is the coefficient of heat transfer. The ambient temperature at this point was $T_{\infty }$. The critical assumption of this study is that the stagnation point happens when the plate temperature equals the ambient air temperature, denoted by $T_w\left(x\right)=T_{\infty }$. In the case of a mixed convection fluid flow, buoyancy force can arise due to density differences in the fluid. The temperature gradient difference in the fluid causes density variations that lead to buoyancy forces. As a result, the fluid can flow asymmetrically due to the buoyancy force.

The ferroparticles in the working fluid are a mix of magnetite $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$ and cobalt ferrite $\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$, making it a hybrid ferrofluid $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$ with water as its base fluid. Here, the first ferroparticle volume fraction parameter, ${\varphi }_1$ in this case, is represented by $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$, while the second ferroparticle volume fraction parameter ${\varphi }_2$ is represented by $\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$. Although the agglomeration effects are ignored, the hybrid ferrofluid takes on a stable spherical shape that is nanoparticle-sized throughout. The nanofluid model suggested by Tiwari and Das [69] is implemented in this instance. It is essential to acknowledge the single-phase approach employed in this nanofluid model as well as the assumption that all the particles are uniform in shape and size. In addition, the interactions between the particles and the fluid with which they come into contact are not considered [72,73]. Under the assumption of the boundary layer and Boussinesq approximations mentioned before, the governing equations may be expressed (Jamaludin et al. [8]; Khashi’ie et al. [52]; Tiwari and Das [69]; Lund et al. [74])

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=u_e\frac{\partial u_e}{\partial x}+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\sigma }_{hnf}{B}_y^2}{{\rho }_{hnf}}\left(u_e-u\right)+\frac{{\left(\rho \beta \right)}_{hnf}}{{\rho }_{hnf}}\left(T-T_{\infty }\right)g,$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\mu }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{{\sigma }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}{B}_y^2{\left(u_e-u\right)}^2,$$

(3)

with respect to the boundary conditions (Jamaludin et al. [8]; Zainal et al. [65])

$$\begin{gathered} u=u_w\left(x\right),v=v_w\left(x\right),-k_{hnf}\frac{\partial T}{\partial y}=h_f\left(T_w\left(x\right)-T\right) \mathrm{a}\mathrm{t} y=0, \\ u\to u_e\left(x\right)=a{x}^n,T\to T_{\infty } \mathrm{a}\mathrm{s} y\to \infty . \end{gathered}$$

(4)

At this point, the $x$-axis velocity component is denoted by $u$, while the $y$-axis velocity component is expressed by $v$. Additionally, the acceleration resulting from gravity is defined as $g$, the temperature of the hybrid ferrofluid is referred to by $T$, $T_0$ denotes the hybrid nanofluid temperature characteristic with $T_0>0$ signifies assisting flow, whereas $T_0<0$ signifies opposing flow, and the mass flux velocity $S$ is categorised into $S>0$ for suction while $S<0$ for injection. Moreover, $a$, $b,$ and $n$ represent a positive constant. Furthermore, ${\left(\rho C_p\right)}_{hnf},{{\left(\rho \beta \right)}_{hnf},{\rho }_{hnf},\mu }_{hnf},$ and ${\sigma }_{hnf}$ illustrate the heat capacitance, thermal expansion coefficient, density, dynamic viscosity, as well as the electrical conductivity of the hybrid ferrofluid ($\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$), respectively. Furthermore, the thermal diffusivity of the hybrid ferrofluid is defined as ${\alpha }_{hnf}=k_{hnf}/{\left({\rho C}_p\right)}_{hnf}$, where $k_{hnf}$ expresses the thermal conductivity of the hybrid ferrofluid. In addition,

Table 1. The theoretical model for properties of the nanofluid and hybrid nanofluid (Devi and Devi [24]; Jamaludin et al. [27]; Oztop and Abu Nada [75]).

Properties	Nanofluid	Hybrid Nanofluid
Viscosity	${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}$	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}}$
Density	${\rho }_{nf}=\left(1-{\varphi }_1\right){\rho }_f+{\varphi }_1{\rho }_{s1}$	${\rho }_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\rho }_f+{\varphi }_1{\rho }_{s1}\right]+{\varphi }_2{\rho }_{s2}$
Electrical Conductivity	$\frac{{\sigma }_{nf}}{{\sigma }_f}=1+\frac{3({\sigma }_{s1}/{\sigma }_f-1){\varphi }_1}{{\sigma }_{s1}/{\sigma }_f+2-({\sigma }_{s1}/{\sigma }_f-1){\varphi }_1}$	$\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_{s2}+2{\sigma }_{nf}-2{\varphi }_2\left({\sigma }_{nf}-{\sigma }_{s2}\right)}{{\sigma }_{s2}+2{\sigma }_{nf}+{\varphi }_2\left({\sigma }_{nf}-{\sigma }_{s2}\right)}$             where $\frac{{\sigma }_{nf}}{{\sigma }_f}=\frac{{\sigma }_{s1}+2{\sigma }_f-2{\varphi }_1\left({\sigma }_f-{\sigma }_{s1}\right)}{{\sigma }_{s1}+2{\sigma }_f+{\varphi }_1\left({\sigma }_f-{\sigma }_{s1}\right)}$
Thermal Diffusivity	${\alpha }_{nf}=\frac{k_{nf}}{{\left({\rho C}_p\right)}_{nf}}$	${\alpha }_{hnf}=\frac{k_{hnf}}{{\left({\rho C}_p\right)}_{hnf}}$
Thermal Expansion Coefficient	${\left(\rho \beta \right)}_{nf}=\left(1-{\varphi }_1\right){\left(\rho \beta \right)}_f+{\varphi }_1{\left(\rho \beta \right)}_{s1}$	${\left(\rho \beta \right)}_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left(\rho \beta \right)}_f+{\varphi }_1{\left(\rho \beta \right)}_{s1}\right]+{\varphi }_2{\left(\rho \beta \right)}_{s2}$
Heat Capacity	${\left({\rho C}_p\right)}_{nf}=\left(1-{\varphi }_1\right){\left({\rho C}_p\right)}_f+{\varphi }_1{\left({\rho C}_p\right)}_{s1}$	${\left({\rho C}_p\right)}_{hnf}=\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right){\left({\rho C}_p\right)}_f+{\varphi }_1{\left({\rho C}_p\right)}_{s1}\right]+{\varphi }_2{\left({\rho C}_p\right)}_{s2}$
Thermal Conductivity	$\frac{k_{nf}}{k_f}=\frac{k_{s1}+2k_f-2{\varphi }_1\left(k_f-k_{s1}\right)}{k_{s1}+2k_f+{\varphi }_1\left(k_f-k_{s1}\right)}$	$k_{hnf}=\frac{k_{s2}+2k_{nf}-2{\varphi }_2\left(k_{nf}-k_{s2}\right)}{k_{s2}+2k_{nf}+{\varphi }_2\left(k_{nf}-k_{s2}\right)}\times k_{nf}$

lists the theoretical models that were employed to describe the thermophysical properties of nanofluid as well as hybrid nanofluid, and

Table 2. Thermophysical properties of ferroparticles and water (base fluid) (Jamaludin et al. [8]; Kumar et al. [33]).

