Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity

Mathematical formulation of the flow problem

We examine the ternary hybrid nanofluid is laminar boundary layer, two dimensional, time-dependent, incompressible flow of an electrically conducting Tangent hyperbolic fluid over a permeable stretching sheet, surface with velocity $U_w(x,t)=\mathit{ax}/(1-\mathit{ct})$ , where a is a positive constant, $t$ is time, and $c$ is the time-dependent parameter of this problem. The wall mass suction velocity is assumed to be $,V_w(x,t)$ . The x-axis is oriented along the direction of the stretching sheet, whereas the y-axis is perpendicular to it. The velocity components are represented as $u$ and $v$ , corresponding to the x- and y-axes, respectively. The well temperature, $T_w$ , ambient temperature $,T_{\infty }$ , well concentration, $C_w$ , and ambient concentration, $C_{\infty }$ are considered constants. Figure 1 graphically illustrate the physical flow model in this study, using Cartesian coordinate. The model for momentum, mass, and heat transfer incorporate thermophoresis, Brownian motion, viscous dissipation, chemical reactions, variable thermal conductivity, non-linear thermal radiation, heat generation, magnetic dissipation, and wall mass suction. When a constant magnetic field is applied perpendicular to the surface in the positive y-direction, the induced magnetic field is considered negligible and ignored. Assuming that the porosity of the porous material remains constant, we also take Darcy-Forchheimer flow into account in our investigation. The energy calculations are made simpler by treating all fluid parameters as constants and assuming that the system enters thermal equilibrium. The energy equation takes into consideration the effects of viscous dissipation, in which viscosity causes kinetic energy to be transformed into thermal energy. Through the dispersion of Al₂O₃, Cu, and TiO₂ nanoparticles in C₂H₆O₂ medium, the ternary hybrid nanofluid is created. According to Refs. 40, the heat conductivity of the ternary hybrid nanomaterial are proportional and linearly connected to the distributed temperature.

Figure 1. Flow configuration and coordinate system.

Constitutive equation of tangential hyperbolic

The tangential hyperbolic model is used to describe the flow behavior of shear-thinning fluids. In this model, the relationship between the shear stress tau and shear rate $\dot{\gamma }$ is given by a hyperbolic tangent function. The following formulas provide the basic equation for the tangential hyperbolic fluid^4,7,26,27:

The shear rate γ˙ is assumed

The governing equations for the flow problem are:^7,21,39

Subject to boundary condition given by³⁹

The ternary hybrid nanofluids thermophysical characteristics are described as^14,18,39:

Here, the subscripts $f,\mathit{nf},\mathit{hnf}$ , and $\mathit{\text{thnf}}$ stand for traditional fluid, NF, HNF, and THNs, respectively, and the letters 1,2, and 3 stand for the three selected nanoparticles.

Since the governing equations are in partial differential form, they are reduced to a system of ordinary differential equations through the use of similarity transformations. Further, to obtain similarity solutions of Eq. (5) to Eq. (8), the unsteady magnetic field B₀ is in the form B₀=B/(1−ct)^1/2 where B is a constant. The unsteady magnetic field B₀=B/(1−ct)^1/2 can induce an electric field according to Maxwell’s equations. Specifically, Faraday’s law of electromagnetic induction states that a time-dependent magnetic field generates a circulating electric field. Since B₀ varies with time due to the presence of (1−ct)^1/2 its time derivative is nonzero, leading to an induced electric field.² This induced electric field can influence charge transport, modify current distributions, and potentially alter the overall nanofluid dynamics, making it an important factor in magnetohydrodynamic (MHD) applications.

The similarity transformations are given by^21,39

In this context, the parameter $\lambda ,$ representing the velocity ratio, assumes various values: $\lambda =0$ corresponds to the static sheet, $\lambda <0$ signifies the shrinking sheet, and $\lambda >0$ denotes the stretching sheet. The Skin friction coefficient $\left(C_f\right)$ , local Nusselt number $\left({\mathit{Nu}}_x\right)$ , and local Sherwood number $\left({\mathit{Sh}}_x\right)$ are given by²¹:

The dimensionless skin friction coefficient, Nusselt number, and the Sherwood number are given as follows:

