Hall-Effect-Modulated Thermal Transport Enhancement in Hybrid Nanofluid Flow over a Stretching Surface Using Taguchi Optimization

In an era of rising energy demands, hybrid nanofluids have become a groundbreaking development in thermal engineering. They provide high thermal conductivity and heat transfer by synergistically combining more than one nanoparticle in a base fluid. Their revolutionary applications include solar energy systems, aerospace cooling, microelectronics, biomedical therapeutics and industrial heat exchangers, pushing the frontiers of contemporary thermal management in multiphysics engineering systems. As a result, the scientific community has seen a phenomenal increase in research interest in hybrid nanofluids, which they have realized as a necessary pillar of future energy-efficient technologies. Sharma et al. [1] focused on magnetohydrodynamic flow of hybrid nanofluid through a damaged artery using a numerical method-based solver. They showed the impact of nanoparticles on wall shear stress, velocity, and impedance, which could be of use in drug delivery systems and biomedical applications. The thermosolutal Marangoni stagnation hybrid nanofluid flow hybrid nanofluid across an extending sheet was demonstrated by Mohanty et al. [2]. The research by Manimegalai and Kameswaran [3] on the hybrid nanofluid flow in solar collectors indicated that radiation, magnetic field, and amount of nanoparticles influence the heat transmission, enhancing thermal conduction. Parida et al. [4] studied a magnetohydrodynamic flow of hybrid nanofluid containing gold and multiwalled carbon nanotubes in blood. It was found that thermal radiation raises temperature while magnetic fields reduce the velocity. Various studies [5], [6], [7] analyzed the hybrid nanofluid flow past a different geometry considering the effects of magnetic fields with thermal effects such as radiation, dissipative heat, and internal heat generation.

A magnetic field is employed perpendicular to an electrically conducting fluid and charge carriers are deflected sideways, leading to a transverse current, which causes a Hall current. The phenomenon is critical to modern applications in electromagnetic engineering since it has a substantial influence on the velocity distribution and thermal transfer behavior in magnetohydrodynamic flows. Rana et al. [8] studied the flow of nanofluids over a revolving disk, considering the effects of Hall current and nanoparticle aggregation using the finite difference method. Hakeem et al. [9] studied the hybrid nanofluid flow over a spinning cone. They numerically solved the governing equations using the Runge-Kutta shooting method. Kanimozhi et al. [10] investigated mixed convective Jeffrey fluid flow across an oblique upright plate. The perturbation approach was used to solve the problem analytically. Many investigations [11], [12], [13], [14] incorporated the influence of magnetic fields and Hall currents in a variety of geometries using porous media.

Thermal radiation can be described as the emission of electromagnetic waves that are produced exclusively by the thermal movements of particles in a body. This radiative mechanism of heat transfer becomes more and more dominant at high temperatures, having a significant effect on the temperature of fluids and becoming an extremely important consideration in the analysis of heat transfer in fluids. Banakar et al. [15] investigated radiative nanofluid flow over an off centric disk considering micro inertial effect. Madhukesh et al. [16] used the Runge–Kutta–Fehlberg method to obtain numerical solutions of nanofluid flow over a Riga surface in a porous material, focusing on radiation and nanoparticle aggregation. The results showed that porosity reduces fluid velocity, whereas aggregation enhances heat dispersion. A numerically analyzed Maxwell hybrid nanofluid flow over a porous stretching plate was discussed by Jayaprakash et al. [17] They revealed that magnetic flux and thermal radiation significantly influence heat transfer, while nanoparticle shape strongly affects temperature, concentration, and flow behavior. Furthermore, several researchers [18], [19], [20], [21] considered the impact of thermal radiation in hybrid nanofluid flows through various geometries such as microchannels, porous Riga surfaces, stretching sheets, and spinning disks. Ramadevi et al. [22] illustrated the magnetohydrodynamic Carreau fluid flow over a widening surface with the emphasis on the effects of dissipative heat and a variable heat source. It was demonstrated that non-uniform sources significantly influence the dynamics of thermal and mass transport. Govindarajulu et al. [23] investigated the magnetohydrodynamic hybrid nanofluid flow through an upright channel with the influence of a non-uniform heat source using a numerical method. Madhukesh et al. [24] applied the Runge–Kutta method to address the governing equations of hybrid nanofluid flow in a microchannel with the effects of waste discharge and space- and time-dependent heat sources. In addition, several recent studies [25], [26], [27], [28] investigated the behavior of thermal transfer in hybrid nanofluid flows over Riga surfaces, extending sheets, and tubes, with the effects of non-uniform heat sources.

