The investigation of nanofluids and their thermal physical characteristics has gained the attention of researchers in recent times due to their ability to enhance and improve the rate of heat transfer in manufacturing, heat exchangers, automobiles, and solar energy systems. Heat moves from the region of high temperature to the region of low temperature in a system with the aid of heat carrier fluid. The traditional heat carrier fluids are water, oils, salt water, and ethyl glycol. However, these fluids meant to carrier heat possess low thermal conductivity [1] . The implication of this challenge is the short life span of the system due to overheating or destruction of the engines. Researchers and scientists have solved this issue by introducing extremely tiny particles called nanoparticles (NPs) like metals (Copper (Cu), Aluminum (Al), Silver (Ag)), metal oxides (Copper oxide, (CuO), Aluminum oxide (Al₂O₃), carbon nanotubes (Diamond, Graphene), organic materials (Titanium oxide (TiO₂), Silicon oxide (SiO₂), Iron oxide (Fe₂O₃)) that have high thermal conductivity into the convectional base fluid to improve its rate of heat transfer. The resulting fluid is called Nanofluid (NF) [2] . This upgraded heat carrier fluid serves better as a lubricant, coolant agent and prolongation of the systems in industries, automobiles, heat exchangers and more. Researchers like [3] - [6] have thoroughly studied NF.

Nanofluid, as it is full of potential, its steadiness depends on the size of the particles, thermophysical properties, temperature, and other dimensionless parameters [3] . The thermophysical properties of NF need to be focused on to meet the worldwide demand for smaller sizes and yet highly effective heat exchanger systems in the industry [6] . Because the addition of appropriate NPs into the base fluid usually increases its thermophysical properties. However, the focus of many researchers is thermal conductivity because it is the main target of any improved heat carrier fluids [7] . The study of thermal conductivity is widely done by [8] - [14] . As mentioned earlier, other properties along with thermal conductivity also increased as NPs are included in the traditional heat carrier fluids. For example, viscosity is a feature of fluid that causes a restriction on the fluid to move easily and quickly to the vital parts of the system [15] . It is important to focus on this characteristic as it can damage the engine entirely due to friction and wear [16] . A series of research has been made on the viscosity of nanofluid in aiding the improvement of heat transfer efficiency [17] - [22] and many more.

Another crucial property of improved fluid is density. It is being affected by the inclusion of the NPs by upsurging the mass per volume of fluid [23] . This property is as important as other properties because changes in mass per volume of fluid influence so many aspects of fluid compartments within the engines and machines [24] [25] . It could affect buoyancy which is the measure of upward forces exerted by the fluid on an object immersed in a fluid which can influence the stability of the systems [26] .

Similarly, the compactness of the heat exchanger systems can be influenced by the specific heat capacity of the fluid used within them [27] . This is another characteristic of fluid that is meant to be carefully considered in the process of improving the rate of heat transfer of the improved fluid [28] . Changes in specific heat capacity can alter the overall thermal performance of the fluid. This characteristic of NF can influence the effectiveness of heat transfer in the system, its changes due to the inclusion of NPs can affect the heat exchanger unit, sizes, and designs of the system [29] . Thus, nanofluids with boosted thermophysical features are seen as a significant potential resolution to augmentation of heat transmission performance in heat exchanger systems [6] .

The issues of sedimentation, aggregation, and instability of nanofluid led to the existence of Hybrid Nanofluid (HNF). This fluid exhibits outstanding heat transfer improvement than NF [30] . It involves the dispersion of more than one NP in the base fluid, and it has gained the attention of researchers [31] . The challenge of hybrid nanofluid is the high viscosity causing an increased drag leading to more power for fluid movement leading to less effective and efficient systems [32] . Researchers have done numerous works on Hybrid nanofluids [33] [34] and Trihybrid Nanofluids [35] - [39] .

Tetra-hybrid nanofluid which comprises four different NPs being introduced into the base fluid to improve the rate of heat transfer of heat carrier fluid is another interesting fluid [40] [41] [42] . It can be broadly used in heat exchangers, automobiles, and biotechnology [43] [44] . In literature, the tetra-hybrid nanofluid started gaining popularity among researchers because of its outstanding improvement in the realm of transmission of heat from the region with higher temperature to the region with lower temperature.