Numerical method

Equations (11) to (13) and with the boundary condition (14) are numerically solved by using the MAT LAB software (https://github.com/asfawmat/Tan_22BVP) utilizing the bvp5c algorithm.³⁹ BVP5C is a numerical algorithm in MATLAB designed to solve boundary value problems (BVPs) for ordinary differential equations. It extends the capabilities of previous algorithms, allowing for the efficient handling of complex systems that require multiple boundary conditions. BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. BVP5C offers high accuracy for stiff boundary value problems, adaptive mesh refinement, and robustness in handling nonlinearities. Compared to finite difference methods, it achieves faster convergence with minimal grid dependence. BVP5C is widely used in structural analysis, heat transfer problems, and fluid dynamics to model systems governed by differential equations with boundary conditions. The algorithm is applicable in fields such as quantum mechanics and thermodynamics, where BVPs arise naturally in modeling physical phenomena. In order to apply the BVP5C with the shooting technique, the boundary value problems must be reduced to a system of first-order initial value problems (IVPs), as explained below. We use the following relations to convert the nonlinear higher-order boundary value problems into a system of first-order initial value problems.

Now let us defined the new variable by the equation

Using Eq.(18), the three coupled higher order differential equations Eq.(11) to Eq. (13) can be written as

The system is solved in MATLAB using the BVP5C function, with a step size of 0.1 and a mesh size of 101. The shooting method employed provides greater precision, achieving a tolerance of ${10}^{-7}$ compared to earlier numerical methods. A comparison of our results with those from previous research is presented in Table 2, and the findings indicate a generally excellent agreement with earlier studies.

Velocity characteristics

Figures 2-10 depict the influence of several factors, like: magnetic field parameter $\left(M\right)$ , porous permeability parameter $\left(K\right)$ , Weissenberg number $\left(\mathit{\text{We}}\right)$ , unsteady parameter $\left(A\right)$ , power law index $\left(n\right)$ , Forchheimer number $\left(\mathit{Fr}\right)$ , suction parameter $\left(S\right)$ , velocity ratio parameter $\left(\lambda \right)$ , and nanoparticle volume fraction on scaled velocity $\left(f^{\prime }\left(\eta \right)\right)$ . Figure 2 shows how the scaled velocity profile is affected by the magnetic field parameter $\left(M\right)$ . Physically, as the magnetic field effect increases, the interplay between the Lorentz force and the fluid’s motion leads to a decline in the velocity distribution, primarily due to increased resistance, damping effects, and changes in the dynamics of the flow. Figure 3 depict the influence of the porous permeability parameter $\left(K\right)$ , on the velocity profile. When the value of the porous permeability parameter increases, the velocity distribution in a fluid flow can declines. Physically, in porous media, as the permeability increases, the inertial effects may become more pronounced, particularly in non-Darcy flow regimes. This can lead to a situation where the increased inertia counteracts the driving forces, resulting in a decline in the overall velocity. Higher permeability typically allows for easier flow through the porous medium. However, if the permeability increases too much, it can lead to a decrease in the pressure gradient needed to drive the flow. This results in reduced fluid velocity as the flow becomes less confined and more dispersed within the pores. Figure 4 illustrate the impact of power law index $\left(n\right)$ on the velocity profile. As the power law index $\left(n\right)$ , increases in a tangent hyperbolic fluid, the resulting increase in viscosity and flow resistance, combined with the fluid’s non-Newtonian characteristics, leads to a decline in the velocity distribution. Figure 5 display the effect of unsteady parameter on $\left(f^{\prime }\left(\eta \right)\right)$ . As the unsteady parameter increases, the balance between inertial and viscous forces shifts, leading to higher velocities due to enhanced acceleration, reduced damping, potential turbulence, and increased momentum transfer. The impact of Forchheimer parameter (Fr) on the velocity profile illustrated in Figure 6. As the Forchheimer parameter increases, the combination of heightened inertial resistance, nonlinear flow behavior, increased energy losses, and complex flow distributions contributes to a diminished velocity profile in porous media. This highlights the interplay between viscous and inertial effects in determining flow behavior. The effect of suction parameter on the velocity distribution demonstrated in Figure 7. When suction is applied to the stretching sheet, the fluid will be pulled closer to the sheet, resulting in the thinning of the boundary layer. As the suction parameter increases, the combined effects of increased resistance to flow, enhanced viscous effects, flow redistribution, boundary layer dynamics, and reduced kinetic energy contribute to a diminished velocity profile. These factors illustrate how suction influences the overall flow behavior and velocity distribution in the system. The Weissenberg number (We) is a dimensionless parameter that characterizes the relative importance of elastic (viscoelastic) effects compared to viscous effects in a fluid flow, particularly for non-Newtonian fluids such as tangent hyperbolic fluids. As the Weissenberg number increases, it indicates that the elastic effects become more significant, as depicted in Figure 8. As the Weissenberg number increases in a tangent hyperbolic fluid, the combined effects of increased elastic resistance, nonlinear flow behavior, thicker boundary layers, increased energy dissipation, and flow stabilization contribute to a diminished velocity profile. These factors highlight the complex interplay between elasticity and viscosity in determining the flow characteristics of non-Newtonian fluids. Figure 9 show the effect of the velocity ratio parameter (λ) on the velocity distribution. When the velocity ratio (λ) is high, the interactions between fluid layers (or between phases in a multiphase flow) become more dynamic. This enhances the momentum exchange, leading to higher velocities being transmitted through the flow field, thus increasing the overall velocity profile. As in Figure 10, as nanoparticle volume fraction (ϕ₁) increases, the interplay of increased viscosity, enhanced drag, altered fluid structure, and boundary effects collectively results in a decrease in the velocity profile of the fluid. Figure 11 depict the comparison of nanofluid (Al₂O₃/C₂H₆O₂), hybrid nanofluid (Al₂O₃ − Cu/C₂H₆O₂), and ternary hybrid nanofluid (Al₂O₃ − Cu − TiO₂/C₂H₆O₂) on the velocity distribution. When the concentration of nanoparticles in nanofluids, hybrid nanofluids, and ternary hybrid nanofluids increases, the velocity pro- file typically decreases due to increased viscosity, enhanced drag forces, complex microstructural changes, potential shear-thickening effects, and shifts in flow regime. These factors collectively hinder the smooth flow of the fluid, resulting in lower velocities. Ternary hybrid nanofluids often achieve the best velocity distribution among the three categories. The synergistic effects of multiple nanoparticles can enhance dispersion stability and thermal performance while managing viscosity better. This results in improved flow characteristics and potentially higher velocities compared to the other two types.