Genichi Taguchi created the reliable statistical technique known as Taguchi optimization to methodically determine the ideal process parameters with the fewest possible experimental trials. It effectively reduces variability and increases performance output by utilizing orthogonal arrays and signal-to-noise ratio analysis, making it a very potent instrument in contemporary engineering and thermal research optimization. By optimizing discrete V-down baffle settings in a solar air heater using the Taguchi–Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) approach, Sharma et al. [29] improved thermo-hydraulic performance and heat transfer efficiency through Taguchi design analysis. Kumar et al. [30] studied the blood flow of gold and zinc nanoparticles in an inclined microchannel. To optimize entropy formation under radiation and heat source impacts, they used Taguchi and ANOVA techniques. Cicek and Johnson [31] used Taguchi and Gray relational analysis to study polycarbonate three-dimensional printing parameters in order to maximize tensile strength, material consumption, and build time through multi-objective optimization. Numerous studies [32], [33], [34], [35] used Taguchi-based optimization in conjunction with numerical methods to analyse non-Newtonian and nanofluid flows by incorporating various effects like radiation, magnetic fields, and chemical effects.

This study aims to:

• Investigate steady, incompressible flow and thermal transfer of tricalcium phosphate (Ca$_3$(PO$_4$)$_2$)–molybdenum disulfide (MoS$_2$)/water hybrid nanofluid over a porous extending sheet.

• Incorporate the Hall current effect under strong magnetic fields into the governing equations, capturing realistic charge transport alterations typically neglected in classical magnetohydrodynamic nanofluid studies.

• Rigorously compare the thermal performance of Ca$_3$(PO$_4$)$_2$–MoS$_2$/water hybrid nanofluid against its mono-nanofluid counterpart, quantifying the heat transfer enhancement from dual nanoparticle synergy.

•Apply the Taguchi technique to evaluate the significance of key non-dimensional parameters and identify optimal conditions for maximum heat transfer, connecting mathematical modeling to practical engineering design.

In accordance with the objectives outlined, this research offers the following contributions to the existing body of knowledge. The use of Ca$_3$(PO$_4$)$_2$–MoS$_2$/water as a hybrid nanofluid is itself a distinctive and uncommon choice that separates this work from the mainstream literature. Building upon this, the study uniquely couples Hall current effects with hybrid nanofluid dynamics over a porous extending sheet, a combination rarely attempted before. Further novelty lies in the concurrent treatment of thermal radiation alongside a non-uniform heat source, reflecting far more realistic thermal conditions than those typically assumed. Most remarkably, the deliberate integration of the Taguchi optimization methodology into this multi-physics nanofluid framework transforms the study from a purely theoretical exercise into a practically actionable engineering tool, making this work both scientifically rigorous and industrially meaningful.

This study investigates the continuous flow of an incompressible hybrid nanofluid near an upward porous extending sheet at the $xz$-plane in the presence of Hall current effects and an electric field ( Figure 1). The sheet is stretched linearly in both positive and negative directions when two opposite and equal forces are assumed to act next to the x-axis while the origin remains constant. The research takes into account a non-Newtonian hybrid nanofluid that generates or absorbs heat and has electrical conductivity. The hybrid nanofluid contains Ca$_3$(PO$_4$)$_2$ and MoS$_2$ nanoparticles in addition to water as the base fluid. To account for the Hall current effect, high-frequency electron collisions are taken into consideration, and an electric field is supposed to be applied across the z-axis. A strong magnetic flux density $B_0$ is used to introduce the Hall effect, and a low magnetic Reynolds number is assumed to discard the generated magnetic field. Additionally, it is assumed that there are no changes in heat transfer or velocity along the z-axis, which is supported by taking into account an infinitely wide sheet.