Polyalphaolefin (PAO) is widely recognized for its outstanding thermal stability and is extensively used as a synthetic oil in industrial applications [45] . It does not contain nitrogen components and possesses high viscosity which makes it excellently applicable where thick and durable lubricant oil is required. The incorporation of NPs such as TiO₂, SiO₂, ZnO and Fe₂O₃ has shown a potential to enhance engine efficiency. [46] - [49] concluded that the rate of heat transfer of fluid is enhanced with the addition of hybrid nanoparticles in the base fluid. However, careful formulation is crucial to prevent issues related to excessive viscosity that could impede lubricant movement and potentially damage the engine. Therefore, achieving proper thermophysical properties in nano lubricants across a range of temperatures is essential for optimal performance.

While previous research has extensively studied nanofluids and hybrid nanofluids, the specific composition and thermal behaviour of Tetra hybrid nanofluids remain relatively unexplored. This research therefore aims to bridge the gap by numerically investigating the temperature-dependent thermophysical properties of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid in the presence of suction along a vertical porous surface. Understanding these properties is crucial for optimizing heat transfer processes in various applications, including heat exchangers, automobiles, and solar energy systems. The objectives of this study are twofold: 1) to determine the effects of Tetra hybrid nanofluid’s changeable temperature-dependent thermophysical parameters, including thermal conductivity, viscosity, density, and specific heat capacity, on the system’s heat transmission rate, and 2) to analyze the rate of change of dependent variables—skin friction and Nusselt number—in relation to the independent variables (thermophysical parameters). By systematically examining these aspects, the research seeks to provide valuable insights into the behaviour and performance of characteristics of Tetra hybrid nanofluid, thus contributing to the advancement of nanofluid technology.

A two-dimensional steady incompressible flow of viscous and dielectric TiO₂-SiO₂-ZnO-Fe₂O₃ Tetra hybrid nanofluid moving through a porous vertical surface with suction, variable thermal conductivity, variable density, variable viscosity parameter, and variable specific heat capacity of tetra hybrid nanofluid was considered. The x-axis and y-axis are taken along the direction of the plate and normal respectively as shown in Figure 1 . The left surface of the plate is heated by convection from a hot temperature, $T_w^{*}$ which provides a heat transmission coefficient $h_w^{*}$ while a flow of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid at the free stream temperature $T_{\infty }^{\ast }$ moves along the right surface of the plate with a uniform free stream velocity $U_{\infty }^{\ast }$. Equations (1)-(3) are continuity, momentum, and energy equations respectively.

$\frac{\partial u^{\ast }}{\partial x^{\ast }}+\frac{\partial v^{\ast }}{\partial x^{\ast }}=0$(1)

$u^{\ast }\frac{\partial u^{\ast }}{\partial x^{\ast }}+v^{\ast }\frac{\partial u^{\ast }}{\partial y^{\ast }}=\frac{\partial }{\partial y^{\ast }}\left[\frac{{\mu }_{TetraNF}\left(T^{\ast }\right)}{{\rho }_{TetraNF}\left(T^{\ast }\right)}\frac{\partial u^{\ast }}{\partial y^{\ast }}\right]-\frac{{\mu }_{TetraNF}\left(T^{\ast }\right)}{{\rho }_{TetraNF}\left(T^{\ast }\right)\bar{k}}{u}^{\ast }+g{\beta }_{TetraNF}\left(T^{\ast }-T_{\infty }^{\ast }\right)$(2)

$u^{\ast }\frac{\partial T^{\ast }}{\partial x^{\ast }}+v^{\ast }\frac{\partial T^{\ast }}{\partial y^{\ast }}=\frac{\partial }{\partial y^{\ast }}\left[\frac{K_{TetraNF}\left(T^{\ast }\right)}{{\left(\rho C_p\right)}_{TetraNF}\left(T^{\ast }\right)}\frac{\partial T^{\ast }}{\partial y^{\ast }}\right]+\frac{{\mu }_{TetraNF}\left(T^{\ast }\right)}{{\left(\rho C_p\right)}_{TetraNF}\left(T^{\ast }\right)}{\left(\frac{\partial u^{\ast }}{\partial y^{\ast }}\right)}^2$(3)

On the right surface of the plate, the boundary condition is as follows:

$u^{\ast }\left(x^{\ast },0\right)=0,v^{\ast }\left(x^{\ast },0\right)=-v_w^{\ast },-k\frac{\partial T^{\ast }}{\partial y^{\ast }}\left(x^{\ast },0\right)=h_w^{\ast }\left[T_w^{\ast }-T^{\ast }\left(x^{\ast },0\right)\right]$ at $y^{\ast }=0$(4)

At the surface of the porous vertical plate, fluid temperature and velocity are specified, while suction influences fluid flow.

