On Prats' problem with anomalous diffusion

TL;DR

The paper studies buoyancy-driven instability in a horizontal porous channel when mass diffusion is anomalous, modeled by a time-dependent diffusivity $\mathcal{D}(t)$. Using a non-autonomous linear stability framework with normal-mode perturbations and wavepacket/steepest-descent analysis, it shows that the long-time growth of perturbations depends on the time-averaged diffusivity and the throughflow parameter $Pe$. Key findings include that superdiffusion stabilizes for all $Ra$ and $Pe$, subdiffusion destabilizes for $Ra>0$ (neutral at $Ra=0$, stable for $Ra<0$), and normal diffusion recovers classical Prats results; absolute-instability thresholds for subdiffusion scale as $Ra_{abs} = 1.270325\, n\pi\, |Pe\cos\phi|$ with orientation $\phi$. The work highlights how anomalous diffusion can markedly shift the onset and detectability of convection in porous media and motivates extensions to more complex flows like Rayleigh–Bénard–Poiseuille systems.

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.

Figures (3)

• Figure 1: Subdiffusion: results of the linear analysis of instability
• Figure 2: Check of the holomorphy requirement: isolines of $\IfNoValueTF{-NoValue-}{\operatorname{Re}}{ \mathopen{}\mathclose{\left{}\right}}{} \IfNoValueTF{}{ \IfNoValueTF{}{ \IfNoValueTF{}{ \IfNoValueTF{} {()} {\left\lvert{}\right\rvert} } {\left(\right) \IfNoValueTF{}{}{||}} } {\left[\right] \IfNoValueTF{}{}{()} \IfNoValueTF{}{}{||}} } {\left\lbrace\right\rbrace \IfNoValueTF{}{}{[]} \IfNoValueTF{}{}{()} \IfNoValueTF{}{}{||}} {} )$ in the complex $k$ plane, for $Ra = Ra_{abs}$ with $n=1$ and $Pe \cos\phi > 0$. The blue dots correspond to the saddle points $k=k_0$, while the black dot denotes the singularity $k = -i \pi$. The red lines show the paths of steepest descent crossing the saddle points
• Figure 3: Transverse modes $(\phi = 0)$ with $n=1$: comparison between the absolute instability threshold for normal diffusion delache2007spatiobarletta2019routes and that for subdiffusion