Multigrid and machine learning for natural convection heat transfer enhancement of nanofluids in snowflake-inspired heat sink structures

The increasing global energy demand and environmental sustainability goals have intensified the need for efficient thermal management systems across engineering applications, including aerodynamics. Nanofluids offer significant potential for enhancing heat transfer due to their improved thermophysical properties. This study investigates natural convection of a copper–water Casson nanofluid within a porous medium under the combined effects of thermal radiation and inclined magneto-hydrodynamics (MHD). Snowflake-inspired heat sink geometries with varying branching complexities are examined to assess the role of structural design on thermal performance. The governing equations are solved using the finite element method (FEM) for Rayleigh numbers ranging from 103 to 106, multiple radiator configurations, and nanoparticle volume fractions (ϕ = 0–0.003). Results indicate that nanoparticle inclusion increases the average Nusselt number by up to 13%, while the dense-branch snowflake structure (DB) delivers a 23% improvement over the circular design at high Ra. Enhanced convection is observed with increasing radiation effects, stronger buoyancy forces, and reduced Casson yield stress. An Artificial Neural Network (ANN) using the Backpropagation Levenberg–Marquardt algorithm is developed to predict the nonlinear thermal behavior, achieving excellent convergence (1.4835 × 10−8) and strong agreement with numerical results. By integrating radiation, inclined MHD, porous media, Casson rheology, and complex heat-sink geometries, this study provides valuable insights for designing high-performance thermal systems relevant to energy technologies, cooling devices, and aerodynamic thermal management applications.