At the far right of the surface of the plate, the boundary condition of Tetra-PAO nanofluid is:

$u^{\ast }\left(x^{\ast },\infty \right)\to U_{\infty }^{\ast },T^{\ast }\left(x^{\ast },\infty \right)\to T_{\infty }^{\ast }$ at $y^{\ast }\to \infty$(5)

At the far right of the plate, the fluid temperature remains constant, and a no-slip boundary condition is enforced for fluid velocity ( Table 1 ).

Thermal Conductivity:

$\frac{K_{TetraNF}}{K_{TriNF}}=\frac{\left(k_{S4}+2k_{TriNF}\right)-2{\varphi }_4\left(k_{TriNF}-k_{S4}\right)}{\left(k_{S4}+2k_{TriNF}\right)+{\varphi }_4\left(k_{TriNF}-k_{S4}\right)}$

where

$\frac{K_{TriNF}}{K_{hNF}}=\frac{\left(k_{S3}+2k_{hNFF}\right)-2{\varphi }_3\left(k_{hNF}-k_{S3}\right)}{\left(k_{S3}+2k_{hNF}\right)+{\varphi }_3\left(k_{hNF}-k_{S3}\right)}$,

$\frac{K_{hNF}}{K_{NF}}=\frac{\left(k_{S2}+2k_{NF}\right)-2{\varphi }_3\left(k_{hNF}-k_{S2}\right)}{\left(k_{S2}+2k_{NF}\right)+{\varphi }_3\left(k_{NF}-k_{S2}\right)}$,

$\frac{K_{NF}}{K_{BF}}=\frac{\left(k_{S1}+2k_{BF}\right)-2{\varphi }_1\left(k_{BF}-k_{S1}\right)}{\left(k_{S1}+2k_{BF}\right)+{\varphi }_1\left(k_{BF}-k_{S1}\right)}$

Density:

$\begin{array}{c}{\rho }_{TetraNF}=\left({\rho }_{BF}\right)[\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\rho }_{S4}}{{\rho }_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\rho }_{S3}}{{\rho }_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\rho }_{S2}}{{\rho }_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\rho }_{S1}}{{\rho }_{BF}}\right)]\end{array}$

Specific Heat Capacity:

$\begin{array}{c}{\left(\rho C_P\right)}_{TetraNF}=\left(\rho C_P{}_{BF}\right)\left(1-{\varphi }_{S1}\right)[\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\left(\rho C_P\right)}_{S4}}{{\left(\rho C_P\right)}_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\left(\rho C_P\right)}_{S3}}{{\left(\rho C_P\right)}_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\left(\rho C_P\right)}_{S2}}{{\left(\rho C_P\right)}_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\left(\rho C_P\right)}_{S1}}{{\left(\rho C_P\right)}_{BF}}\right)]\end{array}$

Viscosity:

$\frac{{\mu }_{TetraNF}}{{\mu }_{BF}}=\frac{1}{{\left(1-{\varphi }_{S1}\right)}^{2.5}{\left(1-{\varphi }_{S2}\right)}^{2.5}{\left(1-{\varphi }_{S3}\right)}^{2.5}{\left(1-{\varphi }_{S4}\right)}^{2.5}}$

Thermal expansion:

$\begin{array}{c}{\beta }_{TetraNF}=\left({\beta }_{BF}\right)\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\beta }_{S4}}{{\beta }_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\beta }_{S3}}{{\beta }_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\beta }_{S2}}{{\beta }_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\beta }_{S1}}{{\beta }_{BF}}\right)]\end{array}$

According to [2] , the temperature-dependent thermal conductivity and temperature-dependent viscosity of fluid are given, similarly, the temperature-dependent specific heat capacity and temperature density following [3] are written as:

$\begin{array}{l}{\mu }_{Bf}\left(T^{\ast }\right)={\mu }_{Bf}\left({\text{e}}^{-a\left(\frac{T^{\ast }-T_{\infty }^{\ast }}{T_0^{\ast }-T_{\infty }^{\ast }}\right)}\right)\\ k_{Bf}\left(T^{\ast }\right)=k_{Bf}\left(1-b\left(\frac{T^{\ast }-T_{\infty }^{\ast }}{T_0^{\ast }-T_{\infty }^{\ast }}\right)\right)\\ {\rho }_{Bf}\left(T^{\ast }\right)={\rho }_{Bf}\left[1+d\left(T^{\ast }-T_{\infty }^{\ast }\right)\right]\\ {\left(\rho C_p\right)}_{Bf}\left(T^{\ast }\right)={\left(\rho C_p\right)}_{Bf}\left[1+c\left(T^{\ast }-T_{\infty }^{\ast }\right)\right]\end{array}\}$(6)

Following [1] and [2] the temperature-dependent thermophysical properties of Tetra-PAO nanofluid can be written as follows:

$\begin{array}{l}{\mu }_{TetraNF}\left(T^{\ast }\right)=\frac{{\mu }_{BF}\left(T^{\ast }\right)}{{\left(1-{\varphi }_{S1}\right)}^{2.5}{\left(1-{\varphi }_{S2}\right)}^{2.5}{\left(1-{\varphi }_{S3}\right)}^{2.5}{\left(1-{\varphi }_{S4}\right)}^{2.5}}\\ K_{TetraNF}\left(T^{\ast }\right)=\frac{K_{TriNF}\left(T^{\ast }\right)\left(k_{S4}+2k_{TriNF}\right)-2{\varphi }_4\left(k_{TriNF}-k_{S4}\right)}{\left(k_{S4}+2k_{TriNF}\right)+{\varphi }_4\left(k_{TriNF}-k_{S4}\right)}\\ {\left(\rho C_P\right)}_{TetraNF}\left(T^{\ast }\right)=\left(\rho C_P{}_{BF}\right)\left(T^{\ast }\right)\left(1-{\varphi }_{S1}\right)[\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\left(\rho C_P\right)}_{S4}}{{\left(\rho C_P\right)}_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\left(\rho C_P\right)}_{S3}}{{\left(\rho C_P\right)}_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\left(\rho C_P\right)}_{S2}}{{\left(\rho C_P\right)}_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\left(\rho C_P\right)}_{S1}}{{\left(\rho C_P\right)}_{BF}}\right)]\\ {\rho }_{TetraNF}\left(T^{\ast }\right)=\left({\rho }_{BF}\right)\left(T^{\ast }\right)[\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\rho }_{S4}}{{\rho }_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\rho }_{S3}}{{\rho }_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\rho }_{S2}}{{\rho }_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\rho }_{S1}}{{\rho }_{BF}}\right)]\end{array}\}$

(7)

Equation (6) into Equation (7) gives:

$\begin{array}{l}{\mu }_{TetraNF}\left(T^{\ast }\right)=\frac{{\mu }_{BF}\left({\ell }^{-\xi \theta \left({\eta }^{\ast }\right)}\right)}{{\left(1-{\varphi }_{S1}\right)}^{2.5}{\left(1-{\varphi }_{S2}\right)}^{2.5}{\left(1-{\varphi }_{S3}\right)}^{2.5}{\left(1-{\varphi }_{S4}\right)}^{2.5}}\\ K_{TetraNF}\left(T^{\ast }\right)=\frac{k_{TriNF}\left(k_{S3}+2k_{hNF}\right)-2{\varphi }_3\left(k_{hNF}-k_{S3}\right)}{\left(k_{S3}+2k_{hNF}\right)+{\varphi }_3\left(k_{hNF}-k_{S3}\right)}{k}_{BF}\left(1-\Omega \theta \left({\eta }^{\ast }\right)\right)\\ {\left(\rho C_P\right)}_{TetraNF}\left(T^{\ast }\right)=[\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\left(\rho C_P\right)}_{S4}}{{\left(\rho C_P\right)}_{BF}}\right)\right]+{\varphi }_{S3}\left(\frac{{\left(\rho C_P\right)}_{S3}}{{\left(\rho C_P\right)}_{BF}}\right)\\ +{\varphi }_{S2}\left(\frac{{\left(\rho C_P\right)}_{S2}}{{\left(\rho C_P\right)}_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\left(\rho C_P\right)}_{S1}}{{\left(\rho C_P\right)}_{BF}}\right)]{\left(\rho C_P\right)}_{BF}\left(1+\varpi \theta \left({\eta }^{\ast }\right)\right)\\ {\rho }_{TetraNF}\left(T^{\ast }\right)=[\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\rho }_{S4}}{{\rho }_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\rho }_{S3}}{{\rho }_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\left(\rho C_P\right)}_{S2}}{{\left(\rho C_P\right)}_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\rho }_{S1}}{{\rho }_{BF}}\right)]{\rho }_{BF}\left(1+\alpha \theta \left({\eta }^{\ast }\right)\right)\end{array}\}$

