Nanotechnology is the field of science that deals with nanometer-sized particles with a size range of 1–100 nm. Nanofluids have a higher thermal conductivity than ordinary fluids due to the addition of nanoparticles. They are mostly used in the smart computing and medical fields. In 1995, Choi and Eastman (1995) first revealed the presence of nanofluids to the scientific world. The Tiwari–Das and Buongiorno models (Buongiorno, 2005; Tiwari and Das, 2007) are the two types of nanofluid models used widely in current academic research. Hatami and Safari (2016) and Makinde and Aziz (2011) completed extensive studies on the nature of nanofluid flows over various geometries. Kumari et al. (2001), Chakraborty and Janapatla (2023) and Gorla et al. (2010) used vertical wedge geometry to study the steady and mixed convective flows of nanofluids.

With a structure mimicking a honeycomb, graphene (Gr) comprises single-layered s p 2 hybridized carbon atoms. After the oxidation process with the oxygen (O₂) atom, a multidimensional compound, graphene oxide (GO), is formed (Natalini and Sciubba, 1999). This compound was first prepared by oxidizing graphite in the presence of H N O 3 and K C l , by Baronent Benjamin Colline Brodie in 1859. At present, the modified Hummers method is implemented for the synthesis of GO (Kock and Herwig, 2004). The electrical conductivity of graphene was measured to be approximately 7200   S / m and at room temperature, and the thermal conductivity varies between 1800 – 5800   W / m K (Kuilla et al. (2010). Graphene is considered to be one of the strongest materials, with an intrinsic strength of 130 G P a and a breaking strength of 42 N / m (Lee et al., 2008). Graphene oxide is one of the most important additives for cement. Small amounts (0.03%) of G O can cause a 39%–57% increase in the flexural strength, increased compressive strength, and increased ductility, and the corrosion caused by microbes might be avoided by using G O (Mangadlao et al., 2015). Graphene oxide is a good emulsion stabilizer because it behaves as a colloidal surfactant due to its amphoteric nature (Kim et al., 2010).

The study of the radiation effect is one of the most important effects studies in academia for its wide range of applications in science and technology concerning heat and mass transfer of flows. Being implemented in nuclear waste extraction and separation processes, the study of convective heat transfer within fluid flow has gained importance among researchers. Some extensive and comprehensive studies were performed on the convective nature of the flows (Sivakumar et al., 2017). One study examined a magnetohydrodynamics (MHD) flow of ferro-liquid in the presence of two types of external effects, viz, viscous dissipative radiation effects with slip and convective boundary conditions along with the thermal radiation effect (Cheng, 1979). TiO₂ is a non-toxic, economical, stable ceramic material with a relatively high thermal conductivity (4.0–11.8 W m^–1 K⁻¹). The thermal behavior was studied in the presence of a heat source/sink for a copper-titanium oxide (Cu–TiO₂) hybrid nanofluid (Leong et al., 2018), and the results were concurrently compared to a conventional (Cu and TiO₂) nanofluid. The mathematical model of a nanofluid with based fluid (engine oil) and titanium dioxide nanoparticles (Vasheghani et al., 2013).

The tool used by the present researchers to examine the extent of the effect of any parameter is the sensitivity analysis, which gained its importance for the wide range of applications in control theory and nuclear industries. Empirical relationships are formed to correlate the input and the output responses with the help of ANOVA using the response surface methodology (RSM). The primary focus of the study was to implement a sensitivity analysis for the Newtonian nanofluid study. In this context, using triangle-shaped obstacles, Rashidi et al. (2015) conducted a sensitivity analysis using the RSM. It was observed that the wedge angle parameter proved to be more sensitive to the Nusselt number than the skin friction coefficient. Darbari et al. (2016) studied the flow through a channel and evaluated the sensitivity analysis of the nanofluid flow properties. Reynold’s number was found to be most sensitive to the entropy generation. The RSM was utilized to investigate the Casson fluid flow, and, as expected, Abdelmalek et al. (2020) found that positive sensitivity prevails for a Nusselt number with increasing magnetic parameters.

The combined effects of magnetic effects, the Falkner–Skan parameter m , and thermal radiation for the hybrid nanofluid of graphene oxide and titanium in water over a wedge has been studied in this article and has not yet been addressed in the literature, indicating the novelty of our work. MATLAB bvp4c has been used to solve the set of ordinary differential equations obtained by similarity transformations. The results were compared with the previously published results and found to be in good agreement. The applications of graphene and its derivatives as emulsion stabilizers, anti-corrosion coatings, etc., in the oil and petroleum industries are the motivation for conducting the present study.