Thermophysical Properties	$\mathbf{F}{\mathbf{e}}_3{\mathbf{O}}_4$	$\mathbf{CoF}{\mathbf{e}}_2{\mathbf{O}}_4$	${\mathbf{H}}_2\mathbf{O}$
$C_p\left(Jk{g}^{-1}{K}^{-1}\right)$	670	700	4179
$\rho \left(kg{m}^{-3}\right)$	5180	4907	997.1
$k\left(W{m}^{-1}{K}^{-1}\right)$	9.7	3.7	0.613
$\beta \times {10}^{-5}\left(K^{-1}\right)$	0.5	1.3	21
$\sigma \left(S{m}^{-1}\right)$	$0.74\times {10}^6$	$1.1\times {10}^7$	$5.5\times {10}^{-6}$

compiles the thermophysical properties of the base fluid as well as suspended nanoparticles. It is worth mentioning that subscripts $nf$, $f$, as well as $hnf$ symbolise the nanofluid, fluid, and hybrid ferrofluid, respectively.

Following Jamaludin et al. [8] and Shen et al. [68], this study now presents the relevant similarity variables, which are defined as follows:

$$\begin{gathered} u=a{x}^n{f}^{\prime }\left(\eta \right),v=-\sqrt{a{v}_f}{x}^{\left(n-1\right)/2}\left[\frac{n+1}{2}f\left(\eta \right)+\frac{n-1}{2}\eta f^{\prime }\left(\eta \right)\right], \\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\eta =\sqrt{\frac{a}{v_f}}{x}^{\left(n-1\right)/2}y. \end{gathered}$$

(5)

In this context, “prime” denotes differentiation conditioned on $\eta$. When the similarity variables from Equation (5) are substituted into the governing equations from Equations (1)$-$(3) with boundary condition Equation (4), the ODEs are obtained as follows:

$$\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{‴}+\frac{{\rho }_{hnf}}{{\rho }_f}\left[\frac{n+1}{2}f{f}^{″}+n-n{\left(f^{\prime }\right)}^2\right]+\frac{{\sigma }_{hnf}}{{\sigma }_f}M\left(1-f^{\prime }\right)+\lambda \frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}\theta =0,$$

(6)

$$\begin{gathered} \frac{1}{\mathrm{Pr}}\frac{k_{hnf}}{k_f}{\theta }^{″}+\frac{{\left({\rho C}_p\right)}_{hnf}}{{\left({\rho C}_p\right)}_f}\left(\frac{n+1}{2}f{\theta }^{\prime }-\left(2n-1\right)f^{\prime }\theta \right)+\frac{{\mu }_{hnf}}{{\mu }_f}\mathrm{Ec}{\left(f^{″}\right)}^2 \\ +\frac{{\sigma }_{hnf}}{{\sigma }_f}M\mathrm{Ec}{\left(1-f^{\prime }\right)}^2=0, \end{gathered}$$

(7)

subject to

$$\begin{gathered} f\left(0\right)=S,f^{\prime }\left(0\right)=c,-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right)=\mathrm{B}\mathrm{i}\left(1-\theta \left(0\right)\right)\mathrm{at}\eta =0, \\ {f}^{\prime }\left(\eta \right)\to 1,\theta \left(\eta \right)\to 0 as \eta \to \infty , \end{gathered}$$

(8)

in which $M$ describes the magnetic parameter, $\mathrm{P}\mathrm{r}$ resembles the Prandtl number, $\mathrm{Ec}$ expresses the Eckert number, $c$ denotes the stretching/shrinking parameter for $c>0$ (shrinking) and $c<0$ (stretching), $\lambda$ represents the mixed convection parameter having conditions that $\lambda <0$ (opposing flow) or $\lambda >0$ (assisting flow), $S$ indicates as suction/injection parameter, where $S>0$ refers to for the suction and $S<0$ represents the injection, while $\mathrm{B}\mathrm{i}$ refers to the Biot number. The governing parameter stated above may be expressed as given below:

$$M=\frac{{{\sigma }_{f}B}_0^2}{a{\rho }_f}, \mathrm{P}\mathrm{r}=\frac{{\mu }_f{\left(C_p\right)}_f}{k_f},\mathrm{Ec}=\frac{u_e^2}{{\left(C_p\right)}_f\left(T_w-T_{\infty }\right)},c=\frac{b}{a},\lambda =\frac{G{r}_x}{\mathrm{R}{\mathrm{e}}_x^2}, \mathrm{B}\mathrm{i}=\frac{h_0}{k_f}\sqrt{\frac{v_f}{a}},$$

(9)

in which the local Grashof number, $\mathrm{G}{\mathrm{r}}_x$, as well as the local Reynold number, $\mathrm{R}{\mathrm{e}}_x$, are described as:

$$\mathrm{G}{\mathrm{r}}_x=\frac{g{\beta }_f\left(T_w-T_{\infty }\right)x^3}{v_f^2}, \mathrm{R}{\mathrm{e}}_x=\frac{u_{e}x}{v_f}.$$

(10)

The physical quantities regarding practical interest are the skin friction coefficient, $C_f$ with the local Nusselt number, $N{u}_x$ expressed as given below:

$$C_f=\frac{{\mu }_{hnf}}{{\rho }_f{u}_e^2}{\left(\frac{\partial u}{\partial y}\right)}_{y=0},N{u}_x=-\frac{x}{\left(T_w-T_{\infty }\right)}\frac{k_{hnf}}{k_f}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}.$$

(11)

Replacing Equation (5) with Equation (10) yields

$$\mathrm{R}{\mathrm{e}}_x^{1/2}{C}_f=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right),\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x=-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right).$$

(12)

3. Stability Analysis

According to the numerical results in Equations (6)$-$(8) and their respective governing parameters, it appears that there is a dual solution. As a direct result of this, a stability analysis was conducted to examine unstable and stable solutions. Equation (1) was maintained before the beginning of the stability analysis, but Equations (2)$-$(4) were deemed unsteady as follows:

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=u_e\frac{\partial u_e}{\partial x}+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\sigma }_{hnf}{B}_y^2}{{\rho }_{hnf}}\left(u_e-u\right)+\frac{{\left(\rho \beta \right)}_{hnf}}{{\rho }_{hnf}}\left(T-T_{\infty }\right)g,$$