Figure 2. Velocity with M.

Figure 3. Velocity with K.

Figure 4. Velocity with n.

Figure 5. Velocity with A.

Figure 6. Velocity with Fr.

Figure 7. Velocity with S.

Figure 8. Velocity with We.

Figure 9. Velocity with λ.

Figure 10. Velocity with ϕ₁.

Thermal characteristics

The effects of the nonlinear thermal radiation parameter (R) on fluid temperatures are illustrated in Figure 12. When the values of nonlinear thermal radiation increase, the temperature profile in a fluid or material can also increase. In materials where radiation is a primary mode of heat transfer, an increase in nonlinear thermal radiation means that more energy is available to heat the material. This can lead to higher thermal gradients within the material, resulting in an overall increase in the temperature profile. Figure 13 depicts the thermal distribution against the temperature ratio $\left({\theta }_w\right)$ . The temperature ratio often refers to the ratio of the temperature of a fluid to that of a boundary or a reference temperature. An increased temperature ratio implies a greater difference between the fluid temperature and a reference or surrounding temperature. This enhanced temperature gradient can lead to more efficient heat transfer from the hotter regions to the cooler ones, resulting in higher temperatures throughout the fluid or material. When the values of the unsteady parameter increase, the temperature profile in a fluid or material can also increase due to several interrelated mechanisms, as shown in Figure 14. The unsteady parameter often reflects the importance of transient effects relative to steady-state conditions, such as in heat transfer or fluid dynamics. An increase in the unsteady parameter can lead to a higher temperature profile through enhanced energy transport, rapid response to thermal gradients, and increased effective thermal conductivity. These factors illustrate the critical role that unsteady conditions play in influencing the thermal behavior of fluids and materials. Figure 15 present the effect of the thermophoresis diffusion on the temperature profile. As the thermophoresis diffusion parameter increases, particles within the fluid experience a stronger thermophoresis force, which causes them to migrate from regions of lower temperature to regions of higher temperature. This movement can lead to a concentration of particles in hotter areas, effectively increasing the local temperature in those regions. The influence of Brownian motion on the temperature distribution, as shown in Figure 16. As the Brownian diffusion parameter increases, the microscopic motion of particles (due to Brownian motion) becomes more significant. This enhanced movement leads to improved mixing within the fluid, promoting more uniform temperature distribution and potentially raising the average temperature profile by allowing hotter regions to mix with cooler ones more effectively. As we can see from the temperature distribution graphs in Figure 17, the temperature functions get better as the Eckert number (Ec) increases in value. The Eckert number (Ec) is a dimensionless parameter that characterizes the relative importance of kinetic energy to thermal energy in a fluid flow, particularly in situations involving heat transfer. As the Eckert number rises, the kinetic energy associated with the flow increases. This kinetic energy can contribute to the overall thermal energy of the fluid, leading to higher temperatures, especially if the kinetic energy is converted into internal energy during viscous dissipation. The significance of the porous permeability media parameter (K) on temperature profiles is depicted in Figure 18. When the porous permeability parameter (K) increases, the temperature profile in a fluid flowing through a porous medium can also increase. In porous media, higher permeability often means a more interconnected pore structure. This can provide a larger surface area for heat exchange between the fluid and the solid matrix of the medium, enhancing heat transfer efficiency and contributing to a higher temperature profile. As Figure 19 illustrate, the temperature increases as the magnetic field parameter (M) increases. As the magnetic field strength increases, it can exert a damping effect on the fluid flow. This damping can reduce the velocity of the fluid, leading to increased viscous heating. The kinetic energy of the fluid being converted into thermal energy due to viscous dissipation results in a higher