Memory dependent time fractional analysis of unsteady magnetohydrodynamic shear thinning fluid flow over an oscillating sheet

This study numerically investigates the two dimensional unsteady magnetohydrodynamic flow of a shear thinning Carreau fluid over an oscillating sheet using a time fractional derivative framework to account for memory effects. The mathematical model incorporates thermal radiation, space and velocity dependent internal heat generation or absorption, chemical reaction, and nanoparticle transport driven by Brownian motion and thermophoresis. The governing fractional partial differential equations are solved using an explicit finite difference scheme implemented in MATLAB. A detailed parametric analysis is performed to examine the influence of key physical parameters on velocity, temperature, and concentration fields, supported by advanced flow visualisations including Q criterion based vortex structures and heat maps. The results reveal that increasing the fractional order weakens memory effects and enhances momentum and thermal diffusion, whereas stronger magnetic fields suppress velocity due to Lorentz forces and elevate thermal energy near the surface. Quantitatively, the skin friction coefficient increases by approximately 3.8% as the Darcy number rises from 0.01 to 0.5, and its magnitude increases by about 5.5% when the magnetic parameter increases from 0.5 to 2.0. Conversely, the Nusselt number decreases by nearly 14.5% with increasing Darcy number, indicating reduced heat transfer in highly permeable media. These findings provide physically meaningful benchmarks for controlling drag, heat, and mass transfer in polymer processing, thermal management systems, and industrial magnetohydrodynamic flow applications.