(13)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\mu }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{{\sigma }_{hnf}}{{\left(\rho C_p\right)}_{hnf}}{B}_y^2{\left(u_e-u\right)}^2,$$

(14)

where $t$ represents the time. Following the similarity solutions (Equation (5)), a new similarity transformation will be introduced for the unsteady issue as follows:

$$\begin{gathered} u=a{x}^n\frac{\partial f}{\partial \eta }\left(\eta ,\tau \right),v=-\sqrt{a{v}_f}{x}^{\left(n-1\right)/2}\left[\frac{n+1}{2}f\left(\eta ,\tau \right)+\frac{n-1}{2}\eta \frac{\partial f}{\partial \eta }\left(\eta ,\tau \right)+\left(n-1\right)\tau \frac{\partial f}{\partial \tau }\left(\eta ,\tau \right)\right],\theta \left(\eta ,\tau \right)=\frac{T-T_{\infty }}{T_w\left(x\right)-T_{\infty }}, \\ \eta =y\sqrt{\frac{a}{v_f}},\tau =a{x}^{n-1}t. \end{gathered}$$

(15)

As a result, the modified Equations (13) and (14) can be expressed as follows:

$$\begin{gathered} \frac{{\mu }_{hnf}}{{\mu }_f}\frac{{\partial }^{3}f}{\partial {\eta }^3}+\frac{{\rho }_{hnf}}{{\rho }_f}\left[\frac{n+1}{2}f\frac{{\partial }^{2}f}{\partial {\eta }^2}+n-n{\left(\frac{\partial f}{\partial \eta }\right)}^2+\left(n-1\right)\tau \left[\frac{\partial f}{\partial \tau }\frac{{\partial }^{2}f}{\partial {\eta }^2}-\frac{\partial f}{\partial \eta }\frac{{\partial }^{2}f}{\partial \tau \partial \eta }\right]-\frac{{\partial }^{2}f}{\partial \tau \partial \eta }\right]+ \\ \frac{{\sigma }_{hnf}}{{\sigma }_f}M\left(1-\frac{\partial f}{\partial \eta }\right)+\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}\lambda \theta =0, \end{gathered}$$

(16)

$$\begin{gathered} \frac{1}{\mathrm{P}\mathrm{r}}\frac{k_{hnf}}{k_f}\frac{{\partial }^2\theta }{\partial {\eta }^2}+\frac{{\left({\rho C}_p\right)}_{hnf}}{{\left({\rho C}_p\right)}_f}\left[\frac{n+1}{2}f\frac{\partial \theta }{\partial \eta }-\left(2n-1\right)\theta \frac{\partial f}{\partial \eta }+\left(n-1\right)\tau \left[\frac{\partial f}{\partial \tau }\frac{\partial \theta }{\partial \eta }-\frac{\partial f}{\partial \eta }\frac{\partial \theta }{\partial \tau }\right]-\frac{\partial \theta }{\partial \tau }\right]+\frac{{\mu }_{hnf}}{{\mu }_f}\mathrm{E}\mathrm{c}{\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2+ \\ \frac{{\sigma }_{hnf}}{{\sigma }_f}M\mathrm{E}\mathrm{c}{\left(1-\frac{\partial f}{\partial \eta }\right)}^2=0, \end{gathered}$$

(17)

subject to

$$\begin{gathered} \frac{\partial f}{\partial \eta }\left(0,\tau \right)=c, f\left(0,\tau \right)=S,-\frac{k_{hnf}}{k_f}\frac{\partial \theta }{\partial \eta }\left(0,\tau \right)=\mathrm{B}\mathrm{i}\left(1-\theta \left(0,\tau \right)\right), \\ \frac{\partial f}{\partial \eta }\left(\eta ,\tau \right)\to 1, \theta \left(\eta ,\tau \right)\to 0\mathrm{as} \eta \to \infty . \end{gathered}$$

(18)

Weidman and Turner’s [71] work established the foundation for our investigation into the dual solutions’ stability, which may be summarised as follows:

$$f\left(\eta ,\tau \right)=f_0\left(\eta \right)+e^{-\gamma \tau }F\left(\eta \right), \theta \left(\eta ,\tau \right)={\theta }_0\left(\eta \right)+e^{-\gamma \tau }G\left(\eta \right),$$

(19)

in which $\gamma$ refers to the unknown eigenvalue determining the stability of the solution. Concurrently, $G\left(\eta \right)$ and $F\left(\eta \right)$ are relatively opposed to ${\theta }_0\left(\eta \right)$ and $f_0\left(\eta \right)$, respectively. The exponential disturbance demonstrates the disruption’s rapid growth or onset. Thus, by employing the time-dependent solutions of Equation (19) in Equations (16)$-$(18), the subsequent expressions can be obtained:

$$\frac{{\mu }_{hnf}}{{\mu }_f}{F}_0^{‴}+\frac{{\rho }_{hnf}}{{\rho }_f}\left[\frac{n+1}{2}\left(f_0{F}_0^{″}+f_0^{″}{F}_0\right)-\left[2n{f}_0^{\prime }-\gamma \right]F_0^{\prime }\right]-\frac{{\sigma }_{hnf}}{{\sigma }_f}M{F}_0^{\prime }+\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}\lambda G_0=0,$$

(20)

$$\begin{gathered} \frac{1}{\mathrm{Pr}}\frac{k_{hnf}}{k_f}{G}_0^{″}+\frac{{\left({\rho C}_p\right)}_{hnf}}{{\left({\rho C}_p\right)}_f}\left[\frac{n+1}{2}\left(f_0{G}_0^{\prime }+F_0{\theta }_0^{\prime }\right)-\left(2n-1\right)\left(F_0^{\prime }{\theta }_0+f_0^{\prime }{G}_0\right)+\gamma G_0\right]+ \\ 2\frac{{\mu }_{hnf}}{{\mu }_f}\mathrm{Ec}{f}_0^{″}{F}_0^{″}-2\frac{{\sigma }_{hnf}}{{\sigma }_f}M\mathrm{Ec}{F}_0^{\prime }\left(1-f_0^{\prime }\right)=0, \end{gathered}$$

(21)

subject to

$$\begin{gathered} F_0^{\prime }\left(0\right)=0, F_0\left(0\right)=0,-\frac{k_{hnf}}{k_f}{G^{\prime }}_0\left(0\right)+\mathrm{B}\mathrm{i}{G}_0\left(0\right)=0, \\ {F}_0^{\prime }\left(\eta \right)\to 0, G_0\left(\eta \right)\to 0\mathrm{as} \eta \to \infty . \end{gathered}$$

(22)

The infinite set of eigenvalues ${\gamma }_1<{\gamma }_2<{\gamma }_3<$… were given by Equations (20)$-$(22), in which ${\gamma }_1$ represents the smallest eigenvalue which is crucial in determining the stability with respect to the dual solution. As per Weidman and Turner [71], either one of the boundary conditions $F_0^{\prime }\left(\infty \right)=0$ as well as $G_0\left(\infty \right)=0$ will be relaxed to demonstrate the infinite set of eigenvalues ${\gamma }_1<{\gamma }_2<{\gamma }_3<$…. Furthermore, the boundary condition $F_0^{\prime }\left(\infty \right)=0$ was taken into consideration to be relaxed and then substituted with a new boundary condition $F_0^{″}\left(0\right)=1$ in solving the linearised equation stated before.

