The Dynamic Motion of Hybrid Nanofluid Intertwined with Viscous Dissipation and Nonlinear Radiative Heat Flux Gliding Across a Disk

The quest for hybrid nanofluids has ignited extraordinary intrigue due to their unparalleled thermal attributes and the vast array of potential applications in an abundance of engineering endeavours, particularly in mechanisms of heat transfer these extraordinary fluids birth from the union of multiple nano particles suspended in a foundational liquid, showcase remarkable thermal conductivity and heightened heat transfer capabilities when Juxtaposed which traditional single component liquids. This rise in thermal efficiency makes hybrid nanofluids particularly enticing for the use in cooling systems, energy storage solutions and various industrial domains where adept heat management is crucial. The incorporation of viscous dissipation alongside non linear radioactive heat flux introduces additional complexities to the flow dynamics, necessitating advanced mathematical modelling to accurately predict their behaviour across the range of operational context. Such modelling not only aids in fine tuning the design of heat transfer systems but also enriches our understanding of the core physical principles governing the efficacy of hybrid nanofluids paving the way of revolutionary paces in thermal management. These breakthroughs hold the promise of introducing next generation cooling systems that significantly boost energy efficiency and reduce operational costs across a wide spectrum of industries including automotive aerospace and electronics. The synthesis of these innovative cooling strategies could also reinforce sensible bolster sustainability initiatives by minimising energy consumption and lowering greenhouse gas emissions, thus nurturing more environmentally conscious industrial landscape.

The exploration of hybrid nano fluids has emerged as an essential frontier in the domain of thermal management fuelled by their remarkable thermal properties and wide ranging applications in engineering pursuits, particularly within heat transfer systems. Hybrid nanofluids characterised by the amalgamation of various types of nano fluids suspended in a fluid medium, demonstrate increased thermal conductivity and improved heat transfer capabilities when compared to traditional single component fluids. This remarkable surge in thermal performance positions hybrid nanofluids as enticing candidates for a multitude of applications such as cooling systems energy storage technologies and numerous industrial process where good heat control is crucial. The inclusion of factors like viscous dissipation and non linear radioactive heat flux introduces layers of complexity to flow dynamics of these fluids, demanding sophisticated mathematical modelling to accurately capture their behaviour in diverse operational context. Such modelling initiatives not only felicitate the better heat transfer system designs but also deepen our understanding of the core physical principles that dictate the effectiveness of hybrid nanofluids. This research could ultimately pave the way for the development of state of the art cooling technologies that enhance energy potency and reduce operational costs across various fields including automotive aerospace and electronics. Furthermore the implementation of these revolutionary cooling advancements could significantly support sustainability initiatives by lowering energy usage consumption and alleviating greenhouse gas emissions thereby fostering more environmentally conscious industrial landscape. The intricate dance of boundary layer dynamics within an electrically charged fluid profoundly influences the transfer magnetic field. Such explanation notably significantly enriched the fabric of collaborative research, thanks to their vast applications spanning engineering, commercial physics, astrophysics and aerodynamics.

The intricate dance of boundary layer dynamics within an electrically charged fluid profoundly influences the transverse magnetic field. Such explorations significantly enrich the fabric of collaborative research, thanks to their vast applications spanning engineering, geophysics, astrophysics, and aerodynamics.

Mansur et al. [1] set forth and exhilarating journey into the captivating and intricate world of p permeable sheets that exhibit both contracting and expanding behaviours in the realm of nano fluid, while providing a sharp analysis of the heat transfer mechanisms that fuel these captivating phenomena. Esfe et al. [2] performed an experimental investigation on the thermal conductivity of Cu/TiO₂–water/EG hybrid nanofluid and formulated predictive models employing artificial neural networks and empirical correlations. Their findings revealed that the hybrid nanofluid exhibits superior thermal conductivity compared to mono nanofluids, with thermal performance significantly influenced by nanoparticle concentration and base fluid composition. Mahanta and Shaw [3] unveiled the wonders of a 3D linear stretchable sheet, intricately linked with convective boundary conditions. Shahzad et al. [4] Immersed themselves in a remarkably intricate 3 dimensional flow of magnetohydrodynamics, skillfully intervening the elements of heat generation within a permeable medium, thus illuminating the intricate dynamics of thermal and fluid interaction in this elaborate terrain. Alao et al. [5] look into the complex dance of heat and mass transfer within a chemically reactive fluid cascading down a semi-infinite vertical plate. Their study explored the underlying physical mechanisms governing the transport phenomena, considering the effects of chemical reactions, thermal diffusion, and fluid dynamics on the overall system behaviour.

