A Free-Boundary Problem Arising from a Galvanizing Process
Thomas I. Vogel · SIAM Journal on Mathematical Analysis · 1985
A free-boundary problem which arises from a galvanizing process is studied. The physical problem is that of an infinite cylinder $\Omega ' \times \mathbb{R}$ withdrawn from a fluid bath. Formally, this is a gravity-driven unidirectional viscous fluid flow on the exterior of the cylinder $\Omega ' \times \mathbb{R}$. The mathematical problem is to find a function u with compact support in the exterior of $\Omega '$ satisfying: \[\begin{gathered} \Delta u = \chi _{\{ u > 0\} } \quad {\text{in }}\mathbb{R}^n - \Omega ', \hfill \\ u = c\quad {\text{on }}\partial \Omega ' \hfill \\ \end{gathered} \] where $\chi _U $ is the characteristic function of U. The existence of a unique classical solution is shown under certain conditions on $\Omega '$, and asymptotic results for the thickness of the coat are obtained for large and small withdrawal speeds. If $\Omega '$ is a convex set, then the region bounded by the free surface of the fluid is shown to be convex, using level curve techniques. Finally, level curve techniques are used to bound the curvature of the free boundary in terms of that of the fixed boundary.