Time-Evolving to Space-Evolving Thermal Instability of a Porous Medium Flow

TL;DR

This paper investigates the stability of a porous medium flow under time-periodic perturbations, focusing on how localized disturbances evolve spatially, providing new insights into the transition to instability in such systems.

Contribution

It introduces a novel approach by analyzing time-evolving perturbations in the spatial stability of porous medium flow, extending classical Fourier mode analysis.

Findings

Identifies conditions for amplification or decay of localized perturbations.

Determines the spatial stability/instability threshold for the flow.

Provides a framework for analyzing time-periodic disturbances in porous media.

Abstract

The Rayleigh-B\'enard instability of the stationary throughflow in a horizontal porous layer, also known as Prats' problem, is here analysed in a fresh new perspective. In fact, the classical analysis of the linear instability, carried out by employing space-periodic Fourier modes, is here reconsidered by focussing on the effects of time-periodic and space-evolving modes. The basic stationary flow is assumed to be perturbed by a localised source of perturbation which is steady-periodic in time. Then, the streamwise development of such perturbations is monitored in order to detect their possible amplification or decay in the direction of propagation. Accordingly, the spatial stability/instability threshold is determined. The study is carried out by employing a Fourier transform formalism, where the transformed variable is time.