A Mathematica‐based algorithm to trace stable/unstable solutions in transient free convection of non‐Newtonian fluids

Abstract This paper presents a Mathematica‐based computational framework for tracing stable and unstable solutions in the transient free convection of power‐law fluids governed by Stokes' second problem. The method combines high‐order finite difference discretization with an event‐driven stability detector implemented through WhenEvent , enabling nonlinear stability assessment without relying on linearization or eigenvalue analysis. Stability maps constructed in the extended parameter space reveal that the critical Grashof number increases with the power‐law index, from for shear‐thinning fluids to for shear‐thickening fluids . Variations in the magnetic parameter and Eckert number shift the stability boundaries significantly, with magnetic damping stabilizing flows for and destabilizing flows for , while viscous dissipation produces strong thermal amplification, thereby narrowing the stable regime. Long‐time periodic solutions and phase‐plane portraits further demonstrate that oscillation intensity and waveform deformation increase with . Wall‐based heat‐transfer and shear‐stress quantities show a consistent hierarchical ordering , confirming the dominant role of shear‐dependent viscosity in organizing momentum–thermal coupling. The proposed algorithm provides a reproducible and efficient tool for exploring nonlinear stability in oscillatory non‐Newtonian convection.