4. Results and Discussion

The boundary value problem that was provided, in addition to the boundary conditions in Equation (8), was numerically assessed using Equations (6) and (7) and the MATLAB bvp4c solver. Several governing parameters are considered in this work, including the volume percentage of nanoparticles ${\varphi }_1$, ${\varphi }_2$, magnetic parameter $M$, mixed convection parameter $\lambda$, Biot number Bi, Eckert number Ec, stretching/shrinking parameter $c$, as well as suction parameter $S$. Values ${\varphi }_1=0.01$, ${\varphi }_2=0.01$, $\mathrm{Ec}=0.1$, $\mathrm{Bi}=0.2$, $M=1$, and $S=1$ were implemented in MATLAB during numerical simulations and may be altered relying on a variety of governing parameters. Here, the Prandtl number was set to 6.2 (water) while the surface was shrunk for $c=-0.5$ and $n$ was set to 3 (nonlinear) for all calculations in this work. The findings for nanofluid, $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$/water, between skin friction coefficient, ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$, including Nusselt number, $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$, were contrasted with those from Jamaludin et al. [8] as well as Nazar et al. [76] to confirm the existing model. To fulfil this validation, a necessary change was made by replacing the boundary condition in Equation (8) with $\theta \left(0\right)=1$ at $\eta =0$. These were calculated in accordance with

Table 3. Comparison values of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ for $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3/\mathrm{water}$ nanofluid with various values of $c$ for $\varphi =0.1$, $\mathrm{Pr}=1$, $n=1$, $M=S=\lambda =\mathrm{Ec}=\mathrm{Bi}=0$. The results that are denoted by parentheses ( ) pertain to the second solution.

$\mathit{c}$	${\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{1/2}\mathit{C}}_{\mathit{f}}$			$\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$
	Nazar et al. [76]	Jamaludin et al. [8]	Present Result	Nazar et al. [76]	Jamaludin et al. [8]	Present Result
−1.1	1.54239 (0.06399)	1.54239 (0.06399)	1.542389 (0.063985)	−0.06258 (−3.69342)	−0.06258 (−3.69356)	−0.062582 (−3.693556)
−1.15	1.40663 (0.15168)	1.40663 (0.15168)	1.406631 (0.151684)	−0.18285 (−2.41407)	−0.18287 (−2.41407)	−0.182867 (−2.414073)
−1.2	1.21198 (0.30369)	1.21198 (0.30369)	1.211984 (0.303686)	−0.35356 (−1.65139)	−0.35359 (−1.65140)	−0.353586 (−1.651395)

, which displayed a favourable comparison result. For the case of shrinking sheet $\left(c<0\right)$ with the absence of mixed convection $\left(\lambda =0\right)$, magnetic field $\left(M=0\right)$, suction $\left(S=0\right)$, Eckert number $\left(\mathrm{Ec}=0\right)$ and Biot number $\left(\mathrm{Bi}=0\right)$, it was found that the value of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ are reduced as the strength of the shrinking case drop from $-$1.1 to $-$1.2 as presented in Table 3. Moreover, this current model for the hybrid ferrofluid problem is reliable in producing dual solutions as this current model can produce and validate the dual solution with Jamaludin et al. [8] as well as Nazar et al. [76]. Additionally, the present results in Table 3 were extended until 6 decimal places for the accuracy of the numerical result.

The volume fraction of the nanoparticles is a critical physical metric in examining how well nanoparticles impact the rate of heat transfer and fluid flow.

Table 4. ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ values for various values of nanoparticle volume fraction, ${\varphi }_1,{\varphi }_2$.

$\mathit{\lambda }$	${\mathit{\varphi }}_1=0,{\mathit{\varphi }}_2=0$	${\mathit{\varphi }}_1=0.01,{\mathit{\varphi }}_2=0.01$	${\mathit{\varphi }}_1=0.02,{\mathit{\varphi }}_2=0.02$
3	5.130316	5.49515	5.862622
2	5.092475	5.45530	5.820690
1	5.054645	5.41547	5.778787
−1	4.979006	5.335855	5.695051
−2	4.941192	5.296066	5.653213
−3	4.903377	5.256285	5.611391

and

Table 5. $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ values for various values of nanoparticle volume fraction, ${\varphi }_1,{\varphi }_2$.

$\mathit{\lambda }$	${\mathit{\varphi }}_1=0,{\mathit{\varphi }}_2=0$	${\mathit{\varphi }}_1=0.01,{\mathit{\varphi }}_2=0.01$	${\mathit{\varphi }}_1=0.02,{\mathit{\varphi }}_2=0.02$
3	0.154154	0.151289	0.148388
2	0.154311	0.151457	0.148567
1	0.154467	0.151624	0.148745
−1	0.154777	0.151956	0.149100
−2	0.154931	0.152121	0.149275
−3	0.155084	0.152284	0.149450

present the influence of nanoparticle volume fraction on ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$, respectively, for $\mathrm{water} \left({\varphi }_1={\varphi }_2=0\right)$, $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0.01\right)$ and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid $\left({\varphi }_1=0.02,{\varphi }_2=0.02\right)$. For both cases of assisting $\left(\lambda >0\right)$ and opposing $\left(\lambda <0\right)$ flow regions, it can be seen in Table 4 that $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid $\left({\varphi }_1=0.02,{\varphi }_2=0.02\right)$ has the highest value of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ when compared with $\mathrm{water} \left({\varphi }_1={\varphi }_2=0\right)$ and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0.01\right)$. This signature represents a colloid consisting of two solid nanoparticles suspended in a base fluid, which results in a higher value for ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$. Physically speaking, the fluid viscosity was enhanced due to the increment value of the nanoparticle volume fraction. Hence, by boosting the value of nanoparticle volume fraction, it seems the shear stress also enhanced and contributes to the surge up values of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$. It is also observable that the value of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ reduced in opposing flow region $\left(\lambda <0\right)$ but enhanced in assisting flow region $\left(\lambda >0\right)$. This phenomenon happens may be due to the direction in which fluid is flowing along with the direction of the sheet being stretchable $\left(c=-0.5\right)$ give an impact on the behaviour of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$. Beyond that, the heat transfer measured by the values of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x,$ as presented in Table 5 is observed to reduce as the nanoparticle volume fraction increase from 0 to 0.02 for both ${\varphi }_1$ and ${\varphi }_2$. Physically, the viscosity enhanced with the increment value of nanoparticle volume fraction due to the strong interaction between the nanoparticle and the fluid molecules. This phenomenon can create a more thermal boundary layer around a heated surface and thicken the thermal boundary layer thickness. Consequently, the temperature gradient will drop, reduce the heat flux, and decrease the heat transfer rate.