(8)

In this section, the similarity technique was used to transform the governing PDEs into ODEs. A similarity solution of the equation is usually obtained by defining an independent variable ${\eta }^{\ast }$ and a dependent variable f in terms of the Stream function, ${\Psi }^{\ast }$.

The continuity Equation (1) is satisfied by defining the stream function:

$\begin{array}{l}{\Psi }^{\ast }=\sqrt{{\nu }_{BF}{x}^{\ast }{U}_{\infty }^{\ast }}f\left({\eta }^{\ast }\right)\\ {\eta }^{\ast }=y^{\ast }\sqrt{\frac{U_{\infty }^{\ast }}{{\nu }_{BF}{x}^{\ast }}}\\ T^{\ast }=\left(T_w^{\ast }-T_{\infty }^{\ast }\right)\theta \left({\eta }^{\ast }\right)+T_{\infty }^{\ast }\\ u^{\ast }=\frac{\partial {\Psi }^{\ast }}{\partial y^{\ast }}=U_{\infty }^{\ast }{f}^{\prime }\left({\eta }^{\ast }\right)\\ v^{\ast }=-\frac{\partial {\Psi }^{\ast }}{\partial x^{\ast }}=-\frac{1}{2}\sqrt{\frac{U_{\infty }^{\ast }{\nu }_{BF}}{x^{\ast }}}\left({\eta }^{\ast }{f}^{\prime }-f\right)\end{array}\}$(9)

After differentiating Equations (2)-(4) with Equation (5) by the quotient, product, and chain rules, the following equations were obtained:

$\begin{array}{l}\frac{\left(1+\alpha \theta \right){\ell }^{-\zeta \theta }{f}^{‴}-\left(\zeta {\ell }^{-\zeta \theta }\left(1+\alpha \theta \right)+\alpha {\ell }^{-\zeta \theta }\right)f^{″}{\theta }^{\prime }}{A^{\ast }{B}^{\ast }{\left(1+\alpha \theta \right)}^2}\\ +\frac{1}{2}f{f}^{″}+C^{\ast }G{r}_{T^{\ast }}\theta -\frac{{\ell }^{-\zeta \theta }{K}_{\ast }}{A^{\ast }{B}^{\ast }\left(1+\alpha \theta \right)}{f}^{\prime }=0\end{array}$(10)

$\begin{array}{l}\frac{D^{\ast }}{E^{\ast }}\left[\frac{\left(\left(1+\varpi \theta \right)\left(1-\Omega \theta \right)\right){\theta }^{″}-\left(\Omega \left(1+\varpi \theta \right)+\varpi \left(1-\Omega \theta \right)\right){{\theta }^{\prime }}^2}{{\left(1+\varpi \theta \right)}^2}\right]\\ +\frac{1}{2}\overline{Pr}f{\theta }^{\prime }+\frac{\overline{Br}{\ell }^{-\zeta \theta }}{B^{\ast }{E}^{\ast }\left(1+\varpi \theta \right)}{\left(f^{″}\right)}^2=0\end{array}$(11)

The boundary conditions are:

$f^{\prime }\left(0\right)=0,f\left(0\right)=S^{\ast },{\theta }^{\prime }\left(0\right)=-Bi\left(1-\theta \left(0\right)\right)$ at ${\eta }^{\ast }=0$

$f^{\prime }\left(\infty \right)\to 1,\theta \left(\infty \right)\to 0$ as ${\eta }^{\ast }\to \infty$(12)

where; $\begin{array}{c}{A}^{\ast }=\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\rho }_{S4}}{{\rho }_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\rho }_{S3}}{{\rho }_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\rho }_{S2}}{{\rho }_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\rho }_{S1}}{{\rho }_{BF}}\right)\end{array}$