The magnetic field follows the simplified version of Ohm's law below when the magnetic field is strong.

$ J=\sigma(\mathrm{E}+q \times B) $

where, $\sigma=\frac{e^2 n_e t_e}{m_e}$ is the electrical conductivity, $n_e$ is the electron number density, $m_e$ is the electron mass, and $e$ is the electron charge. In addition, the current density vector is given by $J=\left(j_x j_y j_z\right)$, and the current can also be expressed using the relation: $j=\sigma \times(E+V \times B)$, where $B=(0,0,0)$ and the fluid velocity vector is $V=(u, v, w)$. The Hall parameter is defined as $m=\frac{\sigma B_0}{e n_e}$. Using the aforementioned assumptions, the standardized Ohm's law for weakly ionized gases yields $\left(J_y=0\right)$ in the flow field. Thus, the elements of current density $j_x$ and $j_z$ are determined as:

$ j_x=\sigma \frac{B_0^2}{1+m^2}(u+m w) \text { and } j_z=\sigma \frac{B_0^2}{1+m^2}(m u-w) $

Since the Rosseland approach takes into consideration both self-absorption and emission as important elements impacting heat transmission, it is appropriate for predicting the radiative flux vector $qr$ in dense fluids. The Rosseland approximation [36] applies because the absorption coefficient is usually significant and changes with wavelength. The governing equations representing the flow problem are as follows:

The boundary conditions are as follows:

Similarity transformations are as follows:

Eqs. (1)-(5) that control the system can be converted to the following expressions:

The transformed boundary conditions are as follows:

The important dimensionless parameters are as follows:

$M F=\frac{\sigma_f B_0^2}{\rho_f a}, \mathrm{~K}=\frac{v_f}{k_p^* a}, \operatorname{Pr}=\frac{\kappa_f}{\left(\rho C_p\right)_f v_f}, R d=\frac{16 \sigma^* T_{\infty}^3}{3 \kappa_f k^*}$

The crucial physical characteristics of heat transmission rate and drag force can be expressed as:

$C f_x=\frac{\left.\mu_{\mathrm{hnf}} \frac{\partial u}{\partial y}\right|_{y=0}}{\rho_f u_w^2}, \quad C f_z=\frac{\left.\mu_{\mathrm{hnf}} \frac{\partial w}{\partial y}\right|_{y=0}}{\rho_f u_w^2}, \quad N u_x=\frac{-\left.x \kappa_{\mathrm{hnf}} \frac{\partial T}{\partial y}\right|_{y=0}+\left.q_r\right|_{y=0}}{\kappa_f\left(T_w-T_{\infty}\right)}$.

The non-dimensional forms of these physical quantities using Eq. (5) are as follows:

where, $R e_x^2=\frac{x u_w}{v}$.

This study employs a fourth-order Runge–Kutta method with a shooting strategy to address this problem numerically, focusing on a system of intricate equations. This method yields a sophisticated mathematical system characterized by high precision. The simulations transform the higher-order issue into a set of first-order equations: $f=\vartheta_1, f^{\prime}=\vartheta_2, f^{\prime \prime}=\vartheta_3, g=\vartheta_4, g^{\prime}=\vartheta_5, \theta=\vartheta_6, \theta^{\prime}=\vartheta_7$.