Figure 2 and Figure 3 depict the behaviour of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$, respectively, against $\lambda$ with the comparison between $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0\right),$ $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0.01\right)$, as well as $\mathrm{water} \left({\varphi }_1={\varphi }_2=0\right)$. It can be observed in Figure 2 that $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid has a slightly better value of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ for both ranges in assisting $\left(\lambda >0\right)$ as well as opposing $\left(\lambda <0\right)$ flow region compared to $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$. According to the findings, the presence of one or more solid nanoparticles is likely the primary contributor to the observed disparity in ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ values between the $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid, $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$ systems. This pattern unambiguously depicts a colloid possessing two solid nanoparticles dispersed in the base fluid ($\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid), which exhibits greater ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$, values than the $\mathrm{water}$ and mono nanoparticle ($\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid). Higher values for the nanoparticle volume fraction proved to physically enhance fluid viscosity performance. As a result, the momentum boundary layer’s thickness is decreased, which boosts fluid flow and increases surface shear stress. Subsequently, the graphical finding in Figure 3 illustrated that $\mathrm{water}$ enhanced heat transfer rate in comparison to the $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid as well as $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid for a higher value in $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$. This indicates $\mathrm{water}$ can enhance the heat transfer rate given to reduced thickness in the thermal boundary layer that sparks greater heat flux compared to the $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid. Furthermore, this finding proposes that raising the temperature gradient can occur from the existence of other effects or factors included in the flow. Figure 2 and Figure 3 also illustrate the critical value ${\lambda }_c$ for $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid is lower than $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$. When compared to $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$, this suggests that hybrid ferrofluid $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid can postpone the boundary layer separation as well as improve the range of solution.

The magnetic parameter, $M$ effect on ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ of hybrid ferrofluid ($\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{C}\mathrm{o}\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$) can be seen in Figure 4. Additionally, in the opposing flow region $(\lambda <0)$, Figure 4 demonstrates a decrease in values of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ with the increment of $M$ until it reached $\lambda ={\lambda }_c$ for $M=1,2,3,4$. The occurrence of a magnetic field in a flow of magnetohydrodynamic (MHD) fluid provides a countervailing force called the Lorentz force, which endures the motion of fluid particles and slows the flow. This occurs because the force acts to push the fluid away from the surface, effectively creating a resistance that reduces the interaction between the fluid and the surface. Physically, buoyancy acts in contrast to the fluid flow in an opposing flow region. This finding points to the magnetic field’s role in reducing fluid flow velocity by bolstering buoyant force with resistance built by Lorentz force. Because of this, the surface has less shear stress and reduced momentum in the boundary layer. Concurrently, contrary results were obtained in the assisting flow region, in which the surge-up values of $M$ improved the ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ values. In the physical aspect, buoyancy acts in the same direction as the flow of a fluid. Hence, it will boost the fluid’s flow velocity in opposition to the Lorentz force’s resistance in MHD fluid flow and assist in increasing the surface shear stress. Further observations prove the rise of $M$ slightly lowered the $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ values, as illustrated in Figure 5. Here, it may be recognised that the occurrence of $M$ decreased the heat transfer rate due to the increase thermal boundary layer thickness and contributed to less heat flux produced. Physically, energy is released from the frictional force between the fluid flow and the resistance form the Lorentz force. Hence, it may contribute to enhancing the thickness of the thermal boundary layer with a load of energy released from the frictional force. Additionally, the presence of $M$ also induced electrical currents in the fluid, which can dissipate energy and reduce the temperature gradient between the fluid and the surface, which eventually reduced the value of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ and drops the heat transfer rate. Moreover, the critical value ${\lambda }_c$ decreases with the growth of $M$ as visualised in Figure 4 and Figure 5. Based on this result, it seems that incorporating a magnetic field into the boundary layer flow may be able to accelerate boundary layer separation as well as widen the range of solutions.

Figure 6 and Figure 7 illustrate the plots of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$, accordingly, against the mixed convection parameter, $\lambda$ for a broad range of values concerning the Biot number, Bi. The trend of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ is shown to decrease slightly as the value of Bi rises for the first solution in the opposing flow region $\left(\lambda <0\right)$, but the trend illustrated for the assisting flow region $\left(\lambda >0\right)$ is shown to grow slightly. This suggests the influence of Bi in opposing flow regions may slightly increase the momentum at the boundary layer and adjourn the fluid flow. In contrast, the fluid flow may slightly enhance due to slight decreases in momentum boundary layer thickness. In this instance, it is likely that the presence of Bi only possesses a tiny impact on the momentum at the boundary layer, with only modest variations occurring due to the rise in the value of Bi in the flow. Physically, heat is conducted more quickly within a body than it is dissipated to the surrounding fluid when the Biot number is low $\left(\mathrm{Bi}<1\right)$. In this instance, the temperature inside the plate is almost uniform, and there is also a minimal temperature difference across the fluid border layer. As a result, the behaviour of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ is not considerably impacted by heat transfer, and the heat transfer coefficient at the fluid boundary layer is essentially constant. Nonetheless, the outcome demonstrated in Figure 7 proves the main impact of Bi on the heat transfer rate, as the $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ values grow with the spike-up value of Bi. Physically, since it is directly admissible to the heat transfer coefficient, $h_f$, the Biot number has an opposite relation with thermal resistance. Due to this, the surface heat transfer will rise since the heat resistance will decrease as Bi increases, as demonstrated in Figure 7. Moreover, the heat flux was enhanced since the presence of Bi reduced the thickness concerning the thermal boundary layer. Additionally, regarding the duration for boundary layer separation, both Figure 6 and Figure 7 show that the critical value ${\lambda }_c$ rises corresponding to the increasing Bi value, which indicates that the boundary layer quickly separates as Bi increases.

