On Prats' problem with anomalous diffusion

TL;DR

This paper revisits Prats' problem by replacing thermal diffusion with anomalous mass diffusion, including superdiffusion and subdiffusion, significantly affecting the stability analysis of buoyancy-driven flow in porous channels.

Contribution

It introduces a model incorporating anomalous mass diffusion into Prats' problem, extending the classical framework to include superdiffusion and subdiffusion effects on flow stability.

Findings

Anomalous diffusion alters the stability eigenvalue problem.

Subdiffusion can induce transition from convective to absolute instability.

Non-autonomous differential equations are key in analyzing the modified problem.

Abstract

The classical Prats' problem of flow instability in a horizontal porous channel saturated by a fluid subject to a buoyancy force is reconsidered. In the original formulation, the driving buoyancy force results from thermal diffusion. This study, however, substitutes thermal diffusion with mass diffusion. Furthermore, the usual scheme of mass diffusion is extended to comprehend also the anomalous phenomena of superdiffusion or subdiffusion. Such phenomena are modelled via a time-dependent mass diffusivity which yields a significant change in the formulation of the stability eigenvalue problem. In particular, the ordinary differential equations governing the time evolution of the perturbations acting on the base throughflow become non-autonomous. This makes a significant difference in the discussion of the conditions leading to instability, with a marked effect of the anomaly in the mass diffusion process. The transition from convective to absolute instability for subdiffusion processes is also addressed.