Transition to Absolute Instability in Porous Media: Analytical Solutions

Abstract

The emergence of a supercritical transition from convective to absolute instability is illustrated. The presence of a horizontal flow is the cause of the delayed onset of absolute instability with respect to convective instability. The focus of this chapter is on situations giving rise to an analytical dispersion relation. In these cases, simple algorithms of numerical root finding are sufficient to carry out the evaluation of the saddle points which are relevant for the onset of absolute instability. The flow system examined in this chapter is a classical variant of the original Horton–Rogers–Lapwood problem, called the Prats problem, the difference being a basic horizontal flow with a given rate. The first analysis relies on Darcy’s law. An improvement of this study is carried out by adopting the more general Darcy–Forchheimer’s model. The transition from convective to absolute instability is initially studied by assuming a purely two-dimensional description. Then, a three-dimensional analysis is also developed by devising a specific lateral confinement of the flow.