Apart from that, Figure 8 and Figure 9 delineate the Eckert number, Ec’s impact on ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$, respectively, with $\lambda$. The rising value of Ec illustrates a reduced value of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ in the opposing flow region $\left(\lambda <0\right)$ but improvements in assisting the flow region $\left(\lambda >0\right),$ as illustrated in Figure 8. In essence, the Eckert number may be referred to as the possible ratio of the advective transport and the heat dissipation. Physically, the combined actions of the Joule heating, viscous dissipation process, as well as the magnetic field accelerate the Eckert number. Moreover, energy may be stored in the fluid area because of dissipation owing to viscosity and elastic deformation, leading to frictional heating as the Eckert number increases. Generally, buoyancy acts counter to fluid flow in an opposing flow region. This proposes that the heat energy produced from frictional heating is capable of supporting buoyant force, which contributes to slowing down the fluid flow velocity. Thus, this consequence reduced the momentum at the boundary layer and reduced the shear stress, as depicted in Figure 8. Concurrently, a contrary result was obtained in assisting the flow region because, in terms of the physical aspect, the buoyancy effect follows the fluid flow’s direction. This recommends that the heat energy produced from frictional heating may support buoyant force, which contributes to speeding up the fluid flow velocity. Because of this effect, as seen in Figure 8, the shear stress and momentum in the boundary layer were both increased. Beyond that, the findings demonstrated in Figure 9 depict that the value of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ decreases as Ec increases. Physically, when an electric current source is passed through a conducting moving plate, Joule heating (which includes magnetic fields) produces heat on the moving plate. Moreover, a higher Ec value produces greater heat because of fluid-particle friction forces. This suggests that more heat was built up during the process and increased the thermal boundary layer thickness, which results in reduced convective heat transfer and leads to a lower value of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$. Therefore, in this specific situation, the Ec value tends to reduce the effectiveness of heat transfer as Ec grows in value. Both Figure 8 and Figure 9 reveal that the critical value ${\mathsf{\lambda }}_{\mathrm{c}}$ expands with the increased value of Ec, which indicates that the boundary layer separates more quickly as Ec increases and reduces the range of solution.

Figure 10 and Figure 11 highlight the distribution of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ towards dimensionless $\lambda$ with various values of $S$. The appearance of the dual solution is recognised within the range of $\lambda >{\lambda }_c$ in opposing and assisting flow regions, as displayed in Figure 10 and Figure 11, accordingly. From the observation in Figure 10, it can be seen the value of ${\mathrm{R}{\mathrm{e}}_x^{1/2}C}_f$ rose with the boost in the value of $S$ in both opposing and assisting flow regions. Physically, the nanofluid’s motion slows down and the surface’s velocity gradient increases because of the suction’s effect at the boundary. Thus, the heated fluid is drawn toward the wall under the action of suction, in which the buoyant forces are reduced by the strong effect in viscosity, resulting in an abrupt gradient of velocity. Hence, this phenomenon enhanced the shear stress at the wall and increased the momentum at the boundary layer. Moreover, Figure 10 illustrates the critical value ${\lambda }_c$ increased as $S$ grows. This indicates the growth strength of suction may delay the boundary layer or flow separation. By encasing the slowly moving molecules within the shrinking sheet, the permeable sheet facilitates the maintenance of the laminar flow. Apart from that, the value of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ is also enhanced with the increment strength of $S$ as portrayed in Figure 11. Physically, a higher rate of heat transmission is achieved due to suction’s role in inducing fluid motion, which simultaneously aids heated particles in moving closer to the wall. More ferrofluid particles can diffuse through the sheet as the $S$ value rises, enhancing the permeability of the sheet. Hence, this phenomenon contributes more heat flux and reduces the thickness of the thermal boundary layer. Hence, growing $S$ accelerates the rate of heat transmission.

The profiles of velocity $f^{\prime }\left(\eta \right)$ as well as temperature $\theta \left(\eta \right)$ in Figure 12 and Figure 13, accordingly, highlight the contrast between $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0\right)$, $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0.01\right)$, as well as $\mathrm{water} \left({\varphi }_1={\varphi }_2=0\right)$ when $\lambda =-70$ (opposing flow zone). Figure 12 impressively portrayed that $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid has a slightly higher $f^{\prime }\left(\eta \right)$ in comparison to $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$. This shows that the $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid minimised the momentum boundary layer thickness and contributed to a higher fluid flow, which enhanced the shear stress at the surface compared with $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$. This finding agrees with those in Figure 2 when $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid has a higher ${\mathrm{R}\mathrm{e}}_x^{1/2}{C}_f$ than $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{water}$ in $\lambda =-70$ (opposing flow region). Concerning the $\theta \left(\eta \right)$ behaviour illustrated in Figure 13, $\mathrm{water}$ can be shown to have a slightly reduced $\theta \left(\eta \right)$ than both $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid. This finding concludes that the thickness of the thermal boundary layer concerning $\mathrm{water}$ is marginally small in comparison to $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid, which contributes more heat flux and improves the heat transfer performance.

In the region of opposing flow $(\lambda =-70)$, the behaviours of $f^{\prime }\left(\eta \right)$ and $\theta \left(\eta \right)$ as a consequence of adjusting the magnetic parameter, $M$ are demonstrated in Figure 14 and Figure 15, accordingly. Interestingly, it is observed that the values of $f^{\prime }\left(\eta \right)$ increase as $M$ grows in Figure 14. This illustrates the fluid flow velocity was enhanced since the momentum boundary layer thickness was diminished with the increasing $M$ values. Physically, the Lorentz force causes the fluid to flow in a particular direction and at a certain velocity. As the value of $M$ grows, the Lorentz force acting on the fluid will become stronger, which will cause the fluid to flow with a greater velocity. This phenomenon raised the shear stress at the wall surface due to the speed-up of fluid flow velocity. Moreover, this result is aligned with the finding in Figure 4 as $M$ grows at $\lambda =-70$ (opposing flow region), the value of skin friction also increases. Apart from that, a build-up of temperature in the thermal boundary layer contributes to an increase in temperature profile with an increment value of $M$, as illustrated in Figure 15. Physically speaking, the effect of $M$ along with the several factors in this fluid flow problem, such as Joule heating, convective boundary condition, and viscous dissipation, boost up the amount of heat energy, which thickens more the thermal boundary layer thickness. Contributing to that, the heat flux was dropped, and the heat transfer rate was lowered, and the proof may be seen in the result from Figure 5, which depicts the value of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ reduced as $M$ increased at $\lambda =-70$ (opposing flow region).

Next, the profiles of velocity $f\prime \left(\eta \right)$ as well as temperature $\theta \left(\eta \right)$ in Figure 16 and Figure 17, accordingly, highlight the influence of Biot number, Bi in opposing flow regions $(\lambda =-70)$. As illustrated in Figure 16, the surge of Bi slightly reduced the velocity profile, which then decreased the thickness of the momentum boundary layer. This phenomenon slightly impedes the fluid flow velocity and contributes to less shear stress at the surface wall. Beyond that, this suggests the presence of Bi does not significantly affect the velocity profile, where this outcome agrees with the finding in Figure 6, indicating the increment of Bi increased the skin friction value slightly in the opposing flow region $(\lambda =-70)$. In contrast, the significance of Bi may be emphasised in Figure 17, in which the surge-up Bi value greatly influences the temperature profile from $\eta =0$ until $\eta \to \infty$. Generally, the Biot number is a measurement of the relationship between convection at the surface and conduction within a solid. It can be seen as Bi grows, the temperature profile also increases. Physically, when the Biot number rises, the surface’s thermal resistance gradually decreases. The increase in convection is what leads to the greater surface temperature, including the built thermal boundary layer thickness. Moreover, the heat flux was diminished, contributing to the thickening of the thermal boundary layer, which reduced the heat transfer performance as Bi grows.