Mushtaq and Mustafa [6] analyzed the flow and heat transfer characteristics of a nanofluid near a stretchable rotating disk under the influence of an axial magnetic field and convective boundary conditions, solving the transformed equations using the Runge-Kutta-Fehlberg method. Their findings highlight that increasing magnetic field strength suppresses velocity while enhancing thermal boundary layer thickness, significantly affecting heat transfer rates.

Hayat and Nadeem [7] investigated the heat transfer enhancement of Ag-CuO/water hybrid nanofluid, emphasizing its superior thermal conductivity compared to conventional nanofluids. Their study demonstrated that the presence of hybrid nanoparticles significantly improves heat transfer efficiency, with increasing solid volume fraction leading to higher Nusselt numbers and enhanced thermal performance.

Ghadikolaei et al. [8] analyse the nonlinear effects of thermal radiation in Magneto nano fluid dynamics, highlighted by Joule heating as it is glided past and inclined penetrable stretchable sheet. Raza [9] assesses the importance of radiative heat transfer and slip effects on the stagnation point dynamics flowing elegantly past a convective stretchable sheet. Alsagria et al. [10] ventured into the enchanting domain of MHD nanofluid dynamics simulation, emphasizing the significant role of dissipative heating. Shoaib et al. [11] journeyed into the captivating three-dimensional landscape of magnetohydrodynamic flow characteristics linked to second-grade fluids as they gracefully traverse an absorbent plate, unveiling the intricacies of fluid behaviour under such distinctive conditions. Sarkar et al. [12] probed into the radiative characteristics of nanofluids streaming over an inclined cylindrical surface while addressing the chemically reactive behaviours of nanofluids on a stretchable sheet in the backdrop of joule heating influences.

The fascinating impact of fusion on MHD Casson fluid dynamics moving across a stretchable sheet within a permeable medium was thoroughly investigated by Mabood and Das [13]. Their findings revealed that increasing the magnetic field strength and nanoparticle concentration enhances thermal conductivity while influencing the boundary layer flow dynamics, making the study significant for advanced cooling and thermal management applications.

Idowu and Falodun [14] investigated the systematic flow of viscous and chemically active non-Newtonian fluids, deftly adjusting viscosity and thermal conductivity. Acharya et al. [15] investigated The Thermal-fluid process behavior of a magnetized TiO₂–CoFe₂O₄ water-based hybrid nanofluid in steady-state flow over a radiative rotating disk, emphasizing the influence of magnetohydrodynamics and thermal radiation on heat transfer efficiency. Waqas et al. [16] conducted a comprehensive study on the influence of magnetohydrodynamic (MHD) radiative Flow dynamics of a hybrid nanofluid over a rotating disk focusing on the interplay between electromagnetic forces, thermal radiation, and heat transfer mechanisms. Their research examined how variations in magnetic field intensity and radiative heat flux impact fluid motion, energy distribution, and overall thermal efficiency. By analysing key governing parameters, such as nanoparticle concentration, magnetic field strength, and radiation effects, the study provided valuable insights into optimizing heat transfer performance in advanced thermal systems. The findings contribute to a deeper understanding of MHD-based thermal management techniques, which are crucial in engineering applications such as energy systems, aerospace technology, and industrial cooling processes.

Their study demonstrated that increasing magnetic field strength suppresses velocity while enhancing temperature distribution, leading to improved thermal efficiency in hybrid nanofluid applications.

