Abstract
This study investigates the linear stability of natural convection in a vertical layer of viscoelastic Navier—Stokes—Voigt (NSV) fluid flow within a Darcy-Brinkman porous medium. The coupled Orr-Sommerfeld equations (OS-EQ) governing the stability of the flow are derived and solved using two prominent numerical methods: the Galerkin method and the Chebyshev collocation method. The basis functions and their derivatives were constructed through symbolic integration to enhance computational accuracy. The resulting eigenvalue problem was solved using advanced eigenvalue solvers provided by the IMSL library (PyIMSL version 2020.0.1) and high-performance computing packages PETSc (petsc4py version 3.21.4) and SLEPc (slepc4py version 3.21.1). The study focuses on capturing the most unstable modes that trigger the onset of instability. The influence of the Grashof number (G), porous parameter (M), Kelvin-Voigt elasticity (Λ), and Prandtl number (Pr) on flow stability is extensively analyzed. The eigenvalue spectra, neutral stability curves, critical triplet analysis, and streamline-isotherm distributions collectively demonstrate that the porous parameter exhibits a strong stabilizing effect on both stationary and traveling wave modes by suppressing oscillatory disturbances and enhancing flow stability. On the other hand, the Kelvin-Voigt parameter shows a dual nature, although its influence is less significant in stationary modes compared to traveling wave modes. The Prandtl number primarily acts as a destabilizing agent by increasing thermal diffusion, thereby promoting the rapid growth of disturbances.
