Throughflow and imperfect heat transfer effects on the stability of natural convection in a vertical porous layer

Abstract

The Gill stability problem of parallel buoyant flow in a differentially heated vertical porous channel (A.E. Gill, J. Fluid Mech., vol. 35, 1969, pp. 545–547) is revisited under the influence of uniform horizontal throughflow and imperfect heat transfer at the vertical boundaries. The boundary imperfections are modelled using Robin-type temperature conditions. The base flow solution is derived analytically, followed by a linear stability analysis that results in a fourth-order eigenvalue problem. The validity of Squire's theorem is established; therefore, two-dimensional motions are considered. Given the limitations of Gill's linear stability proof, a numerical solution to the eigenvalue problem is provided across a broad spectrum of input parameters. The findings suggest that convective instability is precluded for isothermal boundaries, even when horizontal throughflow is present. This conclusion is substantiated by an analysis of the eigenvalue spectrum, which reveals a negative amplification rate for normal mode perturbations, thereby affirming the asymptotic stability of the basic flow. For non-isothermal boundaries, the study traces neutral stability curves that define the threshold for linear instability and identifies the critical Darcy-Rayleigh number at which instability arises, considering various values of the Prandtl-Darcy number, Péclet number, and Biot number. Notably, the magnitude of the Biot number discloses a new pathway for instability, with throughflow not only influencing its onset but also giving rise to different onset modes.