Heat transfer analysis of Cu-Water nanofluid in a square enclosure using Caputo fractional derivative and machine learning

Fractional-based fluid transport modeling provides a deeper and more comprehensive understanding of the thermophysical properties of small-scale fluids and polymer solutions. Moreover, the fractional-order fluid dynamics framework finds applications in various fields, including renewable energy systems, thermal energy storage, and oil storage tanks. Inspired by these developments, the current study addresses the computational challenges inherent in modeling nanofluid dynamics within a square enclosure by introducing a fractional-order approach to enhance the simulation of buoyant flow and heat transport. Our model integrates the linear interpolation (L1 approach) and finite difference approximation for temporal and spatial derivatives by utilizing the Caputo time-fractional derivative instead of the conventional time derivative. This integration facilitates the use of the Backward Difference Alternating Direction Implicit (BD-ADI) method, which was specifically chosen to reduce the computational burden associated with the non-local characteristics of fractional models. Our results show that by altering the fractional order, significant improvements in heat transfer efficiency are achieved compared to classical models. For instance, compared to the integer order, a lower fractional-order parameter of γ=0.80 with a 4% nanoparticle volume fraction at a Rayleigh number of 103 increases the heat transfer rate by 15.77%. However, at a Rayleigh number of 106, the enhancement reduces to 4.15%, indicating that fractional adjustments’ influence diminishes at higher thermal buoyancy levels. This behavior of the fractional-order nanofluid model, influenced by thermal buoyancy force, has been investigated by using a Multiple Linear Regression(MLR) analysis with 258 numerical samples. Besides, MLR analysis further identifies the nanoparticle volume fraction as a critical factor, influencing the heat transfer rate by 10.48% at lower fractional-order settings.