$B^{\ast }={\left(1-{\varphi }_{S1}\right)}^{2.5}{\left(1-{\varphi }_{S2}\right)}^{2.5}{\left(1-{\varphi }_{S3}\right)}^{2.5}{\left(1-{\varphi }_{S4}\right)}^{2.5}$

$\begin{array}{c}{C}^{\ast }=\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\beta }_{S4}}{{\beta }_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\beta }_{S3}}{{\beta }_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\beta }_{S2}}{{\beta }_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\beta }_{S1}}{{\beta }_{BF}}\right)\end{array}$

$D^{\ast }=\frac{k_{TriNF}\left(k_{S4}+2k_{TriNF}\right)-2{\varphi }_{S4}\left(k_{TriNF}-k_{S4}\right)}{\left(k_{S4}+2k_{TriNF}\right)+{\varphi }_{S4}\left(k_{TriNF}-k_{S4}\right)}$

$\begin{array}{c}{E}^{\ast }=\left(1-{\varphi }_{S1}\right)\{\left(1-{\varphi }_{S2}\right)\left(1-{\varphi }_{S3}\right)\left[\left(1-{\varphi }_{S4}\right)+{\varphi }_{S4}\left(\frac{{\left(\rho C_P\right)}_{S4}}{{\left(\rho C_P\right)}_{BF}}\right)\right]\\ +{\varphi }_{S3}\left(\frac{{\left(\rho C_P\right)}_{S3}}{{\left(\rho C_P\right)}_{BF}}\right)+{\varphi }_{S2}\left(\frac{{\left(\rho C_P\right)}_{S2}}{{\left(\rho C_P\right)}_{BF}}\right)\}+{\varphi }_{S1}\left(\frac{{\left(\rho C_P\right)}_{S1}}{{\left(\rho C_P\right)}_{BF}}\right)\end{array}$

$\zeta =a\left(\frac{T_w^{\ast }-T_{\infty }^{\ast }}{T_0^{\ast }-T_{\infty }^{\ast }}\right)$ is variable viscosity, $\Omega =b\left(\frac{T_w^{\ast }-T_{\infty }^{\ast }}{T_0^{\ast }-T_{\infty }^{\ast }}\right)$ is the variable thermal conductivity, $\varpi =c\left(T_w^{\ast }-T_{\infty }^{\ast }\right)$ is the variable specific heat capacity, $\alpha =d\left(T_w^{\ast }-T_{\infty }^{\ast }\right)$ is the variable density, $\overline{Pr}=\frac{{\mu }_{BF}{C}_{PBF}}{k_{BF}}$ is Prandtl number, $\overline{Br}=\frac{{\mu }_{BF}{U}^{\ast }{}_{\infty }^2}{k_{BF}\left(T_w^{\ast }-T_{\infty }^{\ast }\right)}$ is the Brinkmann number, $K_{\ast }=\frac{{\mu }_{BF}{x}^{\ast }}{\bar{k}{\rho }_{BF}{U}_{\infty }^{\ast }}$ is the permeability parameter, $G{r}_{T^{\ast }}=\frac{{\beta }_{BF}g\left(T_w^{\ast }-T_{\infty }^{\ast }\right)x^{\ast }}{U^{\ast }{}_{\infty }^2}$ is the Grashof number, $S^{\ast }=\frac{v_w^{\ast }}{2}\sqrt{\frac{U_{\infty }^{\ast }{\upsilon }_{BF}}{x^{\ast }}}$ is the suction, $Re=\frac{U_{\infty }^{\ast }{x}^{\ast }}{{\upsilon }_{BF}}$ is the Reynolds number, $B{i}_{\ast }=\frac{h_w}{k}\sqrt{\frac{{\upsilon }_{BF}{x}^{\ast }}{U_{\infty }^{\ast }}}$ is the Biot number. The skin friction coefficient, ( $\overline{C_f}$), and Nusselt number, ( $\overline{Nu}$) are the imperative parameters which are defined as;

$\overline{C_f}=\frac{{\tau }_w}{{\rho }_{BF}{U}^{\ast }{}_{\infty }^2}$ and $\overline{Nu}=\frac{x^{\ast }{q}_w}{k_{BF}\left(T_w^{\ast }-T_{\infty }^{\ast }\right)}$(13)

The shear stress, ${\tau }_w$ and wall heat flux, $q_w$ are given as:

${\tau }_w={\mu }_{TetraNF}\frac{\partial u^{\ast }}{\partial y^{\ast }}$ and $q_w=-k_{TetraNF}\frac{\partial T^{\ast }}{\partial y^{\ast }}$(14)