temperature profile. Figure 20 shows a graph of the temperature profile as a result of the increasing the power law index $\left(n\right)$ . An increase in the power law index parameter $\left(n\right)$ , can lead to a higher temperature profile through increased viscosity and viscous heating, reduced flow velocity resulting in enhanced energy dissipation, altered heat transfer characteristics, greater thermal energy retention, improved mixing and flow stability, and interactions with temperature gradients. These factors collectively illustrate how non-Newtonian fluid behavior influences thermal dynamics within the flow. The effect of the Forchheimer number (Fr) on the temperature distribution, demonstrated in Figure 21. When the Forchheimer number increases, it indicates that inertial forces are becoming more significant compared to viscous forces. As inertial forces become more dominant, the mixing and transport of thermal energy are enhanced. The increased movement of the fluid helps distribute heat more uniformly, resulting in a higher average temperature within the porous medium. As in Figure 22, an increase in the Prandtl number indicates that the kinematic viscosity is relatively high compared to thermal diffusivity. This means that momentum diffuses more slowly than heat. Higher viscosity can lead to reduced flow velocities, which can decrease the fluid’s ability to carry heat away from hot regions, resulting in a less effective temperature distribution. Figure 23 show the effect of the heat generation on the temperature profile. Increasing the rate of heat generation results in a direct increase in thermal energy, leading to higher temperatures in the fluid or material. This effect is reinforced by heat transfer mechanisms, thermal gradients, and material properties, all contributing to a rising temperature profile. As a result, regions closer to the heat source experience significant temperature increases, influencing the overall thermal behavior of the system. As in Figure 24 when the variable thermal conductivity parameter (Γ) increases, the temperature profile generally increases due to enhanced heat transfer capabilities of the material or fluid. As heat is conducted more effectively, the temperature can rise in regions further away from the heat source, resulting in an increased overall temperature profile. As the Weissenberg number (We) increases for a tangent hyperbolic fluid, the fluids viscoelastic properties become more pronounced, leading to increased energy storage and viscous dissipation, as seen Figure 25. This results in enhanced heat generation, steeper temperature gradients, and improved heat transfer characteristics, collectively contributing to an increase in the temperature profile. The balance between heat generation and dissipation is crucial in determining the extent of this temperature rise. As the nanoparticle volume fraction $({\varphi }_1,{\varphi }_2,{\varphi }_3)$ increases, the overall temperature profile of the nanofluid tends to rise due to enhanced thermal conductivity, increased heat generation through viscous dissipation, reduced specific heat capacity, improved thermal boundary layer effects, and enhanced convection, as seen Figures 26, 27, 28. Collectively, these factors contribute to more effective heat transfer and distribution, resulting in higher temperatures within the nanofluid. Figure 29 show a graph of the comparison of the nanofluid (Al₂O₃/C₂H₆O₂), hybrid nanofluid (Al₂O₃ − Cu/C₂H₆O₂), and ternary hybrid nanofluid (Al₂O₃ − Cu − TiO₂/C₂H₆O₂) on the temperature distribution. As the volume fraction of nanoparticles $({\varphi }_1,{\varphi }_2,{\varphi }_3)$ increases in nanofluids, hybrid nanofluids, and ternary hybrid nanofluids, the temperature profile increases due to improved thermal conductivity, enhanced viscous dissipation, and better convective heat transfer. The combination of these factors leads to higher localized and overall average temperatures in the fluid. Generally, while all three types of fluids (nanofluids, hybrid nanofluids, and ternary hybrid nanofluids) show increased temperature profiles and enhanced heat transfer capabilities with higher nanoparticle concentrations, ternary hybrid nanofluids generally provide the best performance in terms of thermal conductivity and heat transfer efficiency. This is due to their ability to leverage the synergistic effects of multiple types of nanoparticles, leading to superior thermal management in various applications.