A steady, laminar, and 2D incompressible flow is considered in the present study with an aqueous solution of the G O − T i O 2 /water hybrid nanofluid over a static or moving wedge. Graphene oxide is the first nanoparticle denoted by the subscript 1, and titanium oxide is the second nanoparticle denoted by 2 in the subscript. The thermophysical properties of the nanoparticles are presented in Table 1 as considered in the temperature range of 25 ℃ − 30 ℃ (Dinarvand et al., 2019; Sundar et al., 2020; Verma et al., 2022).

Figure 1 shows the schematic representation of the problem considered for our study. The x-axis is taken along the wedge surface, and the free stream velocity is considered U x = a x m , while for the moving wedge, the velocity is considered u w = U w x m . Along with the boundary layer approximations, the Tiwari–Das model for nanofluids and Bernoulli’s equations have been implemented in the governing set of partial differential equations (PDEs) for our problem. The thermophysical properties of the hybrid nanofluids can be calculated from the properties of the individual nanoparticles and base fluids from the information provided in Table 2 (Maïga et al., 2004).

The assumptions considered for our problem are as follows:

• We see a variable surface temperature of the wedge as T w x = T ∞ + T 0 x 2 m − 1 , and the ambient temperature of the hybrid nanofluid is given by T ∞ .

In Table 2, we have computed the thermal conductivity of the nanofluids k n f and k h n f using the available value of thermal conductivity of the base fluid k f using the Hamilton–Crosser model (Ghadikolaei et al., 2017). Table 3 denotes the empirical shape factor values for the nanoparticles.

The equations governing the flow are as follows: ∂ u ∂ x + ∂ v ∂ y = 0 . (1) u ∂ u ∂ x + v ∂ u ∂ y = U x d U x d x + μ h n f ρ h n f ∂ 2 u ∂ y 2   . (2) u ∂ T ∂ x + v ∂ T ∂ y = k h n f ρ C p h n f ∂ 2 T ∂ y 2 − ∂ ∂ y 1 ρ C p h n f q y r + Q ′ x ρ C p h n f T − T ∞ . (3)

The boundary conditions are as follows (Dinarvand et al., 2019;Maïga et al., 2004): v = 0 ;   u = u w x ;   T = T w   a t   y = 0 . (4) u → U x ;   T → T ∞   a s   y → ∞ . (5)

Using Rosseland’s approximation, q y r is the radiative heat flux given by q y r = − 4 σ * 3 k * ∂ T 4 ∂ y . (6)

Here, k * is Rosseland’s mean absorption coefficient, and σ * is the Stefan–Boltzmann constant. While assuming negligible temperature differences, Eq. (10) reduces to ∂ q y r ∂ y = − 16 σ * T ∞ 3 3 k * ∂ 2 T ∂ y 2 . (7)

The set of nondimensional similarity transformations is given by Kock and Herwig (2004) and Madhu et al. (2024): ψ = 2 x ν f U x m + 1 f η ;   η = y m + 1 U x 2 x ν f ;   θ η = T − T ∞ T w x − T ∞ ; (8)

The stream function ψ x , y can be defined as u = ∂ ψ ∂ y ;   v = − ∂ ψ ∂ x . (9)

On incorporating Eqs 11, 13 into Eqs 5–9, we obtain   A 1 f ‴ + 2 m m + 1 1 − f ′ 2 + f f ″ = 0 . (10)   θ ″ k h n f k f + R d + ⁡ Pr 1 A 2   f θ ′ − 3 β 1 − 2 f ′ θ + 2 m + 1 Q A 2 θ = 0 . (11)

along with the boundary conditions f η = 0 ;   f ′ η = λ ;   θ η = 1   a t   η = 0 . (12) f ′ η → 1 ;   θ η → 0   a s   η → ∞ . (13)

The quantities can be defined as Pr ⁡ = ν α ;   R d = 16 σ * T ∞ 3 3 k f k * ;   Q = Q 0 ρ C p f a   ;   λ = U w a . (14)

Here, Pr is the Prandtl number, R d is the radiation parameter, Q is the heat generation parameter, and λ is the velocity ratio parameter.

In addition, we have   A 1 = μ h n f μ f 1 − ϕ 2 1 − ϕ 1 + ϕ 1 ρ 1 / ρ f + ϕ 2 ρ 2 / ρ f − 1 .   A 2 = 1 − ϕ 2 1 − ϕ 1 + ϕ 1 ρ C p 1 / ρ C p f + ϕ 2 ρ C p 2 / ρ C p f − 1 .