$\frac{\partial u^{\ast }}{\partial y^{\ast }}=\sqrt{\frac{U^{\ast }{}_{\infty }^3}{{\upsilon }_{BF}{x}^{\ast }}}{f}^{″}\left(0\right)$, and $\frac{\partial T^{\ast }}{\partial y^{\ast }}=\sqrt{\frac{U_{\infty }^{\ast }}{{\upsilon }_{BF}{x}^{\ast }}}\left(T_w^{\ast }-T_{\infty }^{\ast }\right){\theta }^{\prime }\left(0\right)$(15)

Substituting Equation (15) into Equation (14) and the resulting Equation into Equation (13) gives Equation (16)

$\overline{C_f}=\frac{1}{\sqrt{Re}{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}{\left(1-{\varphi }_3\right)}^{2.5}{\left(1-{\varphi }_4\right)}^{2.5}}{f}^{″}\left(0\right)$, and $\overline{Nu}=-\frac{k_{TetraNF}}{k_{BF}}\sqrt{Re}{\theta }^{\prime }\left(0\right)$ (16)

where; $\frac{1}{\sqrt{Re}}=\sqrt{\frac{{\upsilon }_{BF}}{U_{\infty }^{\ast }{x}^{\ast }}}$.

Equations (10) and (11) are coupled nonlinear third-order boundary value problems (BVPs), which cannot be easily solved analytically. Hence, the shooting technique is used to convert the governing BVPs into corresponding IVPs with specified initial value conditions. The unknown initial value conditions are guessed continuously until the boundary condition is satisfied in other to solve the IVPs.

The coupled nonlinear BVP ODEs, i.e. Equations (10) and (11) with the boundary conditions Equation (12) were reformed into the system of first-order ODEs as shown in the following:

Let $\begin{array}{l}f=m_1,f^{\prime }={m^{\prime }}_1=m_2,f^{″}={m^{\prime }}_2=m_3,f^{‴}={m^{\prime }}_3=m_4,\\ \theta =m_5,{\theta }^{\prime }={m^{\prime }}_5=m_6,{\theta }^{″}={m^{\prime }}_6=m_7\end{array}$(17)

Putting Equation (17) into Equations (10)-(11) yields:

$\begin{array}{l}{f}^{‴}={m^{\prime }}_3=m_4=\left(\zeta \left(1+\alpha m_5\right)+\alpha \right)m_3{m}_6-\frac{A^{\ast }{B}^{\ast }\left(1+\alpha m_5\right)}{2{\ell }^{-\zeta m_5}}{m}_1{m}_2\\ \text{}\text{}\text{}-\frac{A^{\ast }{B}^{\ast }{C}^{\ast }G{r}_{T^{\ast }}\left(1+\alpha m_5\right)}{{\ell }^{-\zeta m_5}}{m}_5+K_{\ast }{m}_2\end{array}$

$\begin{array}{l}{\theta }^{″}={m^{\prime }}_6=m_7=\frac{\left[\Omega \left(1+\varpi m_5\right)+\varpi \left(1-\Omega m_5\right)\right]m_6^2}{\left(1+\varpi m_5\right)\left(1-\Omega m_5\right)}-\frac{E^{\ast }\left(1+\varpi m_5\right)\overline{Pr}}{2D^{\ast }\left(1-\Omega m_5\right)}{m}_1{m}_6\\ \text{}\text{}\text{}\text{}\text{}\text{}-\frac{\overline{Br}{\ell }^{-\zeta m_5}}{B^{\ast }{D}^{\ast }\left(1-\Omega m_5\right)}{m}_3^2\end{array}$

subject to: $m_1\left(0\right)=S^{\ast },m_2\left(0\right)=0,m_3\left(0\right)=p,m_5\left(0\right)=q,m_6\left(0\right)=-Bi\left(1-q\right)$(18)

where p and q are the unknown values of $f^{″}\left(0\right)$ and $\theta \left(0\right)$ which are related to the wall shear stress and the temperature of the surface. These unknowns were chosen by guessing repeatedly until the boundary conditions $f^{\prime }\left(\infty \right)\to 1$ and $\theta \left(\infty \right)\to 0$ were satisfied. Taking ${\eta }_{\infty }^{\ast }\approx 5$ arbitrarily, where $p=f^{″}=\text{0}\text{.350914274724478}$ and $q=\theta \left(0\right)=\text{1}\text{.40636461202945}$ were later gotten. The Runge-Kutta method of 4^th order computations was done by substituting the values of the thermo-physical properties of TiO₂, SiO₂, ZnO, Fe₂O₃ and PAO with the aid of the Maple 18.0 software bundle.

