Fractal time numerical modelling of radiative heat and mass transfer in Carreau–Yasuda mixed convective flow

This paper demonstrates a computational process in simulating the unsteady radiative mixed convective flow of a Carreau-Yasuda nanofluid by a porous material, when subjected to a magnetic field. Fractal time derivatives are used to model an approach that captures the memory-dependent behaviour of complex transport phenomena. A novel three-stage explicit time integration scheme is developed, delivering third-order temporal accuracy tailored to fractal-time partial differential equations. For spatial discretization, a compact sixth-order scheme is used to improve numerical precision in the interior domain. The proposed framework incorporates thermal and solutal buoyancy effects, nonlinear rheology, Brownian motion, thermophoresis, and contributions from a heat source. It also accounts for the influence of oscillatory boundary conditions and Darcy-Forchheimer drag within porous structures. Rigorous stability and convergence analyses confirm the robustness of the scheme. Quantitative comparisons reveal that at a time step of Δt=0.12250, the proposed scheme achieves an L2 error of 6.72×10−5 and consumes approximately 97.62 s, while the second-order scheme reaches an error of 1.06×10−4 with a runtime of 173.82 s under the same compact discretization, highlighting both its efficiency and stability. Numerical experiments demonstrate that the method outperforms existing first- and second-order schemes in both accuracy and computational efficiency. Furthermore, a velocity reduction of over 30% observed when increasing the power-law index from 0.6 to 1.2, highlighting enhanced shear-thinning behaviour. The proposed methodology not only ensures numerical stability under fine discretization but also demonstrates robustness in capturing complex flow behaviour in porous, radiatively influenced environments. This fractal-based computational framework offers a valuable tool for simulating non-Newtonian nanofluid systems in emerging thermal technologies. Fractal time derivatives effectively capture memory-dependent, scale-invariant transport phenomena, offering computational advantages and localised accuracy over traditional fractional operators.