Interplay of stationary and oscillatory instabilities and onset of capillary waves

Determine the interplay between the curvature-driven stationary instability and the memory-driven oscillatory instability in the conserved active emulsion model (Eqs. (8a–b)), derive a precise onset criterion for capillary wave formation, and quantify how slow propagation of the chemical field (large relative mobility) alters lateral mass redistribution and overall interface stability.

Background

The authors identify two interface-destabilization mechanisms: (i) a stationary instability akin to Mullins–Sekerka that amplifies curvature perturbations via chemotactic fluxes, and (ii) an oscillatory instability that relies on memory effects from slow chemical-field dynamics and can produce capillary waves.

While a heuristic criterion predicts the stationary instability threshold, the precise conditions for capillary wave onset and the interaction between the two mechanisms are not resolved within the heuristic framework.