The purpose of this examination is to assess the effect of Stephan blowing and thermal radiation on chemical reactive flow of Williamson nanofluid flow across a sheet with Marangoni convection. Rapid advancements in technology have led to tremendous growth in the domains of machine learning and artificial intelligence. In order to solve the mathematical formulation including heat sources and chemical reactive flow using the Bayesian-Regularization approach, this study creates a machine learning model based on ANN (artificial neural networks). With error estimates of 2.51 × 10⁻¹², 1.51 × 10⁻¹², and 7.41 × 10⁻¹³ across all three scenarios, the model achieves remarkable test performance by utilizing the BRO-DNN (Bayesian-Regularization Optimizer-Deep Neural Network), exhibiting great accuracy and dependability. Numerous industrial and technical domains where heat and mass movement are important have substantial uses for the suggested paradigm. Williamson nanofluid dynamics' incorporation of Stefan blowing and thermal radiation effects is very helpful for improving heating and cooling systems, such as those used in sophisticated industrial processes, thermal energy storage, and electronic device cooling. In applications involving porous media and composite materials, the thermal non-equilibrium approach improves forecast accuracy. The precision of numerical solutions is also increased by combining the Bayesian-Regularization Optimizer with a Deep Neural Network, which makes it advantageous for machine learning-based predictive modelling in biomedical applications, aerospace thermal management, and renewable energy systems like solar collectors.

The flow and heat transfer characteristics of a tri-hybrid nanofluid in a porous medium are investigated under the influence of magnetohydrodynamics (MHD) and a ramped wall temperature. A Maxwell fluid is employed as the base fluid, in which three types of spherical nanoparticles, tungsten trioxide (WO₃), silver (Ag), and titanium dioxide (TiO₂), are suspended. The physical model is formulated using a system of partial differential equations subject to appropriate initial and boundary conditions. To enhance the novelty of the analysis, fractional derivatives are incorporated into the Maxwell fluid model along with porosity effects. Among the various definitions of fractional derivatives, the Caputo fractional derivative is preferred for its wide applicability in physical problems. The fractional-order derivatives are evaluated using the Caputo formulation, while the Crank–Nicolson numerical scheme is employed to discretize the time-dependent terms and solve the governing equations under ramped heating conditions. The proposed framework, which combines the Caputo fractional derivative with the Crank–Nicolson method to analyze tri-hybrid nanofluid flow, is a distinctive feature of this work. The Caputo derivative effectively captures memory-dependent behavior, allowing the model to account for the system’s dependence on its past states. This capability is particularly important for nanofluids exhibiting nonlocal and anomalous interactions, where classical integer-order models based on simple linear stress–strain relationships fail to accurately represent the complex rheological behavior. Overall, the adopted numerical approach provides improved accuracy and flexibility in modeling complex heat transfer processes, making the present study relevant to a wide range of biomedical and industrial applications.

This research examines the dynamics of heat transfer while highlighting the crucial role of Fourier heat flux concerning the thermodynamic and hydrodynamic characteristics of ternary hybrid nanofluids (HNFs) traversing a disk. The physical model and flow configuration were thoroughly analyzed under the influences of various parameters. The major equations that characterize the flow dynamics are formulated as partial differential equations (PDEs). By utilizing appropriate correspondence variables, the system of PDEs was altered keen on ordinary differential equations (ODEs). The coordination of coupled nonlinear equations is resolved arithmetically utilizing the “bvp4c function in MATLAB.” The influence of the principal appropriate factors on the radial speed, axial speed, and warmth is illustrated realistically. Ultimately, a table is constructed to demonstrate the interrelationships of numerous perilous issues on the Skin friction and Nusselt number. It was observed that an enhancement in the attractive constraint significantly diminishes the speed outline, attributable to the Lorentz influence caused by the applied attractive subject. Additionally, an enhancement in thermal transfer was observed as a consequence of an increase in thermal radiation.