Stabilising effect of generic anomalous diffusion independent of the Rayleigh number

TL;DR

The paper addresses how generic anomalous diffusion, captured by a memory function $h(t/\tau)$ in solute transport, alters the stability of mass convection in a porous medium. By formulating a dimensionless Darcy-type model with a time-nonlocal diffusion term and performing nonlinear energy estimates and linear stability analysis, the authors show that if the memory integral $H(t)=\int_0^t h(\tau)\,d\tau$ grows faster than linearly, the perturbations decay for all mass-diffusion Rayleigh numbers $Ra$, yielding robust asymptotic stability. The linear analysis reveals a disturbance evolution equation $\frac{\partial \Gamma_t}{\partial t} + C_1 h(t)\Gamma_t - C_2\Gamma_t = 0$, leading to $\Gamma_t=\Gamma_0 e^{C_2 t - C_1 H(t)}$ and enabling precise characterization of transient growth via $t_{gmax}$ for various memory forms (power-law, exponential, logarithmic). Across the memory functions studied, asymptotic stability is preserved, but pronounced transient growth can occur depending on the relation between $C_1$ and $C_2$ and the memory function, with stronger or faster diffusion memories generally dampening transient peaks. These results illuminate how non-Fickian transport modulates stability and transient behaviours, offering a framework for controlling convection in porous systems through tailored diffusion memory.

Abstract

This work investigates the influence of a generic anomalous diffusion model on mass convection in a fluid-saturated porous medium, focusing on superdiffusive regimes. A mathematical model is developed, and tability analyses - both linear and nonlinear - are performed. Results demonstrate that the specific form of the time function describing anomalous diffusion significantly affects system stability, allowing stability to persist beyond the classical Rayleigh-Bénard neutral threshold. Furthermore, transient perturbation growth is observed under certain conditions, followed by eventual decay. The paper systematically examines various memory functions, including power-law, exponential, and logarithmic forms, highlighting their impact on the dynamics of disturbances. The findings underscore the importance of anomalous diffusion in modulating stability and provide new insights into the transient behaviours induced by non-Fickian mass transport.

Figures (7)

• Figure 1: Temporal disturbance behaviour for power-law memory function, with $r=1.2$
• Figure 2: Temporal disturbance behaviour for power-law memory function, with $r=1.4$
• Figure 3: Temporal disturbance behaviour for power-law memory function $r=1.6$
• Figure 4: Temporal disturbance behaviour for exponential memory function
• Figure 5: Temporal disturbance behaviour for logarithmic memory function with $a=1$