Figure 11. Velocity with ϕ₁, ϕ₂, ϕ₃.

Figure 12. Temperature with R.

Figure 13. Temperature with θ _w.

Figure 14. Temperature with A.

Figure 15. Temperature with Nt.

Figure 16. Temperature with Nb.

Figure 17. Temperature with Ec.

Figure 18. Temperature with K.

Figure 19. Temperature with M.

Figure 20. Temperature with n.

Figure 21. Temperature with Fr.

Figure 22. Variation of temperature with Pr.

Figure 23. Temperature with Q.

Figure 24. Temperature with Γ.

Figure 25. Temperature with We.

Figure 26. Temperature with ϕ₁.

Figure 27. Temperature with ϕ₂.

Figure 28. Variation of temperature with ϕ₃.

Figure 29. Temperature with ϕ₁, ϕ₂, ϕ₃.

Concentration characteristic

The effect of the thermophoresis diffusion parameter (Nt) on the concentration profile, as illustrated in Figure 30. As the thermophoresis parameter increases, the transport of particles within the fluid becomes more effective. This enhanced transport allows for a greater number of particles to move towards and concentrate in regions of higher temperature, further increasing the concentration profile. Figure 31 illustrate the impact of the Brownian diffusion Parameter (Nb) on the concentration profile. When the Brownian diffusion parameter (Nb) increases, it indicates a stronger influence of Brownian motion on the transport of particles suspended in a fluid. As Brownian diffusion increases, particles undergo more vigorous random motion. This enhanced diffusion can cause particles to spread out more rapidly within the fluid, leading to a decrease in local concentration as particles move away from regions of higher concentration. Figure 32 depict the effect of the Schmidt number (Sc) on the concentration profile. The Schmidt number (Sc) is a dimensionless parameter that characterizes the relative importance of momentum diffusion (viscous effects) to mass diffusion (concentration effects) in a fluid. When the Schmidt number increases, it typically indicates that momentum diffusion is more significant compared to mass diffusion. When momentum diffusion dominates, it can lead to smaller concentration gradients because the particles are less able to respond to local variations in concentration. This smaller gradient reduces the driving force for diffusion, causing a decrease in the concentration profile over time as the particles do not accumulate effectively. Figure 33 show the effect of the unsteady parameter (A) on the concentration profile. When the unsteady parameter (A) increases, it typically indicates that transient effects in the system are becoming more significant compared to steady-state effects. The unsteady parameter affects how particles are transported in the fluid. With higher values of the unsteady parameter, the transport mechanisms (like diffusion and convection) can operate differently, potentially leading to increased mixing or accumulation of particles in certain areas, thereby increasing the concentration profile. Figure 34 present that when the chemical reaction parameter increases, the solute’s boundary layer thickness decreases. In reactions where the concentration of a reactant decreases as it transforms into products, an increase in the chemical reaction parameter means that the reactants are being converted to products more rapidly. This results in a lower concentration of the reactant remaining in the system, causing the overall concentration profile to decline. Figure 35 depict the effect of suction parameter (S) on the concentration profile. As the values of the suction parameter increase, the concentration profile decreases due to the removal of fluid that carries solutes, leading to a reduction in local concentration. This effect is further amplified by the dynamics of the concentration boundary layer and enhanced mass transfer away from the surface that prioritize the extraction of fluid over the retention of solutes. Consequently, the overall concentration profile in the fluid becomes lower with increased suction.

