A Class of Moving Boundary Problems Arising in Industry

Abstract

There is a large class of moving boundary problems of industrial relevance in which, as in the Signorini problem, the moving boundary has spatial codimension two. Examples include saturated- unsaturated porous medium flows when the unsaturated region is narrow, various types of supercooled solidification problems, impact problems in ship hydrodynamics, models for electrical painting, and contact problems. Providing that time is increasing, a surprising number of these problems are amenable to a smoothing similar to the Biaocchi transformation. This means they can be reduced to well-posed families of variational inequalities, parameterised by time, for which simple numerical algorithms are available. Even though a formal time reversal is possible in several of the problems, the failure of the smoothing transformations suggests that such time-reversed problems may be ill-posed. Corroboration of this conjecture might be provided by a formal linear stability analysis but this requires an approach different from the usual stability analysis of moving boundaries with codimension unity.