The transformed equations are as follows:

$f^{\prime \prime \prime}=\frac{\left(\vartheta_2\right)^2-\left(\vartheta_1\right)\left(\vartheta_3\right)+\frac{\frac{\sigma_{\mathrm{hnf}}}{\sigma_f}}{\frac{\rho_{\mathrm{hnf}}}{\rho_f}} \frac{\mathrm{Ha}}{1+m^2}\left(\left(\vartheta_2\right)+m\left(\vartheta_4\right)\right)+\frac{\frac{\mu_{\mathrm{hnf}}}{\mu_f}}{\frac{\rho_{\mathrm{hnf}}}{\rho_f}} K_p\left(\vartheta_2\right)}{\frac{\frac{\mu_{\mathrm{hnf}}}{\mu_f}}{\frac{\rho_{\mathrm{hnf}}}{\rho_f}}}$

$g^{\prime \prime}=\frac{\left(\vartheta_2\right)\left(\vartheta_4\right)-\left(\vartheta_1\right)\left(\vartheta_5\right)-\frac{\frac{\sigma_{\mathrm{hnf}}}{\sigma_f}}{\frac{\rho_{\mathrm{hnf}}}{\rho_f}} \frac{\mathrm{Ha}}{1+m^2}\left(m\left(\vartheta_2\right)-\left(\vartheta_4\right)\right)+\frac{\frac{\mu_{\mathrm{hnf}}}{\mu_f}}{\frac{\rho_{\mathrm{hnf}}}{\rho_f}} K_p\left(\vartheta_4\right)}{\frac{\frac{\mu_{\mathrm{hnf}}}{\mu_f}}{\frac{\rho_{\mathrm{hnf}}}{\rho_f}}}$

$\theta^{\prime \prime}-\frac{\frac{\frac{\kappa_{\mathrm{hnf}}}{\kappa_f}}{\frac{\left(\rho C_p\right)_{\mathrm{hnf}}}{\left(\rho C_p\right)_f}}\left[\alpha^*\left(\vartheta_2\right)+\left(\vartheta_6\right) \beta^*\right]-\operatorname{Pr}\left(\vartheta_1\right)\left(\vartheta_7\right)}{\frac{\left(\frac{\kappa_{\mathrm{hnf}}}{\kappa_f}+\mathrm{Rd}\right)}{\frac{\left(\rho C_p\right)_{\mathrm{hnf}}}{\left(\rho C_p\right)_f}}}$

Transformed boundary conditions are as follows:

$\begin{aligned} & \left(\vartheta_1\right)(\eta)=\left(\vartheta_4\right)(\eta)=0,\left(\vartheta_2\right)(\eta)=1,\left(\vartheta_6\right)(\eta)=1 \text { at } \eta=0 \\ & \left(\vartheta_2\right)(\eta)=\left(\vartheta_4\right)(\eta)=\left(\vartheta_6\right)(\eta)=0 \text { as } \eta \rightarrow \infty\end{aligned}$

The required solution necessitates an initial estimate using the boundary value problem solver bvp4c in MATLAB, which satisfies the prescribed boundary conditions. The value of $\eta$, originally defined as $\infty$, is designated as $\eta=7$ to meet software requirements.

This section explores the transport properties of a hybrid nanofluid by analyzing the influence of critical governing factors on momentum and temperature in an extending sheet configuration. The flow problem is formally represented by partial differential equations and, with the assistance of similarity variables, converted into ordinary differential equations. The fourth-order Runge–Kutta method is employed for the numerical solution of the ordinary differential equations. The influence of the Hall effect, thermal radiation, porosity effect and non-uniform heat source is taken into account. The Taguchi method is employed to optimize the thermal transfer rate. Table 1 describes the thermophysical properties of hybrid nanofluids.