Impacts of parameters on the coefficient of skin friction, local Nusselt and Sherwood number

In Table 3, we present the numerical results for the skin friction $\left(C_{\mathit{fx}}{\mathit{Re}}^{\frac{1}{2}}\right),$ the Nusselt number $\left({\mathit{Nu}}_x{\mathit{Re}}^{-\frac{1}{2}}\right)$ , and the Sherwood number $\left({\mathit{Sh}}_x{\mathit{Re}}^{-\frac{1}{2}}\right)$ . These key parameters are evaluated using various dimensionless parameters for ternary hybrid nanofluids. As the Weissenberg number increases, the skin friction coefficient decreases due to the dominance of elastic effects leading to lower wall shear rates and potentially more stable flow and the Sherwood number decreases as the flow becomes less turbulent and diffusion rates drop, resulting in less effective mass transfer. Also, the Nusselt number increases because the tangent hyperbolic fluid properties can still enhance heat transfer through improved convective mechanisms, allowing for better thermal management despite the reduced mixing and diffusion. The increase in the unsteady parameter leads to a reduction in the effectiveness of both mass and heat transfer, resulting in lower values of the skin friction coefficient, Sherwood number, and Nusselt number. Skin friction coefficient tends to increase with higher magnetic field parameter (M), Forchheimer number (Fr), and velocity ratio parameter (λ), while it decrease with Weissenberg number (We), unsteady parameter (A), and power law index (n). Nusselt numbers decrease as unsteady parameter (A), magnetic field parameter (M), Forchheimer number (Fr), Eckert number (Ec), and velocity ratio parameter (λ) values increase, whereas they increase with the increase in the values of Weissenberg number (We), power law index (n), variable thermal conductivity (Γ), and nonlinear thermal radiation (R). Increases with enhanced thermal conductivity, allowing for better heat transfer. Also, increases as nonlinear thermal radiation enhances heat transfer, particularly at high temperatures. Sherwood numbers increase with the increased values of chemical reaction (C_r) and velocity ratio parameter (λ), whereas they decrease with the increasing values of Weissenberg number (We), unsteady parameter (A), and power law index $\left(n\right)$ . Increases as the chemical reaction can enhance concentration gradients, improving mass transfer.

Figure 30. Concentration with Nt.

Figure 31. Concentration with Nb.

Figure 32. Concentration with Sc.

Figure 33. Concentration with A.

Figure 34. Concentration with Cr.

Figure 35. Concentration with S.

Table 4 explores the effects of ${\varphi }_1$ , ${\varphi }_2$ , and ${\varphi }_3$ on the skin friction coefficient, Nusselt number, and Sherwood number for ternary hybrid nanofluids, hybrid nanofluids, and mono nanofluids. The results indicate that increasing the nanoparticle volume fractions $({\varphi }_1,{\varphi }_2,{\varphi }_3)$ leads to enhancements in the skin friction coefficient, local Nusselt number, and local Sherwood number. The skin friction coefficient is a measure of the frictional resistance experienced by a fluid flowing over a surface. With increased nanoparticle concentration, the skin friction coefficient generally increases, indicating greater resistance due to higher fluid density and viscosity. As nanoparticle concentration rises, both the Nusselt and Sherwood numbers rise, indicating improved heat and mass transfer efficiency, with ternary hybrid nanofluids being the most effective in enhancing heat transfer and thermal conductivity, along with an increase in skin friction coefficient.

The present study has significant real-world applications in various engineering and industrial fields. The enhanced heat transfer properties of tangent hyperbolic ternary hybrid nanofluids make them highly effective for cooling systems, such as electronic device cooling, heat exchangers, and thermal management in power plants. Additionally, their superior thermal conductivity and stability are beneficial in biomedical applications, including targeted drug delivery and hyperthermia treatments. The findings also have implications in polymer processing industries, where precise thermal control is required for extrusion and casting. Furthermore, these nanofluids can be utilized in aerospace and automotive industries for efficient thermal regulation and improved fuel efficiency. The insights gained from this research contribute to optimizing fluid flow and heat transfer processes in advanced engineering systems.