Nature of instability in flow-driven porous anodic oxide

Self-organized porous anodic oxide films are formed by the electrochemical oxidation of reactive metal aluminum in acidic solutions in which the oxide is soluble. Recently, viscous flow models have shown using linear stability analysis that the instability results from a trade-off between the destabilizing effect of viscous flow of oxide and the stabilizing effect of oxide formation, which provides the wavelength selection mechanism for pattern formation. Anion adsorption on surface growth sites causes nonuniform compressive stress at the oxide-solution interface, which drives the flow. This anodic instability is analogous to the classical Marangoni instability induced by surface tension gradients. In this work, nature of the instability beyond the stability threshold is determined using a weakly nonlinear analysis. For the growth of well-developed pores beyond the threshold, a subcritical nature of the instability is essential. However, our weakly nonlinear analysis shows that the solutions emerging from neutral stability are supercritical in nature at all wavenumbers for the practical range of anodizing control parameters investigated. We also determine the region where the model is Hadamard stable, a necessary condition for well-posedness.