Table 2 shows the validation of this research with previous studies. The incorporation of particle concentrations plays an essential role in the flow profiles. The concentration of the nanoparticle

(Ca$_3$(PO$_4$)$_2$) is designated as $\Phi_{1}$, while the concentration of the MoS$_2$ particle is denoted as $\Phi_{2}$. The significance of the factors arises from the examination of the thermophysical models relevant to flow processes, with each model of density, viscosity, thermal conductivity, etc., depending upon the variations in volume fraction. Particle concentration, also referred to as volume fraction, denotes the percentage of particles present in the total volume of the liquid. Specifically, Figure 2 and Figure 3 illustrate the fluctuation of the concentrations $\Phi_{1}$ and $\Phi_{2}$ across the flow profiles, respectively. In both figures, the concentration range is limited to 2%, specifically $0<\Phi_{1},\Phi_{2}<0.03$. The graphical representation of these figures demonstrates that as $\Phi_{1}$ and $\Phi_{2}$ increase, both $f'(\eta)$ and $g(\eta)$ decrease. The dominance of the bi-hybrid nanofluid's density diminishes momentum, while an increase in conductivity leads to a retardation of the profile.

The influence of $Ha$ on $f'(\eta)$ and $g(\eta)$ is demonstrated in Figure 4 and Figure 5. The increasing values of $Ha$ decreases $f'(\eta)$ whereas the opposite behavior is observed for $g(\eta)$. The cross-flow velocity is absent when the Hall current value is zero, correlating with Figure 6 which shows no cross-flow velocity when there is no magnetic field present. The physical interpretation of this result is that the magnetic field exerts a force on the moving charge carriers, thereby contributing to the induction of the Hall current. The Hall effect becomes significant when the magnetic field intensity is adequate. An extra potential difference is generated, resulting in an induced current that flows perpendicularly to the direction of the current as well as in the direction of the magnetic field. The variation of $m$ with $f'(\eta)$ and $g(\eta)$ is visualized in Figure 6 and Figure 7. The increasing values of $m$ decreases $f'(\eta)$ and the opposite behavior is observed for $g(\eta)$.

Figure 8 and Figure 9 display the significant consequences of permeability on $f'(\eta)$ and $g(\eta)$. The factor's distribution is depicted within a broad range of $0.5 context, $K_{p}$ denotes the behavior of the flow profile as the fluid traverses a transparent fluid substrate, whereas the variation illustrates the flow through a porous substrate. The medium's resistivity tends to diminish the velocity profile in both axial and transverse directions, resulting in layer thinning. The negligible variance arises from the restricted range selection for the factor; however, the findings are illustrated with an enlarged view to clarify the differences between the profiles. The effect of permeability on temperature distribution is inverse, as fluid temperature increases with increased permeability. When the fluid velocity diminishes along the profile, the fluid produces energy at the surface's base, resulting in a considerable increase in temperature due to the accumulated energy.

Figure 10 examines the influence of solid nanoparticles $\Phi_{1}$ and $\Phi_{2}$ on the distribution of fluid temperature. The concentration of particles is a critical element that profoundly affects the thermal characteristics of nanofluids and hybrid nanoliquids. The precise number of particles distributed within the base fluid is termed volume concentration. The solid volume concentrations significantly influence the increase in fluid temperature for the recommended values of the constructive factors. The variation of time-dependent ($\alpha^{*}$) and space-dependent ($\beta^{*}$) heat source parameters with $\theta(\eta)$ is demonstrated in Figure 11 and Figure 12. The increasing values of $\alpha^{*}$ and $\beta^{*}$ intensifies $\theta(\eta)$. The incorporation of a non-uniform heat source elevates the surface temperature due to the accumulated energy at the surface. This stored energy induces movement toward the fluid region, which promotes an increase in fluid temperature, ensuring the exit temperature from the surface. This results in the cooling of the thermal boundary surface. The inclusion of additional nanoparticles in the base liquid of the hybrid nanofluid markedly enhances its thermal conductivity, resulting in a considerable rise in the fluid's temperature.

The impact of $Rd$ on the thermal profile is detailed in Figure 13. The quantity of electromagnetic waves that are transferred from the fluid element and subsequently transformed into thermal energy is commonly referred to as $Rd$. It shows that the increased radiative heat significantly raises the fluid temperature because the radiating heat from the surface causes more energy to flow from the surface to the fluid medium. Furthermore, the enhanced thermal characteristics of the combined influence of the crystalline structure nanoparticles significantly increase the hybrid nanoparticles' considerable contribution compared to the nanofluid condition.