The drag coefficient C f and Nusselt number N u x are given by C f = τ w ρ f U 2 ;   N u x = x q w k f T w − T ∞ . (15)

In this context, τ w is the surface shear stress, and q w is the surface heat flux. These quantities can be defined as τ w = μ h n f ∂ u ∂ y y = 0 ;   q w = − k h n f ∂ T ∂ y y = 0 + q y r y = 0 . (16)

Now, using the similarity transformation (8) into Eq. 16, we get the following reduced form as follows: 2 m + 1 1 / 2 C f R e x 1 / 2 = μ h n f μ f f ″ 0 .   2 m + 1 1 / 2 N u x R e x − 1 / 2 = − k h n f k f + R d θ ′ 0 . (17)

The governing set of Eqs 1–3 is converted to a set of coupled nonlinear ordinary differential equations (ODEs) Eqs 10, 11 using the similarity transformations and the theory concerning the boundary layer. The MATLAB bvp4c method implements the Lobatto IIIA method as the base method used to obtain C 1 solutions. The uniform accuracy of the solutions up to the fourth order in the chosen interval of integration is one of the key reasons for using the bvp4c method. The error of tolerance chosen for the present method is 10 − 10 . In Table 4, we have compared the f ″ 0 values for water as the base fluid, in the absence of heat generation and radiation terms Q = R d = 0 and assuming ϕ 1 = ϕ 2 = 0 , for a static wedge λ = 0 . The results had a very good correlation to the results of Yih (1998), White and Majdalani (2006), Ishak et al. (2007), Yacob et al. (2011), and Nadeem et al. (2018), which validates our code.

The present study has been conducted in the presence of a magnetic field and thermal radiation for a G r O –TiO₂/VPO hybrid nanofluid in the light of empirical shape factors that vary from spherical to lamina. The entire study has been conducted with values of the governing parameters: β 1 = 1.0 ;   Q = 0.5 ;   R d = 1.0 , ϕ 1 = ϕ 2 = 0.05 in the context of a static wedge λ = 0 if not mentioned otherwise. The ranges of the parameters used are 1.0 ≤ β 1 ≤ 1.4 ;   0.1 ≤ Q ≤ 0.5 ;   0.6 ≤ R d ≤ 1.0 , 0.01 ≤ ϕ 1 , ϕ 2 ≤ 0.05 .

In Figure 2A, the effects of the Falkner–Skan parameter and the nanoparticle volume fraction values on the velocity profiles for a static wedge have been upheld. In the presence of spherical-shaped nanoparticles n 1 = n 2 = 3 , we can observe that the dimensionless velocity increases with an increasing parameter m , along with a reverse trend for the case of increasing ϕ 1 , ϕ 2 values. The increase in the Falkner–Skan parameter causes the pressure gradient to increase, and it increases both the momentum boundary layer thickness and the velocity profile of the fluid. Increasing the nanoparticle volume fraction values causes an increase in the nanoparticle concentration in the fluid, reducing the overall velocity profile of the fluid.

Increasing the values of the Falkner–Skan parameter causes an increase in the thermal boundary layer thickness, which, in turn, causes the temperature of the fluid to decrease, and this phenomenon has been upheld in Figure 2B. Increased heat generation parameter values cause more thermal energy to dissipate in the fluid, and this causes the thickness of the thermal boundary layer to decrease. This results in an increase in the temperature profile of the fluid; the validation of this fact is shown in Figure 2B.

In our study λ represents the velocity with which the wedge moves in the fluid. Hence, as the velocity of the wedge increases, due to the no-slip conditions, the layer of the fluid adjacent to the wedge surface also starts to accelerate, resulting in superposing the velocity along with the existing velocity of the fluid. Thus, for a nanofluid-saturated medium, as the velocity of the wedge increases, the velocity profile for the fluid flow system increases, as shown in Figure 3A.

The variation in the temperature profile for the increasing radiation parameter values and empirical shape factor of the nanoparticles has been represented in Figure 3B. Increasing the thermal radiation parameter Rd increases the convective flow, which in turn increases the velocity of the fluid. We observe that the thermal boundary layer thickness increases because the heat transfer increases. The temperature distribution is enhanced with an increase in the thermal radiation parameter. Owing to a higher surface area exposed in the fluid flow process, the thermal conductivity increases more for brick and platelet shapes than for spherical shapes. Hence, the temperature profiles are higher for higher shape values.

The increasing radiation parameter also increases the temperature profile of the fluid because, for a higher thermal conductivity, the fluid temperature increases, and, hence, the θ values are maximum for platelet and minimum for spherical shapes. Physically, the velocity of the wedge increases within the fluid flow system, and the thermal boundary layer thickness increases rapidly, causing the temperatures to fall subsequently. This is shown in Figure 4.

Figure 5 shows that the Nusselt number decreases as the Falkner–Skan parameter and nanoparticle volume fraction increase. As the pressure gradient increases, the flow velocity increases, and the density of the flow medium is reduced. This, in turn, increases the thermal diffusivity of the system. The heat transfer rate is found to be inversely proportional to the thermal diffusivity of the system, and thus, the Nusselt number decreases with increasing parameter m .

We see the increasing Nusselt number values when the heat generation parameter increases. Figure 6A, B represents the skin friction coefficient variations for an increasing Falkner–Skan parameter along with increasing nanoparticle volume fraction values for both nanoparticles with spherical shapes. The increasing pressure gradient parameter increases the wall shear stress, while a reverse effect is observed for increasing volume fractions.

