Time-fractional analysis of free convection in a heated square cavity

Abstract This study presents a comprehensive numerical analysis of time-fractional free convection in a square cavity with a heated vertical wall, governed by the time-fractional Navier–Stokes equations incorporating Caputo derivatives to account for memory-dependent and non-local effects. A fourth-order finite difference scheme is employed for spatial discretization, while implicit time-integration schemes of orders $2 - \alpha ,\,\,3 - \alpha ,$ and $4 - \alpha $ are used to approximate the fractional time derivatives, with $0 < \alpha \le 1$. After comparative evaluation, the $2 - \alpha \,\,$ scheme is selected for its balance between accuracy and computational cost. The resulting system is solved using a linear solver implemented in Mathematica 14, utilizing the ‘Pardiso’ method, a parallel direct sparse solver, to ensure efficient and stable computation. Results show that the fractional-order parameter $\alpha $ has a significant impact on the flow structure and heat transfer characteristics. Higher values of $\alpha $ lead to classical behaviour with faster stabilization and organized convection cells, while lower values introduce pronounced memory effects, delayed transitions, and oscillatory dynamics. The proposed method is validated against classical benchmark solutions, demonstrating high accuracy and convergence. This work not only underscores the potential of fractional models in capturing transient convection behaviour but also establishes a benchmark framework for future studies on fractional-order heat transfer systems.