Linear regression estimation analysis for Nusselt number as well as skin friction coefficient was done in this segment. The mathematical formula of linear regression was adopted from [20] as shown below. It describes the line of best fit for the relationship between $\stackrel{\leftrightarrow }{y}$ (dependent variable) and $\stackrel{\leftrightarrow }{x}$ (independent variable)

$\stackrel{\leftrightarrow }{y}=b\stackrel{\leftrightarrow }{x}+a$(19)

where $a=\frac{\left({\displaystyle \sum \stackrel{\leftrightarrow }{y}}\right)\left({\displaystyle \sum {\stackrel{\leftrightarrow }{x}}^2}\right)-\left({\displaystyle \sum \stackrel{\leftrightarrow }{x}}\right)\left({\displaystyle \sum \stackrel{\leftrightarrow }{x}\stackrel{\leftrightarrow }{y}}\right)}{n\left({\displaystyle \sum {\stackrel{\leftrightarrow }{x}}^2}\right)-{\left({\displaystyle \sum \stackrel{\leftrightarrow }{x}}\right)}^2}$ is the intercept and $b=\frac{n\left({\displaystyle \sum \stackrel{\leftrightarrow }{x}\stackrel{\leftrightarrow }{y}}\right)-\left({\displaystyle \sum \stackrel{\leftrightarrow }{x}}\right)\left({\displaystyle \sum \stackrel{\leftrightarrow }{y}}\right)}{n\left({\displaystyle \sum {\stackrel{\leftrightarrow }{x}}^2}\right)-{\left({\displaystyle \sum \stackrel{\leftrightarrow }{x}}\right)}^2}$ is the slope of the line.

The thermophysical parameters of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid taken into consideration here are suction, $S^{\ast }$, permeability parameter, $K_{\ast }$ Grashof number, $G{r}_{T^{\ast }}$, Prandtl number, $\overline{Pr}$, Brinkmann number, $\overline{Br}$, Biot number, Bi, thermal conductivity constraint, Ω variable viscosity parameter, $\zeta$, variable density parameter, $\alpha$, and variable specific heat capacity parameter, $\varpi$. The exploration of their effects on the velocity and temperature profiles as well as on the drag friction $f^{″}\left(0\right)$ and the rate of heat transfer, ${\theta }^{\prime }\left(0\right)$ was considered. The volume fraction of the four NPs is in the range $0\le \varphi \le 0.2$. Table 2 represents the values of the thermophysical properties of PAO and the four NPs.

Comparison of this research with the work of [50] for the plate surface temperature $\theta \left(0\right)$ and the rate of heat transfer $-{\theta }^{\prime }\left(0\right)$ varying Biot number, Bi showed that the values agree up to the fourth place of decimal as shown in Table 3 .

The numerical and statistical techniques were explored to investigate the influence of the combined temperature thermophysical properties on the dissipative movement of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid in the presence of suction along a vertical porous surface. The study demonstrates that the thermophysical properties of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid are crucial in the enhancement of the rate of heat transfer in the system.

1) An increase in the variable thermal conductivity of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid increases the skin friction, Nusselt number, and temperature profile.

2) The skin friction and the Nusselt number increase as the variable viscosity increases, whereas the temperature profiles decrease.

3) As the variable density increases, the velocity profile increases, similarly, both the skin friction and Nusselt number increase.

4) An increase in the variable specific heat capacity increases the temperature profile and both the skin friction coefficient and Nusselt number.

Compared to previous literature, this research provides deeper insights into Tetra hybrid nanofluid behaviour, offering substantial advancements for heat transfer optimization. Future research avenues may explore broader parameter variations, experimental validation, and applications in diverse industrial contexts, further enhancing the field’s understanding and practical implementation. It can also be concluded that linear regression is an excellent approach to give more insights into the relationship between the dependent variables and the independent variables of the boundary layer flow of TiO₂-SiO₂-ZnO-Fe₂O₃/PAO tetra hybrid nanofluid.