Thermal performance optimization in CuO-water nanofluid enclosure with sinusoidal heating using deep learning and multi-expression programming

Natural convection in enclosures containing nanofluids has attracted significant attention due to its relevance in thermal management systems. In this context, this study presents a comprehensive numerical investigation of flow and heat transfer in a square cavity saturated with water-based CuO nanofluid having a centrally placed sinusoidal-shaped heated element. All the enclosure walls satisfy the no-slip velocity condition. Thermally, the vertical walls are kept at a cold reference temperature, the lower wall is partially heated at its center, and the remaining portions of the lower and entire upper walls are adiabatic. The internal sinusoidal element is also uniformly heated. The flow dynamics and thermal fields are governed by the two-dimensional steady-state Navier-Stokes and energy equations, solved using the Galerkin finite element method. Additionally, a novel hybrid approach integrating multi-expression programming (MEP) technique with a convolutional neural network bidirectional gated recurrent unit (CNN-BiGRU) deep learning network is also applied to enhance flow and thermal prediction accuracy. This hybrid approach enables precise evaluation of how heater waviness, magnetic field orientation, and nanoparticle dispersion influence flow structure and heat transfer. Results reveal stronger convection at high Rayleigh numbers, magnetic damping at increased Hartmann numbers, and higher temperatures with reduced velocity at greater nanoparticle concentrations. Among the analyzed situations, increasing heater waviness improves heat-transfer performance. Both the MEP and CNN-BiGRU models accurately capture the key features of flow and heat transport trends, indicating that the hybrid approach provides enhanced predictive capability for complex convection-driven nanofluid systems.