Stability of a falling viscous sheet

Falling films can be found in various processes of the food, glass and polymer industry. We study thin viscous films flowing vertically under the action of gravity, when poured from a slit. The lateral sides are unconstrained and the stretching effect of gravity induces a narrowing of the film in the horizontal direction, by Poisson's effect. This leads to compressive stress for some range of parameters, and we study the associated viscous buckling instabilities. A local stability analysis is used to characterized the flow parameters leading to potential instabilities. A global stability analysis is carried out, and an eigenvalue problem is solved numerically. This is implemented using the finite-element method with high order derivatives.