Thin viscous sheets with inhomogeneous viscosity

Abstract

We derive the equations governing the dynamics of thin viscous sheets having non-homogeneous viscosity, via asymptotic expansion methods. We consider distributions of viscosity that are inhomogeneous in the longitudinal and transverse directions and arbitrary (bulk and surface) external forces. Two specific problems are solved as an illustration. In a first example, we study the effects of purely in-plane variations of viscosity, which lead to thickness modulations when the sheet is stretched or compressed. In a second example, we study a stretched viscous sheet whose viscosity varies both across thickness and in-plane; in that case, we find that in-plane strain leads to out-of-plane displacement as the in-plane forces become coupled to transverse ones.