Necking in coating flow over periodic substrates

Abstract

The free-boundary problem determining the shape of a layer of viscous fluid coating a substrate while draining steadily under gravity is solved analytically for substrates taking the form of a periodic array of long plates of arbitrary width and spacing. The mathematical problem involves solving Poisson’s equation with constant forcing term in the fluid layer subject to vanishing Neumann and Dirichlet conditions on the free boundary. By considering the problem in a potential plane and using conformal mapping, a two-parameter family of solutions is obtained in the form of an infinite series. Explicit, closed-form solutions are derived in the limiting cases of a single gap perforating an infinitely wide plate, and for an array of evenly spaced point plates. In these cases explicit expressions are obtained for the thinning, or necking, of the fluid layer in the gap regions between plates.