Figure 18 illustrates the behaviour of $f^{\prime }\left(\eta \right)$ with Ec adjusted in the opposing flow region $(\lambda =-70)$. It may be observed in the trend of $f^{\prime }\left(\eta \right)$ broaden with the surge of Ec. Physically, the existence of the magnetic field in Joule heating combined with viscous dissipation built up resistance from the Lorentz force in a flow. This suggests the presence of resistance from the growth of Ec reduced the fluid flow velocity, which contributes to diminishing the momentum boundary layer thickness, as portrayed in Figure 18. Therefore, this phenomenon reduced the shear stress at the wall surface. As illustrated in Figure 19, the surge-up value of Ec enhances the $\theta \left(\eta \right)$ in the opposing flow region $(\lambda =-70)$. Physically speaking, Eckert number properties are associated with the dissipation as well as Joule heating terms in the energy equation. The magnetic resistance built from the Lorentz force as well as the internal friction of the fluid, transforms the mechanical with electrical energy into heat, accordingly. Therefore, as Ec is increased, the $\theta \left(\eta \right)$ is also increased, resulting in a thicker thermal boundary layer and lowered heat flux. This phenomenon reduced the performance of heat transfer as Ec grows, and this result aligns with the outcome in Figure 9, in which the Nusselt number was reduced with the acceleration of Ec in the opposing flow region $(\lambda =-70)$.

Figure 20 and Figure 21 exemplify the behaviour of $f^{\prime }\left(\eta \right)$ as well as $\theta \left(\eta \right)$, accordingly, displaying a variety of values regarding the suction parameter $S$ in the opposing flow region $(\lambda =-70)$. This may be observed from Figure 20, where the value of $f^{\prime }\left(\eta \right)$ significantly enhanced with the surge-up value of $S$. Physically, an increase in $S$ leads to a higher rate of fluid extraction from the boundary, which creates a stronger pressure gradient within the boundary layer. This, in turn, causes an increase in the velocity near the boundary, and, thus, an increase in the velocity gradient or slope of the velocity profile. This phenomenon led to a decrease in momentum boundary layer thickness and improved the shear stress at the wall. Moreover, this result aligns with the finding in Figure 10, where skin friction increases as $S$ grows in the opposing flow region $(\lambda =-70)$. Additionally, the thermal boundary layer thickness as well as the temperature profile, also drop with the increased strength of $S$, as depicted in Figure 21. Physically speaking, a decrement in the thickness of the thermal boundary layer will assist in producing more heat flux. Because of this phenomenon, heat transfer performance was also enhanced. This finding is parallel with the outcome and observation in Figure 11, in which the Nusselt number is raised with the enhanced strength of $S$.

It is possible to see the presence of a non-unique solution, also known as a dual solution, in the region represented by Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 that contains both the opposing flow $\left(\lambda <0\right)$ as well as the assisting flow $\left(\lambda >0\right)$. In multiple cases, the dual answer will represent itself somewhere within the range of $\lambda >{\lambda }_c$. The critical value ${\lambda }_c$ is a unique solution that occurs at the intersection of the first as well as second solutions. Bifurcation points of the first as well as the second solutions can be said to have occurred at the point denoted by $\lambda ={\lambda }_c$. In the boundary layer, separation happens when $\lambda ={\lambda }_c$, and when utilising the boundary layer approximation, one cannot achieve a solution if the range is between $\lambda <{\lambda }_c$. Therefore, an analysis of the flow’s stability is essential. The same numerical technique that was addressed in the prior section was implemented to solve the linearised Equations (20)–(22) numerically to execute the stability analysis. In addition, to validate the results of stability analysis, it is necessary to collect eigenvalues such as $\gamma$ that are relatively modest. Here,

Table 6. The smallest eigenvalue $\gamma$ having the variation of $\lambda$ when Pr = 6.2, ${\varphi }_1=0.01$, ${\varphi }_2=0.01$, $c=-0.5$, $\mathrm{E}\mathrm{c}=0.1$ and $S=1$ for $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$.

M	Bi	λ	γ1	γ2
1	0.2	−85 −85.1 −85.11 −85.111	0.4569 0.3345 0.3198 0.3183	−0.4457 −0.3284 −0.3142 −0.3127
1	0.4	−83 −83.9 −83.99 −83.999	1.0211 0.3653 0.2135 0.1920	−0.9672 −0.3582 −0.2111 −0.1900

was created using hybrid ferrofluid $\left({\varphi }_1=0.01,{\varphi }_2=0.01\right)$, to illustrate how small eigenvalues $\gamma$ behave for different Bi = 0.2, 0.4 when $M=1$, $S=1$, $n=3$, Ec = 0.1, and $c=-0.5.$ The first solution provides a positive number concerning the eigenvalue ${\gamma }_1$, which, in accordance with Merkin et al. [66] and Weidman and Turner [67], resembles the initial drop of disturbance for steady and genuine flow through time. At the same time, the second solution’s negative eigenvalue ${\gamma }_2$ indicates an early increase in disturbance for an unrealistic and unstable flow as time progresses. Table 6 makes it clear that when $\lambda$ moves closer to the critical values represented by ${\lambda }_c$, the values of the small eigenvalues expressed by ${\gamma }_1$ as well as ${\gamma }_2$ grow closer to zero. As a result, it is possible to conclude that the second solution is unstable while the first solution is stable.