Abdul Hakeem et al. [17] analyzed the three-dimensional viscous dissipative flow of nanofluids over a Riga plate, emphasizing the combined effects of magnetohydrodynamics and heat dissipation on fluid dynamics. Their study demonstrated that the incorporation of nanoparticles improves thermal conductivity while the Lorentz force generated by the Riga plate significantly influences velocity and temperature profiles, making the findings relevant for advanced thermal engineering applications. Al Mamun et al. [18] explored a numerical simulation to Inspect the periodic magnetohydrodynamic (MHD) flow of Casson nanofluid over a permeable extending surface, focusing on the Joint influences of magnetic fields and fluid rheology. Their findings revealed that increasing the magnetic field intensity and Casson parameter enhances flow resistance, while nanoparticle concentration significantly improves heat transfer efficiency, making the study valuable for biomedical and industrial fluid applications. Thamaraikannan et al. [19] examined the behaviours of nanofluids as they traversed a permeable channel shaped by MHD and periodic body acceleration.

Muntzir et al. [20] undertaken a numerical study on the magnetohydrodynamic (MHD) flow of Carreau nanofluid with gyrotactic microorganisms over various geometries including a plate, wedge, and stagnation point. Their analysis highlighted the influence of non-Newtonian fluid properties, magnetic fields, and bioconvection on heat and mass transfer, demonstrating the potential for enhanced microbial transport and thermal efficiency in engineering and biomedical applications. The subtle intricacies of heat transfer in magnetohydrodynamics, especially concerning the behaviour of non-Newtonian fluids interacting with a permeable shrinking and stretching sheet, were carefully dissected and explored by Vishalakshi et al. [21], illuminating the nuances of this compelling topic. Anwar Saeed et al. [22] explored the flow dynamics of ternary hybrid nanofluids over a spinning disk, incorporating the effects of nonlinear thermal radiation on heat transfer performance. Their study demonstrated that the inclusion of multiple nanoparticles enhances thermal conductivity and energy transport efficiency, while the nonlinear radiation significantly influences temperature distribution, making the findings crucial for high-performance thermal systems and industrial cooling.

Alqawasmi et al. [23] presented a numerical study on the flow of ternary hybrid nanofluids over a disk, considering the influence of a nonlinear heat source-sink and the Fourier heat flux model. Their findings demonstrated that the inclusion of three different nanoparticles enhances thermal conductivity and energy transport, while the nonlinear heat source-sink mechanism plays a crucial role in temperature regulation, making the study valuable for optimizing thermal management in industrial and engineering applications. Algehyne et al. [24] conducted an in-depth analysis of the Mechanical properties of magnetohydrodynamic (MHD) flow in a non-Newtonian, Maxwell fluid across a surface heated by bi-directional convection under conditions of mass flux. Their research highlighted the complex interactions between viscoelastic effects, magnetic field influences, and convective heating, demonstrating their significant impact on velocity distribution and thermal boundary layer formation. The findings provide valuable insights into optimizing heat transfer mechanisms in various advanced engineering applications, including industrial thermal management systems, polymer processing, and energy conversion technologies. Irshad et al. [25] Numerical simulations were conducted to analyze, the effects of magnetohydrodynamics (MHD) on the flow of a generalized- Newtonian fluid over a stretching sheet within a permeable medium. The analysis was performed using the finite difference method to capture flow behavior and thermal characteristics. Their findings highlighted that permeability and magnetic field strength significantly alter velocity and temperature profiles, demonstrating crucial implications for industrial coating processes and advanced heat transfer applications.

Ullah et al. [26] Examined the improvement in thermal performance of a ternary hybrid nanofluid (SiO₂ + Cu + MoS₂/H₂O) in symmetric flow over a nonlinear stretching surface using a hybrid Cuckoo Search-based artificial neural network approach. Their study revealed that the combination of multiple nanoparticles significantly improves heat transfer efficiency, while the AI-based optimization technique provides accurate predictions for fluid behavior, making the research valuable for advanced thermal management and engineering applications.

