The Singular Perturbation of Surface Tension 3

Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidiication. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity, what is then the role played by small surface tension in the dynamic formation and selection of these patterns? What is the asymptotic behavior of the interface in the limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension problem and the singular nature of surface tension pose challenging diiculties to the investigation of these questions. Here, we design a novel numerical method that greatly reduces the impact of noise, and allows us to accurately capture and identify the singular contributions of extremely small surface tensions. The numerical method combines the use of a compact interface parametrization, a rescaling of the governing equations, and very high precision. Our numerical results demonstrate clearly that the zero-surface-tension limit is indeed singular. The impact of a surface-tension induced complex singularity is revealed in detail. The singular eeects of surface tension are rst felt at the tip of the interface and subsequently spread around it. The numerical simulations also indicate that surface tension deenes a length scale in the ngers developing in a later stage of the interface evolution.