Figures 7A–C represent the streamlines for m = 0.5 , 0.8 , 1.0 , that is, increasing the Falkner–Skan parameter during the fluid flow in a medium saturated with hybrid nanofluid. The increasing values of the streamlines denote that the flow is heavier away from the surface of the wedge than the flow near the wedge surface. In the stream plots for two different parameter values, we can observe that the corresponding streamline values are increasing, validating the fact that the velocity profile increases with an increasing pressure gradient parameter.

Analyzing any particular data set for its significance in influencing any response is a key role of any boundary layer mode experimental design. Such experimental models can be seen in the literature, such as response surface methodology and factorial designing using the central composite design (CCD) or the Box–Behnken design (BBD). In accordance with the numerical data evaluated, it can be observed that two of three quantities, namely, radiation parameter and pressure gradient, affect and increase the Nusselt number R e x − 1 / 2 N u x but the heat source parameters reduce it. Among them, the most significant factor affecting the heat transfer rate can be determined by statistical data analysis. In this article, we have implemented the BBD model with three continuous factors, R d , Q , β . The general form of correlation between the input parameters R d , Q , β and the response parameter R e x − 1 / 2 N u x can be written as: R e x − 1 / 2 N u x = α 0 + α 1 A + α 2 B + α 3 C + α 11 A 2 + α 22 B 2 + α 33 C 2 + α 12 A B + α 13 A C + α 23 B C . (18)

Here, A , B , C are the coded symbols corresponding to the input parameters shown in Table 3. These α i ′ s , α i j ′ s are the regression coefficients to be determined by RSM using 20 experimental runs and 19 degrees of freedom. The coefficients will be determined using MINITAB software.

The goal of this study is to determine how the uncertainties corresponding to the system inputs can be correlated to the response to the physical problem. It provides an effective and concrete justification of how much the continuous factors influence the response parameter. The flow chart of the analysis is provided in Figure 13. In this analysis, we need to evaluate the partial derivatives of Eq. 19 with respect to the input-independent parameters R d , Q , β   and evaluate the derivatives at the three different levels mentioned in Table 4. The reduced expressions are ∂ R e x − 1 / 2 N u x ∂ A = 0.17367 + 0.04874 C . (20) ∂ R e x − 1 / 2 N u x ∂ B = − 0.47577 − 0.04796 B . (21) ∂ R e x − 1 / 2 N u x ∂ C = 0.80386 + 0.04874 A + 0.15566 C . (22)

Using Eqs (20)-(22), we evaluate the sensitivity analysis for three possible levels for each parameter and infer using the obtained results in Table 5. The sensitivity graphs shown in Figure 14 clearly portray the behavior of each of the input parameters in influencing the Nusselt number response. In Table 8, we see that the highest sensitivity value 0.95952 occurs for B = − 1 , 0 , 1 and C = 1 corresponding to the magnetic parameter value A = 0 , that is, R d = 0.8 , while the least value − 0.52373 occurs for B = 1 and C = − 1 , 0 , 1 corresponding to the magnetic parameter value A = 0 , that is, R d = 0.8 . So we conclude that the positive sensitivity becomes less intense with increased input parameter values. Hence, we can conclude that the sensitivity of R e x − 1 / 2 N u x increases as we increase the values of all input parameters.

In this present study, we carried out a numerical investigation studying the effects of Rd, the heat absorption parameter, and the Falkner–Skan parameter for a 2D, hybrid nanofluid flow on a wedge geometry and conducted a sensitivity analysis using BBD. The PDEs were obtained using the Tiwari–Das model to define the problem chosen in this article. Later, they were converted to nondimensional ODEs using similarity transformations. Some of the major conclusions drawn from the results are:

• There is a decrease in the heat transfer coefficient for an increased Falkner–Skan parameter, and increasing the empirical shape factor values results in a decrease of the Nusselt number values.

The factors that affect the heat transfer and the skin friction coefficients for the concerning flow have been thoroughly investigated in this article, and the results are represented graphically. The motivation for this study is its contributions in the fields of magnetic drug targeting, medical sciences, and oil and petroleum industries. As a scope for future work, RSM can be paired with multiple regression analysis for three or more independent input parameters and various other effects on a wedge model with different nanoparticle shapes and in the presence of nanoparticle aggregation. The various applications of the present study in the field of oil drilling, emulsion stabilizers, oxalic acid removal, and additives of multi-grade oils provide additional motivation for our present study. Furthermore, this problem can also be extended to apply the distinct schemes like ANN, fractional derivatives and ARA- Sumudu decomposition method etc., see [Saadeh et al. (2023a, 2023b), Chandan et al. (2024), and Qazza et al. (2023)].