Ramzan et al. [27] carried out an analysis of ternary hybrid nanofluid flows over multiple geometries with non-isothermal and non-isosolutal boundary conditions. Their study demonstrated that the presence of three distinct nanoparticles enhances thermal and solutal transport properties, while variations in temperature and concentration gradients significantly impact flow dynamics, making the findings relevant for advanced heat transfer and fluid engineering applications.

Sudarmozhi et al. [28] investigated the boundary layer characteristics of a Maxwell fluid over a porous inclined vertical plate, emphasizing the effects of magnetohydrodynamics (MHD) and thermal radiation on heat transfer. Their findings indicated that the fluid’s viscoelastic nature, along with magnetic and convective heat transfer influences, plays a crucial role in controlling temperature distribution and velocity profiles, making the study significant for industrial cooling and energy systems. Ayele et al. [29] examined the unsteady magnetohydrodynamic (MHD) flow of a hybrid nanofluid over a rotating disk, incorporating the effects of viscous dissipation and the Cattaneo–Christov heat flux model. Their study revealed that thermal relaxation time significantly influences heat transfer behavior, while the presence of hybrid nanoparticles enhances thermal conductivity, making the findings valuable for optimizing high-performance thermal systems and industrial cooling applications.

Pau et al. [30] examined the flow of an engine oil-based Casson tri-hybrid nanofluid over a rotating disk. Their study focused on analyzing the fluid dynamics and heat transfer characteristics. It focuses on effects like viscous dissipation and radiative flux, enhancing heat and mass transfer rates compared to Casson hybrid and nanofluids. The studies of Anjum et al. [31] and Anjum et al. [32] presents a non-similar Keller Box analysis of magnetochemically radiative Buongiorno’s nanofluid flows over a stretched surface, addressing the unexplored effects of thermal radiation, chemical reaction, and magnetic parameters on heat and mass transfer. By employing the Keller Box method, the research offers a comprehensive multi-physics analysis, demonstrating that increasing magnetic fields enhance nanoparticle concentration, while rising thermal radiation and chemical reactions significantly influence velocity, temperature, and concentration profiles, with results showing a 99.9% compatibility rate with previous studies. Parige et al. [33] explored the interplay between Fourier heat flux and Joule heating in ternary hybrid nanofluids. Their study focused on analyzing the combined effects on heat transfer and fluid dynamics. Particularly in the context of a rotating circular porous disk, reveals significant insights into heat transfer dynamics.

All these existing studies have extensively explored the thermal and fluid dynamics of nanofluids and hybrid nanofluids, focusing on their heat transfer characteristics, convective behavior, and radiative effects. Research has also addressed the impact of viscous dissipation, magnetic fields, and nonlinear radiation in Newtonian and some non-Newtonian fluids. However, a significant gap remains in analyzing the symmetric behavior of non-Newtonian hybrid nanofluids under spinning disk motion with integrated nonlinear radiative effects and viscous dissipation. While previous studies have investigated flow over stretching/shrinking sheets and rotating disks, the coupled effects of these parameters on heat transfer efficiency, Nusselt number variation, and local skin friction reduction in symmetric thermal modeling are not well understood. Additionally, the influence of hybrid nanofluids in such a setting, particularly their interaction with external magnetic fields and nonlinear heat radiation, requires further investigation to optimize their industrial and engineering applications.

The current investigation stands out for its innovative approach, analysing the flow characteristics of both hybrid nanofluids and standard nanofluids concurrently, while factoring in nonlinear thermal radiation, viscous dissipation, and the Cattaneo-Christov model. This study meticulously assessed the thermophysical properties of nanomaterials, including Ag, MnZnFe₂O₄, and Cu, utilizing pure water as the foundational fluid. The thermal enhancement of the Fourier heat flux was evaluated within the context of a nonlinear radiative transfer scenario. A MATLAB program was deployed to analyze the parametric elements, shedding light on the physics underlying the issue. To the best of our knowledge, no existing literature has ventured into such an analysis.