5. Conclusions

Investigations are being performed into the steady stagnation point of an MHD mixed convection hybrid ferrofluid flow performed by the implantation of a nonlinearly permeable moving surface. The self-similar equations produced by the similarity transformation are then numerically solved by MATLAB’s built-in solver (bvp4c). We examine in detail the relevance of the Eckert number, the magnetic parameter, the Biot number, the suction parameter, along with the nanoparticle volume fraction. The following are the study’s primary findings:

• The occurrence of dual solutions (first and second solutions) may be seen in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 within the range $\lambda >{\lambda }_c$. The bifurcation point $\left(\lambda ={\lambda }_c\right)$ happens in the opposing flow region $\left(\lambda <0\right)$.
• Stability analysis results imply the first solution is stable. On the other hand, the second solution is unstable.
• Varying nanoparticle volume fraction from ${\varphi }_1,{\varphi }_2=0$ until ${\varphi }_1,{\varphi }_2=0.02$ enhanced the skin friction by about 14.44% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1={\varphi }_2=0.02}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1=0.01,{\varphi }_2=0}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1={\varphi }_2=0}}\right|\times 100\%=14.44\%\right)$ for case opposing flow region $\left(\lambda =-3\right)$ and 14.27% for case assisting flow region $\left(\lambda =3\right)$.
• The heat transfer rate drops by about 3.63% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1={\varphi }_2=0.02}-{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1=0,{\varphi }_2=0}}{{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1={\varphi }_2=0}}\right|\times 100\%=3.63\%\right)$ for case opposing flow region $\left(\lambda =-3\right)$ and 3.74% for case assisting flow region $\left(\lambda =3\right)$ with various nanoparticle volume fraction from ${\varphi }_1,{\varphi }_2=0$ until ${\varphi }_1,{\varphi }_2=0.02$
• $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid has a better ${\mathrm{R}\mathrm{e}}_x^{1/2}{C}_f$ compares with water and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid when adding the $\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$ nanoparticle due to higher values for the nanoparticle volume fraction proved to physically enhance fluid viscosity performance and shear stress. As for case $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid (${\varphi }_1={\varphi }_2=0.01)$ in both assisting $\left(\lambda =2\right)$, and opposing flow $\left(\lambda =-2\right)$ regions, the magnitude of ${\mathrm{R}\mathrm{e}}_x^{1/2}{C}_f$ enhanced by about 7.13% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1={\varphi }_2=0.01}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1=0,{\varphi }_2=0}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1={\varphi }_2=0}}\right|\times 100\%=7.13\%\right)$ and 7.18%, respectively, as compared to water (${\varphi }_1={\varphi }_2=0)$. For the case $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid (${\varphi }_1={\varphi }_2=0.01)$ against $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid (${\varphi }_1=0.01,{\varphi }_2=0)$, the addition of 1% of $\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$ in $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid enhanced the magnitude of ${\mathrm{R}\mathrm{e}}_x^{1/2}{C}_f$ in both assisting $\left(\lambda =2\right)$, and opposing flow $\left(\lambda =-2\right)$ regions by about 3.35% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1={\varphi }_2=0.01}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1=0.01,{\varphi }_2=0}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{{\varphi }_1=0.01,{\varphi }_2=0}}\right|\times 100\%=3.35\%\right)$ and 3.37%, respectively.
• In terms of heat transfer rate, $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid has a lower $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ compared with water and $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid due to an external heat source from Joule heating and energy release from the friction between fluid flow and resistance in Lorentz force. As for case $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid (${\varphi }_1={\varphi }_2=0.01)$ in both assisting $\left(\lambda =2\right)$, and opposing flow $\left(\lambda =-2\right)$ regions, the magnitude of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ reduced by about 1.85% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1={\varphi }_2=0.01}-{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1=0,{\varphi }_2=0}}{{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1={\varphi }_2=0}}\right|\times 100\%=1.85\%\right)$ and 1.81%, respectively, as compared to water (${\varphi }_1={\varphi }_2=0)$. For the case $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$–$\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid (${\varphi }_1={\varphi }_2=0.01)$ against $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$/$\mathrm{water}$ ferrofluid (${\varphi }_1=0.01,{\varphi }_2=0)$, the addition of 1% of $\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4$ in $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4–\mathrm{CoF}{\mathrm{e}}_2{\mathrm{O}}_4/\mathrm{water}$ hybrid ferrofluid enhanced the magnitude of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ in both assisting $\left(\lambda =2\right)$, and opposing flow $\left(\lambda =-2\right)$ regions by about 0.90% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1={\varphi }_2=0.01}-{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1=0.01,{\varphi }_2=0}}{{\mathrm{R}\mathrm{e}}_x^{-1/2}{\left.N{u}_x\right|}_{{\varphi }_1=0.01,{\varphi }_2=0}}\right|\times 100\%=0.90\%\right)$ and 0.92%,
• As the strength of the suction parameter $S$ increases, both skin friction and heat transfer rate significantly improve and facilitate reducing the boundary layer separation. With the increase in the strength of $S$ from 1.0 to 1.4, the magnitude of ${\mathrm{R}\mathrm{e}}_x^{1/2}{C}_f$ and $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ rise by about 49.54% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{S=1.4}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{S=1.0}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.C_f\right|}_{S=1.0}}\right|\times 100\%=49.54\%\right)$ and 3.6%, accordingly, in the opposing flow region $\left(\lambda =-50\right)$. However, it can be seen the percentage of enhancement for the magnitude of ${\mathrm{R}\mathrm{e}}_x^{1/2}{C}_f$ continue to reduce to 12.68% while the magnitude of $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ grows to 6.82% as in assisting flow region $\left(\lambda =20\right)$.
• The existence of magnetic parameter $M$ lowered the heat transfer performance and built up the thermal boundary layer thickness due to the energy release from the frictional force between fluid flow and the resistance from the Lorentz force as well as improved the boundary layer separation. For the case in opposing flow region $\left(\lambda =-70\right)$ and assisting flow region $\left(\lambda =20\right)$, the magnitude $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ reduced by about 10.65% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{M=4}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{M=1}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{M=1}}\right|\times 100\%=10.65\%\right)$ and 8.75%, accordingly, as $M$ grows from 1 to 4.
• An increment of Eckert number, Ec (arising from Joule heating) reduced the heat transfer performance while the rise of Biot number, Bi, accelerates the heat transfer rate for both assisting and opposing flow regions. For the case in opposing flow region $\left(\lambda =-20\right)$ and assisting flow region $\left(\lambda =20\right)$, the magnitude $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ reduced by about 7.27% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{Ec=0.12}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{Ec=0.09}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{Ec=0.09}}\right|\times 100\%=7.27\%\right)$ and 10.24%, respectively, as Ec grows from 0.09 to 0.12. At the same time, for the case in opposing flow region $\left(\lambda =-20\right)$ and assisting flow region $\left(\lambda =20\right)$, the magnitude $\mathrm{R}{\mathrm{e}}_x^{-1/2}N{u}_x$ enhanced by about 280.98% $\left(\left|\frac{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{Bi=0.8}-{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{Bi=0.2}}{{\mathrm{R}\mathrm{e}}_x^{1/2}{\left.N{u}_x\right|}_{Bi=0.2}}\right|\times 100\%=280.98\%\right)$ and 280.64%, respectively, as Bi grows from 0.2 to 0.8. Interestingly, both Ec and Bi are responsible for the build-up of the temperature profile.
• The boundary layer separates more quickly, in which the range of solutions is reduced as Ec is increased. The boundary layer separation decreases as Bi increases while the variety of potential solutions increases.