The combined influence of Hall current, thermal radiation, porosity effect and non-uniform heat source provides a trade-off between axial and cross-flow velocities, crucial for efficient flow regulation. Growing values of Hall current amplifies cross-flow velocity, whereas permeability alters resistance and axial transport, offering prospects for enhanced system design. The augmentation of $\Phi_1$ and $\Phi_2$ significantly elevates thermal distribution due to the enhancement of thermal conductivity of nanoparticles in the system. The analysis indicates that increased radiation parameter values boost thermal profile, underscoring its significance in high-thermal environments. Collectively, these insights provide the practical guidance for improving and designing efficient advanced heat transfer systems.

Traditional computational design methods cannot handle complex computational problems, especially when there are many process elements. The traditional numerical method is excessively costly and complicated [26], [27], [28]. The Taguchi technique uses orthogonal arrays to reduce the number of computations while accounting for the whole parameter space. The minimization of experimental duration and expenses, together with the swift discovery of essential components, and the preservation of research resources are the primary advantages of using the Taguchi technique. As indicated in Table 3, the four primary parameters utilized in this investigation are the volume fraction, radiation parameter, and time- and space-dependent heat source parameters.

For this investigation, four elements and four levels are first determined. A complete factorial would require 256 trials. The Taguchi statistical technique recommends utilizing an L$_{16}$ orthogonal array to maintain statistical coverage and balance, which may greatly minimize the amount of computational and experimental effort required. The L$_{16}$ array for the Taguchi approach is shown in Table 4. Furthermore, Figure 14 represents the residual plot of $Nu$ .

The equation below is employed to ascertain the minimal number of viable numerical runs.

where, $N_Z$ denotes the minimal quantity of calculations required, dictated by the number of control parameters ($N_X$) and levels ($L$). Four control features ($N_X=4$) and four levels ($L=4$) are considered. According to the aforementioned equation, a minimum of 13 numerical computations must be performed. For factors with three levels, the accessible orthogonal arrays are L$_{16}$, L$_{32}$, and alike formations. Therefore, the L$_{16}$ orthogonal array is selected for the present work as it best matches the computational requirements and accommodates four levels for each parameter. The current study employs the “larger-the-better” criteria as its evaluative standard to achieve the goal of optimization of $N u$.

In order to make the required predictions and explain experimental data, analysis of variance (ANOVA) is utilized [29], [30], [31], [32]. Table 5 shows the findings of the ANOVA on the Nusselt number. Table 6 displays the R-squared value of 85.62% and the adjusted R-squared value of 28.09%. The regression equation that ought to be applied to the model is as follows:

This work examined the flow and thermal transmission properties of a water-based hybrid nanofluid over a porous extending sheet, considering the effects of Hall current, a non-uniform heat source, and thermal radiation. The dimensionless system of equations was numerically solved using the MATLAB bvp4c solver environment. Additionally, the Taguchi optimization method was utilized to identify ideal conditions for enhancing heat transfer efficiency in engineering applications. The main outcome of the present investigation is as follows:

• The $f'(\eta)$ profile is markedly affected by the resistive forces associated with the Hartmann number, while the $g(\eta)$ profile displays a contrasting pattern.

• Thermal radiation generates maximal intensity in heat transport phenomena at all regions.

• The thermal profile of the hybrid nanofluid demonstrates a more significant change than that of the traditional nanofluid.

• An R-squared value of 85.62% signifies that the dataset is robust and the model is considerable.

• The Taguchi-based $L_{16}$ orthogonal array optimization indicates that $\mathrm{Rd}$ is the most influential parameter, contributing 51.73% to the heat transfer rate; therefore, it should be prioritized in system design and optimization.

• According to Taguchi optimization, engineers should primarily control $\mathrm{Rd}$, followed by $\beta^{*}$ and $\alpha^{*}$, to maximize heat transfer efficiency in Hall-induced nanofluid systems.

• Hybrid nanomaterials enhance thermal performance and support efficient multiphysics system design.