Therefore, the primary aim of this study is to investigate the profound effects of magnetic fields, viscous dissipation, nonlinear thermal radiation, and Cattaneo-Christov heat flux on the motion of hybrid nanofluids traversing past a rotating disk. The current analysis was numerically resolved using the Runge-Kutta method combined with a shooting technique. The substantial influence of flow parameters is illustrated through graphical representations, while the effects of engineering relevance are systematically tabulated.

The vector forms of the governing equation are [6].

$\begin{gathered}\nabla \cdot u=0 \\ \rho_f(u . \Delta) u=-\nabla p+\mu_f \nabla^2 u+JXB, \\ (u . \Delta) T=\alpha_f \nabla^2 T+\tau\left[D_B(\nabla T)+\frac{D_T}{T_{\infty}}\left(\nabla T^2\right)\right]\end{gathered}$

The approximation of the boundary layer is deemed to be valid, and consequently, the governing equation pertinent to the issue is expressed as follows [23]:

$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0$    (1)

$\begin{aligned} \rho_{h n f}\left(u \frac{\partial u}{\partial r}-\frac{v^2}{r}\right. & \left.+w \frac{\partial u}{\partial z}\right)+\frac{\partial p}{\partial r} =\mu_{h n f}\left(\frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \frac{\partial u}{\partial r}-\frac{u}{r^2}+\frac{\partial^2 u}{\partial z^2}\right) -\sigma_{h n f} B_0^2 u\end{aligned}$    (2)

$\begin{aligned} \rho_{h n f}\left(u \frac{\partial v}{\partial r}-\frac{u v}{r}+\right. & \left.w \frac{\partial v}{\partial z}\right) =\mu_{h n f}\left(\frac{\partial^2 v}{\partial r^2}+\frac{1}{r} \frac{\partial v}{\partial r}-\frac{v}{r^2}+\frac{\partial^2 v}{\partial z^2}\right) -\sigma_{h n f} B_0^2 u\end{aligned}$    (3)

$\rho_{h n f}\left(u \frac{\partial w}{\partial r}+w \frac{\partial w}{\partial z}\right)+\frac{\partial p}{\partial r}=\mu_{h n f}\left(\frac{\partial^2 w}{\partial r^2}+\frac{1}{r} \frac{\partial w}{\partial r}+\frac{\partial^2 w}{\partial z^2}\right)$    (4)

$\begin{aligned}\left(\rho c_p\right)_{h n f}\left(u \frac{\partial T}{\partial r}+\right. & \left.w \frac{\partial T}{\partial z}\right)+\frac{\partial p}{\partial r} \\ & =K_{h n f}\left(\frac{\partial^2 T}{\partial r^2}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}\right) \\ & -\frac{\partial q_r}{\partial z}+Q_0\left(T-T_{\infty}\right) \\ & +\frac{\mu_{h n f}}{\left(\rho c_p\right)_{h n f}}\left(\frac{\partial u}{\partial z}\right)^2 \\ & -\Gamma_1\left(u^2 \frac{\partial^2 T}{\partial r^2}+w^2 \frac{\partial^2 T}{\partial r^2}\right. \\ & +2 u w \frac{\partial^2 T}{\partial r \partial z}+\left(u \frac{\partial u}{\partial r}+w \frac{\partial w}{\partial r}\right) \frac{\partial T}{\partial r} \\ & \left.+\left(u \frac{\partial u}{\partial r}+w \frac{\partial w}{\partial r}\right) \frac{\partial T}{\partial z}\right)\end{aligned}$    (5)

A comprehensive numerical methodology was formulated based on the transformed Eqs. (1)-(13) subject to the conditions outlined in (6), (7) and (14)-(15) The boundary value problem was addressed through the application of the Runge-Kutta method in conjunction with the shooting technique to resolve the transformed ordinary differential equations (ODEs). This methodology was implemented in MATLAB to solve the system of equations, demonstrating notable simplicity and a rapid convergence rate. An iterative framework was employed to handle the nonlinear system, ensuring computational efficiency. The shooting method was utilized by first transforming the third-order differential equation into an equivalent system of first-order equations, as outlined below.

Table 2 presents a comprehensive set of parameters essential for analyzing fluid dynamics and thermal transfer. It includes viscosity (M) values ranging from 0.2 to 3, along with corresponding entries for additional parameters such as Po, Pr, Nr, Δr, and Ec, which play crucial roles in characterizing the thermophysical behavior of the system. The table also incorporates dimensionless quantities, including Qr, which likely represents the heat transfer rate, and −f′(0), which is presumed to be a velocity gradient or shear stress evaluated at a specific boundary. Moreover, the Nusselt number (Nu) is presented as a critical parameter that characterizes the ratio of convective to conductive heat transfer within the system. The dataset facilitates the identification of trends in Qr and Nu as functions of M and other governing parameters, thereby offering insights into their interdependencies within the broader framework of fluid dynamics and heat transfer analysis. By examining these variations, the table helps elucidate the underlying physical mechanisms that influence heat and momentum transport in the system.

Table 2. Comparison outcomes for and Prandtl numbers pertinent to the Nusselt number (Nu)

We meticulously illuminated the intricate dynamics that govern the multifaceted motion of hybrid nanofluids, delving deeply into the profound and far-reaching implications that are brought about by the Cattaneo-Christov heat flux, which introduces an exceptionally fascinating layer of complexity, alongside a nonlinear thermal radiation phenomenon that, without a doubt, plays a pivotal and essential role in shaping our comprehensive understanding of these intricate fluid behaviour’s within this specific and nuanced context. The influence of heat transmission on the intricate behaviour and motion of the hybrid nanofluid was thoroughly examined, while we subjected this complex system to the intricate influence of a uniformly distributed and carefully considered magnetic field, which not only complicates the analysis significantly but also provides a rich and vibrant tapestry of interactions that we sought to unravel and understand during our detailed investigation. In our comprehensive and methodical approach, we adopted a meticulously crafted two-dimensional steady flow model, which greatly facilitated our exploration into the systematic and nuanced characterization of the flow dynamics under consideration, thereby allowing us to capture the subtle nuances and complexities of this multifaceted system. This flow model culminated in the emergence of a complex and challenging system of partial differential equations (PDEs), which we later adeptly transformed into ordinary differential equations (ODEs) by employing similarity transformation variables that significantly streamline the mathematical treatment process and enhance the clarity of our insightful findings. To derive meaningful and deeply insightful solutions that could effectively inform our understanding of these intricate dynamics, we resorted to robust and sophisticated numerical methodologies, specifically implementing the highly regarded Runge-Kutta method in conjunction with innovative shooting techniques that greatly enhance the accuracy and efficiency of our computations, ensuring that our results are both reliable and richly informative. The pivotal and groundbreaking discoveries emerging from our extensive investigation are succinctly summarized as follows:

a) An increase in the magnetic parameter intensifies the Lorentz force, which acts as a resistive force opposing the motion of the hybrid nanofluid. This results in a noticeable reduction in velocity profiles, demonstrating the complex interaction between electromagnetic forces and fluid dynamics. The suppression of velocity due to the magnetic field influence highlights the significance of magnetohydrodynamic (MHD) effects in controlling flow behavior, particularly in applications involving electromagnetic flow regulation and thermal management systems.

b) A rise in thermal radiation enhances the energy exchange within the fluid, leading to an increased thermal distribution and a subsequent expansion of the thermal boundary layer. The thickening of the thermal boundary layer significantly influences heat transfer characteristics. It directly affects the temperature gradient near the surface. The interaction between radiative heat transfer and convection is crucial for thermal performance. This interplay is essential in applications such as high-temperature fluid transport, energy conversion systems, and industrial cooling processes.

c) Furthermore, a rise in the Eckert number (Ec) significantly expands both the velocity and temperature profiles, showcasing the enhanced kinetic and thermal behaviour of the hybrid nanofluid under varying conditions.

d) Lastly, an elevated Prandtl number effectively reduces the thickness of both the momentum and thermal boundary layers, indicating a refined balance between momentum transport and thermal diffusion in the fluid system.